508Px-Bendixen - Carl Friedrich Gauß, 1828480Px-Leonhard Euler 2
Mathematicians
              Gauss          and          Euler

12/20/2017

   
   
 

 

MATHEMATICIANS  (Also see http://aleph0.clarku.edu/~djoyce/mathhist/chronology.html )

      Pythagoras of Samos (c. 570-c. 495 BC): (Greek)  He was an Ionian Greek philosopher, mathematician, and founder of the religious movement called Pythagoreanism. Most of the information about Pythagoras was written down centuries after he lived, so that very little reliable information is known about him.  Around 530 BC, he moved to Croton, a Greek colony in southern Italy, and there set up a religious sect. His followers pursued the religious rites and practices developed by Pythagoras, and studied his philosophical theories. The society took an active role in the politics of Croton, but this eventually led to their downfall. The Pythagorean meeting-places were burned, and Pythagoras was forced to flee the city. He is said to have ended his days in Metapontum. He is often revered   as a great mathematician, mystic and scientist, and he is best known for the Pythagorean theorem which bears his name.

Pythagorean theorem
The sum of the areas of the two squares on the legs (a and b) equals the area of the square on the hypotenuse (c).

The Pythagorean theorem, also known as Pythagoras's theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. The theorem can be written as an equation relating the lengths of the sides a, b and c, often called the "Pythagorean equation":

Although it is often argued that knowledge of the theorem predates him, the theorem is named after the ancient Greek mathematician Pythagoras who is credited with its first recorded proof.There is some evidence that Babylonian mathematicians understood the formula, although little of it indicates an application within a mathematical framework. Mesopotamian, Indian and Chinese mathematicians all discovered the theorem independently and, in some cases, provided proofs for special cases. See 108 proofs and other mathematical curiosities at https://www.pinterest.com/pin/544020829962250556/                                                                            Also see https://www.cut-the-knot.org/pythagoras/

A Pythagorean triple consists of three positive integers a, b, and c, such that a2 + b2 = c2. Such a triple is commonly written (a, b, c), and a well-known example is (3, 4, 5). If (a, b, c) is a Pythagorean triple, then so is (ka, kb, kc) for any positive integer k.

A primitive Pythagorean triple is one in which a, b and c are coprime (that is, they have no common divisor larger than 1).  A triangle whose sides form a Pythagorean triple is called a Pythagorean triangle, and is necessarily a right triangle.

There are 16 primitive Pythagorean triples with c ≤ 100:
(3, 4, 5) (5, 12, 13) (8, 15, 17) (7, 24, 25)
(20, 21, 29) (12, 35, 37) (9, 40, 41) (28, 45, 53)
(11, 60, 61) (16, 63, 65) (33, 56, 65) (48, 55, 73)
(13, 84, 85) (36, 77, 85) (39, 80, 89) (65, 72, 97)

Note, for example, that (6, 8, 10) is not a primitive Pythagorean triple, as it is a multiple of (3, 4, 5). Each of these low-c points forms one of the more easily recognizable radiating lines in the scatter plot.

Additionally these are all the primitive Pythagorean triples with 100 < c ≤ 300:

(20, 99, 101) (60, 91, 109) (15, 112, 113) (44, 117, 125)
(88, 105, 137) (17, 144, 145) (24, 143, 145) (51, 140, 149)
(85, 132, 157) (119, 120, 169) (52, 165, 173) (19, 180, 181)
(57, 176, 185) (104, 153, 185) (95, 168, 193) (28, 195, 197)
(84, 187, 205) (133, 156, 205) (21, 220, 221) (140, 171, 221)
(60, 221, 229) (105, 208, 233) (120, 209, 241) (32, 255, 257)
(23, 264, 265) (96, 247, 265) (69, 260, 269) (115, 252, 277)
(160, 231, 281) (161, 240, 289) (68, 285, 293)

Euclid of Alexandria (300 BC-       ):  (Greek)  He was often referred to as the "Father of Geometry."             His "Elements" is one of the most influential works in mathematics, serving as the main textbook for teaching mathematics, especially geometry, from the time of its publication until the late 19th or early 20th century.           Euclid  would love the following web sites:   http://www.flixxy.com/missing-square-puzzle.htm#.UXvz1q7D8fc   and https://en.wikipedia.org/wiki/Missing_square_puzzle and https://www.brainbashers.com/missinganswer.asp  and http://mathusiasts.wordpress.com/2011/04/16/missing-square-puzzle-solution/ but Fibonacci would have preferred http://jaysdesktop.blogspot.com/2009/07/solution-to-missing-square-problem.html        

Geometry  Are the eight balls moving in a circle or a straight line?
http://showyou.com/v/y-pNe6fsaCVtI/crazy-circle-illusion?u=multimotion
   

Archimedes of Syracuse (c.287-c.212 B.C):  (from Sicily) A mathematician and inventor. He determined the exact value of pi, is also known for his strategic role in ancient war and the development of military techniques. "Give me a place to stand and I will move the earth" was his boast when he discovered the laws of levers and pulleys.  His mechanical inventions defeated the Roman fleet of Marcellus.  The word "eureka" comes from the story that when Archimedes figured out a way to determine whether the king (Hiero II of Syracuse), a possible relative, had been duped by measuring the buoyancy of the king's supposedly solid gold crown in water, he became very excited and exclaimed the Greek (Archimedes' native language) for "I have found it": Eureka. Archimedes requested that his tombstone be decorated with a sphere contained in the smallest possible cylinder and inscribed with the ratio of the cylinder's volume to that of the sphere. Archimedes considered the discovery of this ratio the greatest of all his accomplishments.  

  Heron of Alexandria    Not sure of when he lived. Probably around 62 AD. Spent time in the Library of Alexandria.  Heron also proved his famous formula of the area, A, of a triangle knowing the three sides is:

Heron's Formula

triangle sss

Area of a Triangle from Sides

You can calculate the area of a triangle if you know the lengths of all three sides, using a formula that has been known for nearly 2000 years.

It is called "Heron's Formula" after Hero of Alexandria 

Step 1: Calculate "s" (half of the triangles perimeter):   s = (a+b+c)/2
     
Step 2: Then calculate the Area:   herons formula A = sqrt( s(s-a)(s-b)(s-c) )

 

It has been suggested that Archimedes knew the formula over two centuries earlier.    Other references:  www.math.umn.edu/~rejto/1151/1151_heron.pdf and mathworld.wolfram.com/HeronsFormula.html

    Diophantus of Alexandria (200 and 214 -- 284 and 298): (Greek) Sometimes called "the father of algebra", was an Alexandrian Greek mathematician and the author of a series of books called Arithmetica. These texts deal with solving algebraic equations, many of which are now lost. In studying Arithmetica. Pierre de Fermat concluded that a certain equation considered by Diophantus had no solutions, and noted without elaboration that he had found "a truly marvelous proof of this proposition," now referred to as Fermat's Last Theorem. This led to tremendous advances in number theory, and the study of Diophantine equations ("Diophantine geometry") and of Diophantine approximations remain important areas of mathematical research. Diophantus was the first Greek mathematician who recognized fractions as numbers; thus he allowed positive rational numbers for the coefficients and solutions. In modern use, Diophantine equations are usually algebraic equations with integer coefficients, for which integer solutions are sought. Diophantus also made advances in mathematical notation.  

Read about Diophantine Equations and also The Pirate, Monkey, and Coconuts.

  Liu Hui (fl. 3rd century CE) was a Chinese mathematician who lived in the state of Cao Wei during the Three Kingdoms period (220–280) of China. In 263, he edited and published a book with solutions to mathematical problems presented in the famous Chinese book of mathematics known as The Nine Chapters on the Mathematical Art, in which he was possibly the first mathematician to discover, understand and use negative numbers.Liu provided commentary on a mathematical proof of a theorem identical to the Pythagorean theorem. Liu called the figure of the drawn diagram for the theorem the "diagram giving the relations between the hypotenuse and the sum and difference of the other two sides whereby one can find the unknown from the known".

In the field of plane areas and solid figures, Liu Hui was one of the greatest contributors to empirical solid geometry. For example, he found that a wedge with rectangular base and both sides sloping could be broken down into a pyramid and a tetrahedral wedge.  He also found that a wedge with trapezoid base and both sides sloping could be made to give two tetrahedral wedges separated by a pyramid. In his commentaries on the Nine Chapters, he presented:

bulletAn algorithm for calculation of pi (π) in the comments to chapter 1. He calculated pi to  with a 192 (= 64 × 3) sided polygon. Archimedes used a circumscribed 96-polygon to obtain the inequality , and then used an inscribed 96-gon to obtain the inequality . Liu Hui used only one inscribed 96-gon to obtain his π inequality, and his results were a bit more accurate than Archimedes'.[9] But he commented that 3.142074 was too large, and picked the first three digits of π = 3.141024 ~3.14 and put it in fraction form . He later invented a quick method and obtained , which he checked with a 3072-gon(3072 = 512 × 6). Nine Chapters had used the value 3 for π, but Zhang Heng (78-139 AD) had previously estimated pi to the square root of 10.
bullet Gaussian elimination.
bullet Cavalieri's principle to find the volume of a cylinder and the intersection of two perpendicular cylinders  although this work was only finished by Zu Chongzhi and Zu Gengzhi. Liu's commentaries often include explanations why some methods work and why others do not. Although his commentary was a great contribution, some answers had slight errors which was later corrected by the Tang mathematician and Taoist believer Li Chunfeng.
bulletThrough his work in the Nine Chapters, he could have been the first mathematician to discover and compute with negative numbers; definitely before Ancient Indian mathematician Brahmagupta started using negative numbers.

Liu Hui's information about surveying was known to his contemporaries as well. The cartographer and state minister Pei Xiu (224–271) outlined the advancements of cartography, surveying, and mathematics up until his time. This included the first use of a rectangular grid and graduated scale for accurate measurement of distances on representative terrain maps.Liu Hui provided commentary on the Nine Chapter's problems involving building canal and river dykes, giving results for total amount of materials used, the amount of labor needed, the amount of time needed for construction, etc. Although translated into English long beforehand, Liu's work was translated into French by Guo Shuchun, a professor from the Chinese Academy of Sciences, who began in 1985 and took twenty years to complete his translation.

    Ryabhata (476 - 550): (Indian)  Was the first of the major Indian mathematician astronomers. His works include the Aryabhatiya when he was 23 years old and the Arya-siddhantaHe could be thought of as one of the first actuaries, as he showed how to calculate the present value of a series of financial transactions.  See  Ryabhata

    Bhāskara (c. 600 – c. 680) (commonly called Bhaskara I:  (Indian) A 7th-century mathematician, who was the first to write numbers in the Hindu decimal system with a circle for the zero, and who gave a unique and remarkable rational approximation of the sine function in his commentary on Aryabhatta's work. This commentary, Āryabhaṭīyabhāṣya, written in 629 CE, is among the oldest known prose works in Sanskrit on mathematics and astronomy. He also wrote two astronomical works in the line of Aryabhata's school, the Mahābhāskarīya and the Laghubhāskarīya.[ On 7 June 1979 the Indian Space Research Organisation launched Bhaskara I honouring the mathematician.

 

    Sridharacharya (c.750 CE – c. 930 CE): (Indian) A mathematician, Sanskrit pandit and philosopher. He was known for 2 treatises: Trisatika (Sometimes called the Patiganitasara) and the Patiganita. His major work Patiganitasara was named Trisatika because it was written in three hundred slokas. The book discusses counting of numbers, measures, natural number, multiplication, division, zero, squares, cubes, fraction, rule of three, interest- calculation, joint business or partnership and mensurations.

bulletHe gave an exposition on the zero. He wrote, "If zero is added to any number, the sum is the same number; if zero is subtracted from any number, the number remains unchanged; if zero is multiplied by any number, the product is zero".
bulletIn the case of dividing a fraction he has found out the method of multiplying the fraction by the reciprocal of the divisor.
bulletHe wrote on the practical applications of algebra
bulletHe separated algebra from arithmetic
bulletHe was one of the first to give a formula for solving quadratic equations.

 

     Muḥammad ibn Mūsā al-Khwārizmī (Persian: محمد بن موسى خوارزمی‎; c. 780 – c. 850), formerly Latinized as Algoritmi, was a Persian] scholar in the House of Wisdom in Baghdad who produced works in mathematics, astronomy, and geography during the Abbasid Caliphate.

In the 12th century, Latin translations of his work on the Indian numerals introduced the decimal positional number system to the Western world.[5] Al-Khwārizmī's The Compendious Book on Calculation by Completion and Balancing presented the first systematic solution of linear and quadratic equations in Arabic. Because he is the first to teach algebra as an independent discipline and introduced the methods of "reduction" and "balancing" (the transposition of subtracted terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation), he has been described as the father[6][7][8] or founder[9][10] of algebra. His work on algebra was used until the sixteenth century as the principle mathematical text-book of European universities.[11]

He revised Ptolemy's Geography and wrote on astronomy and astrology.

Some words reflect the importance of al-Khwārizmī's contributions to mathematics. "Algebra" is derived from al-jabr, one of the two operations he used to solve quadratic equations. Algorism and algorithm stem from Algoritmi, the Latin form of his name.[12] His name is also the origin of (Spanish) guarismo[13] and of (Portuguese) algarismo, both meaning digit.

     Leonardo Pisano Fibonacci (1170?-1250):  (Italian)  Fibonacci is considered to be one of the most talented mathematicians for the Middle Ages. Few people realize that it was Fibonacci that gave us our decimal number system (Hindu-Arabic numbering system) which replaced the Roman Numeral system. When he was studying mathematics, he used the Hindu-Arabic (0-9) symbols instead of Roman symbols which didn't have 0's and lacked place value. In fact, when using the Roman Numeral system, an abacus was usually required. There is no doubt that Fibonacci saw the superiority of using Hindu-Arabic system over the Roman Numerals. He shows how to use our current numbering system in his book Liber abbaci. And he gave us the Fibonacci Series. Fibonacci was known as Leonardo of Pisa. He was born in Pisa, home of the famous leaning tower and his statue is located there.

According to William Goetzmann Professor of Finance at Yale University, evidence in Fibonacci's famous Liber abbaci,  Fibonacci was the first to develop present value analysis for comparing the economic value of alternative contractual cash flows.  The modern present-value formula was developed by Irving Fisher in 1930. We can consider Fibonacci one of the first actuaries based on his writings in liber "actuary"   And  Was Fibonacci an Actuary?  Although the latter article was written in fun, Fibonacci could have been considered an actuary. 

In his famous "Rabbit Problem" he produces the Fibonacci Series as the answer: 1 1 2 3 5 8 13 21 34 55 etc., where each term is equal to the sum of the two previous terms.  The Fibonacci sequence obeys the recursion relation F(n) = F(n-1) + F(n-2). The ratio of the current term to the previous term approaches the golden ratio or (1 + sq rt of 5)/2, about 1.618...  This ratio is called the "golden ratio".  The German Adolph Zeising claimed the front of the Parthenon is in proportion to the golden ratio. There is no documentary evidence that Phidias, used the golden ratio in any of his work related to the Parthenon.  However around 1909, the American mathematician Mark Barr, named the golden ratio the Greek letter "phi" for Phidias..  When phi is expressed as a continued fraction it looks like this:

Continued fractions provide mathematicians with a way of rating how irrational a number might be. Since the expression for phi contains only 1s, it is the purest continued fraction that there is, and hence is considered the most irrational number.

IRRATIONAL NUMBERS See http://www.ams.org/samplings/feature-column/fcarc-irrational1

An irrational number by definition is one which cannot be written as the ratio of whole numbers. So it would seem that all irrational numbers are equally irrational. All pigs are equal, Orwell said, but some are more equal than others. And in fact there is a precise sense in which some irrational numbers are more irrational than others. This phenomenon has had important consequences in the organization of the natural world. In packing seeds around a core, many plants choose the strategy of placing each one at the most irrational angle possible to the one directly below it.

The Pythagoreans did not believe in irrational numbers or incommensurability.  But in a^{2}+b^{2}=c^{2}\!\,  we can look at a square with a and b equal to 1 and c = square root of 2.  One rectangle has a =1, c = 2, and b = square root of 3.  Another rectangle has a =1, b = 2, and c = square root of 5.  

These irrational numbers are also found in other geometrical shapes.  The pentagram which contains  sqr5 .  Sir Thomas Mallory in La Morte d'Arthur placed it on Sir Gawain's shield. In Dan Brown'sThe Da Vinci Code, the dying curator of the Louvre drew a pentagram in his own blood on his abdomen as a clue to identify his murderer.  The hexagram was also mentioned by Dan Brown in the same novel.  It contains 3, which is the length of the shorter diagonal of a hexagram whose sides equal 1.

In studying "Sacred Geometry" (http://sacredgeometryinternational.com/the-meaning-of-sacred-geometry-part-3-the-womb-of-sacred-geometry# )we see the Vessel of Fish. This diagram is usually shown as two identical circles in which the right hand circle passes thru the center of the first circle on the left.  The square roots of 3 and 5 are included. The many objects of Sacred Geometry may be achieved by means of the figure known as the Vesica Piscis.

The reconciliation of six and five and many of the other objects of sacred geometry may be achieved by means of the figure known as the vesica piscis

The most irrational number

The most irrational number turns out to be a number already well known in geometry. It is the number 

Phi = phi = (sqr5 + 1)/2 = 1.618033...

which is the length of the diagonal in a regular pentagon of side length 1. This number, known as the "golden mean," has played a large role in mathematical aesthetics. It is not clear whether its supreme irrationality has anything to do with its artistic applications.

It is the only number that is 1 more than its reciprocal. x = 1/x  -1.

The golden mean satisfies the equation x2 - x - 1 = 0, so its continued fraction expansion is the simplest of all:

A representation in terms of a nested radical is

 phi=sqrt(1+sqrt(1+sqrt(1+sqrt(1+...))))

phi is the "worst" real number for rational approximation because its continued fraction is:

phi =

      1+1/(1+1/(1+1/(1+...)))

The Fibonacci numbers are 0, 1, 1, 2, 3, 5, 8, 13, ... (add the last two to get the next)

The Golden Section numbers are ±0·61803 39887... and ±1·61803 39887...  

Phi is the only number that is 1 less than its reciprocal.  Phi -1 = 1/Phi

The Golden String is a fractal string of 0s and 1s that grows in a Fibonacci-like way as follows: 1                                                                                                                                                                                                  10                                                                                                                                                                                               101                                                                                                                                                                                                                        10110                                                                                                                                                                                                          10110101                                                                                                                                                                                           1011010110110                                                                                                                                                                                             101101011011010110101   After the first two lines, all the others are made from the two latest lines in a similar way to each Fibonacci numbers being a sum of the two before it. Each string (list of 0s and 1s) is a copy of the one above it followed by the one above that. The resulting infinitely long string is the Golden String or Fibonacci Word or Rabbit Sequence.  It is is closely related to the Fibonacci numbers and the golden section. There is a relationship between fractals and the "golden string".         See http://www.youtube.com/watch?v=ZDGGEQqSXew

Leonardo Da Vinci called the golden ratio  the "divine proportion" and featured it in many of his paintings.

    Madhava of Sangamagrama (c. 1340 – c. 1425): (Indian) He was a mathematician and astronomer from the town of Sangamagrama (believed to be present-day Aloor, Irinjalakuda in Thrissur District), Kerala, India. He is considered the founder of the Kerala school of astronomy and mathematics. He was the first to use infinite series approximations for a range of trigonometric functions, which has been called the "decisive step onward from the finite procedures of ancient mathematics to treat their limit-passage to infinity".[1] One of the greatest mathematician-astronomers of the Middle Ages, Madhava made pioneering contributions to the study of infinite series, calculus, trigonometry, geometry, and algebra.

     Nicolaus Copernicus (1473-1543):  (Prussia) He was a Renaissance astronomer and the first person to formulate a comprehensive heliocentric cosmology, which displaced the Earth from the center of the universe.  Copernicus' epochal book, De revolutionibus orbium coelestium (On the Revolutions of the Celestial Spheres), published just before his death in 1543, is often regarded as the starting point of modern astronomy and the defining epiphany that began the scientific revolution. His heliocentric model, with the Sun at the center of the universe, demonstrated that the observed motions of celestial objects can be explained without putting Earth at rest in the center of the universe. His work stimulated further scientific investigations, becoming a landmark in the history of science that is often referred to as the Copernican Revolution.

    Niccolò Fontana "Tartaglia" (1499/1500-1557): (Italian)  He published many books, including the first Italian translations of Archimedes and Euclid, and an acclaimed compilation of mathematics. Tartaglia was the first to apply mathematics to the investigation of the paths of cannonballs, known as ballistics, in his Nova Scientia, “A New Science;” his work was later validated by Galileo's studies on falling bodies. He also published a treatise on retrieving sunken ships.

When the French invaded Brescia, a French soldier sliced Niccolò's jaw and palate with a saber. This made it impossible for Niccolò to speak normally, prompting the nickname "Tartaglia" ("stammerer"), which he adopted.

His best known work is his treatise General Trattato di numeri, et misure published in Venice 1556–1560. This has been called the best treatise on arithmetic that appeared in the sixteenth century. Not only does Tartaglia have complete discussions of numerical operations and the commercial rules used by Italian arithmeticians in this work, but he also discusses the life of the people, the customs of merchants and the efforts made to improve arithmetic in the 16th century.

Tartaglia is known for his conflicts with Gerolamo Cardano. Cardano cajoled Tartaglia into revealing the solution to the cubic equations, by promising not to publish them. Tartaglia divulged the secrets of the solutions of three different forms of the cubic equation in verse.  Even though Cardano credited his discovery, Tartaglia was extremely upset. He responded by publicly insulting Cardano.  Mathematical historians now credit both with the paternity of the formula to solve cubic equations, referring to it as the "Cardano-Tartaglia Formula".

    Gerolamo Cardano (1501-1576): (French)  He was an Italian Renaissance mathematician, physician, astrologer and gambler. Today, he is best known for his achievements in algebra. He published the solutions to the cubic and quartic equations in his 1545 book Ars Magna. The solution to one particular case of the cubic, x3 + ax = b (in modern notation), was communicated to him by Niccolo Fontana Tartaglia (who later claimed that Cardano had sworn not to reveal it, and engaged Cardano in a decade-long fight), The quartic was solved by Cardano's student Lodovico Ferrari. Both were acknowledged in the foreword of the book, as well as in several places within its body. In his exposition, he acknowledged the existence of what are now called imaginary numbers, although he did not understand their properties (Mathematical field theory was developed centuries later). In Opus novum de proportionibus he introduced the binomial coefficients and the binomial theorem

Cardano was notoriously short of money and kept himself solvent by being an accomplished gambler and chess player. His book about games of chance, Liber de ludo aleae ("Book on Games of Chance") , written in 1526, but not published until 1663, contains the first systematic treatment of probability, as well as a section on effective cheating methods. Cardano invented several mechanical devices including the combination lock, the gimbal consisting of three concentric rings allowing a supported compass or gyroscope to rotate freely, and the Cardan shaft with universal joints, which allows the transmission of rotary motion at various angles and is used in vehicles to this day. He studied hypocycloids, published in de proportionibus 1570. The generating circles of these hypocycloids were later named Cardano circles or cardanic circles and were used for the construction of the first high-speed printing presses.

      Franciscus Vieta (1540-1603): (French) His work on new algebra was an important step towards modern algebra, due to its innovative use of letters as parameters in equations. He was a lawyer by trade, and served as a privy councillor to both Henry III and Henry IV.

    Galileo Galilei(1564-1642): (Italian)  A physicist, mathematician, astronomer and philosopher who played a major role in the Scientific Revolution. His achievements include improvements to the telescope and consequent astronomical observations, and support for Copernicanism.  Galileo has been called the "father of modern observational astronomy", the "father of modern physics", the "father of science", and "the Father of Modern Science". Stephen Hawking says, "Galileo, perhaps more than any other single person, was responsible for the birth of modern science."  Read about his "square cube" law:    http://dinosaurtheory.com/scaling.html

    Johannes Kepler (1571-1630): (German)  A mathematician, astronomer and astrologer, and key figure in the 17th century scientific revolution. He is best known for his eponymous laws of planetary motion, codified by later astronomers, based on his works Astronomia nova, Harmonices Mundi, and Epitome of Copernican Astronomy. These works also provided one of the foundations for Isaac Newton's theory of universal gravitation.

    Marin Mersenne, Marin Mersennus or le Père Mersenne (September 1588 – 1 September 1648): French polymath, whose works touched a wide variety of fields. He is perhaps best known today among mathematicians for Mersenne prime numbers, those which can be written in the form Mn = 2n − 1 for some integer n. He also developed Mersenne's laws, which describe the harmonics of a vibrating string (such as may be found on guitars and pianos), and his seminal work on music theory, Harmonie universelle, for which he is referred to as the "father of acoustics". Mersenne, an ordained priest, had many contacts in the scientific world and has been called "the center of the world of science and mathematics during the first half of the 1600s." He was also a member of the Minim religious order, and wrote and lectured on theology and philosophy.
Marin mersenne.jpg

Marin Mersenne was born of peasant parents near Oizé, Maine (present day Sarthe, France). He was educated at Le Mans and at the Jesuit College of La Flèche. On 17 July 1611, he joined the Minim Friars, and, after studying theology and Hebrew in Paris, was ordained a priest in 1613.  See  https://en.wikipedia.org/wiki/Mersenne_prime

Between 1614 and 1618, he taught theology and philosophy at Nevers, but he returned to Paris and settled at the convent of L'Annonciade in 1620. There he studied mathematics and music and met with other kindred spirits such as René Descartes, Étienne Pascal, Pierre Petit, Gilles de Roberval, Thomas Hobbes, and Nicolas-Claude Fabri de Peiresc. He corresponded with Giovanni Doni, Constantijn Huygens, Galileo Galilei, and other scholars in Italy, England and the Dutch Republic. He was a staunch defender of Galileo, assisting him in translations of some of his mechanical works.

For four years, Mersenne devoted himself entirely to philosophic and theological writing, and published Quaestiones celeberrimae in Genesim (1623); L'Impieté des déistes (1624); La Vérité des sciences (Truth of the Sciences against the Sceptics, 1624). It is sometimes incorrectly stated that he was a Jesuit. He was educated by Jesuits, but he never joined the Society of Jesus. He taught theology and philosophy at Nevers and Paris.

In 1635 he set up the informal Académie Parisienne (Academia Parisiensis) which had nearly 140 correspondents including astronomers and philosophers as well as mathematicians and was the precursor of the Académie des sciences established by Jean-Baptiste Colbert in 1666. He was not afraid to cause disputes among his learned friends in order to compare their views, notable among which were disputes between Descartes and Pierre de Fermat and Jean de Beaugrand.[4] Peter L. Bernstein in his book Against the Gods: the Remarkable story of risk writes: "The Académie des Sciences in Paris and the Royal Society in London, which were founded about twenty years after Mersenne's death, were direct descendants of Mersenne's activities."[5]

In 1635 Mersenne met with Tommaso Campanella, but concluded that he could "teach nothing in the sciences (...) but still he has a good memory and a fertile imagination." Mersenne asked if René Descartes wanted Campanella to come to Holland to meet him, but Descartes declined. He visited Italy fifteen times, in 1640, 1641 and 1645. In 1643–1644 Mersenne also corresponded with the German Socinian Marcin Ruar concerning the Copernican ideas of Pierre Gassendi, finding Ruar already a supporter of Gassendi's position.  Among his correspondents were Descartes, Galilei, Roberval, Pascal, Beeckman and other scientists.

He died September 1 through complications arising from a lung abscess. Some history scientists suggest he died for having drunk a huge quantity of fresh water, along with Descartes, on a hot summer day.

He wrote a commentary on the Book of Genesis and comprises uneven sections headed by verses from the first three chapters of that book. At first sight the book appears to be a collection of treatises on various miscellaneous topics. However Robert Lenoble has shown that the principle of unity in the work is a polemic against magical and divinatory arts, cabalism, animistic and pantheistic philosophies. He mentions Martin Del Rio's Investigations into Magic and criticises Marsilio Ficino for claiming power for images and characters. He condemns astral magic and astrology and the anima mundi, a concept popular amongst Renaissance neo-platonists. Whilst allowing for a mystical interpretation of the Cabala, he wholeheartedly condemned its magical application—particularly to angelology. He also criticises Pico della Mirandola, Cornelius Agrippa and Francesco Giorgio with Robert Fludd as his main target. Fludd responded with Sophia cum moria certamen (1626), wherein Fludd admits his involvement with the Rosicrucians. The anonymous Summum bonum (1629), another critique of Mersenne, is an openly Rosicrucian text. The cabalist Jacques Gaffarel joined Fludd's side, while Pierre Gassendi defended Mersenne.

L'Harmonie universelle is perhaps Mersenne's most influential work. It is one of the earliest comprehensive works on music theory, touching on a wide range of musical concepts, and especially the mathematical relationships involved in music. The work contains the earliest formulation of what has become known as Mersenne's laws, which describe the frequency of oscillation of a stretched string. This frequency is:

  1. Inversely proportional to the length of the string (this was actually known to the ancients, and is usually credited to Pythagoras himself).
  2. Proportional to the square root of the stretching force, and
  3. Inversely proportional to the square root of the mass per unit length.

The formula for the lowest frequency is

where f is the frequency, L is the length, F is the force and μ is the mass per unit length.

In this book, Mersenne also introduced several innovative concepts that can be considered as the basis of modern reflecting telescopes:

bulletMuch earlier than Laurent Cassegrain, he found the fundamental arrangement of the two-mirror telescope combination, a concave primary mirror associated with a convex secondary mirror, and discovered the telephoto effect that is critical in reflecting telescopes, although it is obvious that he was far from having understood all the implications of that discovery.
bulletMersenne invented the afocal telescope and the beam compressor that is useful in many multiple-mirror telescope designs.[7]
bulletHe recognized also that he could correct the spherical aberration of the telescope by using aspherical mirrors and that in the particular case of the afocal arrangement he could do this correction by using two parabolic mirrors, though a hyperboloid is required.

Because of criticism that he encountered, especially that of Descartes, Mersenne made no attempt to build a telescope of his own.

Mersenne is also remembered today thanks to his association with the Mersenne primes. The Mersenne Twister, named for Mersenne prime, is frequently used in computer engineering, and in related fields such as cryptography.

However, Mersenne was not primarily a mathematician; he wrote about music theory and other subjects. He edited works of Euclid, Apollonius, Archimedes, and other Greek mathematicians. But perhaps his most important contribution to the advance of learning was his extensive correspondence (in Latin) with mathematicians and other scientists in many countries. At a time when the scientific journal had not yet come into being, Mersenne was the centre of a network for exchange of information.

It has been argued that Mersenne used his lack of mathematical specialty, his ties to the print world, his legal acumen, and his friendship with the French mathematician and philosopher René Descartes (1596–1650) to manifest his international network of mathematicians.[9]

Mersenne's philosophical works are characterized by wide scholarship and the narrowest theological orthodoxy. His greatest service to philosophy was his enthusiastic defence of Descartes, whose agent he was in Paris and whom he visited in exile in the Netherlands. He submitted to various eminent Parisian thinkers a manuscript copy of the Meditations on First Philosophy, and defended its orthodoxy against numerous clerical critics.

In later life, he gave up speculative thought and turned to scientific research, especially in mathematics, physics and astronomy. In this connection, his best known work is Traité de l'harmonie universelle (also referred to as Harmonie universelle) of 1636, dealing with the theory of music and musical instruments. It is regarded as a source of information on 17th-century music, especially French music and musicians, to rival even the works of Pietro Cerone.

He made many contributions to musical tuning theory.  The ratio for an equally-tempered semitone (t was more accurate (0.44 cents sharp) than Vincenzo Galilei's 18/17 (1.05 cents flat), and could be constructed using straightedge and compass. Mersenne's description in the 1636 Harmonie universelle of the first absolute determination of the frequency of an audible tone (at 84 Hz) implies that he had already demonstrated that the absolute-frequency ratio of two vibrating strings, radiating a musical tone and its octave, is 1 : 2. The perceived harmony (consonance) of two such notes would be explained if the ratio of the air oscillation frequencies is also 1 : 2, which in turn is consistent with the source-air-motion-frequency-equivalence hypothesis.

He also performed extensive experiments to determine the acceleration of falling objects by comparing them with the swing of pendulums, reported in his Cogitata Physico-Mathematica in 1644. He was the first to measure the length of the seconds pendulum, that is a pendulum whose swing takes one second, and the first to observe that a pendulum's swings are not isochronous as Galileo thought, but that large swings take longer than small swings.

See  https://www.youtube.com/watch?v=XNI0Lpjjdiw

     Rene Descartes (1596-1650): (French)  The inventor of Analytical Geometry.  He was a philosopher, mathematician, physicist and writer. He has been dubbed the "Father of Modern Philosophy".     If you want interesting info on conic sections, see  https://mysite.du.edu/~jcalvert/    (James B Calvert is a retired professor of Engineering at the University of Denver and has an excellent web site).

     Pierre de Fermat (1601-1665): (French)  A lawyer and amateur mathematician who contributed to Number Theory and known for "Fermat's Last Theorem".  Fermat was the first person known to have evaluated the integral of general power functions. Using an ingenious trick, he was able to reduce this evaluation to the sum of geometric series. The resulting formula was helpful to both Newton and Leibnitz in developing calculus.    See https://www.youtube.com/watch?v=ua1K3Eo2PQc

    Evangelista Torricelli (15 October 1608 – 25 October 1647): (Italian) A physicist and mathematician, best known for his invention of the barometer, but is also known for his advances in optics and work on the method of indivisibles.  If a rectangular tank contains water that is H feet high, takes M minutes to empty when a plug is removed, then when that tank is filled to the height of 2H, it will take the square root of 2 times M to empty when the plug is removed, per "Torricelli's Law".

   Torricelli's law, also known as Torricelli's theorem, is a theorem in fluid dynamics relating the speed of fluid flowing out of an orifice to the height of fluid above the opening. The law states that the speed of efflux, v, of a fluid through a sharp-edged hole at the bottom of a tank filled to a depth h is the same as the speed that a body (in this case a drop of water) would acquire in falling freely from a height h, i.e The law was discovered (though not in this form) by the Italian scientist Evangelista Torricelli, in 1643. It was later shown to be a particular case of Bernoulli's principle.

    John Wallis (1616-1703): (English) A  mathematician who is given partial credit for the development of infinitesimal calculus. Between 1643 and 1689 he served as chief cryptographer for Parliament and, later, the royal court. He is also credited with introducing the symbol ∞ for infinity.  Wallis made significant contributions to trigonometry, calculus, geometry, and the analysis of infinite series. In his Opera Mathematica I (1695) Wallis introduced the term "continued fraction".  He is generally credited as the originator of the idea of the number line where numbers are represented geometrically in a line with the positive numbers increasing to the right and negative numbers to the left.  In 1655, Wallis published a treatise on conic sections in which they were defined analytically. This was the earliest book in which these curves are considered and defined as curves of the second degree. It helped to remove some of the perceived difficulty and obscurity of Rene Descartes' work on analytic geometryArithmetica Infinitorum, the most important of Wallis's works, was published in 1656. In this treatise the methods of analysis of Descartes and Cavalieri were systematised and extended.  in 1659, Wallis published a tract containing the solution of the problems on the cycloid which had been proposed by Blaise Pascal. In this he incidentally explained how the principles laid down in his Arithmetica Infinitorum could be used for the rectification of algebraic curves; and gave a solution of the problem to rectify (i.e. find the length of) the semi-cubical parabola x3 = ay2, which had been discovered in 1657 by his pupil William Neile. Since all attempts to rectify the ellipse and hyperbola had been (necessarily) ineffectual, it had been supposed that no curves could be rectified, as indeed Descartes had definitely asserted to be the case. The logarithmic spiral had been rectified by Evangelista Torricelli, and was the first curved line (other than the circle) whose length was determined, but the extension by Neil and Wallis to an algebraic curve was novel. The cycloid was the next curve rectified; this was done by Wren in 1658.

    Blaise Pascal (1623-1662): (French) He helped create two major new areas. He wrote a significant treatise on  projective geometry  at the age of sixteen.   Pascal's development of probability theory was his most influential contribution to mathematics, a subject on which he corresponded with Fermat.  Pascal continued to influence mathematics throughout his life. In 1653 he described a convenient tabular presentation for binomial coefficients, now called Pascal's triangle.  See http://www.storyofmathematics.com/17th_pascal.html  and https://www.pinterest.com/pin/456271005969597610/?lp=true

He was a French mathematician, physicist, inventor, writer and Catholic theologian. He was a child prodigy who was educated by his father, a tax collector in Rouen. Pascal's earliest work was in the natural and applied sciences where he made important contributions to the study of fluids, and clarified the concepts of pressure and vacuum by generalising the work of Evangelista Torricelli. Pascal also wrote in defence of the scientific method.

In 1642, while still a teenager, he started some pioneering work on calculating machines. After three years of effort and 50 prototypes,[4] he built 20 finished machines (called Pascal's calculators and later Pascalines) over the following 10 years,[5] establishing him as one of the first two inventors of the mechanical calculator.[

Pascal was an important mathematician, helping create two major new areas of research: he wrote a significant treatise on the subject of projective geometry at the age of 16, and later corresponded with Pierre de Fermat on probability theory, strongly influencing the development of modern economics and social science. Following Galileo Galilei and Torricelli, in 1647, he rebutted Aristotle's followers who insisted that nature abhors a vacuum. Pascal's results caused many disputes before being accepted.

In 1646, he and his sister Jacqueline identified with the religious movement within Catholicism known by its detractors as Jansenism. His father died in 1651. Following a religious experience in late 1654, he began writing influential works on philosophy and theology. His two most famous works date from this period: the Lettres provinciales and the Pensées, the former set in the conflict between Jansenists and Jesuits. In that year, he also wrote an important treatise on the arithmetical triangle. Between 1658 and 1659 he wrote on the cycloid and its use in calculating the volume of solids.

Pascal had poor health, especially after the age of 18, and he died just two months after his 39th birthday.

      Christian Goldbach (March 18, 1690 – November 20, 1764): (German) A mathematician who also studied law. He is remembered today for Goldbach's conjecture: "Every even integer greater than 2 can be expressed as the sum of two primes."   See https://en.wikipedia.org/wiki/Goldbach%27s_conjecture

Born in the Duchy of Prussia's capital Königsberg, part of Brandenburg-Prussia, Goldbach was the son of a pastor. He studied at the Royal Albertus University. After finishing his studies he went on long educational voyages from 1710 to 1724 through Europe, visiting other German states, England, Holland, Italy, and France, meeting with many famous mathematicians, such as Gottfried LeibnizLeonhard Euler, and Nicholas I Bernoulli. Back in Königsberg he got acquainted with Georg Bernhard Bilfinger and Jakob Hermann.

He went on to work at the newly opened St Petersburg Academy of Sciences in 1725, as a professor of mathematics and historian of the academy.  In 1728, when Peter II became Tsar of Russia, Goldbach became his tutor. In 1742 he entered the Russian Ministry of Foreign Affairs. He died on November 20, 1764 at age of 74, in Moscow.

 

The even integers from 4 to 28 as sums of two primes: Even integers correspond to horizontal lines. For each prime, there are two oblique lines, one red and one blue. The sums of two primes are the intersections of one red and one blue line, marked by a circle. Thus the circles on a given horizontal line give all partitions of the corresponding even integer into the sum of two primes. Number of ways to write an even number n as the sum of two primes (4 ≤ n ≤ 1,000,000)

     Sir Isaac Newton (1643-1727):  (British) His theory of gravity unified the force  that keeps our feet on the ground, with the force that holds planets in their orbits. His 1687 publication of the Philosophiae Naturalis Principia Mathematica  is considered to be among the most influential books in the history of science.  In this work, Newton described universal gravitation and the three laws of motion. Newton shares the credit with Leibnitz for the development of differential and integral calculus. He also demonstrated the generalized binomial theorem and contributed to the study of power series.

     Gottfried Wilhelm Leibnitz (1646-1716): (German) He invented infinitesimal calculus independently of Newton, and his notation has been in general use since then. He also invented the binary system, the foundation of virtually all modern computer architectures.

   Bernoulli Family of Swiss Mathematicians: Three were : Jacob Bernoulli (1654-1705), his brother Johann Bernoulli (1667-1748) and Johann's son Daniel Bernoulli (1700-1787). 

   Jacob wrote the Art of Conjecture.  In this work, he described the known results in probability theory and in enumeration, often providing alternative proofs of known results. This work also includes the application of probability theory to games of chance and his introduction of the theorem known as the law of large numbers. The terms Bernoulli trial and Bernoulli numbers result from this work.  He.chose a figure of a logarithmic spiral  and the motto Eadem mutata resurgo ("Changed and yet the same, I rise again") for his gravestone. He called it the spiral mirabilis, the wonderful spiral. The spiral executed by the stonemasons was, however, an Archimedean spiral.  Just like a fractal, a logarithmic spiral is self similar:  That is, any smaller piece of a larger spiral is identical in shape to the larger piece.

   Johann studied the function y = xx and he also investigated series using the method of integration by parts. Integration to Bernoulli was simply viewed as the inverse operation to differentiation and with this approach he had great success in integrating differential equations. He summed series, and discovered addition theorems for trigonometric and hyperbolic functions using the differential equations they satisfy. Johann was known as the "Archimedes of his age" and this is indeed inscribed on his tombstone. 

   Daniel was a Dutch Swiss mathematician. He is particularly remembered for his applications of mathematics to mechanics, especially fluid mechanics and for his pioneering work in probability and statistics. Bernoulli's work is still studied at length by many schools of science throughout the world. The Bernoulli Principle that was used to explain lift applicable to airplane wings was developed by Daniel Bernoulli.  

   Christian Goldbach (3.08.1690 -11.20.1764) born in Königsberg, Prussia [now Kaliningrad, Russia]—died in Moscow, Russia), Russian mathematician whose contributions to number theory including the Goldbach conjecture. In 1725 Goldbach became professor of mathematics and historian of the Imperial Academy at St. Petersburg.

    Thomas Bayes (1702-1761):  An English mathematician, statistician and Presbyterian minister, known for having formulated a specific case of the theorem that bears his name.   Bayes never published what would become his most famous accomplishment.  His notes were edited and published after his death by Richard Price.   Bayes Theorem deals with "conditional probabilities".  http://plato.stanford.edu/entries/bayes-theorem/ 

There were prominent mathematicians that did not embrace the logic of Bayes Theorem .

ead Sharon Birtsch McGraynes book on "The Theory That Would Not Die" and see her on  http://www.youtube.com/watch?v=8oD6eBkjF9o     Also her book is summarized on  http://www.nytimes.com/2011/08/07/books/review/the-theory-that-would-not-die-by-sharon-bertsch-mcgrayne-book-review.html?pagewanted=all&_r=0    

Also read History:  http://lesswrong.com/lw/774/a_history_of_bayes_theorem/

    Leonhard Euler (1707-1783):  One of his many contributions was called "Euler's Formula".  The formula states that, for any real number  xe^{ix} = \cos x + i\sin x \!   where e is the base of the natural logarithm, i is the imaginary unit, and cos and sin are the trigonometric functions, with the argument x given in radians. The formula is still valid if x is a complex number.   Richard Feynman called Euler's formula "our jewel" and "one of the most remarkable, almost astounding, formulas in all of mathematics".   

Euler Line

In the 18th century, the Swiss mathematician Leonhard Euler noticed that three of the  centers of a triangle are always collinear (they always lie on a straight line). The three centers that have this surprising property are the triangle's centroid (where the three medians of the triangle's sides meet), circumcenter (where the perpendicular bisectors of the triangle's sides meet) and the orthocenter (where the three altitudes to the vertices of the triangle meet).  The distance from the orthocenter to the centroid is two times the distance from the centroid to the circumcenter.  (Another center, the incenter, where the bisectors of the three angles meet, is not on this line.)

Acute Triangle
Obtuse Triangle
Equilateral Triangle

Euler and the Nine Point Circle

The nine-point circle is a circle that can be constructed for any given triangle. It is so named because it passes through nine significant concyclic points defined from the triangle. These nine points are:

bulletThe midpoint of each side of the triangle
bulletThe foot of each altitude
bulletThe midpoint of the line segment from each vertex of the triangle to the orthocenter (where the three altitudes meet; these line segments lie on their respective altitudes).

The nine-point circle is also known as Feuerbach's circle, Euler's circle,  and Terquem's circle.

To construct the nine point circle of a triangle, see http://jwilson.coe.uga.edu/EMT668/EMT668.Folders.F97/Anderson/geometry/geometry1project/construction/construction.html

.1. Draw a triangle ABC and construct the midpoints of the three sides. Label them as L, M, N.

2. Construct the feet of the altitudes of the triangle ABC. Label them as D, E, F. Label the point of intersection of the three altitudes as H. This is also called the orthocenter.

 

3. Construct the midpoints of the segments AH, BH, CH. Label them as X, Y, Z.

4. Notice the nine points, L,M,N,D,E,F,X,Y, Z, lie in a circle called the Nine-Point Circle..

5. Construct the circumscribed circle for triangle LMN. Label the center of that circle U.

The center U of the circumscribed circle for triangle LMN will also be the center of the Nine-Point Circle.

More on Nine-Point Circle at http://en.wikipedia.org/wiki/Triangle_center where we learn  the following:

Let A, B, C denote the vertex angles of the reference triangle, and let x : y : z be a variable point in trilinear coordinates; then an equation for the Euler line is

\sin 2A \sin(B - C)x + \sin 2B \sin(C - A)y + \sin 2C \sin(A - B)z = 0.\,
Center Trilinear Coordiates On Euler Line?
Orttocenter Secant A: Secant B : Secant C Yes
Centroid Cosecant A: Cosecant B: Cosecant C Yes
Circumcenter Cosine A: Cosine B: Cosine C Yes
Nine Point Circle Cosine (B - C): Cosine (C - A): Cosine (A - B) Yes
In Center 1:1:1 Only if Isosceles

Euler and polyhedrons

A platonic solid is a regular, convex polyhedron with congruent faces of regular polygons and the same number of faces meeting at each vertex.  There are five regular polyhedrons that  meet those criteria, and each is named after its number of faces.:

Tetrahedron Hexahedron Octahedron Dodecahedron Icosahedron
Tetrahedron.svg Hexahedron.svg Octahedron.svg POV-Ray-Dodecahedron.svg Icosahedron.svg
4 Triangles 4 Squares 8 Triangles 12 Pentagons 20 Triangles

Euler's formula for polyhedrons is:   V - E + F = 2  That is the number of vertices, minus the number of edges, plus the number of faces, is equal to two.

More on Euler

In analytical mathematics, Euler's identity (also known as Euler's equation) is the equality:

eiπ + 1 = 0
e is Euler's number, the base of natural logarithms
i is the imaginary unit, which satisfies i2 = −1
π is pi, the ratio of the circumference of a circle to its diameter

Euler's  number e is an important mathematical constant, approximately equal to 2.71828, that is the base of the natural logarithm.  It is the limit of (1 + 1/n)n as n becomes large, an expression that arises in the study of compound interest, and can also be calculated as the sum of the infinite series:

e =  \displaystyle\sum\limits_{n = 0}^{ \infty} \dfrac{1}{n!} = 1 + \frac{1}{1} + \frac{1}{1\cdot 2} + \frac{1}{1\cdot 2\cdot 3} + \cdots

e is the unique real number such that the value of the derivative (slope of the tangent line) of the function f(x) = ex at the point x = 0 is equal to 1. The function ex so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm to base e.

The number e is of eminent importance in mathematics, alongside 0, 1, π and i. All five of these numbers play important and recurring roles across mathematics, and are the five constants appearing in one formulation of Euler's identity. Like the constant π, e is irrational: it is not a ratio of integers; and it is transcendental: it is not a root of any non-zero polynomial with rational coefficients. The numerical value of e truncated to 50 decimal places is 2.71828182845904523536028747135266249775724709369995...

The Euler–Mascheroni constant (also called Euler's constant) is a mathematical constant recurring in analysis and number theory, usually denoted by the lowercase Greek letter \gamma(Gamma). It is defined as the limiting difference between the harmonic series and the natural logarithm:

\gamma = \lim_{n \rightarrow \infty } \left(
\sum_{k=1}^n \frac{1}{k} - \ln(n) \right)=\lim_{b \rightarrow \infty } \int_1^b\left({1\over\lfloor x\rfloor}-{1\over x}\right)\,dx.

Here, \lfloor x\rfloor represents the floor function. The numerical value of this constant, to 50 decimal places, is 0.57721566490153286060651209008240243104215933593992 ...  Euler established this formula in 1734. It is outrageous that we cannot decide if \gamma is a rational number or not. Even though over 1,000,000 digits of this number have been calculated, it is not yet known if it is a rational number (the ratio of two integers a/b). But if it is rational, the denominator (b) must have more than 244,663 digits!

The sum of the reciprocals of all integers: 1/1 +1/2 + 1/3+ 1/4 + 1/5 + 1/6+ 1/7  etc equals Ln(N) + Gamma                                          The sum of the reciprocals of the odd integers: 1 + 1/3 + 1/5 + 1/7 etc equals  {Ln(N) + Ln(2) + Gamma}/2                                           The sum of the reciprocals of the even integers:  1/2 +1/4+1/6+1/8 etc equals {Ln(N) -Ln(2) - Gamma}/2

The Basel problem is a famous problem in mathematical analysis with relevance to number theory, first posed by Pietro Mengoli in 1644 and solved by Leonhard Euler in 1735. Since the problem had withstood the attacks of the leading mathematicians of the day, Euler's solution brought him immediate fame when he was twenty-eight. Euler generalised the problem considerably, and his ideas were taken up years later by Bernhard Riemann in his seminal 1859 paper On the Number of Primes Less Than a Given Magnitude, in which he defined his zeta function and proved its basic properties. The problem is named after Basel, hometown of Euler as well as of the Bernoulli family who unsuccessfully attacked the problem.

The Basel problem asks for the precise summation of the reciprocals of the squares of the natural numbers, i.e. the precise sum of the infinite series:


\sum_{n=1}^\infin \frac{1}{n^2} =
\lim_{n \to +\infty}\left(\frac{1}{1^2} + \frac{1}{2^2} + \cdots + \frac{1}{n^2}\right).

The series is approximately equal to 1.644934. The Basel problem asks for the exact sum of this series (in closed form), as well as a proof that this sum is correct. Euler found the exact sum to be π2/6 and announced this discovery in 1735. His arguments were based on manipulations that were not justified at the time, and it was not until 1741 that he was able to produce a truly rigorous proof.

Read: "All about e" at http://www.maa.org/sites/default/files/pdf/upload_library/22/Chauvenet/mccartin.pdf   

Joseph-Louis Lagrange (1736-1813): (Italian) Lagrange was one of the creators of the calculus of variations, deriving the Euler Lagrange equations. Lagrange invented the method of solving differential equations known as variation of parameters, applied differential calculus to the theory of probabilities and attained notable work on the solution of equations. He proved that every natural number is a sum of four squares. His treatise Theorie des fonctions analytiques laid some of the foundations of group theory, anticipating Galois. In calculus, Lagrange developed a novel approach to interpolation and Taylor series. He studied the three-body problem for the Earth, Sun, and Moon and the movement of Jupiter's satellites. In 1772 found the special-case solutions to this problem that are now known as Lagrangian points. He transformed Newtonian mechanics into a branch of analysis, Lagrangian mechanics as it is now called. One of Lagrange's more famous books is the Analytical Mechanics, which, he boasted proudly, contains no pictures.

    Caspar Wessel (1745-1818): (Danish-Norwegian)  Wessel was a mathematician who was born in Norway. In 1763, having completed secondary school, he went to Denmark for further studies (Norway having no university at  the time). In 1778 he acquired the degree of candidatus juris. From 1794, however, he was employed as at ttp://www.maa.org/sites/default/files/pdf/upload_library/22/Chauvenet/mccartin.pdf surveyor (from 1798 as Royal inspector of Surveying).

It was the mathematical aspect of surveying that led him to exploring the geometrical significance of complex numbers. His fundamental paper, Om directionens analytiske betegning, was published in 1799 by the Royal Danish Academy of Sciences and Letters. Since it was in Danish, it passed almost unnoticed, and the same results were later independently found by Argand and Gauss.

One of the more prominent ideas presented in "On the Analytical Representation of Direction" was that of vectors. Even though this wasn't Wessel's main intention with the publication, he felt that a geometrical concept of numbers, with length and direction, was needed. Wessel's approach on addition was: "Two straight lines are added if we unite them in such a way that the second line begins where the first one ends and then pass a straight line from the first to the last point of the united lines. This line is the sum of the united lines". This is the same idea as used today when summing vectors.

Wessel's priority to the idea of a complex number as a point in the complex plane is today universally recognized. His paper was re-issued in French translation in 1899, and in English in 1999 as "On the analytic representation of direction".

    Pierre-Simon, marquis de Laplace (1749-1827):  (French) He is remembered as one of the greatest scientists of all time, sometimes referred to as a French Newton or Newton of France, with a phenomenal natural mathematical faculty superior to any of his contemporaries.  Laplace's writing of Celestial Mechanics, an enormous, five volume tome of celestial mechanics, established him as the Prince of Celestial Mechanicians. When presented with a copy of some of the initial volumes, Napoleon is said to have remarked, "I see no mention of God in this work". Laplace is said to have replied, "Sir, I have no need of that hypothesis." (In an addition to the story, the tale was related to Lagrange, who added "Ah, but it is such a beautiful hypothesis; it explains a great many things!" 

    Jean Baptiste Joseph Fourier (1768-1830): (French)  A mathematician and physicist  best known for initiating the investigation of Fourier series and their applications to problems of heat transfer and vibrations. The Fourier transform and Fourier's Law are also named in his honour. Fourier is also generally credited with the discovery of the greenhouse effect.

    Marie-Sophie Germain (1776-1831):  (French) A French mathematician, physicist, and philosopher. Despite initial opposition from her parents and difficulties presented by society, she gained education from books in her father's library including ones by Leonhard Euler and from correspondence with famous mathematicians such as Lagrange, Legendre, and Gauss. One of the pioneers of elasticity theory, she won the grand prize from the Paris Academy of Sciences for her essay on the subject. Her work on Fermat's Last Theorem provided a foundation for mathematicians exploring the subject for hundreds of years after.  Because of prejudice against her sex, she was unable to make a career out of mathematics, but she worked independently throughout her life.   At the centenary of her life, a street and a girls' school were named after her. The Academy of Sciences established The Sophie Germain Prize in her honor.

    Carl Friedrich Gauss (1777-1855): (German) Called the Prince of Mathematicians and the greatest mathematician since antiquity. He is ranked as one of history's most influential mathematicians. He referred to mathematics as the Queen of Sciences.  Gauss proved the Fundamental Theorem of Algebra. Gauss claimed to have discovered the possibility of non Euclidean Geometries but never published it.    

     Simeon Denis Poisson (1781-1840): (French)  A mathematician, geometer, and physicist.  In probability theory and statistics, the Poisson distribution (or Poisson law of small numbers) is a discrete probability distribution that expresses the probability of a number of events occurring in a fixed period of time if these events occur with a known average rate and independently of the time since the last event. (The Poisson distribution can also be used for the number of events in other specified intervals such as distance, area or volume.)

    Baron Augustin-Louis Cauchy (1789-1857): ( French) He was an early pioneer of analysis. He started the project of formulating and proving the theorems of infinitesimal calculus in a rigorous manner. He also gave several important theorems in complex analysis and initiated the study of permutation groups in abstract algebra. A profound mathematician, Cauchy exercised a great influence over his contemporaries and successors. His writings cover the entire range of mathematics and mathematical physics.

    Michael Faraday (1791-1867) and James Clerk Maxwell (1831-1879):  They proved that electric and magnetic forces are the same force in different guises.

    Nikolai Ivanovich Lobachevsky (Никола́й Ива́нович Лобаче́вский) (1792-1856):  (Russian)  A mathematician and geometer, renowned primarily for his pioneering works on hyperbolic geometry.

    Niels Henrik Abel (1802-1829):  (Norwegian)  At the age of 16, Abel gave a proof of the binomial theorem valid for all numbers, extending Euler's result which had only held for rational numbers. At age 19, he showed there is no general algebraic solution for the roots of a quintic equation, or any general polynomial equation of degree greater than four, in terms of explicit algebraic operations. To do this, he invented (independently of Galois) an extremely important branch of mathematics known as group theory, which is invaluable not only in many areas of mathematics, but for much of physics as well. Among his other accomplishments, Abel wrote a monumental work on elliptic functions which, however, was not discovered until after his death. When asked how he developed his mathematical abilities so rapidly, he replied "by studying the masters, not their pupils." 

    Janos Bolyai (1802-1860): (Hungarian) He was known for his work in non-Euclidean geometryBetween 1820 and 1823 he prepared a treatise on a complete system of non-Euclidean geometry. Bolyai's work was published in 1832 as an appendix to a mathematics textbook by his father.

Gauss, on reading the Appendix, wrote to a friend saying "I regard this young geometer Bolyai as a genius of the first order". In 1848 Bolyai discovered not only that Lobachevsky had published a similar piece of work in 1829, but also a generalization of this theory. As far as is known, Lobachevsky published his work a few years earlier than Bolyai, but it contained only hyperbolic geometry. Bolyai and Lobachevsky did not know each other or each other's works. In addition to his work in the geometry, Bolyai developed a rigorous geometric concept of complex numbers as ordered pairs of real numbers. Although he never published more than the 24 pages of the Appendix, he left more than 20,000 pages of mathematical manuscripts when he died.

    Carl Gustav Jacob Jacobi (1804-1851) (German) A mathematician, widely considered to be the most inspiring teacher of his time and one of the greatest mathematicians of all time.  One of Jacobi's greatest accomplishments was his theory of elliptic functionsHe also made fundamental contributions in the study of differential equations.  It was in algebraic development that Jacobi's peculiar power mainly lay, and he made important contributions of this kind to many areas of mathematics, as shown by his long list of papers in Crelle's Journal and elsewhere from 1826 onwards. One of his maxims was: 'Invert, always invert' ('man muss immer umkehren'), expressing his belief that the solution of many hard problems can be clarified by re-expressing them in inverse form.  He was also one of the early founders of the theory of determinants.

     Johann Peter Gustav Lejeune Dirichlet (1805-1859): (German) He was credited with the modern formal definition of a functionDirichlet's brain is preserved in the anatomical collection of the University of Gottingen, along with the brain of Gauss.

    Sir William Rowan Hamilton (1805-1865): (Irish) A  physicist, astronomer, and mathematician, who made important contributions to classical mechanics, optics, and algebra. His studies of mechanical and optical systems led him to discover new mathematical concepts and techniques. His greatest contribution is perhaps the reformulation of Newtonian mechanics, now called Hamiltonian mechanics. This work has proven central to the modern study of classical field theories such as electromagnetism, and to the development of quantum mechanics. In mathematics, he is perhaps best known as the inventor of quaternionsA striking feature of quaternions is that the product of two quaternions is noncommutative, meaning that the product of two quaternions depends on which factor is to the left of the multiplication sign and which factor is to the right. Hamilton defined a quaternion as the quotient of two directed lines in a three-dimensional space or equivalently as the quotient of two vectors. It can also be represented as the sum of a scalar and a vector. http://en.wikipedia.org/wiki/Quaternion  In four-dimensioal space the tesseract, or hype, is the four-dimensional analog of the cube. The tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of 6 square faces, the hypersurface of the tesseract consists of 8 cubical cells. The tesseract is one of the six convex regular 4-polytopes.  A generalization of the cube to dimensions greater than three is called a "hypercube", "n-cube" or "measure polytope". The tesseract is the four-dimensional hypercube'  See http://en.wikipedia.org/wiki/Fourth_dimension

    Joseph Liouville (1809-1882): (French) Liouville worked in a number of different fields in mathematics, including number theory, complex analysis, differential geometry and topology, but also mathematical physics and even astronomy. He is remembered particularly for Liouville's theorem, a nowadays rather basic result in complex analysis. In number theory, he was the first to prove the existence of transcendental numbers in 1844 by a construction using continued fractions (Liouville numbers). In mathematical physics, Liouville made two fundamental contributions: the Sturm Liouville theory, which was joint work with Charles Francois Sturm, and is now a standard procedure to solve certain types of integral equations by developing into eigenfunctions, and the fact (also known as Liouville's theorem) that time evolution is measure preserving for a Hamiltonian system. In Hamiltonian dynamics, Liouville also introduced the notion of action-angle variables as a description of completely integrable systems. The modern formulation of this is sometimes called the Liouville-Arnold theorem, and the underlying concept of integrability is referred to as Liouville integrability. 

 The following number is known as Liouville's constant.  (The exponent is negative j factorial.)

Liouville's constant is a decimal fraction with  1"s and 0"s in each decimal place. In1844 he constructed an infinite class of transcendental numbers using continued fractions, but the above number was the first decimal constant to be proven  by Liouville in 1850 to be  transcendental.  Cantor subsequently proved that "almost all" real numbers are in fact transcendental.

The crater Liouville on the Moon is named after him. So is the Liouville function, an important function in number theory. See http://mathworld.wolfram.com/LiouvilleFunction.html

     Ernst Eduard Kummer (1810 – 1893): (German)  Skilled in applied mathematics, Kummer trained German army officers in ballistics; afterwards, he taught for 10 years in a gymnasium, the German equivalent of high school, where he inspired the mathematical career of Leopold Kronecker. Kummer made several contributions to mathematics in different areas; he codified some of the relations between different hypergeometric series, known as contiguity relations. The Kummer surface results from taking the quotient of a two-dimensional abelian variety by the cyclic group {1, −1} (an early orbifold: it has 16 singular points, and its geometry was intensively studied in the nineteenth century). Kummer also proved Fermat's last theorem for a considerable class of prime exponents (see regular primeideal class group). His methods were closer, perhaps, to p-adic ones than to ideal theory as understood later, though the term 'ideal' arose here. He studied what were later called Kummer extensions of fields: that is, extensions generated by adjoining an nth root to a field already containing a primitive nth root of unity. This is a significant extension of the theory of quadratic extensions, and the genus theory of quadratic forms (linked to the 2-torsion of the class group). As such, it is still foundational for class field theory.

     Evariste Galois (1811-1832):  A symmetry of an object is what you can do to an object to leave it essentially looking like it did before you touched it.  Galois was interested in the collection of all symmetries and seeing what happens if you do one symmetry after another.  He discovered that it is the interactions between the symmetries in a group that encapsulate the essential qualities of the symmetry of an object.  Mathematicians struggled for centuries to find formulas for the solutions of equations of higher degree, but despite the efforts of Euler, Bezout, Malfatti, Lagrange, and others, no general solutions were found. Finally, Ruffini (1799) and Abel (1826) showed that the solution of the general quintic cannot be written as a finite formula involving only the four arithmetic operations and the extraction of roots. Galois developed the theory of Galois groups and described exactly when a polynomial equation is solvable.

    Augusta Ada King-Noel, Countess of Lovelace (née Byron; 10 December 1815 – 27 November 1852):  She was an English mathematician and writer, chiefly known for her work on Charles Babbage's proposed mechanical general-purpose computer, the Analytical Engine. She was the first to recognise that the machine had applications beyond pure calculation, and published the first algorithm intended to be carried out by such a machine. As a result, she is often regarded as the first to recognise the full potential of a "computing machine" and the first computer programmer.

Ada Lovelace was the only legitimate child of the poet Lord Byron, and his wife Anne Isabella Milbanke("Annabella"), Lady Wentworth. All of Byron's other children were born out of wedlock to other women.  Byron separated from his wife a month after Ada was born and left England forever four months later. He died of disease in the Greek War of Independence when Ada was eight years old. Her mother remained bitter and promoted Ada's interest in mathematics and logic in an effort to prevent her from developing her father's perceived insanity. Despite this, Ada remained interested in Byron and was, upon her eventual death, buried next to him at her request. She was often ill in her childhood. Ada married William King in 1835. King was made Earl of Lovelace in 1838, and Ada in turn became Countess of Lovelace.

Her educational and social exploits brought her into contact with scientists such as Andrew CrosseSir David BrewsterCharles WheatstoneMichael Faraday and the author Charles Dickens, which she used to further her education. Ada described her approach as "poetical science"  and herself as an "Analyst (& Metaphysician)".

When she was a teenager, her mathematical talents led her to a long working relationship and friendship with fellow British mathematician Charles Babbage, also known as "the father of computers", and in particular, Babbage's work on the Analytical Engine. Lovelace first met him in June 1833, through their mutual friend, and her private tutor, Mary Somerville.

Between 1842 and 1843, Ada translated an article by Italian military engineeLuigi Menabrea on the engine, which she supplemented with an elaborate set of notes, simply called Notes. These notes contain what many consider to be the first computer program—that is, an algorithm designed to be carried out by a machine. Lovelace's notes are important in the early history of computers. She also developed a vision of the capability of computers to go beyond mere calculating or number-crunching, while many others, including Babbage himself, focused only on those capabilities. Her mindset of "poetical science" led her to ask questions about the Analytical Engine (as shown in her notes) examining how individuals and society relate to technology as a collaborative tool.

She died of uterine cancer in 1852 at the age of 36.

     Karl Theodor Wilhelm Weierstrass (1815-1897): (German) He is often cited as the "father of modern analysis".

     George Boole (1815-1864): (English) A mathematician and philosopher. As the inventor of Boolean logic, the basis of modern digital computer logic, Boole is regarded in hindsight as a founder of the field of computer science. Boole said:  " ... no general method for the solution of questions in the theory of probabilities can be established which does not explicitly recognise ... those universal laws of thought which are the basis of all reasoning".

    Arthur Cayley (1821-1895): (British) He helped found the modern British school of pure mathematics. He proved the Cayley-Hamilton theorem: that every square matrix is a root of its own characteristic polynomial. He was the first to define the concept of a group in the modern way: as a set with a binary operation satisfying certain laws. Formerly, when mathematicians spoke of "groups", they had meant permutation groups.

    Charles Hermite (1822-1901): (French) He did research on number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra Hermite polynomials, Hermite interpolation, Hermite normal form, Hermitian operators, and cubic Hermite splines are named in his honor. One of his students was Henri Poincare In 1873, Hermite was the first to prove that e, the base of natural logarithms, is a transcendental number. His methods were later used by Ferdinand von Lindemann  to prove in 1882 π is transcendental.  In a letter to Thomas Stieltjes in 1893, Hermite famously remarked: "I turn with terror and horror from this lamentable scourge of continuous functions with no derivatives."       See http://www.pi314.net/eng/lindemann.php

     Leopold Kronecker (1823 – 1891): (German) He worked on number theory and algebra. He criticized Cantor's work on set theory, and was quoted by Weber (1893) as having said, "Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk" ("God made the integers, all else is the work of man."). Kronecker was a student and lifelong friend of Ernst Kummer.  For several years Kronecker focused on business, and although he continued to study mathematics as a hobby and corresponded with Kummer, he published no mathematical results. For several years Kronecker focused on business, and although he continued to study mathematics as a hobby and corresponded with Kummer, he published no mathematical results. In 1853 he wrote a memoir on the algebraic solvability of equations extending the work of Évariste Galois on the theory of equations.

    Bernhard Riemann (1826-1866):  If the "Riemann Hypothesis" is true,  the exact number of primes less than a given number N, or  Pi(N), can be calculated exactly.   Although thought to be correct, this hypothesis is unproven.    Karl Friedrich Gauss (1777-1855) had an approximation to Pi(N), equal to N/ln(N), where ln is the natural logarithm.   Adrien-Marie Legendre (1752-1833) improved on Gauss's estimate using  Pi(N) = N/{ln(N) - 1.08366}   Gauss then improved upon that estimate using Li(N) , which he called the logarithmic integral. (not shown here)  Leonard Euler (1707-1783) showed that the Riemann Zeta Function: Z(s) =  The sum of 1/n raised to the s power for n = 1 to infinity, is also equal to a product series involving primes.  Z(s) = The product of (1 + 1/p to the s + 1/p to the 2s + 1/p to the 3s  + 1/p to the 4s + 1/p to the 5s +...) over all primes.  It is important  to note: "s" is a "complex number". Riemann then hypothesized that Z(s) = 0 for only complex numbers where the real part = 1/2.  The Riemann Hypothesis has not been proven, but computers have shown the first 6.3 billion zeros all lie on the line s = 1/2 +ki.  If the Riemann Hypothesis is correct, then Riemann has a formula for calculating Pi(N) exactly!    Pi(N) =  R(N) minus an Adjustment.  R(N) is a formula involving the logarithmic integral and the Adjustment is expressed in terms of the zeros of the Zeta Function. The function R(N) was named in honor of Riemann.  

The Riemann zeta function ζ(s) is a function of a complex variable s = σ + it. (The notation with s, σ, and t is traditionally used in the study of the ζ-function)

The following infinite series converges for all complex numbers s with real part greater than 1, and defines ζ(s) in this case:


\zeta(s) =
\sum_{n=1}^\infty n^{-s} =
\frac{1}{1^s} + \frac{1}{2^s} + \frac{1}{3^s} + \cdots \;\;\;\;\;\;\; \sigma = \mathfrak{R}(s) > 1.
\!

The Riemann zeta function is defined as the analytic continuation of the function defined for σ > 1 by the sum of the preceding series.  When s =2 , this function equals

Riemann's hypothesis in 1859 is a deep mathematical conjecture which states that the nontrivial Riemann zeta function zeros, i.e., the values of s other than -2, -4, -6, ... such that zeta(s)=0 (where zeta(s) is the Riemann zeta function) all lie on the "critical line" sigma=R[s]=1/2 (where R[s] denotes the real part of s).  It has never been proved or disproved.

Riemann zeta function ζ(s) in the complex plane. The color of a point s encodes the value of ζ(s): colors close to black denote values close to zero, while hue encodes the value's argument. The white spot at s = 1 is the pole of the zeta function; the black spots on the negative real axis and on the critical line Re(s) = 1/2 are its zeros. Values with arguments close to zero including positive reals on the real half-line are presented in red.

    Julius Wilhelm Richard Dedekind (1831-1916): ( German) He did important work in abstract algebra (particularly ring theory), algebraic number theory and the foundations of the real numbers. 

    James Clerk Maxwell (1831-1879): (Scottish) A physicist and mathematician. His most prominent achievement was formulating classical electromagnetic theory. This united all previously unrelated observations, experiments and equations of electricity, magnetism and even optics into a consistent theory. Maxwell's equations demonstrated that electricity, magnetism and even light are all manifestations of the same phenomenon, namely the electromagnetic field. Subsequently, all other classic laws or equations of these disciplines became simplified cases of Maxwell's equations. Maxwell's achievements concerning electromagnetism have been called the "second great unification in physics", after the first one realized by Isaac Newton.

Maxwell demonstrated that electric and magnetic fields travel through space in the form of waves, and at the constant speed of light. In 1864 Maxwell wrote A Dynamical Theory of the Electromagnetic Field. It was with this that he first proposed that light was in fact undulations in the same medium that is the cause of electric and magnetic phenomena. His work in producing a unified model of electromagnetism is one of the greatest advances in physics.     

    Samuel Loyd (1841-1911): (American)  He was an American chess player, chess composer, puzzle author, and recreational mathematician

As a chess composer, he authored a number of chess problems, often with interesting themes. At his peak, Loyd was one of the best chess players in the US, and was ranked 15th in the world, according to chessmetrics.com.  Loyd was inducted into the US Chess Hall of Fame.

Following his death, his book Cyclopedia of 5000 Puzzles was published (1914) by his son.

Loyd is widely acknowledged as one of America's great puzzle-writers and popularizers, often mentioned as the greatest—Martin Gardner called him "America's greatest puzzler", and The Strand in 1898 dubbed him "the prince of puzzlers". As a chess problemist, his composing style is distinguished by wit and humour.

However, he is also known for lies and self-promotion, and criticized on these grounds—Martin Gardner's assessment continues "but also obviously a hustler". Canadian puzzler Mel Stover called Loyd "an old reprobate", and Matthew Costello called him "puzzledom's greatest celebrity ... popularizer, genius", but also "huckster ... and fast-talking snake oil salesman".

He collaborated with puzzler Henry Dudeney for a while, but Dudeney broke off the correspondence and accused Loyd of stealing his puzzles and publishing them under his own name. Dudeney despised Loyd so intensely he equated him with the Devil.

    William James (January 11, 1842 – August 26, 1910)  An American philosopher and psychologist who was also trained as a physician. The first educator to offer a psychology course in the United States, James was one of the leading thinkers of the late nineteenth century and is believed by many to be one of the most influential philosophers the United States has ever produced, while others have labeled him the "Father of American psychology"  See https://en.wikipedia.org/wiki/William_James

Three turtles of varying sizes stacked on top of each other with the largest at the bottom 

The saying holds that the world is supported by a chain of increasingly large turtles. Beneath each turtle is yet another: it is "turtles all the way down".  Read more about Turtles All The Way Down.

   Georg Ferdinand Ludwig Philipp Cantor (1845-1918): (German) He  is best known as the inventor of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between sets, defined infinite and well-ordered sets, and proved that the real numbers are "more numerous" than the natural numbers. In fact, Cantor's theorem implies the existence of an "infinity of infinities". He defined the cardinal and ordinal numbers and their arithmetic. Cantor's work is of great philosophical interest, a fact of which he was well aware.  Cantor's theory of transfinite numbers was originally regarded as so counter-intuitive, even shocking, that it encountered resistance from mathematical contemporaries such as Leopold Kronecker and Henri Poincare and later from Hermann Weyl and L. E. J. Brouwer

 A transcendental number is a (possibly complex) number that is not algebraic—that is, it is not a root of a non-zero polynomial equation with rational coefficients. The most prominent examples of transcendental numbers are π and e. Though only a few classes of transcendental numbers are known (in part because it can be extremely difficult to show that a given number is transcendental), transcendental numbers are not rare. Indeed, almost all real and complex numbers are transcendental, since the algebraic numbers are countable while the sets of real and complex numbers are both uncountable. All real transcendental numbers are irrational, since all rational numbers are algebraic. The converse is not true: not all irrational numbers are transcendental; e.g., the square root of 2 is irrational but not a transcendental number, since it is a solution of the polynomial equation x2 − 2 = 0.

    Seth Carlo Chandler, Jr. (1846-1913): (American astronomer and actuary).

He was born in Boston, Massachusetts to Seth Carlo and Mary (née Cheever) Chandler.During his last year in high school he performed mathematical computations for Benjamin Peirce, of the Harvard College Observatory.

After graduating, he became the assistant of Benjamin A. Gould. Gould was director of the Longitude Department of the U.S. Coast Survey program, a geodetic survey program. When Gould left to become director of the national observatory in Argentina, Chandler also left and became an actuary for Continental Life in NY City.  However, he continued to work in astronomy as an amateur affiliated with Harvard College Observatory.

 

In 1876 Chandler moved his family to Boston where he continued his actuarial work as a consultant to Union Mutual Life Insurance Company.  In 1880 he renewed his association with the Harvard College Observatory.  In 1881 he moved to Cambridge.  He mounted  a telescope to the cupola atop the roof and carried on the duties of Associate Editor of the Astronomical Journal while B. A. Gould was editor.  He later became editor and used his own funds to publish this journal in difficult financial times.

Chandler is best known for his discovery (1884–85) of the Chandler Wobble, a movement in Earth’s axis of rotation that causes latitude to vary with a period of about 433 days. A wandering of the rotation axis had been predicted by Swiss mathematician Leonhard Euler in 1765. Chandler’s detection of this effect was facilitated by his invention of the almucantar, a device for measuring the positions of stars relative to a circle centered at the zenith rather than to the meridian. The North Pole of Earth’s rotation axis wanders in an irregular, quasi-circular path with a radius of about 8–10 metres (26–33 feet).

 From 1896 to 1909 he edited The Astronomical Journal.

Chandler also made contributions to other areas of astronomy, including variable stars. He independently co-discovered the nova T Coronae Borealis, improved the estimate of the constant of aberration, and computed the orbital parameters of asteroids and comets.

Chandler was awarded the Gold Medal of the Royal Astronomical Society in 1896 and the James Craig Watson Medal in 1894.  The crater Chandler on the Moon is named after him.

See his bio at: http://www.nasonline.org/publications/biographical-memoirs/memoir-pdfs/chandler-seth-c.pdf

See more on the Chandler Wobble: http://curiosity.discovery.com/question/what-is-the-chandler-wobble

    Thomas Alva Edison (1847-1931): (American)  An  inventor, scientist, and businessman who developed many devices that greatly influenced life around the world, including the phonograph, the motion picture camera, and a long-lasting, practical electric light bulb. Dubbed "The Wizard of Menlo Park" . He was born in Milan, Ohio.  His quotations include: "There's a way to do it better - find it!"   "Genius is one percent inspiration and ninety-nine percent perspiration."   "I have not failed. I've just found 10,000 ways that won't work."  "I never did a day's work in my life. It was all fun."   Also: "We will make electricity so cheap that only the rich will burn candles." Edison became the owner of his Milan, Ohio, birthplace in 1906. On his last visit, in 1923, he was shocked to find his old home still lit by lamps and candles. Al Bolisha, a Canadian humorist, said, "Do you realize if it weren't for Edison, we'd be watching TV by candelight?"

    Sofia Vasilyevna Kovalevskaya, born Sofia Vasilyevna Korvin-Krukovskaya (1850–1891): She was a Russian mathematician who made noteworthy contributions to analysis, partial differential equations and mechanics. She was the first major Russian female mathematician and a pioneer for women in mathematics around the world. She was the first woman appointed to a full professorship in Northern Europe and was also one of the first women to work for a scientific journal as an editor.  Her sister was the socialist Anne Jaclard.

    Prime Number Theorem states that if you select a large number N, the probability of  it being prime is about 1/Ln(N) was solved independently in 1896 by Jacques-Solomon Hadamard (1865-1963) and Charles de la Vallee Poisson (1866-1962) by showing that the Riemann Zeta Function has no zeros of the form (1 + ki)

     Carl Louis Ferdinand von Lindemann (1852-1939): (German) He was a  noted for his proof, published in 1882, that π (pi) is a transcendental number, i.e., it is not a zero of any polynomial with rational coefficients.

     Jules Henri Poincare (1854-1912): (French) A mathematician,  theoretical physicist, and a philosopher of science.  Poincare is often described  in mathematics as The Last Universalist, since he excelled in all fields of the discipline as it existed during his lifetime. 

    George Eastman (1854-1932):  (American) An  inventor and philanthropist. He founded the Eastman Kodak Company and invented roll film    In his final two years, Eastman was in intense pain, caused by a degenerative disorder affecting his spine. He had trouble standing and his walking became a slow shuffle. Today it might be diagnosed as lumbar spinal stenosis, a narrowing of the spinal canal caused by calcification in the vertebrae. Eastman grew depressed, as he had seen his mother spend the last two years of her life in a wheelchair from the same condition. On March 14, 1932, Eastman died by suicide with a single gunshot to the heart, leaving a note which read, "My work is done.  Why wait?"

The logo from 1987 to 2006.  The letter "K" had been a favorite of Eastman's, he is quoted as saying, "it seems a strong, incisive sort of letter." He and his mother devised the name Kodak with an anagram set. He said that there were three principal concepts he used in creating the name: it should be short, one cannot mispronounce it, and it could not resemble anything or be associated with anything but Kodak.

    Andrey (Andrei) Andreyevich Markov (Андрей Андреевич Марков) (1856-1922): (Russian)  He is best known for his work on theory of stochastic processes. His research later became known as Markov chains.

    Henry Ernest Dudeney (1857-1930):  (English) An author and mathematician who specialized in logic puzzles and mathematical games. He is known as one of the foremost creators of puzzles.

    Max Karl Ernst Ludwig Planck (1858-1947): (German)  A physicist who is regarded as the founder of the quantum theory, for which he received the Nobel Prize in Physics in 1918.

    David Hilbert (1862-1943): (German) He was recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. He discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of geometry. He also formulated the theory of Hilbert spaces, one of the foundations of functional analysis.

    Bertrand Russell (1872-1970):  "Physics is mathematical not because we know so much about the physical world, but because we know so little; it is only its mathematical properties we can discover." 

    Pierre Joseph Louis Fatou (1878 – 1929):  (French): He was a French mathematician and astronomer. He is known for major contributions to several branches of analysis. The Fatou lemma and the Fatou set are named after him.

    Albert Einstein (1879-1955):  The speed of light is the same, irrespective of how the source of light or the observer is moving.  Furthermore, space and time cannot be treated as separate entities, rather they are inseparably tethered together by symmetry.  One of the known results of special relativity is that the length of moving bodies, as measured by observers at rest, contracts along their direction of motion.  The contraction is larger,  the higher the speed. Gravity warps and bends spacetime.  One of the key predictions of general relativity was the bending of light rays under the influence of gravity. Guided by principles of symmetry Einstein showed that acceleration and gravity are two sides of the same coin.(If a train is moving very fast to the north and a man in a boxcar drops his keys, they fall to the south.)(If a man in a stationary box car drops his keys, the keys would fall to the south, if gravity was tilted to the south.)

   Amalie Emmy Noether (1882-1935): (German) A German mathematician known for her groundbreaking contributions to abstract algebra and theoretical physics. Described by Pavel Alexandrov, Albert Einstein, Jean Dieudonné, Hermann Weyl, Norbert Wiener and others as the most important woman in the history of mathematics, she revolutionized the theories of rings, fields, and algebras. In physics, Noether's theorem explains the fundamental connection between symmetry and conservation laws.

    Max Born (1882-1970): (German)  physicist and mathematician who was instrumental in the development of quantum mechanics. He also made contributions to solid-state physics and optics and supervised the work of a number of notable physicists in the 1920s and 30s. Born won the 1954 Nobel Prize in Physics, shared with Walther Bothe.  

    Frank Albert Benford, Jr. (1883-1948): (American)  Benford's law, also called the first-digit law, states that in lists of numbers from many (but not all) real-life sources of data, the leading digit is distributed in a specific, non-uniform way. According to this law, the first digit is 1 almost one third of the time, and larger digits occur as the leading digit with lower and lower frequency, to the point where 9 as a first digit occurs less than one time in twenty. This distribution of first digits arises logically whenever a set of values is distributed logarithmically. Measurements of real world values are often distributed logarithmically (or equivalently, the logarithm of the measurements is distributed uniformly). This counter-intuitive result has been found to apply to a wide variety of data sets, including electricity bills, street addresses, stock prices, population numbers, death rates, lengths of rivers, physical and mathematical constants, and processes described by power laws (which are very common in nature). The result holds regardless of the base in which the numbers are expressed, although the exact proportions change. It is named after physicist Frank Benford, who stated it in 1938, although it had been previously stated by Simon Newcomb in 1881.    

     John Maynard Keynes  (1883-1946):  British Economist and Mathematician.  On the Law of Large Numbers or "long run calculations", he said: "This long run is a misleading guide to current affairs.  In  the long run we are all dead.  Economists set themselves too easy, too useless a task if in tempestuous seasons they can only tell us that when the storm is long past the ocean is flat again."  Other quotes:  "I do not know which makes a man more conservative - to know nothing but the present, or nothing but the past."  " It would be foolish, in forming our expectations, to attach great weight to matters which are very uncertain."  " It is generally agreed that casinos should, in the public interest, be inaccessible and expensive. And perhaps the same is true of Stock Exchanges."  "The outstanding faults of the economic society in which we live are its failure to provide for full employment and its arbitrary and inequitable distribution of wealth and incomes."

    Niels Henrik David Bohr (1885-1962):  (Danish) A physicist who made fundamental contributions to understanding atomic structure and quantum mechanics, for which he received the Nobel Prize in Physics in 1922. Bohr mentored and collaborated with many of the top physicists of the century at his institute in Copenhagen. He was part of a team of physicists working on the Manhattan Project. Bohr married Margrethe Norlund in 1912, and one of their sons, Aage Bohr, grew up to be an important physicist who in 1975 also received the Nobel prize. Bohr has been described as one of the most influential scientists of the 20th century.

    Godfrey Harold "G. H." Hardy (1877 – 1947)(English)  A mathematician, known for his achievements in number theory and mathematical analysis In biology, Hardy is known for the Hardy–Weinberg principle, a basic principle of population genetics.

In addition to his research, Hardy is remembered for his 1940 essay on the aesthetics of mathematics, titled A Mathematician's Apology. He was the mentor of the Indian mathematician Srinivasa Ramanujan (also  born in 1887).  See Hardy  background

    Srīnivāsa Aiyangār Rāmānujan (1887-1920):  (Indian) He was a self taught genius, with almost no formal training in pure mathematics, made substantial contributions to mathematical analysis, number theory, infinite series and continued fractions Ramanujan's talent was said, by the prominent English mathematician G.H. Hardy, to be in the same league as legendary mathematicians such as Euler, Gauss, Newton and Archimedes.  

    Gaston Maurice Julia (1893 – 1978):  (French) He was a French mathematician who devised the formula for the Julia set. His works were popularized by French mathematician Benoit Mandelbrot; the Julia and Mandelbrot fractals are closely related.

    Enrico Fermi (1901-1954): (Italian-American) A physicist particularly known for his work on the development of the first nuclear reactor, Chicago Pile-1, and for his contributions to the development of quantum theory, nuclear and particle physics, and statistical mechanics. He was awarded the 1938 Nobel Prize in Physics for his work on induced radioactivity.

Fermi is widely regarded as one of the leading scientists of the 20th century, highly accomplished in both theory and experiment.  Along with J. Robert Oppenheimer, he is frequently referred to as "the father of the atomic bomb". He also held several patents related to the use of nuclear power.    

     Andre John von Neumann (1903-1957): (Hungarian-born) An  American mathematician who made major contributions to a vast range of fields, including set theory, functional analysis, quantum mechanics, ergodic theory, continuous geometry, economics and game theory, computer science, numerical analysis, hydrodynamics (of explosions), and statistics, as well as many other mathematical fields. He is generally regarded as one of the greatest mathematicians in modern history. The mathematician Jean Dieudonne called von Neumann "the last of the great mathematicians",while Peter Lax described him as possessing the most "fearsome technical prowess" and "scintillating intellect" of the century.Weil (1906-1998):  (from France) "God exists since mathematics is consistent, and the Devil exists since we cannot prove it."

    Julius Robert Oppenheimer (1904-1967): (American) A theoretical physicist and professor of physics at the University of California, Berkeley. He is often called the "father of the atomic bomb" for his role as the scientific director of the Manhattan Project, the World War II project that developed the first nuclear weapons. The first atomic bomb was detonated in July 1945 in the Trinity test in New Mexico; Oppenheimer remarked later that it brought to mind words from the Bhagavad Gita: "Now, I am become Death, the destroyer of worlds."

After the war he became a chief adviser to the newly created United States Atomic Energy Commission and used that position to lobby for international control of nuclear power to avert nuclear proliferation and an arms race with the Soviet Union. After provoking the ire of many politicians with his outspoken opinions during the Second Red Scare, he had his security clearance revoked in a much-publicized hearing in 1954. Though stripped of his direct political influence he continued to lecture, write and work in physics. A decade later President John F. Kennedy awarded (and Lyndon B. Johnson presented) him with the Enrico Fermi Award as a gesture of political rehabilitation.

Oppenheimer's notable achievements in physics include the Born Oppenheimer approximation for molecular wavefunctions, work on the theory of electrons and positrons, the Oppenheimer Phillips process in nuclear fusion, and the first prediction of quantum tunneling. With his students he also made important contributions to the modern theory of neutron stars and black holes, as well as to quantum mechanics, quantum field theory, and the interactions of cosmic rays. As a teacher and promoter of science, he is remembered as a founding father of the American school of theoretical physics that gained world prominence in the 1930s. After World War II, he became director of the Institute for Advanced Study in Princeton.

    Kurt Goedel (1906-1978):  (Austrian  and later American)  A logician, mathematician, and philosopher. Considered with Aristotle and Gottlob Frege to be one of the most significant logicians in history) Goedel's Incompleteness Theorem: Any consistent axiom system is necessarily incomplete in that there will be true statements that can't be deduced from the axioms. 

Gödel published his two incompleteness theorems in 1931 when he was 25 years old, one year after finishing his doctorate at the University of Vienna. The first incompleteness theorem states that for any self-consistent recursive axiomatic system powerful enough to describe the arithmetic of the natural numbers (for example Peano arithmetic), there are true propositions about the naturals that cannot be proved from the axioms. To prove this theorem, Gödel developed a technique now known as Gödel numbering, which codes formal expressions as natural numbers.

He also showed that neither the axiom of choice nor the continuum hypothesis can be disproved from the accepted axioms of set theory, assuming these axioms are consistent. The former result opened the door for mathematicians to assume the axiom of choice in their proofs. He also made important contributions to proof theory by clarifying the connections between classical logic, intuitionistic logic, and modal logic.

     Olga Taussky-Todd (1906 – 1995):  (Austrian and later Czech-American)  She was born into a Jewish family. She worked first in algebraic number theory, with a doctorate at the University of Vienna supervised by Philipp Furtwängler. During that time in Vienna she also attended the meetings of the Vienna Circle.

According to Gian-Carlo Rota, as a young mathematician she was hired by a group of German mathematicians to find and correct the many mathematical errors in the works of David Hilbert, so that they could be collected into a volume to be presented to him on his birthday. There was only one paper, on the continuum hypothesis, that she was unable to repair. 

Later, she started to use matrices to analyze vibrations of airplanes during World War II, at the National Physical Laboratory in the United Kingdom. She became the torchbearer for matrix theory. In 1935, she moved to England and became a Fellow at Girton College, Cambridge University, as well as at Bryn Mawr College. In 1938 she married the British mathematician John Todd (1911-2007), a colleague at the University of London.  In 1945 the Todds emigrated to the United States and worked for the National Bureau of Standards. In 1957 they joined the faculty of California Institute of Technology (Caltech) in Pasadena, California.  

     Andre Weil (1906-1998): (French) He was an influential mathematician of the 20th century, renowned for the breadth and quality of his research output, its influence on future work, and the elegance of his exposition. He is especially known for his foundational work in number theory and algebraic geometry

    James R Newman (1907-1966):  (American) Von Neumann's first significant contribution to economics was the minimax theorem in1928.  He eventually improved and extended the minimax theorem to include games involving imperfect information and games with more than two players. This work culminated in the 1944 classic Theory of Games and Economic Behavior. Von Neumann was one of the pioneers of computer science making significant contributions to the development of logical design. "The Theory of Groups is a branch of mathematics in which one does something to something  and  then compares the result with the result obtained from doing the same thing to something else, or something else to the same thing." 

    Edward Teller (1908-2003):  (Hungarian-born American) A theoretical physicist, known colloquially as "the father of the hydrogen bomb," even though he did not care for the title. In 1942, Teller was invited to be part of Robert Oppenheimer's summer planning seminar at the University of California, Berkeley for the origins of the Manhattan Project, the Allied effort to develop the first nuclear weapons. A few weeks earlier, Teller had been meeting with his friend and colleague Enrico Fermi about the prospects of atomic warfare, and Fermi had nonchalantly suggested that perhaps a weapon based on nuclear fission could be used to set off an even larger nuclear fusion reaction. Even though he initially explained to Fermi why he thought the idea would not work, Teller was fascinated by the possibility and was quickly bored with the idea of "just" an atomic bomb (even though this was not yet anywhere near completion). At the Berkeley session, Teller diverted discussion from the fission weapon to the possibility of a fusion weapon, what he called the "Super" (an early version of what was later known as a hydrogen bomb).

    Stanislaw Marcin Ulam (1909-1984): (Polish-Jewish) He participated in America's Manhattan Project, originated the Teller–Ulam design of thermonuclear weapons, invented the Monte Carlo method of computation, and suggested nuclear pulse propulsion. In pure and applied mathematics, he produced many results, proved many theorems, and proposed several conjectures.

    Alan Turing 1912-1954):  A British mathematician, logician, cryptanalyst and computer scientist. He was highly influential in the development of computer science, giving a formalisation of the concepts of "algorithm" and "computation" with the Turing machine, which can be considered a model of a general purpose computer. Turing is widely considered to be the father of theoretical computer science and artificial intelligence.

During World War II, Turing worked for the Government Code and Cypher School (GC&CS) at Bletchley Park, Britain's code breaking centre. For a time he led Hut 8, the section responsible for German naval cryptanalysis. He devised a number of techniques for breaking German ciphers, including the bombe method, an electromechanical machine that could find settings for the Enigma machine.  His algorithms used Bayes Theorem.

Turing had something of a reputation for eccentricity at Bletchley Park. He was known to his colleagues as 'Prof' and his treatise on Enigma was known as 'The Prof's Book'. Jack Good, a cryptanalyst who worked with him, is quoted by Ronald Lewin as having said of Turing:

In the first week of June each year he would get a bad attack of hay fever, and he would cycle to the office wearing a service gas mask to keep the pollen off. His bicycle had a fault: the chain would come off at regular intervals. Instead of having it mended he would count the number of times the pedals went round and would get off the bicycle in time to adjust the chain by hand. Another of his eccentricities is that he chained his mug to the radiator pipes to prevent it being stolen.

While working at Bletchley, Turing, a talented long-distance runner, occasionally ran the 40 miles (64 km) to London when he was needed for high-level meetings, and he was capable of world-class marathon standards.

Turing was prosecuted for homosexual acts in 1952, when such acts were still criminalised in the UK. He accepted being chemically castrated as an alternative to prison. He died in 1954, 42 days before his 42nd birthday from cyanide poison ing.  An inquest determined his death  a suicide, his mother and some others believe it was accidental.  See http://www.cbsnews.com/news/wwii-codebreaker-alan-turing-gets-royal-pardon-for-gay-conviction/

There were prominent mathematicians that did not embrace the logic of Bayes Theorem.  Alan Turing used it to decode the German Enigma Cipher.  http://lesswrong.com/lw/774/a_history_of_bayes_theorem/

 

Turing worked from 1952 until his death in 1954 on mathematical biology, specifically morphogenisis.  He published one paper on the subject called The Chemical Basis of Morphogenesis in 1952, putting forth the Turing hypothesis of pattern formation (the theory was experimentally confirmed 60 years after his death).  His central interest in the field was understanding Fibonacci phyllotaxis, the existence of Fibonacci numbers in plant structures. He used reaction-diffusion equations which are central to the field of pattern formation.  Later papers went unpublished until 1992 when Collected Works of A. M. Turing was published.  His contribution is considered a seminal piece of work in this field. Removal of Hox genes causes an increased number of digits (up to 14) in mice, demonstrating a Turing-type mechanism in the development of the hand.

 

Note: In October 2016 Great Britain plans to posthumously pardon thousands of men who were convicted of what decades ago was a crime: having or seeking gay sex.

The measure has been nicknamed Turing’s Law, after Alan Turing, the mathematician central to the development of the computer. He committed suicide in 1954, after being convicted on charges of “gross indecency” with another man

    Paul Erdős (1913-1996):  (from Budapest, Hungary)  Erdős published more papers than any other mathematician in history, working with hundreds of collaborators.  His colleague Alfred Renyi said, "a mathematician is a machine for turning coffee into theorems", and Erdős drank copious quantities.

Because of his prolific output, friends created the Erdős number as a humorous tribute; Erdős alone was assigned the Erdős number of 0 (for being himself), while his immediate collaborators could claim an Erdős number of 1, their collaborators have Erdős number at most 2, and so on. Approximately 200,000 mathematicians have an assigned Erdős number, and some have estimated that 90 percent of the world's active mathematicians have an Erdős number smaller than 8.

It is said that Hank Aaron  has an Erdős number of 1 because they both autographed the same baseball when Emory University awarded them honorary degrees on the same day. Erdős numbers have also been assigned to an infant, a horse, and several actors.

    Martin Gardner (1914-2010):  (American)  A mathematics and science writer specializing in recreational mathematics, but with many interests (especially the writings of Lewis Carroll. He wrote the Mathematical Games column in Scientific American from 1956 to 1981, the Notes of a Fringe-Watcher column in Skeptical Inquirer from 1983 to 2002, and published over 70 books. See: http://en.wikipedia.org/wiki/Martin_Gardner

    Ivan Niven (1915-1999):  (Canadian) Ivan published over sixty papers, some with well-known co-authors such as Samuel Eilenberg, Paul Erdos (6 times), Nathan J. Fine, R. D. James, and H. S. Zuckerman (7 times). His areas of expertise were number theory, especially the areas of diophantine approximation and questions of irrationality and transcendance of numbers, and combinatorics. Ken Ross found two of the articles of special interest: his famous 1947 paper containing a simple proof that pi is irrational and his 1969 Monthly article on formal power series, for which he received the Lester R. Ford Award. Ivan viewed his most significant paper to be, "Uniform distribution of sequences of integers".

    Charles Frederick Mosteller (1916 - 2006)  American)  He was known as Frederick Mosteller, one of the most eminent statisticians of the 20th century. He was the founding chairman of Harvard's statistics department, from 1957 to 1971, and served as the president of several professional bodies including the Psychometric Society, the American Statistical Association, the Institute of Mathematical Statistics, the American Association for the Advancement of Science, and the International Statistical Institute.  Mosteller and David Wallace studied the historical problem of who wrote each of the disputed Federalist papers, James Madison or Alexander Hamilton. The Federalist Papers study was conducted in order to demonstrate the power of Bayesian inference and required a great deal of computational power for that time. It was featured in Time Magazine in the September 21st, 1962 edition.  See article at http://www.slideshare.net/FritzFerran/frederick-mosteller-fiftychallengingproblemsibookseeorg  Also see Bio at http://www.nasonline.org/publications/biographical-memoirs/memoir-pdfs/mosteller-frederick.pdf

    Richard Feynman (1918-1988):  Feynman said:  "Mathematics is looking for patterns".  "Mathematics is only patterns".  "Nature uses only the longest threads to weave her patterns, so that each small piece of her fabric reveals the organization of the entire tapestry."  Also: "Physics is like sex.  Sure, it may give some practical results, but that's not why we do it".

Murray Gell-Mann commented to the New York Times that the Feynman Algorithm to solve a problem is:
1. Write down the problem
2. Think very hard
3. Write down the answer.

     Benoit Mandelbrot (1924- 2010):  (from France)  The Father of Fractal Geometry.  He was a Polish-born, French and American mathematician, noted for developing a "theory of roughness" in nature and the field of fractal geometry to help prove it, which included coining the word "fractal". He later discovered the Mandelbrot set of intricate, never-ending fractal shapes, named in his honor. While he was a child, his family fled to France in 1936 to escape the growing Nazi persecution of Jews. From 1945 to 1947 attended the École Polytechnique, where he studied under Gaston Julia and Paul Lévy. From 1947 to 1949 he studied at California Institute of Technology, where he earned a master's degree in aeronautics. Returning to France, he obtained his PhD degree in Mathematical Sciences at the University of Paris in 1952.   

In 1945 Mandelbrot's uncle had introduced him to Julia's important 1918 paper claiming that it was a masterpiece and a potential source of interesting problems, but Mandelbrot did not like it. Indeed he reacted rather badly against suggestions posed by his uncle since he felt that his whole attitude to mathematics was so different from that of his uncle. Instead Mandelbrot chose his own very different course which, however, brought him back to Julia's paper in the 1970s after a path through many different sciences which some characterize as highly individualistic or nomadic. In fact the decision by Mandelbrot to make contributions to many different branches of science was a very deliberate one taken at a young age. It is remarkable how he was able to fulfill this ambition with such remarkable success in so many areas.

As a visiting professor at Harvard University, Mandelbrot began to study fractals called Julia sets that were invariant under certain transformations of the complex plane. Building on previous work by Gaston Julia and Pierre Fatou, Mandelbrot used a computer to plot images of the Julia sets. While investigating the topology of these Julia sets, he studied the Mandelbrot set fractal that is now named after him.

With the aid of computer graphics, Mandelbrot who then worked at IBM's Watson Research Center, was able to show how Julia's work is a source of some of the most beautiful fractals known today. To do this he had to develop not only new mathematical ideas, but also he had to develop some of the first computer programs to print graphics.  See his bio at http://www-history.mcs.st-and.ac.uk/Biographies/Mandelbrot.html

There are many beautiful pictures to view on the web.  For example:  http://sprott.physics.wisc.edu/fractals.htm  Also there are terrific videos to be found at:  http://www.fractal-animation.net/ufvp.html and http://www.ericbigas.com/fractalanimation/index.html and http://www.fractal-animation.net/ufvp.html and http://fractalanimations.com/ and  http://www.google.com/images?hl=&q=fractal+animation&rlz=1B3GGLL_enUS405US405&um=1&ie=UTF-8&source=univ&ei=2-dFTfmfPI-p8AaSw42EAg&sa=X&oi=image_result_group&ct=title&resnum=6&ved=0CEoQsAQwBQ&biw=1045&bih=404 and http://video.google.com/videoplay?docid=1619313842463920970#docid=8570098277666323857  and http://video.google.com/videoplay?docid=1619313842463920970#docid=6460130356432628677  and   http://www.youtube.com/watch?v=34zPvmNXTYQ and http://www.youtube.com/watch?v=G_GBwuYuOOs .

Learn from Robert Devaney at: http://video.google.com/videoplay?docid=1619313842463920970#docid=-6460544449138143366   Fractal art is shown at: http://www.lifesmith.com/art2008.html and at http://www.lifesmith.com/art2006.html and http://www.lifesmith.com/art2007.html 

I presented some pictures of fractals and some background at the Annual Meeting of the Society of Actuaries when I was president in 1992. I may have used Robert Devaney to develop the script.  See Chaos video, but start about 40% of the way through. (OOPs is copywrite protected so will only play on my computer)

Also see http://www.yalescientific.org/2010/04/the-frame-work-of-fractals/

 

Klaus Friedrich Roth, (born October 29, 1925, Breslau, Germany [now Wrocław, Poland]—died November 10, 2015, Inverness, Scotland), German-born British mathematician who was awarded the Fields Medal in 1958 for his work in number theory.
Roth
Klaus Roth.jpeg
Alma mater Peterhouse, Cambridge
Known for diophantine approximation
discrepancy theory
Awards Sylvester Medal (1991)
LMS De Morgan Medal (1983)
Fellow of the Royal Society (1960)
Fields Medal (1958)

Roth attended Peterhouse College, Cambridge, England (B.A., 1945), and the University of London (M.Sc., 1948; Ph.D., 1950). From 1948 to 1966 he held an appointment at University College, London, and then he became professor of pure mathematics at Imperial College of Science, Technology and Medicine, London, a position he held until 1988.

Roth was awarded the Fields Medal at the International Congress of Mathematicians in Edinburgh in 1958. His major work has been in number theory, particularly the analytic theory of numbers, and the work that led to his receiving the Fields Medal had to do with rational approximations to algebraic numbers. If α is any irrational number, algebraic or not, there are infinitely many rational numbers p/q such that | p/qα | < 1/q2 since the convergents of the continued fraction for α will suffice. The extension of this is the question of describing irrational numbers in terms of the exponent μ for which there are infinitely many approximations p/q satisfying | p/qα | < 1/qμ. If μ̄ is the upper bound for such exponents the question of the value of μ̄ when a is algebraic was attacked in 1844 by Joseph Liouville, who showed that μ̄ < n if α is an algebraic number of degree n. In 1908 Axel Thue showed that μ̄ < n/2 + 1, and in 1921 Carl Ludwig Siegel showed that μ̄ < 2Square root ofn essentially. In 1947 Freeman J. Dyson improved that to μ̄ < Square root of2n. In 1955 Roth showed that μ̄ = 2 for any algebraic number α. It was a solution of considerable difficulty. Roth is also known for his work on integer sequences and, in particular, his use of Selberg sieves and investigations in analytic number theory.

Roth’s publications include, with Heini Halberstam, Sequences (1966).                                                                                                            See http://www.telegraph.co.uk/news/obituaries/12172026/Klaus-Roth-mathematician-obituary.html

    Alexander Grothendieck (1928--2014): (German) He is one of the most influential mathematicians of the 20th century. He is known principally for his revolutionary advances in algebraic geometry, and also for major contributions to number theory, category theory and homological algebra, and his early achievements in functional analysis. He was awarded the Fields Medal in 1966. He became the leading figure in the creation of modern algebraic geometry.  His research extended the scope of the field and added elements of commutative algebra, homological algebra, sheaf theory and category theory to its foundations, while his so-called "relative" perspective led to revolutionary advances in many areas of pure mathematics.

    Murray Gell-Mann (1929-        ): (American) A physicist and polymath who received the 1969 Nobel Prize in physics for his work on the theory of elementary particles. He is a Distinguished Fellow and co-founder of the Santa Fe Institute and the Presidential Professor of Physics and Medicine at the University of Southern California.

He formulated the quark model of hadronic resonances, and identified the SU(3) flavor symmetry of the light quarks, extending isospin to include strangeness, which he also discovered. He developed the V-A theory of the weak interaction in collaboration with Richard Feynman. He created current algebra in the 1960s as a way of extracting predictions from quark models when the fundamental theory was still murky, which led to model-independent sum rules confirmed by experiment.

   John Hendricks (9.04.1929--7.07.2007): John Hendricks was born in Regina, Canada, on the 4th of September 1929. He received a degree in mathematics from the University of British Columbia in 1951.

John R. Hendricks in 1951

He worked for the Canadian Meteorological Service for 33 years and took early retirement in 1984, and  resided in Victoria, B.C.

At the beginning of his career, he was a NATO training instructor. He worked at various forecast offices in Canada and eventually became a supervisor. Throughout his career, he was known for his many contributions to statistics and to climatology. While employed, he also participated in volunteer service groups. He was Chairman, Manitoba branch and earlier Saskatchewan Branch, the Monarchist League of Canada.

He was the founding President, Manitoba Provincial Council, The Duke of Edinburgh’s Award in Canada. He received the Canada 125 medal, in recognition of significant contributions to community and to Canada, from the Lieutenant Governor of Manitoba on October the 19th, 1993.

John is known for his many published articles in meteorology, statistics and statistical climatology. But the greatest preponderance of work was devoted to the study of magic squares, cubes and hypercubes.

John started collecting magic squares and cubes when he was 13 years old. This became a hobby with him and eventually an obsession. He never thought that he would ever do anything with it. But soon, he became the first person in the world to successfully make and publish five and six-dimensional magic hypercubes. He also became the first person to make inlaid magic cubes and a wide variety of inlaid magic squares. He has written prolifically on the subject in the Journal of Recreational Mathematics. His impressive bibliography can be found at http://members.shaw.ca/johnhendricksmath/bibliography.htm

His major discoveries:

bulletan inlaid magic tesseract
bulletthe placement of numbers for a perfect magic tesseract of order 16
bulletthe placement of numbers in a perfect five-dimensional magic hypercube of order 32
bulleta new method of making bimagic squares of order 9
bulletthe world's first bimagic cube of order 25

He had Parkinson’s Disease, and passed away in Victoria, Vancouver Island, B.C., Canada, on the 7th of July 2007.

      Donald R Sondergeld (1930-    ) An American Actuary.  BA and BS in Ed in 1952 at Bowling Green State University, MS in theoretical Math at U of Michigan in 1955.  Fellow of the Society of Actuaries in 1962.  President of the Society of Actuaries in 1992.. See  Presidential Address   Read about various roles actuaries have played in Movies and in Literature. Also Actuaries and The Rule Of Eleven  and Crossing The Finishing Line  See an article on "The Tragedy of the Commons":   Page 1   Page 2  and   Page 3. And  Was Fibonacci an Actuary?  Although the latter article was written in fun, Fibonacci could have been considered an actuary as a result of some of his work.    Don's actuarial paper titled "Profitability As A Return On Total Capital" of 1982  and Earnings  an the Internal Rate of Return Measurement of Profit  See Chaos video, but start about 40% of the way through. (OOPs is copywrite protected so will only play on my computer) 

      Roger Penrose (1931 -         ):  (British)  A mathematician and physicist . www.en.wikipedia.org/wiki/Roger_Penrose 

     Harry Lewis Nelson (1.08.1932 -    ):  (American)   Mathematician and computer programmer. He was a member of the team that won the World Computer Chess Championship in 1983 and 1986, and was a co-discoverer of the 27th Mersenne prime in 1979 (at the time, the largest known prime number). He also served as editor of the Journal of Recreational Mathematics for five years. Most of his professional career was spent at Lawrence Livermore National Laboratory where he worked with some of the earliest supercomputers. He was particularly noted as one of the world's foremost experts in writing optimized assembly language routines for the Cray-1 and Cray X-MP computers. Nelson has had a lifelong interest in puzzles of all types, and since his retirement in 1991 he has devoted his time to his own MiniMax Game Company, a small venture that helps puzzle inventors to develop and market their products.

In 1994, Nelson donated his correspondence from his days as editor of the Journal of Recreational Mathematics to the University of Calgary Library as part of the Eugène Strens Recreational Mathematics Special Collection

See https://en.wikipedia.org/wiki/Harry_L._Nelson

      Edward Oakley "Ed" Thorp ( 1932): (American) A mathematics professor, author, hedge fund manager, and blackjack player best known as the "father of the wearable computer" after inventing the world's first wearable computer in 1961.[1] He was a pioneer in modern applications of probability theory, including the harnessing of very small correlations for reliable financial gain.

He is the author of Beat the Dealer, the first book to mathematically prove, in 1962, that the house advantage in blackjack could be overcome by card counting.[2] He also developed and applied effective hedge fund techniques in the financial markets, and collaborated with Claude Shannon in creating the first wearable computer.

Thorp received his Ph.D. in mathematics from the University of California, Los Angeles in 1958, and worked at the Massachusetts Institute of Technology (MIT) from 1959 to 1961. He was a professor of mathematics from 1961 to 1965 at New Mexico State University, and then joined the University of California, Irvine where he was a professor of mathematics from 1965 to 1977 and a professor of mathematics and finance from 1977 to 1982.

      See 1966 book Beat The Dealer and his 2017 book A Man For All Markets.

      Robert Phelan Langlands (1936 -        ): (Canadian) He is best known as the founder of the Langlands Program, a vast web of conjectures and results connecting representation theory and automorphic forms to the study of Galois groups in number theoryHe is an emeritus professor and occupies Albert Einstein's office at the Institute for Advanced Study in Princeton.
See projects.thestar.com/math-the-canadian-who-reinvented-mathematics/ and www.math.ubc.ca/Dept/Newsletters/Robert_Langlands_interview_2010.pdf  and www.math.duke.edu/langlands/ and Frenkel's discription: https://math.berkeley.edu/~frenkel/review.pdf  Also, Frenkel writes: The Langlands Program was launched in the late 60s with the goal of relating Galois representations and automorphic forms. In recent years a geometric version has been developed which leads to a mysterious duality between certain categories of sheaves on moduli spaces of (flat) bundles on algebraic curves. Three years ago, in a groundbreaking advance, Kapustin and Witten have linked the geometric Langlands correspondence to the S-duality of 4D supersymmetric gauge theories. This and subsequent works have already led to striking new insights into the geometric Langlands Program, which in particular involve the Homological Mirror Symmetry of the Hitchin moduli spaces of Higgs bundles on algebraic curves associated to two Langlands dual Lie groups.     

      Langlands Program        

A grand unified theory of mathematics which includes the search for a generalization of Artin reciprocity (known as Langlands reciprocity) to non-Abelian Galois extensions of number fields. In a January 1967 letter to André Weil, Langlands proposed that the mathematics of algebra (Galois representations) and analysis (automorphic forms) are intimately related, and that congruences over finite fields are related to infinite-dimensional representation theory. In particular, Langlands conjectured that the transformations behind general reciprocity laws could be represented by means of matrices (Mackenzie 2000).

In 1998, three mathematicians proved Langlands' conjectures for local fields, and in a November 1999 lecture at the Institute for Advanced Study at Princeton University, L. Lafforgue presented a proof of the conjectures for function fields. This leaves only the case of number fields as unresolved (Mackenzie 2000).
www.ams.org/journals/bull/1984-10-02/S0273-0979-1984-15237-6/S0273-0979-1984-15237-6.pdf

      James Harris Simons (1939 -       ): (American)  Jim is an American mathematician, hedge fund manager, and philanthropist. He is a code breaker and studies pattern recognition. Simons is the co-inventor, with Shiing-Shen Chern, of the Chern–Simons form - Chern and Simons (1974), and contributed to the development of string theory by providing a theoretical framework to combine geometry and topology with quantum field theory. Simons was a professor of mathematics at Stony Brook University and was also the former chair of the Mathematics Department at Stony Brook.

Dr. Simons received his doctorate at 23; advanced code breaking for the National Security Agency at 26; led a university math department at 30; won geometry’s top prize at 37; founded Renaissance Technologies, one of the world’s most successful hedge funds, at 44; and began setting up charitable foundations at 56.

In 1982, Simons founded Renaissance Technologies, a private hedge fund investment company based in New York with over $25 billion under management. Simons retired at the end of 2009 as CEO of one of the world's most successful hedge fund companies. Simons' net worth is estimated to be $16.5 billion.

Dr. Simons now runs a tidy universe of science endeavors, financing not only math teachers but hundreds of the world’s best investigators, even as Washington has reduced its support for scientific research. His favorite topics include gene puzzles, the origins of life, the roots of autism, math and computer frontiers, basic physics and the structure of the early cosmos.

     Harlan J Brothers (      ):  (American)    In 1997, while examining the sequence of counting numbers raised to their own power          ( {an}=nn ), Brothers discovered some simple algebraic formulas that yielded the number 2.71828..., the universal constant e, also known as the base of the natural logarithm.   https://en.wikipedia.org/wiki/Harlan_J._Brothers

Harlan J. Brothers
Harlan J. Brothers in 2012.jpg
Harlan J. Brothers in 2012
Residence Branford, ConnecticutUnited States
Nationality American
Alma mater Berklee College of Music
Gateway Community College
Scientific career
Fields InventingMathematicsMusicEducation

Having no formal college-level mathematics education, he sent brief descriptions of his findings to the host of the National Public Radio show “Science Friday” and also to a well-known mathematician at Scientific American.

His communication with “Science Friday” led to a fruitful collaboration with meteorologist John Knox. Together they discovered over two dozen new formulas and published two papers on their methods. These methods subsequently found their way into the standard college calculus curriculum by way of two popular textbooks on the subject.

Brothers went back to school to study calculus and differential equations. He went on to publish methods for deriving infinite series that include the fastest known formulas for approximating e.

In the summer of 2001, his professor, Miguel Garcia, introduced him to Benoît Mandelbrot and Michael Frame at Yale University. Brothers soon began working with them to incorporate the study of fractals into core mathematics curricula. His current research, begun in collaboration with Frame, is in the field of fractals and music.

Brothers has earned six U.S. patents and is a trained guitarist and composer. He also appeared as guest editor on the NPR show Bruce Barber's Real Life Survival Guide. He currently works as Consulting Director at Forensic Mathematics Services. “Our work is important because e is important,” says Brothers. “We’re not claiming that these theorems represent an advance in the computation of e—we’ve just come up with alternate formulas that may be easier to use in some circumstances. Regardless of whether our work has any actual practical applications, it is already having an impact by sparking the interest of teachers, students, and math buffs around the world. I find this very exciting.”

Also see https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4115525/ 

I Identifying fractals in music requires a different approach than seeing them in an image. “Unlike a picture, which is all laid out so that you can instantly see the structure, music is fundamentally a serial phenomenon,” Brothers says. “With music, the whole piece takes shape in your mind. This makes it more challenging to identify the self-similarity.”  Mandelbrot, who died in 2010, long suspected that music contained hidden fractal patterns but didn’t have time to devote to the subject, says Brothers.   Brothers contributed a chapter giving an overview of fractal music to Benoit Mandelbrot: A Life in Many Dimensions, a biography of the mathematician scheduled to be published next year   http://www.worldscientific.com/worldscibooks/10.1142/8238

And he worked on Pi.  See https://johncarlosbaez.wordpress.com/2014/02/12/triangular-numbers/


Photograph of Harlan Brothers by Elina Flit

Harlan Brothers is an inventor, musician, mathematician and the founder of Brothers Technology, LLC.  Harlan has received six U.S. patents and has published numerous journal articles related to number theory and the fractal geometry of music.  He is also an accomplished jazz guitarist and composer, having studied composition at the Berklee College of Music in Boston.

Over the course of six years, Harlan worked with Michael Frame and Benoit Mandelbrot at the NSF funded Yale University Fractal Geometry Workshops.  The workshops served to train educators in the subject of fractal geometry with the goal of developing fractal-based mathematics curricula for students in middle school through college.

During an informal discussion in 2003 regarding the general lack of understanding associated with the concept of fractal music, Benoit suggested to Harlan that he undertake a rigorous treatment of the subject.  The first result was a lecture and lab Harlan presented at the 2004 Fractal Geometry Workshop at Yale.  His most recent work on fractals and music, entitled "The Nature of Fractal Music," appears as a chapter in the memorial book Benoit Mandelbrot - A Life in Many Dimensions, published by World Scientific Publishing (May 2015).

     Daniel Hardisky (     -     ):  American  A retired civil engineer from Baltimore MD.  Perhaps see https://www.pinterest.co.uk/meszlenyineroka/pascal-h%C3%A1romsz%C3%B6g/ He found a hidden Pi in Pascal's Triangle. http://curiosamathematica.tumblr.com/post/126317657094/daniel-hardisky-discovered-%CF%80-in-pascals-triangle

 

   Michael Frame (            ):  American  See http://users.math.yale.edu/~bbm3/web_pdfs/FrameBook_Chapter1.pdf  See http://users.math.yale.edu/public_html/People/frame/Fractals/   Email: michael.frame@yale.edu   Also see https://www.barnesandnoble.com/w/fractals-graphics-and-mathematics-education-benoit-b-mandelbrot/1101362851?ean=9780883851692  And see http://www.yalescientific.org/2010/04/the-frame-work-of-fractals/  Frame was awarded the William Clyde DeVane Medal which honors outstanding scholarship and undergraduate teaching, and has been conferred annually since 1966 by the Yale Chapter of Phi Beta Kappa. The medal’s namesake, DeVane, was dean of Yale College from 1938 until 1963, a long-time president of the Yale Chapter of Phi Beta Kappa, and former president of the United Chapters of Phi Beta Kappa.  Frame's award related his teachinga course on "Fractals" See http://yaleadmissions.tumblr.com/post/56332668903/michael-frame-professor-of-mathematics-at-yale 

How can creative pedagogy bring to life an abstract field like math? "Sharpen up your imagination," suggests Michael Frame, Yale Professor of Mathematics, who has embraced the Web as a way to motivate students to learn. Because some students feel some anxiety about mathematics, Professor Frame introduces the subject through "a visual approach to math by showing them pictures that really excite them using pictures that they want to understand."

Professor Frame contacted the Instructional Technology Group for initial support in the creation of Web resources for Math 190b, a course on fractal geometry aimed at students not majoring in science. Over the years he has developed the Fractals project into a comprehensive online learning tool for Yale students, which is also being used by dozens of high school teachers during math education workshops.

    Stanislaw Ulam (1909- 1984)(Polish) https://en.wikipedia.org/wiki/Stanislaw_Ulam

A smiling man in a hat and heavy winter coat and scarf, carrying a portfolio tucked under his arm
Stanisław Ulam
Born Stanisław Marcin Ulam
13 April 1909
Lemberg, Austria-Hungary (now Lviv, Ukraine)
Died 13 May 1984(1984-05-13) (aged 75)
Santa Fe, New Mexico
Nationality Polish
Citizenship Poland, United States (awarded 1941)
Education Lwów Polytechnic Institute
Known for Key mathematical formulations in the fields of Physics, Computer Science, and Biology
Teller–Ulam design
Monte Carlo method
Fermi–Pasta–Ulam–Tsingou problem
Nuclear pulse propulsion
Scientific career
Fields Mathematics
Institutions Institute for Advanced Study
Harvard University
University of Wisconsin
Los Alamos National Laboratory
University of Colorado
University of Florida
Doctoral advisor Kazimierz Kuratowski
Doctoral students Paul Kelly
 

Stanisław Marcin Ulam (pronounced ['staɲiswaf 'mart͡ɕin 'ulam]; 13 April 1909 – 13 May 1984) was a Polish-American scientist in the fields of mathematics and nuclear physics. He participated in Manhattan Project, originated the Teller–Ulam design of thermonuclear weapons, discovered the concept of cellular automaton, invented the Monte Carlo method of computation, and suggested nuclear pulse propulsion. In pure and applied mathematics, he proved some theorems and proposed several conjectures.

Born into a wealthy Polish Jewish family, Ulam studied mathematics at the Lwów Polytechnic Institute, where he earned his PhD in 1933 under the supervision of Kazimierz Kuratowski. In 1935, John von Neumann, whom Ulam had met in Warsaw, invited him to come to the Institute for Advanced Study in Princeton, New Jersey, for a few months. From 1936 to 1939, he spent summers in Poland and academic years at Harvard University in Cambridge, Massachusetts, where he worked to establish important results regarding ergodic theory. On 20 August 1939, he sailed for the United States for the last time with his 17-year-old brother Adam Ulam. He became an assistant professor at the University of Wisconsin–Madison in 1940, and a United States citizen in 1941.

In October 1943, he received an invitation from Hans Bethe to join the Manhattan Project at the secret Los Alamos Laboratory in New Mexico. There, he worked on the hydrodynamic calculations to predict the behavior of the explosive lenses that were needed by an implosion-type weapon. He was assigned to Edward Teller's group, where he worked on Teller's "Super" bomb for Teller and Enrico Fermi. After the war he left to become an associate professor at the University of Southern California, but returned to Los Alamos in 1946 to work on thermonuclear weapons. With the aid of a cadre of female "computers", including his wife Françoise Aron Ulam, he found that Teller's "Super" design was unworkable. In January 1951, Ulam and Teller came up with the Teller–Ulam design, which is the basis for all thermonuclear weapons.

Ulam considered the problem of nuclear propulsion of rockets, which was pursued by Project Rover, and proposed, as an alternative to Rover's nuclear thermal rocket, to harness small nuclear explosions for propulsion, which became Project Orion. With Fermi, John Pasta, and Mary Tsingou, Ulam studied the Fermi–Pasta–Ulam–Tsingou problem, which became the inspiration for the field of non-linear science. He is probably best known for realising that electronic computers made it practical to apply statistical methods to functions without known solutions, and as computers have developed, the Monte Carlo method has become a common and standard approach to many problems.

Shiing-Shen Chern (10.26.1911 - 12.03.2004) 陳省身

 
Shiing-Shen Chern.jpg
Shiing-Shen Chern, 1976
Born (1911-10-26)October 26, 1911
Jiaxing, Zhejiang, Qing Empire
Died December 3, 2004(2004-12-03) (aged 93)
Tianjin, People's Republic of China
Nationality Chinese and American
Citizenship Republic of China and United States (multiple citizenship)
Alma mater Nankai University
University of Hamburg
Known for Chern–Simons theory
Chern–Weil theory
Chern class
Awards
Scientific career
Fields Mathematics
Institutions Tsinghua University
Institute for Advanced Study
University of Chicago
University of California, Berkeley
Mathematical Sciences Research Institute
Nankai University
Doctoral advisor Wilhelm Blaschke
Doctoral students Louis Auslander
Thomas Banchoff
Manfredo do Carmo
Robert B. Gardner
Howard Garland
Harold Levine
Katsumi Nomizu
William F. Pohl
Alan Weinstein
Shing-Tung Yau
 
 
Shiing-Shen Chern
Traditional Chinese 陳省身
Simplified Chinese 陈省身
 

Shiing-Shen Chern (/ɜːrn/; Chinese: 陳省身; pinyin: Chén Xǐngshēn, Mandarin: [tʂʰən.ɕiŋ.ʂən]; October 26, 1911 – December 3, 2004) was a Chinese-American mathematician who made fundamental contributions to differential geometry and topology. He was widely regarded as a leader in geometry and one of the greatest mathematicians of the twentieth century, winning numerous awards and recognition including the Wolf Prize and the inaugural Shaw Prize.

Shiing-Shen Chern spent nearly a decade at the University of Chicago before moving to University of California, Berkeley, where he co-founded the world-renowned Mathematical Sciences Research Institute in 1982 and was the institute's founding director.  In memory of Shiing-Shen Chern, the International Mathematical Union established the Chern Medal in 2010 to recognize "an individual whose accomplishments warrant the highest level of recognition for outstanding achievements in the field of mathe

Chern was born in Xiushui County (秀水縣), Jiaxing, in Zhejiang province. The year after his birth, China changed its regime from the Qing Dynasty to the Republic of China. He graduated from Xiushui Middle School (秀水中學) and subsequently moved to Tianjin in 1922 to accompany his father. In 1926, after spending four years in Tianjin, Chern graduated from Fulun High School (扶輪中學).

At age 15, Chern entered the Faculty of Sciences of the Nankai University in Tianjin, but had problems at the laboratory, so he studied mathematics, instead.  Chern graduated with a Bachelor of Science degree in 1930. At Nankai, Chern's mentor was Li-Fu Chiang (姜立夫), a Harvard-trained geometer. Also at Nankai, he was heavily influenced by the physicist Rao Yutai (饶毓泰). Rao is today considered to be one of the founding fathers of modern Chinese informatics.

Chern went to Beiping (now Beijing) to work at the Tsing Hua University Department of Mathematics as a teaching assistant. At the same time he also registered at Tsinghua Graduate School as a student. He studied projective differential geometry under Prof. Sun Guangyuan, a University of Chicago-trained geometer and logician who was also from Zhejiang. Sun is another mentor of Chern who is considered a founder of modern Chinese mathematics. In 1932, Chern published his first research article in the Tsing Hua University Journal. In the summer of 1934, Chern graduated from Tsinghua with a master's degree, the first ever master's degree in mathematics issued in China.

Chen-Ning Yang's father—Yang Ko-Chuen, another Chicago-trained professor at Tsing Hua, but specialising in algebra, also taught Chern. At the same time, Chern was Chen-Ning Yang's teacher of undergraduate maths at Tsinghua.

At Tsinghua, Hua Luogeng, also a mathematician, was Chern's colleague and roommate.

In 1932, Wilhelm Blaschke from the University of Hamburg visited Tsinghua and was impressed by Chern and his research.

In 1934, Chern received a scholarship to study in the United States, but he wanted to study under well-known geometer Wilhelm Blaschke.[10] Co-funded by Tsing Hua and the Chinese Foundation of Culture and Education, Chern went to continue his study in mathematics in Germany with a scholarship.[10] Chern studied at the University of Hamburg and worked under Blaschke's guidance first on the geometry of webs then on the Cartan-Kähler theory. He obtained his Dr. rer.nat. (Doctor of Science, which is equivalent to PhD) degree in February, 1936.[10] Blaschke recommended Chern to study in Paris.

It was at this time that he had to choose between the career of algebra in Germany under Emil Artin, and the career of geometry in France under Élie-Joseph Cartan. Chern was tempted by what he called the "organizational beauty" of Emil Artin's Algebra, but in the end, he decided to go to France in September 1936.[12] He spent one year at the Sorbonne in Paris.

In August 1936, Chern watched the Summer Olympics in Berlin together with Hua Luogeng who paid Chern a brief visit. During that time, Hua was studying at the University of Cambridge in Britain.

In the summer of 1937, Chern accepted the invitation of Tsing Hua's University and returned to China  He was promoted to professor of mathematics at Tsing Hua. However, in August, the Marco Polo Bridge Incident (near Beijing) happened and the Second Sino-Japanese War started, Tsing Hua was forced to move away from Beijing to west China.[13] Three universities including Peking University, Tsing Hua, and Nankai formed the National Southwestern Associated University (NSAU), and relocated to Kunming, Yunnan province. Chern never reached Beijing. In the same year, Hua Luogeng was promoted to professor of mathematics at Tsinghua.

In 1939, Chern married Shih-Ning Cheng, and the couple had two children by the names of Paul a

In July 1943, Chern went to the United States, and worked at the Institute for Advanced Study (IAS) in Princeton on characteristic classes in differential geometry. Shortly afterwards, he was invited by Solomon Lefschetz to be an editor of Annals of Mathemat

Chern returned to Shanghai in 1945 to help found the Institute of Mathematics of the Academia Sinica, which was later moved to Nanking (then-capital of the Republic of China). Chern was the acting president of the institute. Wu Wenjun was Chern's graduate student at the institute.

In 1948, Chern was elected one of the first academicians of the Academia Sinica. He was the youngest academician elected (at age 37).

By the end of 1948, Chern returned to the United States because of the Chinese Civil War.  He then returned to the IAS.  In 1949, Chern became professor of mathematics at the University of Chicago. Coincidentally, Ernest Preston Lane, former Chair at UChicago Department of Mathematics, was the doctoral advisor of Chern's undergraduate mentor at Tsinghua—Sun Guangyuan.

Chern moved to the University of California, Berkeley, in 1960.  He worked and stayed there until he became an emeritus professor in 1979.[14] In 1961, Chern became a naturalized citizen of the United States. In the same year, he was elected member of the United States National Academy of Sciences.

In 1964, Chern was a vice-president of American Mathematical Society (AMS).

Chern retired from Berkeley in 1981. He founded the Mathematical Sciences Research Institute (MSRI) in 1981 and served as the director until 1984. Afterward he became the honorary director of the institute. MSRI now is one of the largest and most prominent mathematical institutes in the world. Shing-Tung Yau was one of his PhD students during this period.

The Shanghai Communiqué was issued by the United States and the People's Republic of China on February 27, 1972. The relationship between these two nations started to normalise, and American citizens were allowed to visit P.R.China. In September 1972, Chern visited Beijing with his wife. During this period of time, Chern visited China 25 times, of which 14 were to his home province Zhejiang.

Chern founded the Nankai Institute for Mathematics (NKIM) at his alma mater Nankai in Tianjin. The institute was formally established in 1984 and fully opened in October 17, 1985. NKIM was renamed the Chern Institute of Mathematics in 2004 after Chern's death.

Based on Chern's advice, a mathematical research center was established in Taipei, Taiwan, whose co-operational partners are National Taiwan University, National Tsing Hua University and the Sinica Academia Institute of Mathematics.[16]

Chern was also a director and advisor of the Center of Mathematical Sciences at Zhejiang University in Hangzhou, Zhejiang.

From 2000 to his death, Chern lived in Tianjin, China. Chern died of heart failure at his home in Tianjin in 2004 at age 93..

Chern's work extends over all the classic fields of differential geometry. It includes areas currently fashionable (the Chern–Simons theory arising from a 1974 paper written jointly with Jim Simons), perennial (the Chern–Weil theory linking curvature invariants to characteristic classes from 1944, after the AllendoerferWeil paper of 1943 on the Gauss–Bonnet theorem), the foundational (Chern classes), and some areas such as projective differential geometry and webs that have a lower profile. He published results in integral geometry, value distribution theory of holomorphic functions, and minimal submanifolds.

He was a follower of Élie Cartan, working on the 'theory of equivalence' in his time in China from 1937 to 1943, in relative isolation. In 1954 he published his own treatment of the pseudogroup problem that is in effect the touchstone of Cartan's geometric theory. He used the moving frame method with success only matched by its inventor; he preferred in complex manifold theory to stay with the geometry, rather than follow the potential theory. Indeed, one of his books is entitled "Complex Manifolds without Potential Theory". In the last years of his life, he advocated the study of Finsler geometry, writing several books and articles on the su

Chern received numerous honors and awards in his life, including:

bullet1970, Chauvenet Prize, of the Mathematical Association of America;
bullet1975, National Medal of Science[17]
bullet1982, Humboldt Prize, Germany;
bullet1983, Leroy P. Steele Prize, of the American Mathematical Society;
bullet1984, Wolf Prize in Mathematics, Israel;
bullet2002, Lobachevsky Medal;
bullet2004 May, Shaw Prize in mathematical sciences, Hong Kong;[18]
bullet1948, Academician, Academia Sinica;
bullet1950, Honorary Member, Indian Mathematical Society;
bullet1961, Member, United States National Academy of Sciences;
bullet1963, Fellow, American Academy of Arts and Sciences;
bullet1971, Corresponding Member, Brazilian Academy of Sciences;
bullet1983, Associate Founding Fellow, TWAS;
bullet1985, Foreign Fellow, Royal Society of London, UK;
bullet1986, Honorary Fellow, London Mathematical Society, UK;
bullet1986, Corresponding Member, Academia Peloritana, Messina, Sicily;
bullet1987, Honorary Life Member, New York Academy of Sciences;
bullet1989, Foreign Member, Accademia dei Lincei, Italy;
bullet1989, Foreign Member, Académie des sciences, France;
bullet1989, Member, American Philosophical Society;
bullet1994, Foreign Member, Chinese Academy of Sciences.

Chern was given a number of honorary degrees, including from The Chinese University of Hong Kong (LL.D. 1969), University of Chicago (D.Sc. 1969), ETH Zurich (Dr.Math. 1982), SUNY Stony Brook (D.Sc. 1985), TU Berlin (Dr.Math. 1986), his alma mater Hamburg (D.Sc. 1971) and Nankai (honorary doctorate, 1985), etc.

Chern was also granted numerous honorary professorships, including at Peking University (Beijing, 1978), his alma mater Nankai (Tianjin, 1978), Chinese Academy of Sciences Institute of Systems Science (Beijing, 1980), Jinan University (Guangzhou, 1980), Chinese Academy of Sciences Graduate School (1984), Nanjing University (Nanjing, 1985), East China Normal University (Shanghai, 1985), USTC (Hefei, 1985), Beijing Normal University (1985), Zhejiang University (Hangzhou, 1985), Hangzhou University (1986, the university was merged into Zhejiang University in 1998), Fudan University (Shanghai, 1986), Shanghai University of Technology (1986, the university was merged to establish Shanghai University in 1994), Tianjin University (1987), Tohoku University (Sendai, Japan, 1987), etc.

 

James Harris "Jim" Simons (/ˈsmənz/; born April 1938) American mathematician, hedge fund manager, and philanthropist. He is known as a quantitative investor and in 1982 founded Renaissance Technologies, a private hedge fund based in New York City. Although Simons retired from the fund in 2009, he remains its non-executive chairman and adviser.
 
James Simons 2007.jpg
Simons speaking at the Differential Geometry, Mathematical Physics, Mathematics and Society conference in 2007 in Bures-sur-Yvette.
Born April 1938 (age 79)
Newton, Massachusetts, U.S.
Alma mater Massachusetts Institute of Technology
University of California, Berkeley
Occupation Mathematician, hedge fund manager, and philanthropist
Known for
bulletFounding and managing Renaissance Technologies
bulletCreating the Chern–Simons form with Shiing-Shen Chern
Net worth US$18 billion (February 2017)
Spouse(s) Barbara Simons
Marilyn Hawrys Simons[1]
Children 3,[1] including Nat Simons
Parent(s) Matthew Simons
Marcia Kantor
Awards Oswald Veblen Prize (1976)

He is also known for his studies on pattern recognition. He developed (with Shiing-Shen Chern) the Chern–Simons form,[4] and contributed to the development of string theory by providing a theoretical framework to combine geometry and topology with quantum field theory.[5] From 1968 to 1978,[6] Simons was a mathematics professor and subsequent chair of the mathematics department at Stony Brook University.[7]

As reported by Forbes, his net worth as of June 2017 is estimated to be $18 billion, while in the previous year, it was $15.5 billion.[8]

In 2016, asteroid 6618 Jimsimons, discovered by Clyde Tombaugh in 1936, was named after Simons by the International Astronomical Union in honor of his numerous contributions to mathematics and philanthro

James Harris Simons was born in April 1938 to an American Jewish family, the only child of Marcia (née Kantor) and Matthew Simons, and raised in Brookline, Massachusetts. His father owned a shoe factory.

He received a Bachelor of Science in mathematics from the Massachusetts Institute of Technology in 1958 and a Ph.D., also in mathematics, from the University of California, Berkeley, under supervision of Bertram Kostant in 1961, at the age of 23. Investment career.

Jim Simons at MSRI.jpg
 
The Carnegie Medal of Philanthropy Ceremony - 17th October 2013, 1:31:40 (Simons section at 52:00-1:05:30), Scottish Parliament

For more than two decades, Simons' Renaissance Technologies' hedge funds, which trade in markets around the world, have employed mathematical models to analyze and execute trades, many automated. Renaissance uses computer-based models to predict price changes in financial instruments. These models are based on analyzing as much data as can be gathered, then looking for non-random movements to make predictions.

Renaissance employs specialists with non-financial backgrounds, including mathematicians, physicists, signal processing experts and statisticians. The firm's latest fund is the Renaissance Institutional Equities Fund (RIEF).  RIEF has historically trailed the firm's better-known Medallion fund, a separate fund that contains only the personal money of the firm's executives.

It's startling to see such a highly successful mathematician achieve success in another field," says Edward Witten, professor of physics at the Institute for Advanced Study in Princeton, NJ, and considered by many of his peers to be the most accomplished theoretical physicist alive...[

In 2006, Simons was named Financial Engineer of the Year by the International Association of Financial Engineers. In 2007, he was estimated to have personally earned $2.8 billion, $1.7 billion in 2006, $1.5 billion in 2005, (the largest compensation among hedge fund managers that year) and $670 million in 2004.

Simons' mathematical work has primarily focused on the geometry and topology of manifolds. His 1962 Berkeley PhD thesis, written under the direction of Bertram Kostant, gave a new and more conceptual proof of Berger's classification of the holonomy groups of Riemannian manifolds, which is now a cornerstone of modern geometry. He subsequently began to work with Shing-Shen Chern on the theory of characteristic classes, eventually discovering the Chern–Simons secondary characteristic classes of 3-manifolds, which are deeply related to the Yang-Mills functional on 4-manifolds, and have had a profound effect on modern physics. These and other contributions to geometry and topology led to Simons becoming the 1976 recipient of the AMS Oswald Veblen Prize in Geometry. In 2014, he was elected to the National Academy of Sciences of the USA.

In 1964, Simons worked with the National Security Agency to break codes. Between 1964 and 1968, he was on the research staff of the Communications Research Division of the Institute for Defense Analyses (IDA) and taught mathematics at the Massachusetts Institute of Technology and Harvard University, ultimately joining the faculty at Stony Brook University. In 1968, he was appointed chairman of the math department at Stony Brook University.

Simons was asked by IBM in 1973 to attack the block cipher Lucifer, an early but direct precursor to the Data Encryption Standard (DES).

Simons founded Math for America, a nonprofit organization, in January 2004 with a mission to improve mathematics education in United States public schools by recruiting more highly qualified teachers. He funds a variety of research projec

Simons shuns the limelight and rarely gives interviews, citing Benjamin the Donkey in Animal Farm for explanation: "God gave me a tail to keep off the flies. But I'd rather have had no tail and no flies." On October 10, 2009, Simons announced he would retire on January 1, 2010 but remain at Renaissance as nonexecutive chairman.

In 1996, his son Paul, aged 34, was riding a bicycle, when he was killed by a car on Long Island. In 2003, his son Nicholas, aged 24, drowned on a trip to Bali, Indonesia. He worked in Nepal for 9 months. His son Nat Simons, from Racine, Wisconsin, is an investor and philanthropist

Simons is a major contributor to Democratic Party political action committees. According to the Center for Responsive Politics, Simons is currently ranked the #5 donor to federal candidates in the 2016 election cycle, coming behind co-CEO Robert Mercer, who is ranked #1 and generally donates to Republicans. Simons has donated $7 million to Hillary Clinton's Priorities USA Action, $2.6 million to the House and Senate Majority PACs, and $500,000 to EMILY's List.  He also donated $25,000 to Republican Senator Lindsey Graham's super PAC. Since 2006 Simons has contributed about $30.6 million to federal campaigns. Since 1990, Renaissance Technologies has contributed $59,081,152 to federal campaigns and since 2001, has spent $3,730,000 on lobb

According to the Wall Street Journal in May 2009, Simons was questioned by investors on the dramatic performance gap of Renaissance Technologies' portfolios. The Medallion Fund, which has been available exclusively to current and past employees and their families, surged 80% in 2008 in spite of hefty fees; the Renaissance Institutional Equities Fund (RIEF), owned by outsiders, lost money in both 2008 and 2009; RIEF declined 16% in 2008.

On July 22, 2014, Simons was subject to bipartisan condemnation by the U.S. Senate Permanent Subcommittee on Investigations for the use of complex barrier options to shield day-to-day trading (usually subject to higher ordinary income tax rates) as long-term capital gains. “Renaissance Technologies was able to avoid paying more than $6 billion in taxes by disguising its day-to-day stock trades as long term investments,” said Sen. John McCain (R., Ariz.), the committee’s ranking Republican, in his opening statement. “Two banks and a handful of hedge funds developed a complex financial structure to engage in highly profitable trades while claiming an unjustified lower tax rate and avoiding limits on trading with borrowed money,” said Sen. Carl Levin (D., Mich.) in his prepared remarks.[35]

An article published in the New York Times in 2015 said that Simons was involved in one of the biggest tax battles of the year, with Renaissance Technologies being "under review by the I.R.S. over a loophole that saved their fund an estimated $6.8 billion in taxes over roughly a decade."

He was named by the Financial Times in 2006 as "the world's smartest billionaire".[37] According to Forbes Magazine Simons has a net worth of $18 billion USD as of February 2017. This makes him #24 on the Forbes 400 richest people list.

In 2011, he was included in the 50 Most Influential ranking of Bloomberg Markets Magazine.  Simons owns a motor yacht, named Archimedes. It was built at the Dutch yacht builder Royal Van Lent and delivered to Simons in 2008.

Simons and his second wife, Marilyn Hawrys Simons, co-founded the Simons Foundation in 1994, a charitable organization that supports projects related to education and health, in addition to scientific research.  In memory of his son Paul, whom he had with his first wife, Barbara Simons, he established Avalon Park, a 130-acre (0.53 km2) nature preserve in Stony Brook. In 1996, 34-year-old Paul was killed by a car driver while riding a bicycle near the Simons home. Another son, Nick Simons, drowned at age 24 while on a trip to Bali in Indonesia in 2003. Nick had worked in Nepal. The Simons have become large donors to Nepalese healthcare through the Nick Simons Institute.

The Simons Foundation established the Simons Foundation Autism Research Initiative (SFARI) in 2003 as a scientific initiative within the Simons Foundation's suite of programs. SFARI's mission is to improve the understanding, diagnosis and treatment of autism spectrum disorders.

In 2004, Simons founded Math for America with an initial pledge of $25 million from the Simons Foundation, a pledge he later doubled in 2006.

Also in 2006, Simons donated $25 million to Stony Brook University through the Stony Brook Foundation, the largest donation ever to a State University of New York school at the time.

On February 27, 2008, then Gov. Eliot Spitzer announced a $60 million donation by the Simons Foundation to found the Simons Center for Geometry and Physics at Stony Brook, the largest gift to a public university in New York state history. In December 2008, it was reported that the Stony Brook University Foundation, of which Simons is chair emeritus, lost $5.4 million in Bernard Madoff’s Ponzi scheme.

Via the foundation, he and Marilyn also funded the renovation of the building housing the mathematics department at MIT, which in 2016 was named after the two of them.

The Simons Foundation established the Flatiron Institute in 2016, to house 4 groups of computational scientists (each with 60 or more PhD level researchers). The institute consists of three cores or departments: CCB (the center for computational biology), CCA (Center for Computational Astrophysics), CCQ (Center for Computational Quantum mechanics). A fourth, yet to be assembled core, will focus on another branch of applied computational science. The new institute is located in Manhattan and represents a major investment in basic computational science.

As reported, the billionaire has given away $2.1 billion to charity, chairs Math for America, and supports autism research.

In 2008, he was inducted into Institutional Investors Alpha's Hedge Fund Manager Hall of Fame along with David Swensen, Louis Bacon, Steven Cohen, Kenneth Griffin, Paul Tudor Jones, George Soros, Michael Steinhardt, Jack Nash, Seth Klarman, Alfred Jones, Leon Levy, Julian Roberston, and Bruce Kovner.

Publications and works

bullet "Minimal Cones, Plateau's Problem, and the Bernstein Conjecture". Proc Natl Acad Sci U S A. 58 (2): 410–411. August 1967. doi:10.1073/pnas.58.2.410. PMC 335649Freely accessible. PMID 16578656. 
bulletwith Shiing-Shen Chern: "Some Cohomology Classes in Principal Fiber Bundles and Their Application to Riemannian Geometry". Proc Natl Acad Sci U S A. 68 (4): 791–794. April 1971. doi:10.1073/pnas.68.4.791. PMC 389044Freely accessible. PMID 16591916. 
bulletwith Jean-Pierre Bourguignon and H. Blaine Lawson: "Stability and gap phenomena for Yang-Mills fields". Proc Natl Acad Sci U S A. 76 (4): 1550–1553. April 1979. doi:10.1073/pnas.76.4.1550. PMC 383426Freely accessible. PMID 16592637. 
bullet "Minimal varieties in riemannian manifolds". Annals of Mathematics. 88 (1): 62–105. July 1968. doi:10.2307/1970556. 
bulletwith Shiing-Shen Chern: "Characteristic forms and geometric invariants". Annals of Mathematics. 99 (1): 48–69. January 1974. doi:10.2307/1971013.

    Stephen Hawkings (1942-      ):  (British) A physicist from Cambridge wrote "A Brief History of Time".  In it he tells the story of a lady commenting on a statement made in a lecture on astronomy.  She said: "Rubbish, The world is really a flat plate supported on the back of a giant tortoise"  When asked what the tortoise was sitting on, her answer would have made Goedel smile: "You're very clever, young man, very clever.  But its turtles all the way down." 

Hawking is an English theoretical physicist, cosmologist, auhor and director of research at the center for Theoretical Cosmology within the University of Cambridge.  Hawking was the first to set forth a cosmology explained by a union of the general theory of relativity and quantum mechantics.  He is a vigorous supporter of the many-worlds interpretation of quantum mechanics.

 

Stephen Hawking has worked on the basic laws which govern the universe. With Roger Penrose he showed that Einstein's General Theory of Relativity implied space and time would have a beginning in the Big Bang and an end in black holes.

 

    Ian Stewart (1945-       ):  A British mathematician.  Stewart came to the attention of the mathematics teacher. The teacher had Stewart sit mock A-level examinations without any preparation along with the upper-sixth students; Stewart placed first in the examination. This teacher arranged for Stewart to be admitted to Cambridge on a scholarship to Churchill College, where he obtained a BA in mathematics. Stewart then went to the University of Warwick for his doctorate, on completion of which in 1969 he was offered an academic position at Warwick, where he presently professes mathematics. He is well known for his popular expositions of mathematics and his contributions to catastrophe theory.

While at Warwick he edited the mathematical magazine Manifold. He also wrote a column called "Mathematical Recreations" for Scientific American magazine for several years. Stewart has published more than 140 scientific papers, including a series of influential papers co-authored with Jim Collins on coupled oscillators and the symmetry of animal gaits.  He lists his recreations as science fiction, painting, guitar, keeping fish, geology, Egyptology and snorkeling.

    Persi Warren Diaconis (1945-       ): He is the statistician who demonstrated that it takes the average card player no fewer than seven shuffles to create a random order in a deck of cards.

    Marilyn vos Savant (1946-       ):  An American magazine columnist, author, lecturer, and playwright. She has written "Ask Marilyn", a Sunday column in Parade magazine in which she solves puzzles and answers questions from readers on a variety of subjects.

Her September 9, 1990 column began with a question now called The Monty Hall problem (Suppose you are on a game show and you are given the choice of three doors.  Behind one door is a car, the others, goats. You pick a door, say #1, and the host, who knows what's behind the doors, opens another door, say #3, which has a goat. He says to you: 'Do you want to pick door #2?' Is it to your advantage to switch doors? 

Marilyn vos Savant answered arguing that the selection should be switched to door #2 because it has a 2/3 chance of success, while door #1 has just 1/3. This response provoked letters of thousands of readers, nearly all arguing doors #1 and #2 each have an equal chance of success. A follow-up column reaffirming her position served only to intensify the debate and soon became a feature article on the front page of The New York Times. Among the ranks of dissenting arguments were hundreds of academics and mathematicians. 

In a subsequent column, vos Savant offered numerous explanations as to why her solution is correct.  She also called upon elementary teachers to simulate the problem in their class. Numerous elementary school math classes devoted themselves to this experiment, playing the game hundreds of times and reporting their results. Nearly 100% of those classes found that your odds of winning were doubled if you switch doors.

Finally, thanks to the diligence of elementary school children, the controversy subsided.

    William Paul Thurston (October 30, 1946 – August 21, 2012): He was an American mathematician. He was a pioneer in the field of low-dimensional topology. In 1982, he was awarded the Fields Medal for his contributions to the study of 3-manifolds. From 2003 until his death he was a professor of mathematics and computer science at Cornell University

Thurston's geometrization conjecture states that certain three-dimensional topological spaces each have a unique geometric structure that can be associated with them. It is an analogue of the uniformization theorem for two-dimensional surfaces, which states that every connected Riemann surface can be given one of three geometries (Euclidean, spherical, or hyperbolic). In three dimensions, it is not always possible to assign a single geometry to a whole topological space. Instead, the geometrization conjecture states that every closed 3-manifold can be decomposed in a canonical way into pieces that each have one of eight types of geometric structure. The conjecture was proposed by William Thurston (1982), and implies several other conjectures, such as the Poincaré conjecture and Thurston's elliptization conjecture.

Thurston's hyperbolization theorem implies that Haken manifolds satisfy the geometrization conjecture. Thurston announced a proof in the 1980s and since then several complete proofs have appeared in print.

Grigori Perelman sketched a proof of the full geometrization conjecture in 2003 using Ricci flow with surgery. There are now several different manuscripts (see below) with details of the proof. The Poincaré conjecture and the spherical space form conjecture are corollaries of the geometrization conjecture, although there are shorter proofs of the former that do not lead to the geometrization conjecture.

    Dr. Keith Devlin (1947-     ): This professor from Stamford defines Mathematics as  the Science of Patterns. 

    Robert L. Devaney (circa 1948-        ):   A native of Methuen, Massachusetts, is currently Professor of Mathematics at Boston University. He received his undergraduate degree from the College of the Holy Cross in 1969 and his PhD from the University of California at Berkeley in 1973 under the direction of Stephen Smale. He taught at Northwestern University and Tufts University before coming to Boston University in 1980.  His main area of research is dynamical systems, primarily complex analytic dynamics, but also including more general ideas about chaotic dynamical systems. Lately, he has become intrigued with the incredibly rich topological aspects of dynamics, including such things as indecomposable continua, Sierpinski curves, and Cantor bouquets.

Devaney developed the 8 minute script contained  in the middle of the presentation on Chaos that Don had presented for the 1992 Annual Meeting of the Society of Actuaries. This link only works on Don's computer.

     Edward Witten (1951-      ):  One of the researchers at Princeton working on "string theory" which may help with the clash between the central ideas of general relativity and quantum mechanics when it comes to extremely small scales.   He is regarded by many of his peers as one of the greatest living physicists, perhaps even a successor to Albert Einstein In 1990 he was awarded a Fields Medal by the International Union of Mathematics, which is the highest honor in mathematics and often regarded as the Nobel Prize equivalent for mathematics. He is the only physicist to have received this honor.

Physicist

Born August 26, 1951, in Baltimore, MD; son of Louis W. Witten (a physicist); married Chiara Nappi (a physicist); children: Daniela, Ilana, Rafael. Education: Brandeis University, history degree, 1971; Princeton University, masters degree (physics), 1974, Ph.D. (physics), 1976.

Addresses: Home —Princeton, NJ. Office —Institute for Advanced Study, School of Natural Sciences, Einstein Dr., Princeton, NJ 08540.

Freelance writer, early 1970s; aide, George McGovern presidential campaign, 1972; junior fellow, Harvard University, Society of Fellows, 1977-80; full professor, Princeton University, Department of Physics, 1980-87; Charles Simonyi professor of mathematical physics, Institute of Advanced Studies, Princeton, NJ, 1987—.

Member: International Centre for Peace in the Middle East, Tel Aviv, Israel; board, Americans for Peace Now; attended Emergency World Jewish Leadership Peace Conference, Jerusalem, Israel.

Awards: MacArthur Fellowship, 1982; Einstein Medal, 1985; New York Academy of Science Award for Physics and Math Science, 1985; National Science Foundation, Alan T. Waterman Award, 1986; Fields Medal, co-winner, 1990; named one of 25 most influential Americans, Time, 1996; named one of 100 most influential in the world, 2004; Dannie Heinemann Prize, 1998; Frederic Esser Nemmers Prize in mathematics, 2000; National Medal of Science, 2002.

In the world of physics, Edward Witten is a super-star, and considered by many to be the savior of the field. His participation in the discovery of the super string theory and his M-string theory has sparked much debate in the science community as physicists and mathematicians everywhere researched to prove these theories right or wrong. Because of his contributions, he has been awarded the Fields Medal and named as one of Time 's 25 Most Influential Americans.

Witten was born on August 26, 1951, in Baltimore, Maryland. He was highly intelligent as a toddler. His father, Louis W. Witten, a gravitational physicist, was talking physics with him at the age of four. His father told Jack Klaff of the Guardian, "I would talk to Ed about science the way I would talk with adults." Witten attended Baltimore Hebrew school as a child. At the age of 12 his letters denouncing the Vietnam War appeared in the local newspaper's editorial section. Although Witten was fascinated by physics, he wanted to become a journalist. He attended Brandeis University, and graduated with a degree in history.

Witten wrote articles for the Nation and the New Republic. He also worked as an aide on George McGovern's presidential campaign in 1972. With his interest in both journalism and politics waning, Witten returned to school. He entered the doctoral program at Princeton University. At first unsure of whether to study mathematics or physics, he chose physics and earned his masters degree in 1974, and his doctorate in 1976.

Witten began his career in physics as a junior fellow of the Society of Fellows at Harvard University. Later he returned to Princeton in 1980 as full professor, one of the youngest to be appointed to that position. He taught in the physics department and many of his students nicknamed him "The Martian" because of his soft-spoken voice and his style of lecturing, which included long pauses as he gathered his thoughts. Despite his nickname, many of his students had the utmost respect for Witten.

When Albert Einstein released his theory of relativity, he breathed new life into the field of physics. Einstein spent his later years expanding his theory and trying to combine relativity with quantum physics, which both contradicted each other. He died in 1955 before finding the solution. Many thought all of the major discoveries in physics and mathematics had been discovered. Three particle theorists developed string theory which theorized that nature is not made up of miniscule particles but of tiny loops and strings, which also vibrated like a violinand instead of four dimensions, there were 26. A number of physicists disproved of the concept and it was later abandoned. A couple of physicists later lowered the number down to ten.

Witten devoted his energies to further developing string theory. He told John Horgan of the Scientific American , "It was very clear that if I didn't spend my life concentrating on string theory, I would simply be missing my life's calling." In 1984 he and a fellow physicist wrote a paper "about anomalies that occur during radioactive decay that could only be studied in terms of topology [shape connection] and only in ten dimensions," according to World of Mathematics. This hypothesis cemented previous findings that stated string theory required the presence of ten dimensions. This theory also became known as the superstring theory.

Witten's papers energized both the mathematics and physics community. Soon five varying ideas were

competing as the string theory. Witten's belief that superstring theory would change the world was so intense he wrote a record 19 papers in one year, making him the chief proponent of string theory. He ended his teaching career at Princeton, and in 1987 became the Charles Simonyi professor of mathematical physics at the Institute of Advanced Studies (IAS), where Einstein spent his last years.

With five varying ideas of string theory, string theory reached a stalemate. However, Witten worked to find which idea was indeed the one that defined string theory. His research soon discovered that all were in fact aspects of the string theory. He combined them all to form the M-string theory. He published his findings in 1995. Again Witten sparked a flurry of debate in the community. Using "the analogy of blind men examining an elephant to explain the course of string theory until 1995," Nathan Seiberg, also working at the IAS, told Alok Jha of theGuardian, "One describes touching a leg, one describes touching a trunk, another describes the ears. They come up with different descriptions but they don't see the big picture. There is only one elephant.…"

Witten's M-theory while bringing together the five various ideas into a workable equation, also added one more dimension and suggested that the strings were membranes or branes. These branes could exist in at least three dimensions and could grow to the size of the universe. Witten also theorized that our universe could be sitting on a brane.

Witten continued to develop new theories, including working on the twistor theory, which was created in 1965. Working this theory with the new discoveries of the day led Witten to conclude that all of the extra dimensions in both string theory and his M-theory were no longer needed. However, he told Jha of the Guardian , "I think twistor string theory is something that only partly works."

In the area of string theory, Witten has been the most prolific contributor. Another colleague at the IAS, Juan Maldacena, told Michael Lemonick in Time, "Most other people have made one or two such contributions. Ed has made ten or 15." Witten and many others believe that string theory is one step toward developing the "Theory of Everything." This theory would provide the answers to nature, the Big Bang theory, and everything else.

Witten was not without detractors. With him being a theoretical physicist, his main focus was on using calculations versus running experiments. As a result, some thought he relied too heavily on mathematics rather than actual physics. Also many believed that string theory was loopy and pure conjecture, since technically, nothing has been proven.

Witten, however, has been the recipient of numerous honors and awards. He won the Einstein Medal and the New York Academy of Science Award for Physics and Math Science in 1985. In 1990 he shared the

 Fields Medal, the most prestigious award given in mathematics, also the closest he could get to a Nobel Prize. He also won the Dannie Heinemann Prize in 1998. In 2000 Witten received the Frederic Esser Nemmers Prize in Mathematics and was also awarded the National Medal of Science for his contributions to mathematics and theoretical physics in 2002.

Witten is considered by many to be a genius or as close as one can get. He is married to Chiara Nappi, a physicist at Princeton University. They have three children. In addition to his many discoveries, Witten is very active in such organizations as the International Centre for Peace in the Middle East and Americans for Peace Now.

Read more: http://www.notablebiographies.com/newsmakers2/2006-Ra-Z/Witten-Edward.html#ixzz4ziumx9Eg

    Sir Andrew John Wiles (1953-      ):  (British)  A professor at Princeton University in Number Theory. He published a flawed proof of Fermat's Last Theorem in 1993.  He corrected the error in 1994.  Royal Society Research Professor at the University of Oxford, specialising in number theory. He is best known for proving Fermat's Last Theorem, for which he received the 2016 Abel Prize. Wiles has received numerous other honours, including the Copley Medal, the Royal Society's highest honour, in 2017.

Wiles states that he came across Fermat's Last Theorem on his way home from school when he was 10 years old. He stopped by his local library where he found a book about the theorem.  Fascinated by the existence of a theorem that was so easy to state that he, a ten-year-old, could understand it, but that no one had proven, he decided to be the first person to prove it. However, he soon realised that his knowledge was too limited, so he abandoned his childhood dream, until it was brought back to his attention at the age of 33 by Ken Ribet's 1986 proof of the epsilon conjecture, which Gerhard Frey had previously linked to Fermat's famous equation.

Wiles earned his bachelor's degree in mathematics in 1974 at Merton College, Oxford, and a PhD in 1980 as a postgraduate student of Clare College, Cambridge. After a stay at the Institute for Advanced Study in New Jersey in 1981, Wiles became a professor at Princeton University. In 1985–86, Wiles was a Guggenheim Fellow at the Institut des Hautes Études Scientifiques near Paris and at the École Normale Supérieure. From 1988 to 1990, Wiles was a Royal Society Research Professor at the University of Oxford, and then he returned to Princeton. He rejoined Oxford in 2011 as Royal Society Research Professor.

Wiles's graduate research was guided by John Coates beginning in the summer of 1975. Together these colleagues worked on the arithmetic of elliptic curves with complex multiplication by the methods of Iwasawa theory. He further worked with Barry Mazur on the main conjecture of Iwasawa theory over the rational numbers, and soon afterward, he generalised this result to totally real fields.[12]

Proof of Fermat's Last Theorem

Starting in mid-1986, based on successive progress of the previous few years of Gerhard Frey, Jean-Pierre Serre and Ken Ribet, it became clear that Fermat's Last Theorem could be proven as a corollary of a limited form of the modularity theorem (unproven at the time and then known as the "Taniyama–Shimura–Weil conjecture"). The modularity theorem involved elliptic curves, which was also Wiles's own specialist area.[13]

The conjecture was seen by contemporary mathematicians as important, but extraordinarily difficult or perhaps impossible to prove.[14]:203–205, 223, 226 For example, Wiles's ex-supervisor John Coates states that it seemed "impossible to actually prove",[14]:226 and Ken Ribet considered himself "one of the vast majority of people who believed [it] was completely inaccessible", adding that "Andrew Wiles was probably one of the few people on earth who had the audacity to dream that you can actually go and prove [it]."[14]:223

Despite this, Wiles, with his from-childhood fascination with Fermat's Last Theorem, decided to undertake the challenge of proving the conjecture, at least to the extent needed for Frey's curve.  He dedicated all of his research time to this problem for over six years in near-total secrecy, covering up his efforts by releasing prior work in small segments as separate papers and confiding only in his wife.[14]:229–230

In June 1993, he presented his proof to the public for the first time at a conference in Cambridge.

He gave a lecture a day on Monday, Tuesday and Wednesday with the title 'Modular Forms, Elliptic Curves and Galois Representations.' There was no hint in the title that Fermat's last theorem would be discussed, Dr. Ribet said. ... Finally, at the end of his third lecture, Dr. Wiles concluded that he had proved a general case of the Taniyama conjecture. Then, seemingly as an afterthought, he noted that that meant that Fermat's last theorem was true. Q.E.D.

In August 1993, it was discovered that the proof contained a flaw in one area. Wiles tried and failed for over a year to repair his proof. According to Wiles, the crucial idea for circumventing, rather than closing this area, came to him on 19 September 1994, when he was on the verge of giving up. Together with his former student Richard Taylor, he published a second paper which circumvented the problem and thus completed the proof. Both papers were published in May 1995 in a dedicated volume of the Annals of Mathematics.

Be sure to read https://www.youtube.com/watch?v=6ymTZEeTjI8

    Yitang "Tom" Zhang (1955 -     ): (Chinese: 张益唐; pinyin: Zhāng Yìtáng) is a Chinese-born American mathematician working in the area of number theory. While working for the University of New Hampshire as a lecturer, Zhang submitted an article to the Annals of Mathematics in 2013 which established the first finite bound on gaps between prime numbers. This work led to a 2014 MacArthur award and his appointment as a professor.

Yitang Zhang
Yitang Zhang.jpg
Zhang in 2014
Born 1955 (age 61–62)
Shanghai, China
Residence United States
Nationality Chinese-American
Alma mater Purdue University (PhD 1991)
Peking University (B.A. 1982; M.A. 1984)
Known for Establishing the existence of an infinitely repeatable prime 2-tuple
Awards Ostrowski Prize (2013)
Cole Prize (2014)
Rolf Schock Prize (2014)
MacArthur Fellowship (2014)
Academia Sinica Fellow (2014)
Scientific career
Fields Mathematics
Institutions University of New Hampshire
University of California, Santa Barbara
Thesis The Jacobian Conjecture And The Degree Of Field Extension (1992)
Doctoral advisor Tzuong-Tsieng Moh (Chinese: 莫宗堅)
 

Zhang was born in Shanghai and lived there until he was 13 years old. At around the age of nine, he found a proof of the Pythagorean theorem.[8] He first learned about Fermat’s last theorem and the Goldbach conjecture when he was 10.[8] During the Cultural Revolution, he and his mother were sent to the countryside to work in the fields. He worked as a laborer for 10 years and was unable to attend high school.[8] After the Cultural Revolution ended, Zhang entered Peking University in 1978 as an undergraduate student and received his B.Sc. degree in mathematics in 1982. He became a graduate student of Professor Pan Chengbiao, a number theorist at Peking University, and obtained his M.Sc. degree in mathematics in 1984.[1]

After receiving his master's degree in mathematics, with recommendations from Professor Ding Shisun, the President of Peking University, and Professor Deng Donggao, Chair of the university's Math Department,[9] Zhang was granted a full scholarship at Purdue University. Zhang arrived at Purdue in January 1985, studied there for seven years, and obtained his Ph.D. in mathematics in December 1991.

Zhang's Ph.D. work was on the Jacobian conjecture. After graduation, Zhang had a hard time finding an academic position. In a 2013 interview with Nautilus magazine, Zhang said he did not get a job after graduation. "During that period it was difficult to find a job in academics. That was a job market problem. Also, my advisor did not write me letters of recommendation." The reason behind this is that Zhang's research pointed out the mistakes made by his advisor Tzuong-Tsieng Moh's previous work. Moh was very unhappy with this and refused to write the job recommendation letter for Zhang.[10] Zhang made this claim again in George Csicsery’s documentary film Counting From Infinity while discussing his difficulties at Purdue and in the years that followed. Tzuong-Tsieng Moh, his Ph.D. advisor at Purdue, said that Zhang never came back to him requesting recommendation letters.[9] In a detailed profile published in The New Yorker magazine in February 2015, Alec Wilkinson wrote Zhang "parted unhappily" with Moh, and that Zhang "left Purdue without Moh’s support, and, having published no papers, was unable to find an academic job". After some years, Zhang managed to find a position as a lecturer at the University of New Hampshire, where he was hired by Kenneth Appel in 1999.

Prior to getting back to academia, he worked for several years as an accountant and a delivery worker for a New York City restaurant. He also worked in a motel in Kentucky and in a Subway sandwich shop.  A profile published in the Quanta Magazine reports that Zhang used to live in his car during the initial job-hunting days.  He served as lecturer at UNH from 1999 until around January 2014, when UNH appointed him to a full professorship as a result of his breakthrough on prime numbers. In Fall 2015, Zhang accepted an offer of full professorship at the University of California, Santa Barbara.

On April 17, 2013, Zhang announced a proof that states there are infinitely many pairs of prime numbers that differ by 70 million or less. This result implies the existence of an infinitely repeatable prime 2-tuple, thus establishing a theorem akin to the twin prime conjecture. Zhang's paper was accepted by Annals of Mathematics in early May 2013, his first publication since his last paper in 2001.  The proof was refereed by leading experts in analytic number theory.[7] Zhang's result set off a flurry of activity in the field, such as the Polymath8 project.

If P(N) stands for the proposition that there is an infinitude of pairs of prime numbers (not necessarily consecutive primes) that differ by exactly N, then Zhang's result is equivalent to the statement that there exists at least one even integer k < 70,000,000 such that P(k) is true. The classical form of the twin prime conjecture is equivalent to P(2); and in fact it has been conjectured that P(k) holds for all even integers k.[16][17] While these stronger conjectures remain unproven, a result due to James Maynard in November 2013, employing a different technique, showed that P(k) holds for some k ≤ 600.  Subsequently, in April 2014, the Polymath project 8 lowered the bound to k ≤ 246.[19] With current methods k ≤ 6 is the best attainable, and in fact k ≤ 12 and k ≤ 6 follow using current methods if the Elliott–Halberstam conjecture and its generalisation, respectively, hold.

He is a recipient of the 2014 MacArthur award, and was elected as an Academia Sinica Fellow during the same year.[1] He was an invited speaker at the 2014 International Congress of Mathematicians.

Zhang was born in Shanghai and lived there until he was 13 years old. At around the age of nine, he found a proof of the Pythagorean theorem.  He first learned about Fermat’s last theorem and the Goldbach conjecture when he was 10.  During the Cultural Revolution, he and his mother were sent to the countryside to work in the fields. He worked as a laborer for 10 years and was unable to attend high school.  After the Cultural Revolution ended, Zhang entered Peking University in 1978 as an undergraduate student and received his B.Sc. degree in mathematics in 1982. He became a graduate student of Professor Pan Chengbiao, a number theorist at Peking University, and obtained his M.Sc. degree in mathematics in 1984.

After receiving his master's degree in mathematics, with recommendations from Professor Ding Shisun, the President of Peking University, and Professor Deng Donggao, Chair of the university's Math Department, Zhang was granted a full scholarship at Purdue University. Zhang arrived at Purdue in January 1985, studied there for seven years, and obtained his Ph.D. in mathematics in December 1991.

Zhang's Ph.D. work was on the Jacobian conjecture. After graduation, Zhang had a hard time finding an academic position. In a 2013 interview with Nautilus magazine, Zhang said he did not get a job after graduation. "During that period it was difficult to find a job in academics. That was a job market problem. Also, my advisor did not write me letters of recommendation." The reason behind this is that Zhang's research pointed out the mistakes made by his advisor Tzuong-Tsieng Moh's previous work. Moh was very unhappy with this and refused to write the job recommendation letter for Zhang.   Zhang made this claim again in George Csicsery’s documentary film Counting From Infinity while discussing his difficulties at Purdue and in the years that followed.  Tzuong-Tsieng Moh, his Ph.D. advisor at Purdue, said that Zhang never came back to him requesting recommendation letters.  In a detailed profile published in The New Yorker magazine in February 2015, Alec Wilkinson wrote Zhang "parted unhappily" with Moh, and that Zhang "left Purdue without Moh’s support, and, having published no papers, was unable to find an academic job".  After some years, Zhang managed to find a position as a lecturer at the University of New Hampshire, where he was hired by Kenneth Appel in 1999.

Prior to getting back to academia, he worked for several years as an accountant and a delivery worker for a New York City restaurant. He also worked in a motel in Kentucky and in a Subway sandwich shop. A profile published in the Quanta Magazine reports that Zhang used to live in his car during the initial job-hunting days.  He served as lecturer at UNH from 1999  until around January 2014, when UNH appointed him to a full professorship as a result of his breakthrough on prime numbers.  In Fall 2015, Zhang accepted an offer of full professorship at the University of California, Santa Barbara

Research  On April 17, 2013, Zhang announced a proof that states there are infinitely many pairs of prime numbers that differ by 70 million or less. This result implies the existence of an infinitely repeatable prime 2-tuple,  thus establishing a theorem akin to the twin prime conjecture. Zhang's paper was accepted by Annals of Mathematics in early May 2013, his first publication since his last paper in 2001. The proof was refereed by leading experts in analytic number theory.Zhang's result set off a flurry of activity in the field, such as the Polymath8 project.

If P(N) stands for the proposition that there is an infinitude of pairs of prime numbers (not necessarily consecutive primes) that differ by exactly N, then Zhang's result is equivalent to the statement that there exists at least one even integer k < 70,000,000 such that P(k) is true. The classical form of the twin prime conjecture is equivalent to P(2); and in fact it has been conjectured that P(k) holds for all even integers k.  While these stronger conjectures remain unproven, a result due to James Maynard in November 2013, employing a different technique, showed that P(k) holds for some k ≤ 600. Subsequently, in April 2014, the Polymath project 8 lowered the bound to k ≤ 246. With current methods k ≤ 6 is the best attainable, and in fact k ≤ 12 and k ≤ 6 follow using current methods if the Elliott–Halberstam conjecture and its generalisation, respectively, hold.

Zhang was awarded the 2013 Morningside Special Achievement Award in Mathematics, the 2013 Ostrowski Prize, the 2014 Frank Nelson Cole Prize in Number Theory, and the 2014 Rolf Schock Prize in Mathematics.

He is a recipient of the 2014 MacArthur award,  and was elected as an Academia Sinica Fellow during the same year. He was an invited speaker at the 2014 International Congress of Mathematicians.

   Simon Kirwan Donaldson (1957-      ):  (British)  An English mathematician famous for his work on the topology of smooth (differentiable) four-dimensional manifolds. He is now Royal Society research professor in Pure Mathematics and President of the Institute for Mathematical Science at Imperial College London.  He used the solutions to the Yang-Mills equations to discover a finger-print which allowed him to distinguish whether two shapes were actually the same.  These finger-prints are called invariants.  

     Steven Strogatz ( 1959 -        ): (American) He is an applied mathematics professor at Cornell.  He has written an interesting book: The Joy of X.

     James Grime (          -      ): http://singingbanana.com/about/ 

James is a mathematician with a personal passion for math communication and the promotion of mathematics in schools and to the general public. He can be mostly found doing exactly that, either touring the world giving public talks, or on YouTube.

James has a PhD in mathematics and his academic interests include group theory (the mathematics of symmetry) and combinatorics (the mathematics of networks and solving problems with diagrams and pictures). James also has a keen interest in cryptography (the mathematics of codes and secret messages), probability (games, gambling and predicting the future) and number theory (the properties of numbers).

James loves math and remembers watching Johnny Ball leaping about on TV explaining the parabolic path of projectiles. And the theme tune of the Royal Institution Christmas Lectures still gives him chills of excitement – yes he was that type of geek.

James went on to study mathematics at Lancaster university. He was attracted by the challenge of the analytical and creative thought required in a math degree, but it was probably the lack of essays and reading list he found most attractive. Later, James went to York University with the aim of getting a PhD and avoiding the real world for at least another three years. He was successful on both counts.

James Grime: a riddle wrapped in a mystery standing next to an enigma.

After working in research in combinatorics and group theory, James joined the Millennium Mathematics Project from the University of Cambridge. On their behalf James ran The Enigma Project, with the aim to bring mathematics to life through the fascinating history and mathematics of codes and code breaking. Spies! Secrets! And secret messages!

James travelled extensively giving public talks and visiting schools, colleges, universities, festivals and other events, and reaching 12,000 people, of all ages, every year. Touring took James all over the UK, and the world, and involved talks for Google, Microsoft, RSA conference, Maths Inspiration, Maths in Action, BrainStem (Perimeter Institute Canada), and various science festivals. James’ aim is to bring not only an in depth knowledge of mathematics to the talk, but also to present it in an accessible and fun style.

After leaving the Millennium Mathematics Project in 2014 James continues to give public talks on code breaking and other topics.

In 2008 James started making maths videos on YouTube on his own channel called “singingbanana” (why not). These were made to entertain a few friends, so James was thrilled to when he reached 100 subscribers, ecstatic when he reached 1000 subscribers, and now exploding from joy with over 100,000 subscribers. The videos are a series of problems, tricks, and whatever mathematical things James happens to find interesting that week.

In 2011 James was contacted by video journalist Brady Haran (periodicvideos, sixtysymbols) to help create a new maths channel called Numberphile. This series uses numbers to introduce people to mathematics, including current news. Numberphile involves a team of contributors and is one of the most popular channels on YouTube, with over 1,000,000 subscribers.

Through his videos, public talks and other work, James hopes to explain to kids and general audiences why he love his maths so much, to challenge some of the public’s misconceptions, and to explain why he considers it a beautiful subject in a way that is closer to an art.

In his spare time James has many hobbies, including juggling, unicycling and a great number of other circus skills, and has finally embraced the fact that his ultimate purpose in life may be simply to make a fool of himself in public.

James was very disappointed when they changed the Royal Institution theme tune.

     Henry Bottomley (1963-      ):  (English)  http://www.se16.info/hgb/cv.htm  Statistician 

 

   

 

 

 

 

 

 

Marcus Peter Francis du Sautoy (1965 -      ): (born in London) A Professor of Mathematics at the University of Oxford. His academic work concerns mainly group theory and number theory. He is known for his books popularizing mathematics. In 2001 he won the Berwick Prize of the London Mathematical Society, which is awarded every two years to reward the best mathematical research by a mathematician under forty. In March, 2006, his article Prime Numbers Get Hitched was published on Seed Magazine's website.   http://seedmagazine.com/content/article/prime_numbers_get_hitched/  In it he explained how the number 42, mentioned in The Hitchhiker's Guide to the Galaxy as the answer to everything, is related to the Riemann zeta function. See http://www.culturenorthernireland.org/article/2836/belfast-festival-marcus-du-sautoy Also:  http://people.maths.ox.ac.uk/dusautoy/newleft.htm and http://people.maths.ox.ac.uk/dusautoy/newright.htm  

    Grigori Yakovlevich Perelman (1966-       ):  (from Russia) The Millennium Prize Problems are seven problems in mathematics that were stated by the Clay Mathematics Institute  in 2000. Currently, six of the problems remain unsolved. A correct solution to any of the problems results in a US$1,000,000 prize (sometimes called a Millennium Prize) being awarded by the institute. One of the problems, the Poincare' conjecture, was solved by Perelman in  2002  He was also awarded  the Fields Medal in 2006.  He has not accepted either prize.   

     Edward Vladimirovich Frenkel (1967 circa -     ): (Russian  A mathematician working in representation theory, algebraic geometry, and mathematical physics. He is a professor of mathematics at University of California, Berkeley and author of the bestselling book Love and Math.  As a high school student he studied higher mathematics privately with Evgeny Evgenievich Petrov, although his initial interest was in quantum physics rather than mathematics. He was not admitted to Moscow State University because of discrimination against Jews and enrolled instead in the applied mathematics program at the Gubkin University of Oil and Gas. While a student there, he attended the seminar of Israel Gelfand and worked with Boris Lwowitsch Feigin (de) and Dmitry Fuchs. After receiving his college degree in 1989, he was first invited to Harvard University as a visiting professor, and a year later he enrolled as a graduate student at Harvard He received his Ph.D. at Harvard University in 1991, after one year of study, under the direction of Joseph Bernstein. He was a Junior Fellow at the Harvard Society of Fellows from 1991 to 1994, and served as an associate professor at Harvard from 1994 to 1997. He has been a professor of mathematics at University of California, Berkeley since 1997.  Frenkel's recent work has focused on the Langlands program and its connections to representation theoryintegrable systems, geometry, and physics. Together with Dennis Gaitsgory and Kari Vilonen, he has proved the 

geometric Langlands conjecture for GL(n). His joint work with Robert Langlands and Ngô Bảo Châu suggested a new approach to the functoriality of automorphic representations and trace formulas. He has also been investigating (in particular, in a joint work with Edward Witten) connections between the geometric Langlands correspondence and dualities in quantum field theory

    

Kannan Soundararajan
Kannan Soundararajan Stanford October 2010.jpg
Soundararajan teaching at Stanford University
Nationality Indian
Alma mater University of Michigan
Princeton University
Awards Ostrowski Prize (2011)
Infosys Prize (2011)
SASTRA Ramanujan Prize (2005)
Salem Prize (2003)
Morgan Prize (1995)
Scientific career
Fields Mathematics
Institutions Stanford University
Doctoral advisor Peter Sarnak
Doctoral students Maksym Radziwill
 

Kannan Soundararajan is a mathematician and a professor of mathematics at Stanford University. Before moving to Stanford in 2006, he was a faculty member at University of Michigan where he pursued his undergraduate studies. His main research interest is in analytic number theory, particularly in the subfields of automorphic L-functions, and multiplicative number theory.

Soundararajan grew up in Chennai and was a student at Padma Seshadri High School in Nungambakkam in Madras (now Chennai), India. In 1989, he attended the prestigious Research Science Institute. He represented India at the International Mathematical Olympiad in 1991 and won a Silver Medal.

Soundararajan joined the University of Michigan, Ann Arbor, in 1991 for undergraduate studies, and graduated with highest honours in 1995. Soundararajan won the inaugural Morgan Prize in 1995 for his work in analytic number theory whilst an undergraduate at the University of Michigan,[1] where he later served as professor. He joined Princeton University in 1995 and did his Ph.D under the guidance of Professor Peter Sarnak.

After his Ph.D. he received the first five-year fellowship from the American Institute of Mathematics, and held positions at Princeton University, the Institute for Advanced Study, and the University of Michigan. He moved to Stanford University in 2006 where he is currently a Professor of Mathematics and the Director of the Mathematics Research Center (MRC) at Stanford.

He proved a conjecture of Ron Graham in combinatorial number theory jointly with Ramachandran Balasubramanian. He made important contributions in settling the arithmetic Quantum Unique Ergodicity conjecture for Maass wave forms and modular fo

He received the Salem Prize in 2003 "for contributions to the area of Dirichlet L-functions and related character sums". In 2005, he won the $10,000 SASTRA Ramanujan Prize, shared with Manjul Bhargava, awarded by SASTRA in Thanjavur, India, for his outstanding contributions to number theory. In 2011, he was awarded the Infosys science foundation prize 2011. He was awarded the Ostrowski prize  in 2011, shared with lb Madsen and David Preiss, for a cornucopia of fundamental results in the last five years to go along with his brilliant earlier work.

He gave an invited talk at the International Congress of Mathematicians in 2010, on the topic of "Number Theory". He was elected to the 2018 class of fellows of the American Mathematical Society

Soundararajan resides in Palo Alto, California with his wife and one son.

Selected publications

bullet R. Holowinsky and K. Soundararajan, "Mass equidistribution for Hecke eigenforms," arXiv:0809.1636v1
bulletK. Soundararajan, "Nonvanishing of quadratic Dirichlet L-functions at s=1/2" arXiv:math/9902163v2

References

External links

bullet Homepage at Stanford University
bullet Making Waves
bullet Kannan Soundararajan at the Mathematics Genealogy Project
bullet "Kannan Soundararajan's results". International Mathematical Olympiad. 
bullet Infosys Prize 2011
bullet Videos of Kannan Soundararajan in the AV-Portal of the German National Library of Science and Technology

     Maryam Mirzakhani (May 3,1977- July 15, 2017.)  (Persian)  was an Iranian mathematician and a professor of mathematics at Stanford University. Her research topics include Teichmüller theory, hyperbolic geometry, ergodic theory, and symplectic geometry.On 13 August 2014, Mirzakhani became both the first woman and the first Iranian honored with the Fields Medal, the most prestigious award in mathematics. The award committee cited her work in "the dynamics and geometry of Riemann surfaces and their moduli spaces".

Maryam Mirzakhani
Maryam Mirzakhani in Seoul 2014.jpg
Mirzakhani at the 2014 International Congress of Mathematicians in Seoul, where she received her Fields Medal
Native name Persian: مریم میرزاخانی‎‎
Born 3 May 1977
Tehran, Iran
Died 14 July 2017 (aged 40)
Stanford, California, U.S.
Nationality Iranian
Education Sharif University of Technology (BSc)
Harvard University (PhD)
Spouse(s) Jan Vondrák
Children 1
Awards
Scientific career
Fields Mathematics
Institutions
Thesis Simple geodesics on hyperbolic surfaces and the volume of the moduli space of curves (2004)
Doctoral advisor Curtis T. McMullen[3]
Other academic advisors Ebadollah S. Mahmoodian[4]

 

Early life and education

Mirzakhani was born on 3 May 1977 in Tehran, Iran. She attended Farzanegan School there, part of the National Organization for Development of Exceptional Talents.

In 1994, Mirzakhani won a gold medal in the International Mathematical Olympiad, the first female Iranian student to do so. In the 1995 International Mathematical Olympiad, she became the first Iranian student to achieve a perfect score and to win two gold medals.

She obtained her BSc in mathematics (1999) from Sharif University of Technology in Tehran. She went to the United States for graduate work, earning a PhD from Harvard University in 2004, where she worked under the supervision of the Fields Medalist Curtis McMullen. She was also a 2004 research fellow of the Clay Mathematics Institute and a professor at Princeton University.

Research work

Mirzakhani has made several contributions to the theory of moduli spaces of Riemann surfaces. In her early work, Mirzakhani discovered a formula expressing the volume of a moduli space with a given genus as a polynomial in the number of boundary components. This led her to obtain a new proof for the formula discovered by Edward Witten and Maxim Kontsevich on the intersection numbers of tautological classes on moduli space,[7] as well as an asymptotic formula for the growth of the number of simple closed geodesics on a compact hyperbolic surface, generalizing the theorem of the three geodesics for spherical surfaces. Her subsequent work has focused on Teichmüller dynamics of moduli space. In particular, she was able to prove the long-standing conjecture that William Thurston's earthquake flow on Teichmüller space is ergodic.

Most recently as of 2014, with Alex Eskin and with input from Amir Mohammadi, Mirzakhani proved that complex geodesics and their closures in moduli space are surprisingly regular, rather than irregular or fractal.The closures of complex geodesics are algebraic objects defined in terms of polynomials and therefore they have certain rigidity properties, which is analogous to a celebrated result that Marina Ratner  arrived at during the 1990s. The International Mathematical Union said in its press release that, "It is astounding to find that the rigidity in homogeneous spaces has an echo in the inhomogeneous world of moduli space."

Mirzakhani was awarded the Fields Medal in 2014 for "her outstanding contributions to the dynamics and geometry of Riemann surfaces and their moduli spaces". The award was made in Seoul at the International Congress of Mathematicians on 13 August.

At the time of the award, Jordan Ellenberg explained her research to a popular audience:

... [Her] work expertly blends dynamics with geometry. Among other things, she studies billiards. But now, in a move very characteristic of modern mathematics, it gets kind of meta: She considers not just one billiard table, but the universe of all possible billiard tables. And the kind of dynamics she studies doesn't directly concern the motion of the billiards on the table, but instead a transformation of the billiard table itself, which is changing its shape in a rule-governed way; if you like, the table itself moves like a strange planet around the universe of all possible tables ... This isn't the kind of thing you do to win at pool, but it's the kind of thing you do to win a Fields Medal. And it's what you need to do in order to expose the dynamics at the heart of geometry; for there's no question that they're there.

Personal life

Mirzakhani was diagnosed with breast cancer in 2013. After four years, it spread to her bone marrow. She died on 15 July 2017. She was survived by her husband Jan Vondrák, a Czech theoretical computer scientist and applied mathematician who is an associate professor at Stanford University, and a daughter named Anahita.

Awards and honors

bullet IPM Fellowship, Tehran, Iran, 1995–99
bulletMerit fellowship Harvard University, 2003
bulletHarvard Junior Fellowship Harvard University, 2003
bulletClay Mathematics Institute Research Fellow 2004
bulletAMS Blumenthal Award 2009
bulletInvited to talk at the International Congress of Mathematicians in 2010, on the topic of "Topology and Dynamical Systems & ODE"
bulletThe 2013 AMS Ruth Lyttle Satter Prize in Mathematics. "Presented every two years by the American Mathematical Society, the Satter Prize recognizes an outstanding contribution to mathematics research by a woman in the preceding six years. The prize was awarded on Thursday, 10 January 2013, at the Joint Mathematics Meetings in San Diego."
bulletNamed one of Nature's ten "people who mattered" of 2014.
bullet Clay Research Award 2014
bullet Plenary speaker at the International Congress of Mathematicians (ICM 2014)
bullet Fields Medal 2014
bulletElected foreign associate to the French Academy of Science in 2015
bulletElected to the American Philosophical Society in 2015.
bullet National Academy of Sciences 2016
bulletElected to the American Academy of Arts and Sciences in 2017.

         Jonas Castillo Tolozo:  (       -     ):  (Amateur mathematician from Columbia) Did work on Triangular Numbers and Pascal's triangle See  https://plus.google.com/+JonasCastilloToloza   Below is his calculation of Pi.  And see https://johncarlosbaez.wordpress.com/2014/02/12/triangular-numbers/  And see https://arxiv.org/abs/1304.5262   and        http://www.cut-the-knot.org/arithmetic/algebra/PiInPascal.shtml

  Pi in Pascal's Triangle via Triangular Numbers

RANKING OF GREAT MATHEMATICIANS  See www.fabpedigree.com/james/mathmen.htm
 


Isaac Newton

Archimedes

Carl Gauss

Leonhard Euler

Bernhard Riemann

Henri Poincaré
 

J.-L. Lagrange

David Hilbert

Euclid

G.W. Leibniz

Alex. Grothendieck

Pierre de Fermat
  1. Isaac Newton
  2. Archimedes
  3. Carl F. Gauss
  4. Leonhard Euler
  5. Bernhard Riemann
 
  1. Henri Poincaré
  2. Joseph-Louis Lagrange
  3. Euclid of Alexandria
  4. David Hilbert
  5. Gottfried W. Leibniz
 
  1. Alexandre Grothendieck
  2. Pierre de Fermat
  3. Évariste Galois
  4. John von Neumann
  5. Niels Abel
 
  1. Karl W. T. Weierstrass
  2. René Descartes
  3. Peter G. L. Dirichlet
  4. Carl G. J. Jacobi
  5. Srinivasa Ramanujan
 
  1. Augustin Cauchy
  2. Brahmagupta
  3. Georg Cantor
  4. Hermann K. H. Weyl
  5. Arthur Cayley
 
  1. Emma Noether
  2. Pythagoras of Samos
  3. Leonardo `Fibonacci'
  4. William R. Hamilton
  5. Aryabhata
 
  1. Charles Hermite
  2. Richard Dedekind
  3. Apollonius of Perga
  4. Pierre-Simon Laplace
  5. Muhammed al-Khowârizmi
 
  1. Kurt Gödel
  2. Diophantus of Alexandria
  3. Bháscara Áchárya
  4. Eudoxus of Cnidus
  5. Blaise Pascal
 
  1. Felix Christian Klein
  2. Jean le Rond d'Alembert
  3. Élie Cartan
  4. Hipparchus of Nicaea
  5. Godfrey H. Hardy
 
  1. Archytas of Tarentum
  2. Alhazen ibn al-Haytham
  3. Carl Ludwig Siegel
  4. Gaspard Monge
  5. Jacques Hadamard
 
  1. Andrey N. Kolmogorov
  2. Johannes Kepler
  3. Joseph Liouville
  4. Hermann G. Grassmann
  5. Julius Plücker
 
  1. F.E.J. Émile Borel
  2. François Viète
  3. Joseph Fourier
  4. Stefan Banach
  5. Jacob Bernoulli
 
  1. F. Gotthold Eisenstein
  2. Giuseppe Peano
  3. Liu Hui
  4. André Weil
  5. Jakob Steiner
 
  1. Jean-Victor Poncelet
  2. M. E. Camille Jordan
  3. Panini of Shalatula
  4. Bonaventura Cavalieri
  5. Christiaan Huygens
 
  1. Jean-Pierre Serre
  2. Pafnuti Chebyshev
  3. L.E.J. Brouwer
  4. James J. Sylvester
  5. Henri Léon Lebesgue
 
bullet Albert Einstein
bullet Galileo Galilei
bullet James C. Maxwell
bullet Aristotle
bullet Girolamo Cardano
 
  1. Alan M. Turing
  2. Michael F. Atiyah
  3. Siméon-Denis Poisson
  4. Pappus of Alexandria
  5. Atle Selberg
 
  1. John E. Littlewood
  2. Johann Bernoulli
  3. Shiing-Shen Chern
  4. Hermann Minkowski
  5. Ernst E. Kummer
 
  1. George Pólya
  2. Felix Hausdorff
  3. F. L. Gottlob Frege
  4. Alfred Tarski
  5. Thales of Miletus
 
  1. Marius Sophus Lie
  2. Johann H. Lambert
  3. John Wallis
  4. George D. Birkhoff
  5. Adrien M. Legendre
 
  1. Omar al-Khayyám
  2. Israel M. Gelfand
  3. Simon Stevin
  4. Daniel Bernoulli
  5. George Boole
 
 
bullet John Willard Milnor
bullet John Horton Conway
bullet Thabit ibn Qurra
bullet Hippocrates of Chios
bullet Nicolai Lobachevsky
 
bullet Jean Gaston Darboux
bullet Andrei A. Markov
bullet Nasir al-Din al-Tusi
bullet Sofia Kovalevskaya
bullet Paul Erdös
 
bullet Oliver Heaviside
bullet James Gregory
bullet Leopold Kronecker
bullet John Napier of Merchiston
bullet Norbert Wiener

horizontal rule

bullet Emil Artin
bullet Georg F. Frobenius
bullet Lennart A.E. Carleson
bullet Tullio Levi-Civita
bullet Gérard Desargues
 
bullet Alexis C. Clairaut
bullet J. Müller `Regiomontanus'
bullet Alfred Clebsch
bullet Oswald Veblen
bullet Abu Rayhan al-Biruni
 
bullet Colin Maclaurin
bullet Ptolemy of Alexandria
bullet S.G. Vito Volterra
bullet Qin Jiushao & Zhu Shiejie
bullet Brook Taylor

horizontal rule

bullet R. Maurice Fréchet
bullet Michel F. Chasles
bullet Henri P. Cartan
bullet Thoralf A. Skolem
bullet Samuel Eilenberg
 
bullet Henry J.S. Smith
bullet Augustus F. Möbius  
bullet Luigi Cremona
bullet Rafael Bombelli
bullet Edmund G.H. Landau  
 
bullet Leonardo da Vinci
bullet Paul A.M. Dirac  
bullet Jamshid Al-Kashi
bullet William K. Clifford
bullet Nicole Oresme
 

Also see http://famous-mathematicians.org/

 

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