

MATHEMATICIANS (Also see
http://aleph0.clarku.edu/~djoyce/mathhist/chronology.html )
Pythagoras of Samos (c. 570c. 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 meetingplaces 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.cuttheknot.org/pythagoras/
A Pythagorean triple consists of
three positive
integers a,
b, and
c, such
that a^{2}
+ b^{2} = c^{2}.
Such a triple is commonly written
(a, b,
c), and a wellknown 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
lowc 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/missingsquarepuzzle.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/missingsquarepuzzlesolution/
but Fibonacci would have preferred
http://jaysdesktop.blogspot.com/2009/07/solutiontomissingsquareproblem.html
Geometry
Are the eight balls
moving in a circle or a straight line?
http://showyou.com/v/ypNe6fsaCVtI/crazycircleillusion?u=multimotion
Archimedes
of Syracuse (c.287c.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
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): 





Step 2: Then calculate the Area: 


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:
 An algorithm for calculation of
pi (π) in
the comments to chapter 1. He calculated pi to
3.141024
<
π
<
3.142074
{\displaystyle 3.141024<\pi
<3.142074}
with a 192 (= 64 × 3) sided
polygon. Archimedes used a
circumscribed 96polygon to obtain the inequality
π
<
22
7
{\displaystyle \pi <{\tfrac
{22}{7}}}
, and then used an inscribed 96gon to obtain the
inequality
223
71
<
π
{\displaystyle {\tfrac
{223}{71}}<\pi }
. Liu Hui used only one inscribed 96gon 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
π
=
157
50
{\displaystyle \pi ={\tfrac
{157}{50}}}
. He later invented a
quick method and obtained
π
=
3.1416
{\displaystyle \pi =3.1416}
, which he checked with a 3072gon(3072 = 512 × 6).
Nine Chapters had used the value 3 for
π, but
Zhang Heng (78139 AD) had previously
estimated pi to the square root of 10. 

Gaussian elimination. 

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. 
 Through 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 Aryasiddhanta. He
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
7thcentury 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.
 He 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". 
 In the case of dividing a fraction he has found out
the method of multiplying the fraction by the reciprocal
of the divisor. 
 He wrote on the practical applications of
algebra 
 He separated
algebra from
arithmetic 
 He was one of the first to give a formula for
solving
quadratic equations. 
Muḥammad ibn Mūsā alKhwārizmī^{
}( 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]}
AlKhwārizmī'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 textbook of European
universities.^{[11]}
He revised
Ptolemy's
Geography and wrote on
astronomy and astrology.
Some words reflect the importance of alKhwārizmī's
contributions to mathematics. "Algebra" is derived from
aljabr, 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 (HinduArabic numbering system) which
replaced the Roman Numeral system. When he was studying mathematics, he used
the HinduArabic (09) 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 HinduArabic 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
presentvalue 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(n1) + F(n2). 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/featurecolumn/fcarcirrational1
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
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
. 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/themeaningofsacredgeometrypart3thewombofsacredgeometry#
)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 most irrational number
The most irrational number turns out to be a number already well known in
geometry. It is the number
Phi =
= (
+ 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 x^{2}  x  1 = 0, so its
continued fraction expansion is the simplest of all:
A representation in terms of a
nested radical
is
is the "worst" real number for rational approximation because its continued
fraction is:
=
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 Fibonaccilike 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 presentday
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
limitpassage to
infinity".^{[1]}
One of the greatest mathematicianastronomers of the
Middle Ages, Madhava made pioneering
contributions to the study of infinite series,
calculus,
trigonometry,
geometry, and
algebra.
Nicolaus Copernicus (14731543):
(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.
Niccolň Fontana "Tartaglia" (1499/15001557):
(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 Niccolň's jaw
and palate with a saber. This made it impossible for Niccolň 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 16^{th} 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 "CardanoTartaglia Formula".
Gerolamo
Cardano (15011576): (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, x^{3} + 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 decadelong 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 highspeed
printing presses.
Franciscus
Vieta (15401603):
(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(15641642):
(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 (15711630):
(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 Pčre 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
M_{n} = 2^{n}
− 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 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
Flčche. 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
NicolasClaude 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
JeanBaptiste 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
neoplatonists.
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:
 Inversely proportional to the
length of the string (this was actually known to the
ancients, and is usually credited to
Pythagoras
himself).
 Proportional to the square root of the stretching
force, and
 Inversely proportional to the square root of the
mass per unit length.
The formula for the lowest frequency is

f = 1
2 L
F μ
,
{\displaystyle f={\frac {1}{2L}}{\sqrt {\frac {F}{\mu
}}},}
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:
 Much earlier than
Laurent Cassegrain, he
found the fundamental arrangement of the twomirror
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. 
 Mersenne invented the
afocal telescope and
the beam compressor that is useful in many
multiplemirror telescope designs.^{[7]} 
 He 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 17thcentury 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
equallytempered
semitone (2
12
{\displaystyle {\sqrt[{12}]{2}}}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 absolutefrequency 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 sourceairmotionfrequencyequivalence
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 PhysicoMathematica 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
(15961650): (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 (16011665):
(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 sharpedged
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 (16161703): (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 geometry.
Arithmetica 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 semicubical parabola x^{3} = ay^{2},
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 (16231662): (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 BrandenburgPrussia,
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
Leibniz, Leonhard
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 (16431727):
(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 (16461716):
(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 (16541705),
his brother
Johann Bernoulli (16671748) and Johann's son
Daniel Bernoulli (17001787).
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 = x^{x}
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 (17021761):
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/bayestheorem/
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/thetheorythatwouldnotdiebysharonbertschmcgraynebookreview.html?pagewanted=all&_r=0
Also read History:
http://lesswrong.com/lw/774/a_history_of_bayes_theorem/
Leonhard Euler
(17071783): One of his many contributions was called
"Euler's Formula". The formula states that, for any real
number 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 ninepoint 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:
The ninepoint 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
NinePoint 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 NinePoint Circle.
More on NinePoint 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

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.:
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:
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
is the unique
real number such that
the value of the
derivative (slope of
the
tangent line) of the
function f(x) = e^{x}
at the point x = 0 is equal to 1. The
function e^{x} 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 nonzero
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).
It is defined as the
limiting difference
between the
harmonic series and
the
natural logarithm:

Here,
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
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 twentyeight. 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:

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 π^{2}/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
JosephLouis Lagrange (17361813): (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
threebody problem
for the Earth, Sun, and Moon and the movement of Jupiter's
satellites. In 1772 found the specialcase 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 (17451818):
(DanishNorwegian) 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 reissued in French translation
in 1899, and in English in 1999 as "On the analytic representation of
direction".
PierreSimon, marquis de Laplace (17491827):
(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 (17681830):
(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.
MarieSophie Germain
(17761831): (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 (17771855): (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
(17811840): (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
AugustinLouis Cauchy (17891857):
(
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 (17911867) and James Clerk Maxwell (18311879):
They proved that electric and magnetic forces are the same force in
different guises.
Nikolai Ivanovich Lobachevsky (Никола́й
Ива́нович Лобаче́вский) (17921856):
(Russian)
A
mathematician and
geometer, renowned
primarily for his pioneering works on
hyperbolic geometry.
Niels Henrik Abel (18021829):
(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 (18021860):
(Hungarian)
He was known for his work in
nonEuclidean geometry.
Between 1820 and 1823 he prepared a treatise on a complete system
of
nonEuclidean 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 (18041851)
(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 functions.
He 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 reexpressing them in
inverse form. He was also one of the early founders of the theory of
determinants.
Johann Peter Gustav Lejeune Dirichlet
(18051859): (German) He was credited with the modern formal
definition of a
function.
Dirichlet's brain is preserved in the anatomical collection of the
University of Gottingen, along with the brain of Gauss.
Sir William Rowan Hamilton
(18051865): (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
quaternions. A
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 threedimensional 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 fourdimensioal space the tesseract, or
hype, is the
fourdimensional
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 4polytopes.
A generalization of the cube to dimensions greater than three is called a
"hypercube",
"ncube" or "measure
polytope". The
tesseract is the fourdimensional hypercube' See
http://en.wikipedia.org/wiki/Fourth_dimension
Joseph Liouville (18091882):
(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
actionangle variables
as a description of completely
integrable systems.
The modern formulation of this is sometimes called the LiouvilleArnold
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 twodimensional 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
prime, ideal
class group).
His methods were closer, perhaps, to padic 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 2torsion of the class group). As such, it is still foundational for class
field theory. Evariste Galois (18111832):
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 KingNoel, 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 generalpurpose
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
Crosse, Sir
David Brewster, Charles
Wheatstone, Michael
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 engineer Luigi
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 numbercrunching, 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
(18151897): (German) He is often cited as the "father of modern
analysis".
George Boole
(18151864): (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 (18211895):
(British) He helped found the modern British school of
pure mathematics. He
proved the
CayleyHamilton 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
(18221901):
(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 (18261866): 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 (17771855) had an approximation to Pi(N), equal to N/ln(N),
where ln is the natural logarithm. AdrienMarie Legendre
(17521833)
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
(17071783)
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 ζfunction)
The following
infinite series converges for all complex numbers s with real
part greater than 1, and defines ζ(s) in this case:

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
other than
,
,
,
... such that
(where
is the
Riemann zeta function) all lie on the "critical
line"
(where
denotes the real part
of
).
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 halfline are presented in
red. 

Julius Wilhelm Richard Dedekind
(18311916): (
German) He did
important work in
abstract algebra
(particularly
ring theory),
algebraic number theory
and the foundations of the
real numbers.
James Clerk Maxwell (18311879): (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 (18411911): (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 puzzlewriters 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 selfpromotion, 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 fasttalking
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
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
(18451918): (German)
He is best known as the inventor of
set theory, which has
become a
fundamental theory in
mathematics. Cantor established the importance of
onetoone correspondence
between sets, defined
infinite and
wellordered 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 counterintuitive, 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 nonzero
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
x^{2} − 2 = 0.
Seth Carlo Chandler, Jr. (18461913):
(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, quasicircular 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
codiscovered 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/biographicalmemoirs/memoirpdfs/chandlersethc.pdf
See more on the Chandler Wobble:
http://curiosity.discovery.com/question/whatisthechandlerwobble
Thomas Alva Edison (18471931):
(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 longlasting, 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 ninetynine 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 KorvinKrukovskaya (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 JacquesSolomon
Hadamard (18651963) and
Charles de la Vallee Poisson (18661962) by showing that the Riemann
Zeta Function has no zeros of the form (1 + ki).
Carl Louis Ferdinand von Lindemann (18521939):
(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 (18541912):
(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 (18541932):
(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) (18561922): (Russian) He is best
known for his work on theory of
stochastic processes.
His research later became known as
Markov chains.
Henry Ernest Dudeney (18571930):
(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 (18581947):
(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 (18621943): (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 (18721970): "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 (18791955): 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 (18821935): (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 (18821970): (German)
A
physicist and
mathematician who was
instrumental in the development of
quantum mechanics. He
also made contributions to
solidstate 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. (18831948):
(American) Benford's law, also called the
firstdigit law, states that in lists of numbers from many (but not
all) reallife sources of
data, the leading
digit is distributed in a specific, nonuniform 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 counterintuitive 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
(18831946):
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 (18851962):
(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 (18871920):
(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 (19011954):
(ItalianAmerican)
A
physicist
particularly known for his work on the development of the first
nuclear reactor,
Chicago Pile1, 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
(19031957): (Hungarianborn)
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 (19061998): (from
France) "God exists since mathematics is consistent, and the Devil exists
since we cannot prove it."
Julius Robert Oppenheimer (19041967):
(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 muchpublicized 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 (19061978): (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 selfconsistent 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 TausskyTodd
(1906
– 1995):
(Austrian and
later CzechAmerican)
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 GianCarlo
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 (19112007),
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
(19061998): (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 (19071966): (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 (19082003):
(Hungarianborn
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
(19091984): (PolishJewish) 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 19121954): 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
longdistance runner, occasionally ran the 40 miles (64 km) to London when
he was needed for highlevel meetings, and he was capable of worldclass
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/wwiicodebreakeralanturinggetsroyalpardonforgayconviction/
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 reactiondiffusion 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
Turingtype 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 (19131996): (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 (19142010): (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 FringeWatcher
column in Skeptical Inquirer from 1983 to 2002, and published over 70 books.
See:
http://en.wikipedia.org/wiki/Martin_Gardner
Ivan Niven
(19151999): (Canadian) Ivan published over sixty papers, some with wellknown
coauthors 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/frederickmostellerfiftychallengingproblemsibookseeorg
Also see Bio at
http://www.nasonline.org/publications/biographicalmemoirs/memoirpdfs/mostellerfrederick.pdf
Richard Feynman
(19181988):
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 GellMann 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
Polishborn, 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,
neverending 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://wwwhistory.mcs.stand.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.fractalanimation.net/ufvp.html and
http://www.ericbigas.com/fractalanimation/index.html and
http://www.fractalanimation.net/ufvp.html and
http://fractalanimations.com/
and
http://www.google.com/images?hl=&q=fractal+animation&rlz=1B3GGLL_enUS405US405&um=1&ie=UTF8&source=univ&ei=2dFTfmfPIp8AaSw42EAg&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/theframeworkoffractals/
Klaus Friedrich Roth, (born October
29, 1925,
Breslau,
Germany [now Wrocław,
Poland]—died November 10, 2015,
Inverness, Scotland),
Germanborn British mathematician who was awarded the
Fields Medal in 1958 for
his work in
number theory.
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/q −
α  < 1/q^{2} 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/q − α  < 1/q^{μ}.
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 of√n
essentially. In 1947 Freeman J. Dyson improved that to
μ̄ <
Square root of√2n.
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/KlausRothmathematicianobituary.html
Alexander Grothendieck (19282014):
(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 socalled "relative" perspective
led to revolutionary advances in many areas
of pure
mathematics.
Murray GellMann (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 cofounder 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 VA 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 modelindependent
sum rules confirmed
by experiment.
John Hendricks (9.04.19297.07.2007): 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 sixdimensional 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:
 an inlaid magic tesseract 
 the placement of numbers for a perfect magic tesseract of order 16 
 the placement of numbers in a perfect fivedimensional magic
hypercube of order 32 
 a new method of making bimagic squares of order 9 
 the 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 codiscoverer 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
Cray1 and
Cray XMP 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 Eugčne 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
theory. He is an emeritus professor and
occupies Albert
Einstein's office at the Institute
for Advanced Study in
Princeton.
See
projects.thestar.com/maththecanadianwhoreinventedmathematics/ 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 Sduality
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 nonAbelian 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 infinitedimensional 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/19841002/S027309791984152376/S027309791984152376.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
coinventor, with
ShiingShen 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
( {a_{n}}=n^{n} ),
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
Having no formal collegelevel mathematics
education, he sent
brief descriptions of his findings to the host of the National
Public Radio show “Science Friday” and also to
a wellknown 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 selfsimilarity.”
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/triangularnumbers/
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 fractalbased 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/pascalh%C3%A1romsz%C3%B6g/
He found a hidden Pi in Pascal's Triangle.
http://curiosamathematica.tumblr.com/post/126317657094/danielhardiskydiscovered%CF%80inpascalstriangle
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/fractalsgraphicsandmathematicseducationbenoitbmandelbrot/1101362851?ean=9780883851692
And see
http://www.yalescientific.org/2010/04/theframeworkoffractals/
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 longtime 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/michaelframeprofessorofmathematicsatyale
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
Stanisław Marcin Ulam (pronounced
['staɲiswaf 'mart͡ɕin 'ulam];
13 April 1909 – 13 May 1984) was a PolishAmerican 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 17yearold 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
implosiontype 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 nonlinear
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.
ShiingShen Chern (10.26.1911 
12.03.2004) 陳省身
ShiingShen Chern (;
Chinese:
陳省身;
pinyin:
Chén Xǐngshēn, Mandarin: [tʂʰən.ɕiŋ.ʂən];
October 26, 1911 – December 3, 2004) was a
ChineseAmerican
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.
ShiingShen Chern spent nearly a decade
at the
University of Chicago
before moving to
University of
California, Berkeley, where he
cofounded the worldrenowned
Mathematical Sciences
Research Institute in 1982 and
was the institute's founding director.^{
}In memory of ShiingShen
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
LiFu
Chiang (姜立夫),
a
Harvardtrained
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 Chicagotrained
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.
ChenNing Yang's
father—Yang
KoChuen, another Chicagotrained
professor at Tsing Hua, but specialising in
algebra,
also taught Chern. At the same time, Chern
was ChenNing 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 wellknown geometer Wilhelm
Blaschke.^{[10]}
Cofunded 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 CartanKähler
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
ÉlieJoseph
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 SinoJapanese
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 ShihNing 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^{
}(thencapital 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 vicepresident 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.
ShingTung 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 cooperational 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
Allendoerfer–Weil
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:
 1970,
Chauvenet Prize,
of the Mathematical Association of
America; 
 1975,
National Medal of
Science^{[17]} 
 1982,
Humboldt Prize,
Germany; 
 1983,
Leroy P. Steele
Prize, of the American
Mathematical Society; 
 1984,
Wolf Prize in
Mathematics, Israel; 
 2002,
Lobachevsky Medal; 
 2004 May,
Shaw Prize
in mathematical sciences, Hong Kong;^{[18]} 
 1948, Academician,
Academia Sinica; 
 1950, Honorary Member,
Indian
Mathematical Society; 
 1961, Member,
United States
National Academy of Sciences; 
 1963, Fellow,
American Academy
of Arts and Sciences; 
 1971, Corresponding Member,
Brazilian Academy
of Sciences; 
 1983, Associate Founding Fellow,
TWAS; 
 1985, Foreign Fellow,
Royal Society of
London, UK; 
 1986, Honorary Fellow,
London
Mathematical Society, UK; 
 1986, Corresponding Member, Academia
Peloritana, Messina, Sicily; 
 1987, Honorary Life Member,
New York Academy
of Sciences; 
 1989, Foreign Member,
Accademia dei
Lincei, Italy; 
 1989, Foreign Member,
Académie des
sciences, France; 
 1989, Member,
American
Philosophical Society; 
 1994, 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 (;
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 nonexecutive chairman
and adviser.
He is also known for his studies on
pattern recognition.
He developed (with
ShiingShen 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.
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 computerbased models to
predict price changes in financial
instruments. These models are based on
analyzing as much data as can be gathered,
then looking for nonrandom
movements to make predictions.
Renaissance employs specialists with
nonfinancial 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 betterknown 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
ShingShen Chern
on the theory of characteristic classes,
eventually discovering the
Chern–Simons
secondary characteristic classes of
3manifolds, which are deeply related to the
YangMills functional on 4manifolds, 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
coCEO
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 daytoday trading (usually
subject to higher ordinary income tax rates)
as longterm capital gains. “Renaissance
Technologies was able to avoid paying more
than $6 billion in taxes by disguising its
daytoday 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, cofounded 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 130acre
(0.53 km^{2}) nature preserve in
Stony Brook.
In 1996, 34yearold 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
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 manyworlds 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
Alevel examinations without any
preparation along with the uppersixth 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 coauthored 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 followup 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
lowdimensional topology. In 1982, he was
awarded the
Fields Medal for his
contributions to the study of
3manifolds. From 2003 until his death he
was a professor of mathematics and
computer science at
Cornell University.
Thurston's geometrization conjecture states that certain
threedimensional
topological spaces each have a unique
geometric structure that can be associated with them. It is an analogue of
the
uniformization theorem for twodimensional
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
3manifold 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, 197780; full
professor, Princeton University, Department of Physics, 198087; 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, cowinner, 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 superstar, and considered by
many to be the savior of the field. His participation in the discovery of
the super string theory and his Mstring 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 softspoken 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 Mstring 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 Mtheory 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 Mtheory 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/newsmakers2/2006RaZ/WittenEdward.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 tenyearold, 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 mid1986, based on successive progress of the
previous few years of
Gerhard Frey,
JeanPierre 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 exsupervisor
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 fromchildhood 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 neartotal 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 Yětáng)^{
}is a Chineseborn 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.
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 TzuongTsieng 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.^{
}TzuongTsieng 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
jobhunting 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 2tuple, 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 TzuongTsieng 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.^{
}TzuongTsieng 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 jobhunting 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 2tuple,^{
} 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) fourdimensional
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 YangMills equations to discover a fingerprint
which allowed him to distinguish whether two shapes were actually the same.
These fingerprints 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.
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/belfastfestivalmarcusdusautoy
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
theory, integrable
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 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
Lfunctions, 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 fiveyear
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 Lfunctions 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
References
External
links
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".
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,
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 longstanding 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 rulegoverned 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
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/triangularnumbers/
And see
https://arxiv.org/abs/1304.5262 and
http://www.cuttheknot.org/arithmetic/algebra/PiInPascal.shtml
RANKING OF GREAT MATHEMATICIANS See
www.fabpedigree.com/james/mathmen.htm
Also see
http://famousmathematicians.org/
