Jules Henri Poincaré (April 29, 1854 – July 17, 1912) was a French mathematician and theoretical physicist, and a philosopher of science. Poincaré is often described as a polymath, and in mathematics as The Last Universalist, since he excelled in all fields of the discipline as it existed during his lifetime.

As a mathematician and physicist, he made many original fundamental contributions to pure and applied mathematics, mathematical physics, and celestial mechanics. He was responsible for formulating the Poincaré conjecture, one of the most famous problems in mathematics. In his research on the three-body problem, Poincaré became the first person to discover a chaotic deterministic system which laid the foundations of modern chaos theory. He is considered to be one of the founders of the field of topology.

Poincaré introduced the modern principle of relativity and was the first to present the Lorentz transformations in their modern symmetrical form. Poincaré discovered the remaining relativistic velocity transformations and recorded them in a letter to Lorentz in 1905. Thus he obtained perfect invariance of all of Maxwell's equations, the final step in the formulation of the theory of special relativity.

Jacques Salomon Hadamard (December 8, 1865 – October 17, 1963) was a French mathematician best known for his proof of the prime number theorem in 1896.

Hadamard studied at the École Normale Supérieure under the direction of Charles Émile Picard. After the Dreyfus affair, which involved him personally (Dreyfus was his brother-in-law), Hadamard, Jewish himself in his historical identity, became politically active and became a staunch supporter of Jewish causes though he professed to be an atheist in his religion.

He introduced the idea of well-posed problem in the theory of partial differential equations. He also gave his name to the Hadamard inequality on volumes, and the Hadamard matrix, on which the Hadamard transform is based. The Hadamard gate in quantum computing uses this matrix.

His students included Maurice Fréchet, Paul Lévy, Szolem Mandelbrojt and André Weil.

George David Birkhoff (21 March 1884, Overisel, Michigan - 12 November 1944, Cambridge, Massachusetts) was an American mathematician, best known for what is now called the ergodic theorem. Birkhoff was one of the most important leaders in American mathematics in his generation, and during his prime he was considered by many to be the preeminent American mathematician.

Birkhoff obtained his A.B. and A.M. from Harvard. He completed his Ph.D. in 1907, on differential equations, at the University of Chicago. While Eliakim Hastings Moore was his supervisor, he was most influenced by the writings of Henri Poincaré. After teaching at the University of Wisconsin and Princeton University, he taught at Harvard University from 1912 until his death.

Awards and honors

In 1923, he was awarded the inaugural Bôcher Memorial Prize by the American Mathematical Society for his paper Birkhoff (1917) containing, among other things, what is now called the Birkhoff curve shortening flow.

He was elected to the National Academy of Sciences, the American Philosophical Society, the American Academy of Arts and Sciences, the Académie des Sciences in Paris, the Pontifical Academy, and the London and Edinburg Mathematical Societies.

Service
Vice-president of the American Mathematical Society, 1919.
President of the American Mathematical Society, 1925-1926.
Editor of Transactions of the American Mathematical Society, 1920-1924.

Andrey Nikolaevich Kolmogorov (April 25, 1903 - October 20, 1987) was a Soviet mathematician who made major advances in different scientific fields (among them probability theory, topology, intuitionistic logic, turbulence, classical mechanics and computational complexity). Kolmogorov is widely considered to be one of the pre-eminent mathematicians of the 20th century.

Dame Mary Lucy Cartwright DBE (December 17, 1900 – April 3, 1998) was a leading 20th-century British mathematician. She was born in Aynho, Northamptonshire where her father was the vicar and died in Cambridge, England.

In October 1919 she entered St Hugh's College, Oxford to study mathematics, one of only five women at that university studying the subject. She graduated from Oxford in 1923 with a first class degree in Final Honours.

She then taught at Alice Ottley School in Worcester and Wycombe Abbey School in Buckinghamshire before returning to Oxford in 1928 to read for her D.Phil.

She was supervised by G. H. Hardy in her doctoral studies. During the academic year 1928–29 Hardy was at Princeton, so it was E. C. Titchmarsh who took over the duties as a supervisor. Her thesis on zeros of entire functions was examined by J. E. Littlewood whom she met for the first time as an external examiner in her oral examination for the D.Phil. She would later become a major collaborator with Littlewood, over many years.

In 1930 Cartwright was awarded a Yarrow Research Fellowship and she went to Girton College, Cambridge, to continue working on the topic of her doctoral thesis. Attending Littlewood's lectures, she solved one of the open problems which he posed. Her theorem, now known as Cartwright's theorem, gives an estimate for the maximum modulus of an analytic function that takes the same value no more than p times in the unit disc. To prove the theorem she used a new approach, applying a technique introduced by Lars Ahlfors for conformal mappings.

She anticipated Ivan Niven's elementary proof of the irrationality of ?. Her version of the proof was published in an appendix to Sir Harold Jeffreys' book Scientific Inference.

In 1936 she became director of studies in mathematics at Girton College, and in 1938 she began work on a new project which had a major impact on the direction of her research. The Radio Research Board of the Department of Scientific and Industrial Research produced a memorandum regarding certain differential equations which came out of modelling radio and radar work. They asked the London Mathematical Society if they could help find a mathematician who could work on these problems and Cartwright became interested in this memorandum. The dynamics lying behind the problems were unfamiliar to Cartwright so she approached Littlewood for help with this aspect. They began to collaborate studying the equations. Littlewood wrote:
"For something to do we went on and on at the thing with no earthly prospect of "results"; suddenly the whole vista of the dramatic fine structure of solutions stared us in the face."

The fine structure which Littlewood describes here is today seen to be a typical instance of the butterfly effect. The collaboration led to important results, and these have greatly influenced the direction that the modern theory of dynamical systems has taken.

In 1947 she was elected to be a Fellow of the Royal Society and, although she was not the first woman to be elected to that Society, she was the first woman mathematician.

Cartwright was appointed Mistress of Girton in 1948 then, in addition, a Reader in the Theory of Functions in Cambridge in 1959, holding this appointment until 1968.

She was the first woman:
to receive the Sylvester Medal
to serve on the Council of the Royal Society
to be President of the London Mathematical Society (in 1961–62)

She also received the De Morgan Medal of the Society in 1968. In 1969 she received the distinction of being honoured by the Queen, becoming Dame Mary Cartwright, Dame Commander of the Order of the British Empire.

John Edensor Littlewood (9 June 1885 – 6 September 1977) was a British mathematician, best known for his long collaboration with G. H. Hardy.

Littlewood was born in Rochester in Kent. He attended St Paul's School in London, where he was taught by F. S. Macaulay, now known for his contributions to ideal theory. He studied at Trinity College, Cambridge and was the Senior Wrangler in the Mathematical Tripos of 1905. He was elected a Fellow of Trinity College in 1908 and, apart from three years as Richardson Lecturer in the University of Manchester, his entire career was spent in the University of Cambridge. He was appointed Rouse Ball Professor of Mathematics in 1928, retiring in 1950. He was elected a Fellow of the Royal Society in 1916, awarded the Royal Medal in 1929, the Sylvester Medal in 1943 and the Copley Medal in 1958. He was president of the London Mathematical Society from 1941 to 1943, and was awarded the De Morgan Medal in 1938 and the Senior Berwick Prize in 1960.

Stephen Smale (born July 15, 1930) is an American mathematician from Flint, Michigan, and winner of the Fields Medal in 1966. He entered the University of Michigan in 1948. Initially, Smale was a good student, placing into an honors calculus sequence taught by Bob Thrall and earning himself A's. However, his sophomore and junior years were marred with mediocre grades, mostly Bs, Cs and even an F in nuclear physics. However, with some luck, Smale was accepted as a graduate student at the University of Michigan's mathematics department. Yet again, Smale performed poorly his first years, earning a C average as a graduate student. It was only when the department chair, Hildebrant, threatened to kick out Smale, that he began to work hard. Smale finally earned his Ph.D. in 1957, under Raoul Bott.

Smale began his career as an instructor at the college at the University of Chicago. In 1958, he astounded the mathematical world with a proof of a sphere eversion. He then cemented his reputation with a proof of the Poincaré conjecture for all dimensions greater than or equal to 5; he later generalized the ideas in a 107 page paper that established the h-cobordism theorem.

After having made great strides in topology, he then turned to the study of dynamical systems, where he made significant advances as well. His first contribution is the Smale horseshoe that jumpstarted significant research in dynamical systems. He also outlined a research program carried out by many others. Smale is also known for injecting Morse theory into mathematical economics, as well as recent explorations of various theories of computation.

In 1998 he compiled a list of 18 problems in mathematics to be solved in the 21st century, known as Smale's problems. This list was compiled in the spirit of Hilbert's famous list of problems produced in 1900. In fact, Smale's list contains some of the original Hilbert problems, including the Riemann hypothesis and the second half of Hilbert's sixteenth problem, both of which are still unsolved. Other famous problems on his list include the Poincaré conjecture, the P = NP problem, and the Navier-Stokes equations, all of which have been designated Millennium Prize Problems by the Clay Mathematics Institute.

Earlier in his career, Smale was involved in controversy over remarks he made regarding his work habits while proving the higher dimensional Poincaré conjecture. He said that his best work had been done "on the beaches of Rio". This led to the withholding of his grant money from the NSF. He has been politically active in various movements in the past, such as the Free Speech movement. At one time he was subpoenaed by the House Un-American Activities Committee.

In 1960 Smale was appointed an associate professor of mathematics at the University of California, Berkeley, moving to a professorship at Columbia University the following year. In 1964 he returned to a professorship at UC Berkeley where he has spent the main part of his career. He retired from UC Berkeley in 1995 and took up a post as professor at the City University of Hong Kong. He also amassed over the years one of the finest private mineral collections in existence. Many of Smale's mineral specimens can be seen in the book - The Smale Collection: Beauty in Natural Crystals.

Smale is currently a professor at the Toyota Technological Institute at Chicago, a research institute closely affiliated with the University of Chicago.

In 2007, Smale was awarded the Wolf Prize in mathematics. He is the last of only eight Fields Medallists to win both prizes.

Edward Norton Lorenz (May 23, 1917 – April 16, 2008) was an American mathematician and meteorologist, and a pioneer of chaos theory. He discovered the strange attractor notion and coined the term butterfly effect.

Lorenz was born in West Hartford, Connecticut. He studied mathematics at both Dartmouth College in New Hampshire and Harvard University in Cambridge, Massachusetts. During World War II, he served as a weather forecaster for the United States Army Air Corps. After his return from the war, he decided to study meteorology. Lorenz earned two degrees in the area from the Massachusetts Institute of Technology where he later was a professor for many years.

Lorenz built a mathematical model of the way air moves around in the atmosphere. As Lorenz studied weather patterns he began to realize that they did not always change as predicted. Minute variations in the initial values of variables in his twelve variable computer weather model (c. 1960) would result in grossly divergent weather patterns. This sensitive dependence on initial conditions came to be known as the butterfly effect.

Lorenz went on to explore the underlying mathematics and published his conclusions in a seminal work titled Deterministic Nonperiodic Flow, in which he described a relatively simple system of equations that resulted in a very complicated dynamical object now known as the Lorenz attractor.

He was awarded the Kyoto Prize in 1991 and cited for "profoundly [influencing] a wide range of basic sciences and brought about one of the most dramatic changes in mankind’s view of nature since Sir Isaac Newton."

Lorenz continued to be active in his work well into his seventies, winning the Kyoto Prize for basic sciences, in the field of earth and planetary sciences, in 1991. In his later years, he lived in Cambridge, Massachusetts. He was an avid outdoorsman, who enjoyed hiking, climbing, and cross-country skiing. He kept up with these pursuits until very late in his life, and managed to continue most of his regular activities until only a few weeks before his death. According to his daughter, Cheryl Lorenz, Lorenz had "finished a paper a week ago with a colleague." On April 16th, 2008, Lorenz died at his home in Cambridge at the age of 90, having suffered from cancer.

Benoît B. Mandelbrot (born November 20, 1924) is a Polish-French-Jewish-American mathematician, best known as the "father of fractal geometry". He was born in Poland, but his family moved to France when he was a child; he is a dual French and American citizen and was educated in France. Mandelbrot now lives and works in the United States. He is Sterling Professor of Mathematical Sciences, Emeritus at Yale University; IBM Fellow Emeritus at the Thomas J. Watson Research Center; and Battelle Fellow at the Pacific Northwest National Laboratory.

The one who discovered chaos!

David Pierre Ruelle (b. August 20, 1935 Ghent, Belgium) is a Belgian-French mathematical physicist. He has worked on statistical physics and dynamical systems. With Floris Takens he coined the term strange attractor, and founded a new theory of turbulence. In 1986, he received the Boltzmann Medal for his outstanding contributions to statistical mechanics. In 2004, he received the Matteucci Medal.

He studied physics at the Université Libre de Bruxelles, obtaining a Ph.D. degree in 1959. He spent two years (1960-1962) at the ETH Zurich, and another two years (1962-1964) at the Institute for Advanced Study in Princeton, New Jersey. In 1964, he became Professor at the Institut des Hautes Études Scientifiques (IHES), in Bures-sur-Yvette, France. Since 2000, he is an Emeritus Professor at IHES and distinguished visiting professor at Rutgers University.

Little is known, only that he was a remarkable man who contributed with vital parts to the chaos theory.

James A. Yorke (born August 3, 1941) is a Distinguished University Professor of Mathematics and Physics and chair of the Mathematics Department at the University of Maryland, College Park. He is also a recipient of the 2003 Japan Prize for his work in chaotic systems.

Born in Plainfield, New Jersey, United States, Yorke attended The Pingry School, then located in Hillside, New Jersey. He and his co-author T.Y. Li coined the mathematical term chaos in a paper they published in 1975 entitled "Period Three Implies Chaos", in which it was proved that any continuous 1-dimensional map
R ? R

that contains a period-3 orbit must have two properties:

(1) For each positive integer P, there is a point in R that returns to where it started after P applications of the map and not before. (Of course this means there are infinitely many periodic points, different points for each period P.) This turned out to be a special case of Sharkovsky's theorem.

The second property requires some definitions. A pair of points x and y is called “scrambled” if as the map is applied repeated to the pair, they get closer together and later move apart and then get closer together and move apart, etc, so that they get arbitrarily close together without staying close together. Picture an egg being scrambled forever. You would expect typical atoms x and y to behave in this way. A set S is called "scrambled" if every pair of distinct points in S is scrambled. Scrambling is a kind of mixing.

(2) There is an uncountably infinite set S that is scrambled.

A map satisfying property 2 is sometimes called "chaotic in the sense of Li and Yorke".