1.
Israel Gelfand
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Israel Moiseevich Gelfand, also written Israïl Moyseyovich Gelfand, or Izrail M. Gelfand was a prominent Soviet mathematician. He made significant contributions to many branches of mathematics, including group theory, representation theory and his legacy continues through his students, who include Endre Szemerédi, Alexandre Kirillov, Edward Frenkel, Joseph Bernstein, as well as his own son, Sergei Gelfand. A native of Kherson Governorate of the Russian Empire, Gelfand was born into a Jewish family in the small southern Ukrainian town of Okny, according to his own account, Gelfand was expelled from high school because his father had been a mill owner. Bypassing both high school and college, he proceeded to study at Moscow State University, where his advisor was the preeminent mathematician Andrei Kolmogorov. He nevertheless managed to attend lectures at the University and began study at the age of 19. The Gelfand–Tsetlin basis is a widely used tool in theoretical physics, Gelfand also published works on biology and medicine. For a long time he took an interest in cell biology and he worked extensively in mathematics education, particularly with correspondence education. In 1994, he was awarded a MacArthur Fellowship for this work, Gelfand was married to Zorya Shapiro, and their two sons, Sergei and Vladimir both live in the United States. A third son, Aleksandr, died of leukemia, following the divorce from his first wife, Gelfand married his second wife, Tatiana, together they had a daughter, Tatiana. The family also includes four grandchildren and three great-grandchildren, the memories about I. Gelfand are collected at the special site handled by his family. Gelfand held several degrees and was awarded the Order of Lenin three times for his research. In 1977 he was elected a Foreign Member of the Royal Society and he won the Wolf Prize in 1978, Kyoto Prize in 1989 and MacArthur Foundation Fellowship in 1994. Israel Gelfand died at the Robert Wood Johnson University Hospital near his home in Highland Park and he was less than five weeks past his 96th birthday. His death was first reported on the blog of his former collaborator Andrei Zelevinsky and confirmed a few hours later by an obituary in the Russian online newspaper Polit. ru. Gelfand, I. M. Lectures on linear algebra, Courier Dover Publications, ISBN 978-0-486-66082-0 Gelfand, I. M. Fomin, Sergei V. Silverman, Richard A. ed. Calculus of variations, Englewood Cliffs, ISBN 978-0-486-41448-5, MR0160139 Gelfand, I. Raikov, D. Shilov, G. Commutative normed rings, Translated from the Russian, with a chapter, New York. ISBN 978-0-8218-2022-3, MR0205105 Gelfand, I. M. Shilov, G. E. Generalized functions. Vol. I, Properties and operations, Translated by Eugene Saletan, Boston, MA, Academic Press, ISBN 978-0-12-279501-5, MR0166596 Gelfand, I. M. Shilov, G. E. Generalized functions

2.
Carl Ludwig Siegel
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Carl Ludwig Siegel was a German mathematician specialising in number theory and celestial mechanics. He is known for, amongst other things, his contributions to the Thue–Siegel–Roth theorem in Diophantine approximation and he was named as one of the most important mathematicians of the 20th century. André Weil, without hesitation, named Siegel as the greatest mathematician of the first half of the 20th century. Atle Selberg said of Siegel and his work, Siegel was born in Berlin, where he enrolled at the Humboldt University in Berlin in 1915 as a student in mathematics, astronomy, and physics. Amongst his teachers were Max Planck and Ferdinand Georg Frobenius, whose influence made the young Siegel abandon astronomy and his best student was Jürgen Moser, one of the founders of KAM theory, which lies at the foundations of chaos theory. Another notable student was Kurt Mahler, the number theorist, Siegel was an antimilitarist, and in 1917, during World War I he was committed to a psychiatric institute as a conscientious objector. According to his own words, he withstood the experience only because of his support from Edmund Landau, after the end of World War I, he enrolled at the Georg-August University of Göttingen, studying under Landau, who was his doctoral thesis supervisor. He stayed in Göttingen as a teaching and research assistant, many of his results were published during this period. In 1922, he was appointed professor at the Johann Wolfgang Goethe-Universität of Frankfurt am Main as the successor of Arthur Moritz Schönflies. Siegel, who was opposed to Nazism, was a close friend of the docents Ernst Hellinger and Max Dehn. This attitude prevented Siegels appointment as a successor to the chair of Constantin Carathéodory in Munich, in Frankfurt he took part with Dehn, Hellinger, Paul Epstein, and others in a seminar on the history of mathematics, which was conducted at the highest level. In the seminar they read only original sources, Siegels reminiscences about the time before World War II are in an essay in his collected works. In 1936 he was a Plenary Speaker at the ICM in Oslo and he returned to Göttingen only after World War II, when he accepted a post as professor in 1951, which he kept until his retirement in 1959. Siegels work on theory, diophantine equations, and celestial mechanics in particular won him numerous honours. In 1978, he was awarded the first Wolf Prize in Mathematics, when the prize committee decided to select the greatest living mathematician, the discussion centered around Siegel and Israel Gelfand as the leading candidates. The prize was split between them. He worked on L-functions, discovering the Siegel zero phenomenon and his work, derived from the Hardy–Littlewood circle method on quadratic forms, appeared in the later, adele group theories encompassing the use of theta-functions. The Siegel modular forms are recognised as part of the theory of abelian varieties

3.
Jean Leray
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Jean Leray was a French mathematician, who worked on both partial differential equations and algebraic topology. He studied at École Normale Supérieure from 1926 to 1929 and he received his Ph. D. in 1933. Leray wrote an important paper that founded the study of solutions of the Navier–Stokes equations. From 1938 to 1939 he was professor at the University of Nancy and he did not join the Bourbaki group, although he was close with its founders. His main work in topology was carried out while he was in a prisoner of war camp in Edelbach and he concealed his expertise on differential equations, fearing that its connections with applied mathematics could lead him to be asked to do war work. Lerays work of this period proved seminal to the development of spectral sequences and sheaves and these were subsequently developed by many others, each separately becoming an important tool in homological algebra. He returned to work on differential equations from about 1950. He was professor at the University of Paris from 1945 to 1947 and he was awarded the Malaxa Prize, the Grand Prix in mathematical sciences, the Feltrinelli Prize, the Wolf Prize in Mathematics, and the Lomonosov Gold Medal. Leray spectral sequence Leray cover Lerays theorem Leray–Hirsch theorem OConnor, John J. Robertson, Jean Leray, MacTutor History of Mathematics archive, University of St Andrews. Jean Leray at the Mathematics Genealogy Project Jean Leray, by Armand Borel, Gennadi M. Henkin, and Peter D. Lax, Notices of the American Mathematical Society, vol

4.
Henri Cartan
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Henri Paul Cartan was a French mathematician with substantial contributions in algebraic topology. He was the son of the French mathematician Élie Cartan and the brother of composer Jean Cartan, Cartan studied at the Lycée Hoche in Versailles, then at the École Normale Supérieure in Paris, receiving his doctorate in mathematics. Cartan is known for work in topology, in particular on cohomology operations, the method of killing homotopy groups. The number of his students was small, but includes Adrien Douady, Roger Godement, Max Karoubi, Jean-Louis Koszul, Jean-Pierre Serre. Cartan also was a member of the Bourbaki group and one of its most active participants. His book with Samuel Eilenberg Homological Algebra was an important text, Cartan used his influence to help obtain the release of some dissident mathematicians, including Leonid Plyushch and Jose Luis Massera. For his humanitarian efforts, he received the Pagels Award from the New York Academy of Sciences, the Cartan model in algebra is named after Cartan. Cartan died on 13 August 2008 at the age of 104 and his funeral took place the following Wednesday on 20 August in Die, Drome. Cartan received numerous honours and awards including the Wolf Prize in 1980 and he was an Invited Speaker at the ICM in 1932 in Zurich and a Plenary Speaker at the ICM in 1950 in Cambridge, Massachusetts and in 1958 in Edinburgh. From 1974 until his death he had been a member of the French Academy of Sciences, Sur les systèmes de fonctions holomorphes à variétés linéaires lacunaires et leurs applications, thèse,1928 Sur les groupes de transformations analytiques,1935. Sur les classes de fonctions définies par des inégalités portant sur leurs dérivées successives,1940, cohomologie des groupes, suite spectrale, faisceaux, 1950-1951. Algèbres dEilenberg - Mac Lane et homotopie, 1954-1955, Homological Algebra, Princeton Univ Press,1956 ISBN 978-0-69104991-5 Séminaires de lÉcole normale supérieure, Secr. IHP, 1948-1964, New York, W. A. Benjamin ed.1967, théorie élémentaire des fonctions analytiques, Paris, Hermann,1961. Differential Forms, Dover 2006 Œuvres — Collected Works,3 vols, ed. Reinhold Remmert & Jean-Pierre Serre, Springer Verlag, Heidelberg,1967. Relations dordre en théorie des permutations des ensembles finis, Neuchâtel,1973, théorie élémentaire des fonctions analytiques dune ou plusieurs variables complexes, Paris, Hermann,1975. Elementary theory of functions of one or several complex variables, Dover 1995 Cours de calcul différentiel, Paris. Correspondance entre Henri Cartan et André Weil, Paris, SMF,2011, oConnor, John J. Robertson, Edmund F. Henri Cartan, MacTutor History of Mathematics archive, University of St Andrews. A17, retrieved 2008-08-25 Cartan, Henri, Eilenberg, Samuel, notices of the American Mathematical Society, Sept.2010, vol

5.
Andrey Kolmogorov
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Andrey Kolmogorov was born in Tambov, about 500 kilometers south-southeast of Moscow, in 1903. Kolmogorova, died giving birth to him, Andrey was raised by two of his aunts in Tunoshna at the estate of his grandfather, a well-to-do nobleman. Little is known about Andreys father and he was supposedly named Nikolai Matveevich Kataev and had been an agronomist. Nikolai had been exiled from St. Petersburg to the Yaroslavl province after his participation in the movement against the czars. He disappeared in 1919 and he was presumed to have killed in the Russian Civil War. Andrey Kolmogorov was educated in his aunt Veras village school, and his earliest literary efforts, Andrey was the editor of the mathematical section of this journal. In 1910, his aunt adopted him, and they moved to Moscow, later that same year, Kolmogorov began to study at the Moscow State University and at the same time Mendeleev Moscow Institute of Chemistry and Technology. Kolmogorov writes about this time, I arrived at Moscow University with a knowledge of mathematics. I knew in particular the beginning of set theory, I studied many questions in articles in the Encyclopedia of Brockhaus and Efron, filling out for myself what was presented too concisely in these articles. Kolmogorov gained a reputation for his wide-ranging erudition, during the same period, Kolmogorov worked out and proved several results in set theory and in the theory of Fourier series. In 1922, Kolmogorov gained international recognition for constructing a Fourier series that diverges almost everywhere, around this time, he decided to devote his life to mathematics. In 1925, Kolmogorov graduated from the Moscow State University and began to study under the supervision of Nikolai Luzin, Kolmogorov became interested in probability theory. In 1929, Kolmogorov earned his Doctor of Philosophy degree, from Moscow State University, in 1930, Kolmogorov went on his first long trip abroad, traveling to Göttingen and Munich, and then to Paris. He had various contacts in Göttingen. His pioneering work, About the Analytical Methods of Probability Theory, was published in 1931, also in 1931, he became a professor at the Moscow State University. In 1935, Kolmogorov became the first chairman of the department of probability theory at the Moscow State University, around the same years Kolmogorov contributed to the field of ecology and generalized the Lotka–Volterra model of predator-prey systems. In 1936, Kolmogorov and Alexandrov were involved in the persecution of their common teacher Nikolai Luzin, in the so-called Luzin affair. In a 1938 paper, Kolmogorov established the basic theorems for smoothing and predicting stationary stochastic processes—a paper that had military applications during the Cold War

6.
Lars Ahlfors
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Lars Valerian Ahlfors was a Finnish mathematician, remembered for his work in the field of Riemann surfaces and his text on complex analysis. Ahlfors was born in Helsinki, Finland and his mother, Sievä Helander, died at his birth. His father, Axel Ahlfors, was a professor of engineering at the Helsinki University of Technology, the Ahlfors family was Swedish-speaking, so he first attended a private school where all classes were taught in Swedish. Ahlfors studied at University of Helsinki from 1924, graduating in 1928 having studied under Ernst Lindelöf and he assisted Nevanlinna in 1929 with his work on Denjoys conjecture on the number of asymptotic values of an entire function. In 1929 Ahlfors published the first proof of this conjecture, now known as the Denjoy–Carleman–Ahlfors theorem and it states that the number of asymptotic values approached by an entire function of order ρ along curves in the complex plane going toward infinity is less than or equal to 2ρ. He completed his doctorate from the University of Helsinki in 1930, Ahlfors worked as an associate professor at the University of Helsinki from 1933 to 1936. In 1936 he was one of the first two people to be awarded the Fields Medal, in 1935 Ahlfors visited Harvard University. He returned to Finland in 1938 to take up a professorship at the University of Helsinki, the outbreak of war led to problems although Ahlfors was unfit for military service. He was offered a post at the Swiss Federal Institute of Technology at Zurich in 1944, Ahlfors was a visiting scholar at the Institute for Advanced Study in 1962 and again in 1966. He was awarded the Wihuri Prize in 1968 and the Wolf Prize in Mathematics in 1981 and his book Complex Analysis is the classic text on the subject and is almost certainly referenced in any more recent text which makes heavy use of complex analysis. Ahlfors wrote several significant books, including Riemann surfaces and Conformal invariants. He made decisive contributions to meromorphic curves, value distribution theory, Riemann surfaces, conformal geometry, quasiconformal mappings, in 1933, he married Erna Lehnert, an Austrian who with her parents had first settled in Sweden and then in Finland. FUNDAMENTAL POLYHEDRONS AND LIMIT POINT SETS OF KLEINIAN GROUPS, proceedings of the National Academy of Sciences. Lars Ahlfors at the Mathematics Genealogy Project Ahlfors entry on Harvard University Mathematics department web site, lars Valerian Ahlfors, Notices of the American Mathematical Society, vol. Lars Valerian Ahlfors, a biographical memoir, National Academy of Sciences Biographical Memoir Author profile in the database zbMATH

7.
Oscar Zariski
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Oscar Zariski (born Oscher Zaritsky was a Russian-born American mathematician and one of the most influential algebraic geometers of the 20th century. Zariski was born Oscher Zaritsky to a Jewish family and in 1918 studied at the University of Kiev, Zariski wrote a doctoral dissertation in 1924 on a topic in Galois theory, which was proposed to him by Castelnuovo. At the time of his publication, he changed his name to Oscar Zariski. Zariski emigrated to the United States in 1927 supported by Solomon Lefschetz and he had a position at Johns Hopkins University where he became professor in 1937. During this period, he wrote Algebraic Surfaces as a summation of the work of the Italian school, the book was published in 1935 and reissued 36 years later, with detailed notes by Zariskis students that illustrated how the field of algebraic geometry had changed. It is still an important reference and it seems to have been this work that set the seal of Zariskis discontent with the approach of the Italians to birational geometry. He addressed the question of rigour by recourse to commutative algebra, the Zariski topology, as it was later known, is adequate for biregular geometry, where varieties are mapped by polynomial functions. That theory is too limited for algebraic surfaces, and even for curves with singular points, a rational map is to a regular map as a rational function is to a polynomial, it may be indeterminate at some points. In geometric terms, one has to work with functions defined on some open, the description of the behaviour on the complement may require infinitely near points to be introduced to account for limiting behaviour along different directions. This introduces a need, in the case, to use also valuation theory to describe the phenomena such as blowing up. After spending a year 1946–1947 at the University of Illinois, Zariski became professor at Harvard University in 1947 where he remained until his retirement in 1969, in 1945, he fruitfully discussed foundational matters for algebraic geometry with André Weil. The two sets of foundations werent reconciled at that point, Zariski himself worked on equisingularity theory. Zariski proposed the first example of a Zariski surface in 1958, Zariski was awarded the Steele Prize in 1981, and in the same year the Wolf Prize in Mathematics with Lars Ahlfors. He wrote also Commutative Algebra in two volumes, with Pierre Samuel and his papers have been published by MIT Press, in four volumes. Zariski, Oscar, Abhyankar, Shreeram S. Lipman, Joseph, Mumford, David, an introduction to the theory of algebraic surfaces, Lecture notes in mathematics,83, Berlin, New York, Springer-Verlag, doi,10. Vol. II, Berlin, New York, Springer-Verlag, ISBN 978-0-387-90171-8, MR0389876 Zariski, Oscar, Kmety, François, Merle, Michel, Lichtin, Ben, the moduli problem for plane branches, University Lecture Series,39, Providence, R. I. American Mathematical Society, ISBN 978-0-8218-2983-7, MR0414561, Le problème des modules pour les branches planes Zariski, Oscar, Collected papers. Vol. I, Foundations of algebraic geometry and resolution of singularities, The MIT Press, Cambridge, Mass. -London, ISBN 978-0-262-08049-1, MR0505100 Zariski, Oscar, robertson, Edmund F. Oscar Zariski, MacTutor History of Mathematics archive, University of St Andrews

8.
Mark Krein
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Mark Grigorievich Krein was a Soviet Jewish mathematician, one of the major figures of the Soviet school of functional analysis. He is known for works in theory, the problem of moments, classical analysis. He was born in Kiev, leaving home at age 17 to go to Odessa and he had a difficult academic career, not completing his first degree and constantly being troubled by anti-Semitic discrimination. He was awarded the Wolf Prize in Mathematics in 1982, but was not allowed to attend the ceremony, david Milman, Mark Naimark, Izrail Glazman, Moshe Livshits and other known mathematicians were his students. On 14 January 2008, the plaque of Mark Krein was unveiled on the main administration building of I. I. Mark Krein at the Mathematics Genealogy Project INTERNATIONAL CONFERENCE Modern Analysis, dedicated to the centenary of Mark Krein

9.
Shiing-Shen Chern
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Shiing-Shen Chern was a Chinese-American mathematician. Shiing-Shen Chern co-founded the world-renowned Mathematical Sciences Research Institute at Berkeley in 1982, Chern was born in Xiushui County, Jiaxing, in Zhejiang province. The year after his birth, China changed its regime from the Qing Dynasty to the Republic of China and he graduated from Xiushui Middle School 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, at age 15, Chern entered the Faculty of Sciences of the Nankai University in Tianjin, studied mathematics there, and graduated with a Bachelor of Science degree in 1930. At Nankai, Cherns mentor was Li-Fu Chiang, a Harvard-trained geometer, also at Nankai, he was heavily influenced by the physicist Rao Yutai. Rao is today considered to be one of the fathers of modern Chinese informatics. Chern went to Beiping to work at the Tsinghua 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 geometry under Prof. Sun Guangyuan, a University of Chicago-trained geometer and logician who was also from Zhejiang, Sun is another mentor of Chern who is considered a founder of modern Chinese mathematics. In 1932, Chern published his first research article in the Tsinghua University Journal, in the summer of 1934, Chern graduated from Tsinghua with a masters degree, the first ever masters degree in mathematics issued in China. Chen-Ning Yangs father — Yang Ko-Chuen, another Chicago-trained professor at Tsinghua, at the same time, Chern was Chen-Ning Yangs teacher of undergraduate maths at Tsinghua. At Tsinghua, Hua Luogeng, also a mathematician, was Cherns colleague, in 1932, Wilhelm Blaschke from the University of Hamburg visited Tsinghua and was impressed by Chern and his research. In 1934, co-funded by Tsinghua and the Chinese Foundation of Culture and Education, Chern studied at the University of Hamburg and worked under Blaschkes guidance first on the geometry of webs then on the Cartan-Kähler theory. Blaschke recommended Chern to study in Paris, in August 1936, Chern watched 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 September 1936, Chern went to Paris and worked with Élie Cartan. Chern spent one year at the Sorbonne in Paris, in 1937, Chern accepted Tsinghuas invitation and was promoted to professor of mathematics at Tsinghua. However, at the time the Marco Polo Bridge Incident happened. Three universities including Peking University, Tsinghua, and Nankai formed the National Southwestern Associated University, in the same year, Hua Luogeng was promoted to professor of mathematics at Tsinghua

10.
Kunihiko Kodaira
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Kunihiko Kodaira was a Japanese mathematician known for distinguished work in algebraic geometry and the theory of complex manifolds, and as the founder of the Japanese school of algebraic geometers. He was awarded a Fields Medal in 1954, being the first Japanese national to receive this honour and he graduated from the University of Tokyo in 1938 with a degree in mathematics and also graduated from the physics department at the University of Tokyo in 1941. During the war years he worked in isolation, but was able to master Hodge theory as it then stood and he obtained his Ph. D. from the University of Tokyo in 1949, with a thesis entitled Harmonic fields in Riemannian manifolds. He was involved in work from about 1944, while holding an academic post in Tokyo. In 1949 he travelled to the Institute for Advanced Study in Princeton, at this time the foundations of Hodge theory were being brought in line with contemporary technique in operator theory. Kodaira rapidly became involved in exploiting the tools it opened up in algebraic geometry and this work was particularly influential, for example on Hirzebruch. In a second phase, Kodaira wrote a long series of papers in collaboration with D. C. Spencer, founding the theory of complex structures on manifolds. This gave the possibility of constructions of moduli spaces, since in such structures depend continuously on parameters. This theory is still foundational, and also had an influence on the theory of Grothendieck. Spencer then continued work, applying the techniques to structures other than complex ones. In a third part of his work, Kodaira worked again from around 1960 through the classification of algebraic surfaces from the point of view of birational geometry of complex manifolds. This resulted in a typology of seven kinds of two-dimensional compact complex manifolds, recovering the five algebraic types known classically and this work also included a characterisation of K3 surfaces as deformations of quartic surfaces in P4, and the theorem that they form a single diffeomorphism class. Again, this work has proved foundational, Kodaira left the Institute for Advanced Study in 1961, and briefly served as chair at the Johns Hopkins University and Stanford University. In 1967, returned to the University of Tokyo and he was awarded a Wolf Prize in 1984/5. He died in Kofu on 26 July 1997, ISBN 978-0-691-08158-8, MR0366598 Kodaira, Kunihiko, Baily, Walter L. ed. Kunihiko Kodaira, collected works, II, Iwanami Shoten, Publishers, Tokyo, Princeton University Press, Princeton, N. J. ISBN 978-0-691-08163-2, MR0366599 Kodaira, Kunihiko, Baily, Walter L. ed. Kunihiko Kodaira, collected works, III, Iwanami Shoten, Publishers, Tokyo, Princeton University Press, Princeton, N. J. Robertson, Edmund F. Kunihiko Kodaira, MacTutor History of Mathematics archive, spencer, Kunihiko Kodaira, Notices of the AMS,45, 388–389

11.
Hans Lewy
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Hans Lewy was a German born American mathematician, known for his work on partial differential equations and on the theory of functions of several complex variables. Lewy was born in Breslau, Germany, on October 20,1904, at Göttingen, he studied both mathematics and physics, his teachers there included Max Born, Richard Courant, James Franck, David Hilbert, Edmund Landau, Emmy Noether, and Alexander Ostrowski. He earned his doctorate in 1926, at time he. After Hitlers election as chancellor in 1933, Lewy was advised by Herbert Busemann to leave Germany again and he was offered a position in Madrid, but declined it, fearing for the future there under Francisco Franco. At the end of term, in 1935, he moved to the University of California. During World War II, Lewy obtained a license. He married Helen Crosby in 1947, in 1950, Lewy was fired from Berkeley for refusing to sign a loyalty oath. He taught at Harvard University and Stanford University in 1952 and 1953 before being reinstated by the California Supreme Court case Tolman v. Underhill and he retired from Berkeley in 1972, and in 1973 became one of two Ordway Professors of Mathematics at the University of Minnesota. He died on August 23,1988, in Berkeley, Lewy was elected to the National Academy of Sciences in 1964, and was also a member of the American Academy of Arts and Sciences. He became a member of the Accademia dei Lincei in 1972. He was awarded a Leroy P. Steele Prize in 1979, in 1986, the University of Bonn gave him an honorary doctorate. A priori limitations for Monge-Ampère equations, on the non-vanishing of the Jacobian in certain one-to-one mappings. Proc Natl Acad Sci U S A.22, 377–381, a priori limitations for Monge-Ampère equations. On the existence of a convex surface realizing a given Riemannian metric. Proc Natl Acad Sci U S A.24, 104–106, on differential geometry in the large. Aspects of the Calculus of Variations, Berkeley, U. of California Press, notes by J. W. Green from lectures by Hans Lewy, vi+96 pp. Lewy, Hans. On the boundary behavior of minimal surfaces, proc Natl Acad Sci U S A.37, 103–110. A note on harmonic functions and a hydrodynamical application, on the reflection laws of second order differential equations in two independent variables

12.
Samuel Eilenberg
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Samuel Eilenberg was a Polish-born American mathematician. He was born in Warsaw, Kingdom of Poland to a Jewish family and died in New York City, United States and he earned his Ph. D. from University of Warsaw in 1936. His thesis advisor was Karol Borsuk and his main interest was algebraic topology. He worked on the treatment of homology theory with Norman Steenrod. In the process, Eilenberg and Mac Lane created category theory, Eilenberg was a member of Bourbaki and with Henri Cartan, wrote the 1956 book Homological Algebra, which became a classic. Later in life he worked mainly in category theory, being one of the founders of the field. The Eilenberg swindle is a construction applying the telescoping cancellation idea to projective modules, Eilenberg also wrote an important book on automata theory. The X-machine, a form of automaton, was introduced by Eilenberg in 1974, Eilenberg was also a prominent collector of Asian art. His collection mainly consisted of sculptures and other artifacts from India, Indonesia, Nepal, Thailand, Cambodia, Sri Lanka. Samuel Eilenberg, Automata, Languages and Machines, Samuel Eilenberg & Tudor Ganea, On the Lusternik-Schnirelmann category of abstract groups, Annals of Mathematics, 2nd Ser. MR0085510 Eilenberg, Samuel, Mac Lane, Saunders, relations between homology and homotopy groups of spaces. Relations between homology and homotopy groups of spaces, limits and spectral sequences, Topology,1, 1–23, doi,10. 1016/0040-938390093-9, ISSN 0040-9383 Eilenberg, Samuel, Niven, Ivan. The fundamental theorem of algebra for quaternions, Eilenberg, Samuel, Steenrod, Norman E. Axiomatic approach to homology theory. Samuel Eilenberg & Norman E. Steenrod, Foundations of algebraic topology, Princeton University Press, Princeton, xv+328 pp. Robertson, Edmund F. Samuel Eilenberg, MacTutor History of Mathematics archive, University of St Andrews. Eilenbergs biography − from the National Academies Press, by Hyman Bass, Henri Cartan, Peter Freyd, Alex Heller and Saunders Mac Lane

13.
Atle Selberg
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Atle Selberg was a Norwegian mathematician known for his work in analytic number theory, and in the theory of automorphic forms, in particular bringing them into relation with spectral theory. He was awarded the Fields Medal in 1950, Selberg was born in Langesund, Norway, the son of teacher Anna Kristina Selberg and mathematician Ole Michael Ludvigsen Selberg. Two of his brothers went on to become mathematicians as well. During the war he fought against the German invasion of Norway and he studied at the University of Oslo and completed his Ph. D. in 1943. During World War II, Selberg worked in isolation due to the German occupation of Norway, after the war his accomplishments became known, including a proof that a positive proportion of the zeros of the Riemann zeta function lie on the line ℜ =12. After the war, he turned to sieve theory, a previously neglected topic which Selbergs work brought into prominence. In a 1947 paper he introduced the Selberg sieve, a well adapted in particular to providing auxiliary upper bounds. In 1948 Selberg submitted two papers in Annals of Mathematics in which he proved by means the theorems for primes in arithmetic progression. This challenged the widely held view of his time that certain theorems are only obtainable with the methods of complex analysis. For his fundamental accomplishments during the 1940s, Selberg received the 1950 Fields Medal, Selberg moved to the United States and settled at the Institute for Advanced Study in Princeton, New Jersey in the 1950s where he remained until his death. During the 1950s he worked on introducing spectral theory into number theory, culminating in his development of the Selberg trace formula and he was awarded the 1986 Wolf Prize in Mathematics. He was also awarded an honorary Abel Prize in 2002, its founding year, Selberg received many distinctions for his work in addition to the Fields Medal, the Wolf Prize and the Gunnerus Medal. He was elected to the Norwegian Academy of Science and Letters, the Royal Danish Academy of Sciences and Letters, in 1972 he was awarded an honorary degree, doctor philos. Honoris causa, at the Norwegian Institute of Technology, later part of Norwegian University of Science, Selberg had two children, Ingrid Selberg and Lars Selberg. Ingrid Selberg is married to playwright Mustapha Matura and he died at home in Princeton on 6 August 2007 of heart failure. Baas, Nils A. Skau, Christian F, the lord of the numbers, Atle Selberg. Doi,10. 1090/S0273-0979-08-01223-8 Interview with Selberg Hejhal, Dennis, notices of the American Mathematical Society. Atle Selberg Archive webpage Obituary at IAS Obituary in The Times Atle Selbergs private archive exists at NTNU University Library Dorabiblioteket

14.
Peter Lax
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Peter David Lax is a Hungarian-born American mathematician working in the areas of pure and applied mathematics. Lax is listed as an ISI highly cited researcher, Lax was born in Budapest, Hungary to a Jewish family. His parents Klara Kornfield and Henry Lax were both physicians, and his uncle, Albert Kornfeld, was a mathematician and a friend of Leó Szilárd, Lax began displaying an interest in mathematics at age twelve, and soon his parents hired Rózsa Péter as a tutor for him. The family left Hungary on November 15,1941, and traveled via Lisbon to the United States, as he was still 17 when he finished high school, he could avoid military service, and was able to study for three semesters at New York University. In a complex analysis class that he had begun in the role of a student, before being able to complete his studies, Lax was drafted into the U. S. Army. After basic training, the Army sent him to Texas A&M University for more studies, then Oak Ridge National Laboratory, at Los Alamos, he began working as a calculator operator, but eventually moved on to higher-level mathematics. Lax returned to NYU for the 1946-1947 academic year, and by pooling credits from the four universities at which he had studied, he graduated that year. He stayed at NYU for his studies, marrying Anneli in 1948. In a 1958 paper Lax stated a conjecture about matrix representations for third order hyperbolic polynomials which remained unproven for over four decades. Interest in the Lax conjecture grew as mathematicians working in different areas recognized the importance of its implications in their field. Lax holds a faculty position in the Department of Mathematics, Courant Institute of Mathematical Sciences and he is a member of the Norwegian Academy of Science and Letters and the National Academy of Sciences, USA. He won a Lester R. Ford Award in 1966 and again in 1973 and he was awarded the National Medal of Science in 1986, the Wolf Prize in 1987, the Abel Prize in 2005 and the Lomonosov Gold Medal in 2013. The American Mathematical Society selected him as its Gibbs Lecturer for 2007, in 2012 he became a fellow of the American Mathematical Society. Some of the present, possibly members of the Weathermen, threatened to destroy the computer with incendiary devices. Complex Proofs of Real Theorems, with Lawrence Zalcman, University Lecture Series,2012,90 pp, softcover, Volume,58, ISBN 978-0-8218-7559-9 Functional Analysis, Wiley-Interscience, linear Algebra and Its Applications, 2nd ed. Wiley-Interscience, New York. Hyperbolic Partial Differential Equations, American Mathematical Society/Courant Institute of Mathematical Sciences, scattering Theory, with R. S. Phillips, Academic Press. Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, decay of Solutions of Systems of Nonlinear Hyperbolic Conservation Laws, with J. Glimm, American Mathematical Society. Recent Mathematical Methods in Nonlinear Wave Propagation, with G. Boillat, C. M. Dafermos, scattering Theory for Automorphic Functions with R. S. Phillips, Princeton Univ

15.
Friedrich Hirzebruch
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Friedrich Ernst Peter Hirzebruch ForMemRS was a German mathematician, working in the fields of topology, complex manifolds and algebraic geometry, and a leading figure in his generation. He has been described as the most important mathematician in Germany of the postwar period, Hirzebruch was born in Hamm, Westphalia in 1927. He studied at the University of Münster from 1945–1950, with one year at ETH Zürich, Hirzebruch then held a position at Erlangen, followed by the years 1952–54 at the Institute for Advanced Study in Princeton, New Jersey. After one year at Princeton University 1955–56, he was made a professor at the University of Bonn, more than 300 people gathered in celebration of his 80th birthday in Bonn in 2007. Hirzebruchs book Neue topologische Methoden in der algebraischen Geometrie was a text for the new methods of sheaf theory. He went on to write the foundational papers on topological K-theory with Michael Atiyah, in his later work he provided a detailed theory of Hilbert modular surfaces, working with Don Zagier. In March 1945, Hirzebruch became a soldier, and in April, in the last weeks of Hitlers rule, when a British soldier found that he was studying mathematics, he drove him home and released him, and told him to continue studying. Hirzebruch died at the age of 84 on 27 May 2012, amongst many other honours, Hirzebruch was awarded a Wolf Prize in Mathematics in 1988 and a Lobachevsky Medal in 1989. The government of Japan awarded him the Order of the Sacred Treasure in 1996, Hirzebruch won an Einstein Medal in 1999, and received the Cantor medal in 2004. In 1980–81 he delivered the first Sackler Distinguished Lecture in Israel

16.
John Milnor
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John Willard Milnor is an American mathematician known for his work in differential topology, K-theory and dynamical systems. Milnor is a professor at Stony Brook University and one of the four mathematicians to have won the Fields Medal, the Wolf Prize. Milnor was born on February 20,1931 in Orange, New Jersey and his father was J. Willard Milnor and his mother was Emily Cox Milnor. As an undergraduate at Princeton University he was named a Putnam Fellow in 1949 and 1950, upon completing his doctorate he went on to work at Princeton. He was a professor at the Institute for Advanced Study from 1970 to 1990 and his students have included Tadatoshi Akiba, Jon Folkman, John Mather, Laurent C. His wife, Dusa McDuff, is a professor of mathematics at Barnard College, one of his published works is his proof in 1956 of the existence of 7-dimensional spheres with nonstandard differential structure. Later, with Michel Kervaire, he showed that the 7-sphere has 15 differentiable structures, an n-sphere with nonstandard differential structure is called an exotic sphere, a term coined by Milnor. Milnors 1968 book on his theory inspired the growth of a huge, in 1961 Milnor disproved the Hauptvermutung by illustrating two simplicial complexes which are homeomorphic but combinatorially distinct. In 1984 Milnor introduced a definition of attractor, the objects generalize standard attractors, include so-called unstable attractors and are now known as Milnor attractors. Milnors current interest is dynamics, especially holomorphic dynamics, Milnor cast his eye on dynamical systems theory in the mid-1970s. By that time the Smale program in dynamics had been completed, milnors approach was to start over from the very beginning, looking at the simplest nontrivial families of maps. The first choice, one-dimensional dynamics, became the subject of his joint paper with Thurston, even the case of a unimodal map, that is, one with a single critical point, turns out to be extremely rich. This work may be compared with Poincarés work on circle diffeomorphisms, milnors work has opened several new directions in this field, and has given us many basic concepts, challenging problems and nice theorems. He was an editor of the Annals of Mathematics for a number of years after 1962 and he has written a number of books. In 1962 Milnor was awarded the Fields Medal for his work in differential topology. He later went on to win the National Medal of Science, in 1991 a symposium was held at Stony Brook University in celebration of his 60th birthday. Milnor was awarded the 2011 Abel Prize, for his discoveries in topology, geometry. Reacting to the award, Milnor told the New Scientist It feels very good, in 2013 he became a fellow of the American Mathematical Society, for contributions to differential topology, geometric topology, algebraic topology, algebra, and dynamical systems

17.
Ennio de Giorgi
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Ennio De Giorgi was an Italian mathematician, member of the House of Giorgi, who worked on partial differential equations and the foundations of mathematics. He solved Bernsteins problem about minimal surfaces and he solved the 19th Hilbert problem on the regularity of solutions of elliptic partial differential equations.29403. The first note published by De Giorgi on his approach to Caccioppoli sets, the first complete exposition of his approach to the theory of Caccioppoli sets by De Giorgi.49014 CS1 maint, Unrecognized language. The first paper on SBV functions and related variational problems.49036 CS1 maint, Unrecognized language. De Giorgi, Ennio, Colombini, Ferruccio, Piccinini, Livio, Frontiere orientate di misura minima e questioni collegate, Quaderni, Pisa, Edizioni della Normale, p.180, MR493669, Zbl 0296.49031. An advanced text, oriented to the theory of surfaces in the multi-dimensional setting. De Giorgi, Ennio, Ambrosio, Luigi, Dal Maso, Gianni, Forti, Marco, Miranda, Mario, Spagnolo, Sergio, selected papers, Berlin–Heidelberg–New York, Springer-Verlag, ISBN 978-3-540-26169-8, MR2229237, Zbl 1096. De Giorgi, Ennio, Homepage, retrieved 21 May 2011 available at the web site of the Research Group in Calculus of Variations and Geometric Measure Theory, Scuola Normale Superiore, Pisa. Ennio De Giorgi at the Mathematics Genealogy Project Emmer, Michele, Ennio De Giorgi, archived from the original on May 25,2011, oConnor, John J. Robertson, Edmund F. Ennio de Giorgi, MacTutor History of Mathematics archive, University of St Andrews. Giornata in ricordo di Ennio De Giorgi, Dipartimento di Matematica L. Tonelli, Faedo Hall, Università di Pisa,30 November 2006, workshop The Mathematics of Ennio De Giorgi, Pisa, Scuola Normale Superiore, 24–27 October 2001, retrieved 21 May 2011

18.
Ilya Piatetski-Shapiro
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Ilya Piatetski-Shapiro was a Soviet-born Israeli mathematician. During a career that spanned 60 years he made contributions to applied science as well as theoretical mathematics. In the last forty years his research focused on mathematics, in particular, analytic number theory, group representations. His main contribution and impact was in the area of automorphic forms, for the last 30 years of his life he suffered from Parkinsons disease. However, with the help of his wife Edith, he was able to continue to work and do mathematics at the highest level, even when he was able to walk. Ilya was born in 1929 in Moscow, Soviet Union, both his father, Iosif Grigorevich, and mother, Sofia Arkadievna, were from traditional Jewish families, but which had become assimilated. His father was from Berdichev, a city in the Ukraine. His mother was from Gomel, a small city in Belorussia. Both parents families were middle-class, but they sank into poverty after the October revolution of 1917, in 1952, Piatetski-Shapiro won the Moscow Mathematical Society Prize for a Young Mathematician for work done while still an undergraduate at Moscow University. His winning paper contained a solution to the problem of the French analyst Raphaël Salem on sets of uniqueness of trigonometric series, the award was especially remarkable because of the atmosphere of strong anti-Semitism in Soviet Union at that time. Ilya was ultimately admitted to the Moscow Pedagogical Institute, where he received his Ph. D. in 1954 under the direction of Alexander Buchstab and his early work was in classical analytic number theory. His contact with Shafarevich, who was a professor at the Steklov Institute, broadened Ilyas mathematical outlook and directed his attention to modern number theory and this led, after a while, to the influential joint paper in which they proved a Torelli theorem for K3 surfaces. Ilyas career was on the rise, and in 1958 he was made a professor of mathematics at the Moscow Institute of Applied Mathematics, by the 1960s, he was recognized as a star mathematician. In 1965 he was appointed to a professorship at the prestigious Moscow State University. He conducted seminars for advanced students, among them Grigory Margulis and he was invited to attend 1962 International Congress of Mathematicians in Stockholm, but was not allowed to go by Soviet authorities. In 1966, Ilya was again invited to ICM in Moscow where he presented a 1-hour lecture on Automorphic Functions, but despite his fame, Ilya was not allowed to travel abroad to attend meetings or visit colleagues except for one short trip to Hungary. The Soviet authorities insisted on one a condition, become a party member, Ilya gave his famous answer, “The membership in the Communist Party will distract me from my work. ”During the span of his career Piatetski-Shapiro was influenced greatly by Israel Gelfand. The aim of their collaboration was to introduce novel representation theory into classical modular forms, together with Graev, they wrote the classic “Automorphic Forms and Representations” book

19.
Lennart Carleson
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Lennart Axel Edvard Carleson is a Swedish mathematician, known as a leader in the field of harmonic analysis. One of his most famous achievements is his proof of Lusins conjecture and he was a student of Arne Beurling and received his Ph. D. from Uppsala University in 1950. Between 1978 and 1982 he served as president of the International Mathematical Union, Carleson married Butte Jonsson in 1953, and they had two children, Caspar and Beatrice. His work has included the solution of some outstanding problems, using techniques from combinatorics, in the theory of Hardy spaces, Carlesons contributions include the corona theorem and establishing the almost everywhere convergence of Fourier series for square-integrable functions. He is also known for the theory of Carleson measures, in the theory of dynamical systems, Carleson has worked in complex dynamics. He is a member of the Norwegian Academy of Science and Letters, in 2012 he became a fellow of the American Mathematical Society. Selected Problems on Exceptional Sets, Van Nostrand,1967 Matematik för vår tid, Prisma 1968 with T. W. Gamelin, Complex Dynamics, Springer,1993

20.
John G. Thompson
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John Griggs Thompson is a mathematician at the University of Florida noted for his work in the field of finite groups. He was awarded the Fields Medal in 1970, the Wolf Prize in 1992 and he received his B. A. from Yale University in 1955 and his doctorate from the University of Chicago in 1959 under the supervision of Saunders Mac Lane. He is currently a Professor Emeritus of Pure Mathematics at the University of Cambridge and he received the Abel Prize 2008 together with Jacques Tits. Thompsons doctoral thesis introduced new techniques, and included the solution of a problem in finite group theory which had stood for sixty years. At the time, this achievement was noted in The New York Times, Thompson became a figure in the progress toward the classification of finite simple groups. In 1963, he and Walter Feit proved that all finite simple groups are of even order. This work was recognised by the award of the 1965 Cole Prize in Algebra of the American Mathematical Society and his N-group papers classified all finite simple groups for which the normalizer of every non-identity solvable subgroup is solvable. This included, as a by-product, the classification of all finite simple groups. This work had some influence on developments in the classification of finite simple groups. The Thompson group Th is one of the 26 sporadic finite simple groups, Thompson also made major contributions to the inverse Galois problem. He found a criterion for a group to be a Galois group. In 1971, Thompson was elected to the United States National Academy of Sciences, in 1982, he was awarded the Senior Berwick Prize of the London Mathematical Society, and in 1988, he received the honorary degree of Doctor of Science from the University of Oxford. Thompson was awarded the United States National Medal of Science in 2000 and he is a Fellow of the Royal Society, and a recipient of its Sylvester Medal in 1985. He is a member of the Norwegian Academy of Science and Letters, John G. Thompson at the Mathematics Genealogy Project List of mathematical articles by John G. Thompson Biography from the Abel Prize center

21.
Mikhail Leonidovich Gromov
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Mikhail Leonidovich Gromov, is a French-Russian mathematician known for important contributions in many different areas of mathematics, including geometry, analysis and group theory. He is a permanent member of IHÉS in France and a Professor of Mathematics at New York University, Gromov has won several prizes, including the Abel Prize in 2009 for his revolutionary contributions to geometry. Mikhail Gromov was born on 23 December 1943 in Boksitogorsk, Soviet Union and his father Leonid Gromov and his Jewish mother Lea Rabinovitz were pathologists. Gromov was born during World War II, and his mother, when Gromov was nine years old, his mother gave him the book The Enjoyment of Mathematics by Hans Rademacher and Otto Toeplitz, a book that piqued his curiosity and had a great influence on him. Gromov studied mathematics at Leningrad State University where he obtained a degree in 1965. His thesis advisor was Vladimir Rokhlin, in 1970, invited to give a presentation at the International Congress of Mathematicians in France, he was not allowed to leave the USSR. Still, his lecture was published in the conference proceedings, disagreeing with the Soviet system, he had been thinking of emigrating since the age of 14. In the early 1970s he ceased publication, hoping that this would help his application to move to Israel and he changed his last name to that of his mother. When the request was granted in 1974, he moved directly to New York where a position had been arranged for him at Stony Brook. In 1981 he left Stony Brook to join the faculty of University of Paris VI, at the same time, he has held professorships at the University of Maryland, College Park from 1991 to 1996, and at the Courant Institute of Mathematical Sciences since 1996. He adopted French citizenship in 1992, Gromovs style of geometry often features a coarse or soft viewpoint, analyzing asymptotic or large-scale properties. In the 1980s, Gromov introduced the Gromov–Hausdorff metric, a measure of the difference between two metric spaces. The possible limit points of sequences of such manifolds are Alexandrov spaces of curvature ≥ c, Gromov was also the first to study the space of all possible Riemannian structures on a given manifold. Gromov introduced geometric group theory, the study of infinite groups via the geometry of their Cayley graphs, in 1981 he proved Gromovs theorem on groups of polynomial growth, a finitely generated group has polynomial growth if and only if it is virtually nilpotent. The proof uses the Gromov–Hausdorff metric mentioned above, along with Eliyahu Rips he introduced the notion of hyperbolic groups. Gromov founded the field of symplectic topology by introducing the theory of pseudoholomorphic curves and this led to Gromov–Witten invariants which are used in string theory and to his non-squeezing theorem. Gromov is also interested in biology, the structure of the brain and the thinking process. Member of the French Academy of Sciences Gromov, M. Hyperbolic manifolds, groups, riemann surfaces and related topics, Proceedings of the 1978 Stony Brook Conference, pp. 183–213, Ann. of Math

22.
Jacques Tits
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Jacques Tits is a Belgium-born French mathematician who works on group theory and incidence geometry, and who introduced Tits buildings, the Tits alternative, and the Tits group. Tits was born in Uccle to Léon Tits, a professor, Jacques attended the Athénée of Uccle and the Free University of Brussels. His thesis advisor was Paul Libois, and Tits graduated with his doctorate in 1950 with the dissertation Généralisation des groupes projectifs basés sur la notion de transitivité. His academic career includes professorships at the Free University of Brussels, the University of Bonn and he changed his citizenship to French in 1974 in order to teach at the Collège de France, which at that point required French citizenship. Because Belgian nationality law did not allow dual nationality at the time and he has been a member of the French Academy of Sciences since then. Tits received the Wolf Prize in Mathematics in 1993, the Cantor Medal from the Deutsche Mathematiker-Vereinigung in 1996, and the German distinction Pour le Mérite. In 2008 he was awarded the Abel Prize, along with John Griggs Thompson, “for their profound achievements in algebra and he is a member of the Norwegian Academy of Science and Letters. He became a member of the Royal Netherlands Academy of Arts. He introduced the theory of buildings, which are structures on which groups act. The related theory of pairs is a tool in the theory of groups of Lie type. Of particular importance is his classification of all buildings of spherical type. In the rank-2 case spherical building are generalized n-gons, and in joint work with Richard Weiss he classified these when they admit a group of symmetries. In collaboration with François Bruhat he developed the theory of affine buildings, the Tits group and the Tits–Koecher construction are named after him. Buildings of spherical type and finite BN-pairs, lecture Notes in Mathematics, Vol.386. MR0470099 Tits, Jacques, Weiss, Richard M. Moufang polygons, MR1938841 J. Tits, Oeuvres - Collected Works,4 vol. J. Tits, Résumés des cours au Collège de France, Jacques Tits at the Mathematics Genealogy Project OConnor, John J. Robertson, Edmund F. Jacques Tits, MacTutor History of Mathematics archive, University of St Andrews. Biography at the Abel Prize site List of publications at the Université libre de Bruxelles

23.
Robert Langlands
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Robert Phelan Langlands is a Canadian mathematician. He is best known as the founder of the Langlands program and he is an emeritus professor and occupies Albert Einsteins office at the Institute for Advanced Study in Princeton. Langlands received a degree from the University of British Columbia in 1957. He then went to Yale University where he received a Ph. D. in 1960 and his academic positions since then include the years 1960–67 at Princeton University, ending up as Associate Professor, and the years 1967–72 at Yale University. He was a Miller Research Fellow at the University of California Berkeley from 1964-65 and he was appointed Hermann Weyl Professor at the Institute for Advanced Study in 1972, becoming Professor Emeritus in January 2007. His Ph. D. thesis was on the theory of semigroups. His first accomplishment in this field was a formula for the dimension of certain spaces of automorphic forms, as a first application, he proved the Weil conjecture on Tamagawa numbers for the large class of arbitrary simply connected Chevalley groups defined over the rational numbers. Previously this had been only in a few isolated cases. As a second application of work, he was able to show meromorphic continuation for a large class of L-functions arising in the theory of automorphic forms. These occurred in the constant terms of Eisenstein series, and meromorphicity as well as a functional equation were a consequence of functional equations for Eisenstein series. This work led in turn, in the winter of 1966–67 and these conjectures were first posed in relatively complete form in a famous letter to Weil, written in January 1967. It was in this letter that he introduced what has become known as the L-group and along with it. Langlandss introduction of these notions broke up large and to some extent intractable problems into smaller, for example, they made the infinite-dimensional representation theory of reductive groups into a major field of mathematical activity. Functoriality is the conjecture that automorphic forms on different groups should be related in terms of their L-groups, in its application to Artins conjecture, functoriality associated to every N-dimensional representation of a Galois group an automorphic representation of the adelic group of GL. In the theory of Shimura varieties it associates automorphic representations of groups to certain l-adic Galois representations as well. This book applied the trace formula for GL and quaternion algebras to do this. Subsequently James Arthur, a student of Langlands while he was at Yale, the functoriality conjecture is far from proved, but a special case was the starting point of Andrew Wiles attack on the Taniyama–Shimura conjecture and Fermats last theorem. In the mid-1980s Langlands turned his attention to physics, particularly the problems of percolation, in recent years he has turned his attention back to automorphic forms, working in particular on a theme he calls beyond endoscopy

24.
Andrew Wiles
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Sir Andrew John Wiles KBE FRS is a British mathematician and a Royal Society Research Professor at the University of Oxford, specialising in number theory. He is most notable for proving Fermats Last Theorem, for which he received the 2016 Abel Prize, Wiles has received numerous other honours. Wiles was born in 1953 in Cambridge, England, the son of Maurice Frank Wiles, the Regius Professor of Divinity at the University of Oxford and his father worked as the Chaplain at Ridley Hall, Cambridge, for the years 1952–55. Wiles attended Kings College School, Cambridge, and The Leys School, Wiles states that he came across Fermats Last Theorem on his way home from school when he was 10 years old. He stopped by his local library where he found a book about the theorem. Fascinated by the existence of a theorem that was so easy to state that he, a ten-year-old, could understand it, Wiles earned his bachelors degree in mathematics in 1974 at Merton College, Oxford, and a PhD in 1980 at Clare College, Cambridge. After a stay at the Institute for Advanced Study in New Jersey in 1981, 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 he rejoined Oxford in 2011 as Royal Society Research Professor. Wiless 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 conjecture of Iwasawa theory over the rational numbers. The modularity theorem involved elliptic curves, which was also Wiless own specialist area, the conjecture was seen by contemporary mathematicians as important, but extraordinarily difficult or perhaps impossible to prove. Despite this, Wiles, with his fascination with Fermats Last Theorem, decided to undertake the challenge of proving the conjecture. In June 1993, he presented his proof to the public for the first time at a conference in Cambridge and 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 Fermats last theorem would be discussed, 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 Fermats last theorem was true, 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 paper which circumvented the problem. Both papers were published in May 1995 in a volume of the Annals of Mathematics

25.
Yakov Sinai
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Yakov Grigorevich Sinai is a mathematician known for his work on dynamical systems. He contributed to the metric theory of dynamical systems and connected the world of deterministic systems with the world of probabilistic systems. He has also worked on physics and probability theory. His efforts have provided the groundwork for advances in the physical sciences, Sinai has won several awards, including the Nemmers Prize, the Wolf Prize in Mathematics and the Abel Prize. Yakov Grigorevich Sinai was born into a Russian Jewish academic family on September 21,1935, in Moscow and his parents, Nadezda Kagan and Gregory Sinai, were both microbiologists. His grandfather, Veniamin Kagan, headed the Department of Differential Geometry at Moscow State University and was an influence on Sinais life. Sinai received his bachelors and masters degrees from Moscow State University, in 1960, he earned his Ph. D. also from Moscow State, his adviser was Andrey Kolmogorov. Together with Kolmogorov, he showed that even for unpredictable dynamic systems, in 1963, Sinai introduced the idea of dynamical billiards, also known as Sinai Billiards. In this idealized system, a particle bounces around inside a square boundary without loss of energy, inside the square is a circular wall, of which the particle also bounces off. It was the first time proved a dynamic system was ergodic. From 1960 to 1971, Sinai was a researcher in the Laboratory of Probabilistic, in 1971, he was promoted to professor and named a senior researcher at the Landau Institute for Theoretical Physics in Russia. Since 1993, Sinai has been a professor of mathematics at Princeton University, for the 1997–98 academic year, he was the Thomas Jones Professor at Princeton, and in 2005, the Moore Distinguished Scholar at the California Institute of Technology. Sinai has supported the poet, mathematician and human rights activist Alexander Esenin-Volpin, in 2002, Sinai won the Nemmers Prize for his revolutionizing work on dynamical systems, statistical mechanics, probability theory, and statistical physics. In 2005, the Moscow Mathematical Journal dedicated an issue to Sinai writing Yakov Sinai is one of the greatest mathematicians of our time and his exceptional scientific enthusiasm inspire several generations of scientists all over the world. In 2013, Sinai received the Leroy P. Steele Prize for Lifetime Achievement, in 2014, the Norwegian Academy of Science and Letters awarded him the Abel Prize, for his contributions to dynamical systems, ergodic theory, and mathematical physics. Presenting the award, Jordan Ellenberg said Sinai had solved real world problems with the soul of a mathematician. He praised the tools developed by Sinai which demonstrate how systems that look different may in fact have fundamental similarities, the prize comes with 6 million Norwegian krone, equivalent at the time to $US1 million or £600,000. He was also inducted into the Norwegian Academy of Science and Letters and he is a member of the United States National Academy of Sciences, the Russian Academy of Sciences, and the Hungarian Academy of Sciences

26.
Elias M. Stein
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Elias Menachem Stein is a mathematician. He is a figure in the field of harmonic analysis. He is an emeritus of Mathematics at Princeton University. Stein was born to Elkan Stein and Chana Goldman, Ashkenazi Jews from Belgium, after the German invasion in 1940, the Stein family fled to the United States, first arriving in New York City. He graduated from Stuyvesant High School in 1949, where he was classmates with future Fields Medalist Paul Cohen, in 1955, Stein earned a Ph. D. from the University of Chicago under the direction of Antoni Zygmund. He began teaching in MIT in 1955, moved to the University of Chicago in 1958 as an assistant professor, and in 1963 became a professor at Princeton. Stein has worked primarily in the field of analysis, and has made contributions in both extending and clarifying Calderón–Zygmund theory. He has written books on harmonic analysis, which are often cited as the standard references on the subject. His Princeton Lectures in Analysis series were penned for his sequence of courses on analysis at Princeton. Stein is also noted as having trained a number of graduate students. They include two Fields medalists, Charles Fefferman and Terence Tao and his honors include the Steele Prize, the Schock Prize in Mathematics, the Wolf Prize in Mathematics, and the National Medal of Science. In addition, he has fellowships to National Science Foundation, Sloan Foundation, Guggenheim Foundation, in 2005, Stein was awarded the Stefan Bergman prize in recognition of his contributions in real, complex, and harmonic analysis. In 2012 he became a fellow of the American Mathematical Society, in 1959, he married Elly Intrator, a former Jewish refugee during World War II. They had two children, Karen Stein and Jeremy C, Stein, and grandchildren named Alison, Jason, and Carolyn. Singular Integrals and Differentiability Properties of Functions, Topics in Harmonic Analysis Related to the Littlewood-Paley Theory. Introduction to Fourier Analysis on Euclidean Spaces, Lectures on Pseudo-differential Operators, Regularity Theorems and Applications to Non-elliptic Problems. Harmonic Analysis, Real-variable Methods, Orthogonality and Oscillatory Integrals, Stein, Elias, Shakarchi, R. Fourier Analysis, An Introduction. Stein, Elias, Shakarchi, R. Complex Analysis, real Analysis, Measure Theory, Integration, and Hilbert Spaces

27.
Raoul Bott
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Raoul Bott, ForMemRS was a Hungarian-American mathematician known for numerous basic contributions to geometry in its broad sense. He is best known for his Bott periodicity theorem, the Morse–Bott functions which he used in context. Bott was born in Budapest, Hungary, the son of Margit Kovács and his father was of Austrian descent, and his mother was of Hungarian Jewish descent, Bott was raised a Catholic by his mother and stepfather. Bott grew up in Czechoslovakia and spent his life in the United States. His family emigrated to Canada in 1938, and subsequently he served in the Canadian Army in Europe during World War II, Bott later went to college at McGill University in Montreal, where he studied electrical engineering. He then earned a Ph. D. in mathematics from Carnegie Mellon University in Pittsburgh in 1949 and his thesis, titled Electrical Network Theory, was written under the direction of Richard Duffin. Afterward, he began teaching at the University of Michigan in Ann Arbor, Bott continued his study at the Institute for Advanced Study in Princeton. He was a professor at Harvard University from 1959 to 1999, in 2005 Bott died of cancer in San Diego. With Richard Duffin at Carnegie Mellon, Bott studied existence of electronic filters corresponding to given positive-real functions, in 1949 they proved a fundamental theorem of filter synthesis. Duffin and Bott extended earlier work by Otto Brune that requisite functions of complex frequency s could be realized by a network of inductors and capacitors. Bott met Arnold S. Shapiro at the IAS and they worked together and he studied the homotopy theory of Lie groups, using methods from Morse theory, leading to the Bott periodicity theorem. In the course of work, he introduced Morse–Bott functions. This led to his role as collaborator over many years with Michael Atiyah and he is also well known in connection with the Borel–Bott–Weil theorem on representation theory of Lie groups via holomorphic sheaves and their cohomology groups, and for work on foliations. He introduced Bott–Samelson varieties and the Bott residue formula for complex manifolds, in 1964, he was awarded the Oswald Veblen Prize in Geometry by the American Mathematical Society. In 1983, he was awarded the Jeffery–Williams Prize by the Canadian Mathematical Society, in 1987, he was awarded the National Medal of Science. In 2000, he received the Wolf Prize, in 2005, he was elected an Overseas Fellow of the Royal Society of London. Bott had 26 Ph. D. students, including Stephen Smale, Lawrence Conlon, Daniel Quillen, Peter Landweber, Robert MacPherson, Robert W. Brooks, Robin Forman, András Szenes, birkhäuser Boston, xx+485 pp. ISBN 0-8176-3648-X MR13218901995, Collected Papers. Birkhäuser, xxxii+610 pp. ISBN 0-8176-3647-1 MR13218861994, Collected Papers, birkhäuser, xxxiv+802 pp. ISBN 0-8176-3646-3 MR12903611994, Collected Papers

28.
Jean-Pierre Serre
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Jean-Pierre Serre is a French mathematician who has made contributions to algebraic topology, algebraic geometry, and algebraic number theory. He was awarded the Fields Medal in 1954, the Wolf Prize in 2000, born in Bages, Pyrénées-Orientales, France, to pharmacist parents, Serre was educated at the Lycée de Nîmes and then from 1945 to 1948 at the École Normale Supérieure in Paris. He was awarded his doctorate from the Sorbonne in 1951, from 1948 to 1954 he held positions at the Centre National de la Recherche Scientifique in Paris. In 1956 he was elected professor at the Collège de France and his wife, Professor Josiane Heulot-Serre, was a chemist, she also was the director of the Ecole Normale Supérieure de Jeunes Filles. Their daughter is the former French diplomat, historian and writer Claudine Monteil, the French mathematician Denis Serre is his nephew. Serres thesis concerned the Leray–Serre spectral sequence associated to a fibration, together with Cartan, Serre established the technique of using Eilenberg–MacLane spaces for computing homotopy groups of spheres, which at that time was one of the major problems in topology. In his speech at the Fields Medal award ceremony in 1954, Hermann Weyl gave high praise to Serre, Serre subsequently changed his research focus. However, Weyls perception that the place of classical analysis had been challenged has subsequently been justified. In the 1950s and 1960s, a collaboration between Serre and the two-years-younger Alexander Grothendieck led to important foundational work, much of it motivated by the Weil conjectures. Two major foundational papers by Serre were Faisceaux Algébriques Cohérents, on coherent cohomology, even at an early stage in his work Serre had perceived a need to construct more general and refined cohomology theories to tackle the Weil conjectures. The problem was that the cohomology of a coherent sheaf over a finite field couldnt capture as much topology as singular cohomology with integer coefficients, amongst Serres early candidate theories of 1954–55 was one based on Witt vector coefficients. Around 1958 Serre suggested that isotrivial principal bundles on algebraic varieties — those that become trivial after pullback by a finite étale map — are important and this acted as one important source of inspiration for Grothendieck to develop étale topology and the corresponding theory of étale cohomology. These tools, developed in full by Grothendieck and collaborators in Séminaire de géométrie algébrique 4 and SGA5, from 1959 onward Serres interests turned towards group theory, number theory, in particular Galois representations and modular forms. In his paper FAC, Serre asked whether a finitely generated module over a polynomial ring is free. This question led to a deal of activity in commutative algebra. This result is now known as the Quillen-Suslin theorem, Serre, at twenty-seven in 1954, is the youngest ever to be awarded the Fields Medal. He went on to win the Balzan Prize in 1985, the Steele Prize in 1995, the Wolf Prize in Mathematics in 2000 and he has been awarded other prizes, such as the Gold Medal of the French National Scientific Research Centre. He is a member of several scientific Academies and has received many honorary degrees

29.
Vladimir Arnold
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Vladimir Igorevich Arnold was a Soviet and Russian mathematician. Arnold was also known as a popularizer of mathematics, through his lectures, seminars, and as the author of several textbooks and popular mathematics books, he influenced many mathematicians and physicists. Many of his books were translated into English, Vladimir Igorevich Arnold was born on 12 June 1937 in Odessa, Soviet Union. His father was Igor Vladimirovich Arnold, a mathematician and his mother was Nina Alexandrovna Arnold, an art historian. This is the Kolmogorov–Arnold representation theorem, after graduating from Moscow State University in 1959, he worked there until 1986, and then at Steklov Mathematical Institute. He became an academician of the Academy of Sciences of the Soviet Union in 1990, Arnold can be said to have initiated the theory of symplectic topology as a distinct discipline. The Arnold conjecture on the number of fixed points of Hamiltonian symplectomorphisms, Arnold worked at the Steklov Mathematical Institute in Moscow and at Paris Dauphine University up until his death. As of 2006 he was reported to have the highest citation index among Russian scientists, to his students and colleagues Arnold was known also for his sense of humour. In accordance with this principle I shall formulate some problems. ”Arnold died of pancreatitis on 3 June 2010 in Paris. He was buried on June 15 in Moscow, at the Novodevichy Monastery, in a telegram to Arnolds family, Russian President Dmitry Medvedev stated, “The death of Vladimir Arnold, one of the greatest mathematicians of our time, is an irretrievable loss for world science. It is difficult to overestimate the contribution made by academician Arnold to modern mathematics, teaching had a special place in Vladimir Arnolds life and he had great influence as an enlightened mentor who taught several generations of talented scientists. The memory of Vladimir Arnold will forever remain in the hearts of his colleagues, friends and students, as well as everyone who knew and admired this brilliant man. ”Arnold is well known for his writing style, combining mathematical rigour with physical intuition. His defense is that his books are meant to teach the subject to those who wish to understand it. Arnold was a critic of the trend towards high levels of abstraction in mathematics during the middle of the last century. Arnold was very interested in the history of mathematics and he liked to study the classics, most notably the works of Huygens, Newton and Poincaré, and many times he reported to have found in their works ideas that had not been explored yet. Arnold worked on systems theory, catastrophe theory, topology, algebraic geometry, symplectic geometry, differential equations, classical mechanics, hydrodynamics. Moser and Arnold expanded the ideas of Kolmogorov and gave rise to what is now known as KAM Theory, KAM theory shows that, despite the perturbations, such systems can be stable over an infinite period of time, and specifies what the conditions for this are. In 1965, Arnold attended René Thoms seminar on catastrophe theory, after this event, singularity theory became one of the major interests of Arnold and his students

30.
Saharon Shelah
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Saharon Shelah is an Israeli mathematician. He is a professor of mathematics at the Hebrew University of Jerusalem, Shelah was born in Jerusalem on July 3,1945. He is the son of the Israeli poet and political activist Yonatan Ratosh and he received his PhD for his work on stable theories in 1969 from the Hebrew University. Shelah is married to Yael, and has three children, Shelah wanted to be a scientist while at primary school, but initially was attracted to physics and biology, not mathematics. At the age of 15, he decided to become a mathematician, a choice cemented after reading Abraham Halevy Fraenkels book An Introduction to Mathematics. He received a B. Sc. from Tel Aviv University in 1964, Shelah was a Lecturer at Princeton University during 1969-70, and then worked as an Assistant Professor at the University of California, Los Angeles during 1970-71. He became a professor at Hebrew University in 1974, a position he continues to hold and he has been a Distinguished Visiting Professor at Rutgers University since 1986. Shelahs archiv, Template, As of February lists 1103 mathematical papers including joint papers with over 220 co-authors and his main interests lie in mathematical logic, model theory in particular, and in axiomatic set theory. In model theory, he developed classification theory, which led him to a solution of Morleys problem, in set theory, he discovered the notion of proper forcing, an important tool in iterated forcing arguments. With PCF theory, he showed that in spite of the undecidability of the most basic questions of cardinal arithmetic, Shelah constructed a Jonson group, an uncountable group for which every proper subgroup is countable. He showed that Whiteheads problem is independent of ZFC and he gave the first primitive recursive upper bound to van der Waerdens numbers V. He extended Arrows impossibility theorem on voting systems, Shelahs work has had a deep impact on model theory and set theory. Following that he has extended the work far beyond first order theories and this work also has had important applications to algebra by works Zilber. The Leroy P. Steele Prize, for Seminal Contribution to Research, in 2013 Honorary member of the Hungarian Academy of Sciences, advanced Grant of the European Research Council. Classification Theory for Abstract Elementary Classes, Volume 2, College Publications 2009, cardinal Arithmetic, Oxford University Press 1994 List of Israel Prize recipients Sauer–Shelah lemma Shelah cardinal Archive of Shelahs mathematical papers, shelah. logic. at John T. Baldwin. Abstract Elementary Classes, Some Answers, More Questions, a survey of recent work on AECs

31.
John Tate
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John Torrence Tate, Jr. is an American mathematician, distinguished for many fundamental contributions in algebraic number theory, arithmetic geometry and related areas in algebraic geometry. He is professor emeritus at Harvard University and he was awarded the Abel Prize in 2010. His father, John Tate Sr. was a professor of physics at the University of Minnesota, and his mother, Lois Beatrice Fossler, was a high school English teacher. Tate Jr. received his bachelors degree in mathematics from Harvard University and he later transferred to the mathematics department and received his PhD in 1950 as a student of Emil Artin. Tate taught at Harvard for 36 years before joining the University of Texas in 1990 and he retired from the Texas mathematics department in 2009, and returned to Harvard as a professor emeritus. He currently resides in Cambridge, Massachusetts with his wife Carol and he has three daughters with his first wife Karin Tate. Together with his teacher Emil Artin, Tate gave a cohomological treatment of class field theory, using techniques of group cohomology applied to the idele class group. Subsequently, Tate introduced what are now known as Tate cohomology groups, in the decades following that discovery he extended the reach of Galois cohomology with the Poitou–Tate duality, the Tate–Shafarevich group, and relations with algebraic K-theory. With Jonathan Lubin, he recast local class field theory by the use of formal groups and he found a p-adic analogue of Hodge theory, now called Hodge–Tate theory, which has blossomed into another central technique of modern algebraic number theory. Other innovations of his include the Tate curve parametrization for certain p-adic elliptic curves, many of his results were not immediately published and some of them were written up by Serge Lang, Jean-Pierre Serre, Joseph H. Silverman and others. Tate and Serre collaborated on a paper on good reduction of abelian varieties, the classification of abelian varieties over finite fields was carried out by Taira Honda and Tate. The Tate conjectures are the equivalent for étale cohomology of the Hodge conjecture and they relate to the Galois action on the l-adic cohomology of an algebraic variety, identifying a space of Tate cycles that conjecturally picks out the algebraic cycles. A special case of the conjectures, which are open in the case, was involved in the proof of the Mordell conjecture by Gerd Faltings. Tate has also had a influence on the development of number theory through his role as a Ph. D. advisor. His students include Benedict Gross, Robert Kottwitz, Jonathan Lubin, Stephen Lichtenbaum, James Milne, V. Kumar Murty, Carl Pomerance, Ken Ribet, Joseph H. Silverman, in 1956 Tate was awarded the American Mathematical Societys Cole Prize for outstanding contributions to number theory. In 1995 he received the Leroy P. Steele Prize for Lifetime Achievement from the American Mathematical Society and he was awarded a Wolf Prize in Mathematics in 2002/03 for his creation of fundamental concepts in algebraic number theory. In 2012 he became a fellow of the American Mathematical Society, in 2010, the Norwegian Academy of Science and Letters, of which he is a member, awarded him the Abel Prize, citing his vast and lasting impact on the theory of numbers. He has truly left an imprint on modern mathematics

32.
Grigory Margulis
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Gregori Aleksandrovich Margulis is a Russian-American mathematician known for his work on lattices in Lie groups, and the introduction of methods from ergodic theory into diophantine approximation. He was awarded a Fields Medal in 1978 and a Wolf Prize in Mathematics in 2005, in 1991, he joined the faculty of Yale University, where he is currently the Erastus L. DeForest Professor of Mathematics. Margulis was born in Moscow, Soviet Union and he received his PhD in 1970 from the Moscow State University, starting research in ergodic theory under the supervision of Yakov Sinai. Early work with David Kazhdan produced the Kazhdan–Margulis theorem, a result on discrete groups. His superrigidity theorem from 1975 clarified an area of classical conjectures about the characterisation of arithmetic groups amongst lattices in Lie groups and he was awarded the Fields Medal in 1978, but was not permitted to travel to Helsinki to accept it in person. In 1991, Margulis accepted a position at Yale University. Margulis was elected a member of the U. S. National Academy of Sciences in 2001, in 2012 he became a fellow of the American Mathematical Society. In 2005, Margulis received the Wolf Prize for his contributions to theory of lattices and applications to ergodic theory, representation theory, number theory, combinatorics, and measure theory. Marguliss early work dealt with Kazhdans property and the questions of rigidity and arithmeticity of lattices in semisimple algebraic groups of rank over a local field. It had been known since the 1950s that a certain simple-minded way of constructing subgroups of semisimple Lie groups produces examples of lattices and it is analogous to considering the subgroup SL of the real special linear group SL that consists of matrices with integer entries. Margulis proved that under suitable assumptions on G, any lattice Γ in it is arithmetic, thus Γ is commensurable with the subgroup G of G, i. e. they agree on subgroups of finite index in both. Unlike general lattices, which are defined by their properties, arithmetic lattices are defined by a construction, therefore, these results of Margulis pave a way for classification of lattices. Arithmeticity turned out to be related to another remarkable property of lattices discovered by Margulis. Superrigidity for a lattice Γ in G roughly means that any homomorphism of Γ into the group of invertible n × n matrices extends to the whole G. While certain rigidity phenomena had already known, the approach of Margulis was at the same time novel, powerful. Margulis solved the Banach–Ruziewicz problem that asks whether the Lebesgue measure is the only normalized rotationally invariant finitely additive measure on the n-dimensional sphere, Margulis gave the first construction of expander graphs, which was later generalized in the theory of Ramanujan graphs. In 1986, Margulis gave a resolution of the Oppenheim conjecture on quadratic forms. He has formulated a program of research in the same direction

33.
Stephen Smale
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Stephen Smale is an American mathematician from Flint, Michigan. His research concerns topology, dynamical systems and mathematical economics and he was awarded the Fields Medal in 1966 and spent more than three decades on the mathematics faculty of the University of California, Berkeley. Smale entered the University of Michigan in 1948, initially, he was a good student, placing into an honors calculus sequence taught by Bob Thrall and earning himself As. However, his sophomore and junior years were marred with mediocre grades, mostly Bs, Cs, however, with some luck, Smale was accepted as a graduate student at the University of Michigans mathematics department. Yet again, Smale performed poorly in his first years, earning a C average as a graduate student and it was only when the department chair, Hildebrandt, threatened to kick Smale out 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 world with a proof of a sphere eversion. After having made great strides in topology, he turned to the study of dynamical systems. His first contribution is the Smale horseshoe that started significant research in dynamical systems and he also outlined a research program carried out by many others. Smale is also known for injecting Morse theory into mathematical economics, in 1998 he compiled a list of 18 problems in mathematics to be solved in the 21st century, known as Smales problems. This list was compiled in the spirit of Hilberts famous list of problems produced in 1900, in fact, Smales list contains some of the original Hilbert problems, including the Riemann hypothesis and the second half of Hilberts sixteenth problem, both of which are still unsolved. Earlier in his career, Smale was involved in controversy over remarks he made regarding his work habits while proving the higher-dimensional Poincaré conjecture and 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 and he has been politically active in various movements in the past, such as the Free Speech movement and the movement against the Vietnam War. 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 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 and he also amassed over the years one of the finest private mineral collections in existence. Many of Smales mineral specimens can be seen in the book—The Smale Collection, since 2002 Smale is a Professor at the Toyota Technological Institute at Chicago, starting August 1,2009, he is also a Distinguished University Professor at the City University of Hong Kong. In 2007, Smale was awarded the Wolf Prize in mathematics, generalized Poincarés conjecture in dimensions greater than four

34.
Hillel Furstenberg
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He is known for his application of probability theory and ergodic theory methods to other areas of mathematics, including number theory and Lie groups. Hillel Furstenberg was born in Germany, in 1935, and the family emigrated to the United States in 1939 and he attended Marsha Stern Talmudical Academy and then Yeshiva University, where he concluded his BA and MSc studies in 1955. He obtained his Ph. D. under Salomon Bochner at Princeton University in 1958, after several years at the University of Minnesota he became a Professor of Mathematics at the Hebrew University of Jerusalem in 1965. He gained attention at a stage in his career for producing an innovative topological proof of the infinitude of prime numbers. He proved unique ergodicity of horocycle flows on compact hyperbolic Riemann surfaces in the early 1970s, in 1977, he gave an ergodic theory reformulation, and subsequently proof, of Szemerédis theorem. The Furstenberg boundary and Furstenberg compactification of a symmetric space are named after him. 1993 – Furstenberg received the Israel Prize, for exact sciences,1993 – Furstenberg received the Harvey Prize from Technion. 2006/7 – He received the Wolf Prize in Mathematics, Furstenberg, Harry, Stationary processes and prediction theory, Princeton, N. J. Furstenberg, Harry, Recurrence in ergodic theory and combinatorial number theory, Princeton, compactification Ratners theorems List of Israel Prize recipients OConnor, John J. Robertson, Edmund F. Hillel Furstenberg, MacTutor History of Mathematics archive, University of St Andrews. Mathematics Genealogy page Press release Israel Academy of Sciences and Humanities

35.
Pierre Deligne
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Pierre René, Viscount Deligne is a Belgian mathematician. He is known for work on the Weil conjectures, leading to a proof in 1973. He is the winner of the 2013 Abel Prize,2008 Wolf Prize and he was born in Etterbeek, attended school at Athénée Adolphe Max and studied at the Université libre de Bruxelles. In 1968, he worked with Jean-Pierre Serre, their work led to important results on the l-adic representations attached to modular forms. Delignes also focused on topics in Hodge theory and he introduced weights and tested them on objects in complex geometry. He also collaborated with David Mumford on a new description of the spaces for curves. Their work came to be seen as an introduction to one form of the theory of algebraic stacks, perhaps Delignes most famous contribution was his proof of the third and last of the Weil conjectures. This proof completed a programme initiated and largely developed by Alexander Grothendieck, as a corollary he proved the celebrated Ramanujan–Petersson conjecture for modular forms of weight greater than one, weight one was proved in his work with Serre. From 1970 until 1984, when he moved to the Institute for Advanced Study in Princeton, during this time he did much important work outside of his work on algebraic geometry. He received a Fields Medal in 1978 and this idea allows one to get around the lack of knowledge of the Hodge conjecture, for some applications. All this is part of the yoga of weights, uniting Hodge theory, the Shimura variety theory is related, by the idea that such varieties should parametrize not just good families of Hodge structures, but actual motives. This theory is not yet a finished product – and more recent trends have used K-theory approaches and he was awarded the Fields Medal in 1978, the Crafoord Prize in 1988, the Balzan Prize in 2004, the Wolf Prize in 2008, and the Abel Prize in 2013. In 2006 he was ennobled by the Belgian king as viscount, in 2009, Deligne was elected a foreign member of the Royal Swedish Academy of Sciences. He is a member of the Norwegian Academy of Science and Letters, Quantum fields and strings, a course for mathematicians. Material from the Special Year on Quantum Field Theory held at the Institute for Advanced Study, Princeton, NJ, edited by Pierre Deligne, Pavel Etingof, Daniel S. Freed, Lisa C. Jeffrey, David Kazhdan, John W. Morgan, David R. Morrison, american Mathematical Society, Providence, RI, Institute for Advanced Study, Princeton, NJ,1999. Vol.1, xxii+723 pp. Vol.2, pp. i--xxiv, Deligne wrote multiple hand-written letters to other mathematicians in the 1970s. These include Delignes letter to Piatetskii-Shapiro and it was proved by Kontsevich–Soibelman, McClure–Smith and others

36.
Phillip Griffiths
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Phillip Augustus Griffiths IV is an American mathematician, known for his work in the field of geometry, and in particular for the complex manifold approach to algebraic geometry. He was a developer in particular of the theory of variation of Hodge structure in Hodge theory. He received his B. S. from Wake Forest College in 1959, since then, he has held positions at Berkeley, Princeton, Harvard University, and Duke University. From 1991 to 2003 he was the Director of the Institute for Advanced Study at Princeton and he has published on algebraic geometry, differential geometry, geometric function theory, and the geometry of partial differential equations. Griffiths serves as the Chair of the Science Initiative Group and he is co-author, with Joe Harris, of Principles of Algebraic Geometry, a well-regarded textbook on complex algebraic geometry. In 2008 he was awarded the Wolf Prize and the Brouwer Medal, in 2012 he became a fellow of the American Mathematical Society. Moreover, in 2014 Griffiths was awarded the Leroy P. Steele Prize for Lifetime Achievement by the American Mathematical Society, also in 2014, Griffiths was awarded the Chern Medal for lifetime devotion to mathematics and outstanding achievements. Proc Natl Acad Sci U S A.48, 780–783, some remarks on automorphisms, analytic bundles, and embeddings of complex algebraic varieties. Proc Natl Acad Sci U S A.49, 817–820, on the differential geometry of homogeneous vector bundles. The residue calculus and some results in algebraic geometry, I. Proc Natl Acad Sci U S A.55, 1303–1309, the residue calculus and some transcendental results in algebraic geometry, II. Proc Natl Acad Sci U S A.55, 1392–1395, some results on locally homogeneous complex manifolds. Proc Natl Acad Sci U S A.56, 413–416, a transcendental method in algebraic geometry. Periods of integrals on algebraic manifolds, with Joe Harris, A Poncelet theorem in space. With S. S. Chern, Abels Theorem and Webs, introduction to Algebraic Curves, American Mathematical Society, Providence, RI,1989, ISBN0821845306 Differential Systems and Isometric Embeddings, with Gary R

37.
David Mumford
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David Bryant Mumford is an American mathematician known for distinguished work in algebraic geometry, and then for research into vision and pattern theory. He won the Fields Medal and was a MacArthur Fellow, in 2010 he was awarded the National Medal of Science. He is currently a University Professor Emeritus in the Division of Applied Mathematics at Brown University, Mumford was born in Worth, West Sussex in England, of an English father and American mother. His father William started a school in Tanzania and worked for the then newly created United Nations. In high school, he was a finalist in the prestigious Westinghouse Science Talent Search, after attending the Phillips Exeter Academy, Mumford went to Harvard, where he became a student of Oscar Zariski. At Harvard, he became a Putnam Fellow in 1955 and 1956 and he completed his Ph. D. in 1961, with a thesis entitled Existence of the moduli scheme for curves of any genus. He met his first wife, Erika Jentsch, at Radcliffe College, after Erika died in 1988, he married his second wife, Jenifer Gordon. He and Erika had four children, Mumfords work in geometry combined traditional geometric insights with the latest algebraic techniques. He published on moduli spaces, with a theory summed up in his book Geometric Invariant Theory, on the equations defining an abelian variety and his books Abelian Varieties and Curves on an Algebraic Surface combined the old and new theories. His lecture notes on scheme theory circulated for years in unpublished form, at a time when they were, beside the treatise Éléments de géométrie algébrique and they are now available as The Red Book of Varieties and Schemes. Other work that was less thoroughly written up were lectures on varieties defined by quadrics, and this work on the equations defining abelian varieties appeared in 1966–7. He published some books of lectures on the theory. He also was one of the founders of the toroidal embedding theory and these pathologies fall into two types, bad behavior in characteristic p and bad behavior in moduli spaces. This second example is developed further in Mumfords third paper on classification of surfaces in characteristic p, worse pathologies related to p-torsion in crystalline cohomology were explored by Luc Illusie. Further such examples arise in Zariski surface theory and he also conjectures that the Kodaira vanishing theorem is false for surfaces in characteristic p. In the third paper, he gives an example of a surface for which Kodaira vanishing fails. The first example of a surface for which Kodaira vanishing fails was given by Michel Raynaud in 1978. In the second Pathologies paper, Mumford finds that the Hilbert scheme parametrizing space curves of degree 14, in the fourth Pathologies paper, he finds reduced and irreducible complete curves which are not specializations of non-singular curves

38.
Dennis Sullivan
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Dennis Parnell Sullivan is an American mathematician. He is known for work in topology, both algebraic and geometric, and on dynamical systems and he holds the Albert Einstein Chair at the City University of New York Graduate Center, and is a professor at Stony Brook University. He received his B. A. in 1963 from Rice University and his Ph. D. thesis, entitled Triangulating homotopy equivalences, was written under the supervision of William Browder, and was a contribution to surgery theory. He was a permanent member of the Institut des Hautes Études Scientifiques from 1974 to 1997, Sullivan is one of the founders of the surgery method of classifying high-dimensional manifolds, along with Browder, Sergei Novikov and C. T. C. In homotopy theory, Sullivan put forward the concept that spaces could directly be localised. This area has generated considerable further research, in 1985, he proved the No wandering domain theorem. The Parry–Sullivan invariant is named after him and the English mathematician Bill Parry, in 1987, he proved Thurstons conjecture about the approximation of the Riemann map by circle packings together with Burton Rodin. 47, 269–331, MR0646078 OConnor, John J. Robertson, Edmund F. Dennis Sullivan, MacTutor History of Mathematics archive, Dennis Sullivan at the Mathematics Genealogy Project Sullivans homepage at CUNY Sullivans homepage at SUNY Stony Brook Dennis Sullivan International Balzan Prize Foundation