1.
Clay Mathematics Institute
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The Clay Mathematics Institute is a private, non-profit foundation, based in Peterborough, New Hampshire, United States. CMIs scientific activities are managed from the Presidents office in Oxford, the institute is dedicated to increasing and disseminating mathematical knowledge. It gives out awards and sponsorships to promising mathematicians. The institute was founded in 1998 through the sponsorship of Boston businessman Landon T. Clay, harvard mathematician Arthur Jaffe was the first president of CMI. While the institute is best known for its Millennium Prize Problems, it out a wide range of activities, including a postdoctoral program, conferences, workshops. Nicholas Woodhouse is the current president of CMI, the institute is best known for establishing the Millennium Prize Problems on May 24,2000. These seven problems are considered by CMI to be important classic questions that have resisted solution over the years, for each problem, the first person to solve it will be awarded $1,000,000 by the CMI. In announcing the prize, CMI drew a parallel to Hilberts problems, which were proposed in 1900, of the initial 23 Hilbert problems, most of which have been solved, only the Riemann hypothesis is included in the seven Millennium Prize Problems. For each problem, the Institute had a professional mathematician write up a statement of the problem. In recognition of major breakthroughs in mathematical research, the institute has an annual prize - the Clay Research Award, the institute also has a yearly Clay Research Award, recognizing major breakthroughs in mathematical research. Finally, the institute organizes a number of schools, conferences, workshops, public lectures. CMI publications are available in PDF form at most six months after they appear in print, the Episode of Elementary entitled Solve for X mentions the Clay Mathematics Institute in reference to their involvement in the P versus NP Problem. Keith J. Devlin, The Millennium Problems, The Seven Greatest Unsolved Mathematical Puzzles of Our Time, Basic Books, ISBN 0-465-01729-0

2.
Timothy Gowers
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Sir William Timothy Gowers, FRS is a British mathematician. In 1998 he received the Fields Medal for research connecting the fields of functional analysis, after his PhD, Gowers was elected to a Junior Research Fellowship at Trinity College. From 1991 until his return to Cambridge in 1995 he was lecturer at University College London and he was elected to the Rouse Ball Professorship at Cambridge in 1998. During 2000–2 he was visiting professor at Princeton University, Gowers attended Kings College School, Cambridge, as a choirboy in the Kings College choir, and then Eton College as a Kings Scholar. He completed his PhD, with a dissertation entitled Symmetric Structures in Banach Spaces, at Trinity College, Cambridge in 1990, Gowers initially worked on Banach spaces. After this, Gowers turned to combinatorics and combinatorial number theory, in 1997 he proved that the Szemerédi regularity lemma necessarily comes with tower-type bounds. In 1998 he proved the first effective bounds for Szemerédis theorem, one of the ingredients in Gowerss argument is a tool now known as the Balog–Szemerédi–Gowers theorem, which has found many further applications. He also introduced the Gowers norms, a tool in arithmetic combinatorics and this work was further developed by Ben Green and Terence Tao, leading to the Green–Tao theorem. In 2003, Gowers established a regularity lemma for hypergraphs, analogous to the Szemerédi regularity lemma for graphs, in 2005, he introduced the notion of a quasirandom group. More recently Gowers has worked on Ramsey theory in random graphs and random sets with David Conlon and he has also developed an interest, in joint work with Mohan Ganesalingam, in automated problem solving. In 1996 he received the Prize of the European Mathematical Society, in 1999 he became a Fellow of the Royal Society and in 2012 was knighted by the British monarch for services to mathematics. He also sits on the committee for the Mathematics award. Gowers has written several works popularising mathematics, including Mathematics, A Very Short Introduction and he was consulted about the 2005 film Proof, starring Gwyneth Paltrow and Anthony Hopkins. He edited The Princeton Companion to Mathematics, which traces the development of various branches, for his work on this book, he won the 2011 Euler Book Prize of the Mathematical Association of America. After asking on his blog whether massively collaborative mathematics was possible, the first problem in what is called the Polymath Project, Polymath1, was to find a new combinatorial proof to the density version of the Hales–Jewett theorem. After 7 weeks, Gowers wrote on his blog that the problem was probably solved, in 2009, with Olof Sisask and Alex Frolkin, he invited people to post comments to his blog to contribute to a collection of methods of mathematical problem solving. Contributors to this Wikipedia-style project, called Tricki. org, include Terence Tao, in 2012, Gowers posted to his blog to call for a boycott of the publishing house Elsevier. A petition ensued, branded the Cost of Knowledge project, in which researchers commit to stop supporting Elsevier journals, commenting on the petition in The Guardian, Alok Jha credited Gowers with starting an Academic Spring

3.
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

4.
Michael Atiyah
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Sir Michael Francis Atiyah OM FRS FRSE FMedSci FREng is an English mathematician specialising in geometry. Atiyah grew up in Sudan and Egypt and spent most of his life in the United Kingdom at Oxford and Cambridge. He has been president of the Royal Society, master of Trinity College, Cambridge, chancellor of the University of Leicester, since 1997, he has been an honorary professor at the University of Edinburgh. Atiyahs mathematical collaborators include Raoul Bott, Friedrich Hirzebruch and Isadore Singer, together with Hirzebruch, he laid the foundations for topological K-theory, an important tool in algebraic topology, which, informally speaking, describes ways in which spaces can be twisted. His best known result, the Atiyah–Singer index theorem, was proved with Singer in 1963 and is used in counting the number of independent solutions to differential equations. Some of his recent work was inspired by theoretical physics, in particular instantons and monopoles. He was awarded the Fields Medal in 1966, the Copley Medal in 1988, Atiyah was born in Hampstead, London, to a Lebanese father, the academic, Eastern Orthodox, Edward Atiyah and Scot Jean Atiyah. Patrick Atiyah is his brother, he has one brother, Joe. He returned to England and Manchester Grammar School for his HSC studies and his undergraduate and postgraduate studies took place at Trinity College, Cambridge. He was a student of William V. D. Hodge and was awarded a doctorate in 1955 for a thesis entitled Some Applications of Topological Methods in Algebraic Geometry. Atiyah married Lily Brown on 30 July 1955, with whom he has three sons, in 1961, he moved to the University of Oxford, where he was a reader and professorial fellow at St Catherines College. He became Savilian Professor of Geometry and a fellow of New College, Oxford. He was president of the London Mathematical Society from 1974 to 1976, Atiyah has been active on the international scene, for instance as president of the Pugwash Conferences on Science and World Affairs from 1997 to 2002. He also contributed to the foundation of the InterAcademy Panel on International Issues, the Association of European Academies, within the United Kingdom, he was involved in the creation of the Isaac Newton Institute for Mathematical Sciences in Cambridge and was its first director. He was President of the Royal Society, Master of Trinity College, Cambridge, Chancellor of the University of Leicester, since 1997, he has been an honorary professor in the University of Edinburgh. Atiyah has collaborated with other mathematicians. His later research on gauge field theories, particularly Yang–Mills theory, other contemporary mathematicians who influenced Atiyah include Roger Penrose, Lars Hörmander, Alain Connes and Jean-Michel Bismut. Atiyah said that the mathematician he most admired was Hermann Weyl, the six volumes of Atiyahs collected papers include most of his work, except for his commutative algebra textbook and a few works written since 2004

5.
David Hilbert
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David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th, Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of geometry. He also formulated the theory of Hilbert spaces, one of the foundations of functional analysis, Hilbert adopted and warmly defended Georg Cantors set theory and transfinite numbers. A famous example of his leadership in mathematics is his 1900 presentation of a collection of problems set the course for much of the mathematical research of the 20th century. Hilbert and his students contributed significantly to establishing rigor and developed important tools used in mathematical physics. Hilbert is known as one of the founders of theory and mathematical logic. In late 1872, Hilbert entered the Friedrichskolleg Gymnasium, but, after a period, he transferred to. Upon graduation, in autumn 1880, Hilbert enrolled at the University of Königsberg, in early 1882, Hermann Minkowski, returned to Königsberg and entered the university. Hilbert knew his luck when he saw it, in spite of his fathers disapproval, he soon became friends with the shy, gifted Minkowski. In 1884, Adolf Hurwitz arrived from Göttingen as an Extraordinarius, Hilbert obtained his doctorate in 1885, with a dissertation, written under Ferdinand von Lindemann, titled Über invariante Eigenschaften spezieller binärer Formen, insbesondere der Kugelfunktionen. Hilbert remained at the University of Königsberg as a Privatdozent from 1886 to 1895, in 1895, as a result of intervention on his behalf by Felix Klein, he obtained the position of Professor of Mathematics at the University of Göttingen. During the Klein and Hilbert years, Göttingen became the preeminent institution in the mathematical world and he remained there for the rest of his life. Among Hilberts students were Hermann Weyl, chess champion Emanuel Lasker, Ernst Zermelo, john von Neumann was his assistant. At the University of Göttingen, Hilbert was surrounded by a circle of some of the most important mathematicians of the 20th century, such as Emmy Noether. Between 1902 and 1939 Hilbert was editor of the Mathematische Annalen, good, he did not have enough imagination to become a mathematician. Hilbert lived to see the Nazis purge many of the prominent faculty members at University of Göttingen in 1933 and those forced out included Hermann Weyl, Emmy Noether and Edmund Landau. One who had to leave Germany, Paul Bernays, had collaborated with Hilbert in mathematical logic and this was a sequel to the Hilbert-Ackermann book Principles of Mathematical Logic from 1928. Hermann Weyls successor was Helmut Hasse, about a year later, Hilbert attended a banquet and was seated next to the new Minister of Education, Bernhard Rust

6.
Grigori Perelman
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Grigori Yakovlevich Perelman is a Russian mathematician. He was the winner of the all-Russian mathematical olympiad and he made a landmark contribution to Riemannian geometry and geometric topology. In 1994, Perelman proved the soul conjecture, in 2003, he proved Thurstons geometrization conjecture. This consequently solved in the affirmative the Poincaré conjecture, on 18 March 2010, it was announced that he had met the criteria to receive the first Clay Millennium Prize for resolution of the Poincaré conjecture. Hamilton, the mathematician who pioneered the Ricci flow with the aim of attacking the conjecture and he also turned down the prestigious prize of the European Mathematical Society. Grigori Perelman was born in Leningrad, Soviet Union on 13 June 1966, grigoris mother Lyubov gave up graduate work in mathematics to raise him. Grigoris mathematical talent became apparent at the age of ten, and his mathematical education continued at the Leningrad Secondary School #239, a specialized school with advanced mathematics and physics programs. Grigori excelled in all subjects except physical education and he continued as a student of School of Mathematics and Mechanics at the Leningrad State University, without admission examinations and enrolled to the university. In the late 1980s and early 1990s, with a recommendation from the celebrated geometer Mikhail Gromov. In 1991 Perelman won the Young Mathematician Prize of the St. Petersburg Mathematical Society for his work on Aleksandrovs spaces of curvature bounded from below. In 1992, he was invited to spend a semester each at the Courant Institute in New York University, from there, he accepted a two-year Miller Research Fellowship at the University of California, Berkeley in 1993. Until late 2002, Perelman was best known for his work in comparison theorems in Riemannian geometry, among his notable achievements was a short and elegant proof of the soul conjecture. The Poincaré conjecture, proposed by French mathematician Henri Poincaré in 1904, was one of key problems in topology, any loop on a 3-sphere—as exemplified by the set of points at a distance of 1 from the origin in four-dimensional Euclidean space—can be contracted into a point. The Poincaré conjecture asserts that any closed three-dimensional manifold such that any loop can be contracted into a point is topologically a 3-sphere, the analogous result has been known to be true in dimensions greater than or equal to five since 1960 as in the work of Stephen Smale. The four-dimensional case resisted longer, finally being solved in 1982 by Michael Freedman, but the case of three-manifolds turned out to be the hardest of them all. Roughly speaking, this is because in topologically manipulating a three-manifold there are too few dimensions to move problematic regions out of the way without interfering with something else, the most fundamental contribution to the three-dimensional case had been produced by Richard S. Hamilton. The role of Perelman was to complete the Hamilton program, Perelman modified Richard S. Hamiltons program for a proof of the conjecture. The central idea is the notion of the Ricci flow, the heat equation describes the behavior of scalar quantities such as temperature

7.
Riemann-hypothese
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In mathematics, the Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1/2. It was proposed by Bernhard Riemann, after whom it is named, the name is also used for some closely related analogues, such as the Riemann hypothesis for curves over finite fields. The Riemann hypothesis implies results about the distribution of prime numbers, along with suitable generalizations, some mathematicians consider it the most important unresolved problem in pure mathematics. The Riemann zeta function ζ is a function whose argument s may be any complex number other than 1 and it has zeros at the negative even integers, that is, ζ =0 when s is one of −2, −4, −6. These are called its trivial zeros, However, the negative even integers are not the only values for which the zeta function is zero. The other ones are called non-trivial zeros, the Riemann hypothesis is concerned with the locations of these non-trivial zeros, and states that, The real part of every non-trivial zero of the Riemann zeta function is 1/2. Thus, if the hypothesis is correct, all the non-trivial zeros lie on the line consisting of the complex numbers 1/2 + i t. There are several books on the Riemann hypothesis, such as Derbyshire, Rockmore. The books Edwards, Patterson, Borwein et al. and Mazur & Stein give mathematical introductions, while Titchmarsh, Ivić, furthermore, the book Open Problems in Mathematics, edited by John Forbes Nash Jr. and Michael Th. Rassias, features an essay on the Riemann hypothesis by Alain Connes. The convergence of the Euler product shows that ζ has no zeros in this region, the Riemann hypothesis discusses zeros outside the region of convergence of this series and Euler product. To make sense of the hypothesis, it is necessary to continue the function to give it a definition that is valid for all complex s. This can be done by expressing it in terms of the Dirichlet eta function as follows. If the real part of s is greater than one, then the function satisfies ζ = ∑ n =1 ∞ n +1 n s =11 s −12 s +13 s − ⋯. However, the series on the right converges not just when the part of s is greater than one. Thus, this alternative series extends the function from Re >1 to the larger domain Re >0. The zeta function can be extended to these values, as well, by taking limits, in the strip 0 < Re <1 the zeta function satisfies the functional equation ζ =2 s π s −1 sin Γ ζ. If s is an even integer then ζ =0 because the factor sin vanishes

8.
Vermoeden van Birch en Swinnerton-Dyer
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In mathematics, the Birch and Swinnerton-Dyer conjecture describes the set of rational solutions to equations defining an elliptic curve. It is a problem in the field of number theory and is widely recognized as one of the most challenging mathematical problems. The conjecture was chosen as one of the seven Millennium Prize Problems listed by the Clay Mathematics Institute and it is named after mathematicians Bryan Birch and Peter Swinnerton-Dyer who developed the conjecture during the first half of the 1960s with the help of machine computation. As of 2016, only cases of the conjecture have been proved. The conjecture relates arithmetic data associated with an elliptic curve E over a number field K to the behaviour of the Hasse–Weil L-function L of E at s =1, mordell proved Mordells theorem, the group of rational points on an elliptic curve has a finite basis. This means that for any elliptic curve there is a finite sub-set of the points on the curve. If the number of points on a curve is infinite then some point in a finite basis must have infinite order. The number of independent basis points with infinite order is called the rank of the curve, if the rank of an elliptic curve is 0, then the curve has only a finite number of rational points. On the other hand, if the rank of the curve is greater than 0, although Mordells theorem shows that the rank of an elliptic curve is always finite, it does not give an effective method for calculating the rank of every curve. The rank of elliptic curves can be calculated using numerical methods. An L-function L can be defined for an elliptic curve E by constructing an Euler product from the number of points on the curve modulo each prime p and this L-function is analogous to the Riemann zeta function and the Dirichlet L-series that is defined for a binary quadratic form. It is a case of a Hasse–Weil L-function. The natural definition of L only converges for values of s in the plane with Re > 3/2. Helmut Hasse conjectured that L could be extended by analytic continuation to the complex plane. This conjecture was first proved by Deuring for elliptic curves with complex multiplication and it was subsequently shown to be true for all elliptic curves over Q, as a consequence of the modularity theorem. Finding rational points on an elliptic curve is a difficult problem. Finding the points on an elliptic curve modulo a prime p is conceptually straightforward. However, for large primes it is computationally intensive, initially this was based on somewhat tenuous trends in graphical plots, this induced a measure of skepticism in J. W. S. Cassels