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Clay Research Award
__NOTOC__ The Clay Research Award is an annual award given by the Oxford-based Clay Mathematics Institute to mathematicians to recognize their achievement in mathematical research. The following mathematicians have received the award: {, class="wikitable sortable" , - ! Year !! Winner !! Citation , - , 2022 , , Søren Galatius and Oscar Randal-Williams John Pardon , , "for their profound contributions to the understanding of high dimensional manifolds and their diffeomorphism groups; they have transformed and reinvigorated the subject." "in recognition of his wide-ranging and transformative work in geometry and topology, particularly his groundbreaking achievements in symplectic topology." , - , 2021 , , Bhargav Bhatt , , "For his groundbreaking achievements in commutative algebra, arithmetic algebraic geometry, and topology in the p-adic setting." , - , 2020 , , not awarded , - , 2019 , , Wei Zhang Tristan Buckmaster, Philip Isett and Vlad Vicol , , "In recog ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting poin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bernd Siebert
Bernd Siebert (born 5 March 1964 in Berlin-Wilmersdorf) is a German mathematician who researches in algebraic geometry. Life Siebert studied mathematics starting 1984 at the University of Erlangen. In 1986, he changed to the University of Bonn and in 1987 to the University of Göttingen where he finished his Diplom in 1989 under the supervision of Hans Grauert with distinction. He became Grauert's PhD student and assistant in Göttingen. He received his PhD in 1992 (''Faserzykelräume, geometrische Plattifikation und meromorphe Äquivalenzrelationen''). A stay at the Courant Institute followed in 1993–94 after which we went to Bochum. In 1997–98 he spent some time at the MIT as a visiting scholar before completing his habilitation in Bochum in 1998 (''Gromov–Witten invariants for general symplectic manifolds''). As a DFG-Heisenberg Fellow, he went to the Universität Paris VI/ Universität Paris VII from 2000 to 2002. From there, he was called to a professorship at the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rahul Pandharipande
Rahul Pandharipande (born 1969) is a mathematician who is currently a professor of mathematics at the Swiss Federal Institute of Technology Zürich (ETH) working in algebraic geometry. His particular interests concern moduli spaces, enumerative invariants associated to moduli spaces, such as Gromov–Witten invariants and Donaldson–Thomas invariants, and the cohomology of the moduli space of curves. His father Vijay Raghunath Pandharipande was a renowned theoretical physicist who worked in the area of nuclear physics. Educational and professional history He received his A.B. from Princeton University in 1990 and his PhD from Harvard University in 1994 with a thesis entitled `''A Compactification over the Moduli Space of Stable Curves of the Universal Moduli Space of Slope-Semistable Vector Bundles. His thesis advisor at Harvard was Joe Harris. After teaching at the University of Chicago and the California Institute of Technology, he joined the faculty as Professor of Mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arithmetic Algebraic Geometry
In mathematics, arithmetic geometry is roughly the application of techniques from algebraic geometry to problems in number theory. Arithmetic geometry is centered around Diophantine geometry, the study of rational points of algebraic varieties. In more abstract terms, arithmetic geometry can be defined as the study of schemes of finite type over the spectrum of the ring of integers. Overview The classical objects of interest in arithmetic geometry are rational points: sets of solutions of a system of polynomial equations over number fields, finite fields, p-adic fields, or function fields, i.e. fields that are not algebraically closed excluding the real numbers. Rational points can be directly characterized by height functions which measure their arithmetic complexity. The structure of algebraic varieties defined over non-algebraically closed fields has become a central area of interest that arose with the modern abstract development of algebraic geometry. Over finite fiel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ratner's Theorems
In mathematics, Ratner's theorems are a group of major theorems in ergodic theory concerning unipotent flows on homogeneous spaces proved by Marina Ratner around 1990. The theorems grew out of Ratner's earlier work on horocycle flows. The study of the dynamics of unipotent flows played a decisive role in the proof of the Oppenheim conjecture by Grigory Margulis. Ratner's theorems have guided key advances in the understanding of the dynamics of unipotent flows. Their later generalizations provide ways to both sharpen the results and extend the theory to the setting of arbitrary semisimple algebraic groups over a local field. Short description The Ratner orbit closure theorem asserts that the closures of orbits of unipotent flows on the quotient of a Lie group by a lattice are nice, geometric subsets. The Ratner equidistribution theorem further asserts that each such orbit is equidistributed in its closure. The Ratner measure classification theorem is the weaker statement that eve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ergodic Theory
Ergodic theory (Greek: ' "work", ' "way") is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, statistical properties means properties which are expressed through the behavior of time averages of various functions along trajectories of dynamical systems. The notion of deterministic dynamical systems assumes that the equations determining the dynamics do not contain any random perturbations, noise, etc. Thus, the statistics with which we are concerned are properties of the dynamics. Ergodic theory, like probability theory, is based on general notions of measure theory. Its initial development was motivated by problems of statistical physics. A central concern of ergodic theory is the behavior of a dynamical system when it is allowed to run for a long time. The first result in this direction is the Poincaré recurrence theorem, which claims that almost all points in any subset of the ph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peter Scholze
Peter Scholze (; born 11 December 1987) is a German mathematician known for his work in arithmetic geometry. He has been a professor at the University of Bonn since 2012 and director at the Max Planck Institute for Mathematics since 2018. He has been called one of the leading mathematicians in the world. He won the Fields Medal in 2018, which is regarded as the highest professional honor in mathematics. Early life and education Scholze was born in Dresden and grew up in Berlin. His father is a physicist, his mother a computer scientist, and his sister studied chemistry. He attended the in Berlin-Friedrichshain, a gymnasium devoted to mathematics and science. As a student, Scholze participated in the International Mathematical Olympiad, winning three gold medals and one silver medal. He studied at the University of Bonn and completed his bachelor's degree in three semesters and his master's degree in two further semesters. He obtained his Ph.D. in 2012 under the supervisi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maryam Mirzakhani
Maryam Mirzakhani ( fa, مریم میرزاخانی, ; 12 May 1977 – 14 July 2017) was an Iranian mathematician and a professor of mathematics at Stanford University. Her research topics included Teichmüller space, Teichmüller theory, hyperbolic geometry, ergodic theory, and symplectic geometry. In 2005, as a result of her research, she was honored in ''Popular Science's'' fourth annual "Brilliant 10" in which she was acknowledged as one of the top 10 young minds who have pushed their fields in innovative directions. On 13 August 2014, Mirzakhani was honored with the Fields Medal, the most prestigious award in mathematics, becoming the first Iranian to be honored with the award and the first of only two women to date. The award committee cited her work in "the dynamics and geometry of Riemann surfaces and their moduli spaces". On 14 July 2017, Mirzakhani died of breast cancer at the age of 40. Early life and education Mirzakhani was born on 12 May 1977 in Tehran, Ir ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Incidence Geometry
In mathematics, incidence geometry is the study of incidence structures. A geometric structure such as the Euclidean plane is a complicated object that involves concepts such as length, angles, continuity, betweenness, and incidence. An ''incidence structure'' is what is obtained when all other concepts are removed and all that remains is the data about which points lie on which lines. Even with this severe limitation, theorems can be proved and interesting facts emerge concerning this structure. Such fundamental results remain valid when additional concepts are added to form a richer geometry. It sometimes happens that authors blur the distinction between a study and the objects of that study, so it is not surprising to find that some authors refer to incidence structures as incidence geometries. Incidence structures arise naturally and have been studied in various areas of mathematics. Consequently, there are different terminologies to describe these objects. In graph theory th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Erdős Distinct Distances Problem
In discrete geometry, the Erdős distinct distances problem states that every set of points in the plane has a nearly-linear number of distinct distances. It was posed by Paul Erdős in 1946 and almost proven by Larry Guth and Nets Katz in 2015. The conjecture In what follows let denote the minimal number of distinct distances between points in the plane, or equivalently the smallest possible cardinality of their distance set. In his 1946 paper, Erdős proved the estimates :\sqrt-1/2\leq g(n)\leq c n/\sqrt for some constant c. The lower bound was given by an easy argument. The upper bound is given by a \sqrt\times\sqrt square grid. For such a grid, there are O( n/\sqrt) numbers below ''n'' which are sums of two squares, expressed in big O notation; see Landau–Ramanujan constant. Erdős conjectured that the upper bound was closer to the true value of ''g''(''n''), and specifically that (using big Omega notation) g(n) = \Omega(n^c) holds for every . Partial results Paul Erd� ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nets Katz
Nets Hawk Katz is the IBM Professor of Mathematics at the California Institute of Technology. He was a professor of Mathematics at Indiana University Bloomington Indiana University Bloomington (IU Bloomington, Indiana University, IU, or simply Indiana) is a public research university in Bloomington, Indiana. It is the flagship campus of Indiana University and, with over 40,000 students, its largest camp ... until March 2013. Katz earned a B.A. in mathematics from Rice University in 1990 at the age of 17. He received his Ph.D. in 1993 under Dennis DeTurck at the University of Pennsylvania, with a dissertation titled "Noncommutative Determinants and Applications". He is the author of several important results in combinatorics (especially additive combinatorics), harmonic analysis and other areas. In 2003, jointly with Jean Bourgain and Terence Tao, he proved that any subset of \Z/p\Z grows substantially under either addition or multiplication. More precisely, if A is a set such ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Larry Guth
Lawrence David Guth (born 1977) is a professor of mathematics at the Massachusetts Institute of Technology. Education and career Guth graduated from Yale in 2000, with BS in mathematics. In 2005, he got his PhD in mathematics from the Massachusetts Institute of Technology, where he studied geometry of objects with random shapes under the supervision of Tomasz Mrowka. After MIT, Guth went to Stanford as a postdoc, and later to the University of Toronto as an Assistant Professor. In 2011, New York University's Courant Institute of Mathematical Sciences hired Guth as a professor, listing his areas of interest as "metric geometry, harmonic analysis, and geometric combinatorics." In 2012, Guth moved to MIT, where he is Claude Shannon Professor of Mathematics. Research In his research, Guth has strengthened Gromov's systolic inequality for essential manifolds and, along with Nets Katz, found a solution to the Erdős distinct distances problem. His wide-ranging interests include ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
