|
Andrei Okounkov
Andrei Yuryevich Okounkov (russian: Андре́й Ю́рьевич Окунько́в, ''Andrej Okun'kov'') (born July 26, 1969) is a Russian mathematician who works on representation theory and its applications to algebraic geometry, mathematical physics, probability theory and special functions. He is currently a professor at the University of California, Berkeley and the academic supervisor of HSE International Laboratory of Representation Theory and Mathematical Physics. In 2006, he received the Fields Medal "for his contributions to bridging probability, representation theory and algebraic geometry.""Information about Andrei Okounkov, Fields Medal winner" ICM Press Release Education and career He received his doctorate at |
Moscow
Moscow ( , US chiefly ; rus, links=no, Москва, r=Moskva, p=mɐskˈva, a=Москва.ogg) is the capital and largest city of Russia. The city stands on the Moskva River in Central Russia, with a population estimated at 13.0 million residents within the city limits, over 17 million residents in the urban area, and over 21.5 million residents in the metropolitan area. The city covers an area of , while the urban area covers , and the metropolitan area covers over . Moscow is among the world's largest cities; being the most populous city entirely in Europe, the largest urban and metropolitan area in Europe, and the largest city by land area on the European continent. First documented in 1147, Moscow grew to become a prosperous and powerful city that served as the capital of the Grand Duchy that bears its name. When the Grand Duchy of Moscow evolved into the Tsardom of Russia, Moscow remained the political and economic center for most of the Tsardom's history. When th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of the sample space is called an event. Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes (which provide mathematical abstractions of non-deterministic or uncertain processes or measured quantities that may either be single occurrences or evolve over time in a random fashion). Although it is not possible to perfectly predict random events, much can be said about their behavior. Two major results in probability ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
International Congress Of Mathematicians
The International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union (IMU). The Fields Medals, the Nevanlinna Prize (to be renamed as the IMU Abacus Medal), the Carl Friedrich Gauss Prize, Gauss Prize, and the Chern Medal are awarded during the congress's opening ceremony. Each congress is memorialized by a printed set of Proceedings recording academic papers based on invited talks intended to be relevant to current topics of general interest. Being List of International Congresses of Mathematicians Plenary and Invited Speakers, invited to talk at the ICM has been called "the equivalent ... of an induction to a hall of fame". History Felix Klein and Georg Cantor are credited with putting forward the idea of an international congress of mathematicians in the 1890s.A. John Coleman"Mathematics without borders": a book review ''CMS Notes'', vol 31, no. 3, April 1999 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Varieties
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Modern definitions generalize this concept in several different ways, while attempting to preserve the geometric intuition behind the original definition. Conventions regarding the definition of an algebraic variety differ slightly. For example, some definitions require an algebraic variety to be irreducible, which means that it is not the union of two smaller sets that are closed in the Zariski topology. Under this definition, non-irreducible algebraic varieties are called algebraic sets. Other conventions do not require irreducibility. The fundamental theorem of algebra establishes a link between algebra and geometry by showing that a monic polynomial (an algebraic object) in one variable with complex number coefficients is determined ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gromov–Witten Invariant
In mathematics, specifically in symplectic topology and algebraic geometry, Gromov–Witten (GW) invariants are rational numbers that, in certain situations, count pseudoholomorphic curves meeting prescribed conditions in a given symplectic manifold. The GW invariants may be packaged as a homology or cohomology class in an appropriate space, or as the deformed cup product of quantum cohomology. These invariants have been used to distinguish symplectic manifolds that were previously indistinguishable. They also play a crucial role in closed type IIA string theory. They are named after Mikhail Gromov and Edward Witten. The rigorous mathematical definition of Gromov–Witten invariants is lengthy and difficult, so it is treated separately in the stable map article. This article attempts a more intuitive explanation of what the invariants mean, how they are computed, and why they are important. Definition Consider the following: *''X'': a closed symplectic manifold of dimensi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nikita Nekrasov
Nikita Alexandrovich Nekrasov (russian: Ники́та Алекса́ндрович Некра́сов; born 10 April 1973) is a mathematical and theoretical physicist at the Simons Center for Geometry and Physics and C.N.Yang Institute for Theoretical Physics at Stony Brook University in New York, and a Professor of the Russian Academy of Sciences. Career Nekrasov studied at the Moscow State 57th School in 1986–1989. He graduated with honors from the Moscow Institute of Physics and Technology in 1995, and joined the theory division of the Institute for Theoretical and Experimental Physics. In parallel, in 1994–1996 Nekrasov did his graduate work at Princeton University, under the supervision of David Gross. His Ph.D. thesis on ''Four Dimensional Holomorphic Theories'' was defended in 1996. He joined Harvard Society of Fellows at Harvard University as a Junior Fellow 1996–1999. He was then a Robert. H. Dicke Fellow at Princeton University from 1999 to 2000. In 2000 ... [...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 M ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the imaginary numbers. The complex plane allows a geometric interpretation of complex numbers. Under addition, they add like vectors. The multiplication of two complex numbers can be expressed more easily in polar coordinates—the magnitude or ''modulus'' of the product is the product of the two absolute values, or moduli, and the angle or ''argument'' of the product is the sum of the two angles, or arguments. In particular, multiplication by a complex number of modulus 1 acts as a rotation. The complex plane is sometimes known as the Argand plane or Gauss plane. Notational conventions Complex numbers In complex analysis, the complex numbers are customarily represented by the symbol ''z'', which can be separated into its real (''x'') and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert Scheme
In algebraic geometry, a branch of mathematics, a Hilbert scheme is a scheme that is the parameter space for the closed subschemes of some projective space (or a more general projective scheme), refining the Chow variety. The Hilbert scheme is a disjoint union of projective subschemes corresponding to Hilbert polynomials. The basic theory of Hilbert schemes was developed by . Hironaka's example shows that non-projective varieties need not have Hilbert schemes. Hilbert scheme of projective space The Hilbert scheme \mathbf(n) of \mathbb^n classifies closed subschemes of projective space in the following sense: For any locally Noetherian scheme , the set of -valued points :\operatorname(S, \mathbf(n)) of the Hilbert scheme is naturally isomorphic to the set of closed subschemes of \mathbb^n \times S that are flat over . The closed subschemes of \mathbb^n \times S that are flat over can informally be thought of as the families of subschemes of projective space parameterized by . Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Cohomology
In mathematics, specifically in symplectic topology and algebraic geometry, a quantum cohomology ring is an extension of the ordinary cohomology ring of a closed symplectic manifold. It comes in two versions, called small and big; in general, the latter is more complicated and contains more information than the former. In each, the choice of coefficient ring (typically a Novikov ring, described below) significantly affects its structure, as well. While the cup product of ordinary cohomology describes how submanifolds of the manifold intersect each other, the quantum cup product of quantum cohomology describes how subspaces intersect in a "fuzzy", "quantum" way. More precisely, they intersect if they are connected via one or more pseudoholomorphic curves. Gromov–Witten invariants, which count these curves, appear as coefficients in expansions of the quantum cup product. Because it expresses a structure or pattern for Gromov–Witten invariants, quantum cohomology has importan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Plane Partition
In mathematics and especially in combinatorics, a plane partition is a two-dimensional array of nonnegative integers \pi_ (with positive number, positive integer indices ''i'' and ''j'') that is nonincreasing in both indices. This means that : \pi_ \ge \pi_ and \pi_ \ge \pi_ for all ''i'' and ''j''. Moreover, only finitely many of the \pi_ may be nonzero. Plane partitions are a generalization of Partition (number theory), partitions of an integer. A plane partition may be represented visually by the placement of a stack of \pi_ unit cubes above the point (''i'', ''j'') in the plane, giving a three-dimensional solid as shown in the picture. The image has matrix form : \begin 4 & 4 & 3 & 2 & 1\\ 4 & 3 & 1 & 1\\ 3 & 2 & 1\\ 1 \end Plane partitions are also often described by the positions of the unit cubes. From this point of view, a plane partition can be defined as a finite subset \mathcal of positive integer lattice points (''i'', ''j'', ''k'') in \mathbb^3, such that if (''r'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
.svg "Click for more on -> Probability Theory")
