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Alexander Varchenko
Alexander Nikolaevich Varchenko (russian: Александр Николаевич Варченко, born February 6, 1949) is a Soviet and Russian mathematician working in geometry, topology, combinatorics and mathematical physics. Education and career From 1964 to 1966 Varchenko studied at the MoscoKolmogorov boarding school No. 18for gifted high school students, where Andrey Kolmogorov anYa. A. Smorodinskywere lecturing mathematics and physics. Varchenko graduated from Moscow State University in 1971. He was a student of Vladimir Arnold. Varchenko defended his Ph.D. thesis ''Theorems on Topological Equisingularity of Families of Algebraic Sets and Maps'' in 1974 and Doctor of Science thesis ''Asymptotics of Integrals and Algebro-Geometric Invariants of Critical Points of Functions'' in 1982. From 1974 to 1984 he was a research scientist at the Moscow State University, in 1985–1990 a professor at the Gubkin Institute of Gas and Oil, and since 1991 he has been the Ernest Eliel Pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Russia
Russia (, , ), or the Russian Federation, is a List of transcontinental countries, transcontinental country spanning Eastern Europe and North Asia, Northern Asia. It is the List of countries and dependencies by area, largest country in the world, with its internationally recognised territory covering , and encompassing one-eighth of Earth's inhabitable landmass. Russia extends across Time in Russia, eleven time zones and shares Borders of Russia, land boundaries with fourteen countries, more than List of countries and territories by land borders, any other country but China. It is the List of countries and dependencies by population, world's ninth-most populous country and List of European countries by population, Europe's most populous country, with a population of 146 million people. The country's capital and List of cities and towns in Russia by population, largest city is Moscow, the List of European cities by population within city limits, largest city entirely within E ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Newton Polygon
In mathematics, the Newton polygon is a tool for understanding the behaviour of polynomials over local fields, or more generally, over ultrametric fields. In the original case, the local field of interest was ''essentially'' the field of formal Laurent series in the indeterminate ''X'', i.e. the field of fractions of the formal power series ring K X, over K, where K was the real number or complex number field. This is still of considerable utility with respect to Puiseux expansions. The Newton polygon is an effective device for understanding the leading terms aX^r of the power series expansion solutions to equations P(F(X)) = 0 where P is a polynomial with coefficients in K /math>, the polynomial ring; that is, implicitly defined algebraic functions. The exponents r here are certain rational numbers, depending on the branch chosen; and the solutions themselves are power series in K Y with Y = X^ for a denominator d corresponding to the branch. The Newton polygon gives an effective, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Algebraic Geometry
In mathematics, real algebraic geometry is the sub-branch of algebraic geometry studying real algebraic sets, i.e. real-number solutions to algebraic equations with real-number coefficients, and mappings between them (in particular real polynomial mappings). Semialgebraic geometry is the study of semialgebraic sets, i.e. real-number solutions to algebraic inequalities with-real number coefficients, and mappings between them. The most natural mappings between semialgebraic sets are semialgebraic mappings, i.e., mappings whose graphs are semialgebraic sets. Terminology Nowadays the words 'semialgebraic geometry' and 'real algebraic geometry' are used as synonyms, because real algebraic sets cannot be studied seriously without the use of semialgebraic sets. For example, a projection of a real algebraic set along a coordinate axis need not be a real algebraic set, but it is always a semialgebraic set: this is the Tarski–Seidenberg theorem. Related fields are o-minimal theory and r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boris Shapiro (mathematician)
Boris Shapiro (born 1957, Moscow, Soviet Union) is a Russian-Swedish mathematician, whose research concerns differential equations, commutative algebra and Schubert calculus. The Shapiro–Shapiro conjecture (or simply the Shapiro conjecture) was named after Michael Shapiro and him (it is now the well-known Mukhin–Tarasov– Varchenko theorem). Shapiro enrolled in the Ph.D. program at Moscow State University, Soviet Union in 1985 as a student of Vladimir Arnold, but his thesis defense was rejected by the examining committee. He then defended the same thesis at Stockholm University, Sweden in 1990, and was awarded his Ph.D. Ironically, he became the most prolific Ph.D. student of Arnold, in terms of academic descendance. He has been a professor at Stockholm University since 1993.According to Google Scholar, as of 21 August 2019, Shapiro's works have been cited 1638 times, and his h-index is 20: https://scholar.google.se/citations?user=V2gZ4SsAAAAJ Selected papers *A. Postni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annals Of Mathematics
The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as the founding editor-in-chief. It was "intended to afford a medium for the presentation and analysis of any and all questions of interest or importance in pure and applied Mathematics, embracing especially all new and interesting discoveries in theoretical and practical astronomy, mechanical philosophy, and engineering". It was published in Des Moines, Iowa, and was the earliest American mathematics journal to be published continuously for more than a year or two. This incarnation of the journal ceased publication after its tenth year, in 1883, giving as an explanation Hendricks' declining health, but Hendricks made arrangements to have it taken over by new management, and it was continued from March 1884 as the ''Annals of Mathematics''. The n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pavel Etingof
Pavel Ilyich Etingof (russian: Павел Ильич Этингоф; born 1969) is an American mathematician of Russian-Ukrainian origin. Biography Etingof was born in Kyiv, Ukrainian SSR, and studied in the Kyiv Natural Science Lyceum No. 145 in 1981–1984, and at the Department of Mathematics and Mechanics of the Taras Shevchenko National University of Kyiv in 1984–1986. He received his M.S. in applied mathematics from the Oil and Gas Institute in Moscow in 1989 and then went to the US in 1990. In 1994, he received his PhD in mathematics at Yale University under Igor Frenkel with thesis ''Representation Theory and Holonomic Systems''. After his PhD, he became Benjamin Peirce Assistant Professor at Harvard University and in 1998 an assistant professor at MIT. Since 2005 he is a professor at MIT. He is married to Tanya Javits-Etingof (1997–present) and has two daughters; Miriam (1998) and Ariela (2004). Etingof does research on the intersection of mathematical physics (exa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 |
Quantum KZ Equations
In mathematical physics, the quantum KZ equations or quantum Knizhnik–Zamolodchikov equations or qKZ equations are the analogue for quantum affine algebras of the Knizhnik–Zamolodchikov equations for affine Kac–Moody algebras. They are a consistent system of difference equations satisfied by the ''N''-point functions, the vacuum expectations of products of primary fields. In the limit as the deformation parameter ''q'' approaches 1, the ''N''-point functions of the quantum affine algebra tend to those of the affine Kac–Moody algebra and the difference equations become partial differential equations. The quantum KZ equations have been used to study exactly solved models in quantum statistical mechanics. See also *Quantum affine algebras *Yang–Baxter equation *Quantum group *Affine Hecke algebra *Kac–Moody algebra *Two-dimensional conformal field theory A two-dimensional conformal field theory is a quantum field theory on a Euclidean two-dimensional space, that is invar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gauss–Manin Connection
In mathematics, the Gauss–Manin connection is a connection on a certain vector bundle over a base space ''S'' of a family of algebraic varieties V_s. The fibers of the vector bundle are the de Rham cohomology groups H^k_(V_s) of the fibers V_s of the family. It was introduced by for curves ''S'' and by in higher dimensions. Flat sections of the bundle are described by differential equations; the best-known of these is the Picard–Fuchs equation, which arises when the family of varieties is taken to be the family of elliptic curves. In intuitive terms, when the family is locally trivial, cohomology classes can be moved from one fiber in the family to nearby fibers, providing the 'flat section' concept in purely topological terms. The existence of the connection is to be inferred from the flat sections. Intuition Consider a smooth morphism of schemes X\to B over characteristic 0. If we consider these spaces as complex analytic spaces, then the Ehresmann fibration theorem te ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Knizhnik–Zamolodchikov Equations
In mathematical physics the Knizhnik–Zamolodchikov equations, or KZ equations, are linear differential equations satisfied by the correlation functions (on the Riemann sphere) of two-dimensional conformal field theories associated with an affine Lie algebra at a fixed level. They form a system of complex partial differential equations with regular singular points satisfied by the ''N''-point functions of affine primary fields and can be derived using either the formalism of Lie algebras or that of vertex algebras. The structure of the genus-zero part of the conformal field theory is encoded in the monodromy properties of these equations. In particular, the braiding and fusion of the primary fields (or their associated representations) can be deduced from the properties of the four-point functions, for which the equations reduce to a single matrix-valued first-order complex ordinary differential equation of Fuchsian type. Originally the Russian physicists Vadim Knizhnik and A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Limit Cycle
In mathematics, in the study of dynamical systems with two-dimensional phase space, a limit cycle is a closed trajectory in phase space having the property that at least one other trajectory spirals into it either as time approaches infinity or as time approaches negative infinity. Such behavior is exhibited in some nonlinear systems. Limit cycles have been used to model the behavior of a great many real-world oscillatory systems. The study of limit cycles was initiated by Henri Poincaré (1854–1912). Definition We consider a two-dimensional dynamical system of the form x'(t)=V(x(t)) where V : \R^2 \to \R^2 is a smooth function. A ''trajectory'' of this system is some smooth function x(t) with values in \mathbb^2 which satisfies this differential equation. Such a trajectory is called ''closed'' (or ''periodic'') if it is not constant but returns to its starting point, i.e. if there exists some t_0>0 such that x(t + t_0) = x(t) for all t \in \R. An orbit is the image of a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert's Sixteenth Problem
Hilbert's 16th problem was posed by David Hilbert at the Paris conference of the International Congress of Mathematicians in 1900, as part of his list of 23 problems in mathematics. The original problem was posed as the ''Problem of the topology of algebraic curves and surfaces'' (''Problem der Topologie algebraischer Kurven und Flächen''). Actually the problem consists of two similar problems in different branches of mathematics: * An investigation of the relative positions of the branches of real algebraic curves of degree ''n'' (and similarly for algebraic surfaces). * The determination of the upper bound for the number of limit cycles in two-dimensional polynomial vector fields of degree ''n'' and an investigation of their relative positions. The first problem is yet unsolved for ''n'' = 8. Therefore, this problem is what usually is meant when talking about Hilbert's sixteenth problem in real algebraic geometry. The second problem also remains unsolved: no upp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
