|
Fukaya Category
In symplectic topology, a Fukaya category of a symplectic manifold (M, \omega) is a category \mathcal F (M) whose objects are Lagrangian submanifolds of M, and morphisms are Floer chain groups: \mathrm (L_0, L_1) = FC (L_0,L_1). Its finer structure can be described in the language of quasi categories as an ''A''∞-category. They are named after Kenji Fukaya who introduced the A_\infty language first in the context of Morse homology, and exist in a number of variants. As Fukaya categories are ''A''∞-categories, they have associated derived categories, which are the subject of the celebrated homological mirror symmetry conjecture of Maxim Kontsevich.Kontsevich, Maxim, ''Homological algebra of mirror symmetry'', Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), 120–139, Birkhäuser, Basel, 1995. This conjecture has been computationally verified for a number of comparatively simple examples. Formal definition Let (X, \omega) be a sy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symplectic Topology
Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form. Symplectic geometry has its origins in the Hamiltonian formulation of classical mechanics where the phase space of certain classical systems takes on the structure of a symplectic manifold. The term "symplectic", introduced by Weyl, is a calque of "complex"; previously, the "symplectic group" had been called the "line complex group". "Complex" comes from the Latin ''com-plexus'', meaning "braided together" (co- + plexus), while symplectic comes from the corresponding Greek ''sym-plektikos'' (συμπλεκτικός); in both cases the stem comes from the Indo-European root *pleḱ- The name reflects the deep connections between complex and symplectic structures. By Darboux's Theorem, symplectic manifolds are isomorphic to the standard symplectic vector space locally, hence only ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homological Mirror Symmetry
Homological mirror symmetry is a mathematical conjecture made by Maxim Kontsevich. It seeks a systematic mathematical explanation for a phenomenon called mirror symmetry first observed by physicists studying string theory. History In an address to the 1994 International Congress of Mathematicians in Zürich, speculated that mirror symmetry for a pair of Calabi–Yau manifolds ''X'' and ''Y'' could be explained as an equivalence of a triangulated category constructed from the algebraic geometry of ''X'' (the derived category of coherent sheaves on ''X'') and another triangulated category constructed from the symplectic geometry of ''Y'' (the derived Fukaya category). Edward Witten originally described the topological twisting of the N=(2,2) supersymmetric field theory into what he called the A and B model topological string theories. These models concern maps from Riemann surfaces into a fixed target—usually a Calabi–Yau manifold. Most of the mathematical predictions of mi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
MathOverflow
MathOverflow is a mathematics question-and-answer (Q&A) website, which serves as an online community of mathematicians. It allows users to ask questions, submit answers, and rate both, all while getting merit points for their activities. It is a part of the Stack Exchange Network. It is primarily for asking questions on mathematics research – i.e. related to unsolved problems and the extension of knowledge of mathematics into areas that are not yet known – and does not welcome requests from non-mathematicians for instruction, for example homework exercises. It does welcome various questions on other topics that might normally be discussed among mathematicians, for example about publishing, refereeing, advising, getting tenure, etc. It is generally inhospitable to questions perceived as tendentious or argumentative. Origin and history The website was started by Berkeley graduate students and postdocs Anton Geraschenko, David Zureick-Brown, and Scott Morrison on 28 Septe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs. The society is one of the four parts of the Joint Policy Board for Mathematics and a member of the Conference Board of the Mathematical Sciences. History The AMS was founded in 1888 as the New York Mathematical Society, the brainchild of Thomas Fiske, who was impressed by the London Mathematical Society on a visit to England. John Howard Van Amringe was the first president and Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance, due to concerns about competing with the American Journal of Mathematics. The result was the ''Bulletin of the American Mathematical Society'', with Fiske as editor-in-chief. The de facto journal, as intended, was influential in in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paul Seidel
Paul Seidel (born December 30, 1970) is a Swiss-Italian mathematician. He is a faculty member at the Massachusetts Institute of Technology. Career Seidel attended Heidelberg University, where he received his Diplom under supervision of Albrecht Dold in 1994. He then pursued his Ph.D. studies at the University of Oxford under supervision of Simon Donaldson (Thesis: ''Floer Homology and the Symplectic Isotopy Problem'') in 1998. He was a chargé de recherche at the CNRS from 1999 to 2002, a professor at Imperial College London from 2002 to 2003, a professor at the University of Chicago from 2003 to 2007, and then a professor at the Massachusetts Institute of Technology from 2007 onwards. Awards In 2000, Seidel was awarded the EMS Prize. In 2010, he was awarded the Oswald Veblen Prize in Geometry "for his fundamental contributions to symplectic geometry and, in particular, for his development of advanced algebraic methods for computation of symplectic invariants." In 2012, he becam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Denis Auroux
Denis Auroux (born April 1977 in Lyon) is a French mathematician working in geometry and topology. Education and career Auroux was admitted in 1993 to the École normale supérieure. In 1994, he received a licentiate and ''maîtrise'' in mathematics from Paris Diderot University (Paris 7). In 1995, he received a licentiate in physics from Pierre and Marie Curie University (Paris 6) and passed the ''agrégation''. In 1995, he received a master's degree in mathematics from Paris-Sud University with a thesis on Seiberg-Witten invariants of symplectic manifolds. In 1999, he received his doctorate from the École polytechnique with supervisors Jean-Pierre Bourguignon and Mikhael Gromov for a thesis on structure theorems for compact symplectic manifolds via almost-complex techniques. In 2003, he completed his habilitation at Paris-Sud University with a thesis on approximately holomorphic techniques and monodromy invariants in symplectic topology. As a postdoc, he was a Moore Instruc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopy Associative Algebra
In mathematics, an algebra such as (\R,+,\cdot) has multiplication \cdot whose associativity is well-defined on the nose. This means for any real numbers a,b,c\in \R we have :a\cdot(b\cdot c) - (a\cdot b)\cdot c = 0. But, there are algebras R which are not necessarily associative, meaning if a,b,c\in R then :a\cdot(b\cdot c) - (a\cdot b)\cdot c \neq 0 in general. There is a notion of algebras, called A_\infty-algebras, which still have a property on the multiplication which still acts like the first relation, meaning associativity holds, but only holds up to a homotopy, which is a way to say after an operation "compressing" the information in the algebra, the multiplication is associative. This means although we get something which looks like the second equation, the one of inequality, we actually get equality after "compressing" the information in the algebra. The study of A_\infty-algebras is a subset of homotopical algebra, where there is a homotopical notion of associative algeb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Almost Complex Structure
In mathematics, an almost complex manifold is a smooth manifold equipped with a smooth linear complex structure on each tangent space. Every complex manifold is an almost complex manifold, but there are almost complex manifolds that are not complex manifolds. Almost complex structures have important applications in symplectic geometry. The concept is due to Charles Ehresmann and Heinz Hopf in the 1940s. Formal definition Let ''M'' be a smooth manifold. An almost complex structure ''J'' on ''M'' is a linear complex structure (that is, a linear map which squares to −1) on each tangent space of the manifold, which varies smoothly on the manifold. In other words, we have a smooth tensor field ''J'' of degree such that J^2=-1 when regarded as a vector bundle isomorphism J\colon TM\to TM on the tangent bundle. A manifold equipped with an almost complex structure is called an almost complex manifold. If ''M'' admits an almost complex structure, it must be even-dimensional. This can ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maxim Kontsevich
Maxim Lvovich Kontsevich (russian: Макси́м Льво́вич Конце́вич, ; born 25 August 1964) is a Russian and French mathematician and mathematical physicist. He is a professor at the Institut des Hautes Études Scientifiques and a distinguished professor at the University of Miami. He received the Henri Poincaré Prize in 1997, the Fields Medal in 1998, the Crafoord Prize in 2008, the Shaw Prize and Fundamental Physics Prize in 2012, and the Breakthrough Prize in Mathematics in 2014. Academic career and research He was born into the family of Lev Kontsevich, Soviet orientalist and author of the Kontsevich system. After ranking second in the All-Union Mathematics Olympiads, he attended Moscow State University but left without a degree in 1985 to become a researcher at the Institute for Information Transmission Problems in Moscow. While at the institute he published papers that caught the interest of the Max Planck Institute in Bonn and was invited for three ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Derived Categories
In mathematics, the derived category ''D''(''A'') of an abelian category ''A'' is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on ''A''. The construction proceeds on the basis that the objects of ''D''(''A'') should be chain complexes in ''A'', with two such chain complexes considered isomorphic when there is a chain map that induces an isomorphism on the level of homology of the chain complexes. Derived functors can then be defined for chain complexes, refining the concept of hypercohomology. The definitions lead to a significant simplification of formulas otherwise described (not completely faithfully) by complicated spectral sequences. The development of the derived category, by Alexander Grothendieck and his student Jean-Louis Verdier shortly after 1960, now appears as one terminal point in the explosive development of homological algebra in the 1950s, a decade in which it had made remarkab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symplectic Manifold
In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, M , equipped with a closed nondegenerate differential 2-form \omega , called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology. Symplectic manifolds arise naturally in abstract formulations of classical mechanics and analytical mechanics as the cotangent bundles of manifolds. For example, in the Hamiltonian formulation of classical mechanics, which provides one of the major motivations for the field, the set of all possible configurations of a system is modeled as a manifold, and this manifold's cotangent bundle describes the phase space of the system. Motivation Symplectic manifolds arise from classical mechanics; in particular, they are a generalization of the phase space of a closed system. In the same way the Hamilton equations allow one to derive the time evolution of a system from a set of differential equations, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Morse Homology
In mathematics, specifically in the field of differential topology, Morse homology is a homology theory defined for any smooth manifold. It is constructed using the smooth structure and an auxiliary metric on the manifold, but turns out to be topologically invariant, and is in fact isomorphic to singular homology. Morse homology also serves as a model for the various infinite-dimensional generalizations known as Floer homology theories. Formal definition Given any (compact) smooth manifold, let ''f'' be a Morse function and ''g'' a Riemannian metric on the manifold. (These are auxiliary; in the end, the Morse homology depends on neither.) The pair (f, g) gives us a gradient vector field. We say that (f, g) is Morse–Smale if the stable and unstable manifolds associated to all of the critical points of ''f'' intersect each other transversely. For any such pair (f, g), it can be shown that the difference in index between any two critical points is equal to the dimension of th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
