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Arnold–Givental Conjecture
The Arnold–Givental conjecture, named after Vladimir Arnold and Alexander Givental, is a statement on Lagrangian submanifolds. It gives a lower bound in terms of the Betti numbers of a Lagrangian submanifold on the number of intersection points of with another Lagrangian submanifold which is obtained from by Hamiltonian isotopy, and which intersects transversally. Statement Let (M, \omega) be a compact 2n-dimensional symplectic manifold. An anti-symplectic involution is a diffeomorphism \tau: M \to M such that \tau^* \omega = -\omega. The fixed point set L \subset M of \tau is necessarily a Lagrangian submanifold. Let H_t\in C^\infty(M), 0 \leq t \leq 1 be a smooth family of Hamiltonian functions on M which generates a 1-parameter family of Hamiltonian diffeomorphisms \varphi_t: M \to M. The Arnold–Givental conjecture says, suppose \varphi_1(L) intersects transversely with L, then \# (\varphi_1(L) \cap L) \geq \sum_^n H_*(L; _2). Status The Arnold–Givental conjecture ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vladimir Arnold
Vladimir Igorevich Arnold (alternative spelling Arnol'd, russian: link=no, Влади́мир И́горевич Арно́льд, 12 June 1937 – 3 June 2010) was a Soviet and Russian mathematician. While he is best known for the Kolmogorov–Arnold–Moser theorem regarding the stability of integrable systems, he made important contributions in several areas including dynamical systems theory, algebra, catastrophe theory, topology, algebraic geometry, symplectic geometry, differential equations, classical mechanics, hydrodynamics and singularity theory, including posing the ADE classification problem, since his first main result—the solution of Hilbert's thirteenth problem in 1957 at the age of 19. He co-founded two new branches of mathematics—KAM theory, and topological Galois theory (this, with his student Askold Khovanskii). Arnold was also known as a popularizer of mathematics. Through his lectures, seminars, and as the author of several textbooks (such as the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alexander Givental
Alexander Givental (russian: Александр Борисович Гивенталь) is a Russian-American mathematician working in symplectic topology and singularity theory, as well as their relation to topological string theories. He graduated from Moscow Phys-Math school number 2 (later renamed into Lyceum ) and then the Gubkin Russian State University of Oil and Gas, and he finally his Ph.D. under the supervision of V. I. Arnold in 1987. He emigrated to the USA in 1990. He provided the first proof of the mirror conjecture for Calabi–Yau manifolds that are complete intersections in toric ambient spaces, in particular for quintic hypersurfaces in P4. He is now Professor of Mathematics at the University of California, Berkeley. As an extracurricular activity, he translates Russian poetry into English and publishes books, including his own translation of a textbook () in geometry by Andrey Kiselyov and poetry of Marina Tsvetaeva Marina Ivanovna Tsvetaeva (russian: � ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lagrangian Submanifold
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, t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Betti Number
In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of ''n''-dimensional simplicial complexes. For the most reasonable finite-dimensional spaces (such as compact manifolds, finite simplicial complexes or CW complexes), the sequence of Betti numbers is 0 from some point onward (Betti numbers vanish above the dimension of a space), and they are all finite. The ''n''th Betti number represents the rank of the ''n''th homology group, denoted ''H''''n'', which tells us the maximum number of cuts that can be made before separating a surface into two pieces or 0-cycles, 1-cycles, etc. For example, if H_n(X) \cong 0 then b_n(X) = 0, if H_n(X) \cong \mathbb then b_n(X) = 1, if H_n(X) \cong \mathbb \oplus \mathbb then b_n(X) = 2, if H_n(X) \cong \mathbb \oplus \mathbb\oplus \mathbb then b_n(X) = 3, etc. Note that only the ranks of infinite groups are considered, so for example if H_n(X) \cong \mathbb^k \oplus \mathbb/(2) , where \ ... [...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 equatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diffeomorphism
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable. Definition Given two manifolds M and N, a differentiable map f \colon M \rightarrow N is called a diffeomorphism if it is a bijection and its inverse f^ \colon N \rightarrow M is differentiable as well. If these functions are r times continuously differentiable, f is called a C^r-diffeomorphism. Two manifolds M and N are diffeomorphic (usually denoted M \simeq N) if there is a diffeomorphism f from M to N. They are C^r-diffeomorphic if there is an r times continuously differentiable bijective map between them whose inverse is also r times continuously differentiable. Diffeomorphisms of subsets of manifolds Given a subset X of a manifold M and a subset Y of a manifold N, a function f:X\to Y is said to be smooth if for all p in X there is a neighb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamiltonian Function
Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momenta''. Both theories provide interpretations of classical mechanics and describe the same physical phenomena. Hamiltonian mechanics has a close relationship with geometry (notably, symplectic geometry and Poisson structures) and serves as a link between classical and quantum mechanics. Overview Phase space coordinates (p,q) and Hamiltonian H Let (M, \mathcal L) be a mechanical system with the configuration space M and the smooth Lagrangian \mathcal L. Select a standard coordinate system (\boldsymbol,\boldsymbol) on M. The quantities \textstyle p_i(\boldsymbol,\boldsymbol,t) ~\stackrel~ / are called ''momenta''. (Also ''generalized momenta'', ''conjugate momenta'', and ''canonical momenta''). For a time instant t, the Legendre transfor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Givental
Alexander Givental (russian: Александр Борисович Гивенталь) is a Russian-American mathematician working in symplectic topology and singularity theory, as well as their relation to topological string theories. He graduated from Moscow Phys-Math school number 2 (later renamed into Lyceum ) and then the Gubkin Russian State University of Oil and Gas, and he finally his Ph.D. under the supervision of V. I. Arnold in 1987. He emigrated to the USA in 1990. He provided the first proof of the mirror conjecture for Calabi–Yau manifold In algebraic geometry, a Calabi–Yau manifold, also known as a Calabi–Yau space, is a particular type of manifold which has properties, such as Ricci flatness, yielding applications in theoretical physics. Particularly in superstring ...s that are complete intersections in toric ambient spaces, in particular for quintic equation, quintic hypersurfaces in P4. He is now Professor of Mathematics at the University of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kenji Fukaya
Kenji Fukaya (Japanese: 深谷賢治, ''Fukaya Kenji'') is a Japanese mathematician known for his work in symplectic geometry and Riemannian geometry. His many fundamental contributions to mathematics include the discovery of the Fukaya category. He is a permanent faculty member at the Simons Center for Geometry and Physics and a professor of mathematics at Stony Brook University. Biography Fukaya was both an undergraduate and a graduate student in mathematics at the University of Tokyo, receiving his BA in 1981, and his PhD in 1986. In 1987, he joined the University of Tokyo faculty as an associate professor. He then moved to Kyoto University as a full professor in 1994. In 2013, he then moved to the United States in order to join the faculty of the Simons Center for Geometry and Physics at Stony Brook. The Fukaya category, meaning the A_\infty category of whose objects are Lagrangian submanifolds of a given symplectic manifold, is named after him, and is intimately rel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arnold Conjecture
The Arnold conjecture, named after mathematician Vladimir Arnold, is a mathematical conjecture in the field of symplectic geometry, a branch of differential geometry. Statement Let (M, \omega) be a compact symplectic manifold. For any smooth function H: M \to , the symplectic form \omega induces a Hamiltonian vector field X_H on M, defined by the identity \omega( X_H, \cdot) = dH. The function H is called a Hamiltonian function. Suppose there is a 1-parameter family of Hamiltonian functions H_t: M \to , 0 \leq t \leq 1, inducing a 1-parameter family of Hamiltonian vector fields X_ on M. The family of vector fields integrates to a 1-parameter family of diffeomorphisms \varphi_t: M \to M. Each individual of \varphi_t is a Hamiltonian diffeomorphism of M. The Arnold conjecture says that for each Hamiltonian diffeomorphism of M, it possesses at least as many fixed points as a smooth function on M possesses critical points. Nondegenerate Hamiltonian and weak Arnold conjecture A H ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
