Arnold conjecture
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The Arnold conjecture, named after mathematician
Vladimir Arnold Vladimir Igorevich Arnold (or Arnol'd; , ; 12 June 1937 – 3 June 2010) was a Soviet and Russian mathematician. He is best known for the Kolmogorov–Arnold–Moser theorem regarding the stability of integrable systems, and contributed to s ...
, is a mathematical conjecture in the field of
symplectic geometry 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 ...
, a branch of
differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...
.


Strong Arnold conjecture

Let (M, \omega) be a closed (compact without boundary)
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 sy ...
. For any smooth function H: M \to , the symplectic form \omega induces a
Hamiltonian vector field Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...
X_H on M defined by the formula :\omega( X_H, \cdot) = dH. The function H is called a
Hamiltonian function In physics, Hamiltonian mechanics is a reformulation of Lagrangian mechanics that emerged in 1833. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (gene ...
. Suppose there is a smooth 1-parameter family of Hamiltonian functions H_t \in C^\infty(M), t \in ,1/math>. This family induces a 1-parameter family of Hamiltonian vector fields X_ on M. The family of vector fields integrates to a 1-parameter family of
diffeomorphisms In mathematics, a diffeomorphism is an isomorphism of differentiable manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are continuously differentiable. Defini ...
\varphi_t: M \to M. Each individual \varphi_t is a called a Hamiltonian diffeomorphism of M. The strong Arnold conjecture states that the number of fixed points of a Hamiltonian diffeomorphism of M is greater than or equal to the number of critical points of a smooth function on M. See also comments, pp. 284–288.


Weak Arnold conjecture

Let (M, \omega) be a closed symplectic manifold. A Hamiltonian diffeomorphism \varphi:M \to M is called nondegenerate if its graph intersects the diagonal of M\times M transversely. For nondegenerate Hamiltonian diffeomorphisms, one variant of the Arnold conjecture says that the number of fixed points is at least equal to the minimal number of critical points of a
Morse function In mathematics, specifically in differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a typical differenti ...
on M, called the Morse number of M. In view of the Morse inequality, the Morse number is greater than or equal to the sum of
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 simplicia ...
s over a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
, namely \sum_^ \dim H_i (M; ). The weak Arnold conjecture says that :\# \ \geq \sum_^ \dim H_i (M; ) for \varphi : M \to M a nondegenerate Hamiltonian diffeomorphism.


Arnold–Givental conjecture

The Arnold–Givental conjecture, named after Vladimir Arnold and
Alexander Givental Alexander Givental () is a Russian-American mathematician who is currently Professor of Mathematics at the University of California, Berkeley. His main contributions have been in symplectic topology and singularity theory, as well as their relati ...
, gives a lower bound on the number of intersection points of two Lagrangian submanifolds and L' in terms of the Betti numbers of L, given that L' intersects transversally and L' is Hamiltonian isotopic to . Let (M, \omega) be a compact 2n-dimensional symplectic manifold, let L \subset M be a compact Lagrangian submanifold of M, and let \tau : M \to M be an anti-symplectic involution, that is, a diffeomorphism \tau : M \to M such that \tau^* \omega = -\omega and \tau^2 = \text_M, whose fixed point set is L. Let H_t\in C^\infty(M), t \in ,1/math> be a smooth family of
Hamiltonian function In physics, Hamiltonian mechanics is a reformulation of Lagrangian mechanics that emerged in 1833. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (gene ...
s on M. This family generates a 1-parameter family of diffeomorphisms \varphi_t: M \to M by flowing along the
Hamiltonian vector field Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...
associated to H_t. The Arnold–Givental conjecture states that if \varphi_1(L) intersects transversely with L, then :\# (\varphi_1(L) \cap L) \geq \sum_^n \dim H_i(L; \mathbb Z / 2 \mathbb Z).


Status

The Arnold–Givental conjecture has been proved for several special cases. *
Alexander Givental Alexander Givental () is a Russian-American mathematician who is currently Professor of Mathematics at the University of California, Berkeley. His main contributions have been in symplectic topology and singularity theory, as well as their relati ...
proved it for (M, L) = (\mathbb^n, \mathbb^n). * Yong-Geun Oh proved it for real forms of compact Hermitian spaces with suitable assumptions on the Maslov indices. * Lazzarini proved it for negative monotone case under suitable assumptions on the minimal Maslov number. * Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, and Kaoru Ono proved it for (M, \omega) semi-positive. * Urs Frauenfelder proved it in the case when (M, \omega) is a certain symplectic reduction, using gauged Floer theory.


See also

* Symplectomorphism#Arnold conjecture *
Floer homology In mathematics, Floer homology is a tool for studying symplectic geometry and low-dimensional topology. Floer homology is an invariant that arises as an infinite-dimensional analogue of finite-dimensional Morse homology. Andreas Floer intro ...
* Spectral invariants *
Conley–Zehnder theorem In mathematics, the Conley–Zehnder theorem, named after Charles C. Conley and Eduard Zehnder, provides a lower bound for the number of fixed points of Hamiltonian diffeomorphisms of standard symplectic tori in terms of the topology of the unde ...


References


Citations


Bibliography

*. * * ** *. *{{citation, doi=10.1007/978-3-0348-9217-9_23 , chapter=Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks, III: Arnold-Givental Conjecture , title=The Floer Memorial Volume , date=1995 , last1=Oh , first1=Yong-Geun , pages=555–573 , isbn=978-3-0348-9948-2 Conjectures Symplectic geometry Hamiltonian mechanics Unsolved problems in mathematics