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.be a closed (compact without boundary) symplectic manifold.induces a Hamiltonian vector fielddefined by the formula The functionis called a Hamiltonian function.Suppose there is a smooth 1-parameter family of Hamiltonian functionsThe family of vector fields integrates to a 1-parameter family of diffeomorphismsis a called a Hamiltonian diffeomorphism ofThe strong Arnold conjecture states that the number of fixed points of a Hamiltonian diffeomorphism ofis greater than or equal to the number of critical points of a smooth function onbe a closed symplectic manifold.is called nondegenerate if its graph intersects the diagonal ofFor 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 on, called the Morse number ofIn view of the Morse inequality, the Morse number is greater than or equal to the sum of Betti numbers over a fieldThe weak Arnold conjecture says that fora nondegenerate Hamiltonian diffeomorphism.[2][3] The Arnold–Givental conjecture, named after Vladimir Arnold and Alexander Givental, gives a lower bound on the number of intersection points of two Lagrangian submanifolds L andin terms of the Betti numbers ofbe a compact Lagrangian submanifold ofbe an anti-symplectic involution, that is, a diffeomorphism, whose fixed point set isbe a smooth family of Hamiltonian functions onby flowing along the Hamiltonian vector field associated to, then The Arnold–Givental conjecture has been proved for several special cases.