The Conley conjecture, named after mathematician Charles Conley, is a mathematical conjecture in the field of symplectic geometry, a branch of differential geometry.
is exact (i.e., equals to the differential of a function
is the integration of a 1-parameter family of Hamiltonian vector fields
Conley first made such a conjecture for the case that
[2] The Conley conjecture is false in many simple cases.
, which is a Hamiltonian diffeomorphism, has only 2 geometrically different periodic points.
[1] On the other hand, it is proved for various types of symplectic manifolds.
The Conley conjecture was proved by Franks and Handel for surfaces with positive genus.
[3] The case of higher dimensional torus was proved by Hingston.
[4] Hingston's proof inspired the proof of Ginzburg of the Conley conjecture for symplectically aspherical manifolds.
Later Ginzburg--Gurel and Hein proved the Conley conjecture for manifolds whose first Chern class vanishes on spherical classes.
Finally, Ginzburg--Gurel proved the Conley conjecture for negatively monotone symplectic manifolds.