Arnold's fixed-point conjecture for Hamiltonian symplectomorphisms

Let $(M,\omega)$ be a compact symplectic manifold, and let $f\colon M\to M$ be a symplectomorphism exactly homotopic to the identity. A fixed point of $f$ is nondegenerate if its differential is nonsingular. A Morse function on $M$ is a smooth function whose critical points are all nondegenerate.

Arnold's conjecture. The number of fixed points of $f$ satisfies

$$\left|\{\text{fixed points of }f\}\right|\geq \min\left|\{\text{critical points of a smooth function on }M\}\right|.$$

If all fixed points are nondegenerate, then

$$\left|\{\text{nondegenerate fixed points of }f\}\right|\geq \min\left|\{\text{critical points of a Morse function on }M\}\right|\geq \sum_{i=0}^{2n}\dim H^i(M,\mathbb{R}).$$

This is a fundamental fixed-point problem in symplectic geometry. The conjecture has been proven using Floer homology by Conley--Zehnder, Floer, Hofer--Salamon, Ono, Fukaya--Ono, and Liu--Tian.