Arnold's Conjecture and Morava $K$-theories

After a brief review of the Arnold's Conjecture, I will give an overview of the proof of the following joint result with Blumberg: for every closed symplectic manifold, the number of time-$1$ periodic orbits of a non-degenerate Hamiltonian is bounded below by the rank of the cohomology with coefficients in any field. The case of characteristic $0$ was proved by Fukaya and Ono as well as Li and Tian. The new ingredient in our proof is the construction of generalized Floer cohomology groups with coefficients in Morava $K$-theory. This means that we have to use higher dimensional moduli spaces of pseudo-holomorphic curves, and extract “fundamental chains” in generalized homology.