The deveolpment of symplectic topology was motivated by Hamiltonian mechanics. It has been particularly successful in addressing one specific aspect, namely fixed points and periodic points of discrete-time Hamiltonian systems. I will explain how such applications work, both in older and more recent examples.
Title: Applications of the Ky Fan inequality to random (and almost periodic) operators
Abstract: We shall discuss the Ky Fan inequality for the eigenvalues of the sum of two Hermitian matrices. As an application, we shall derive a sharp version of a recent result of Hislop and Marx pertaining to the dependence of the integrated density of states of random Schroedinger operators on the distribution of the potential. Time permitting, we shall also discuss an application to quasiperiodic operators.