I will make a survey of recent results on the spectrum of periodic and, to a smaller extent, almost-periodic operators. I will consider two types of results:

1. Bethe-Sommerfeld Conjecture. For a large class of multidimensional periodic operators the numbers of spectral gaps is finite.

2. Asymptotic behaviour of the integrated density of states of periodic and almost-periodic operators for large energies.

For closed surfaces and for surfaces with boundary there are natural eigenvalue extremal problems whose solutions, when they exist, determine minimal surfaces in the sphere or the ball with a natural boundary condition. We will discuss the existence problem and describe some geometric properties of extremal metrics.

Consider a Brownian particle in a prescribed time-intependent incompressible flow in a bounded domain. We investigate how the strength of the flow and its geometric properties affect the expected exit time of the particle. The two main questions we analyze in this talk are as follows.  1. Incompressible flows are known to enhance mixing in many contexts, but do they also always decrease the exit time? We prove that the answer is no, unless the domain is a disk. 2.  Suppose the flow is cellular with amplitude A, and the domain is of size L.   What could be said about the exit time when both L and A are large? We prove that there are two characteristic regimes: a) if L << A^4, then the exit time from the entire domain is compatible with the exit time from a single flow cell, and it can be determined from the Freidlin–Wentzell theory; b) if L>> A^4, then the problem `homogenizes' and the exit time is determined by the effective diffusivity of cellular flows.

A Hodge class on a smooth complex projective variety gives rise to an associated hermitian line bundle on a Zariski open subset of a complex projective space P^n.  I will discuss recent work with P. Brosnan which shows that the Hodge conjecture is equivalent to the existence of a particular kind of degenerate behavior of this metric near the boundary.