In this lecture I will give an overview of string topology. This is a theory that studies the
differential and algebraic topology of spaces of paths and loops in manifolds. I will describe the
algebraic topological structure of this theory, as well as its motivation from physics.
I will then discuss some applications.

Multiscale asymptotic analysis is a particularly useful tool for studying nonlinear waves when exact solutions are not available. This is demonstrated in concrete problems: reaction diffusion front speeds in random shear flows, and localized propagating pulses in nonlinear scalar wave equations, both in two space dimensions. Complementary numerical results will also be shown.

We study the evolutionary game dynamics of a two-strategy game. In infinite populations, the well-known replicator equations describe the deterministic evolutionary dynamics. There are three generic selection scenarios. The dynamics of a finite group of players has received little attention. We provide a framework for studying stochastic evolutionary game dynamics in finite populations. We define a Moran process with frequency dependent fitness. We find that there are eight selection scenarios. And for a given payoff matrix, a number of these sceanrios can occur for different population size. Our results have interesting applications in biology and economics. In particular, we obtain new results on the evolution of cooperation in the classic repeated Prisoner's Dilemma game. We show that a single cooperator using a reciprocal strategy such as Tit-For-Tat can invade a population of defectors with a probability that corresponds to a net selective advantage. We specify the conditions for natural selection to favor the emergence of cooperation and derive conditions for evolutionary stability in finite populations.

It is well-known (Liouville's Theorem) that a complex projective manifold X does not admit any non-constant algebraic (or holomorphic) functions. We explain how the collection of all algebraic vector bundles on X and morphisms between them gives rise to a structure (the derived category of X) which serves as a replacement -in many interesting ways - of the L^2 space of functions in analysis. Several results describing the properties of this structure will be explained.

In 1967 Furstenberg discovered a very surprising phenomenon:
while both $T: x \to 2 x \bmod 1$ and $s: x \to 3 x \bmod 1$ on $\R / \Z$ have many closed invariant sets, closed sets which are invariant under both $T$ and $S$ are very rare (indeed, are either finite sets of rationals or $\R / \Z$). Furstenberg also conjectured that a similar result holds for invariant measures. This conjecture is of course still open.
As has been shown by several authors, including Katok-Spatzier and Margulis this phenomenon is not an isolated curiosity but rather a deep property of many natural $\Z ^ d$ and $\R ^ d$ actions ($d > 1$) with many applications.
Recently, there has been substantial progress in the study of measures invariant under such actions. While we are at present still far from full resolution of this intriguing question, the partial results we currently
have are already powerful enough to prove results in other fields. In particular, these techniques enable proving a special but important case of Rudnick and Sarnak's Quantum Unique Ergodicity Conjecture, as well as a partial result towards Littlewood's Conjecture on simultaneous diophantine approximations (the later is in a joint paper with M. Einsiedler and A. Katok).