Serre duality is a twisted form of Poincare' duality satisfied by a very special class of projective schemes. For a variety of reasons the moduli stack of elliptic curves doesn't meet the hypotheses needed and doesn't quite exhibit Serre duality; however, when we consider a derived version of this stack -- replacing the structure sheaf by a sheaf of ring spectra -- suddenly and mysteriously the duality reappears. The purpose of this talk is to explain these ideas and calculations. This is a meditation on work of Hopkins, Miller, Lurie, Behrens, and many others.

We will discuss a method of carrying out integration on moduli spaces defined from Fredholm systems. The main point is that we do away with the regularity assumptions that is generally used in such a setup to obtain smooth manifolds (or orbifolds). We will also discuss some applications for such an integration. Most of the constructions will be elementary and will be motivated from finite dimensional examples. This is based on works of Chen-Li-Tian and works in progress with Chen and Hyvrier.

Iterating a map on the plane can produce complicated self-similar patterns resulting in fractal pictures. The map that produces the do-it-yourself fractal operates just on points with integer coordinates and is simple enough to be worked out by hand. I will use it to explain how self-similarity arises naturally in fractal pictures.

When our simple map is extended to operate on the whole plane, things become much more complicated. A bit of Calculus, a bit of Number Theory, and plenty of Linear Algebra will be needed to look at the question, "What Happens Eventually?"

In this informal talk, I will outline new work using elementary ideas by means of which an algebraic analogue of the usual KZ equation is exhibited. This difference equation arises via the introduction of Hurwitz polyzeta functions and suitable regularizations. A difference equation analogue of the notion of connection will also be presented, built around the algebraic KZ equation, for which a generating series for Hurwitz polyzeta functions is a "flat section". Also, the algebraic independence of the Hurwitz polyzeta functions will be demonstrated, along with a construction of regularizations of polyzeta functions using limits.

Consider six points in the plane and connect (or not) each pair of those at random. Many graphs can be obtained that way but all of them share the following property: there are three points where the pairs are all connected or never connected. This simple combinatorial result is at the base of what is called Ramsey theory, a theory that describes the appearance of unavoidable patterns and nowadays touches areas of mathematics that are seemingly very far from combinatorics. The purpose of this talk will be to present some of those connections taken from metric geometry and topological group theory.

Let M be a closed symplectic manifold and denote by Ham its group of Hamiltonian diffeomorphisms. When equipped with the standard smooth topology, this is an infinite dimensional Fréchet Lie group. It is generally believed that Ham is "tamer" than the diffeomorphism group Diff(M) and constitutes an intermediate object between compact Lie groups and more general diffeomorphism groups. To develop a better understanding of this principle, one may look at maximal Hamiltonian actions by tori or, in other words, to classify symplectic conjugacy classes of maximal compact tori in Ham. In this talk, we will show that for 4-dimensional symplectic manifolds, there are at most finitely many of those conjugacy classes. As a by-product, we will also prove that the rational cohomology algebra of the symplectomorphism group of a generic blow-up is not finitely generated.