Georgia Institute of Technology College of Sciences

Four dimensions is unique in many ways. For examplen-dimensional Euclidean space has a unique smooth structure if andonly if n is not equal to four. In other words, there is only one wayto understand smooth functions on R^n if and only if n is not 4. Thereare many other way that smooth structures on 4-dimensional manifoldsbehave in surprising ways. In this talk I will discuss this and I willsketch the beautiful interplay of ideas (you got algebra, analysis andtopology, a little something for everyone!) that go into proving R^4has more that one smooth structure (actually it has uncountably manydifferent smooth structures but that that would take longer toexplain).

Gauge theory is a beautiful subject that studies the space of connections on a vector bundle. It is also the natural language in which theories of particle physics are formulated. In fact, the word "gauge" has its origins in electromagnetism, and in this talk, we explore the basic geometric objects of gauge theory and show how one explicitly recovers the classical Maxwell's equations as a special case of the equations of gauge theory . Next, generalizing Maxwell's equations to a ``nonabelian" setting, we obtain the Yang-Mills equations, which describe the electroweak force in nature. Surprisingly, these equations were used by Simon Donaldson in the 1980s to prove spectacular results for the topology of smooth four-manifolds. We conclude this talk by describing some of the beautiful geometry and analysis behind gauge theory that goes into the work of Donaldson (for which we awarded a Fields Medal), and time permitting, we hope to say a bit about other gauge-theoretic applications to low-dimensional topology, for instance, instanton Floer homology.

Suppose you want to stir a pot of soup with several spoons. What is the most efficient way to do this? Thurston's theory of surface homeomorphisms gives us a concrete way to analyze this question. That is, to each mixing pattern we can associate a real number called the entropy. We'll start from scratch with a simple example, state the Nielsen-Thurston classification of surface homeomorphisms, and give some open questions about entropies of surface homeomorphisms.

Modern Economic Theory is largely based on the concept of Nash Equilibrium. In its simplest form this is an essentially statics notion. I'll introduce a simple model for the origin of money (Kiotaki and Wright, JPE 1989) and use it to introduce a more general (dynamic) concept of Nash Equilibrium and my understanding of its relation to Dynamical Systems Theory and Statistical Mechanics.