The outer automorphism group Out(F) of a free group F of finite rank shares many properties with the mapping class group of a surface, however the techniques for studying these groups are generally quite different. Analogues of the pseudo-Anosov elements of the mapping class group are the so-called fully irreducible automorphisms, which exhibit north-south dynamics on Culler-Vogtmann's Outer Space. We will explain a method for constructing these automorphisms and suggest why this construction should be useful. This is joint work with Matt Clay (University of Oklahoma).

I'll show that every torsion-free delta-hyperbolic group supports a non-trivial topological action on a high-dimensional ball. Aside from a bad limit set in the boundary of the ball, this action is well-behaved. This was joint work with Tom Farrell.

Let X be a symplectic manifold and G a Lie group acting in a Hamiltonian fashion with a moment map f. Let P denote a principal G-bundle over a surface with area form V. A pair (A,u) of a connection A on P and a section u of the associated bundle P(X):=P imes_G X is a gauged pseudo-holomorphic map if it satisfies the A-twisted Cauchy-Rieman equation. The space of vortices is the quotient of gauged pseudo-holomorphic map by Aut(P). We will give a brief introduction to moduli spaces of curves arising from Gromov-Witten theory, including some Fredholm theory. We will show that under some good choices this moduli space can be compactified and get an orbifold structure. This is work in progress with A. Ott, C. Woodward and F. Ziltiner.

Gromov-Witten theory for projective varieties has been the subject of intense research since its intruduction. Many important results in the area (eg. Givental's mirror theorem) were proven using equivariant localization techniques from topology, using natural group actions. For a symplectic manifold X, GW invariants are defined using moduli spaces pseudoholomophic maps u:Σ o X from a Riemann surface Σ to X. Suppose that a compact Lie group is acting on X in a Hamiltonian way. After an introduction to the general theory, I will introduce the "space of vortices" which are pairs (A,u) of a connection over a principal bundle P, and a section u:Σ satisfying certain "gauged" equations. Using these spaces one can define gauged Gromov-Witten invariants, which are an equivariant version of Gromov-Witten invariants. This theory depends on a choice of area form on Σ as well as Σ. I will describe joint work with C. Woodward regarding the dependency on the area form, as well as joint work with Ott, Ziltener and Woodward on punctured surfaces. I will also give relations with other equivariant theories and I will discuss a conjecture related to the Gromov-Witten theory of symplectic quotients.

Simplicial volume is an invariant of closed manifolds that measures how efficiently the manifold can be "triangulated over the reals". I will discuss various geometric and topological consequences that result from positivity of the simplicial volume. Finally, I will end by giving an indication of the proof (joint with Ben Schmidt) that locally symmetric spaces of non-compact type have positive simplicial volume.

Generalized cohomology theories have been studied continuously since the late 1950's. My lecture of several weeks ago was devoted to a brief survey of topological K-theory, the first generalized cohomology theory to be developed and utilized. The present lecture will begin by discussing another family of generalized cohomology theories which have also been extensively studied - cobordism theory and its associated theories, notably Brown-Peterson theory and Morava K-theory. This work includes major contributions by Thom, Novikov and Quillen. A major theme of this discussion will be the algebra of cohomology operations associated with each of these theories.

As we will see, topological K-theory can also be fitted into the framework of this family of cohomology theories. In addition we will return to K-theory and discuss both operations and Adams operations. However these operations will turn out out be rather distinct from the ones considered for cobordism and BP theory.