Georgia Institute of Technology College of Sciences

A sequence of remarkable results in recent decades have shown that for a surface group H there are many Lie groups G and connected components C of Hom(H,G) consisting of discrete and faithful representations. These are known as higher Teichmüller spaces. With two exceptions, all known constructions of higher Teichmüller spaces work only for surface groups. This is an expository talk on the remarkable paper Convexes Divisibles III (Benoist ‘05), in which the first construction of higher Teichmüller spaces that works for some non-surface-groups was discovered. The paper implies the fundamental group H’ of any closed hyperbolic n-manifold has a higher Teichmüller space C’ in PGL(n+1,R). This is proved by showing any element of C’ preserves a convex domain in RP^n with a group-invariant tiling.

Vassiliev knot invariants, or finite-type invariants, are a broad class of knot invariants resulting from extending usual invariants to knots with transverse double points. We will show that the Conway and Jones polynomials are fully described by Vassiliev invariants. We will discuss the fundamental theorem of Vassiliev invariants, relating them to the algebra of chord diagrams and weight systems. Time permitting, we will also discuss the Kontsevich integral, the universal Vassiliev invariant.

Mapping class groups of surfaces in general have cohomology that is hard to compute. Meanwhile, within something called the cohomologically-stable range, a family of characteristic classes called the MMM classes (of surface bundles) is enough to generate this cohomology and thus plays an important role for understanding both the mapping class group and surface bundles. Moreover, constructing the so-called Atiyah-Kodaira manifold provides the setting to prove that these MMM classes are non-trivial. Most of this beginner-friendly talk will be dedicated to proving the non-triviality of the first MMM class. Maybe as a side quest, we will also give a crash course on the geometric viewpoint of (co)homology and then apply this viewpoint to understand the constructions and the proofs.

Morse theory analyzes the topology of a smooth manifold by studying the behavior of its real-valued functions. From this, one obtains a well-behaved homology theory which provides further information about the manifold and places constraints on the smooth functions it admits. This idea has proven to be useful in approaching topological problems, playing an essential role in Smale's solution to the generalized Poincare conjecture in dimensions greater than 4. Morse theory has been adapted to study complex manifolds, and even algebraic varieties over more general fields, but the underlying principles remain the same. In this talk, we will define the basic notions of Morse theory and describe some of the fundamental results. Then we will define Morse homology and discuss some important corollaries and applications.