Georgia Institute of Technology College of Sciences

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. 

You are probably familiar with the concept of a knot in 3 space: a tangled string that can't be pushed and pulled into an untangled one. We briefly show how to prove mathematical knots are in fact knotted, and discuss some conditions which guarantee unknotting. We then give explicit examples of knotted 2-spheres in 4 space, and discuss 2-sphere version of the familiar theorems. A large part of the talk is practice with visualizing objects in 4 dimensional space. We will also prove some elementary facts to give a sense of what working with these objects feels like. Time permitting we will describe know knotted 2-spheres were used to give evidence for the smooth 4D Poincare conjecture, one of the guiding problems in the field.

Say you’ve got an (orientable) surface S and you want to do geometry with it. Well, the complex plane C has dimension 2, so you might as well try to model S on C and see what happens. The objects you get from following this thought are called complex structures. It turns out that most surfaces have a rich but manageable amount of genuinely different complex structures. I’ll focus in this talk on how to think about comparing and deforming complex structures on S. I’ll explain the remarkable result that there are highly structured “best” maps between (marked) complex structures, and how this can be used to show the right space of complex structures on S is a finite-dimensional ball. This is known as Teichmüller’s theorem, and I’ll be following Bers’ proof.