Georgia Institute of Technology College of Sciences

Every four-dimensional Stein domain has a Morse function whoseregular level sets are contact three-manifolds. This allows us to studycomplex curves in the Stein domain via their intersection with thesecontact level sets, where we can comfortably apply three-dimensional tools.We use this perspective to understand links in Stein-fillable contactmanifolds that bound complex curves in their Stein fillings.

In this series of talks, I will introduce basic concepts and results in singularity theory of smooth and holomorphic maps. In the first talk, I will present a gentle introduction to the elements of singularity theory and give a proof of the well-known Morse Lemma that illustrates key geometric and algebraic principles of singularity theory.

We give "visual descriptions" of cut points and non-parabolic cut pairs in the Bowditch boundary of a relatively hyperbolic right-angled Coxeter group. We also prove necessary and sufficient conditions for a relatively hyperbolic right-angled Coxeter group whose defining graph has a planar flag complex with minimal peripheral structure to have the Sierpinski carpet or the 2-sphere as its Bowditch boundary. We apply these results to the problem of quasi-isometry classification of right-angled Coxeter groups.

The mapping class group of the plane minus a Cantor set naturally appears in many dynamical contexts, including group actions on surfaces, the study of groups of homeomorphisms on a Cantor set, and complex dynamics. In this talk, I will present the 'ray graph', which is a Gromov-hyperbolic graph on which this big mapping class group acts by isometries (it is an equivalent of the curve graph for this surface of infinite topological type).

There has been a recent interest in studying surfaces of infinite type, i.e. surfaces with infinitely-generated fundamental groups. In this talk, we will focus on their mapping class groups, often called big mapping class groups. In contrast to the finite-type case, there are many open questions regarding the basic algebraic and topological properties of big mapping class groups. I will discuss several such questions and provide some answers.

I will talk about homomorphisms between surface braid groups. Firstly, we will see that any surjective homomorphism from PB_n(S) to PB_m(S) factors through a forgetful map. Secondly, we will compute the automorphism group of PB_n(S). It turns out to be the mapping class group when n>1.