Georgia Institute of Technology College of Sciences

The Torelli group of a surface is a natural yet mysterious subgroup of the mapping class group.  We will discuss a few recent results about finiteness properties of the Torelli group, as well as a result about the cohomological dimension of the Johnson filtration.  

In the 20th century, Thurston proved two classification theorems, one for surface homeomorphisms and one for branched covers of surfaces.  While the theorems have long been understood to be analogous, we will present new work with Belk and Winarski showing that the two theorems are in fact special cases of one Ubertheorem.  We will also discuss joint work with Belk, Lanier, Strenner, Taylor, Winarski, and Yurttas on algorithmic aspects of Thurston’s theorem.  This talk is meant to be accessible to a wide audience.

We introduce a decomposition of a 4-manifold called a multisection, which is a mild generalization of a trisection. We show that these correspond to loops in the pants complex and provide an equivalence between closed smooth 4-manifolds and loops in the pants complex up to certain moves. In another direction, we will consider multisections with boundary and show that these can be made compatible with a Weinstein structure, so that any Weinstein 4-manifold can be presented as a collection of curves on a surface.

Given a Legendrian knot L in a contact 3 manifold, one can associate a so-called LOSS invariant to L which lives in the knot Floer homology group. We proved that the LOSS invariant is natural under the positive contact surgery. In this talk I will review some background and definition, try to get the ideal of the proof and talk about the application which is about distinguishing Legendrian and Transverse knot.