Georgia Institute of Technology College of Sciences

Abstract: Fundamental to our understanding of Teichm\"uller space T(S) of a closed oriented genus $g \geq 2$ surface S are two different perspectives: one as connected  component in the  PSL(2,\R) character variety  \chi(\pi_1S, PSL(2,\R)) and one as the moduli space of marked hyperbolic structures on S. The latter can be thought of as a moduli space of (PSL(2,\R), \H^2) -structures. The G-Hitchin component, denoted Hit(S,G), for G a split real simple Lie group, is a connected component in \chi(\pi_1S, G) that is a higher rank generalization of T(S). In this talk, we discuss a new geometric structures (i.e., (G,X)-structures) interpretation of Hit(S, G_2'), where G_2' is the split real form of the exceptional complex simple Lie group G_2.

After discussing the motivation and background, we will present some of the main ideas of the theorem, including a family of almost-complex curves
that serve as bridge between the geometric structures and representations.

I will discuss a mixture of results and conjectures related to the Khovanov homology and Knot Floer homology for ribbon knots. We will explore relationships with fusion numbers (a measure of complexity on ribbon disks) and particular families of symmetric unions (ribbon knots given by particular diagrams). This is joint work with Nathan Dunfield, Sherry Gong, Tom Hockenhull, and Marco Marengon.

The Burau representation is a kind of homological representation of braid groups that has been around for around a century. It remains mysterious in many ways and is of particular interest because of its relation to quantum invariants of knots and links such as the Jones polynomial. In recent work, I came across a relationship between this representation and a moduli space of Euclidean cone metrics on spheres (think e.g. convex polyhedra) first examined by Thurston. After introducing the relevant definitions, I'll explain a bit about this connection and how I used the geometric structure on this moduli space to exactly identify the kernel of the Burau representation after evaluating its formal parameter at complex roots of unity. There will be many pictures!

The talk is based on my joint works with Maxim Prasolov and Vladimir Shastin, where we studied the relation between rectangular diagrams of links and Legendrian links. This relation allows for a complete classification of exchange classes of rectangular diagrams in terms of equivalence classes of Legendrian links and their symmetry groups. Since all rectangular diagrams of given complexity can be searched, this yields a method to algorithmically compare Legendrian links. Of course, the general algorithm has too high complexity for a practical implementation, but in some situations, the most time consuming parts can be bypassed, which allows us to confirm the non-equivalence of Legendrian knots in several previously unresolved cases.