Georgia Institute of Technology College of Sciences

Quantum topology is a collection of ideas and techniques for studying knots and manifolds using ideas coming from quantum mechanics and quantum field theory. We present a gentle introduction to this topic via Kauffman bracket skein algebras of surfaces, an algebraic object that relates "quantum information" about knots embedded in the surface to the representation theory of the fundamental group of the surface. In general, skein algebras are difficult to compute. We associate to every triangulation of the

I will give 2 or 3 lectures on Legendrian contact homology. This invariant has played a big role in our understanding of Legendrian submanifolds of contact manifolds in all dimensions. We will discuss the general definition but focus on the 3-dimensional setting where it easiest to compute (and describe Legendrian knots). I will also discuss the A^\infty structure associated to the linearized co-chain groups of contact homology.

I will give 2 or 3 lectures on Legendrian contact homology. This invariant has played a big role in our understanding of Legendrian submanifolds of contact manifolds in all dimensions. We will discuss the general definition but focus on the 3-dimensional setting where it easiest to compute (and describe Legendrian knots). I will also discuss the A^\infty structure associated to the linearized co-chain groups of contact homology.

We will discuss some facts about intersection forms on closed, oriented 4-manifolds. We will also sketch the proof that for two closed, oriented, simply connected manifolds, they are homotopy equivalent if and only if they have isomorphic intersection forms.

Trisections of 4-manifolds relative to their boundary were introduced by Gay and Kirby in 2012. They are decompositions of 4-manifolds that induce open book decomposition in the bounding 3-manifolds. This talk will focus on diagrams of relative trisections and will be divided in two. In the first half I will focus on trisections as fillings of open book decompositions and I will present different fillings of different open book decompositions of the Poincare homology sphere.

Dehn surgery is a fundamental tool for constructing oriented 3-Manifolds. If we fix a knot K in an oriented 3-manifold Y and do surgeries with distinct slopes r and s, we can ask under which conditions the resulting oriented manifold Y(r) and Y(s) might be orientation preserving homeomorphic. The cosmetic surgery conjecture state that if the knot exterior is boundary irreducible then this can't happen. My talk will be about the case where Y is an homology sphere and K is an hyperbolic knot.

In this lecture series, held jointly (via video conference) with the University of Buffalo and the University of Arkansas, we aim to understand the lecture notes by Vincent Guirardel on geometric small cancellation. The lecture notes can be found here: https://perso.univ-rennes1.fr/vincent.guirardel/papiers/lecture_notes_pcmi.pdf This week we will begin Lecture 4.

Given a surface, intersection information about the simple closed curves on the surface is encoded in its curve graph. Vertices are homotopy classes of curves, and edges connect vertices corresponding to curves with disjoint representatives. We can wonder what subgraphs of the curve graph are possible for a given surface. For example, if we fix a surface, then a graph with sufficiently large clique number cannot be a subgraph of its curve graph. This is because there are only so many distinct and mutually disjoint curves in a given surface.

For every surface (sphere, torus, etc.) there is an associated graph called the curve graph. The vertices are curves in the surface and two vertices are connected by an edge if the curves are disjoint. The curve graph turns out to be very important in the study of surfaces. Even though it is well-studied, it is quite mysterious. Here are two sample problems: If you draw two curves on a surface, how far apart are they as edges of the curve graph? If I hand you a surface, can you draw two curves that have distance bigger than three?