Georgia Institute of Technology College of Sciences

How good of an invariant is the Jones polynomial? The question is closely tied to studying braid group representations since the Jones polynomial can be defined as a (normalized) trace of a braid group representation.

In this talk, I will present my work developing a new theory to precisely characterize the entries of classical braid group representations, which leads to a generic faithfulness result for the Burau representation of B_4 (the faithfulness is a longstanding question since the 1930s). In forthcoming work, I use this theory to furthermore explicitly characterize the Jones polynomial of all 3-braid closures and generic 4-braid closures. I will also describe my work which uses the class numbers of quadratic number fields to show that the Jones polynomial detects the unknot for 3-braid links - this work also answers (in a strong form) a question of Vaughan Jones.

I will discuss all of the relevant background from scratch and illustrate my techniques through simple examples.

https://gatech.zoom.us/my/margalit?pwd=b3RhY3pVZUdlRUR3S1FLZzhFR1RVUT09

The ring of multivariate polynomials carries a natural action of the symmetric group. Quotienting by the ideal generated by the polynomials which are invariant under this action yields the "coinvariant algebra," an object with many beautiful algebraic and combinatorial properties. We will survey these properties and then discuss recent generalizations where the multivariate polynomials may contain anti-commuting ("superspace") variables. This talk is based on joint work with Brendon Rhoades.

One of the most beautiful aspects of math is the interplay between its different fields. We will discuss one such interaction by studying topology using tools from combinatorics and group theory. In particular, given a surface (two-dimensional manifold) S, we construct the curve complex of S, which is a graph that encodes topological data about the surface. We will then state a seminal result of Ivanov: the symmetries of a surface S are in a natural bijection with the symmetries of its curve complex. In the direction of the proof of Ivanov's result, we will touch on some tools we have when working with infinite graphs.

In recent years, there has been increased interest in using quantum computing for the purposes of solving problems in combinatorial optimization. No prior knowledge of quantum computing is necessary for this talk; in particular, the talk will be divided into three parts: (1) a gentle high-level introduction to the basics of quantum computing, (2) a general framework for solving combinatorial optimization problems with quantum computing (the Quantum Approximate Optimization Algorithm introduced by Farhi et al.), (3) and some recent results that my colleagues and I have found. Our group has looked at the Max-Cut problem and have developed a new quantum algorithm that utilizes classically-obtained warm-starts in order to improve upon already-existing quantum algorithms; this talk will discuss both theoretical and experimental results associated with our approach with our main results being that we obtain a 0.658-approximation for Max-Cut, our approach provably converges to the Max-Cut as a parameter (called the quantum circuit depth) increases, and (on small graphs) are approach is able to empirically beat the (classical) Goemans-Williamson algorithm at a relatively low quantum circuit-depth (i.e. using a small amount of quantum resources). This work is joint with Jai Moondra, Bryan Gard, Greg Mohler, and Swati Gupta.