Tour of Knots & Theta Functions

Seminar presented as part of the Postgraduate (PG) Seminar series of Stellenbosch University's mathematics department.

Overview

This talk closely follows the material covered in Theta Functions and Knots by Răzvan Gelca. The main goal is to build up to the punchline of chapter 5 from that book - namely, the isomorphism between the space of theta functions defined by a Riemann surface, and the skein space of its enclosed handlebody.

This is a remarkable result - reaching it takes us on a lightning tour through quantum mechanics, complex analysis, low-dimensional topology, and representation theory.

Why this matters

The isomorphism in question, aside from being interesting in its own right, is especially relevant to my current research - indeed, my research aims to improve a certain analogue of this very isomorphism.

To be precise, it turns out that the theory of theta functions as given in Gelca can be formulated as a specific instance of Chern-Simons theory, namely 'abelian' Chern-Simons theory, where the gauge group is U(1) instead of the usual SU(2). My research topic deals with an isomorphism analogous to the one described above, but arising within the SU(2) version of the theory. We hope to "improve" this isomorphism by stating it in a *basis-free* manner, which has currently not been done. This is of interest because expanding with respect to the preferred basis is generally hard.