Topics in Modern Mathematical Physics II
Course on Moduli of Curves, Ribbon Graphs, and Witten’s Conjecture
General Description
This course provides a tour among theory of the moduli spaces of curves, matrix models and integrable systems. Later (hopefully) we will proceed to Chekov-Eynard-Orantin’s topological recursion, an elegant theory at the interplay of these topics.
We will begin with the analytic construction of the Deligne-Mumford compactified moduli space from the perspective of Teichmüller theory. We will then develop the combinatorial theory of ribbon graphs, which provides a cellular decomposition of these spaces. These materials lead to a detailed exposition of Maxim Kontsevich’s celebrated proof of Edward Witten’s conjecture, which shows that a generating function for intersection numbers on the moduli space is a $\tau$-function for the KdV hierarchy, by identifying it with the partition function of a matrix model.
From the theory of matrix models, if time permits, we will introduce Chekov-Eynard-Orantin’s topological recursion (TR). TR is a powerful algorithm motivated by matrix models, providing solutions to many enumerative geometry problems and beyond.