# Pure Mathematics Seminar 2020 Semester 1

Pre-seminar introductory session: 10:30-11am in Room 107 Peter Hall

**Organisers**: Jesse Gell-Redman, Ting Xue

**Schedule**

March 20 **Alessandro Giacchetto** (MPI / Trieste)

**Title:** The Kontsevich geometry of the combinatorial Teichmüller spaces

**Abstract**: In the early ’80s, a combinatorial description of the moduli spaces of curves was discovered, which led to remarkable results about its topology. Inspired by analogue arguments in the hyperbolic setting, we describe the combinatorial Teichmüller space parametrising marked metric ribbon graphs on a surface. We introduce global Fenchel–Nielsen coordinates and show a Wolpert-type formula for the Kontsevich symplectic form. As applications of this set-up, we present a combinatorial analogue of Mirzakhani’s identity, resulting in a new proof of Witten–Kontsevich recursion for the symplectic volumes of the combinatorial moduli space, and Norbury’s recursion for the counting of integral points. The talk is based on a joint work in progress with J.E. Andersen, G. Borot, S. Charbonnier, D. Lewański and C. Wheeler.

March 13 **David Ridout** (Melbourne)

**Title:** From vertex algebras to simple Lie algebras (and back again).

**Abstract**: The representation theory of affine vertex algebras is central to mathematicians studying W-algebras and physicists studying 4d/2d correspondences. In many cases, the highest-weight modules have already been well-understood. However, an open question is how to generalise this to larger module categories. I will describe recent work with Kazuya Kawasetsu that answers this question for weight modules with finite-dimensional weight spaces.

March 6 **Johanna Knapp** (Melbourne)

**Title:** Calabi-Yaus and categories from a physics perspective**Abstract:** The aim of the talk is to give an example of how problems in pure mathematics can arise in physics. Concretely, I will discuss the role of Calabi-Yau spaces and categories in string theory and explain how physics methods can be used to analyse them. In particular, I want to introduce a notion of a gauged linear sigma model (GLSM). This theory can be used construct and analyse Calabi-Yaus, their moduli spaces and associated categories. It provides a physics framework to uncover equivalences and connections between different categories associated to Calabi-Yaus.

Feb 21 **Arun Ram** (Melbourne)

**Title: **Level 0 and the Bethe ansatz

**Abstract: **First I will review the mechanics for finding eigenvalues and eigenvectors of the Murphy elements in the group algebra of the symmetric group as a model method for finding the eigenvalues and eigenvectors of the transfer matrices that appear in the Algebraic Bethe ansatz. Second I will review the connection between transfer matrices and pseudoquasitriangular Hopf (psqtH) algebras. Third I will visit the 3 favourite families of pqtH-algebras and hint at their relation to affine Lie algebras. Finally I will explain, via an illustrative (9-dimensional) example for , the construction of the eigenvalues and eigenvectors in a level 0 representation.