The Levine-Tristram signature is a classical link invariant, given by a function $(S^1\setminus \{1\}) \rightarrow \mathbb{Z}$. It admits a multivariable generalization for links, given as a function $(S^1\setminus \{1\})^n \rightarrow \mathbb{Z}$, where $n$ is the number of link components. This multivariable signature was recently further extended by Cimasoni-Markiewicz-Politarczyk to a function defined on the full torus $(S^1)^n$. In this talk I will introduce those new extended signatures, try to explain why they're interesting both for understanding their classical predecessors and as invariants on their own, and discuss some of their properties, in particular concerning concordance invariance and the relationship with the Alexander module. The talk will be introductory in nature, and no familiarity with classical knot theory will be assumed. Based on joint work with D. Cimasoni and I. Popova.

I will talk about conjectural dualities for the moduli of Higgs bundles on
a smooth projective complex curve associated with dual reductive groups.
These dualities are formulated using BPS cohomology and quasi-BPS
categories, which refine the BPS invariants that are of interest in
enumerative geometry. The talk is based on joint work with Chenjing Bu,…

The magnitude of a metric space is a real-valued invariant that counts the "effective number of points" in the space. In this talk, I will describe a relation between the magnitude of a (nice enough) metric space and an invariant from Euclidean geometry: to each space X, one can associate a simplex S such that the magnitude of X is a simple transformation of the circumradius of S. Furthermore, this relation extends to all subspaces X' of X and corresponding faces S' of S, and is uniquely defined by this property. As one application, I will discuss how this geometric perspective identifies a new regime in the asymptotic behaviour of metric spaces when rescaling the metric. Finally, if time permits, I will say a few words on connections to graph theory and potential theory; some keywords which may or may not be covered are: Laplacians, discrete curvature, capacity, second moment.

Abstract: TBA