Algebraic Geometry in Melbourne
8–10 July 2026

Title

Multivariate V-filtrations and the Strong Monodromy Conjecture for hyperplane arrangements

Abstract

For a hypersurface singularity \(f = 0\), the Strong Monodromy Conjecture of Igusa and Denef-Loeser proposes a surprising connection between poles of zeta functions (defined by \(p\)-adic or motivic integration of \(f\)) and zeroes of Bernstein-Sato polynomials (defined by differential equations satisfied by \(f\)). In this talk, I will discuss a proof of this conjecture when \(f\) is the equation of a hyperplane arrangement with arbitrary multiplicities. The main ingredient is a new approach to Sabbah’s theory of \(V\)-filtrations of holonomic D-modules along divisors with normal crossings, which shows that these filtrations have much better properties than was previously known. I will explain how this general theory works, and outline how the additional control it gives over Bernstein-Sato roots is used to prove the conjecture. This is joint work with Ruijie Yang.

Title

On the higher algebraic K-theory of toric varieties

Abstract

(Split) toric varieties are given by combinatorial data that can be encoded, for example, in an object called a monoid scheme (a scheme built out of commutative monoids). I will report on some ongoing work with Weibel regarding the K-theory of monoid schemes, and explain how it determines that of the associated toric varieties.

Title

Quasi-algebraic braids

Abstract

I will talk about some joint work in progress with Konstantin Jakob and Masoud Kamgarpour in which we define quasi-algebraic braids. These are braids coming from maps of formal loops (\(\operatorname{Spec} \mathbf{C}((t))\)) into a hyperplane complement. These arise naturally in two ways: 1) as links of singularities of plane curves 2) as Stokes data for irregular singularities. It turns out that algebraic braids are quite a restrictive class. I’ll give explain a complete

classification/construction of them.

Title

Towards projective coarse moduli spaces of stable super Riemann surfaces

Abstract

Super Riemann surfaces play a central role in algebraic supergeometry. In this talk, I will present a stack-theoretic construction of the moduli spaces of stable super Riemann surfaces. The key ingredient is a generalized Keel–Mori theorem on the existence of coarse moduli spaces in the super setting. I will explain how this applies to stable super Riemann surfaces and discuss ongoing questions regarding the projectivity of the resulting coarse moduli spaces.

Title

The Deligne-Simpson problem, braid stacks, and computers

Abstract

This talk is centred on an old problem about matrices, known as the Deligne–Simpson problem. We will discuss a modern geometric version, called the irregular Deligne–Simpson problem. In joint work with Masoud Kamgarpour, we resolve the irregular Deligne–Simpson problem in many cases by exploiting a connection with the world of braids and braid stacks. I will also discuss how Julia and Codex can contribute to mathematical research.