Algebraic Geometry in Melbourne
8–10 July 2026

On Wed (8 Jul) at 9:00.

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.

On Wed (8 Jul) at 10:30.

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.

On Wed (8 Jul) at 11:30.

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.

On Wed (8 Jul) at 2:30.

Title

Cohomological Hall algebras and line operators

Abstract

We construct, from a general quiver with potential, a triangulated monoidal category, which we call the category of lines. In the case of ADE Dynkin quiver, we establish a monoidal equivalence between the construction above and the category of BPS line operators in \(4d\), \(N=2\) pure gauge theory, constructed by Braverman, Finkelberg, and Nakajima, further developed by Cautis and Williams. Finally, we discuss gauge theories with matters. This talk is based on ongoing joint work with Ryo Fujita, Yan Soibelman, and Gufang Zhao.

On Wed (8 Jul) at 4:00.

Title

Triangular modular curves and regular polyhedra over finite fields

Abstract

Triangular modular curves, introduced by Clark and Voight, give a variant of modular curves by replacing \(\operatorname{PSL}_2(\mathbf{Z})\) with triangle groups \(\Delta(a,b,c)\). In the case that \(a=2\), we give a classification of these triangular modular curves. Using Clifford algebras, such curves are naturally associated with Grothendieck’s regular polyhedra over finite fields. Our classification thus answers one of Grothendieck’s questions regarding these polyhedra.

On Thu (9 Jul) at 09:00.

Title

Moduli construction for higher dimensional varieties through Hodge theory and GIT

Abstract

Deligne–Mumford showed that the geometric notion of stability for curves is intimately connected to GIT‑stability of the corresponding Hilbert‑scheme points (for a suitable choice of linearization). Mumford’s construction of the coarse moduli space of stable curves as a GIT‑quotient follows directly from this relationship. In higher dimensions, KSB‑stable varieties, discovered by Kollár and Shepherd-Barron, play the role of stable curves. However, “standard’’ linearizations are known to fail in realizing their coarse moduli space as a GIT‑quotient. Kollár therefore asks whether alternative linearizations might still produce such quotients. We give an affirmative answer: with an appropriate choice of linearization, KSB‑stability implies GIT‑stability. This is based on joint work with S. Kovács.

On Thu (9 Jul) at 10:30.

Title

New invariants for classifying projective toric Fano manifolds

Abstract

In this talk, I will summarise the results of a series of projects that led to defining two new invariants for projective toric Fano manifolds. One, the minimal projective bundle dimension of \(X\), describes the minimal positive \(m\) such that \(X\) is generically a \(\mathbf{P}^m\)-bundle over a toric base. Its study was motivated by the conjecture that there are no toric 2-Fano manifolds apart from \(\mathbf{P}^n\), \(n \geq 2\). Another invariant, the twisting capacity of \(X\), describes how complicated fibre bundles over \(X\) can be. For example, if \(\operatorname{tcap}(X)=0\), then all toric fibre bundles over \(X\) are split. Using this invariant and the criterion for \(X\) being a toric fibre bundle, we can prove a linear bound for Assarf–Joswig–Paffenholz’s 2014 conjecture about the existence of splitting \(S_3\)-factors, which improves the 2016 quadratic bound achieved by Assarf–Nill. Underlying these stories is the deep connection between classes of curves on X and relations among the generators of the fan of \(X\).

On Thu (9 Jul) at 11:30.

Title

Pseudo-coherent complexes on algebraic stacks.

Abstract

Let \(f \colon X \to Y\) be a proper morphism of locally noetherian schemes. Grothendieck’s coherence theorem is the statement that \(R^if_*\) sends coherent sheaves to coherent sheaves. In this talk, I will discuss suitable adaptations of this result to the setting of non-noetherian algebraic stacks. This is joint work in progress with Jack Hall.

On Thu (9 Jul) at 2:30.

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.

On Thu (9 Jul) at 4:00.

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.

On Fri (10 Jul) at 09:00.

Title

On representation zeta functions in positive characteristic

Abstract

Representation zeta functions are Dirichlet generating functions associated with the representation growth of groups, encoding the numbers of irreducible representations of each dimension. The representation zeta functions of \(p\)-adic and arithmetic groups in characteristic zero have been studied extensively over the past two decades. By contrast, the positive-characteristic case remains comparatively unexplored, owing in part to the absence of the orbit method and the limited availability of resolution-of-singularities techniques. In this lecture, I will survey some of the known results in this area and present recent joint work with Amritanshu Prasad and Pooja Singla on the representation zeta functions of groups of type \(A_2\).

On Fri (10 Jul) at 10:30.

Title

Atiyah Flops and Lie Superalgebras

Abstract

This is based on joint work with Mina Aganagic, Jinghang Miao, Spencer Tamagni and Peng Zhou. McKay correspondence for ADE Lie algebras gives geometric realizations of ADE root systems via the ADE surface singularities. More precisely, the Dynkin diagram can be read off from the intersection matrix of the exceptional divisors in the minimal resolutions of the ADE singularities. In this talk we generalize this picture to the case of Lie superalgebra \(\operatorname{gl}(m|n)\). The new phenomenon is that there are multiple inequivalent Dynkin diagrams associated to \(\operatorname{gl}(m|n)\), related by the action of the Weyl groupoid. We will see that this corresponds geometrically to the various minimal resolutions of the 3-fold singularity \(xy = z^mw^n\), related by Atiyah flops.

On Fri (10 Jul) at 11:30.

Title

Relative period relation of abelian differential on Riemann surfaces

Abstract

A translation surface is defined by an abelian differential on a Riemann surface. Every such pair determines a representation called the absolute period representation or period character. In this seminar we discuss the realisation problem of whether a given representation to arise as the period character of some translation surface, possibly with prescribed data such as the order of singularities, spin structure and hyperelliptic structure. Next we focus on the subtle problem of prescribing the so-called relative periods, thereby answering a question posed by Simion Filip. This is joint work with Dawei Chen.

On Fri (10 Jul) at 2:30.

Title

Arithmetic of moduli spaces: Weighted and unweighted point counts and motivic classes in the Grothendieck ring

Abstract

A natural arithmetic question in moduli theory is how many isomorphism classes of a geometric object (say, curves or principally polarised abelian varieties of a fixed type) there are over a fixed finite field. We begin by unravelling the question, the connections of the answer to the cohomology of the moduli space, and discussing known results. Then we’ll introduce Ekedahl’s Grothendieck ring of stacks, a generalisation of Grothendieck’s original construction for varieties, that not only encodes the answer over all finite fields simultaneously, but encodes the information of all additive invariants of the moduli stack. (This is part of work in progress with Di Lorenzo, Grushevsky, Hulek, and Park.)

On Fri (10 Jul) at 4:00.