Scientific Program

Suppose we are given a regular, flat and proper scheme X over the spectrum of a henselian discrete valuation ring A. Let Y denote the special fiber. Then it is an interesting question to relate the Chow group of relative n-cycles on X to the Chow group of n-cycles on Y. If A is for example the ring of integers in a local field, then one can hope to deduce finiteness results for the generic fiber of X from the special fiber. The situation is well understood (the groups are isomorphic) if one considers either (1) the Chow group of relative zero-cycles with finite coefficients or (2) the Chow group of relative zero-cycles integrally but assumes additionally that the generic fiber of X is rationally connected and that Y is smooth. I will explain the definition of Chow groups and ideas for proofs in both situation. This involves a reduction to curves, deformation theory and moduli of curves of genus zero. Subsequently, I will highlight two open questions in the following variants of situation (2). So assume that the generic fiber of X is rationally connected. Firstly: is it possible to prove a similar result for zero-cycles if the special fiber Y is a simple normal crossing divisor? This will involve studying curves and their deformations and combinatorics in this setting. Secondly: assuming Y is smooth, is the Chow group of relative one-cycles with finite coefficients of X isomorphic to that of Y? This will involve deformation theory of surfaces. Heuristically, the second question suggests that relative zero-cycles with finite coefficients on a general X behave similar to relative one-cycles with finite coefficients for rationally connected X.

The quantum Bruhat graph is a combinatorial object, defined solely from a finite Weyl group, encoding many geometric and algebraic information of reductive groups and associated objects. First introduced to compute the quantum cohomology of flag varieties, it was more recently put to use to study Iwahori-Hecke algebras and affine Deligne-Lusztig varieties. Motivated by the generalization of these applications to (non-reductive) Kac-Moody groups, one could try to define quantum Bruhat groups also for infinite Weyl groups. While the naive definition extends naturally, some of its key combinatorial properties are harder to obtain on the nose. In 2025, L. Dean showed that they still hold in type A1 affine (the smallest non-reductive Kac-Moody type) and deduced a Demazure product formula for the associated Iwahori-Hecke algebra. With a good enough understanding of infinite Weyl groups (and of the associated root systems), one could expect to tackle the problem in full generality.

Tropical geometry studies piecewise-linear, combinatorial shadows of degenerations of algebraic varieties. In many cases, familiar algebro-geometric objects, such as divisors and line bundles on curves, have tropical analogues that are closely tied to their classical counterparts. For instance, the theory of divisors and line bundles on metric graphs has played a central role in recent advances in Brill–Noether theory and in the birational geometry of moduli spaces. In this talk, I will present an elementary theory of tropical principal bundles on metric graphs, generalizing tropical line bundles to bundles with arbitrary reductive structure group. Our approach is based on tropical matrix groups arising from the root datum of the reductive group, and leads to an appealing geometric picture: tropical principal bundles can be described as pushforwards of line bundles along covers equipped with Weyl group symmetry. I will also explain how this construction relates to algebraic principal bundles via tropicalization. Building on Frăţilă’s description of the moduli space of semistable principal bundles on an elliptic curve, we study semistable principal bundles on a Tate curve and relate their moduli to a natural component of the tropical moduli space of principal bundles on the associated dual metric graph. I will conclude by discussing open questions and possible further directions, including extensions to tropicalizations of principal bundles on higher-genus curves. This is joint work with Andreas Gross, Martin Ulirsch, and Dmitry Zakharov.

On a smooth projective complex curve X, Simpson's non abelian Hodge correspondence establishes a homeomorphism between the moduli space of degree 0 Higgs bundles on X and the character variety, parametrising representations of the fundamental group of X. For degree d Higgs bundles, the character variety is replaced by a twisted version. In this talk, we explain an alternative construction of this twisted version. The key insight is that the twist appearing in the variety of representations can be thought of as monodromy around an orbifold point. This insight is formalized by adding a stacky point to the curve X and then showing that the moduli space of degree d Higgs bundles on X can be realized as a component of the moduli space of degree 0 Higgs bundles on the stacky curve. This construction can be used to compute the Hodge weights of the tautological classes on the twisted character variety, following a result by Shende.

Counts of conjugacy classes of matrices over finite fields (and more generally, of quiver representations) enjoy interesting numerical properties. They are polynomials in the cardinality of the residue field, with non-negative coefficients. In the case of quiver representations, the coefficients of these polynomials also have deep Lie-theoretic meaning. Much less is known about counts of conjugacy classes over rings of truncated power series over finite fields. For instance, the problem of classifying conjugacy classes of matrices over such rings remains open in size larger than 3. In this talk, I will present some of the cases where these counts are known and exhibit similar polynomiality and positivity. I will also present a geometric approach to computing these counts in certain new cases, based on embedded resolutions of determinantal singularities.

Hilbert schemes of points on smooth surfaces provide natural compactifications of configuration spaces of points. They arise in many areas of mathematics, including the theory of symmetric functions, representation theory, and hyperkähler geometry. Their cohomology rings have a rich structure and admit a deformation, known as quantum cohomology, which encodes information about curves in the space. In this talk, I will introduce these ideas and explain how tropical geometry, a piecewise-linear counterpart to algebraic geometry, can be used to compute the quantum cohomology ring of the Hilbert scheme of points on an elliptic surface. This is joint work with Georg Oberdieck and Aaron Pixton.

Motivated by number-theoretic questions, we study the problem of classifying the matrices over a field of characteristic p that commute with their Frobenius (entrywise p-th power). These matrices are naturally sorted by their ""type"", whose combinatorial information we encode in a quiver. For each (balanced) quiver, we construct a Grassmannian-like variety that parametrizes the matrices of the corresponding type, and we compute the dimension of that variety. Determining the ""generic"" type of a (semi-simple) matrix commuting with its Frobenius then amounts to finding out the quiver(s) for which the dimension is maximal. As we will see, it turns out that the solution to this combinatorial problem is a quiver shaped like an octopus!

We classify all compactifications of M_{g,n} arising as good moduli spaces of stacks of curves with A_i-type singularities, for i < 4. These are realized as Theta-semistable loci with respect to line bundles on a suitable stack. Varying the line bundle yields an involved wall-crossing picture in the space of stability conditions, which can be described combinatorially using cluster algebras. These compactifications arise naturally in the minimal model program for the Deligne-Mumford compactification of M_{g,n}, realizing the Q-factorializations of a space appearing in the Hassett-Keel program.

Spherical varieties are varieties with a reductive group action for which a Borel subgroup acts with a dense orbit. A special class of such actions arises from the representation spaces of quivers under their base change action. In this talk I will give a classification of these spherical represenation spaces. Time permitting, I will also discuss ongoing work and recent developments related to this classification.

Finite subgroups of automorphisms have been an active area of research since foundational work of Nikulin in 1979. We will discuss current progress in positive characteristic and the important role of supersingular K3 surfaces in this setting.

We define an invariant of the module category of a representation-finite algebra by the magnitude of its Auslander algebra. This invariant will be called the magnitude of the module category. For bound path algebras, it can be computed as the Euler characteristic of the Auslander--Reiten quiver, in a suitable sense. We build on classical results in Auslander--Reiten theory to express the magnitude of module categories of families of algebras in terms of other known quantities. Based on our results, we obtain a conjectural characterisation of representation-finite biserial algebras in terms of the magnitude of the module category and the rank of the Grothendieck group. The talk is based on joint work with Erlend D. Børve and Martin Kalck.

The Feigin-Fourier-Littelmann-Vinberg basis provides a basis for finite-dimensional highest representations of sl(n) in terms of monomials of the universal enveloping algebra. This basis, given by lattice points of a polytope, is not only linked to the Gelfand-Tsetlin basis in a combinatorial fashion, but also allows, using its combinatorial properties, for a toric degeneration of the embedded partial flag variety in the respective highest weight space. In joint work with Pranav Enugandla, we constructed for the Lie superalgebra gl(m|n) a monomial basis again given by lattice points of a polytope for representations that we call 'thin-legged' among the covariant representations. This extends previous results from Fourier and Kus from 2021. Further, this new basis provides us with degenerations of embedded partial flag supermanifolds of the supergroup GL(m|n).

We will introduce an algebra generated by semistandard Young tableaux with a simple operation - row concatenation. Using a minimal example, we will illustrate its connection to partial flag varieties and discuss how it can be used to enumerate the corresponding Plücker relations. This is based on joint work with Spencer Daugherty, Nicolle González, Pablo S. Ocal, Jianping Pan and Jacinta Torres.

Given a Fano variety X and a divisor D on X, one may define a special subset of rational points, called Campana points, that interpolates between the notion of integral points on X \ D and rational points on X. The study of integral points on quasi-affine varieties is notoriously difficult, and thus one may hope to gain useful insights by studying these intermediate sets. Our main goal is to show how the geometric construction encapsulates arithmetically interesting subsets of rational points in a natural framework. We present some results on asymptotic counts for the number of Campana points of bounded height on high-dimensional hypersurfaces given by generalised diagonal forms in the spirit of Manin's conjecture on rational points.

In this talk, we introduce a moduli stack of principal G-bundles with a pair of commuting Higgs fields on a smooth projective curve where G is a reductive algebraic group. These objects are in bijection with certain coherent sheaves on a Calabi-Yau threefold. Therefore, the moduli space of these objects admits a vanishing cycle sheaf known as Donaldson-Thomas sheaf. We propose a conjectural equality between the mixed Hodge numbers of these sheaves for Langlands dual groups G = SLn and PGLn using a torus localization on the SLn-moduli space. By computing the mixed Hodge numbers of the torus fixed loci, we prove this conjecture for n = 2. Our result generalizes the topological mirror symmetry of Hausel and Thaddeus to a class of Calabi-Yau threefolds.

Cluster categories and cluster algebras can be described via triangulations of surfaces. For a cluster category of affine type D, the associated surface is a twice punctured disk. We discuss a geometric model for the rank two tubes of the Auslander-Reiten quiver of the cluster category. We extend the model for the tube of rank n-2 given by Baur, Bittmann, Gunawan, Todorov and Yıldırım to the two tubes of rank 2. In order to do that, we extend the definition of generalized tagged arcs to some particular arcs between the two punctures. This talk is based on arxiv:2510.23280.

A quiver and a dimension vector are associated with a reductive group and a quiver representation space that is a module over that group. In 2000, Derksen and Weyman showed that the semigroup consisting of the weights of the semi-invariants of the polynomial algebra over the representation space is described by a finite set of linear inequalities. This remarkable result implies in particular that the semigroup is saturated, thereby generalising the saturation property of Littlewood-Richardson coefficients demonstrated by Knutson-Tao.

In this talk, I will present some results concerning the regularity properties of linear degenerations of flag varieties. I will provide a classification of the smooth linear degenerations of (partial) flag varieties. Moreover, under certain conditions on the dimension vector, I will prove that flat degenerations are regular in codimension 2, generalizing a result by Cerulli Irelli, Feigin, and Reineke for Feigin degenerations. Finally, I will show how these results can be generalized using suitable algebraic group actions and techniques from deformation theory.