Past editions
Past editions
13:30 - 14:30 Reinder Meinsma (Bruxelles)
15:00 - 16:00 Tijs Buggenhout (Leuven)
16:30 - 17:30 Simon Pepin Lehalleur (Amsterdam)
Reinder Meinsma, L-equivalence for hyperkähler manifolds
Two varieties are called L-equivalent if the difference of their classes in the Grothendieck ring of varieties is annihilated by a power of the affine line. Kuznetsov—Shinder conjectured that simply connected derived equivalent varieties must be L-equivalent. I will present examples of hyperkähler manifolds that are derived equivalent but not L-equivalent, giving counterexamples to the conjecture.
Tijs Buggenhout, Geometric dimension growth
The dimension growth conjecture, raised in 1983 by Heath-Brown, asserts uniform upper bounds on the number of rational points on a variety of bounded height. Here uniform means that the bounds depend only on the degree and dimension of the variety and the dimension of the ambient space. In recent years, the conjecture has been almost fully resolved by Salberger. In joint work with Yotam Hendel and Floris Vermeulen, we explore a geometric analogue of the dimension growth conjecture, where we consider ℂ[t]-points of bounded degree on a variety defined over ℂ(t). In general, there are infinitely many such points; however, they form a ℂ-variety in a natural way, whose dimension we can bound.
Simon Pepin Lehalleur, Real jet schemes and the real log-canonical threshold
The real log-canonical threshold is an invariant of singularities of a real analytic function F which controls the analytic behaviour of various associated objects (integrability of powers of F, archimedean zeta function, volume of sublevel sets, exponential integrals, Gelfand-Leray form...) It is the real counterpart of the log-canonical threshold, a classical invariant in complex singularity theory and birational geometry. It also plays a central role in Bayesian statistics of singular parametric models. I will explain how the real log-canonical threshold is related to the geometry of the real jet schemes of F, generalizing results of Mustață and Ein-Lazarsfeld-Mustață in the complex algebraic setting.
13:30 - 14:30 Céline Fietz, Universiteit Leiden, Categorical Resolutions of A_2 singularities
15:00 - 16:00 Michaël Maex, KU Leuven, Edixhoven jumps of jacobians, weight functions and tropical geometry
16:30 - 17:30 Timothy De Deyn, University of Glasgow, Categorical resolutions of filtered schemes
Céline Fietz, Categorical Resolutions of A_2 singularities
Recently, A. Kuznetsov and E. Shinder proved that the bounded derived category of a projective variety with an isolated A_1 (nodal) singularity admits a particularly “small” categorical resolution. Moreover, they show that the “categorical exceptional divisor” is generated by a special kind of coherent sheaf which is closely connected to the spinor bundles on a smooth quadric. In this talk I will recall basics about categorical resolutions and discuss some of the results of my master thesis. There I proved that in the case of a four-dimensional variety these results can be generalised to A_2 (cuspidal) singularities and more precisely, we obtain two “special” generators of the “categorical exceptional divisor” which are related to the spinor sheaves on a nodal quadric. This generalisation is expected to be true in any even dimension, which is work in progress. Ultimately, the goal is to understand the derived category of any variety with isolated ADE singularities.
Master thesis of Céline is available under the address: https://www.math.leidenuniv.nl/~duttay/pdfs/thesis-fietz.pdf
Michaël Maex, Edixhoven jumps of jacobians, weight functions and tropical geometry
Let A be an abelian variety over a discrete valuation field K. There are stil many open questions about the behaviour A and its Néron model under extensions of K. B. Edixhoven defined a tuple of real numbers known as the jumps (j_i) of A which measure the behaviour under tame base change. It is unknown whether (j_i) are rational numbers. More is understood when A is the jacobian of a curve C. Here the rationality of (j_i) is known, but the existing proofs are lengthy and technical.
In this talk I will discuss my recent work joint with E. Kaya and A. Waeterschoot, which shows that the jumps of the Jacobian of a curve C can be computed from weight functions. These functions are associated to canonical forms on C and live on the Berkovich analytification of C. As a corollary we obtain a short proof of the rationality of (j_i). Everything can be made explicit in the tropical framework of Δ_v-regular curves by T. Dokchitser, giving a large class of examples.
Timothy De Deyn, Categorical resolutions of filtered schemes
A. Kuznetsov and V. Lunts showed that over a field of characteristic zero one can always construct a categorical resolution of singularities. Their approach requires a strong version of Hironaka's resolution of singularities, namely that any variety can be resolved by a sequence of blow-ups along smooth centres. In the first part of the talk I will introduce categorical resolutions and Kuznetsov--Lunts' results. Thereafter I will discuss my recent work in which the use of strong Hironaka is circumvented: only the existence of projective resolutions is needed. For this the framework of filtered schemes is paramount. Finally, if time permits, I will explain ongoing work with M. Van den Bergh on generalising the construction to certain mild noncommutative varieties.
13:30 - 14:30 Boaz Moerman (Utrecht), Generalized integral points and strong approximation
14:30 - 15:00 Coffee
15:00 - 16:00 Dan Bath (Leuven), Polynomials attached to Feynman diagrams have rational singularities
16:00 - 16:30 Coffee
16:30 - 17:30 Jesse Vogel (Leiden), An Arithmetic-Geometric Correspondence for Character Stacks
18:15 - ... Dinner
The Chinese remainder theorem states that given coprime integers p_1, ..., p_n and integers a_1, ..., a_n, we can always find an integer m such that m ~ a_i mod p_i for all i. Similarly given distinct numbers x_1,..., x_n and y_1, ..., y_n we can find a polynomial f such that f(x_i)=y_i. These statements are two instances of strong approximation for the affine line. In this talk we will consider when an analogue of this holds for special subsets of Z and k[x], such as squarefree integers or polynomials without simple roots. We give a precise description for which subsets this holds on a toric variety.
In doing so, we obtain insight into integral points and the embeddings of curves on such varieties.
Given a matroid on ground set E we define and consider a large class of polynomials in C[E], the polynomial ring whose variables correspond to elements of the ground set, assembled via the matroid structure. We show that all of these polynomials have rational singularities. The method involves jet schemes and Mustata's characterization of rationality in terms of irreduciblity of jet spaces.
Within our class of polynomials are three important members: matroid basis polynomials; configuration polynomials; polynomials attached to Feynman diagrams. The latter is constructed out of the former in a straightforward way. In the world of Quantum Field Theory and Feynman diagrams, one instance of our rationality result is the following: given a Feynman diagram with standard simplifying assumptions, the Feynman integral is the Mellin transformation of a polynomial with rational singularities.
Joint with Uli Walther.
The moduli space of G-local systems on a closed oriented surface, also known as the G-character stack of the surface, is an extensively studied object. In the literature, two approaches exist to compute certain algebraic invariants of these spaces. The arithmetic method, initiated by Hausel and Rodriguez-Villegas, computes the point count of these spaces over finite fields, in terms of the representation theory of the finite groups G(𝔽_q). The geometric method, initiated by Logares, Muñoz and Newstead, computes their virtual class in the Grothendieck ring of varieties using clever stratifications. We describe a framework, in terms of Topological Quantum Field Theories, that unifies these approaches.
This talk is based on arXiv:2309.15331, which is joint work with Márton Hablicsek and Ángel Gonzalez-Prieto.