Algebraic Geometry Seminar
@

Location: HFG 7.07

Location: HFG 7.07

Title: $T$-deformations of vertex algebras and tautological stable pair wall-crossing

Abstract: There are two natural ways to deform Joyce's construction of vertex algebras. One is to work equivariantly, and the other is to include tautological insertions. I introduce $T$-deformed vertex algebras that unify these two refinements. For the purpose of wall-crossing, they must be treated differently because the associated Lie algebras use different expansions. I will explain this behavior explicitly by applying it to 2-dimensional stable pair wall-crossing, focusing on conjectures of Bae-Kool-Park.

Location: HFG 7.07

Title: Points of low degree on curves over function fields

Abstract: The degree of a closed point on a curve is the degree of the field extension given by the residue field of the point extending the ground field. Theorems by Faltings, Harris-Silverman, Abramovich-Harris and Kadets-Vogt geometrically classify curves over number fields having a potentially dense set of points of degree 1, 2, 3, 4 and 5. I will explain how to generalize these results to curves over function fields of characteristic zero.

Location: HFG 7.07

Title: An Inverse Galois Problem for minimal del Pezzo surfaces of degree 1 with conic bundles over finite fields

Abstract: We study del Pezzo surfaces of degree 1 over finite fields, which are a class of rational surfaces. These surfaces can be divided into 112 types, based on the Galois action on the surface. We focus on the 7 types corresponding to surfaces which are both minimal and have a conic bundle structure. These types are determined by the configuration of the singular fibers of the conic bundle structure. We discuss an answer to the following question: for which values of $q$ can each of these configurations exist on a del Pezzo surface with conic bundle defined over $\mathbb{F}_q$?

Location: HFG 7.07

Title: Crepant resolution of threefold singularities

Abstract: Terminal threefold singularities do not always admit crepant resolution (even locally!). If we go categorical, such resolutions, once carefully defined, are expected to exist. In this talk, I will show an explicit construction for some of the simplest singularities, namely, $A_n$, i.e. the ones that are locally given by $x^2+y^2+z^2+w^{n+1}$. When $n$ is odd, the resolution has Calabi-Yau kernel category. This is a joint work in progress with Fietz and Shinder.

Location: HFG 7.07

Title: Boundedness of Lagrangian fibrations

Abstract: I will discuss recent work showing that deformation classes of symplectic varieties admitting a Lagrangian fibration, and of fixed dimension, are finite in number. Conditional on the generalized semiampleness conjecture, we deduce a bound on the number of deformation classes of hyperkahler varieties in a fixed dimension, with second Betti number at least 5. This is joint work with Filipazzi, Greer, Mauri, and Svaldi.

Location: BBG 1.61

Title: Tautological Classes on the Moduli Space of Curves via Spin Structures

Abstract: The tautological ring of the moduli space of curves is a subring of its Chow ring. Understanding whether naturally defined classes lie in this subring has been a fundamental problem in the study of the intersection theory of the moduli space of curves. In this talk, we will define Chow classes arising from spin structures on curves, that is, square roots of the canonical line bundle, compute them, and prove that they are tautological. If time permits, we will also discuss applications to the computation of integrals. This talk is based on joint work with Adrien Sauvaget and David Holmes, and with Stijn Velstra and Andrei Bud.

Location: HFG 7.07

Title: Tate-Shafarevich group of polarized K3 surfaces

Abstract: The Tate-Shafarevich group of an elliptic K3 surface $S$ with a section parametrizes twists of $S$, that is, other elliptic K3 surfaces that are étale locally isomorphic to $S$. By a result of Artin and Tate, this group is in fact isomorphic to $\operatorname{Br}(S)$. I will present joint work with D. Huybrechts in which we generalize this story in higher dimensions by defining and studying the Tate-Shafarevich group of fibrations in Jacobians of curves of arbitrary genus on K3 surfaces, and its relation to the (special) Brauer group.

Location: HFG 7.07

Title: Bondal-Orlov’s reconstruction theorem in noncommutative projective geometry

Abstract: The (derived) category of coherent sheaves on a scheme encodes rich information about the underlying geometry. P. Gabriel showed that for noetherian schemes $X$ and $Y$, if $\operatorname{Coh} X$ and $\operatorname{Coh} Y$ are equivalent as abelian categories, then $X$ and $Y$ are isomorphic. Furthermore, A. Bondal and D. Orlov proved that for smooth projective schemes $X$ and $Y$ with (anti-)ample canonical bundles, if $\mathbf{D}^b(\operatorname{Coh} X)$ and $\mathbf{D}^b(\operatorname{Coh} Y)$ are equivalent as triangulated categories, then $X$ and $Y$ are isomorphic. On the other hand, J.-P. Serre showed that the category of coherent sheaves on a projective scheme can be described as the quotient category of finitely generated graded modules over the homogeneous coordinate ring by the subcategory of torsion modules. Motivated by the results of Gabriel and Serre, the quotient category of finitely generated graded modules over a (not necessarily commutative) graded ring by the subcategory of torsion modules is called a noncommutative projective scheme. In this talk, I will present an analogue of Bondal–Orlov’s reconstruction theorem in the setting of noncommutative projective geometry.

Location: HFG 7.07

Location: HFG 7.07

Title: Stable toric sheaves

Abstract: In the seventies, Hartshorne proposed several problems regarding the existence of low rank vector bundles on projective spaces. In rank 2 and characteristic 0, those bundles are notoriously difficult to produce, and Hartshorne conjectured that none should exist from dimension seven. In this talk, I will explain some motivations for this conjecture, and a new approach to the problem, by mean of toric sheaves.

Location: HFG 7.07

Title: Quiver moduli and their geometry

Abstract: Quiver moduli spaces are algebraic varieties encoding the continuous parameters of linear algebra type classification problems. In recent years their topological and geometric properties have been explored, and applications to, among others, Donaldson-Thomas and Gromov-Witten theory have emerged. The first aim of the lectures is to motivate the study of quiver moduli spaces from the point of view of representation theory, to review their construction via Geometric Invariant Theory, and to discuss several classes of examples. We will proceed by reviewing results on the topology and geometry of these moduli spaces, in particular their cohomology.

Location: HFG 7.07

Title: Wild Brauer groups and rational points

Abstract: The behaviour of rational points on a variety is strongly influenced by its geometry. For example, one can ask questions about so-called “local-global” principles, and the answers to these questions depend on the geometry of the variety. One way in which this link manifests itself is through the Brauer group. With Rachel Newton we studied the “wild” part of the Brauer group and proved the surprising fact that every variety over a number field admitting a non-zero global 2-form and having a prime of good ordinary reduction fails weak approximation, possibly after a finite extension of the base field. I will describe this result and more recent developments by Ambrosi, Newton and Pagano.

Location: HFG 7.07

Title: Stability of line bundles and vector bundles on some surfaces

Abstract: Donaldson and Uhlenbeck-Yau established the classical result that on a compact Kahler manifold, an irreducible holomorphic vector bundle admits a Hermitian metric solving the Hermitian-Yang-Mills equation if and only if the vector bundle is Mumford-Takemoto stable. A modern analog of this question was posted by Collins-Yau. In this talk, we will discuss partial answers to this modern analog for a set of line bundles and tangent/cotangent bundles on some surfaces. This is based on joint work/work in progress with Tristan Collins, Jason Lo, and Shing-Tung Yau.

Location: HFG 7.07

Title: Some torsion and divisibility phenomena in higher Chow groups

Abstract: We study the injectivity property of certain actions of higher Chow groups on refined unramified cohomology. As an application for every $p \geq 1$ and for each $d \geq p+4$ and $n \geq 2$, we establish the first examples of smooth complex projective $d$-folds $X$ such that for all $p+3 \leq c \leq d-1$, the higher Chow group $\operatorname{CH}^{c}(X,p)$ contains infinitely many torsion cycles of order $n$ that remain linearly independent modulo $n$. Our bounds for $c$ and $d$ are optimal. A crucial tool for the proof is Lawson homology.

Location: HFG 7.07

Title: When is a polarised abelian variety determined by its $p$-divisible group?

Abstract: We will study the moduli space of abelian varieties in characteristic $p$ and in particular its supersingular locus. We first determine precisely when this locus is geometrically irreducible. Since it was known that the number of components is a class number, this comes down to solving a “class number one problem” or “Gauss problem”. Next, we will show when a polarised abelian variety is determined by its $p$-divisible group. This can be viewed as a Gauss problem for central leaves, which are the loci consisting of points whose associated $p$-divisible groups are isomorphic. Our solution involves mass formulae, computations of automorphism groups, and a careful analysis of Ekedahl-Oort strata in genus 4. This is based on joint work with Ibukiyama and Yu.

Location: HFG 7.07

Title: Quadratic Donaldson-Thomas invariants: computations and a conjecture

Abstract: (Zero-dimensional) Donaldson-Thomas-invariants "count" ideal sheaves of a given length which have zero-dimensional support on a smooth projective complex threefold. In case the threefold is toric, Maulik, Nekrasov, Okounkov and Pandharipande have proven a formula for the generating series of these Donaldson-Thomas invariants in terms of the MacMahon function. We discuss a conjectural quadratically enriched analogue of this result for smooth projective real threefolds satisfying an orientation condition, using a quadratic version of Donaldson-Thomas invariants taking values in Witt rings, which are constructed using work of Levine. We provide evidence for the conjecture coming from computations for $\mathbb{P}^3$ and $(\mathbb{P}^1)^3$. This talk is based on my thesis and on joint work with Marc Levine.

Location: HFG 7.07

Title: Logarithmic generalised Kummer varieties

Abstract: Generalised Kummer varieties, are one of the main examples of hyperkähler varieties. The moduli space of generalised Kummer varieties is far from compact, and finding modular compactifications is an old question. This talk explains how logarithmic and tropical geometry can be used to construct a modular compactification of the generalised Kummer locus inside the moduli space of hyperkähler varieties.

Location: HFG 7.07

Title: The Mahler measure of exact polynomials and special values of $L$-function of motives

Abstract: The Mahler measure of polynomials was introduced by Mahler in 1962 as a tool to study transcendental number theory. Over time, numerous connections have been discovered between Mahler measure and special values of $L$-functions. In this talk, we express the Mahler measure of an exact polynomial in arbitrarily many variables in terms of Deligne–Beilinson cohomology, and study its relationship with Beilinson regulators, and then with special values of $L$-function of motives. As an application, we show that under some conditions, the Mahler measure of three-variable polynomials can be expressed in terms of special values of the elliptic curve $L$-function and the Bloch–Wigner dilogarithm. In the four-variable case, the Mahler measure of certain polynomials can be written as a $\mathbb{Q}$-linear combination of special values of the $L$-function of K3 surfaces and the Riemann zeta function.

Location: HFG 7.07

Title: Abelian varieties with elliptic subgroups

Abstract: The locus of principally polarized abelian varieties with elliptic subgroups provides interesting cycles on the moduli space $A_g$. The cycles provide non-tautological classes. The interaction between these cycles, the locus of Jacobians and the tautological projection (recently developed by Canning, Molcho, Oprea and Pandharipande) gives a prediction for Gromov-Witten invariants of a family of elliptic curves, which now have been verified in joint work with Rahul Pandharipande and Hsian Hua Tseng. In the talk I will recall the definition of the tautological ring of $A_g$, explain the main results on Noether-Lefschetz cycles and discuss the ideas between two proofs of the mentioned prediction. If time allows, I will talk about modularity of these cycles.

Location: HFG 7.07

Title: Degenerations of abelian varieties associated to regular matroids

Abstract: Mumford constructed degenerations of abelian varieties in his influential paper ''An analytic construction of degenerating abelian varieties over complete rings.'' In this second lecture, I will explain the main ideas behind the Mumford construction in the complex analytic setting, in which toric geometry plays a key role. I will also introduce the notion of matroid, and explain how to associate a degeneration of abelian varieties over a higher dimensional base to a regular matroid. This lecture will be based on the paper ''Combinatorics and Hodge theory of degenerations of abelian varieties: A survey of the Mumford construction'' by Engel, Schreieder and myself.

Location: HFG 7.07 See the Mathematical Institute Calendar for rest of the lectures.

Title: Topologically twisted Yang-Mills theories and invariants of moduli spaces

Abstract: I will discuss invariants of moduli spaces of instantons on a four-manifold, and their relation to topologically twisted Yang-Mills theory. After reviewing general aspects, I will explain recent work on Segre invariants and $K$-theoretic Donaldson invariants, and the physical theories whose correlations functions capture these invariants.

Location: HFG 7.07

Title: How to compute enumerative invariants from residues?

Abstract: Many examples of enumerative invariants appearing in the context of algebraic geometry are given by integrating over the moduli space of maps, sheaves, and so on. When there exists a nice group action on it, we may compute such invariants based on the fixed point contributions, i.e., the equivariant localization formula. In this talk, I'd like to explain an alternative use of the localization formula based on the Jeffrey-Kirwan residue formula for multi-variable contour integrals. The key idea is to use the residue of the poles appearing in the integral to classify and evaluate the fixed point contribution under the equivariant action. I'll demonstrate this formalism with several examples, including the equivariant DT/PT invariants for the affine spaces $\mathbb{C}^3$ and $\mathbb{C}^4$.

Location: HFG 7.07

Title: Failure of the integral Hodge conjecture for abelian varieties and the existence of non stably rational cubic threefolds

Abstract: In this first lecture, I will first provide an introduction to the general framework: the integral Hodge conjecture, which is a property that can hold or fail for smooth projective varieties over the complex numbers. Examples of varieties that fail that property were provided for the first time by Atiyah and Hirzebruch. Nonetheless, there are interesting classes of varieties that do satisfy the integral Hodge conjecture. I will also explain in this lecture the connection between the integral Hodge conjecture for abelian varieties and (stable) rationality of cubic threefolds, as discovered by Clemens—Griffiths and Voisin. I will present the two main theorems of our work, which imply that the integral Hodge conjecture for abelian varieties fails, and that not every smooth cubic threefold is stably rational. I will sketch an overview of the proof.

Location: HFG 7.07

Title: Formalising the étale fundamental group of a scheme

Abstract: Formalising mathematics is the process of teaching mathematics to a proof assistant such as Lean. While sometimes this is a mechanical translation process, most often it requires mathematical insights on how to encode the mathematical objects efficiently in a formal language. Lean's mathematical library mathlib aims to provide unified foundations and definitions for all of mathematics. This objective requires every formalisation to fit in a bigger picture, often leading to a gap between pen-and-paper mathematics and its formal counterpart. In my talk, after giving a brief introduction to Lean, I will explain how we bridge this gap in the area of algebraic geometry using the example of the formalisation of the étale fundamental group of a scheme highlighting some of the unforeseen difficulties encountered during the formalisation.

Location: Minnaert 2.01

Title: The projective coinvariant algebra

Abstract: The coinvariant algebra, the quotient of the coordinate ring of $(\mathbb{A}^1)^n=\mathbb{A}^n$ by the ideal generated by positive degree invariant polynomials, plays a basic role in algebraic combinatorics and the representation theory of the symmetric group $S_n$, equipping its regular representation with a graded algebra structure. Using the coordinate ring of $(\mathbb{P}^1)^n$ in its Segre embedding, I will introduce a degeneration of the coinvariant algebra, the projective coinvariant algebra, which gives a bigraded structure on the regular representation of $S_n$ with interesting Frobenius character that generalises a classical result of Lusztig and Stanley. I will also show how this algebra contains bigraded versions of partial coinvariant algebras, coming from coordinate rings of all possible Segre embeddings. Relations to Haiman's diagonal coinvariant algebra, and a certain equivariant Hilbert scheme, will also be discussed.

Location: HFG 7.07

Title: Characterizing random polynomials

Abstract: How likely is a polynomial pulled from a box likely to have certain characteristics? We'll make this question precise in different ways and give some answers using a delightful interplay of tools and techniques. This journey is as much about the statistics as it is about connecting the mathematical landscape.

Location: HFG 7.07

Title: On a theorem of Lafforgue

Abstract: A folklore theorem attibuted to Laurent Lafforgue states that the thin Schubert cell of a rigid matroid is the union of finitely many torus orbits. After a gentle introduction to matroids and thin Schubert cells, bands and matroid representations, we explain a short conceptual proof of this theorem, which traces it back to the Bieri-Groves theorem for tropicalizations. All of this is joint work with Matt Baker.

Location: HFG 7.07

Location: HFG 7.07

Title: Equivariant wall-crossing for Calabi-Yau fourfolds

Abstract: I will present a formulation of equivariant wall-crossing for CY fourfolds in terms of additive deformations of vertex algebras à la Joyce. The proof of wall-crossing formulae will then be sketched on the example of CY4 quivers. Using a quantum-Lefschetz-type argument, I will motivate why the relevant invariants are well-defined. At the end, I will provide immediate and future applications of these results.

Location: HFG 7.07

Title: Stable pair invariants on local curves

Abstract: Curve counting on 3-folds is a rich subject with deep connections to physics and representation theory. There exists a multitude of approaches such as Gromov-Witten and Donaldson-Thomas theory, however all of these approaches suffer from redundancies and it is not known in general whether these yield equivalent answers. More recently, John Pardon found a more universal framework that connects these and reduces their differences to so-called local curve contributions. We give closed formulas for these local curve contributions in the case of stable pair theory. Time permitting, we also sketch the connections of our results to the theory of quantum integrable systems.

Location: HFG 7.07

Title: Ramified periods and field of definition

Abstract: In a joint work with Dragos Fratila and Alberto Vezzani, we construct hyperelliptic curves of large genus, defined over quadratic fields that are isomorphic to their Galois conjugates but do not descend to $\mathbb{Q}$. The obstruction to descent is new and we call it "ramified periods." These are $p$-adic numbers that arise from the comparison between de Rham cohomology and crystalline cohomology (hence the term periods). These numbers can reveal interesting information if $p$ ramifies in the quadratic field.

Location: HFG 7.07

Title: Bounding unlikely intersections on elliptic surfaces

Abstract: Consider a Jacobian elliptic surface $E$ with a section $P$ of infinite order. How often are multiples of $P$ tangent to the zero section? These are unlikely intersections, and over the complex numbers, we bound their number using the "Betti foliation" and a certain real analytic 1-form. It was not evident how to do this over a general base field, but using a descent map we find an analogue in characteristic $p$, and translating back to characteristic zero, we find another class of tangencies and a bound on them. These are joint works with Giancarlo Urzua and Felipe Voloch.

Location: Springer Room (HFG Floor 7)

Title: Motives of moduli spaces of bundles on curves

Abstract: I will give an overview of various motivic results for moduli spaces of (semistable) vector bundles and Higgs bundles of coprime rank and degree on a smooth projective curve. These classical moduli spaces are related to various other interesting moduli spaces in representation theory and mathematical physics, and the fact that their cohomology is tautologically generated played an important role in recent proofs of the $P=W$ conjecture. The aim of this talk is to explain a motivic incarnation of this tautological generation: that the motives of these moduli spaces are generated by the motive of the curve. For $\operatorname{SL}$-Higgs moduli spaces, which are non-tautologically generated, we additionally need motives of certain etale covers of the curve. At the end, I will explain how conservatively properties for abelian motives can be harnessed to obtain motivic formulae and provide motivic lifts of known cohomological phenomena, such as $\chi$-independence and mirror symmetry. This is joint work with Simon Pepin Lehalleur and partially also with Lie Fu.

Location: HFG 7.07

Title: Symmetric products and puncturing Campana-special varieties

Abstract: In 2001, Hassett and Tschinkel posed the following "puncturing problem": If $X$ is a projective variety with at most canonical singularities such that no finite etale cover of $X$ dominates a variety of general type and $Z$ is a closed subset of $X$ of codimension at least 2, does it follow that no finite etale cover of $X \setminus Z$ dominates a variety of log-general type? Following the philosophy that maps to varieties of general type should be the main obstruction to density of rational points, they also suggested the following "arithmetic puncturing problem": With $X$ and $Z$ as above, if the rational points on $X$ are potentially dense, are the integral points on $X - Z$ potentially dense? In this talk, I will explain how symmetric powers of products of curves provide counterexamples to both of these puncturing problems. On the other hand, conjectures of Campana suggest that the arithmetic puncturing problem has a positive answer if we additionally assume $X$ to be smooth. This is joint work with Ariyan Javanpeykar and Aaron Levin.

Location: HFG 7.07

Title: Quasimaps of surfaces

Abstract: After reviewing the relevant concepts in enumerative geometry,I will present a correspondence between Pandharipande-Thomas theory of Calabi-Yau fourfolds, of the form total space of a rank-two vector bundle over a compact Kahler surface, and certain gauge theories living on the surface. The link is via twisted quasimaps with (fixed) domain the surface, which are shown to be isomorphic to PT1 counts. Interpreting quasimaps as the Higgs branch of some gauge theory, I will explain how to compute $K$-theoretic partition functions on the Coulomb branch via supersymmetric localization, and as a result fix all signs in equivariant PT1 theory of such toric fourfolds from first principles. Based on upcoming work with E. Diaconescu.

Location: HFG 7.07

Title: Strong approximation and Brauer-Manin obstruction for homogeneous spaces

Abstract: For an algebraic variety $X$ over a number field $k$, strong approximation studies the density of rational points inside the set $X(\mathbb{A}_k^S)$ of adelic points away from a finite set $S$ of primes, generalizing the Chinese Remainder Theorem. When strong approximation fails, we want to understand the closure of rational points inside $X(\mathbb{A}_k^S)$. For $X$ a homogeneous space under a semisimple simply connected group with commutative stabilizers, we give conditions showing when the closure of the rational points is equal to the set of adelic points cut out by certain elements from the Brauer group.

Location: HFG 7.07

Title: Counting rational curves with prescribed tangency conditions: a motivic analogue via universal torsors

Abstract: Given a smooth projective and geometrically irreducible curve $C$ and a Mori Dream Space $X$, we present a general parametrisation of morphisms from $C$ to $X$ which allows us to express the Grothendieck motive of $\operatorname{Hom}(C,X)$ as a motivic function defined on some power of the scheme of effective divisors of $C$, generalising previous works of Bourqui. Such a parametrisation should be understood as lifting our morphisms to the universal torsor of $X$. As an application, we prove a motivic analogue of a variant of Manin's conjecture for Campana curves on smooth projective split toric varieties (arXiv:2502.11704).

Location: HFG 7.07

Title: Ceresa Cycles of Modular Curves

Abstract: The Ceresa cycle is an algebraic cycle attached to a curve with a marked point. Although it is always homologically trivial, Ceresa proved that for a very general complex curve of genus at least 3, this cycle is algebraically non-trivial. Notably, hyperelliptic curves (with a marked Weierstrass point) have trivial Ceresa cycle, but there are few other explicit examples where triviality/non-triviality is known. In this talk I will discuss the non-vanishing of the Ceresa cycle attached to the modular curve $X_0(N)$. This is joint work with James Rawson.

Location: HFG 7.07

Title: $u$-power torsions of prismatic cohomology

Abstract: In this talk, we will explain relation between $u$-power torsions in Breuil--Kisin prismatic cohomology and various pathologies in $p$-adic cohomology theories, as well as mention some new results. Part of the talk will be based on earlier joint works with Tong Liu, we shall also report some recent ongoing projects with Ofer Gabber and Alexander Petrov separately.

Location: HFG 7.07

Title: Finite subgroups of automorphisms on K3 surfaces

Abstract: Finite order automorphisms of K3 surfaces have been an active area of research since foundational work of Nikulin in 1997. A classification of finite subgroups of automorphisms is known over the complex numbers due to work of Mukai, Brandhorst, Hashimoto and Hofmann. But it has been noted by Dolgachev and Keum that this classification generally cannot hold in positive characteristic due to the existence of supersingular K3 surfaces. In the classification of finite groups of symplectic automorphisms on supersingular K3 surfaces, there has been recent progress by Ohashi, Schütt, Wang and Zheng in 2024. In this talk, we will give a historic overview of the results in this area and discuss progress of ongoing work to extend the results of Ohashi and Schütt to non-symplectic finite groups acting on the supersingular K3 surface of Artin invariant one.

Location: Minnaert 0.14

Title: The Derived Category of the Hilbert Scheme of Three Points

Abstract: After introducing Hilbert schemes of points on smooth varieties $X$, I will motivate the study of their invariants (in terms of invariants of $X$) in the case of up to three points and in arbitrary dimension. The focus lies especially on a description of the bounded derived category of the Hilbert scheme of three points by means of a semi-orthogonal decomposition, for which an explicit conjecture is presented. I will report on proven parts of the conjecture, to appear soon in my doctoral thesis. Special attention is drawn to a semi-orthogonal sequence obtained from Fourier-Mukai transforms which embed the derived category of $X$ into the one of its Hilbert scheme of three points. The proof of fully-faithfulness of these Fourier-Mukai transforms leads back to the geometry of Hilbert schemes of points, in particular to normal bundle computations on Grassmannian bundles.

Location: Minnaert 0.14

Title: Highest weight category structures on $\operatorname{Rep}(B)$ and full exceptional collections on generalized flag varieties

Abstract: Say the base field is algebraically closed. Given a semi-simple simply connected algebraic group $G$ and a parabolic subgroup $P \subseteq G$, we construct $G$-linear semiorthogonal decompositions of the bounded derived category of finite dimensional representations of $P$, with each semiorthogonal component being equivalent to the bounded derived category of finite dimensional representations of $G$. The $G$-linear semiorthogonal decompositions in question are compatible with the Bruhat order on cosets of the Weyl group of $P$ in the Weyl group of $G$. Their construction builds upon the foundational results on $B$-modules from the works of Mathieu, Polo, and van der Kallen, and upon properties of the Steinberg basis of the $T$-equivariant $K$-theory of $G/B$. As a corollary, we obtain full exceptional collections in the bounded derived category of coherent sheaves on generalized flag varieties $G/P$. This is joint work with Alexander Samokhin.

Location: Minnaert 0.14

Title: Locally Recoverable Codes with availability from surfaces with multiple fibrations

Abstract: Locally Recoverable Codes (LRCs) are a type of error-correcting codes with major applications in cloud storage systems, as they allow to retrieve corrupted (or lost) data by only accessing a small portion of the remaining information. Over the last decade, new LRCs with special properties have been obtained with constructions from algebraic varieties over finite fields. In this talk, I will present a new construction of LRCs from surfaces admitting multiple fibrations. These codes have the availability property, which ensures an extra level of data protection. Joint work with Cecília Salgado.

Location: DDW 1.36

Title: Categorical absorption for hereditary orders

Abstract: The idea of categorical absorption of singularities can be used to study the bounded derived category of a flat family of varieties over a smooth curve. We explain how this setup allows the construction of a semiorthogonal decomposition in special situations. Extending this framework to the noncommutative setting, we interpret a hereditary order as a family of finite-dimensional algebras over a curve and construct a semiorthogonal decomposition of the bounded derived category of a hereditary order.

Location: HFG 7.07

Title: Modular Curves and the Asymptotic Fermat Conjecture

Abstract: The modular method became one of the fundamental tools for solving Diophantine equations following Wiles' remarkable work on Fermat's Last Theorem. In this talk, we will provide an overview of Wiles' proof and explore how it can be extended. Additionally, we will discuss the difficulties encountered in this extension.

Location: Minnaert 0.13

Title: On the transcendental part of K3 surfaces associated with 3D Fano polytopes

Abstract: Up to affine transformations over $\mathbb{Z}$ there are 18 different 3D Fano polytopes. The set of vertices of such a polytope is a subset $V$ of $\mathbb{Z}^3$ which can be used as exponents for a Laurent polynomial. The surface in $\mathbb{P}^3$ defined by the homogenization of such a Laurent polynomial is a quartic K3 surface. Varying the coefficients of the Laurent polynomial yields a family of K3 surfaces. The aim of the talk is to demonstrate how the Gelfand-Kapranov-Zelevinsky hypergeometric system associated with $V$ and results on Mirror Symmetry for lattice polarized K3 surfaces lead to simple elegant expressions for the transcendental periods as functions of the coefficients of the Laurent polynomial.

Location: DDW 1.36

Title: Rank growth of elliptic curves over $S_3$ extensions with fixed quadratic resolvents

Abstract: In joint work with Daniel Keliher, we study the probability with which an elliptic curve, subject to some technical conditions, gains rank upon base extension to families of $S_3$ cubic extensions with a fixed quadratic resolvent field, all three types of fields of which are subject to some mild technical conditions. We determine the distribution (under a non-standard ordering) of Selmer ranks of an auxiliary abelian variety associated to the elliptic curve and cubic extensions by adapting previous studies by Klagsbrun, Mazur, and Rubin. One corollary of this distribution is that the elliptic curve gains rank by at most one upon base change to cubic extensions with probability at least 31.95%.

Location: Minnaert 0.14

Title: Shifted symplectic pushforwards

Abstract: Fundamental examples of symplectic varieties are moduli spaces of sheaves on K3 surfaces. This can be extended to higher-dimensional Calabi-Yau varieties through the concept of shifted symplectic structures in derived algebraic geometry. In this talk, I will introduce a general operation of producing shifted symplectic stacks from given ones. Basic examples like cotangent bundles, critical loci, and Hamiltonian reduction can be understood as special cases of this operation. Moreover, this unification enables us to provide an etale local structure theorem for shifted symplectic Artin stacks. I will briefly explain some applications to Donaldson-Thomas theory of Calabi-Yau 3-folds and 4-folds.

Location: Minnaert 0.09

Title: Refined Chabauty--Kim for the thrice-punctured line over $\mathbb{Z}[1/6]$

Abstract: If $X$ is a curve of genus at least 2 defined over the rational numbers, we know by Faltings's Theorem that the set $X(\mathbb{Q})$ of rational points is finite but we don't know how to systematically compute this set. In 2005, Minhyong Kim proposed a new framework for studying rational (or $S$-integral) points on curves, called the Chabauty-Kim method. It aims to produce $p$-adic analytic functions on $X(\mathbb{Q}_p)$ containing the rational points $X(\mathbb{Q})$ in their zero locus. We apply this method to solve the $S$-unit equation for $S=\{2,3\}$ and computationally verify Kim's Conjecture for many choices of the auxiliary prime $p$.

Location: HFG 611

Title: How to tropicalize algebraic groups (and why we should care) - Log edition

Abstract: Tropical geometry studies a piecewise linear combinatorial shadow of classical algebraic geometry. In many ways, tropical geometry describes the extra information that is added to classical scheme theory in logarithmic geometry in order to study moduli-theoretic problems from a unified perspective. In this talk I will explain what little we currently know about the tropical geometry of algebraic groups. Along the way I will outline how different avenues of progress towards an answer to this question have already led to numerous advantages within and beyond tropical and logarithmic geometry. This talk will touch upon joint work with Luca Battistella, Desmond Coles, Andreas Gross, Inder Kaur, Kevin Kühn, Arne Kuhrs, Margarido Melo, Sam Molcho, Annette Werner, Alejandro Vargas, Filippo Viviani, and Dmitry Zakharov.

Location: BBG 065

Title: Noncommutative plane curves through the looking glass

Abstract: Noncommutative plane curves are plane curves on noncommutative projective planes, defined by a homogeneous central element in a 3-dimensional Artin--Schelter regular algebra. I will introduce this approach to noncommutative algebraic geometry, and explain how noncommutative plane curves have been studied and classified in degree 2, and some examples have been considered in degree 3, but a general theory did not exist previously. Introducing the notion of central curves for orders on surfaces and using the fact that the noncommutative projective planes in question are finite over their centers, we can translate the algebraic questions into geometric problems and solve them using geometric methods. This also gives further insight into the dictionary between noncommutative algebra and algebraic stacks. This is joint work with Thilo Baumann and Okke van Garderen.

Location: BBG 061

Title: Geometry of Newton strata via automophic forms

Abstract: In char $p > 0$, Newton strata arise as decomposition of the moduli space of principally polarised abelian varieties where the Newton polygon associated to the abelian variety is constant. More generally, the Newton stratification can be defined on the special fibre of Shimura varieties. In this talk, I will outline an approach to study the geometry of the Newton strata via automorphic forms. This approach involves a comparison of two trace formulas: (1) Leftschetz-Verdier trace formula which relates the actions of Frobenius and Hecke operators on the cohomology of a Newton stratum to fixed points of these operators and (2) Arthur-Selberg trace formula which links automorphic representations of a group to orbital integrals. Kret carried out this approach to describe the zeta function of the basic stratum of Kottwitz varieties at split primes of good reduction in terms of automorphic forms. Kottwitz varieties are a very restrictive class of Shimura varieties since they have no endoscopic contributions to their cohomology. Endoscopic contributions are the source of several complications which arise in establishing the above mentioned comparison of trace formulas for Newton strata of more general classes of Shimura varieties. I will discuss my progress towards understanding the endoscopic contributions and establishing the comparison of trace formulas for Newton strata of abelian type Shimura varieties at places of good reduction.

Location: BBG 061

Title: Cox rings and Mori dream spaces

Abstract: In this talk I will start with the problem to lift maps between Mori dream spaces to homomorphisms between their respective Cox rings. By doing this, quotient stacks appear in a quite natural manner. At that point, one may ask how Cox rings are constructed for stacks; as for varieties the crucial point is to define the multiplication. This will be the second part of the talk. This is joint work with Elena Martinengo and Fabio Tonini.

Location: BBG 061

Title: The weakly special conjecture contradicts orbifold Mordell, and thus abc

Abstract: Lang conjectured that varieties of general type over a number field do not have a dense set of rational points. In 2000, guided by Lang's conjecture and in search of a converse statement, Abramovich, Colliot-Thelene, Harris, and Tschinkel formulated the "Weakly Special Conjecture": every weakly special variety over a number field has a potentially dense set of rational points. In this talk I will explain how this conjecture contradicts the abc conjecture, and more precisely Campana's "Orbifold Mordell" conjecture. Indeed, starting from an Enriques surface over $\mathbb{Q}(t)$ constructed by Lafon, we give the first examples of smooth projective weakly special threefolds which fiber over the projective line in Enriques surfaces with nowhere reduced, but non-divisible, fibers. I will explain that the existence of these threefolds shows that the Weakly Special Conjecture contradicts the abc conjecture. The existence of such threefolds also shows that Enriques surfaces and K3 surfaces can have non-divisible but nowhere reduced degenerations, thereby answering a question raised by Campana in 2005. This is joint work with Finn Bartsch, Frederic Campana, and Olivier Wittenberg.

Location: Daltonlaan 500 Room 1.19

Title: Geometric Class Field Theory and Fargues Fontaine Curve

Abstract: We give an outline of Fargues' proof of local class field theory using a study of line bundles on the curve and its Abel-Jacobi morphism on degree-one divisors, introducing the audience to $p$-adic geometry and the étale cohomology of diamonds in the process. The essence of the proof is a descent theorem for line bundles in this set-up; we study a generalization of this descent result for the functor $\operatorname{Bun}_1$ of line bundles on the absolute Fargues-Fontaine curve.

Location: BBG 061

Title: Moduli space of genus one curves on smooth cubic threefolds

Abstract: Let $X$ be a smooth Fano variety over the complex numbers, a natural object to investigate is the Kontsevitch moduli space of genus $g$ stable maps on $X$. In this talk, I will illustrate a strategy to study the components of this moduli space through the lens of Geometric Manin's Conjecture, and present a result on genus 1 stable maps when $X$ is a smooth cubic threefold.

Location: BBG 061

Title: Power Tower Theory: Foliations and Galois Theory in Positive Characteristic Algebraic Geometry

Abstract: A power tower is an object analogous to a foliation, but in positive characteristic. They generalise fibrations. They include purely inseparable morphisms. We develop a Galois-type correspondence for power towers. This gives a useful framework for working with purely inseparable morphisms. In particular, we prove a formula for a pullback of a canonical divisor with respect to any such morphism.

Location: HFG 7.07

Title: Kodaira dimension of moduli spaces of hyperkähler varieties

Abstract: In a celebrated series of articles from 2007-2011, Gritsenko, Hulek and Sankaran studied the birational geometry of moduli spaces of K3 surfaces and some higher-dimensional hyperkähler varieties. In particular, they showed that the Kodaira dimension of these moduli spaces is often maximal, by reducing the question to the existence of a certain cusp form for an orthogonal modular variety. In this talk, I will sketch the reduction argument and present new results on the Kodaira dimension of many more hyperkähler moduli spaces. This is joint work with I. Barros, P. Beri and L. Flapan.

Location: HFG 7.07

Title: Euler characteristics of moduli of twisted sheaves on Enriques surfaces

Abstract: Let $Y$ be an Enriques surface and let $A$ be an Azumaya algebra corresponding to the non-trivial Brauer class. Let $M$ be the moduli space of stable twisted sheaves on Enriques surfaces with fixed twisted Chern character. The virtual dimension of $M$ is $N$. We show that the virtual Euler characteristic of $M$ only depends on $N$, more precisely, it is 0 when $N$ is odd and it equals to the Euler characteristic of Hilbert scheme of $N/2$ points when $N$ is even.

Location: HFG 7.07

Title: Expansions for simple normal crossing degenerations of Hilbert schemes of points

Abstract: Let $X \to C$ be a semi stable family of surfaces over a curve with special fibre $X_0$. In the case where $X_0$ is simple normal crossing and its singular locus is singular, we construct modular simple normal crossing degenerations of Hilbert schemes of points over $X \to C$. We use the method of expanded degenerations, generalizing some of the work of Li and Wu and provide explicit examples of the logarithmic Hilbert schemes of Maulik and Ranganathan.

Location: HFG 7.07

Title: Nodal Calabi-Yau 3-folds and torsion refined GV-invariants

Abstract: In general, a projective Calabi-Yau threefold with nodal singularities does not admit any Kaehler small resolution. This happens in particular if the exceptional curves are torsion in the homology of some small resolution. I will first introduce a class of examples, constructed as double covers of $\mathbb{P}^3$ with symmetric determinantal ramification locus. The proof that the exceptional curves are torsion is based on a so-called conifold transition to another smooth CY 3-fold. From a physical perspective, this can be interpreted as a Higgs transition that leads to a discrete gauge symmetry in M-theory on the singular CY. The M-theory interpretation then motivates a proposal for torsion refined Gopakumar-Vafa invariants that are associated to the singular CY 3-fold and, in a certain sense, capture the enumerative geometry of non-Kaehler small resolutions. Finally, I will discuss the relation to topological strings on "Calabi-Yau gerbes" and how one can calculate these invariants using mirror symmetry.

Location: HFG 7.07

Title: The equations of an algebraic curve

Abstract: Determining the structure of the equations of an algebraic curve in its canonical embedding (given by its holomorphic forms) has been a central question in algebraic geometry from the beginnings of the subject. In 1984 Mark Green put forward a very elegant conjecture (generalizing classical work of M. Noether and Enriques), linking the complexity of the curve in its moduli space (that is, its gonality) to the structure of its equations in the canonical embedding. I will discuss how novel ideas coming from topology led to a surprising solution of this conjecture for generic curves in arbitrary characteristic.

Location: HFG 7.07

Title: Rational equivalences from hyperelliptic curves

Abstract: The Chow group of zero-cycles of a variety $X$ points can roughly be described as follows: the generators are closed points of $X$, and the relations, also known as rational equivalences, are divisors of rational functions on curves in $X$. In general, it can be very difficult to tell whether a given zero-cycle is trivial in the Chow group, and the structure of the Chow group as a whole is very mysterious. Deep conjectures due to Bloch and Beilinson give some indication of what the structure should be and which zero-cycles should vanish, but very little has been proven in this direction. In this talk we will focus on certain abelian surfaces $A$, and discuss a collection of methods that can take one of the zero-cycles that is predicted to vanish in Chow and verify that it is indeed a rational equivalence. The key idea behind these methods is a relation between hyperelliptic curves in $A$ and rational curves in the Kummer surface of $A$. This is joint work with Evangelia Gazaki.

Location: HFG 7.07

Title: Constructible sheaves on toric varieties

Abstract: Given a reasonable topological space or algebraic variety, its covering spaces are classified by the fundamental group, via the monodromy correspondence. I will review some of the different ways to construct this correspondence, also for the étale fundamental group. I will then show how to extend each of these methods to a classification of constructible sheaves using the stratified fundamental category (or exit path category), leading to an exodromy correspondence. As a concrete example, this gives a computation of the constructible sheaves on a toric variety (with respect to the toric stratification) over an arbitrary field $k$, generalising a result of Braden and Lunts over $\mathbb{C}$.

Location: HFG 7.07

Title: Reduction modulo $p$ of the Noether problem

Abstract: Let $k$ be an algebraically closed field of characteristic $p \geq 0$ and $V$ a faithful $k$-rational representation of an $\ell$-group $G$. The Noether's problem asks whether $V/G$ is (stably) birational to a point. If $\ell$ is equal to $p$, then Kuniyoshi proved that this is true, while, if $\ell$ is different from $p$, Saltman constructed $\ell$-groups for which $V/G$ is not stably rational. Hence, the geometry of $V/G$ depends heavily on the characteristic of the field. We show that for all the groups $G$ constructed by Saltman, one cannot interpolate between the Noether problem in characteristic 0 and $p$. More precisely, we show that it does not exist a complete valuation ring $R$ of mixed characteristic $(0,p)$ and a smooth proper $R$-scheme $X \to \operatorname{Spec}(R)$ whose special fiber and generic fiber are both stably birational to $V/G$. The proof combines the integral $p$-adic Hodge theoretic results of Bhatt-Morrow-Scholze, with the study of the Cartier operator on differential forms in positive characteristic. This is a joint work with Domenico Valloni.

Location: BBG 385

Title: Powers of an abelian variety isogenous to a Jacobian and the Coleman--Oort conjecture

Abstract: I will prove that, for a very general principally polarized complex abelian variety of dimension at least four, no power is isogenous to a Jacobian. This confirms cases of the Coleman--Oort conjecture on special subvarieties in $A_g$. This is joint work with Stefan Schreieder.

Location: BBG 017

Title: Cohomological integrality isomorphisms

Abstract: The theory of cohomological Hall algebras has proven powerful in studying Donaldson-Thomas invariants of some 3-Calabi-Yau categories. It is in particular crucial to obtain cohomological integrality identities. Roughly speaking, the cohomological integrality results concern the finiteness of some invariants associated to the category. They also give cohomologically refined invariants. In my talk, I will give a brief overview of such results, explain their meaning, and explain how to obtain them in various contexts. I will concentrate on a new situation given by symmetric representations of reductive groups.

Location: Minnaert building Room 0.14

Title: An integral structure on the sheaf of differentials

Abstract: We explain how to define an integral structure on the sheaf of differentials $\Omega_X$ of an adic space $X$. This should be thought of as an analogue of the subsheaf $\mathcal{O}_X^+$ of the structure sheaf $\mathcal{O}_X$. This integral structure $\Omega_X^+$ can be described in terms of logarithmic differentials on a log regular model (if such a model exists). Possibly assuming resolution of singularities this gives a strategy for transferring results on log étale cohomology to tame cohomology.

Location: BBG 201

Title: Rank 2 local systems and abelian varieties

Abstract: Motivated by work of Corlette-Simpson over the complex numbers, we conjecture that all irreducible rank 2 $\ell$-adic local systems with cyclotomic determinant and infinite image on a smooth variety over a finitely generated field come from families of abelian varieties. We will survey partial results on this conjecture. Time permitting, we will provide indications of the proofs, which involve the work of Hironaka and Hartshorne on positivity, Drinfeld's first work on the Langlands correspondence over function fields, space filling curves, and the pigeonhole principle. The results are all joint work with either Ambrus Pál or Jinbang Yang and Kang Zuo.

Location: BBG 083

Title: Moduli of roots of universal line bundles

Abstract: Let $L$ be a line bundle on the universal curve over $M_{g,n}$. The moduli space $S^0$ of $r$-th roots of $L$ admits a natural compactification $S$ over the Deligne-mumford compactification of $M_{g,n}$. When $r$ is 2 and $L$ is the canonical bundle, $S^0$ is the well-studied moduli space of spin curves, which is known to have two connected components: one parametrizing even spin curves, and one parametrizing odd spin curves. I will describe a natural partition of $S$ into connected locally closed subspaces for arbitrary $r$ and $L$. This partition is compatible with the partition of the Deligne-Mumford compactification indexed by stable graphs. This is joint work in progress with Margarida Melo.

Location: BBG 083

Title: Shifted Lagrange multipliers method

Abstract: Lagrange multipliers method relates critical points on a submanifold with those on the ambient manifold. In derived algebraic geometry, we are allowed to consider a more general type of functions called shifted functions and their critical loci. In this talk, I will discuss how Lagrange multipliers method adapted to derived algebraic geometry naturally produces quantum Lefschetz principle of Chang-Li and more.

Location: BBG 083

Title: Higher genus GW theory of the Hilbert scheme of points of the plane

Abstract: The quantum cohomology of the $\operatorname{Hilb}(\mathbb{C}^2,n)$ is determined by the operator of quantum multiplication by the divisor class (and was computed 20 years ago in work with Okounkov). I will explain how to think about the higher genus theory from several perspectives: CohFT, MNOP, and, in the case of genus 1, the intersection theory of the moduli space $A_g$ of PPAVs. The talk is connected to recent joint and disjoint work of several mathematicians: S. Canning, F. Greer, A. Iribar Lopez, C. Lian, S. Molcho, D. Oprea, A. Pixton,and H.-H. Tseng.

Location: KBG - ATLAS

Title: Infinitesimal rational actions

Abstract: For any finite $k$-group scheme $G$ acting rationally on a $k$-variety $X$, if the action is generically free then the dimension of $\operatorname{Lie}(G)$ is upper bounded by the dimension of the variety. This inequality turns out to be also a sufficient condition for the existence of such actions, when $k$ is a perfect field of positive characteristic and $G$ is infinitesimal commutative trigonalizable. These group schemes are non-reduced and arise only in positive characteristic. After presenting the main objects involved and overviewing the motivation for this problem, we willl explain the result in the case of actions of the $p$-torsion of a supersingular elliptic curve.

Location: BBG 083

Title: The Mumford conjecture

Abstract: The Mumford conjecture is an important result that computes the cohomology of the moduli space of curves in the stable range and was originally proved by Madsen and Weiss about 20 years ago. Recently, a new alternative proof of the Mumford conjecture was given by Bianchi. In this talk, I will discuss a version of Bianchi's proof which was further streamlined by Das and Petersen.

Location: BBG 161

Title: Minimal exponent of a hypersurface

Abstract: Recently, the minimal exponent of a hypersurface over complex numbers has been understood as a quite useful refined invariant of the log canonical threshold. It has found many new applications including deformation of Calabi-Yau 3-folds (Friedman-Laza), higher rational and higher du Bois singularities (Mustata-Popa) and geometric Schottky problem (Schnell-Yang). However, some properties of this invariant remain mysterious. In this talk I will discuss the conjecture of Mustata and Popa on birational characterization of the minimal exponent via log resolutions, which is the main obstruction for the computation in practice. I will explain the heuristic of the Mustata-Popa conjecture from Igusa's work on counting integer solutions of congruence equations and the monodromy conjecture. Then I will discuss how several ideas from Hodge theory and geometry representation theory can lead to a better understanding of the minimal exponent (birationally). This is based on the joint work with Christian Schnell on higher multiplier ideals and the joint work in progress with Dougal Davis on a new description of the Kashiwara-Malgrange $V$-filtration of mixed Hodge modules.

Location: BBG 083

Title: The motive of the Hilbert scheme of points

Abstract: The geometry of Hilbert schemes of points is largely unknown, or known to be pathological in a precise sense. This should in principle make most (naive) invariants essentially inaccessible. In this talk we explain how to obtain a closed formula for the generating function of the motives (classes in the Grothendieck ring of varieties) of Hilbert schemes of points $\operatorname{Hilb}(X,d)$ for $X$ a smooth variety of arbitrary dimension, and for fixed number of points $d$

Location: BBG 017

Title: Generalized Faber-Zagier relations on relative Jacobian

Abstract: Let $M_g$ be the moduli space of smooth genus $g$ algebraic curves. By Madsen-Weiss, the rational cohomology of $M_g$ is a free algebra generated by tautological classes in the “stable range”. Outside the “stable range”, there are relations among tautological classes. Faber-Zagier conjectured that there is a structure of relations among tautological classes outside the “stable range”. This conjecture, and its extension to the moduli space of stable curves, is proven by Pandharipande-Pixton-Zvonkine and Janda. Similar stability results extends to the relative Jacobian over $M_g$ by Ebert-Randal-Williams. It is an interesting question to ask whether we have a generalised Faber-Zagier relations on the relative Jacobian. I will give a candidate of those relations using the stable quotients over the relative Jacobian. This is a joint work in progress with H. Lho.

Location: BBG 017

Title: The Logarithmic Jacobian and extension of torsors over families of degenerating curves

Abstract: Molcho and Wise constructed the log Picard group, a canonical compactification of the Picard group of families of nodal curves that is smooth, proper and possesses a group structure, but which can only be represented in the category of logarithmic spaces. Afterwards, it was shown that the restriction of this log Picard group to the degree zero log line bundles - which gives the so called Logarithmic Jacobian- is in fact the log Néron model of the Jacobian of the smooth locus. In this talk, I will first explain the construction of this material. Then, I will show that the Logarithmic Jacobian classifies finite log torsors over families of nodal curves, generalizing the classical situation for smooth curves. Finally, we will see that the Néron property of the Log Jacobian allows to get a result on extending fppf torsors into log torsors over families of nodal curves. This is a joint work with Thibault Poiret.

Location: KBG 224

Title: Fourier transform and class of sections

Abstract: Fourier transformation is an important tool to study the Chow group (or relative Chow motive) of abelian schemes. By Arinkin, the Fourier transforamtion extends to relative compactified Jacobian for family of integral local planar curves. Using Arinkin’s Fourier transformation, we will see how to compute the class of Abel-Jacobi sections on the relative Jaocbian over a moduli space of integral nodal curves. This computation recovers Pixton’s formula without $r$-polynomial and partially recovers the universal double cycle formula by Bae-Holmes-Pandharipande-Schmitt-Schwartz. This is a joint work with S. Molcho.

Location: BBG 061

Title: Intersection theory on compactified Jacobians over the moduli spaces of stable curves

Abstract: I will give three talks on compactified Jacobians over the moduli space of stable curves. Three talks will be independent and loosely related. Over the moduli space of stable curves, integrals of tautological classes have been an important subject. For example, Witten-Kontsevich theorem says that the generating series of integrals of $\psi$ classes satiesfies the KdV hierarchy. What can we say about relative compactified Jacobian? Using the quasi-stable model, there is a natural choice of tautological classes on compactified Jacobian. In this talk, we will see that the pushforward along the forgetful morphism from the compactified Jacobian to the moduli space of stable curves preserves tautological classes. Our main ingredient is the universal double ramification cycle formula. Using the Witten-Kontsevich theorem, one can compute integrals of tautological classes on compactified Jacobian. Along the way, I will explain how this idea can be adapted to study the logarithmic Picard group constructed by Molcho-Wise. This is a joint work in progress with S. Molcho.

Location: Minnaert building Room 0.13

Title: The Manin conjecture for toric stacks

Abstract: A basic question in the Diophantine geometry is how many integer solutions to a system of polynomial equations are there, or, in other words, how many rational points are there on algebraic varieties. When there are “many” solutions, the precise asymptotic behaviours are predicted by the Manin conjecture. Recently, the Manin conjecture has been generalized to Deligne–Mumford stacks. Recall that the Deligne–Mumford stacks are the algebro-geometric objects which arise as solutions to the questions of parametrizing objects having finite (not necessarily trivial) automorphism groups. In this talk, we study the Manin conjecture for toric stacks, which are Deligne–Mumford stacks, whose geometry can be described using combinatorial data similar to that describing toric varieties.

Location: HFG 409

Title: Toroidal $b$-divisors and applications in differential and arithmetic geometry

Abstract: We define toroidal $b$-divisors on a quasi projective variety. These can be seen as conical functions on a balanced polyhedral space. We show the existence of an intersection pairing for so called nef toroidal $b$-divisors, which gives rise to a Monge-Ampére type measure on the polyhedral space. We then show some applications of this theory. On the one hand side, $b$-divisors are used to encode singularities of psh metrics and we derive Chern-Weil type formulae for such metrics on line bundles. On the other hand, using a Hilbert-Samuel formula, we compute asymptotic dimension formulae of spaces of automorphic forms on mixed Shimura varieties. If time permits, we connect our work to the notion of adelic line bundles of Yuan and Zhang, and outline some current research directions.

Location: BBG 061

Title: Generalized Beauville decomposition

Abstract: By the work of Beauville and Deninger-Murre, the rational cohomology group (in fact, the relative Chow motive) of abelian scheme over a regular base has a canonical decomposition into pure weight pieces. The essential ingredient is the Fourier-Mukai transformation relative to the base. It is a natural question to ask what happens if there exist singular fibers. For a family of integral local planar curves, the relative compactified Jacobian is an example of such “degenerate abelian fibration”. In this talk, we consider perverse filtration on the rational cohomology of the relative compactified Jacobian and ask when the filtration has Fourier-stable multiplicative splitting. If the relative compactified Jacobian arise from Beauville-Mukai system, we get such multiplicative splitting. On the other hand, for general family of integral curves, even for family of nodal curves, such multiplicative splitting cannot exist. This is a joint work with D. Maulik, J. Shen and Q. Yin.

Location: BBG 061

Title: Murphy's law on a fixed locus of the Quot scheme

Abstract: The Quot and Hilbert schemes of 0 dimensional sheaves on $\mathbb{A}^d$ are moduli spaces that are known to be highly singular for sufficiently large values of $d$. In particular, starting from $d = 16$, by a result of Jelisiejew they are known to satisfy a form of "Murphy's law", meaning that they have arbitrarily bad singularities. It is however still unknown what is the smallest value of $d$ for which we can expect such pathological behaviour. In an attempt to answer this question, in this talk we will study the singularities of the locus of the Quot scheme consisting of fixed points under the torus action coming from $\mathbb{A}^d$. In particular, I will show that already for $d = 4$ this locus satisfies Murphy's law.

Location: BBG 061

Title: Perverse coherent extensions on Calabi-Yau threefolds and representations of cohomological Hall algebras

Abstract: Motivated by the problem of generalizing the ADHM construction of the moduli space of framed torsion free sheaves on $\mathbb{P}^2$, to describe the entire stack of quiver representations geometrically and moreover to extend this to some other toric surfaces, I'll explain a construction of certain moduli stacks of coherent sheaves on toric Calabi-Yau threefolds which admit a natural equivalence with stacks of representations of a framed quiver with potential, following results of Bridgeland and Van den Bergh. I'll also describe natural representations on the critical cohomology groups of subvarieties of stable objects in these stacks, towards proving a generalization of a conjecture of Alday-Gaiotto-Tachikawa.

Location: HFG 409

Title: Moduli spaces of one dimensional sheaves on the projective plane

Abstract: Moduli spaces of one dimensional sheaves on the projective plane have been studied in connections to enumerative geometry and meromorphic Hitchin system. Cohomology group of the moduli space is equipped with a perverse filtration which plays an important role in these connections. I will explain how to study perverse filtration by combining ideas from tautological ring, BPS integrality and $\chi$-independence. I will finish by proposing a conjectural algebraic characterization of the perverse filtration. This is a joint work in progress with Y. Kononov, M. Moreira, W. Pi.

Location: BBG 115

Title: Non-density of rational points on algebraic varieties over number fields

Abstract: Which varieties over a number field should have a potentially dense set of rational points? A naive guess based on Lang's conjectures was made in the early nineties: a smooth projective variety over a number field which does not dominate a positive-dimensional variety of general type after any etale covering should have a dense set of rational points over some large enough number field. This conjecture is probably false (and the "correct" conjecture was later formulated by Campana). In joint work with Finn Bartsch and Erwan Rousseau we disprove the natural analogue of the aforementioned naive conjecture for transcendental-rational points (a notion I will explain in this talk). In our proof, we develop the theory of moduli spaces of orbifold maps, establish an analogue of Kobayashi-Ochiai's finiteness theorem for dominant maps in Campana's orbifold setting, employ Mori's bend-and-break and ultimately rely on Faltings's finiteness theorem (formerly Mordell's conjecture).

Location: BBG 001

Title: Concerning Weil restrictions

Abstract: We begin with a zoo of examples of Weil restrictions. The main three are: 1) number theoretic, 2) jet spaces, and 3) the Kontsevich space of (pre)stable maps. We discuss each in their own right, leading to an elementary main theorem which applies to all of them. Open questions and doable exercises are emphasized throughout, and input encouraged.

Location: BBG 001

Title: Computing vertical Vafa-Witten invariants

Abstract: I'll present a computation in the algebraic approach to Vafa-Witten invariants of projective surfaces, as introduced by Tanaka-Thomas. The invariants are defined using moduli spaces of stable Higgs pairs on surfaces and are formed from contributions of components. The physical notion of $S$-duality translates to conjectural symmetries between these contributions. One component, the "vertical" component, is a nested Hilbert scheme on a surface. I'll explain work in preparation with M. Kool and T. Laarakker in which we express invariants of this component in terms of a quiver variety, the instanton moduli space of torsion-free framed sheaves on $\mathbb{P}^2$. As a consequence, we deduce constraints on Vafa-Witten invariants, including a formula for the contribution of the vertical component to refined invariants in rank 2.

Location: BBG 001

Title: Quasi-BPS categories for K3 surfaces

Abstract: Hyperkahler varieties are higher dimensional analogues of K3 surfaces. Moduli spaces of semistable sheaves on a K3 surface for a primitive Mukai vector are examples of hyperkahler varieties. Other hyperkahler varieties are obtained as crepant resolutions of such moduli spaces when the Mukai vector is not primitive. However, it is known that there is only one new example produced in this way, constructed by O’Grady. In joint work with Yukinobu Toda, we construct dg categories which are analogues of crepant resolutions of singularities for the moduli space of semistable sheaves on a K3 surface for a generic stability condition and a general Mukai vector. The construction is inspired by the study of BPS invariants in enumerative geometry of Calabi-Yau threefolds.

Location: BBG 001

Title: Equivariant Strings and Gromov-Witten theory

Abstract: In the past three decades some of the most exciting developments in enumerative geometry were inspired by analogies in mathematical physics. After giving you a crash course on how an algebraic geometer should think about (A-model topological) string theory, I will make a proposal for a mathematically rigorous formulation of the so called refined topological string in the framework of equivariant Gromov-Witten theory on Calabi-Yau fivefolds. Moreover, I will present a surprising correspondence between the so called Nekrasov-Shatashvili limit for a local surface $K_S$ and the enumeration of tangents to a smooth anti-canonical curve in $S$ (assuming the latter exist). This is ongoing work with Andrea Brini.

Location: BBG 001

Title: Nekrasov's formula for some orbifolds

Abstract: Nekrasov's formula computes equivariant $K$-theoretic DT invariants for Hilbert schemes of points of toric CY3-folds. We extend this to certain global quotient CY3 orbifolds, refining a result of Young to equivariant $K$-theoretic DT theory and proving a conjecture by Cirafici. Okounkov's proof of Nekrasov's formula combines two techniques. First, the virtual structure sheaves satisfy a certain factorization property, which allows us to simplify the general form of the DT generating series. Secondly, the rigidity principle allows us to determine the remaining unknowns by computing a limit in the equivariant parameters. We will explain these techniques in the case of schemes and describe some of the modifications to make them work for orbifolds.

Location: BBG 001

Title: The cycle class of the supersingular locus

Abstract: Deuring gave a formula for the number of supersingular elliptic curves in characteristic $p$. We generalize this to a formula for the cycle class of the supersingular locus in the moduli space of principally polarized abelian varieties of given dimension $g$. The formula determines the class up to a multiple and shows that it lies in the tautological ring. We also give the multiple for $g$ up to $4$. This is joint work with S. Harashita.

Location: BBG 017

Title: Doubly isogenous genus-2 curves over finite fields

Abstract: Our main question is the following: Can you tell the difference between two curves defined over a finite field if you only know their zeta functions and the zeta functions of certain covers? This distinguishing problem is partially motivated by a possible strategy for developing a deterministic polynomial-time algorithm for factoring polynomials over finite fields due to Kayal and Poonen. Ultimately, this talk is concerned with cases where the distinguishing problem is not solvable due to the existence of "very similar" curves. Specifically, we study a family of genus-2 curves with a dihedral action and show that so-called doubly isogenous pairs of curves are surprisingly common in this family. We also provide an explanation of this phenomena which corrects the naive heuristics. This is joint work with Arul, Booher, Groen, Howe, Li, Matei and Pries.

Location: BBG 161

Title: Unramified Gromov-Witten and Gopakumar-Vafa invariants

Abstract: Kim, Kresch and Oh defined moduli spaces of unramified stable maps, which are natural generalisations of (compactified) Hurwitz spaces for a target of an arbitrary dimension. Just like Hurwitz spaces, which are smooth irreducible varieties after normalisation, moduli spaces of unramified stable maps are 'better' compactifications than moduli spaces of stable maps. Pandharipande conjectured that unramified Gromov-Witten invariants of a projective threefold are equal to Gopakumar-Vafa (BPS) invariants in the case of Fano classes (classes that intersect negatively with the canonical class) and primitive Calabi-Yau classes (trivial intersection). After a gentle introduction to unramified Gromov-Witten theory, we will discuss a work in progress which aims to prove the conjecture. This provides a geometric construction of Gopakumar-Vafa invariants in these cases. The proof is based on a certain wall-crossing technique.

Location: HFG 409

Title: 2d Cohomological Hall Algebras for Cyclic Quivers and and Integral form of Affine Yangian.

Abstract: In 2012, Schiffmann and Vasserot considered a Hall algebra type construction on the cohomology of moduli of sheaves supported on points on a plane and used it to prove AGT conjecture. However due to the mysterious nature of the moduli of representations of pre-projective algebra, these algebras are very hard to study and are often highly non trivial. They are conjectured to be the same as Maulik-Okounkov Yangians which has further applications in Quantum Cohomology. In this talk, my goal is to give an introduction to these algebras and explain how one can use tools from Cohomological DT theory to study these algebras. In particular, I will explain how for the case of cyclic quiver, this algebra turn out to be a half of the universal enveloping algebra of the Lie algebra of matrix differential operators on torus, while its deformation turn out be an explicit integral form of Affine Yangian of $\mathfrak{gl}(n)$.

Location: HFG 409

Title: Periods and o-minimality

Abstract: Periods are numbers obtained by integrating algebraic differential forms over semi-algebraic domains. The set contains interesting numbers like logarithms of algebraic numbers or the values of the Riemann zeta function at integral points. The linear or algebraic relations between them is a classical topic of transcendence theory. Grothendieck gave a more conceptual interpretation in terms of the pairing between singular and algebraic de Rham cohomology of algebraic varieties over the rationals. This leads to the (wide open) Period Conjecture predicting all relations between periods. Kontsevich and Zagier suggested to extend the theory to the so called exponential periods appearing in the theory of irregular connections. The conceptual part of the story has been worked out by Hien and Fresan-Jossen. In joint work with Johan Commelin an Philipp Habegger we show that all exponential periods can be written as volumes in a certain o-minimal structure. This hints at a deeper connection between periods and o-minimiality.

Location: BBG - 223

Title: Ekedahl-Oort types of stable curves

Abstract: In this talk, I will present some invariants of curves in positive characteristic $p$, such as the $p$-rank, the $a$-number, or the Ekedahl-Oort type, and discuss intrinsic ways to define them. The main focus will be on Moonen's definition of Ekedahl-Oort types of smooth curves in terms of Hasse-Witt triples. I will show that we can extend this definition to all stable curves. The description we obtain in this manner enables us to compute the dimensions of certain loci of curves. Finally, I will mention some new examples in characteristics $p = 2$ and $p = 3$.

Location: HFG 610

Title: Hodge theory and projective structures on compact Riemann surfaces

Abstract: A projective structure on a compact Riemann surface is an equivalence class of projective atlases, i.e. an equivalence class of coverings by holomorphic coordinate charts such that the transition functions are all Moebius transformations. Any compact Riemann surface admits two canonical projective structures: one coming from uniformization's theorem, and one from Hodge theory. These yield two (different) families of projective structures over the moduli space $M_g$ of compact Riemann surfaces. We wish to compare them and give a characterization of the Hodge theoretic family.

Location: HFG 610

Title: 4G Networks

Abstract: I'll introduce the equivariant $K$-theoretic Donaldson-Thomas theory for toric Calabi-Yau fourfolds, and construct its four-valent vertex with generic plane partition asymptotics. String-theoretically, this is the count of BPS states of a system of D0-D2-D4-D6-D8-branes in the presence of a large Neveu-Schwarz $B$-field. The talk is based on 2306.12995, 2306.12405 and ongoing work.

Location: BBG 109