Algebraic geometry, integrable systems and automorphic forms
May 26-30, 2025
Lille (France)

Laboratoire Paul Painlevé

Conference Schedule

A weekly overview of all sessions and speakers.

Time Monday 26 Tuesday 27 Wednesday 28 Thursday 29 Friday 30
9h30-10h30 Vanhove de la Ossa 2 Golyshev 2 Ovsienko Fairon
Coffee Break
11h00-12h00 de la Ossa 1 Terpereau Fantini Rota Candelas
Lunch
14h00-15h00 Gritsenko Golyshev 1 Pichon-Pharabod Gaiur Roubtsov
Coffee Break
15h30-16h30 Cléry Gounelas Vesselov Oelen Odesski
16h40-17h40 Fedosova Reider Adler Simon


Titles, abstracts and talks

Dmitrii Adler(Max Planck Institute for Mathematics, Bonn):
Jacobi forms and Kaneko--Zagier type equations
Abstract: Modular forms and their generalization, Jacobi forms, can be viewed as specific types of special functions. Consequently, it is quite natural to study the differential equations that these functions satisfy. For modular forms, the Kaneko--Zagier equation is a second-order differential equation with respect to the Serre derivative. Analogously, for Jacobi forms, there exists an analogue of the Serre derivative that increases the weight of a Jacobi form by 2 while preserving its index. It is not difficult to describe the kernel of this operator, but finding solutions of even second-order differential equations, known as Kaneko--Zagier type equations, is not such a trivial problem. In my talk, I will present some recent results on this topic.
[pdf of the talk]
Philip Candelas (University of Oxford):
L-Functions for Families of Calabi-Yau Varieties
Abstract: The local zeta functions of a Calabi-Yau manifold are gathered together into a global L - function. These L - functions are conjectured to have modular properties and to satisfy certain functional equations. I will review evidence that supports this, in the context of 1-parameter families of Calabi-Yau varieties, and will explain the calculation of the conductors for these families and the data structures that describe the L - functions as functions of the parameter. (This is work in progress with X de la Ossa, N Gegelia and D van Straten.)
Fabien Cléry(Loughborough University):
Picard modular forms via invariant theory
Abstract: In previous work with Carel Faber and Gerard van der Geer, we showed how invariant theory can be used to generate all Siegel modular forms of degree 2 and 3. In this talk, we will explain how invariant theory can again be used to construct all Picard modular forms in the case of signature (2,1). This is a joint work with Gerard van der Geer.
[Generating Picard modular forms by means of invariant theory] by Fabien Cléry and Gerard van der Geer
Xenia de la Ossa(University of Oxford): On the arithmetic and modularity of Calabi-Yau varieties: attractors, periods and counting points over finite fields (minicourse, 2 lectures)
Abstract: The main goal of these talks is to explore questions of common interest for physicists, number theorists and geometers, in the context of Calabi-Yau 3-folds. There are many relations, however we will focus on the rich structure of black hole solutions of superstrings on Calabi-Yau varieties. I will try to give a self contained introduction aimed at a mixed audience of physicists and mathematicians. The main quantities of interest in the arithmetic context are the number of points of the variety, considered as varieties over finite fields, and how these numbers vary with the parameters of the varieties. The generating function for these numbers is the zeta function, about which much is known in virtue of the Weil conjectures. The first surprise, for a physicist, is that the numbers of these points, and so the zeta function, are given by expressions that involve the periods of the manifold. These same periods determine also many aspects of the physical theory, including the properties of black hole solutions. I will discuss a number of interesting topics related to the zeta function, the corresponding L-function, and the appearance of modularity for one parameter families of Calabi-Yau manifolds. We will focus on an example for which the quartic numerator of the zeta function of a one parameter family factorises into two quadrics at special values of the parameter. These special values, for which the underlying manifold is smooth, satisfy an algebraic equation with coefficients in Q, so independent of any particular prime. The significance of these factorisations is that they are due to the existence of black hole attractor points in the sense of type II supergravity and are related to a splitting of the Hodge structure and at these special values of the parameter. Modular groups and modular forms arise in relation to these attractor points, in a way that is familiar to mathematicians as a consequence of the Langland’s Program, but which is a surprise to a physicist. The rank two attractor points were found together with Philip Candelas, Mohamed Elmi and Duco van Straten by the application of these number theoretic techniques, and provide the first explicit examples of such attractor points for Calabi-Yau varieties. I will also include some work in progress with Philip Candelas and Eleonora Svanberg on the relation between periods and counting points. A following up talk by Philip Candelas will discuss in more detail the L-functions and the modularity of one parameter families of Calabi-Yau varieties.
[pdf of the lecture]
Maxime Fairon(Université de Bourgogne):
Quivers and (supersymmetric) integrable systems

References for the talk
[1] Chalykh, O., Silantyev, A.: KP hierarchy for the cyclic quiver. J. Math. Phys. 58, 071702 (2017)
[2] Fairon, M.; Görbe, T.F.: Superintegrability of Calogero-Moser systems associated with the cyclic quiver. VNonlinearity 34, no. 11, 7662-7682 (2021)
[4] Kimura, T.; Pestun, V.: Super instanton counting and localization. Preprint; arXiv:1905.01513.
[5] Okazaki, T.; Smith, D.J.: Matrix supergroup Chern-Simons models for vortex-antivortex systems. JHEP 2018, 119 (2018)
[6] Sergeev, A.N.; Veselov, A.P.: Deformed quantum Calogero-Moser problems and Lie superalgebras. Commun. Math. Phys. 245, No. 2, 249-278 (2004)
[7] Wilson, G.: Collisions of Calogero-Moser particles and an adelic Grassmannian (With an appendix by I. G. Macdonald). Invent. Math. 133(1), 1-41 (1998)
Veronica Fantini(Université Paris-Saclay):
Local weighted projective spaces and their resurgent invariants
Abstract. In the framework of the Topological String/Spectral Theory correspondence of Grassi-Hatsuda-Mariño, a new source of invariants (resurgent invariants) for local Calabi-Yau threefolds has been constructed using the theory of resurgence. These invariants are analytic, i.e., they are defined from the analytic continuation of a given function (the first fermionic trace). However, they are conjectured to be related to Gromov-Witten invariants, and they have interesting arithmetic properties. In this talk, I will focus on the arithmetic properties of the resurgent invariants for local weighted projective spaces, showing that in a few cases, these invariants are coefficients of certain L- functions. This is part of a joint project with C. Rella. [Strong-weak symmetry and quantum modularity of resurgent topological strings on local P^2] by Veronica Fantini, Claudia Rella
[Modular resurgence, q-Pochhammer symbols, and quantum operators from mirror curves] by Veronica Fantini, Claudia Rella
Ksenia Fedosova(Universität Münster):
Convolution identities for divisor functions
[pdf of the talk] and[article]
Ilia Gaiur(University of Geneva):
Kernels In-depth: Higher Heun Equations
Abstract: I will report on joint work in progress with Vasily Golyshev (IHES). Following a program laid out by Golyshev- Mellit-Roubtsov-van Straten, we compute Betti multiplication kernels for the Heun local systems. I will describe ideas standing behind our approach and underline important techniques developed in the framework of this project. In addition, I will show how the analogues of the Heun equations, which we refer to as Higher Heun equations, appear naturally within our framework.
References:
On L-derivatives and biextensions of Calabi-Yau motives, and The arithmetic of Calabi-Yau motives and mobile higher regulators
Vassily Golyshev(IHES, Bures-sur-Yvette):
Hodge conjectures for Calabi-Yau motives: a survey (minicourse, 2 lectures)
Abstract: In the first lecture, I will give a survey of cases where Hodge(-type) conjectures have been proved for Calabi-Yau motives and explain some arithmetic consequences. In the second lecture, I will discuss some challenging unknown cases and present supporting evidence.
Frank Gounelas(Bonn University):
Smooth isotrivial families of curves on K3 surfaces
Abstract: I will survey recent results with Chen and Dutta regarding existence of smooth curves on K3 surfaces which deform in an isotrivial family. In the particular case of Picard rank one, I will prove these do not occur.
Valery Gritsenko(NRU HSE):
Elliptic genus of Calabi-Yau varieties and modular differential equations
[pdf of the talk]
Alexandre Odesski (Brock University, St. Catharines, Canada):
Explicit formulas for arithmetic support of differential and difference operators
[Explicit formulas for arithmetic support of differential and difference operators] by Maxim Kontsevich, Alexander Odesskii
Casper Oelen (Heriot-Watt University, Edinburgh):
Elliptic automorphic Lie algebras and integrable systems Abstract: Automorphic Lie algebras are a class of infinite-dimensional Lie algebras over the complex numbers that naturally arise in integrable systems, in particular in the context of reduction of Lax pairs. They can be thought of as Lie algebras of meromorphic maps (usually with prescribed poles) from a compact Riemann surface X into a finite-dimensional Lie algebra g which are equivariant with respect to a finite group $G$ acting on $X$ and on g, both by automorphisms. Independently of their origins in integrable systems, they show up in algebra as examples of equivariant map algebras. In this talk, we will highlight some motivations from integrable systems to study these algebras. We will mainly focus on elliptic automorphic Lie algebras, which, for example, prominently appear in the context of Landau-Lifshitz type of equations. We show that well-known algebras - such as Holod's hidden symmetry algebra of the Landau-Lifshitz equation and the Wahlquist-Estabrook prolongation algebra of the same equation - admit a particularly simple description arising from the automorphicity perspective. They turn out to be isomorphic to a current algebra sl(2,C)xR, or to its direct sum with the two-dimensional abelian Lie algebra C2, in the latter case, where R is a suitable ring of elliptic functions invariant under a particular action of the dihedral group D2 of order 4. This talk is based on joint work with Sara Lombardo and Vincent Knibbeler.
[pdf of the talk]
Valentin Ovsienko(Université de Reims Champagne-Ardenne):
From Catalan numbers to integrable dynamics: continued fractions and Hankel determinants for q-numbers
Abstract: The classical Catalan and Motzkin numbers have remarkable continued fraction expansions, the corresponding sequences of Hankel determinants consist of -1, 0 and 1 only. We find an infinite family of power series corresponding to q-deformed real numbers that have very similar properties. Moreover, their sequences of Hankel determinants turn out to satisfy Somos and Gale-Robinson recurrences known to be remarkable examples of discrete integrable systems. (Partially based on a joint work with Emmanuel Pedon.)
[pdf of the talk]
Eric Pichon-Pharabod(Max Planck institute for Mathematics in Sciences, Leipzig):
Periods of fibre product of elliptic surfaces and the Gamma conjecture
[pdf of the talk]
Igor Reider(Université d'Angers):
Refinement of IVHS invariants: the case of canonical curves
Abstract:
[pdf of the talk]
Franco Rota(Université Paris-Saclay):
Towards Homological Mirror Symmetry for log del Pezzo surfaces
Abstract: Motivated by Homological Mirror Symmetry, we study a series of singular surfaces called log del Pezzo. I will describe the derived category of a series of log del Pezzo's, using the McKay correspondence and explicit birational geometry. If time permits, I'll mention early mirror results, focusing mostly on the special case of smooth low degree del Pezzo surfaces. For these, we will compare several mirror constructions using the language of pseudolattices. This is joint work with Giulia Gugiatti.
[pdf of the talk] and the article Full Exceptional Collections for Anticanonical Log del Pezzo Surfaces by Giulia Gugiatti and Franco Rota
Volodya Roubtsov(Université d'Angers):
Bessel and beyond. (Old songs with new motives.)
[pdf of the talk]
Damien Simon (Université Paris-Saclay):
Chiral differential operators on a reductive group and representation theory
Abstract: Vertex algebras of chiral differential operators on a complex reductive group G are "Kac-Moody" versions of the usual algebra of differential operators on G. Their categories of modules are especially interesting because they are related to the theory of algebraic D-modules on the loop group of G. That allows one to reformulate some conjectures of the (quantum) geometric Langlands program in the language of vertex algebras. For instance, in view of the geometric Satake equivalence, one may expect the appearance of the category of representations of the Langlands dual group of G. I will explain why this is a reasonable expectation and give some basic statements.
Ronan Terpereau(Université de Lille):
Maximal connected algebraic subgroups of the real Cremona group
Abstract. In this talk, we will explain how we were able to achieve a (partial?) classification of the maximal connected algebraic subgroups of the real Cremona group in rank 3 by determining the rational real forms of certain complex Mori fiber spaces. This work was carried out in collaboration with Susanna Zimmermann.
[pdf of the talk]
Pierre Vanhove(IPhT CEA):
Motives for Feynman integrals
Abstract. We explain how to attach a motive to a Feynman diagram, and present a classification of motives that appear depending on the number of loops of the graphs. We show how to connect this classification with the understanding of Feynman integrals as relative periods of twisted cohomology.
[pdf of the talk]
Alexander Vesselov (Loughborough University, UK):
Markov fractions and slopes of exceptional bundles on P2
Abstract :We show that the Markov fractions introduced recently by Boris Springborn are precisely the slopes of the exceptional vector bundles on P2 studied in 1980s by Drézet and Le Potier and by Rudakov. In particular, we provide a simpler proof of Rudakov's result claiming that the ranks of the exceptional bundles on P2 are Markov numbers.
[pdf of the talk]