Schedule    

Titles and abstracts

Indranil Biswas (TIFR Mumbai): On positivity for parabolic vector bundles

Fabien Cléry (Institute of Computational and Experimental Research in Mathematics, Providence): Modular forms and invariant theory

Abstract: I will explain how invariant theory can be used to construct modular forms. This talk is based on joint work with Gerard van der Geer and Carel Faber.

Evgeny Ferapontov (Loughborough University): On ODEs satisfied by modular forms

Abstract: It is well known that every modular form f on a discrete subgroup of SL(2, R) satisfies a third-order nonlinear ODE that expresses algebraic dependence of the functions f, f', f'' and f'''. These ODEs are automatically invariant under the Lie group SL(2, R), which acts on the solution spaces thereof with an open orbit. Similarly, every modular form satisfies a fourth-order nonlinear ODE that is invariant under the Lie group GL(2, R) acting on its solution space with an open orbit. ODEs for modular forms can be compactly expressed in terms of the differential invariants of these actions. The invariant forms of both ODEs define plane algebraic curves naturally associated with every modular form; the corresponding ODEs can be seen as modular parametrisations of the associated curves.

Valery Gritsenko (University of Lille/International laboratory of mirror symmetry and automorphic forms, NRU HSE): On МDEs satisfied by the elliptic genus of hyperkähler varieties

Abstract: The elliptic genus (EG) of a compact complex variety of dimension d with the trivial first Chern class is a weak Jacobi form of weight 0 and index d/2. In 2022 we proved that EG of a Calabi-Yau 3-fold satisfies the simplest possible modular differential equation of order one in terms of the heat operator. In this talk we present much more complicated MDE of the elliptic genus of hyperkähler varieties of dimension 2, 4 and 6. Based on joint work with Dimitri Adler (State University of St. Petersburg).

Dimitri Markushevich (University of Lille): On the complex crystallographic group associated to Klein's simple group of order 168

Abstract: A general conjecture of Bernstein-Schwarzman suggests that the quotient of the affine complex space Cⁿ by an irreducible complex crystallogrpahic reflection group Γ is a weighted projective space; this is a natural generalization of Shephard-Todd-Chevalley theorem for finite linear groups generated by reflections. I will explain in the talk a particular case of this conjecture when the projectivized group of linear parts of transformations from Γ is Klein's simple group of order 168 in its 3-dimensional representation. In this case C³/ Γ can be otherwise described as the quotient of the Jacobian of Klein's quartic curve by the full automorphism group of the Jacobian as a PPAV, and this quotient is isomorphic to P(1, 2, 4, 7). This is a joint work with Anne Moreau.

Donihakalu Nagaraj (IISER Tirupati): On nef and big vector bundles

Werner Nahm (IAS Dublin): On vanishing theorems for vector bundles

Alexandre Odesski (Brock University, St. Catharines): Nonabelian elliptic Poisson structures on projective spaces

Abstract: We review noncommutative Poisson structures on affine and projective spaces over C. We also construct a class of examples of noncommutative Poisson structures on CP^n–1 for n>2. These noncommutative Poisson structures depend on a modular parameter τ∊C and an additional descrete parameter k∊Z, where 1≤k<n and k, n are coprime. The abelianization of these Poisson structures can be lifted to the quadratic elliptic Poisson algebras q_n,k(τ). This is a joint work with Vladimir Sokolov.

Stanislav Opanasenko (Loughborough University): Involutive differential systems for Jacobi forms

Abstract: After reviewing examples of nonlinear ODEs satisfied by classical modular forms (such as Eisenstein series, modular forms on congruence subgroups of level two and three, theta constants, and some newforms of weight two), we generalise these results to Jacobi forms; these satisfy involutive third-order PDE systems that are invariant under the semidirect product of SL(2, R) with the Heisenberg group.

Valentin Ovsienko (Reims University): Modular invariant q-deformed numbers: first steps

Abstract: The goal of this talk is to explain the main ideas of the emerging new theory of “quantum rationals” based on modular, or PSL(2,Z)-invariance, and that of “quantum irrationals”. Quantum rationals are rational functions in one variable q, it turns out that every coefficient of the polynomials in the numerator and denominator has a combinatorial meaning. Quantum irrationals are Laurent series in q that have nice analytic properties. I will outline relations of these q-numbers to various subjects of combinatorics, knot theory, Markov-Hurwitz approximations, and enumerative geometry.

Vladimir Roubtsov (University of Angers): Multiplicative kernels: from Bessel to Kontsevich, Buchstaber and Calabi-Yau

Abstract: We discuss a few very recent results of a work in progress (in collaboration with I. Gaiur and D. Van Straten and with V. Buchstaber and I. Gaiur) about interesting properties of multiplication Bessel kernels, which includes well-known Clausen and Sonin-Gegenbauer formulae of XIX century, special examples of Kontsevich discriminant loci polynomials, raised as addition laws for special two-valued formal groups (Buchshtaber-Novikov-Veselov) and period functions for some CY and Landau–Ginzburg models.

Nils Skoruppa (University of Siegen): Dessins d'enfants, network flows and modular forms

Alexander Veselov (Loughborough University): Complex cobordisms, theta divisors and permutohedra

Abstract: It was known since 1960s (Novikov, Mischenko) that the logarithm of the formal group in complex cobordisms can be written explicitly in terms of the complex projective spaces, but the algebro-geometric nature of the coefficients of the corresponding exponential was not clear until recently. In the talk I will explain that the answer can be given by the smooth theta divisors of principally polarised abelian varieties.

It will be shown that the topological characteristics of the theta divisors and their intersections can be expressed in terms of the combinatorics of permutohedra. We reveal also interesting relations between the theta divisors, permutohedral varieties and Tomei manifolds from the Toda lattice theory. The talk is based on joint work with V.M. Buchstaber.

Wadim Zudilin (Radboud University of Nijmegen): Magnetic modular forms

Abstract: Given positive even integer k, let E_k(τ) stands for the normalised Eisenstein series of weight k; denote Δ(τ) = q∏_m≥1(1−q^m)²⁴ = (E₄³−E₆²)/1728 with q=exp(2πiτ), and δ = q(d/dq) is the differentiation. The Eisenstein series and (quasi-)modular forms in general have integer coefficients in their q-expansions. They are also known to satisfy nonlinear differential equations; one example usually attributed to Ramanujan is a simple algebraic system of differential equations for E₂, E₄ and E₆.

The recent work of Broadhurst and myself (responsible for the adjective "magnetic") followed by the work of Li and Neururer bring to life new examples of non-modular q-series with integral coefficients whose derivatives are modular. This theme is systematically explored in our joint work with Paşol, on which this talk is based. Examples include the anti-derivatives δ⁻¹(Δ/E₄²) and δ⁻¹(E₄Δ/E₆²) which have integral coefficients in their q-expansions, while at the same time these series are transcendental over the field of (quasi-)modular functions. I plan to discuss this phenomenon (in a greater generality) and ideas behind proofs in my talk.

Working Groups

WG1: Algebraic geometry and automorphic forms

WG2: Automorphic forms and integrable models

WG3: New perspective on cooperative research on the subject of the conference