Relations and derivatives of multiple Eisenstein series

Henrik Bachmann, Nagoya University

Multiple Eisenstein series are holomorphic functions on the upper half-plane that can be viewed as hybrids of classical Eisenstein series and multiple zeta values. The latter satisfy the double shuffle relations, which are conjectured to generate all linear relations among them. In this talk, we present an analogue of the double shuffle conjecture for multiple Eisenstein series. The conjectured relations arise from an explicit formula for the derivative of multiple Eisenstein series, expressed via the Drop1 operator of Hirose–Maesaka–Seki–Watanabe. We further give evidence that the space of multiple Eisenstein series carries an $\mathfrak{sl}_2$-algebra structure, and prove that this structure holds for the algebra of formal multiple Eisenstein series. This is joint work with Hayato Kanno and Takumi Maesaka.

A reinforcement learning approach to computing zeta functions of groups and algebras

Anton Baykalov, University of Galway

In this talk, I will report on ongoing work on explicit computations of zeta functions associated with various types of counting problems attached to groups, algebras, and related algebraic structures.

The goal of this project is to combine systematic methods (which can be very computationally involved and limited in scope) and ad hoc approaches driven by human insight: various manipulations with $p$-adic integrals (which traditionally are chosen and performed by a human researcher) can be formulated as moves in an interactive “game” of computing the integral. Neural networks then are trained through repeated play of this game.

Birational Zeta Functions

Tom Biesbrouck, KU Leuven

The motivic zeta function is a classical rich invariant in singularity theory. In this talk, I will report on joint work with N. Budur, J. Nicaise and W. Veys where we define a birational analog of the motivic zeta function of a reduced polynomial f in terms of minimal models. It admits an intrinsic meaning in terms of contact loci of arcs, an analog of a result of Denef and Loeser in the motivic case. We show that for local plane curve singularities the poles of the birational zeta function essentially coincide with the poles of the motivic zeta function.

Local zeta functions, Mellin transforms and counting points on hypersurfaces over finite fields

Francis Brown, University of Oxford

I will give an overview of the topics in the title, starting with a very classical and elementary formula for the number of points of a hypersurface over $\mathbb{F}_p$ modulo p, where $p$ is a prime. This naturally segues to a discussion of the sequence of diagonal coefficients of a polynomial, the notion of P-recursive functions and the theory of Mellin transforms.

I will then mention recent work with Erik Panzer where we extend this formula to prime powers, and if time permits, discuss congruences, Mellin-Feynman integrals, Apéry sequences, and applications to the completion conjecture for graph hypersurfaces.

An update on archimedean, birational, and motivic zeta functions

Nero Budur, KU Leuven

In this talk we present an overview of recents results on archimedean, birational, and motivic zeta functions related to singularities.

Lie algebras for multiple $q$-zeta values

Annika Burmester, Bielefeld University

Multiple $q$-zeta values are certain $q$-series which degenerate to multiple zeta values under the limit $q$ to 1 and which have strong connections to multiple Eisenstein series. In this talk, we investigate the algebraic structure of these multiple $q$-zeta values by studying their associated Lie algebras. Our approach to these Lie algebras is inspired by Racinet’s work on multiple zeta values, as well as the study of derivations on (formal) multiple Eisenstein series by Bachmann and van Ittersum.

Degree of (twisted) commutativity and subgroup growth

Laura Ciobanu, TU Berlin

The degree of commutativity of a group $G$ measures the proportion of pairs of commuting elements in $G$, so the number of $(x,y)$ in $G\times G$ such that $xy=yx$, among all pairs of elements in $G$. This is a concept that connects to the standard and conjugacy growth of $G$, and has been studied asymptotically in many classes of infinite groups.

In this talk I will introduce a generalization of the above called the `degree of twisted commutativity’, which is concerned with the proportion of $(x,y)$ such that $xy=f(y)x$, where $f$ is an automorphism of $G$. I will show how this connects to twisted conjugacy growth, the Reidemeister number of an automorphism and relative subgroup growth, and present a few results for groups of both polynomial and exponential growth. This is joint work with Corentin Bodart, Gemma Crowe, and Pieter Senden.

Zeta functions for point counting and some of their roles in mathematics

Raf Cluckers, University of Lille and KU Leuven

In this survey talk, I will sketch various uses of zeta functions in the context of point (or other) counting and how they play roles in mathematics, from counting and bounding to sums and Birch’s local global principle from the early ‘60 leading to Igusa’s dream. This brings us naturally to integrals which of course play important roles in almost all fields of mathematics and are studied from many angles. Already the basic notion of integrability itself leads to many concepts, results, and open questions, think for example of the log canonical threshold. If time permits, I will mention several open questions along the way and possibly reflect on the question why to count and bound.

Ask Zeta Functions of Unitary Lie Algebras

David Cormican, University of Galway

Let $K$ be a quadratic algebraic number field and $\mathfrak{O}_K$ its ring of integers. For prime $p$ inert in $O_K$ and almost all $p$ which split in $O_K$, we present recent work giving the ask zeta function of $\mathfrak{gu}_d(\mathfrak{O}_K/\mathbb{Z}) \otimes \mathbb{Z}_p$ and $\mathfrak{su}_d(\mathfrak{O}_K/\mathbb{Z}) \otimes \mathbb{Z}_p$ for the general and special unitary Lie algebras of dimension $d > 1$ over $\mathfrak{O}_K$. We note that these local ask zeta functions are identical, except in the case of the $d = 2$ where the prime $p$ splits. We will also explain the links between this work and earlier investigations of ask squared zeta functions for $\mathfrak{gl}_d(\mathbb{Z}_p)$ and conjugacy class zeta functions for certain unipotent group schemes.

Multiple zeta values, symmetries, and graphs

Clément Dupont, University of Montpellier

In this talk I will report on work in progress (joint with Erik Panzer and Brent Pym) on the graphical structures underlying the symmetries of multiple zeta values. Our work aims at understanding:

  • the arithmetic symmetries of Kontsevich’s deformation quantization formula, which involves multiple zeta values by a result of Banks—Panzer—Pym;

  • the relationship between the motivic Galois group of multiple zeta values and Kontsevich’s graph complex.

The main ingredient is an algebro-geometric model for the little disks operad, which uses the language and technology of logarithmic geometry.

A Motivic Poisson Formula for Split Algebraic Tori and Application To Motivic Height Zeta Functions

Loïs Faisant, KU Leuven

Over the past ten or twenty years, a number of works—by Bilu & Browning, Browning & Sawin, Browning & Vishe, Bourqui, Chambert-Loir & Loeser, Glas & Hase-Liu, Peyre, among others—have demonstrated how number theory, particularly analytic number theory, can provide new insights into the study of moduli spaces of curves.

Notably, the recent development of motivic versions of number-theoretic tools (harmonic analysis on adèles, the circle method, lifting to universal torsors) has paved the way for a motivic version of Manin’s program: the dictionary between number fields and function fields allows one to move from the fine study of the distribution of rational points on Fano varieties to predictions about the virtual motive of the moduli space of morphisms from a given curve to a Fano variety over the complex numbers.

The purpose of this talk is to present the results of a collaboration with Margaret Bilu (CNRS/École Polytechnique), in which we develop a multiplicative version of the motivic Poisson formula.

This allows us to demonstrate a motivic stabilization phenomenon concerning the moduli space of morphisms from an algebraic curve (projective, smooth, of arbitrary genus) to a toric variety, a problem which is encoded in a so-called motivic height zeta function.

Hopf algebras and multiple zeta values in positive characteristic

Bo-Hae Im, KAIST (Korea Advanced Institute of Science and Technology)

Multiples zeta values (MZV’s for short) in positive characteristic were introduced by Thakur as analogues of classical multiple zeta values of Euler. In this talk we give a systematic study of algebraic structures of MZV’s in positive characteristic. We construct both the stuffle algebra and the shuffle algebra of these MZV’s and alternating MZV’s and equip them with algebra and Hopf algebra structures. In particular, we completely solve a problem suggested by Deligne and Thakur in 2017 and establish Shi’s conjectures. The construction of the stuffle algebra is based on our work on Zagier-Hoffman’s conjectures in positive characteristic. Also we have extened our result for alternating MZV’s. This is a joint work with Kim, Le, Ngo Dac, and Pham.

Multiple Dirichlet series and naive duality

Adam Keilthy, Trinity College Dublin

The standard example of multiple Dirichlet series are multiple zeta values, which can equivalently be defined via an iterated sum or an iterated integral. These representations naturally lead to quadratic algebraic relations coming from the decomposition of products of simplices - the double shuffle relations - and a duality relation coming from an automorphism of the punctured plane.

We consider two approaches to extending a general Dirichlet series to multiple variables and consider what conditions must be imposed to obtain analogous quadratic and duality relations. In particular, we define a notion of a naive duality, and show that, under some analytic assumptions, multiple zeta values are the unique multiple Dirichlet series satisfying a duality relation.

On the abscissa of convergence of Weil representation zeta functions

Steffen Kionke, University of Hagen

The Weil representation zeta function of a group $G$ enumerates the absolutely irreducible representations of $G$ over all finite fields in a way that resembles the Hasse-Weil zeta function of an arithmetic scheme. Examples suggest that the zeta function behaves nicely for a rich class of groups. However, studying basic properties of the Weil representation zeta function can be subtle as one quickly runs into deep problems concerning the distribution of primes.

In this talk I will introduce the Weil representation zeta function, discuss the abscissa of convergence and explain recent results from joint work with Ged Corob Cook and Martin Jann.

Zeta functions of groups, rings, and modules

Joshua Maglione, University of Galway

We discuss key themes surrounding zeta functions arising from algebra. A central focus is on the question of how the local factors of a global zeta function vary with the prime. In general, this question reflects the difficulty of counting rational points on algebraic varieties over finite fields. Of particular interest are cases involving polynomial such point counts, leading to uniform variation with the prime. We present recent results establishing uniformity for large classes of zeta functions obtained via substitutions into so-called combinatorial Denef-type formulae. These substitutions reveal a striking connection: on the one hand, they produce zeta functions; on the other, they recover familiar rational functions from algebraic combinatorics, such as Hilbert series of Stanley–Reisner rings.

On multiple zeta values with combinatorial structure

Maki Nakasuji, Sophia University

Multiple zeta values have been studied at least since Euler, who found many of their algebraic properties.

In contrast, this talk introduces a combinatorial generalization of Euler-Zagier multiple zeta values, namely Schur multiple zeta values. These values are defined from a new perspective as sums over combinatorial objects called semi-standard Young tableaux.

In this talk, we will present a determinant formula for Schur multiple zeta values derived from a combinatorial approach, and discuss how this formula leads to the duality formula for these values.

Furthermore, if time permits, we will introduce a further extension, namely multiple zeta values endowed with the structure of symmetric functions.

On Representation Zeta Functions in Positive Characteristic

Uri Onn, Australian National University

Representation zeta functions are Dirichlet generating functions associated with the representation growth of groups, encoding the numbers of irreducible representations of each dimension. The representation zeta functions of $p$-adic and arithmetic groups in characteristic zero have been studied extensively over the past two decades. By contrast, the positive-characteristic case remains comparatively unexplored, owing in part to the absence of the orbit method and the limited availability of resolution-of-singularities techniques. In this lecture, I will survey some of the known results in this area and present recent joint work with Amritanshu Prasad and Pooja Singla on the representation zeta functions of groups of type $A_2$.

Representation zeta functions of split extensions of $\mathrm{SL}_2^m(O)$

Margherita Piccolo, Fernuniversität in Hagen

The representation growth of a group $G$ measures the asymptotic distribution of its irreducible representations. Whenever the growth is polynomial, a suitable vehicle for studying it is a Dirichlet generating series called the representation zeta function of $G$. One of the key invariants in this context is the abscissa of convergence of the representation zeta function. The spectrum of all abscissae arising across a given class of groups is of considerable interest and has been studied in some cases. In the realm of $p$-adic analytic groups (with perfect Lie algebra), the abscissae of convergence are explicitly known only for groups of small dimensions. But there are interesting asymptotic results for “simple” p-adic analytic groups of increasing dimension.

In this talk, I will give an overview of the main tools and ingredients in this area and I will report on recent work joint with Moritz Petschick to enlarge the class of groups to split extensions of $\mathrm{SL}_2^m(O)$.

$p$-Adic asymptotics of nilpotent algebras

Tomas Reunbrouck, Universität Bielefeld

Given a finite-dimensional algebra $L$, the poles of its local zeta function correspond to terms of exponential growth in the number of finite-index subalgebras. The largest (real) pole therefore dominates the asymptotic behaviour. The other poles, for their part, relate to the algebra’s combinatorial structure in more subtle ways. In particular, the smallest pole describes the $p$-adic asymptotics of the coefficients. A conjecture by Tobias Rossmann determines this pole and its residue for nilpotent algebras, resulting in a very strong $p$-adic asymptotic description of the number of finite-index subalgebras.

Subgroup zeta functions of higher Heisenberg groups

Jianhao Shen, Bielefeld University

Subgroup zeta functions encode subgroup growth in finitely generated nilpotent groups, but are difficult to compute explicitly, even for one of the simplest families of nonabelian nilpotent groups: the higher Heisenberg groups. I will discuss explicit formulae for these groups, based on recent joint work with C. Voll.

Tensor Products of Regular Representations of General Linear Groups of Degree Two over Finite Principal Ideal Local Rings

Pooja Singla, IIT Kanpur

The tensor product problem, a classical question in representation theory, concerns the decomposition of the tensor product (or Kronecker product) of two irreducible representations into a direct sum of irreducible constituents. This problem arises naturally in several areas of mathematics and has been extensively studied for polynomial representations of $\mathrm{GL}_n$ over the complex numbers, as well as for symmetric and alternating groups. Although multiplicity-free tensor products have been characterised in certain cases, the tensor product problem remains widely open even for symmetric groups.

In this talk, we study the tensor product problem for general linear groups of degree two over finite principal ideal local rings. Restricting attention to regular representations, we extend several known results from the finite field setting to this broader context. In particular, we classify pairs of regular irreducible representations whose tensor product is multiplicity free. We use the structure of the representations of groups over finite local rings with a few character-theoretic techniques. This talk is based on joint work with Archita Gupta and M Hassain.

Poles of zeta functions & finite (ramified) covers of normal surfaces

Juan Viu-Sos, Universidad Politécnica de Madrid

Motivic and topological zeta functions are analytic invariants of singularities. In particular, one can attach them to a pair $(D,W)$ of Weil divisors on a normal surface germ, where $D$ plays the role of a function and $W$ encodes a differential form.

In this work, we study the behavior of these zeta functions under finite morphisms of normal surfaces and we relate the sets of poles on the source and the target, focusing in the case of quotient maps induced by an action of a finite abelian group. We illustrate by examples how the hypotheses and the resulting relations are sharp.

This is a joint work with E. León-Cardenal (U. Zaragoza), J. Martín-Morales (U. Zaragoza) & W. Veys (KU Leuven).

Motivic volumes and reductive group actions

Dimitri Wyss, EPFL

Motivated by work of Batyrev on the McKay correspondence, Denef and Loeser studied motivic integration on algebraic varieties with finite quotient singularities. In particular they proved the orbifold formula which allows to compute certain motivic volumes without the need of a resolution of singularities. This has also been used by Léon-Cardenal-Martin-Morales-Veys-Viu-Sos to simplify certain computations of Igusa zeta functions.

In joint work with Arthur Forey and François Loeser we extend this theory to quotients of smooth varieties by reductive groups and prove a generalization of this orbifold formula. I will explain some direct applications to the computations of stringy Hodge numbers and how one can give a cohomological interpretation similar to Chen-Ruan orbifold cohomology for finite quotients. The last part is joint work in progress with Sebastian Schlegel-Mejia.

Local Zeta Functions, String Amplitudes, and Graphs

Wilson A. Zúñiga-Galindo, University of Texas Rio Grande Valley

The talk aims to discuss the connections between local zeta functions and physics. The first part of the talk is dedicated to the Koba-Nielsen amplitudes defined on any local field of characteristic zero; we discuss their meromorphic continuations in the kinematic parameters. In the regularization process, we use techniques of local zeta functions and embedded resolution of singularities. As a consequence, we introduced the notion of the Koba-Nielsen local zeta function, which is a mathematical version of the string amplitudes with the same name. In the second part, we present a generalization of the Koba-Nielsen zeta functions where the integration is over a variety of (bounded or unbounded) convex subsets; the resulting integrals also admit meromorphic continuations in the complex parameters. We describe the meromorphic continuation’s polar locus explicitly, using the technique of embedded resolution. This type of integral includes, as a particular case, many integrals used in physics. In the third part, we introduce the local zeta functions associated with graphs and discuss their connections to Coulomb gases. The final part is dedicated to discussing the limit when $p$ tends to one of the $p$-adic Koba-Nielsen zeta functions, which connects ordinary string amplitudes with the $p$-adic ones.

References
  1. Veys Willem, Zúñiga-Galindo W. A., “Koba-Nielsen local zeta functions, convex subsets, and generalized Selberg-Mehta-Macdonald and Dotsenko-Fateev-like integrals. To appear in Number Theory and Physics.
  2. Fuquen-Tibatá A. R., García-Compeán H., Zúñiga-Galindo W. A., Euclidean quantum field formulation of p-adic open string amplitudes, Nuclear Phys. B 975 (2022), Paper No. 115684, 27 pp.
  3. Zúñiga-Galindo W. A., Zambrano-Luna B. A., León-Cardenal E., “Graphs, local zeta functions, log-Coulomb gases, and phase transitions at finite temperature,” J. Math. Phys. 63 (2022), no. 1, Paper No. 013506, 21 pp.
  4. Bocardo-Gaspar M., Veys W. & Zúñiga-Galindo W.A., “Meromorphic continuation of Koba-Nielsen string amplitudes,” J. High Energ. Phys. 2020, 138 (2020).
  5. García-Compeán H., López Edgar Y., Zúñiga-Galindo W. A., “p-adic open string amplitudes with Chan-Paton factors coupled to a constant B-field,” Nuclear Phys. B 951 (2020), 114904, 33 pp.
  6. Bocardo-Gaspar M., Garcia-Compean H., Zúñiga-Galindo W. A., “p-Adic string amplitudes in the limit p approaches to one,” J. High Energy Phys. 2018, no. 8, 043, front matter +22 pp.
  7. Bocardo-Gaspar M, Garcia-Compean H., Zúñiga-Galindo W. A., “Regularization of p-adic String Amplitudes, and Multivariate Local Zeta Functions,” Lett. Math. Phys. 109 (2019), no. 5, 1167–1204.