Priority Program Annual Conference

The second Annual Conference of the Priority Program Combinatorial Synergies (SPP2458) takes place in Hannover on September 3-5, 2025.
The goal is to bring together members of the SPP working on the different core themes and to foster collaborations.

Speakers

Schedule

Wednesday Thursday Friday
10:00 – 10:30 Arrival and Welcome 09:00 – 10:00 Ana Botero 09:30 – 10:30 Joshua Maglione
10:30 – 11:30 Laura Ciobanu 10:00 – 10:30 Coffee break 10:30 – 11:00 Coffee break
11:30 – 12:30 Aida Maraj 10:30 – 12:00 Discussion of projects 11:00 – 12:00 Eleonore Faber
12:30 – 14:00 Lunch break 12:00 – 14:00 Lunch break 12:00 – 13:30 Lunch break
14:00 – 15:00 Volkmar Welker 14:00 – 15:00 Katharina Jochemko 13:30 – 14:30 Christian Krattenthaler
15:00 – 15:30 Coffee break 15:00 – 15:45 Marius Lindauer
15:30 – 16:30 Martin Ulirsch 16:30 – 17:30 Discussion
18:30 Conference Dinner

Registration

Registration is open now via https://forms.gle/Wu93yrpvumpPG9BeA. There is no conference fee, but registration is mandatory.

Venue

The talks take place in the main building of the Leibniz University Hannover.

Funding + Accommodation

There is some funding available for junior participants. Please indicate in the registration form if you need funding.

We have reserved 40 rooms at CVJM City Hotel Hannover, Limburgstrasse 3, until August 4. Please use the magic word "Jahrestagung" at your registration.

Abstracts

Joshua Maglione: Symplectic Hecke eigenbases from Ehrhart polynomials

We consider the functions that map a lattice polytope in R^n to the l-th coefficient of its Ehrhart polynomial for l in {0, 1, ..., n}. These functions form a basis for the space of so-called unimodular invariant valuations. We show that, in even dimensions, these functions are in fact simultaneous symplectic Hecke eigenfunctions. We leverage this and apply the theory of spherical functions and their associated zeta functions to prove analytic, asymptotic, and combinatorial results about arithmetic functions averaging l-th Ehrhart coefficients.

Joint with Claudia Alfes and Christopher Voll

Christian Krattenthaler: Proofs of Borwein Conjectures

The (so-called) "Borwein Conjecture" arose around 1990 and states that the coefficients in the polynomial

\[(1-q)(1-q^2)(1-q^4)(1-q^5)\cdots(1-q^{3n-2})(1-q^{3n-1})\]
have the sign pattern \(+--+--\dots\). This innocent looking prediction has withstood all proof attempts until two years ago when Chen Wang found a proof that combines asymptotic estimates with a computer verification for "small" \(n\).

However, Borwein made actually in total three sign pattern conjectures of similar character - with the previously mentioned conjecture being just the first one -, and recently Wang discovered a further one. It seemed unlikely that Wang's proof could be adapted to work for these other conjectures since it crucially used identities that are only available for the "First Borwein Conjecture".

I shall start by presenting these conjectures and then review the history of the conjectures and the various attempts that have been made to prove them - as a matter of fact, these attempts concerned exclusively the "First Borwein Conjecture", while nobody had any idea how to attack the other conjectures.

I shall then outline a proof plan that is (in principle) applicable to all these conjectures. Indeed, this leads to a new proof of the "First Borwein Conjecture", the first proof of the "Second Borwein Conjecture", and to a proof of "two thirds" of Wang's conjecture. We are convinced that further work along these lines will lead to - at least - a partial proof of the "Third Borwein Conjecture".

I shall close with further open problems in the same spirit.

This is joint work with Chen Wang.

Laura Ciobanu: Growth in groups, geometry and combinatorics

In this talk I will give an overview of standard and conjugacy growth in groups and their associated formal series. I will highlight how the rationality (or lack thereof) of these series is connected to both the algebraic and the geometric nature of groups such as (relatively) hyperbolic or nilpotent, and how tools from analytic combinatorics can be employed in this context.

Katharina Jochemko: Preservation of Inequalities under Hadamard Products

Formal power series are ubiquitous in enumerative combinatorics and related areas, where the Hadamard product of two generating functions often corresponds to fundamental operations of the structures they are enumerating. A prime example is the Ehrhart series of a lattice polytope: the Hadamard product of two Ehrhart series equals the Ehrhart series of the Cartesian product of the corresponding lattice polytopes. By a result of Wagner (1992), the Hadamard product of two Pólya frequency sequences that are interpolated by polynomials is again a Pólya frequency sequence. In this talk, we discuss the preservation under Hadamard products of related properties of significance in combinatorics, in particular, ultra log-concavity and gamma-positivity, with a focus on Ehrhart theory. Joint work with Petter Brändén and Luis Ferroni.

Eleonore Faber: Frieze patterns from Grassmannian cluster algebras of infinite rank and Penrose tilings

This talk is about certain non-periodic frieze patterns, which can be obtained from a categorification of a Grassmannian cluster algebra of infinite rank: the category of maximal Cohen-Macaulay modules over the so-called A-infinity curve singularity. This Frobenius category has a rich combinatorial structure and was studied in the context of triangulations of the infinity-gon by August, Cheung, Faber, Gratz, and Schroll. Extending the cluster character from work of Paquette and Yildirim to this setting we obtain a new type of infinite friezes that can be related to Penrose tilings. This is joint work with Özgür Esentepe.

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