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 09:00 – 10:00 Joshua Maglione
10:30 – 11:30 Aida Maraj 10:00 – 10:30 Coffee break 10:00 – 10:30 Coffee break
11:30 – 12:30 Laura Ciobanu 10:30 – 12:00 Discussion of projects 10:30 – 11:30 Eleonore Faber
12:30 – 14:00 Lunch break 12:00 – 14:00 Lunch break 11:30 - 12:30 Christian Krattenthaler
14:00 – 15:00 Volkmar Welker 14:00 – 15:00 Katharina Jochemko 12:30  Lunch
15:00 – 15:30 Coffee break 15:00 – 16:00 Marius Lindauer
15:30 – 16:30 Martin Ulirsch 16:00 - 16:30 Coffee break
16:30 – 17:30 Christian Stump
18:30 Conference Dinner

Venue

The talks take place in the main building (Hauptgebäude) of the Leibniz University Hannover in B305 https://standortfinder.uni-hannover.de/

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

Ana Botero: Pluripotential theory on balanced polyhedral spaces

Pluripotential theory is a branch of complex analysis in several variables that studies plurisubharmonic functions and their properties. It is a strong tool with relevant applications to the study of the complex Monge--Ampère equation, a nonlinear, second-order partial differential equation whose solution solves the famous Calabi--Yau problem. The main goal of this talk is to present a similar pluripotential theory on balanced polyhedral spaces and to study a polyhedral Monge--Ampère equation in this setting. This is joint work with Enrica Mazzon and Léonard Pille-Schneider.

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.

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.

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.

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.

Marius Lindauer: TBA

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 Stump: ScienceBench: Learn how to use AI in your research

We do a joint live session to discuss the use of the new project math.science-bench.ai.

Martin Ulirsch: Rethinking tropical linear algebra: Bimatroids, buildings and more

One way to think about tropical geometry is to do geometry over the tropical semifield \(\mathbb{T}\). Applied to the realm of linear algebra, this perspective leads to a calculus of matrices over \(\mathbb{T}\). This naive approach to tropical linear algebra has two major shortcomings:

  • Tropical matrices may not witness determinants of minors correctly.
  • The process of tropicalization is not functorial.

In this talk I will explain how to use valuated bimatroids and affine Bruhat--Tits buildings respectively in order to overcome these two problems. Moreover, along the way, we will also explore several new logarithmic concavity results that (partially) generalize known results, such as Mason's conjecture, to the valuated setting.

This talk, in parts, is based on joint works with - Luca Battistella, Kevin Kühn, Arne Kuhrs, and Alejandro Vargas, - Felix Röhrle, as well as with - Jeffrey Giansiracusa, Felipe Rincon, and Victoria Schleis.

Volkmar Welker: Common bases and decompositions of algebraic and combinatorial structures

The common basis complex was first introduced by Rognes for free modules over suitable rings. We extend the definition to a wide class of algebraic and combinatorial structures and show that the resulting complex is homotopy equivalent to the poset of partial direct sum decompositions of the respective structure. We provide an axiomatic description of the class of algebraic and combinatorial stuctures which satisfy these conditions. We give examples of explicit homotopy types and homology modules for some classes.

We also define ordered partial direct sum decompositions and show that in the case of vectorspaces its order complex is homotopy equivalent to the join of two copies of the building and carries the tensor square of the Steinberg representation. We provide a construction having the same properties for other reductive groups.

This is joint work with various subsets of Banjamin Brück, Kevin Piterman and John Shareshian.

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