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 Federico Ardila 14:00 – 15:00 Katharina Jochemko 13:30 – 14:30 Volkmar Welker
15:00 – 15:30 Coffee break 15:00 – 15:45 Marius Lindauer 14:30 – 15:00 Coffee break
15:30 – 16:30 Martin Ulirsch 16:30 – 17:30 Discussion 15:00 – 16:00 Christian Krattenthaler
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.

Federico Ardila: Inequalities for trees and matroids

In their 1971 study of telephone switching circuitry, Graham and Pollak designed a novel addressing scheme that was better suited for the faster communication required by computers. They introduced the distance matrix of a graph, and used its eigenvalues to bound the size of the addresses in their scheme. We continue their investigation, obtaining more precise spectral information about tree distance matrices. As a consequence, we prove two conjectural inequalities for matroids and valuated matroids.

This is joint work with Sergio Cristancho, Graham Denham, Chris Eur, June Huh, and Botong Wang. The talk will not assume previous knowledge of these topics.

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.

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