Priority Program Kick-Off Meeting

The first Annual Conference of the Priority Program Combinatorial Synergies (SPP2458) takes place in Osnabrück on September 11-13, 2024.
The goal is to bring together members of the SPP working on the different core themes and to foster collaborations.

Group Picture

group picture

Confirmed speakers

Schedule

The regular workshop program starts Wednesday, September 11, at 9 a.m. and ends Friday, September 13, around noon.

The workshop on 'Implicit Biases' takes place Friday, September 13, 13:30-18:00.

Wednesday September 11 Thursday September 12 Friday September 13
09:00-09:30 Arrival and Welcome 09:00-10:00 Christian Stump
09:30-10:30 Karin Baur 10:00-10:30 Coffee Break 9:30-10:30 Mareike Dressler
10:30-11:00 Coffee Break 10:30-12:00 Discussion of the SPP Research Projects 10:30-11:00 Coffee Break
11:00-12:00 Martin Wahl 11:00-12:00 Giulio Salvatori
12:00-14:00 Lunch Break 12:00-14:00 Lunch Break 12:00-13:30 Lunch Break
14:00-15:00 Gennadiy Averkov 14:00-15:00 Alheydis Geiger 13:30-18:00 Workshop on 'Implicit Biases' (Dr. Sabine Müller)
15:00-15:30 Coffee Break 15:00-15:30 Wrap-up of the discussion session
15:30-16:30 Lisa Seccia 15:30-16:00 Coffee Break
16:45-18:00 Mentoring for PhD students (Bernd Sturmfels) 16:00-17:00 Martin Winter
19:00 Dinner at Pizzahaus

Registration

Registration is now open. You can register here.

Dates and Program

Besides nine research talks, covering all core topics of the SPP, there will be special network activities tailored to strengthen and discover connections between different projects. The regular program starts September 11, in the morning and ends September 13 after lunch. In the afternoon of September 13, there will be a workshop on 'Implicit Biases' for Senior participants. (Please note that this workshop has a limited number of 14 participants.)

Workshop Dinner

The workshop dinner takes place at 7 p.m. on September 12, at Pizzahaus. There will be a menu, including starters, main dish and dessert. For the main dish, there will be a meat, fish and a vegan option. Only drinks are self-paid, the food is paid by the funds of the SPP.

Venue

The workshop takes place in the Helikonensaal Bohnenkamp-Haus in the Botanical Garden, Albrechtstraße 29 in Osnabrück.

We suggest to have lunch at the Mensa of Campus Westerberg which is at about a 5 minutes walk from the Bohnenkamp-Haus.

Funding + Accomodation

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

There are various hotels in Osnabrück. We recommend to book a hotel room in walking distance of the workshop venue. Some options for this are:

Please let us know if you need help in finding and/or booking a hotel.

Mentoring Session

There will be a mentoring session for PhD students on Wednesday afternoon. To get started please have a look at the poster with the nine themes and the following exercise sheet with related questions: Mentoring Session

Abstracts

Gennadiy Averkov: Mixed volumes of zonoids and the absolute value of the Grassmannian

Zonoids are Hausdorff limits of zonotopes, while zonotopes are convex polytopes defined as the Minkowski sums of finitely many segments. We present a combinatorial framework that links the study of mixed volumes of zonoids (a topic that has applications in algebraic combinatorics) with the study of the absolute value of the Grassmannian, defined as the image of the Grassmannian under the coordinate-wise absolute value map. We use polyhedral computations to derive new families of inequalities for n zonoids in dimension d, when (n,d)=(6,2) and (6,3). Unlike the classical geometric inequalities, originating from the Brunn-Minkowski and Aleksandrov-Fenchel inequalities, the inequalities we produce have the special feature of being Minkowski linear in each of the n zonoids they involve.

Karin Baur: Frieze patterns, surfaces and representation theory

Cluster categories and cluster algebras can be described via triangulations of surfaces. Polygon triangulations correspond to Coxeter’s frieze patterns of finite type. We explain how infinite frieze patterns arise and discuss their growth behaviour. In particular, we show that in affine types, cluster categories yield friezes with linear growth.

Mareike Dressler: Beyond the Sum of Squares based Approach for Polynomial Optimization

Deciding nonnegativity of polynomials is a key problem in real algebraic geometry since the 19th century with crucial importance to nonlinear polynomial optimization. In recent years, several interrelated approaches for nonnegative polynomials and signomials (weighted sums of exponentials composed with linear functionals) have been proposed, which are aimed at sparse settings. In this talk, I will introduce sums of nonnegative circuit (SONC) polynomials - a recent class of nonnegativity certificates for real, multivariate polynomials, independent of sums of squares. I will present geometrical and structural results of SONC polynomials, and I will provide an overview of polynomial optimization via SONC polynomials. To conclude, I show how these ideas extend to signomial optimization and discuss some applications.

Alheydis Geiger: Self-dual matroids and the tropical Grassmannian

Self-dual point configurations have been studied by Dolgachev and Ortland, Coble and many others. We investigate them in the context of self-dual matroids: Can every identically self-dual matroids be realised by a self-dual point configuration? We classify identically self-dual matroids of rank up to five and determine the dimension of their (self-dual) realization spaces. Self-dual point configurations are parametrized by the self-dual Grassmannian, a subvariety of the Grassmannian Gr(n,2n). When considering degenerations of the point configurations, the tropicalization of this subvariety becomes important. We analyze the case n=3 in detail and point towards the difficulties for n=4.

This talk is based on joined work with Sachi Hashimoto, Bernd Sturmfels and Raluca Vlad.

Giulio Salvatori: Tropical Geometry, Divergent Integrals and Particle Physics

Physical phenomena at very high energies are described through the celebrated Feynman integrals. The computation of these integrals is notoriously plagued by divergences. Rather than being an unpleasant feature of the formalism, as they are often thought of, these divergences actually reflect the vast hierarchies in the energy scales of the physical processes under consideration, and thus encode important physical knowledge. It is therefore crucial to understand the mathematical structures that better describe them. In this talk, I will explain how concepts from 'combinatorics and tropical geometry' illuminate our understanding of the singularities of Feynman integrals, while concretely providing new algorithmic tools for their computation. I will also discuss various open problems, for which we can anticipate significant breakthroughs in the future by further deepening the synergy between physics and discrete mathematics.

Lisa Seccia: Determinantal facet ideals

Given a matrix of indeterminates X of size mxn, we can define J_D to be the ideal generated by a subset D of its maximal minors: such ideals are called determinantal facet ideals. In fact, the generators naturally correspond to the facets of a pure simplicial complex. When m=2, this ideal is well understood, and it has numerous applications in the study of conditional independence statements in algebraic statistics. More generally, several varieties in applied algebraic geometry can be realized as ideals of maximal minors. As it turns out, the variety defined by J_D​ is typically non-reduced. It is reduced for m = 2, and we will see that it is reduced for a large class of D. Further results relating the algebraic properties of J_D with the combinatorial properties of its underlying simplicial complex will be presented, along with open questions in this area. This is joint work with Bruno Benedetti and Matteo Varbaro.

Christian Stump: Poincaré-extended ab-indices of posets

I will introduce the Poincaré-extended ab-index of a finite graded poset. I then present how this polynomial specializes to the coarse flag Hilbert-Poincaré series of (the intersection poset of) a hyperplane arrangement introduced in the context of Igusa local zeta functions and to the Chow ring of (the lattice of flats of) a matroid. We finally show how it is related to the thoery of quasisymmetric functions and how it can be specialized to the Schur function of a partition, together with its Schur-positivity This is in particular based on joint work with Galen Dorpalen-Barry, Elena Hoster, Josh Maglione.

Martin Wahl: Statistical analysis of empirical Hodge Laplacians

Laplacian Eigenmaps and Diffusion Maps are nonlinear dimensionality reduction methods that use the eigenvalues and eigenvectors of (un)normalized graph Laplacians. Both methods are applied when the data is sampled from a low-dimensional manifold, embedded in a high-dimensional Euclidean space. From a mathematical perspective, the main problem is to understand these empirical Laplacians as spectral approximations of the Laplace operator of the underlying Riemannian manifold. In this talk, I will first study empirical Laplacians through the lens of principal component analysis. This leads to novel points of view and allows to leverage recent results from high-dimensional probability. I will then discuss higher-order generalizations of the graph Laplacians, and show how they are connected to Hodge theory on Riemannian manifolds.

Martin Winter: The stress-flex conjecture for coned polytope frameworks

A coned polytope framework (or CPF for short) consists of the 1-skeleton of a convex polytope together with additional edges connecting the polytope's vertices to some common interior point. Deciding rigidity of general bar-joint frameworks is a hard problem, and a common approach is to establish first- or second-order rigidity instead. Quite surprisingly therefore, Wachspress Geometry can show that CPFs are rigid while their order of rigidity remains elusive. In particular the search for second-order rigidity gave rise to an intriguing new conjecture -- the stress-flex conjecture. Roughly, it asserts that the stresses and first-order flexes of a CPF are "orthogonal to each other". While initially stated for polytopes, it now appears to be of far greater generality, going beyond convexity, sphericity and orientability. In this talk I will first recall the essential terminology of rigidity theory on the example of coned polytope frameworks. The eventual goal is to present the stress-flex conjecture, its status, as well as some implications and approaches. This is joined work with Steven Gortler, Robert Connelly and Louis Theran.

Workshop 'Implicit Bias'

Dealing with implicit biases is one challenge for researchers – be it in the research process, in selection processes or team development. This half day training will give participants the opportunity to familiarize themselves with the most common mechanisms and workings of implicit biases within the research context. In particular, the workshop will focus on challenges in selection and recruitment processes as well as on supervision of doctoral and/or postdoctoral researchers. Using examples from the participants, the workshop will provide some concrete strategies to prevent and deal with implicit biases within a researcher’s everyday.

Trainer: Sabine Müller

Dr. Sabine Müller (she/her) is certified diversity trainer and process consultant. Before she became a freelancer, she was senior research manager at one of the four big non-university research organisations in Germany (Leibniz Association), responsible for equal opportunities, career advancement and leadership development. She holds a DPhil from the University of Oxford, where she also worked for nine years in various positions as translation tutor, researcher and lecturer with the responsibilities as head of German literature, philosophy and film at several colleges.

Local Organizers

  • Paul Breiding
  • Martina Juhnke
  • Tim Römer
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