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Events
Combinatorial Synergies East

MPI für Mathematik in den Naturwissenschaften Leipzig

E1 05 (Leibniz-Saal)

conference

19.03.25 21.03.25

Combinatorial Synergies East

This 3-day workshop is dedicated to Geometric Combinatorics, Algebraic Combinatorics, and related fields. It provides a unique opportunity for regional networking within the DFG priority program SPP 2458 "Combinatorial Synergies". Lectures will be given by speakers from the nodes in Berlin, Chemnitz, Cottbus, Greifswald, Jena, Leipzig and Magdeburg. The workshop is open to everyone; the focus will be on current projects and future collaborations.

Program

13:00 - 13:30	Welcome Coffee
13:30 - 14:15	Rainer Sinn (University of Leipzig) Adjoints in Positive Geometry
14:15 - 15:00	Tom Baumbach (TU Berlin) Polytopes with Polynomial Barycentric Coordinates and Low-Degree (Affine) Adjoints (joined Work with Martin Winter) Given a convex polytope P, a system of "generalized barycentric coordinates" (GBCs for short) provides a canonical coordinate system intrinsic to the geometry of P. Many GBCs have been developed with applications in geometric modeling and finite element analysis. In this talk, we study polytopes for which there exist polynomial GBCs or, more generally, rational GBCs of the form f_i/g, where g is a polynomial of low degree. We describe the geometric and combinatorial constraints imposed by low-degree denominators and provide a classification for polytopes with denominator degrees zero and one. Finally, we explain the relation to Wachspress coordinates, which are the rational GBCs that minimize the numerator degree.
15:00 - 15:30	Coffee
15:30 - 16:15	Thomas Wannerer (Friedrich-Schiller-Universität Jena) Hard Lefschetz Theorems for Polytopes and Convex Bodies The hard Lefschetz theorem, as proved by McMullen for the polytope algebra, plays a pivotal role in the characterization of the f-vector of simple polytopes. Together with a closely related theorem by Stanley, it serves as a prototype for a range of recent results in algebraic combinatorics that share a "hard Lefschetz" flavor. In this talk, we present analogous results for smooth convex bodies, which were obtained in recent joint work with A. Bernig and J. Kotrbatý.
16:15 - 17:00	Anna Hofer (Otto-von-Guericke-Universität) Computing Injective Resolutions and Local Cohomology Although injective resolutions are fundamental objects in homological algebra, their computation remains intractable over general Noetherian rings. In the case of affine semigroup rings, Helm and Miller introduced algorithms to compute injective resolutions of finitely generated $\mathbb{Z}^d$-graded modules and, as an application, algorithms to compute local cohomology modules. These algorithms exploit the polyhedral structure of $\mathbb{Z}^d$-graded injective modules over these rings. We provide an OSCAR implementation of these algorithms, enabling the computation of first injective resolutions and local cohomology modules over non-regular rings.

09:00 - 09:45	Marie Brandenburg (Ruhr-Universität Bochum) Decomposition Polyhedra of Piecewise Linear Functions It is well-known that every continuous piecewise linear function can be written as the difference of two convex piecewise linear functions. For applications in optimization and neural network theory, it is crucial to find decompositions where the convex functions have as few linear pieces as possible. But how can we find such a minimal decomposition, and how small can a minimal decomposition be? Under certain assumptions, we can describe the space of all convex decompositions as a polyhedron. In this talk, we will discuss properties of these polyhedra, and how this point of view can contribute to finding a minimal decomposition. This talk is based on joint work with Moritz Grillo and Christoph Hertrich.
09:45 - 10:30	Coffee
10:30 - 11:15	Lakshmi Ramesh (Universität Bielefeld) Convex Bodies and Maximum Likelihood Sets We consider points sampled from a uniform distribution on a convex body in high dimensional real space with unknown location. In this case, the maximum likelihood estimator set is a convex body containing the true location parameter, and hence has a volume and diameter. We estimate these quantities, in terms of dimension and number of samples, by introducing upper and lower bounds. These bounds are different depending on the geometry of the convex body. We arrive at our results by employing algebraic, probabilistic and statistical techniques.
11:15 - 12:00	Ben Hollering (TU Munich) Causal Discovery for Max-Linear Bayesian Networks Directed acyclic graphical models, sometimes called Bayesian networks, are of critical importance in modern data science and statistics through their applications to causality and probabilistic inference. These statistical models use directed acyclic graphs (DAGs) to represent causal relationships between random variables and are often specified by a system of structural equations which is defined using the associated DAG. A fundamental related problem is that of causal discovery where one seeks to determine the causal relationships, i.e. the underlying DAG, from a given empirical distribution. While this problem is statistical in nature, many classical causal discovery algorithms which are used today are fundamentally combinatorial due to a strong link between conditional independence of the observed random variables and the structure of the underlying DAG. We will then show how the first causal discovery algorithm, typically known as the PC algorithm, fails to accurately reconstruct the true DAG if the observed data is generated by a max-linear Bayseian network due to the differences in the combinatorial criteria which govern conditional independence in these models. Lastly, we will discuss how the PC algorithm can be changed to yield a new algorithm which reconstructs the true underyling graph when the input data is generated by a max-linear Bayesian network.
12:00 - 13:00	Lunch
13:00 - 13:45	Gennadiy Averkov (Brandenburg University of Technology) Flatness Constant and its Variations The standard flatness constant is the maximum lattice width among all hollow convex bodies in a given dimension, where hollow means that no lattice points lie in the interior of the body. This value is known exactly in dimensions one and two. There are variations on this constant that arise by restricting the family of the bodies or replacing the hollowness by another condition or replacing the lattice width by other lattice functions. The flatness constant and its variation are useful in integer programming, algebra and algebraic geometry. In this talk I will focus on the role of hollowness in the cutting plane theory for integer optimization problems and highlight the challenge of solving such problems as calculation of the flatness constants or enumeration of special families of lattice polytopes in fixed dimensions.
13:45 - 14:30	Christian Haase (Freie Universität Berlin) Multigraded Betti numbers of Veronese surfaces via simplicial homology In a major computation, Bruce, Erman, Goldstein, and Yang determined the multigraded Betti numbers of the first six Veronese embeddings of the projective plane. Via a well known interpretation of these numbers in terms of simplicial complexes, one can prove (non-)vanishing results. This in ongoing joint work with Zongpu Zhang.
14:30 - 15:30	Coffee and Synergies
15:30 - 16:15	Ansgar Freyer (FU Berlin) Isominwidth problems in non-Euclidean settings Abstract
16:15 - 17:15	Vadym Kurylenko (OvGU Magdeburg) Local Ehrhart Theory and Its Synergies Local h-polynomial is an Ehrhart-theoretic invariant that helps to describe the behaviour of the more familiar h-polynomial under polytope subdivisions. In this talk, we will examine some basic properties of the local h-polynomial and show that its coefficients correspond to certain Hodge numbers. Thus, the local h-polynomial is strongly connected to algebraic geometry, and in particular to the theory of hypergeometric motives. We will explore this connection further and discuss how it naturally leads to the involvement of Gale duality in Ehrhart theory.
	Conference dinner at Ratskeller (self-paid)

09:00 - 09:45	Joscha Diehl (University of Greifswald) Counting permutation patterns and calculating iterated sums
09:45 - 10:15	Coffee
10:15 - 11:00	Alheydis Geiger (MPI for Mathematics in the Sciences) The self-dual Grassmannian, Matroids & Positivity
11:00 - 11:45	Hendrik Süß (Friedrich Schiller University Jena) On operators preserving volume polynomials We are discussing operators on multivariate polynomials, which preserve certain positivity properties. More precisely, we will discuss the diagonalisation operation (i.e. setting several variables equal) for volume polynomials and denormalised volume polynomials, respectively. This is joint work in progress with Julius Ross and Thomas Wannerer.

Participants

Ibrahim Ahmad

RWTH Aachen University

Gennadiy Averkov

Brandenburg University of Technology

Andrei Balakin

Technische Universität Berlin

George Balla

MPI MiS

Daniel Bath

KU Leuven

Tom Baumbach

TU Berlin

Florian Besau

FSU Jena

Barbara Betti

Max Planck Institute MiS Leipzig

Marie Brandenburg

Ruhr-Universität Bochum

Clemens Brüser

TU Dresden

Veronica Calvo Cortes

MPI MiS Leipzig

Laura Casabella

MPI MiS Leipzig

Giulia Codenotti

FU Berlin

Shelby Cox

MPI MiS

Joscha Diehl

University of Greifswald

Matthew Dupraz

FU Berlin

Ansgar Freyer

FU Berlin

Hannah Friedman

UC Berkeley

Alheydis Geiger

MPI for Mathematics in the Sciences

Paul Görlach

Otto-von-Guericke-University Magdeburg

Moritz Leo Grillo

TU Berlin

Lukas Grund

FSU Jena

Christian Haase

Freie Universität Berlin

Anna Hofer

Otto-von-Guericke-Universität

Johannes Hofscheier

University of Nottingham

Ben Hollering

TU Munich

Linda Hoyer

RWTH Aachen University

Kyle Huang

BTU Cottbus

Thomas Kahle

Otto-von-Guericke Universität

Cecilia Knäbchen

Karlsruhe Institute of Technology

Katarina Krivokuca

Freie Universität Berlin

Vadym Kurylenko

OvGU Magdeburg

Alexandros Leivaditis

Ruhr-Universität Bochum

Haydee Lindo

Harvey Mudd College

Georg Loho

FU Berlin & U Twente

Paul Mücksch

TU Berlin

Elke Neuhaus

MPI MiS

Vien Nguyen

Osnabrück University

Benjamin Nill

OVGU Magdeburg

Francesco Nowell

TU Berlin

Niharika Paul

MPI MIS

Dmitrii Pavlov

TU Dresden

Irem Portakal

Max Planck Institute for Mathematics in the Sciences

Lakshmi Ramesh

Universität Bielefeld

Tomas Reunbrouck

Universität Bielefeld

Nikola Sadovek

Freie Universität Berlin

Flavio Salizzoni

MPI-MIS

Leonard Schmitz

TU Berlin

Alec Schmutz

Universität Bielefeld

Christian Sevenheck

TU Chemnitz

Rainer Sinn

University of Leipzig

Luca Sodomaco

MPI MiS

Bernd Sturmfels

Max-Planck-Institut für Mathematik in den Naturwissenschaften

Hendrik Süß

Friedrich Schiller University Jena

Máté Telek

MPI MiS Leipzig

Niraj Velankar

University of Greifswald

Thomas Wannerer

Friedrich-Schiller-Universität Jena

Julian Weigert

MPI MIS/University Leipzig

Lena Weis

TU Berlin

Maximilian Wiesmann

MPI MiS

Francesca Zaffalon

Max Planck Institute for Mathematics in the Sciences

Zongpu Zhang

Berlin Mathematical School

Organizers

Thomas Kahle

Otto-von-Guericke-Universität

Bernd Sturmfels

Max-Planck-Institut für Mathematik in den Naturwissenschaften

Administrative Contact

Saskia Gutzschebauch

Max Planck Institute for Mathematics in the Sciences Contact via Mail

Mirke Olschewski

Max Planck Institute for Mathematics in the Sciences Contact via Mail