Conference Program

Geometry meets Combinatorics Poster


Monday 08:15am Registration
Monday 08:50am Welcome
Monday 09:00am Small topological counter-examples to the Hirsch Conjecture
Francisco Santos (Cantabria)
The Hirsch Conjecture claimed that the combinatorial diameter of a \(d\)-polytope with \(n\) facets should not exceed \(n-d\). It was stated by Hirsch in 1957 and disproved by me in 2010 with a counter-example of dimension 43. In subsequent work with Matschke and Weibel we constructed a smaller counter-example, of dimension 20. Both constructions are based in the use of prismatoids, polytopes with all vertices contained in two disjoint facets, and in a \(d\)-step theorem for prismatoids, showing that any prismatoid of "width" larger than its dimension can be lifted to a non-Hirsch polytope. In this talk I will review this construction and report on recent work with F. Criado (Exp. Math. 2022) in which we look at topological prismatoids and use them to construct about 4000 non-Hirsch simplicial spheres of dimensions ranging from 8 to 20. Unfortunately, many of our spheres, including all those of dimensions less than 10, have been proved to be non-polytopal by Pfeifle and/or by Gouveia, Macchia and Wiebe. Slides
Monday 10:00am Flag Polymatroids
Alexander Black (Davis)
Flag matroids are an analog of matroids for flags of independent sets, where matroids are exactly the flag matroids for which all flags have length 1. For example, given a matroid \(M\), the set of all complete flags of subsets of bases of \(M\) forms a flag matroid that we call the complete flag matroid of \(M\). Polymatroids are a different generalization of matroids that are no longer defined in terms of a set system, which makes it unclear how to generalize flag matroids to flag polymatroids. Like matroids, flag matroids have a corresponding polytope. For complete flag matroids, we show the corresponding polytope arises as a monotone path polytope of a matroid independence polytope and use this to generalize complete flag matroids to polymatroids. I will discuss this generalization and describe new novel descriptions of the normal fans of complete flag matroid polytopes in relation to the greedy algorithm for (poly)matroid optimization, the lattice of flats of a polymatroid, and nestohedra. Based on joint work with Raman Sanyal.
Monday 10:30am Coffee & Tea Break
Monday 11:00am Many polytopes
Arnau Padrol (Paris)
This talk will be an overview of the problem of estimating the number of combinatorial types of \(d\)-dimensional convex polytopes with \(n\) vertices. While in dimensions up to 3 we have a very good understanding on the asymptotic growth of the number of polytopes with respect to the number of vertices, in higher dimensions we only have coarse estimates. Upper bounds arise from results of Milnor and Thom from real algebraic geometry, whereas lower bounds are obtained with explicit constructions. I will present the constructions giving the current best lower bounds for the number of polytopes, found in collaboration with Eva Philippe and Francisco Santos. Slides
Monday 12:00pm Tropical positivity and determinantal varieties
Marie-Charlotte Brandenburg (Leipzig)
A determinantal variety is the set of \((d \times n)\)-matrices of bounded rank. We study the tropicalization of the set of matrices with positive entries and bounded rank, i.e. the positive part of determinant varieties. Given such a matrix of fixed rank \(r\), we can interpret the columns of the tropicalization of this matrix as \(n\) points in \(d\)-dimensional space, lying on a common \(r\)-dimensional tropical linear space. We consider such tropical point configurations, and introduce a combinatorial criterion to characterize which such configurations can be obtained as the tropicalization of matrices with positive entries. No prior knowledge of tropical geometry will be assumed for this talk. This is joint work with Georg Loho and Rainer Sinn. Slides
Monday 12:30pm Lunch Break
Monday 14:00pm Geometry and combinatorics of arrangements of hypersurfaces
Emanuele Delucchi (Lugano)
Over the last 50 years, Geometry has met Combinatorics in the theory of arrangements, leading to a wealth of results in several related fields. For instance, the recently celebrated Hodge theory for matroids has its origins in the study of the geometry of linear arrangements. This talk will be about current developments beyond the linear case, focussing on arrangements of hypersurfaces in connected abelian Lie Groups (including so-called toric and elliptic arrangements). I will explain the definition of such arrangements and I'll summarize the state of the art by presenting a selection of results and open questions. A recurring theme will be the development of an extension of matroid theory that can provide a unified combinatorial framework for the new geometric, topological and algebraic aspects of the theory.
Monday 15:00pm Grassmannians over rings and subpolygons
Michael Cuntz (Hannover)
We investigate special points on the Grassmannian which correspond to friezes with coefficients in the case of rank two. Using representations of arithmetic matroids we obtain a theorem on subpolygons of specializations of the coordinate ring. As a special case we recover the characterization of subpolygons in classic frieze patterns. Moreover, we observe that specializing clusters of the coordinate ring of the Grassmannian to units yields representations that may be interpreted as arrangements of hyperplanes with notable properties. In particular, we get an interpretation of certain Weyl groups and groupoids as generalized frieze patterns. Slides
Monday 15:30pm Coffee & Tea Break
Monday 16:00pm Real Hyperplane Arrangements and the Varchenko-Gelfand Ring
Galen Dorpalen-Barry (Bochum)
For a real hyperplane arrangement \(\mathcal{A}\), Varchenko-Gelfand ring is the ring of functions from the chambers of \(\mathcal{A}\) to the integers with pointwise addition and multiplication. Varchenko and Gelfand gave a simple presentation for this ring, along with a filtration whose associated graded ring has its Hilbert function given by the coefficients of the Poincaré polynomial. Their work was extended to oriented matroids by Gelfand-Rybnikov, who gave an analogous presentation and filtration. We extend this work first to pairs \((\mathcal{A},K)\) consisting of an arrangement \(\mathcal{A}\) in a real vector space and open convex set \(K\), and then to conditional oriented matroids. Time permitting, we will discuss an interesting special case arising in Coxeter theory: Weyl cones of Shi arrangements. In that context, we find that the coefficients of the cone Poincaré polynomial of a Weyl cone are described by antichains in the root poset. This talk contains joint work with Christian Stump, Nicholas Proudfoot, and Jayden Wang.
Monday 17:00pm Pruned inside-out polytopes, combinatorial reciprocity theorems, and generalized permutahedra
Sophie Rehberg (Berlin)
We introduce pruned inside-out polytopes, a generalization of inside-out polytopes introduced by Beck-Zaslavsky (2006), which have many applications such as recovering the famous reciprocity result for graph colorings by Stanley. We study the integer point count of pruned inside-out polytopes by applying classical Ehrhart polynomials and Ehrhart-Macdonald reciprocity. This yields a geometric perspective on and a generalization of several known combinatorial reciprocity results such as a combinatorial reciprocity theorem for generalized permutahedra by Aguiar-Ardila (2017) and Billera-Jia-Reiner (2009), and hypergraphs by Aval-Karaboghossian-Tanasa (2020). Moreover, this can be extended to type B generalized permutahedra. This is joint work with Matthias Beck. Slides


Tuesday 09:00am Conjugacy growth in groups, geometry and combinatorics
Laura Ciobanu (Edinburgh)
Conjugacy growth in groups has been studied, from a geometric perspective, for many decades. Initially, the growth of conjugacy classes naturally occurred while counting closed geodesics (up to free homotopy) on complete Riemannian manifolds, as formulas for the number of such geodesics give, via quasi-isometry, good estimates for the number of conjugacy classes in the manifolds' fundamental groups. More recently, the study of conjugacy growth has expanded to groups of all flavours, ranging from nilpotent to linear to acting on cube complexes, and beyond. In this talk I will give an overview of what is known about conjugacy growth and the formal series associated with it in infinite discrete groups. I will highlight how the rationality (or rather lack thereof) of these series is connected to both the algebraic and the geometric nature of groups such as (relatively) hyperbolic or groups acting on trees, and how tools from analytic combinatorics can be employed in this context. Slides
Tuesday 10:00am Geometric aspects of flag Hilbert-Poincaré series of matroids
Joshua Maglione (Magdeburg)
We define a class of multivariate rational functions associated with matroids called flag Hilbert-Poincaré series, which are connected to local Igusa zeta functions. Maglione and Voll established a connection between Coxeter arrangements of these series and Eulerian polynomials. We generalize this to the setting of oriented matroids. This is joint work with Lukas Kühne.
Tuesday 10:30am Coffee & Tea Break
Tuesday 11:00am High-dimensional rational cohomology of \(\text{SL}_n(\mathbb{Z})\)
Benjamin Brück (Zürich)
In joint work with Miller-Patzt-Sroka-Wilson, we made progress in understanding the high-dimensional cohomology of the arithmetic group \(\text{SL}_n(\mathbb{Z})\): We showed its rational cohomology vanishes in codimension two, i.e. \(H^{{n \choose 2} -2}(\text{SL}_n(\mathbb{Z});\mathbb{Q}) = 0\) for \(n \geq 3\). This is a result at the intersection of topology, group theory and number theory. Our proof however is mostly concerned with studying the combinatorics of certain simplicial complexes whose vertices are lines in \(\mathbb{Z}^n\). These complexes are closely related to Tits buildings and we show that they are highly connected, in fact homotopy Cohen-Macaulay. In my talk, I will firstly outline a general strategy, due to Church-Farb-Putman, for computing these high-dimensional cohomology groups via duality. Secondly, I will give an idea of the structure of the involved simplicial complexes and of the type of questions one needs to understand about these for applying the general strategy.
Tuesday 12:00pm Valuative invariants for large classes of matroids
Luis Ferroni (Stockholm)
Matroid invariants are ubiquitous within algebraic combinatorics, and it is surprising that many of them behave well under polytope subdivisions, even when they are defined in totally unrelated matroidal frameworks. We will give an explanation of why certain invariants, such as the Tutte polynomial, the Kazhdan-Lusztig polynomial or the Hilbert series of the Chow ring of a matroid are valuations. Then, we will explain how one can use this property in practice to obtain explicit formulas of this invariants on large classes of matroids. This framework is particularly useful to find counterexamples to conjectures in matroid theory, or to give good reasons to support conjectures. This is based on joint work with Benjamin Schröter. Slides
Tuesday 12:30pm Lunch Break
Tuesday 14:00pm Reflection length in Coxeter groups
Petra Schwer (Magdeburg)
The length of an element \(w\) in a group with respect to a generating set \(R\) is the minimal number of elements needed to write w as word in \(R\). Reflection length now measures the length of a word in a Coxeter group with respect to the set \(R\) of all reflections. This talk highlights old and new results on reflection length in Coxeter groups.
Tuesday 15:00pm Lower bounds on neural network depth via lattice polytopes
Christoph Hertrich (London)
We study the set of functions representable by ReLU neural networks, a standard model in the machine learning community. It is an open question whether this set strictly increases with the number of layers used. We prove that this is indeed the case if one considers neural networks with only integer weights. More precisely, we show that at least \(\log(n)\) many layers are required to compute the maximum of n numbers, matching known upper bounds. To show our result, we first use previously discovered connections between neural networks and tropical geometry to translate the problem into the language of Newton polytopes. These Newton polytopes are lattice polytopes arising from alternatingly taking convex hulls and Minkowski sums. Our depth lower bounds then follow from a parity argument for the volume of faces of such polytopes, which might be of independent interest. This is joint work with Christian Haase and Georg Loho. Slides
Tuesday 15:30pm Coffee & Tea Break
Tuesday 16:00pm Length and reflections: odds and ends
Angela Carnevale (Galway)
It is well known that the Coxeter length can be described in terms of reflections. In this talk I will discuss interesting consequences of restricting our attention to special subsets of reflections. We will focus on so-called odd reflections and those of bounded length, discussing in particular some unexpected connections between the former and the Bruhat order, and new poset structures associated with the latter. This is based on joint projects with Francesco Brenti and Bridget Tenner, and with Matthew Dyer and Paolo Sentinelli, respectively. Slides
Tuesday 17:00pm Problem session
If you want to present a problem at the problem session please get in touch with the organizers.


Wednesday 09:00am Flatness constants and lattice-reduced convex sets
Giulia Codenotti (Frankfurt)
The lattice width of a convex body measures how "thin" the body is in lattice directions. The so-called flatness theorem states that in each fixed dimension there is an upper bound for the lattice width of a special class of convex bodies, those which are hollow. In this talk we introduce these definitions and certain generalizations thereof and look at the few known extremal hollow convex bodies achieving maximum lattice width. This leads us to a conjecture about these extremal examples, namely that they are lattice-reduced. This class of convex bodies has not been previously studied, and we present (many) questions and (some) answers about such lattice-reduced convex bodies. Slides
Wednesday 10:00am Enumerating all Triangulations up to Symmetry
Jörg Rambau (Bayreuth)
TOPCOM has been the standard software since around 2000 to enumerate a connected component of the flip-graph of all triangulations of a point configuration up to symmetry. As such, it has meanwhile been integrated in most distributions of the computer algebra dashboard SAGEMATH, which is standard in most Linux-distributions. In 2018 Jordan, Joswig, and Kastner have pushed the concept of flipping algorithms in their multi-processor code MPTOPCOM even further to enumerate larger instances on computing clusters using parallel reverse search on orbits, which is based on ideas by Imai et al. from 2002. There has always been a client in TOPCOM to enumerate all triangulations, no matter whether connected by flips or not. Since Santos's examples of point configurations with disconnected flip-graphs in 2000 we know that this makes a difference in general. TOPCOM's "alltriangulations" client, or any other software, for that matter, has never really been able to handle instances as large as the flip-type algorithms. This has significantly changed recently: TOPCOM's multi-threaded implementation of a special version of Symmetric Lexicographic Subset Reverse Search (based on a generic concept by Pech and Reichard from 2009) is now able to enumerate up to symmetry, in particular, all triangulations of the regular dodecahedron (12,775,757,027 in 11h/16 threads) with 120 symmetries and the pyritohedron (1,363,918,758,719 in 693h/16 threads) with 24 symmetries. These are, to the best of my knowledge the largest sets of triangulations that have been enumerated by a general-purpose code (i.e., accepting any point configuration in any dimension as input) to date, flip-based or not. In this talk, the generic algorithm as well as the two simple but crucial ideas making this possible are presented. Slides Code and Paper
Wednesday 10:30am Coffee & Tea Break
Wednesday 11:00am Convex Polytopes: Examples and Counterexamples, Problems and Conjectures
Günter M. Ziegler (Berlin)
My plan is to talk about some of my favourite (open and solved) polytope problems and on this occasion discuss - the value of examples - the value and the limits of enumeration - our limited tool-box of construction methods - the challenges of explicit constructions - the value of counter-examples in polytope theory. Slides
Wednesday 12:00pm What are thin polytopes?
Benjamin Nill (Magdeburg)
In this talk I would like to present the novel notion of thin polytopes as lattice polytopes whose local \(h^*\)-polynomials vanish. The local \(h^*\)-polynomial is a fundamental invariant in modern Ehrhart theory. Its definition goes back to Stanley with beautiful results achieved by Karu, Borisov & Mavlyutov, Schepers, and Katz & Stapledon. The study of thin simplices was originally proposed in the book by Gelfand, Kapranov and Zelevinsky, where in this case the local \(h^*\)-polynomial simply equals its so-called box polynomial. In joint work with Christopher Borger and Andreas Kretschmer, we classify all thin polytopes up to dimension 3 and give a complete characterization of thin Gorenstein polytopes in any dimension. Slides
Wednesday 12:30pm Lunch Break
Wednesday 14:00pm Hike
Wednesday 19:00pm Conference Dinner at Jivino


Thursday 09:00pm Top-heaviness and the Kazhdan--Lusztig theory of matroids
Jacob P. Matherne (Bonn)
If you take a collection of planes in \(\mathbf{R}^3\), then the number of lines you get by intersecting the planes is at least the number of planes. This is an example of a more general statement, called the "Top-Heavy Conjecture", that Dowling and Wilson conjectured in 1974. On the other hand, given a hyperplane arrangement, I will explain how to uniquely associate a certain polynomial (called its Kazhdan-Lusztig polynomial) to it. The problems of proving the "Top-Heavy Conjecture" and the non-negativity of the coefficients of these Kazhdan-Lusztig polynomials are related, and they are controlled by the Hodge theory of a certain singular projective variety. The "Top-Heavy Conjecture" was proven for hyperplane arrangements by Huh and Wang in 2017, and the non-negativity was proven by Elias, Proudfoot, and Wakefield in 2016. I will discuss work, joint with Tom Braden, June Huh, Nicholas Proudfoot, and Botong Wang, on these two problems for arbitrary matroids.
Thursday 10:00am Geometry and combinatorics meet zonoids
Chiara Meroni (Leipzig)
A classical problem in convex geometry asks for a way to recognize zonoids. The latter are limits of zonotopes, namely Minkowski sums of segments. The Zonoid Problem is a hard problem from many points of view, but it is solved for the polytopal case. What happens more in general? We will focus on the family of discotopes, i.e., Minkowski sums of discs. We will see how methods from algebraic geometry can be used to get information on the boundary of a discotope. Slides
Thursday 10:30am Coffee & Tea Break
Thursday 11:00am Beyond positivity for lattice polytopes and unimodality of \(h^*\)
Karim Adiprasito (Jerusalem & Copenhagen)
The semigroup algebra of an IDP reflexive lattice polytope was shown to be Gorenstein by Hochster. We compute the fundamental class and conclude a Lefschetz property in char 0. In particular, the \(h^*\) polynomial has unimodular coefficients. This talk will be a general survey of the methods beyond positivity. More details of the proof are presented by J. Steinmeyer
Thursday 12:00pm Lefschetz and unimodality for Lattice Polytopes
Johanna Steinmeyer (Jerusalem & Copenhagen)
One of the fundamental questions of Ehrhart theory lies in characterizing the possible \(h^*\)-polynomials. Given a lattice polytope with the integer decomposition property, whose dual is also a lattice polytope, Hibi and Ohsugi conjectured that the coefficients of the \(h^*\)-polynomial are always unimodal. Following K. Adiprasito's survey, I will give the proof of this conjecture via anisotropy and Lefschetz properties for the associated semigroup algebra. As time permits, I will also sketch the boundary of the methods, including what happens when we relax the duality condition.
Thursday 12:30pm Lunch Break
Thursday 14:00pm Directions in graph rigidity
Eran Nevo (Jerusalem)
Given a graph \(G\) and an embedding of its vertices in \(\mathbf{R}^d\), what continuous motions of the vertices preserve all edge lengths? Clearly all motions induced by an isometry of \(\mathbf{R}^d\) do, these are the trivial motions; are there any others? If the answer is NO for all (equivalently, for one) generic embedding, \(G\) is called \(d\)-rigid. What are the \(d\)-rigid graphs? I will discuss various directions and open problems in graph rigidity, focusing on a quantitative version of rigidity via spectral analysis of the related stiffness matrix.
Thursday 15:00pm The anticanonical complex — a combinatorial tool for Fano varieties
Milena Wrobel (Oldenburg)
The anticanonical complex has been introduced as a combinatorial tool, extending the correspondence between toric Fano varieties and Fano polytopes, and has so far been developed for certain classes of Fano varieties. In this talk, we introduce a general construction for the anticanonical complex and discuss its applications in the classification of Fano varieties.
Thursday 15:30pm Coffee & Tea Break
Thursday 16:00pm Inequalities for \(f^*\)-vectors of lattice polytopes
Danai Deligeorgaki (Stockholm)
The Ehrhart polynomial \(L_P(m)\) of a lattice polytope \(P\) counts the number of integer points in the \(n\)-th integral dilate of \(P\). Ehrhart polynomials of polytopes are often described in terms of the vector of coefficients of \(L_P(m)\) with respect to different binomial bases, under which they have non-negative coefficients. Such vectors give rise to the \(h^*\) and \(f^*\)-vector of \(P\), which coincide with the \(h\) and \(f\) vectors of a regular unimodular triangulation of \(P\), whenever it exists. In particular, \(f^*\)-vectors were introduced in 2012 by Felix Breuer as the coefficients of \(L_P(m)\) expressed in the basis \({m-1\choose 0}\), \({m-1\choose 1}\), \({m-1\choose 2}\), ... . We will see some examples of \(f^*\)-vectors of lattice polytopes, including a family of simplices whose \(f^*\)-vectors are not unimodal. Even though \(f^*\)-vectors of lattice polytopes are not necessarily unimodal, there are several interesting inequalities that can hold among their coefficients. We will discuss some results in this direction. This talk is based on work with Matthias Beck, Max Hlavacek and Jerónimo Valencia. Slides


Friday 09:00am Symmetries and local-global principles in discrete geometry
Tim Römer (Osnabrück)
We consider cones and monoids in infinite dimensional spaces which are invariant under actions of symmetric groups. A key idea is to study these objects via associated symmetric chains of cones/monoids in finite dimensional spaces and to formulate related local-global principles. In this talk we discuss recent results and open questions on these topics.
Friday 10:00am Double Schubert polynomials do have saturated Newton polytopes
Yairon Cid-Ruiz (Ghent)
We prove that double Schubert polynomials have the Saturated Newton Polytope property. This settles a conjecture by Monical, Tokcan and Yong. Our ideas are motivated by the theory of multidegrees. We introduce a notion of standardization of ideals that enables us to study non-standard multigradings. This allows us to show that the support of the multidegree polynomial of each Cohen-Macaulay prime ideal, and in particular, that of each Schubert determinantal ideal is a discrete polymatroid. This is joint work with Federico Castillo, Fatemeh Mohammadi and Jonathan Montaño.
Friday 10:30am Coffee & Tea Break
Friday 11:00am 2-LC triangulated manifolds are exponentially many
Marta Pavelka (Miami)
We introduce \(t\)-LC triangulated manifolds as those triangulations obtainable from a tree of \(d\)-simplices by recursively identifying two boundary \((d-1)\)-faces whose intersection has dimension at least \(d - t - 1\). The \(t\)-LC notion interpolates between the class of LC manifolds introduced by Durhuus-Jonsson (corresponding to the case \(t = 1\)), and the class of all manifolds (case \(t = d\)). Benedetti-Ziegler proved that there are at most \(2^{N d^2}\) triangulated 1-LC \(d\)-manifolds with \(N\) facets. Here we show that there are at most \(2^{N/2 d^3}\) triangulated 2-LC \(d\)-manifolds with \(N\) facets. We also introduce \(t\)-constructible complexes, interpolating between constructible complexes (the case \(t = 1\)) and all complexes (case \(t = d\)). We show that all \(t\)-constructible pseudomanifolds are \(t\)-LC, and that all \(t\)-constructible complexes have (homotopical) depth larger than \(d - t\). This extends the famous result by Hochster that constructible complexes are (homotopy) Cohen-Macaulay. This is joint work with Bruno Benedetti. Slides
Friday 11:30am Matroid and delta-matroid tautological classes
Alex Fink (London)
Relationships between matroids and the permutahedral toric variety are central to matroid Hodge theory. One might wish to generalise these relationships, and the Hodge theory, to delta-matroids, which are Coxeter type B objects. I'll introduce delta-matroids and present one such relationship, inspired by the work of Berget-Eur-Spink-Tseng. Its consequences include volume polynomial formulae and positivity results for invariants like the interlace polynomial. This talk is based on work in progress with Chris Eur, Matt Larson and Hunter Spink.
Friday 12:30pm Lunch Break