Contributed talks

Below you may find the abstract for the contributed talks. Each talk should be at most 20 minutes long (strictly).

Monday

First Session

• Laura Casabella - The variety of orthogonal frames

  An orthogonal \(n\)-frame is an ordered set of \(n\) pairwise orthogonal vectors. The set of all orthogonal \(n\)-frames in a \(d\)-dimensional quadratic vector space is an algebraic variety \(V(d,n)\). In this talk, we investigate the geometric and algebraic properties of \(V(d,n)\). This variety has a combinatorial interpretation as the zero set of the Lovasz-Saks-Schrijver ideal of the complete graph on n vertices. We also explain how studying \(V(d,n)\) also gives information on Lovasz-Saks-Schrijver ideal of general graphs. This is joint work with Alessio Sammartano.

• Pia Herkenrath - Listing Spanning Trees of Graphs

  Combinatorial generation is concerned with the task of enumerating all combinatorial objects of a given class. One such instance is the problem of listing all spanning trees of a graph. Adding the requirement that two neighboring spanning trees in a list differ by only some small change, such as a single edge exchange, gives a so-called Gray code listing. This talk aims to give an overview of the problem of generating Gray codes for spanning trees of a given graph, including some ongoing work on generalisations and variations.

• Sebastian Degen - The polytope of all q-rank functions

  A \(q\)-rank function is a real-valued function defined on the subspace lattice of \(F_q^n\) that is non-negative, upper bounded by the dimension function, non-decreasing, and satisfies the submodularity law. Each such function corresponds to the rank function of a \(q\)-polymatroid. Intuitively, we can view these objects as \(q\)-analogues of polymatroids. In this viewpoint the concept of q-analogue can be interpreted as generalizing from finite sets to finite-dimensional vector spaces over finite fields. Their original motivation comes from algebraic coding theory, as the representable q-polymaroids arise from so-called rank-metric codes. In this talk, we identify these functions with points in a polytope. The lattice points of that polytope correspond to integer-valued \(q\)-polymatroids, also called \(q\)-matroids. We show that all lattice points are among the vertices of the polytope and investigate several properties of \(q\)-polymatroids arising by convex combinations of two such vertices.

• Kyle Huang - Computing Flatness Constants in the Plane

  We study a variant of the flatness problem from integer programming, where we study convex bodies in \(\mathrm{R}^d\) with \(k\) interior lattice points. The maximum lattice width of such bodies is denoted \(\mathrm{Flt}(d,k)\) and recovers the classical flatness constants when \(k=0\). We show that \(\mathrm{Flt}(2,1) = 3\), which leads to a discrete isominwidth inequality, as a consequence.

Second Session

• Torben Donzelmann - Edges of random symmetric edge polytopes

  Given a graph one can define its symmetric edge polytope by taking two symmetric points per edge and then taking the convex hull. Their combinatorial and Ehrhart theoretical properties have been intensively studied recently. On the other hand, from a probabilistic perspective random polytopes are usually defined as the convex hull of some geometric distribution. We take a different approach, by choosing a random graph and then studying the (edges of) the random symmetric edge polytope.

• Fenja Mehlan - A bivariate characteristic polynomial for hyperplane arrangements

  We introduce a new bivariate polynomial for hyperplane arrangements. In the case of real simplicial arrangements, this specializes to both the characteristic and the h-polynomial. Its coefficients are nonnegative for simplicial hyperplane arrangements and admit a nice interpretation in terms of permutation statistics for the braid arrangement.

Tuesday

First Session

• Daniel Green Tripp - Counting fibres of the Hadamard product using Bergman fans

  We study the fibre of a general point of the Hadamard product of linear spaces via matroid theory and tropical geometry. To do so, we introduce the flip product, a numerical invariant associated to a pair of matroids defined via the stable intersection of their (flipped) Bergman fans. Our first main result is that the cardinality of a generic fibre for the Hadamard product of linear spaces is exactly the flip product of their matroids. We also provide a recursive algorithm for computing the flip product of any pair of matroids. Finally, we show a number of existing matroid invariants are specialisations of the flip product, including the beta invariant. This is a joint work with Oliver Clarke, Sean Dewar, Matteo Gallet, Georg Grasegger and Ben Smith.

• Veronika Körber - Cross-ratio degrees and triangulations

  Tropical geometry can be seen as a combinatorial version of algebraic geometry. Analogously to algebraic geometry, in tropical geometry, a cross-ratio describes a relation of four ends of the underlying graph of a tropical curve. When considering several cross-ratios at the same time, the answer to the question of how many tropical curves, counted with multiplicity, fulfill all these cross-ratio relations is in general hard to find. Triangulations of \(n\)-gons also can define sets of cross-ratios. When considering these, the answer to the mentioned question only depends on the combinatorics of this triangulation. In this talk, I will explain how to find all the demanded tropical curves and their multiplicities.

• Sergio Alejandro Fernandez de soto Guerrero - Positroidal Magic

  Positroids are a subclass of matroids born in the study of the non-negative Grassmanian by Postnikov in 2006. Since then, there have been a plethora of combinatorial objects indexing positroids, two of these being the families of decorated and bicolored permutations, which are generalizations of classical permutations. These two families can be used to study properties of positroids, and as a byproduct we end up with useful ways to describe a group action on a deck of cards. In this context, we give a definition of invariants under this group action allowing us, as an application, to develop new magic tricks with unusual ways of shuffling cards.

• Zongpu Zhang - Multigraded Betti numbers of Veronese embeddings

  We study the multigraded Betti numbers of Veronese embeddings of projective spaces. Due to Hochster’s formula, we interpret these multigraded Betti numbers in terms of the homology of certain simplicial complexes. By analyzing these simplicial complexes and applying Forman’s discrete Morse theory, we derive vanishing and non-vanishing results for these multigraded Betti numbers. This is joint work with Christian Haase.

Second Session

• Nupur Jain - Trees, Dyck paths, and homological dimensions of Nakayama algebras

  The projective dimension of a module records the length of its shortest projective resolution, and measures how far it is from being projective. The global dimension of an algebra is the supremum of the projective dimensions of its (left) modules. Computing these and related homological invariants is a central problem in the representation theory of finite-dimensional algebras that can often be approached combinatorially. Nakayama algebras are finite-dimensional algebras arising from linearly oriented \(A_n\) quivers or cyclically oriented \(\tilde{A}_n\) quivers. In this talk, we present a combinatorial description of these algebras and a framework to study their homological dimensions. Our playground will consist of quivers, trees, and Dyck paths, which help uncover insights about the distribution of the global dimension. This will be an expository talk; no background in homological algebra or representation theory is assumed. All concepts will be introduced from a combinatorial viewpoint.

• Filip Jonsson Kling - On maximal rank properties for symmetric polynomials in a monomial complete intersection

  It is well known that a monomial complete intersection has the strong Lefschetz property in characteristic zero. This property is equivalent to the statement that any power of the sum of the variables is a maximal rank element on the complete intersection. In this talk, we will examine what happens when this element is replaced by another symmetric polynomial, in an equigenerated complete intersection. We answer the question completely for the power sum symmetric polynomial using a grading technique, and for any Schur polynomial in the case of two variables by deriving a closed formula for the determinants of a family of Toeplitz matrices. If time permits, we will mention partial results in three or more variables for the elementary and the complete homogeneous symmetric polynomials and pose several open questions.

Thursday

First Session

• Lakshmi Ramesh - Convex Bodies in Statistics

  Convex bodies and their intersections are interesting objects in statistics, since they arise as higher dimensional analogs of confidence intervals. The maximum likelihood estimator sets of a family of translates of convex bodies represents a region with 100 percent confidence level. The certainty of the MLE set is complemented by a large estimator. The “size” of the MLE set is of great interest. I provide an overview of 3 (or 4) projects where I study the volume, diameter, and in the case of polytopes, the combinatorics of the MLE set.

• Leonie Mühlherr - The Braid arrangement and friends

  The Braid arrangement is a wellknown object in combinatorics and has been studied through various different lenses: It is an example of a supersolvable Coxeter arrangement and its regions can be enumerated by use of permutations, which is why we encounter the permutahedron as its zonotope. A graph theoretical perspective interprets its subarrangements as arrangements associated to simple finite graphs. The intersection lattice of the braid arrangement is the lattice of set partitions which was recently used to compute its Chow polynomial. In this presentation, I would like to talk about it and related arrangements and present some recent developments involving it.

• Vien Nguyen - On monoids up to symmetry

  In this talk, we focus on monoids in \(\mathbb{Z}^{(\mathbb{N})}\) that are invariant under the action of the infinite symmetric group \(\mathrm{Sym}\). We present fundamental structural results for \(\mathrm{Sym}\)-invariant monoids, including characterizations of positivity and non-positivity, a description of their groups of units, and explicit formulas for the ranks of \(\mathrm{Sym}(n)\)-invariant monoids. Furthermore, we discuss local–global principles that characterize important properties of arbitrary \(\mathrm{Sym}\)-invariant monoids—such as finite generation, positivity, normality, and simplicity—in terms of their associated \(\mathrm{Sym}\)-invariant chains of monoids.

• Lyuhui Wu - Convex Analysis on tropical varieties

  This talk is based on joint works with Omid Amini and Matthieu Piquerez. We study convex functions on polyhedral spaces. We define a convex function on a polyhedral space as a continuous function that admits a local affine support function at each point. This class of convex functions turns out to coincide with the class introduced by Botero-Burgos-Sombra. We present several convex-analytic results, including a regularization theorem stating that every convex function on a polyhedral space can be uniformly approximated by piecewise linear convex functions.

Second Session

• Katarina Krivokuca - Upper Bounds on Covering Minima of Convex Bodies

  We give two new upper bounds on the covering minima of convex bodies, depending on covering minima of certain projections and intersections with linear subspaces. We show one bound to be sharp for direct sums of two convex bodies, generalizing previous results on the covering radius and lattice width of direct sums. We apply our results to standard terminal simplices, reducing the gap between the upper and lower bounds in a conjecture of Gonzaléz Merino and Schymura (2017), which gives insight on a conjecture of Codenotti, Santos and Schymura (2021) on the maximal covering radius of a non-hollow lattice polytope.

• Arne Kuhrs - Tropical principal bundles on metric graphs

  Tropical geometry studies a piecewise linear, combinatorial shadow of degenerations of algebraic varieties. In many cases, usual algebro-geometric objects such as divisors or line bundles on curves have tropical analogues that are closely tied to their classical counterparts. In this talk, I will present an elementary theory of tropical principal bundles on metric graphs, generalizing the case of tropical line bundles to bundles with arbitrary reductive structure group. Our approach is based on tropical matrix groups arising from the root datum of the corresponding reductive group. Building on Fratila’s description of the moduli space of semistable principal bundles on an elliptic curve, we describe a tropicalization procedure for semistable principal bundles on a Tate curve. This is joint work with Andreas Gross, Martin Ulirsch and Dmitry Zakharov.