Poster Abstracts
Aenne Benjes (Goethe University Frankurt) - Type cones and alcoved polytopes
For a fixed matrix $A$ with rows $a_1, \dots, a_n \in \mathbb{R}^d$ we would like to consider the space of all right-hand sides $b \in \mathbb{R}^n$, such that $P_A(b) = \{x \in \R^d: Ax \le b\}$ is a non-empty polytope and all hyperplanes spanned $a_1, \dots,a_n$ are supporting. This space is sometimes called closed inner region or closed irredundancy domain and carries a natural structure. It decomposes into so called type cones. Two right-hand sides $b,b' \in \R^n$ are contained in the same type cone, if the polytopes $P_A(b)$ and $P_A(b')$ are normally equivalent. In this poster, we want to have closer look on these structures in general, and assuming that the polytopes $P_A(b)$ are alcoved. This is joint work with Raman Sanyal and Benjamin Schröter.
Alejandro Martinez Mendez (University of Groningen) - Patchworking in F1 geometry
The field of one element, F_1, is an idea that was first proposed by Jaques Tits as a link between Chevalley groups and their Weyl groups, but it didn't garner serious interest until the late 20th century when its links to other areas, including arithmetic, combinatorics and tropical geometry, was unearthed.
In the last two decades, several approaches to F_1-geometry were developed that generalize algebraic geometry from different perspectives. What is common to most approaches is that F1-scheme is a space with a covering by affine patches.
In this poster, we explain this patchworking from a general and simplified perspective, and we comment on the topological realizations of F1-schemes. This is work in progress, in collaboration with Matt Baker and Oliver Lorscheid.
Alejandro Vargas (University of Bologna) - Tropical linear differential equations
Tropical methods applied to differential equations enable to extract information from the set of solutions expressed as a power series expansion. This is a nascent approach within tropical geometry, which so far has yielded the ability to state results about the supports of the solutions, and more recently about convergence radii. In this poster we focus on the case of linear differential equations. Classically, the solution of a system of linear equations is a linear subspace. Tropically, we get a valuated matroid. Thus, in the power series setting, we get something akin to an infinite valuated matroid, whose geometrical properties are given by the inverse limit of a family of polyhedrons that arise by truncation of the solutions. There are many questions to explore and we present lots of particular examples in this poster.
Alvaro Gutierrez (University of Bristol) - Symmetric functions are symmetric (types A, B, and C)
We show that the combinatorial definitions of King and Sundaram of the symmetric polynomials of types B and C are indeed symmetric, in the sense that they are invariant by the action of the Weyl groups. Our proof is combinatorial and inspired by Bender and Knuth's classic involutions for type A.
Anita Arora (Indian Institute of Science) - The monopole-dimer model for the Cartesian products of plane graphs
The monopole-dimer model is a signed variant of the monomer-dimer model which has determinantal structure. We extend the monopole-dimer model for planar graphs introduced by Prof. Arvind Ayyer (Math. Phys. Anal. Geom., 2015) to Cartesian products thereof and show that the partition function of this model can be expressed as a determinant of a generalised signed adjacency matrix. We then give an explicit product formula for three-dimensional grid graphs a la Kasteleyn and Temperley--Fischer, in which case the partition function turns out to be fourth power of a polynomial when all grid lengths are even. Finally, we generalise this product formula to $d$ dimensions, again obtaining an explicit product formula.
This is a joint work with Prof. Arvind Ayyer.
Barbara Betti (Max Planck Institute MiS Leipzig) - Mukai lifting problem for self-dual points in P^6
A set of 2n+2 points in P^n is self-dual if it is invariant under the Gale transform. Petrakiev showed that a general self-dual set of points in P^6 arises as a linear section of the Grassmannian Gr(2,6) in its Plücker embedding in P^14 with a subspace of dimension 6. In this poster we focus on the problem of recovering such a linear section associated to a general self-dual set of points. This lifting relies on finding the unique 6x6 skew-symmetric matrix whose Pfaffian is a cubic singular at the self-dual points. We use numerical and symbolic methods to approach the problem. This is an ongoing project with Leonie Kayser.
Birte Ostermann (Technische Universität Braunschweig) - A speed-up for Helsgaun's TSP heuristic by relaxing the positive gain criterion
The Traveling Salesman Problem (TSP) is one of the most extensively researched and widely applied combinatorial optimization problems. It is NP-hard even in the symmetric and metric case. The Lin-Kernighan-Helsgaun (LKH) heuristic has led to many best known solutions for instances with up to ten million vertices. LKH repeatedly improves an initially random tour by exchanging edges along alternating circles. Thereby, it respects several criteria designed to quickly find alternating circles that give a feasible improvement of the tour. Among those criteria, the positive gain criterion stayed mostly untouched in previous research. It requires that, while constructing an alternating circle, the total gain has to be positive after each pair of edges. We relax this criterion carefully leading to improvement steps hitherto undiscovered by LKH. We confirm this improvement experimentally via extensive simulations on various benchmark libraries for TSP. Our computational study shows that for large instances our method is on average 13% faster than the latest version of LKH while the quality of solutions stays virtually the same. This is joint work with Sabrina Ammann, Sebastian Stiller and Timo de Wolff.
Bryson Kagy (North Carolina State University) - Equidistant Circular Split Networks
Phylogenetic networks are generalizations of trees that allow for the modeling of non-tree like evolutionary processes. Split networks give a useful way to construct networks with intuitive distance structures induced from the associated split graph. We explore the polyhedral geometry of distance matrices built from circular split systems which have the added property of being equidistant. We give a characterization of the facet defining inequalities and the extreme rays of the cone of distances that arises from an equidistant network associated to any circular split network. We also explain a connection to the Chan-Robbins-Yuen polytope from geometric combinatorics.
Daniel Green Tripp (University of Bristol) - On an equidistribution result in toric varieties
Equidistribution problems are key in the study of holomorphic dynamics. In this case, we study the effect of the map that sends each point in a toric variety to its m-th power. We show that, for "generic" points, the sequences of m-th powers are equidistributed with respect to the Haar measure on the compact torus inside the corresponding torus orbit. A brief introduction to toric varieties and measures will be given before giving a sketch of the proof.
Eline Mannino (University of Oslo) - The rank-nullity ring of a matroid
We introduce the rank-nullity ring of a matroid M. This is the subring of the Chow ring of the free matroid generated by sums of variables corresponding to subsets having the same rank and the same nullity in the matroid M. We present symmetric combinatorial expressions for the tautological Chern classes of M, introduced by Berget, Eur, Spink and Tseng which imply that they live in the rank-nullity ring of M.
George Balla (TU Berlin) - Tropical symplectic flag varieties
The symplectic flag variety is the parameter space of flags of isotropic linear subspaces of a symplectic vector space. We consider the tropicalization of this variety under its Plücker embedding. We identify a particular maximal prime cone in this tropicalization by explicitly giving its facets. For every point in the relative interior of this maximal cone, the corresponding Gröbner degeneration is the toric variety associated with a particular polytope that labels a nice monomial basis of simple modules for the symplectic Lie algebra.
Giacomo Masiero (KU Leuven) - On Circuits of Matroids and Liftability Properties of Point-Line Configurations
For any rank-r matroid M over a finite set {1, .., n}, its circuit variety V is the algebraic variety defined as the realization space of the circuits of M, over a suitable complex vector space. Literature demonstrates that a theoretical irreducible decomposition for circuit varieties is possible as union of matroid varieties (i.e. Zariski closures of realization spaces of matroids).
We consider rank-3 matroids over {1,…,n}, associated with a point-line configuration C with n points. In this setting, the irreducible decomposition of the circuit variety V shows a close relation with the following geometric problem. Given n distinct points on a line l in the projective plane, when are they the image of a projection from a point P, on the line l, of a full realization of C?
We investigate this relation to provide:
- a family of matroids having an irreducible circuit variety, and
- a family of matroids whose circuit variety has two irreducible components.
Joint work with O. Clarke and F. Mohammadi
Giulia Iezzi (RWTH Aachen) - Schubert varieties as quiver Grassmannians
Quiver Grassmannians are projective varieties parametrising subrepresentations of quiver representations. Their geometry is an interesting object of study, due to the fact that many geometric properties can be studied via the representation theory of quivers. For instance, this method was used to study linear degenerations of flag varieties, obtaining characterizations of flatness, irreducibility and normality via rank tuples.
We give a construction for smooth quiver Grassmannians of a specific wild quiver, which realises smooth Schubert varieties and desingularises the singular ones. This also allows us to define linear degenerations of Schubert varieties.
Gökçen Dilaver Tunç (Hacettepe University) - Generalized Splines
In this poster, we introduce a generalized spline, which is a vertex labeling of an edge-labeled graph $G$ by elements of commutative ring $R$ with identity so that the difference between the labels of any two adjacent vertices lies in the corresponding edge ideal. The collection of generalized splines over the edge-labeled graph is denoted by $R_{(G, \alpha)}. $ It has a ring and an $R$- module structure. We define flow-up classes, which is a special type of generalized splines, and give some properties. We especially focus on generalized spline modules on complete graphs whose edges are labeled with proper ideals of $\mathbb{Z} / m\mathbb{Z}$. We compute minimum generating sets of constant flow-up classes for spline modules on edge-labeled complete graphs over $\mathbb{Z} / m\mathbb{Z}$ and determine their rank under some restrictions.
Harper Niergarth (University of Waterloo) - Complements of coalescing sets
We consider matrices of the form qD + A, with D being the diagonal matrix of degrees, A being the adjacency matrix, and q a fixed value. Given a graph H and a subset B of V(G), which we call a coalescent pair (H, B), we derive a formula for the characteristic polynomial where a copy of same rooted graph G is attached by the root to each vertex of B. Moreover, we establish if (H_1, B_1) and (H_2, B_2) are two coalescent pairs which are cospectral for any possible rooted graph G, then (H_1, V (H_1)\B_1) and (H_2, V (H_2)\B_2) will also always be cospectral for any possible rooted graph G.
Jonathan Boretsky (Harvard) - Positroid Boundary Algebras
Boundary algebras are an important tool in the categorification, by Jensen--King--Su and by Pressland, of cluster structures on positroid varieties, defined by Scott and by Galashin--Lam. Each connected positroid has a corresponding boundary algebra. We give a combinatorial way to recover a positroid from its boundary algebra. We then describe the set of algebras which arise as the boundary algebra of some connected positroid. This yields a new characterization of connected positroids which we state in purely combinatorial terms. Finally, we give the first complete description of the minimal relations in the boundary algebra. We expect this description to be helpful in extending results known for Grassmannian boundary algebras to more general settings.
Julian Weigert (University of Konstanz) - Equivariant Tutte polynomial
The usual Tutte polynomial T_M of a matroid M on {0,…,n} can be recovered in a geometrical way by considering certain classes in the cohomology ring of the permutohedral variety of dimension n. One can upgrade this formula to torus-equivariant cohomology and obtain an equivariant version of the Tutte polynomial. We will discuss this construction and look at some of the combinatorial properties of the resulting polynomial. In particular we will look at a universality result of this polynomial with regards to deletion/contraction relations of matroids.
Kaelyn Willingham (University of Minnesota) - Neural Network Theory through the lens of Tropical Geometry
The field of Deep Learning has taken off in popularity in recent years, both in the academic & industrial sectors, and this is largely due to the sheer effectiveness of neural network modeling. However, a theoretical understanding of why neural networks are so effective remains elusive. Recent work has shown the potential for tropical geometry to be useful in building a rigorous theory of neural networks, and this poster will highlight some of that work. In particular, we'll see that ReLU neural networks can be built algebraically using tropical rational functions and geometrically using Newton polygons & zonotopes. We'll also see how these tropical constructions can allow us to study the nature of fitting for neural network models, as well as their geometric complexity.
Karel Devriendt (MPI MiS, Leipzig) - Tropical toric maximum likelihood estimation
The maximum likelihood estimation problem (MLE) asks: given a statistical model and some observed data, which distribution in the model best explains the data? This is a classical problem in algebraic statistics with a rich theory. The tropical MLE problem asks a similar question: given a statistical model and the convergence rates of some data that depends on a parameter, what is the convergence rate of the MLE solution? This problem was solved for linear models by Agostini et al. and Ardila et al. For log-linear or toric models, the tropical MLE problem comes down to finding the intersection points of a (classical) linear space and a tropical affine space. In this poster, I will show that the tropical toric MLE points are piecewise linear functions of the tropical data, indexed by certain triangulations of the Newton polytope of the model. This is joint work with Erick Boniface and Serkan Hosten.
Kevin Kühn (Goethe-Universität Frankfurt) - Graph Curve Matroids
We introduce a new class of matroids, called graph curve matroids. A graph
curve matroid is associated to a graph and defined on the vertices of the graph as a ground set. We prove that these matroids provide a combinatorial description of hyperplane sections of degenerate canonical curves in algebraic geometry. Our focus lies on graphs that are 2-connected and trivalent, which define identically self-dual graph curve matroids, but we also develop generalizations. Finally, we provide an algorithm to compute the graph curve matroid associated to a given graph, as well as an implementation and data of examples that can be used in Macaulay2.
This is joint work with Alheydis Geiger and Raluca Vlad.
Laura Casabella (MPI MiS Leipzig) - Subdivisions of Hypersimplices, Finite Metric Spaces and Eulerian Numbers
In this joint work with Michael Joswig and Lars Kastner, we obtain various computational results concerning secondary fans of hypersimplices (2,7) and (3,6). We then study specific coarsest non-matroidal regular subdivisions which feature interesting combinatorial properties connected to Eulerian numbers. We relate these objects to the study of finite metric spaces.
Leonie Kayser (MPI MiS Leipzig) - Recovering forms from their space of partial derivatives
Calculus tells us that one can recover a multivariate polynomial from its partials by integrating - Euler's formula even gives a very concrete way of doing so. On the other hand there are linearly independent forms whose spaces of partial derivatives are the same (find an example!). We study the problem of recovering a suitably general form from its spaces of (higher) partials. A particularly nice case is that of binary forms, which we understand fairly well. A follow-up question is to classify which linear subspaces of polynomials occur as spaces of partials and to study their locus in the Grassmannian.
This is joint work in progress with Javier Sendra-Arranz (Uni Tübingen).
Lorenzo Campioni (Università degli Studi dell'Aquila) - Unrefinable partitions into distinct parts
A partition into distinct parts of an integer number is called unrefinable if none of the parts can be written as the sum of smaller integers, without introducing a repetition.
Clearly the condition of being unrefinable imposes on the partition a non-trivial limitation on the size of the largest part and on the possible distribution of the parts. We prove an upper bound for the largest part in an unrefinable partition of n, and we call maximal those which reach the bound. We show a complete classification of maximal unrefinable partitions exhibiting a bijection with a suitable partitions into distinct parts, depending on the distance from the successive triangular number.
Maddalena Pismataro (University of Bologna) - Cohomology ring of abelian arrangements
Abelian arrangements are a generalization of hyperplane and toric arrangements, whose complements cohomology ring has been studied since the 70’s. We introduce the complex hyperplane case, proved by Orlik and Solomon (1980), and the real case, Gelfand-Varchenko (1987). Then, we describe toric arrangements, showing results due to De Concini and Procesi (2005) and to Callegaro, D ’Adderio, Delucchi, Migliorini, and Pagaria (2020). We exhibit a new technique to prove the Orlik-Solomon and De Concini-Procesi relations from the Gelfand-Varchenko ring and to provide a presentation of the cohomology of all abelian arrangements. This is a join work with Evienia Bazzocchi and Roberto Pagaria.
Mieke Fink (University Bonn) - Valuative invariants of matroids
Valuative invariants are matroid invariants that behave well under decompositions of matroid polytopes. Understanding matroid invariants on simple pieces of matroid polytopes could be useful to explain their behavior on more complex matroids.
Milo Orlich (Aalto University) - The regularity of almost all edge ideals
A fruitful contemporary paradigm in graph theory is that almost all graphs that do not contain a certain subgraph have common structural characteristics.
An example is the well-known result saying that almost all triangle-free graphs are bipartite. The “almost” is crucial, without it such theorems do not hold. In this
paper we transfer this paradigm to commutative algebra and make use of deep graph theoretic results. A key tool are the critical graphs introduced relatively recently by Balogh and Butterfield, who proved that almost all graphs not containing a critical subgraph have common structural characteristics analogous to being bipartite.
For a graph G, let I_G denote its edge ideal, the monomial ideal generated by x_ix_j for every edge ij of G. In this poster we study the graded Betti numbers of I_G, which are combinatorial invariants that measure the complexity of a minimal free resolution of I_G. The Betti numbers of the form β_{i,2i+2} constitute the “main diagonal” of the Betti table. It is well known that for edge ideals any Betti number to the left of this diagonal is always zero. We identify a certain “parabola” inside the Betti table and call parabolic Betti numbers the entries of the Betti table bounded on the left by the main diagonal and on the right by this parabola. Let β_{i,j} be a parabolic Betti number on the r-th row of the Betti table, for r ≥ 3. We prove that almost all graphs G with β_[i,j}(I_G)=0 can be partitioned into r-2 cliques and one independent set. In particular, for almost all graphs G with β_{i,j}(I_G)=0, the regularity of IG is r-1. This is a joint work with Alexander Engström, published in December 2023 in "Advances in Mathematics". If time allows, I will present some directions for future research based on this work.
Pardis Semnani (University of British Columbia) - Causal Inference in Directed, Possibly Cyclic, Graphical Models
We consider the problem of learning a directed graph G from observational data. We assume that the distribution which gives rise to the samples is Markov and faithful to the graph G and that there are no unobserved variables. We do not rely on any further assumptions regarding the graph or the distribution of the variables. In particular, we allow for directed cycles in G and work in the fully non-parametric setting. Given the set of conditional independence statements satisfied by the distribution, we aim to find a directed graph which satisfies the same d-separation statements as G. We propose a hybrid approach consisting of two steps. We first find a partially ordered partition of the vertices of G by optimizing a certain score in a greedy fashion. We prove that any optimal partition uniquely characterizes the Markov equivalence class of G. Given an optimal partition, we propose an algorithm for constructing a graph in the Markov equivalence class of G whose strongly connected components correspond to the elements of the partition, and which are partially ordered according to the partial order of the partition. Our algorithm comes in two versions -- one which is provably correct and another one which performs fast in practice.
Petra Rubí Pantaleón Mondragón (CCM-UNAM) - Foliations and the Hilbert Scheme
In this poster, I will present a programmable method to identify foliations in the Hilbert Scheme of points.
Rodrigo Iglesias González (Universidad de La Rioja) - Minimal free resolution of the Rees algebra of some monomial ideals
Let $I$ be a monomial ideal in two variables generated by three monomials and let $\Rk(I)$ be its Rees ideal. This kind of ideals generalizes the well known class of ideals associated to the parametrization of a monomial plane curve parametrization. We describe an algorithm that computes the set defining equations of the Rees Algebra of such ideals. Furthermore, utilizing this algorithm, we are able to build a graph that encodes the minimal free resolution of $\Rk(I)$ with explicit descriptions of the modules and differentials on it.
Sarah Eggleston (University of Osnabrück) - Typical ranks of real order-three tensors
While typical ranks of real matrices are well understood, much less is known about typical ranks of higher order tensors. Here, we consider random real order-three tensors, primarily from a geometric perspective. We show that tensors of the m x n x (m-1)(n-1)+1 have more than one typical rank, and for the case m = n = 3, we prove that the probability that a random tensor T has rank 5 is equal to the probability that all 27 lines on a certain cubic surface are real.
Sergio Alejandro Fernandez de soto Guerrero (TU Graz) - MathMagic: A 'positroidal action' over a deck of cards
Positroids are a subclass of Matorids that were born in the study of the non-negative Grassmanian by Postnikov in 2006, since then, there are a plethora of combinatoric objects that index positroids, like Decored and Bicolored permutations, two generalizations of the symmetric group. These generalizations are useful to describe group action over a deck of cards, in this context, we can describe a different definition of invariants under a group action that allows us to explore its application with unusual ways of shuffling cards to develop new magic tricks. We will see how to define the generalization of the permutations and how to use it in magic!
Seungkyu Lee (University of Bonn) - Log-concavity in standard Koszul algebras
Standard Koszul algebras naturally occur in Lie algebra representation theory, especially in the BGG category O.
Though crucial, these algebras are complex to compute, with limited examples known.
However, for the hypertoric category O case, the corresponding standard Koszul algebras called hypertoric convolution algebras can be easily defined from affine hyperplane arrangements.
In this poster presentation, we observe a possible log-concavity of the Hilbert series of these algebras by seeing examples.
In particular, we show that the Hilbert series are log-concave for d = 1 and n = d-1 arrangements.
This point of view potentially gives a new understanding of log-concavity in representation theory and Kazhdan-Lusztig theory.
Sharon Robins (Simon Fraser University) - Versal deformations of smooth complete toric varieties
Deformation theory is a vital tool for understanding the local structure of a moduli space around a fixed object X. A systematic approach to studying infinitesimal deformations of X involves defining a functor, Def_X, that associates, for every local Artin ring, the set of deformations over that ring up to equivalence. Despite a theoretical understanding of Def_X, explicit computations with examples are challenging. In this poster, I will discuss the combinatorial description of Def_X when X is a smooth complete toric variety. This is joint work in progress with Nathan Ilten.
Tom Goertzen (RWTH Aachen) - Combinatorial Surfaces with given Automorphism Group
Frucht shows that, for any finite group G, there exists a cubic graph such that its automorphism group is isomorphic to G. For groups generated by two elements we simplify his construction to a graph with fewer nodes. In the general case, we address an oversight in Frucht’s construction. We prove the existence of cycle double covers of the resulting graphs, leading to simplicial surfaces with given automorphism group, where 'simplicial surfaces' refer to the combinatorial incidence structures of triangulated surfaces. For almost all finite non-abelian simple groups we give alternative constructions based on graphic regular representations. In the general cases of cyclic and dihedral group, we provide alternative constructions of simplicial spheres. Furthermore, we embed these surfaces into the Euclidean 3-Space with equilateral triangles such that the automorphism group of the surface and the symmetry group of the corresponding polyhedron in the orthogonal group O(3) are isomorphic. This is joint work with Reymond Akpanya.
Vadym Kurylenko (SISSA, Trieste) - Ehrhart theory and Gale duality
Let A be a spanning lattice point configuration. Consider a matrix L, whose rows generate the set of the Z-linear relations between the points of A. The columns of this matrix can be taken as a Gale dual of A. I will explain how we can use the matrix L to express Ehrhart-theoretic invariants of the convex hull of A. In particular, I will focus on the case of circuits and its connections to hypergeometric functions.
Victor Wang (Harvard University) - P-partition power sums
We develop the theory of weighted P-partitions, which generalises the theory of P-partitions from labelled posets to weighted labelled posets. We define the related generating functions in the natural way and compute their product, coproduct and other properties. As an application we introduce the basis of combinatorial power sums for the Hopf algebra of quasisymmetric functions and the reverse basis, both of which refine the power sum symmetric functions. These bases share many properties with the type 1 and type 2 quasisymmetric power sums introduced by Ballantine, Daugherty, Hicks, Mason and Niese, and moreover expand into the monomial basis of quasisymmetric functions with nonnegative integer coefficients. We prove formulas for products, coproducts and classical quasisymmetric involutions via the combinatorics of P-partitions, and give combinatorial interpretations for the coefficients when expanded into the monomial and fundamental bases.