Poster Abstracts
Quasihomomorphisms from the integers into matrix groups (Alejandro Vargas, Nantes Université)
We consider functions $f$ with domain $(\mathbb{Z}, +)$ and codomain in a family $\mathcal M$ of matrix groups such that $f$ is \emph{at distance $c$} of being a homomorphism, and ask how well $f$ can be approximated with an actual homomorphism. In this exposition we make these notions precise, and look at two cases: $\mathcal M$ equals diagonal matrices, and $\mathcal M$ equals symmetric matrices with $f$ at distance 1 of being a homomorphism. In both cases there exists a global constant $C$ such that for each $f$ there is a matrix $A$ with $\operatorname{rk}(f(m)-mA) \le C$. Both proofs show that $f$ follows certain cyclical combinatorial patterns. The general case where $\mathcal M$ is $\{ \operatorname{Mat}(K, n\times n) \, | \, n \in \mathbb{N} \}$, with $K$ a fixed field, is open, and we close the exposition with a few remarks, examples, and possible paths towards a proof.
The affine building of PGL_r and linear tropicalizations (Arne Kuhrs and Kevin Kühn, University Frankfurt)
We show that the affine building of PGL_r over a spherically complete non-Archimedean field, is the projective limit of all tropicalizations of \P^r with respect to linear embeddings. This result can be thought of as a linear algebraic version of a result of Sam Payne, which shows that the Berkovich analytification is the limit of all (not necessarily linear) tropicalizations. Along the way we proved a faithful tropicalization result for compactified linear spaces. These results tell us that the building of $\PGL_r$ may be thought of as a universal realizable tropical linear space. This is joint work with Luca Battistella, Martin Ulirsch and Alejandro Vargas.
Bisectors of polyhedral norms (Aryaman Jal, KTH Royal Institute of Technology)
Every symmetric convex body induces a norm on the underlying space. The object of our study is the bisector of two points with respect to this norm. A topological description of bisectors is known in the 2 and 3-dimensional cases and recent work of Criado, Joswig and Santos expanded this to a fuller characterisation of the geometric, combinatorial and topological properties of the bisector. A key object introduced was the bisection fan of a polytope which they were able to explicitly describe in the case of the tropical norm. We discuss the bisector as a polyhedral complex, introduce the notion of bisection cones and describe the bisection fan corresponding to other polyhedral norms. This is joint work with Katharina Jochemko.
Laplacian polytopes of simplicial complexes (Daniel Köhne, Universität Osnabrück)
Given a (finite) simplicial complex, we define its \emph{$i$-th Laplacian polytope} as the convex hull of the columns of its $i$-th Laplacian matrix. This extends Laplacian simplices of finite simple graphs, as introduced by Braun and Meyer. After studying basic properties of these polytopes, we focus on the $d$-th Laplacian polytope of the boundary of a $(d+1)$-simplex $\partial(\sigma_{d+1})$. If $d$ is odd, then as for graphs, the $d$-th Laplacian polytope turns out to be a $(d+1)$-simplex in this case. If $d$ is even, we show that the $d$-th Laplacian polytope of $\partial(\sigma_{d+1})$ is combinatorially equivalent to a $d$-dimensional cyclic polytope on $d+2$ vertices. Moreover, we provide an explicit regular unimodular triangulation for the $d$-th Laplacian polytope of $\partial(\sigma_{d+1})$. This enables us to compute the normalized volume and to show that the $h^\ast$-polynomial is real-rooted and unimodal, if $d$ is odd and even, respectively.
Initial degenerations of the Grassmannian via matroid subdivisions (Dante Luber, TU Berlin)
We discuss initial degenerations of the (3,8) Grassmannian. Nonempty degenerations are induced by a term order coming from a height function in the tropical Grassmannian. On the other hand, such height functions induce regular subdivisions of the (3,8) hypersimplex into matroidal cells. Hence we use data from the subdivision to deduce information about the degeneration. We show that all such initial degerations are smooth, and exhibit a degeneration with two irreducible components. As an application, we resolve a conjecture by Hacking Keel and Tevelev. Our techniques employ algebraic, tropical, and polyhedral geometry, matroid theory, and computation. This is joint work with Daniel Corey.
Degree Types of Stanley-Reisner Ideals with Pure Resolutions (David Carey, University of Sheffield)
We say that a resolution of a graded module over a polynomial ring is pure if the matrices representing its maps all consist of elements of the same degree. We define the degree type of such a resolution to be the sequence of those degrees. In the case of Stanley-Reisner ideals, Hochster’s Formula allows us to reinterpret this data in terms of homological properties of the corresponding simplicial complexes. This research concerns the family of complexes whose dual Stanley-Reisner ideals have pure resolutions. In particular, we have used this combinatorial framework to prove that there exists a Stanley-Reisner ideal with a pure resolution of any given degree type.
Quiver representation varieties (Dzoara Selene Núñez Ramos, Bergische Universität Wuppertal)
One of the main problems of quiver representation theory is to classify all the indecomposable representations for a given quiver Q. There is an approach to this problem via algebraic geometry that turned out to be useful for some wild-type path algebras, known for its difficulty. Motivated by this, we study the main object of this approach, called the variety of representations of quivers. In order to understand the scope of this concept, we explore an example of each type of algebra, namely finite, tame and wild types. In other words, we study its generic classification of its indecomposable representations via orbit classification. In order to study the tangent space of these varieties and their orbits, we study some homological tools and its main properties.
Some properties of Lovász-Saks-Schrijver ideals defined on forests (Eliana Tolosa Villarreal, Universita degli Studi di Genova)
Lovász-Saks-Schrijver ideals are ring polynomial ideals defined by graphs and depending on an integer d. We showed that when the graph G is a forest and for d>= grade(G)+2, the ring defined by such ideal is a UFD. Moreover, for d>= grade(G)+1, the ring is strong F-regular, F-rational, F-injective and F-pure. Finally, we conjectured that the divisor class group of the ring is closely related to the carnality of the set of vertices with maximum degree.
The Geometry of 2-level Polytopes (Jan Stricker, Johann Wolfgang Goethe-University Frankfurt am Main)
2-level polytopes are a special class of polytopes, which are occurring in all kinds of different fields of mathematics. We already know a lot about these polytopes. But also there are some new properties which where proven.
Let us find out about the combinatorics and geometry of 2-level polytopes.
The Duality of SONC (Janin Heuer, TU Braunschweig)
The cone of sums of nonnegative circuits (SONCs) is a subset of the cone of nonnegative polynomials / exponential sums, which has been studied extensively in recent years.
We construct a subset of the SONC cone which we call the DSONC cone. The DSONC cone can be seen as an extension of the dual SONC cone; membership can be tested via linear programming. We show that the DSONC cone is a proper, full-dimensional cone, we provide a description of its extreme rays, and collect several properties that parallel those of the SONC cone. Moreover, we show that functions in the DSONC cone cannot have real zeros, which yields that DSONC cone does not intersect the boundary of the SONC cone. Furthermore, we discuss the intersection of the DSONC cone with the SOS and SDSOS cones. Finally, we show that circuit functions in the boundary of the DSONC cone are determined by points of equilibria, which hence are the analogues to singular points in the primal SONC cone, and relate the DSONC cone to tropical geometry.
Basic properties of a new partition lattice and open problems (Jhon Bladimir Caicedo Portilla, Universität Osnabrück)
The partition lattice is one of the most studied objects in algebraic combinatorics and it is determined by the Stirling numbers of the second kind. There are different characterizations of these numbers by giving certain restrictions on their blocks and they are called incomplete Stirling numbers of the second kind. The idea of this poster is to give the basic definition of the incomplete partition lattice, based on the incomplete Stirling numbers of the second kind, and to show some open problems of this new family of lattices.
A calculus for monomials in Chow group of zero cycles in the moduli space of stable curves (Jiayue Qi, Johannes Kepler University Linz)
We introduce an algorithm for computing the value of all monomials in the Chow group of zero cycles in the moduli space of stable curves.
On the connected blocks polytope (Justus Bruckamp, Osnabrück University)
In this poster, we study the connected blocks polytope, which, apart from its own merits, can be seen as the generalization of certain connectivity based or Eulerian subgraph polytopes. We provide a complete facet description of this polytope, characterize its edges and show that it is Hirsch. We also show that connected blocks polytopes admit a regular unimodular triangulation by constructing a squarefree Gröbner basis. In addition, we prove that the polytope is Gorenstein of index 2 and that its h^*-vector is unimodal.
Smooth Realizations of Simple Polytopes (Kyle Huang, Freie Universitat Berlin)
In Stanley's initial proof of the g-conjecture, a critical error lies in the fact that not every simply polytope can be realized smoothly, ie with lattice points such that the normal fan is unimodular. We discuss some obstructions to smooth realizations and some classes of simple polytopes which can be realized as a smooth polytope.
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Divisional and Inductive Abelian Arrangements (Maddalena Pismataro, University of Bologna)
To generalise the concept of Inductively Free Hyperplane Arrangements, in a joint work with Lorenzo Vecchi and Tan Nhat Tran, we define the class of Divisional Abelian (Lie group) Arrangements and its subclass of Inductive Arrangements. One of the most interesting properties of Divisional Arrangements is the fact that their characteristic polynomials factor over Z, and the roots have nice recursive properties. The class of Inductive Arrangements contains the Strictly Supersolvable one, recently introduced by Bibby and Delucchi (2022). These results can be regarded as an extension of classical results due to Jambu and Terao (1984). As an application, we prove that the toric arrangements defined by an arbitrary ideal of a root system of type A, B or C is Inductive.
Covariance matrices of length power functionals of random geometric graphs (Mandala von Westenholz , Universität Osnabrück)
Asymptotic properties of a vector of length power functionals of random geometric graphs, which arise as the 1-skeleton of considered random simplicial complexes, are investigated. More precisely, its asymptotic covariance matrix is studied as the intensity of the underlying homogeneous Poisson point process increases. This includes a consideration of matrix properties like rank, definiteness, determinant, eigenspaces or decompositions of interest. For the formulation of the results a case distinction is necessary. Indeed, in the three possible regimes the respective covariance matrix is of quite different nature which leads to different statements.
Castelnuovo-Mumford regularity of projective monomial curves via sumsets (Mario Gonzalez Sanchez, University of Valladolid)
Given $A=\{a_0,\ldots,a_{n-1}\}$ a finite set of $n\geq 4$ non-negative integers that we will assume to be in normal form, i.e., such that $0=a_0<a_1<\cdots<a_{n-1}=d$ and relatively prime, the $s$-fold sumset of $A$ is the set $sA$ of integers obtained by collecting all the sums of $s$ elements in $A$. On the other hand, given an infinite field $k$, one can associate to $A$ the projective monomial curve $C_A$ parametrized by $A$: \[C_A=\{(v^d:u^{a_1}v^{d-a_1}:\cdots:u^{a_{n-2}}v^{d-a_{n-2}}:u^d)\}\] where $(u:v)$ covers the whole projective line over $k$. This allows us to establish a bridge between Additive Number Theory and Commutative Algebra and obtain some results connecting the Castelnuovo-Mumford regularity of $C_A$ and the behaviour of the sumsets $sA$.
This poster is based on a joint work with Philippe Gimenez.
Asymptotic results on the regularity of edge ideals of graphs via critical graphs (Milo Orlich, Aalto University)
To any graph G one can associate its edge ideal. One of the most famous results in combinatorial commutative algebra, Hochster's formula, describes the Betti numbers of the edge ideal in terms of combinatorial data of the graph G. More explicitly, each specific Betti number is given in terms of the occurrence of certain induced subgraphs in G. The machinery of critical graphs, relatively recently introduced by Balogh and Butterfield, deals with characterizing asymptotically the structure of graphs based on their induced subgraphs. In a joint work with Alexander Engström, "The regularity of almost all edge ideals", we apply these techniques to Betti numbers and regularity of edge ideals. We introduce parabolic Betti numbers, which constitute a non-trivial portion of the Betti table. One of our main results is that, fixed a parabolic Betti number on row r of the Betti table, for almost all graphs with that Betti number equal to zero, the regularity of the edge ideal is r-1.
Quantifying discontinuity (Nikola Sadovek, Freie Universität Berlin)
The Borsuk--Ulam theorem states that any odd function from an n-sphere to an (n-1)-sphere is discontinuous. Moreover, Dubins and Schwarz proved in 1981 that any such function has ~modulus of discontinuity~ at least r_n, where r_n is the geodesic distance between two vertices of the regular (n + 1)-simplex inscribed in S^n. We view this result as quantitative generalization of the Borsuk--Ulam theorem: ‘not only does there not exist an odd continuous function between higher and lower dimensional sphere, but any odd function between them has to be ~at least r_n discontinuous~ (with respect to the appropriate measurement of discontinuity)’. We then use this as motivation to prove similar quantitative generalizations of a number of known results: 1) Borsuk--Ulam theorems on Z/p-equivariant functions. 2) The nonembeddability of R^k into R^n for k > n. 3) The topological Tverberg theorem. 4) Knaster’s problem. To prove these results, we use a combinatorial-geometric reduction of each of them to the original case. The central objects of the proofs are Vietoris-Rips complexes of the appropriate spaces.
Chain algebras of finite distributive lattices (Oleksandra Gasanova, University of Duisburg-Essen)
This presentation is based on my ongoing project with Lisa Nicklasson. Let L be a finite distributive lattice and let t_1,...,t_n denote the elements of its ground set. To each maximal chain C of L one can associate a squarefree monomial m in K[t_1,...,t_n] which equals the product of all t_i belonging to C. We then consider the subalgebra K[m_1,...,m_s], generated by all such monomials, and call it the chain algebra of L.
I will present some properties of chain algebras in connection to combinatorial properties of the corresponding lattices, and the connection of such algebras to Hibi rings.
Information Geometry for Combinatorial Tensor Decomposition of Multi-Modal Data (Pedro Soto, University of Oxford)
Some datasets possess interesting relationships between their variables that are not well modeled by a vector or a matrix, e.g., genomic data with one index corresponding to individuals, a second index corresponding to RNA sequences, and a third corresponding to the cell type it was extracted from. Such data is naturally modeled as an order-3 tensor, and just as in the order-2 (i.e., matrix) case, there exists a generalization of rank as a measure of complexity. Once we go from linear to bi-linear and beyond (\emph{i.e.,} order-$n$ tensors), computing the rank becomes NP-Hard; however, computing the low-rank approximation of a tensor is feasible for a small enough order. Likewise, another complication is that the only definition of rank that generalizes to the multilinear case is that the rank of a tensor is the smallest number of summations of outer products of vectors needed to compute it. Tensor decompositions generalize many classic linear algebra algorithms, such as SVD and PCA, to the realm of multilinear algebra.
We present a recent method that uses techniques from information theory to efficiently compute tensor decompositions of data that are naturally modeled as discrete variables, e.g., counts. In particular, the algorithm designed to compute low-rank tensor decompositions is inspired by Csiszár and Körner's Method of Types, a method they used to great effect towards extracting strong bounds on channel coding theorems and rate-distortion results, but, as we shall see, can be put to great use in information-theoretic statistical learning. Furthermore, we show how to use the dimensionality reduction of the low rank approximation to perform an interpretable cluster analysis.
Extreme Values of Permutation Statistics (Philip Dörr, OVGU Magdeburg)
We investigate limit theorems for \textit{maxima of statistics on random permutations}. In particular, considering a random permutation $\pi \in S_n$, we are interested in the number of inversions $X_\inv(\pi)$ or the number of descents $X_\des(\pi)$. These concepts can be extended to finite Coxeter groups. The central limit theorem (CLT) is known to hold for the number of inversions or descents. Corresponding results are achieved for the maxima of these random variables in a suitably constructed triangular array.
Multiplication polynomials for elliptic curves over finite local rings (Riccardo Invernizzi, KU Leuven)
For a given elliptic curve $E$ over a finite local ring, we denote by $E^{\infty}$ its subgroup at infinity. Every point $P \in E^{\infty}$ can be described solely in terms of its $x$-coordinate $P_x$, which can be therefore used to parameterize all its multiples $nP$. We refer to the coefficient of $(P_x)^i$ in the parameterization of $(nP)_x$ as the $i$-th multiplication polynomial.
We show that this coefficient is a degree-$i$ rational polynomial without a constant term in $n$. We also prove that no primes greater than $i$ may appear in the denominators of its terms. As a consequence, for every finite field $\mathbb{F}_q$ and any $k\in\mathbb{N}^*$, we prescribe the group structure of a generic elliptic curve defined over $\mathbb{F}_q[X]/(X^k)$, and we show that their ECDLP on $E^{\infty}$ may be efficiently solved.
Efficient algebraic methods for system reliability (Rodrigo Iglesias, Universidad de La Rioja)
The evaluation of system reliability is an NP-hard problem even in the binary case.
There exist several general methodologies to analyze and compute system reliability. The two main ones are the sum-of-disjoint-products (SDP), which expresses the logic function of the system as a union of disjoint terms, and the (Improved) Inclusion-Exclusion (IIE) formulas. The algebraic approach to system reliability, assigns a monomial ideal to the system and obtains information about the structure of the system and computes its reliability by analyzing the properties of the ideal. We present here the algebraic versions of the two methods. The algebraic version of the IIE method makes use of free resolutions and Hilbert series of the ideal. The algebraic version of the SDP method makes use of the combinatorial decomposition of the system’s ideal provided by involutive bases. We also propose a candidate to improve this method by replacing the involutive bases with involutive-like bases which provide more compact decompositions. These algebraic versions are suitable for binary and multi-state systems. At the end, we included some useful information about our implementation of these two algebraic approaches using the C++ computer algebra library CoCoALib together with a discussion on which of the algebraic methods can be more efficient depending on the type of system under analysis.
Magical uses of positroids and Bicolored permutations (Sergio Alejandro Fernandez de soto Guerrero, TU Graz)
In 2006, Postnikov studied a decomposition of the nonnegative Grassmannian (Gr≥0(k, n)) in terms of cells indexed by a particular subclass of matroids known as positroids. In his work, he provides a plethora of objects that are in bijection with positroids showing the combinatorial richness of this theory. One of these families is formed by a {0, 1}-filling of Young (Ferrer) diagrams avoiding a particular pair of patterns. These objects are known as Le-diagrams. Josuat-Vergès found an explicit bijection between the families of Le-diagrams and a different family of {0, 1}-filling of Young diagrams known as X-diagrams. We will show that a new family of objects could be added to the combinatorial “zoo” of positroidal objects. These new additions are called bicolored permutations. Bicolored permutations were introduced recently by González D’León, who showed that this kind of permutations form a basis for the multilineal component of a bicolored exterior algebra. Finally, we will see how the set of bicolored permutations on n elements describes a different kind of shuffles in a deck of cards, and how these new permutations can be applied (or used) for magic tricks and recreational maths!
Enumerative Combinatorics and the RSK Algorithm (Seyyed Ali Mohammadiyeh, University of Kashan)
In this poster, we will explore the connection between combinatorics and the RSK (Robinson-Schensted-Knuth) algorithm, a fundamental tool in algebraic combinatorics. We will begin by introducing some basic notions of combinatorics, such as permutations, Young tableaux, and Schur functions. Then, we will discuss the RSK algorithm, which is used to insert permutations into Young tableaux and produce two tableaux with some special properties. We will explore how the RSK algorithm can be used to prove various combinatorial identities, such as the Littlewood-Richardson rule and the Jacobi-Trudi formula. We will also discuss some algorithmic aspects of the RSK algorithm, such as its computational complexity and practical implementations. Finally, we will briefly mention some recent developments related to the RSK algorithm and its applications in algebra, geometry, and representation theory.
Minors of eigenvector matrices (Tarek Emmrich, Osnabrück University)
We study the connection between the minors of the eigenvector matrix of A and the Galois Group of the characteristic polynomial of A. Asymptotically this will yield a new uncertainty principle for the Graph Fourier transform. As far as possble, we present similar results for symmetric matrices.
Colored quasisymmetric functions and characters of descent representations (Vassilis Dionyssis Moustakas)
The quasisymmetric generating function of an inverse descent class equals a skew Schur function associated to a zigzag diagram. This is known to be Schur-positive and the corresponding representation of the symmetric group is called the descent representation. In this work, we provide an extension of this result to colored permutations which generalizes a recent result of Adin, Athanasiadis, Elizalde and Roichman (2017), who studied the signed case. For this purpose, we introduce a notion of colored zigzag diagrams and prove that the representations associated to these shapes coincide with the descent representations of colored permutation groups introduced by Bagno and Biagioli (2007). Gessel's fundamental quasisymmetric functions are replaced by Poirier's colored quasisymmetric functions.
Khovanskii Bases for fiber products of toric varieties (Viktoriia Borovik, Osnabrück Universität)
We provide an efficient approach to counting roots of polynomial systems, where each polynomial is a general linear combination of prescribed, fixed polynomials. Our tools relay mostly on the theory of Khovanskii bases, combined with toric geometry, BKK theorem and fiber products.
As a direct application of this theory, we solve the problem of counting the number of approximate stationary states for coupled Duffing oscillators. We derive a Khovanskii basis for the corresponding polynomial system and the number of its complex solutions for an arbitrary degree of nonlinearity in the Duffing equation and an arbitrary number of oscillators.