Program and Abstracts

First week

Research Talks

Speaker: Luis Felipe Vargas,
Institution: Centrum Wiskunde & Informatica (CWI)

Title: The Stability Number of a Graph and Sums of Squares of Polynomials

Abstract: Given a graph G, its stability number is the cardinality of the largest subset of vertices without edges between them. Computing the stability number is an NP-hard problem and some approximations via semidefinite optimization have been developed. One of them is a hierarchy proposed by de Klerk and Pasechnik by following an idea given by Parrilo for approximating problems over the copositive cone via sum of squares of polynomials. One open question asks for the finite convergence of this hierarchy. We prove finite convergence for the class of graphs without critical edges. Our analysis includes a characterization of the minimizers of the Motzkin-Straus formulation. As an application, we can show that deciding whether a standard quadratic problem has finitely many global minimizers is hard.

This is a joint work with Monique Laurent

Speaker: Sergio Alejandro Fernandez de Soto Guerrero
Insititution: Universidad del Valle

Title: Una conexión entre el Grasmaniano no negativo y el álgebra exterior coloreada

Abstract: En matemáticas, el grasmaniano es un espacio que parametriza todos los subespacios lineales de un espacio vectorial de cierta dimensión. A su vez el álgebra exterior es el dual Koszul de un algebra simétrica. Estos dos objetos matemáticos tienen una relación peculiar con diferentes llenados de diagramas de Ferrer (Young), el grasmaniano con lo que se conoce como tableaux de permutación y el álgebra exterior con los llamados X-diagramas.

Esta charla se centrará en definir de manera adecuada el grasmaniano y el álgebra exterior para hablar de sus diferentes biyecciones con el mundo de la combinatoria gracias a dichos llenados de diagramas de Ferrer, y finalmente presentar parte de los resultados de mi actual tesis de maestría los cuales dan el puntapié para conectar dichos mundos, del grasmaniano y el álgebra exterior; entre muchos otros.

Speaker: Dzoara Núñez
Institution: Universidade Federal do Amanzonas

Title: An Example of a Quiver Representaion Variety

Abstract: The goal of quiver representation theory, is classify all representations of a given quiver Q and all morphisms between them up to isomorphism. But some of the algebras associated to a quiver are wild, in the sense that the problem of classification of their representations and their irreducible morphism is difficult or sometimes impossible. The main obstacle in this case is the dependence of the isomorphism classes of representations on arbitrarily many continuous parameters, to which many of the classical tools of the representation theory of algebras do not apply. The aim in this poster is to motivate the geometric approach to the classification problem, from the point of view of Reineke.

We will see that the isomorphism classes of representations of a fixed vector dimension v, have a nice geometric structure. They correspond to orbits of a certain algebraic GL-group acting over a certain variety. This structure allows us to study the classification of quiver representations using geometric techniques. We will present the above theory and some examples, specially we will apply this theory to Kronecker of 3 arrows which is a wild example.

Speaker: Theo Douvropoulos

Institution: University of Massachusetts, Amherst

Title: Hurwitz numbers via transportation polytopes

Abstract: The classical Hurwitz numbers H_{g,k} enumerate, up to equivalence, branched coverings of the sphere by a genus g surface all of whose branch points are simple apart from a single one whose monodromy is given by the partition k. These numbers H_{g,k} are given via the ELSV formula as integrals over the moduli space \bar{M}_{g,n}, but explicit calculations are often difficult.

Under an equivalent interpretation, the Hurwitz numbers count certain transitive factorizations in S_n and classical representation-theoretic techniques describe their generating function via a finite sum of exponentials exp(t*m). The coefficients a_m that appear in this sum are integers and completely determine the enumeration.

Computational and theoretical evidence has led us to conjecture a description of these coefficients in terms of the h- and h*-vectors of certain transportation polytopes indexed by the partition k. We will present the conjecture and give proofs for certain cases (in particular, for the identity permutation k=(1^n) which corresponds to the central transportation polytope T_{n,n-1}). We will further give an equivalent formulation, where the answer for all partitions is simultaneously encoded in the equivariant Ehrhart theory of T_{n,n-1}.

Second week

Speaker: Diana Toquica
Institution: Universidad Nacional de Colombia

Title: Algunas Estadísticas sobre los Poliminós asociados a las Palabras de Catalan

Abstract: Las palabras de Catalan están definidas sobre los enteros positivos y son enumeradas por los números de Catalan. Dada una palabra de Catalan, es posible asignarle un poliminó cuya altura de cada columna corresponde al término de la i-ésima posición. En estos diagramas de barras se han definido distintas estadísticas utilizando funciones generatrices y algunos métodos combinatorios; en este póster o charla se mostrarán algunas fórmulas obtenidas para el número de puntos reticulares al interior del poliminó.

Speaker: Marie-Charlotte Brandenburg

Institution: Max Planck Institute for Mathematics in the Sciences

Title: The positive tropicalization of low rank matrices

Abstract: We consider the tropicalization of determinantal varieties of matrices of rank at most 3, and give combinatorial and geometric descriptions of their positive parts.

Moreover, we introduce a tool for detecting positivity in terms of matching multifields and investigate connections to the totally positive part of the Grassmannian Gr(2,n).

This is joint work in progress with Georg Loho and Rainer Sinn.

Speaker: Jodi McWhirter

Institution: Washington University in St. Louis

Title: Oriented Matroid Circuit Polytopes

Abstract: Matroids, combinatorial structures that generalize the idea of linear independence in vector spaces, were introduced in the 1930s and give rise to several natural constructions of polytopes. Oriented matroids, similarly, yield many of these same constructions. In this talk, we will examine polytopes that arise from the signed circuits of an oriented matroid. We are able to give bounds on the dimension of a family of these polytopes coming from graphical oriented matroids. Moreover, when we look at the polytope constructed from the cocircuits of the oriented matroid generated by the positive roots of the classical type A root system, we can give an explicit description of the polytope, including its Ehrhart theory.

Speaker: Philippe Nadeau

Institution: CNRS & University Lyon 1

Title: Local generalizations of the classical parking algorithm

Abstract: We define a class of "local" parking procedures that take as input a sequence of cars with their preferred parking spots in Z, and output the final set of occupied spots. This class contains the classical parking procedure that determines "parking functions", and it extends some of its properties. We will explain that the numbers (n+1)^(n-1) are "universal" for all procedures. We will also show some nice tree combinatorics arising in this context. Time permitting, we will explain how a particular procedure is linked with geometric problems around the permutahedron.

Speaker: Jerónimo Valencia

Institution: Universidad de los Andes

Title: Aspectos de la teoría de Ehrhart de matroides de caminos reticulares

Abstract: Dado un politopo reticular, un problema común es encontrar fórmulas combinatorias para su h*-vector, dado que este codifica información como el volumen, el área superficial y el polinomio de Ehrhart del politopo. En esta charla daremos una interpretación combinatoria para el h*-vector de politopos asociados a matroides de caminos reticulares de rango 2. Esto generaliza el trabajo de Katzman, quien dio una fórmula para las componentes del vector en el caso de matroides uniformes de rango 2. Como resultado de nuestro trabajo, podemos dar fórmulas para el volumen de tales politopos. Además, daremos una caracterización de las matroides de caminos reticulares cuyo politopo asociado es Gorenstein.

Speaker: Alex McDonough

Institution: UC Davis

Title: Rotor-Routing Induces the Only Consistent Sandpile Torsor Structure on Plane Graphs

Abstract: Every finite graph has an associated sandpile group, which can be described in terms of chip-firing. A sandpile torsor algorithm is a map which associates each plane graph (i.e. planar embedding) with a free transitive action of its sandpile group on its spanning trees. We define a notion of consistency, which requires a torsor algorithm to be preserved with respect to a certain class of contractions and deletions. We then show that the rotor-routing sandpile torsor algorithm (which is equivalent to the Bernardi algorithm) is consistent. Furthermore, we demonstrate that there are only three other consistent algorithms, which all have the same structure as rotor-routing. This proves a conjecture of Klivans. Joint work with Ankan Ganguly.

Open problems

  • List of open problems presented at ECCO 2022. Compiled by Daniel Tamayo, Jerónimo Uribe, and Jonathan Niño