Charlas contribuidas

Title: The convex algebraic geometry of higher rank numerical ranges

Abstract: The higher-rank numerical range is a convex compact set generalizing the classical numerical range of square complex matrices, first appearing in the study of quantum error correction. In this talk, I will discuss some of the real algebraic and convex geometry of these sets, including a generalization of Kippenhahn’s theorem, and describe an algorithm to explicitly calculate the higher-rank numerical range of a given matrix. 

Title: On the spectra of Fine polyhedral adjoints

Abstract: Originally introduced by Fine and Reid, the Fine interior of a lattice polytope got recently into the focus of research. Based on the Fine interior, we study a modification of so-called adjoint polytopes and define the Fine adjoint polytope of a polytope P as consisting of the points in P that have lattice distance at least s to all its valid inequalities. In this manner, we obtain a Fine Polyhedral Adjunction Theory that is better behaved than its original analogue. Many existing results in Polyhedral Adjunction Theory carry over. In this talk, we will focus mainly on one of our conclusions obtained with simpler, more natural proofs as is the case of the finiteness of the Fine spectrum. Namely, we introduce the \Q-codegree as an invariant arising in the context of toric geometry, and discuss why it can take only finitely many values for polytopes under certain conditions. 

Title: Log-concavity of the Alexander polynomial

Abstract: The central question of knot theory is that of distinguishing links up to isotopy. The first polynomial invariant of links devised to help answer this question was the Alexander polynomial (1928). Almost a century after its introduction, it still presents us with tantalizing questions, such as Fox's conjecture (1962) that the absolute values of the coefficients of the Alexander polynomial $\Delta_L(t)$ of an alternating link $L$ are unimodal. Fox's conjecture remains open in general with special cases settled by Hartley (1979) for two-bridge knots, by Murasugi (1985) for a family of alternating algebraic links, and by Ozsv\'ath and Szab\'o (2003) for the case of genus $2$ alternating knots, among others. We settle Fox's conjecture for special alternating links. We do so by proving that a certain multivariate generalization of the Alexander polynomial of special alternating links is Lorentzian. As a consequence, we obtain that the absolute values of the coefficients of $\Delta_L(t)$, where $L$ is a special alternating link, form a log-concave sequence with no internal zeros. In particular, they are unimodal. 

Title: Symmetric spaces and shellability 

Abstract: In the realm of algebraic geometry, a spherical variety $Y$ emerges within the framework of a reductive algebraic group $G$ over an algebraically closed field, featuring a Borel subgroup $B$. This characterization delineates $Y$ as a $G$-variety where the action of $B$ yields finitely many orbits. Noteworthy examples of such varieties include toric varieties, flag varieties, symmetric varieties, and their corresponding embeddings. In many instances, orbits of the Borel subgroup can be parametrized combinatorially and endowed with a poset structure derived from the closure-containment (or Bruhat) order. The property of shellability garners significant attention in the realm of poset topology. This property, for instance, implies that a simplicial complex associated with a poset exhibits the homotopy type of a wedge of spheres. Notably, since the 1980s, it has been established that Bruhat orders of flag varieties possess the property of shellability, and recent endeavors have extended these findings to settings involving symmetric spaces. Our discussion will delve into various results and open questions within the posets stemming from the symmetric varieties. This collaborative work is conducted jointly with Aram Bingham (for further details

Title: Soluciones monocromáticas de ecuaciones en los enteros 

Abstract: Ramsey theory is the study of the appearance of unavoidable monochromatic patterns in large enough arbitrarily colored structures. This can be studied on the positive integers, where a natural question arises: how large must the integer n be so that any coloring of the positive integers from 1 to n, contains a certain monochromatic substructure? One such structure can be a solution set of a given equation.

In 1917, Schur proved that there exists a minimum positive integer s such that for any 2-coloring of the integers from 1 to s there is a monochromatic solution to x + y = z. We call (x,y,z) a Schur triple if it is a solution to x + y = z. An interesting problem is to count the number of monochromatic Schur triples in any k-coloring of the integers from 1 to a large enough n. Long-standing computation tools and techniques have been used to optimize this number. In this talk, we will expose improved lower bounds for the number of monochromatic Schur triples in any k-coloring of [n] when k is 3,4,5, and we will discuss our techniques in order to minimize the number of monochromatic triples associated to the equation ax + ay = z in any 2-coloring of [n], where n is large enough and a is at least 2.

Title: Acyclotopes and Tocyclotopes 

Abstract: There is a well-established dictionary between zonotopes, hyperplane arrangements, and (oriented) matroids. Arguably one of the most famous examples is the class of graphical zonotopes, hyperplane arrangements, and matroids. Those also encode subzonotopes of the type A root polytope, the permutahedron. Stanley gave a general interpretation of the coefficients of Ehrhart polynomials of zonotopes via linearly independent subsets of the generators. Applying this to the graphical case shows that Ehrhart coefficients count subforests of the graph with a fixed number of edges. The goal of this talk is twofold. First, we want to extend and popularize the above sketched story to other root systems, which on the combinatorial side is encoded by signed graphs. Many of the pieces in this dictionary can already be found in work by Zaslavsky and Greene, as the signed graphic zonotope, called acylotope, whereas computing the Ehrhart polynomial is novel for types B, C and D. Second, we want to dualize the whole picture, starting from matroid duality. This has well-known implications for hyperplane arrangements. For example, regions in the cographical arrangement now correspond to totally cyclic orientations. The corresponding construction for zonotopes can be found in work by McMullen, but seems to be less known. For the case of graphs and signed graphs, we develop dual zonotopes, called tocyclotopes, that behave well arithmetically. We again establish combinatorial interpretations for the Ehrhart coefficients. This is joint work with Eleonore Bach and Matthias Beck. 

Title: Faces of Parking Function Polytopes 

Abstract: Let $\bbu = (u_1, \dots, u_n) \in \R^n_{\geq 0}$ satisfy $0 \leq u_1 \leq \cdots \leq u_n.$ For $\bba = (a_1, \dots, a_n) \in \R^n_{\geq 0}$, we let $b_1 \leq b_2 \leq \dots \leq b_n$ be the increasing rearrangement of $a_1, \dots, a_n.$ We say $\bba$ is a $\bbu$-\emph{parking function} if $b_i \leq u_i$ for all $i = 1, \dots, n.$ The \emph{parking function polytope} associated to $\bbu$, denoted by $\pf(\bbu)$, is defined to be the convex hull of all $\bbu$-parking functions. We note that $\pf(\bbu)$ defined here generalizes the parking function polytopes previously studied in response to a question posed by Stanley in 2020. 

Title: On the computation of Kronecker coefficients 

Abstract: We give an algorithm that computes, for each triple of partitions, the Kronecker coefficient associated to them. The formula in which this algorithm is based is an alternating sum of numbers of integer points in a new family of convex polytopes depending on the three partitions. The integral points in each of these polytopes are 3-way statistical tables (tensors of order 3) whose 1-marginals (plane sums) depend on the original partitions and that satisfy certain extra inequalities. The dimensions of these families of polytopes differs by a binomial coefficient of the degree of the corresponding quasi-polynomial functions computed by Baldoni and Vergne. The same method yields an algorithm for computing the reduced Kronecker coefficients. 

Title: Inequalities for f^*-vectors of lattice polytopes 

Abstract: The Ehrhart polynomial ehr_P(n) 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 ehr_P(n) 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 ehr_P(n) expressed in a certain binomial basis. In joint work with Matthias Beck, Max Hlavacek and Jerónimo Valencia-Porras, we computed 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. These inequalities resemble striking similarities with existing inequalities for the coefficients of f-vectors of simplicial polytopes; e.g., the first half of the f*-coefficients increases and the last quarter decreases. Even though f*-vectors of polytopes are not always unimodal, there are several families of polytopes that carry the unimodality property. We also show that for any polytope with a given Ehrhart h*-vector, there is a polytope with the same h*-vector whose f*-vector is unimodal. 

Title: Computing the q-Multiplicity of the Positive Roots of sl_(r+1) (C) and Products of Fibonacci Numbers 

Abstract: Using Kostant's weight multiplicity formula, we describe and enumerate the terms contributing a nonzero value to the multiplicity of a positive root μ in the adjoint representation of sl_(r+1) (C), which we denote L(α ̃), where α ̃is the highest root of sl_(r+1) (C). We prove that the number of terms contributing a nonzero value in the multiplicity of the positive root μ=α_i+α_(i+1)+⋯+α_j with 1≤i≤j≤r in L(α ̃) is given by the product F_i∙F_(r-j+1), where F_n is the nth Fibonacci number. Using this result, we show that the q-multiplicity of the positive rootμ=α_i+α_(i+1)+⋯+α_j with with 1≤i≤j≤r in the representation L(α ̃), is precisely q^(r-h(μ)), where h(μ)=j-i+1 is the height of the positive root μ. Setting q=1 recovers the known result that the multiplicity of a positive root in the adjoint representation of sl_(r+1) (C) is one. 

Title: Shifted and threshold matroids 

Abstract: We explore two classes of matroids: shifted matroids and threshold matroids. Shifted matroids are defined by an ordering of their groundset while threshold matroids are defined by a weight function on their groundset. It is a known result that all threshold matroids are shifted matroids. We classify exactly which shifted matroids are threshold using the word structure of their maximal basis. We use this classification to give an example of a shifted matroid that is not threshold and to enumerate the number of shifted matroids on a groundset of fixed size. This enumeration proves that almost all shifted matroids are not threshold. Finally, we use our classification to answer multiple questions posed by Deza and Onn about the relationship between exchangeable and equatable matroids. 

Title: Cohomology ring of abelian arrangements 

Abstract: Abelian arrangements are generalizations 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. 

Title: Probabilistic combinatorics 

Abstract: This time we are going to discuss a topic that lies at the interface of combinatorics, probability theory and statistical physics. Bootstrap percolation is a stochastic process which is meant to model different progressive dynamics, like infection spreading among cells, rumor spreading in a society, fire propagation in a forest, etc, with a random initial condition and a deterministic evolution process. Our main goal will be to understand the so-called percolation threshold, which describes the phase transition in the process. 

Title: The critical group of an orientable ribbon graph

Abstract: The critical group of a connected graph is now a well-established structure in Combinatorics. It is usually defined using the reduced Laplacian of the graph. However, the critical group is also isomorphic to the quotient Z^m/(C+D), where m, C and D are the number of edges, cycle space and cocycle space, respectively, of the graph. We use that an orientable ribbon graph is an even delta-matroid representable by a principal unimodular matrix to associate an abelian group to any orientable ribbon graph. For planar graphs both groups are isomorphic. As a byproduct we obtain a formula for the number of quasi-trees of an orientable ribbon graph by computing a determinant.