Program
Monday 8
Villarreal: a beautiful mathematical journey
Aron Simis
Universidade Federal de Pernambuco, Brazil
Abstract:Having been an acquaintance and friend of Villarreal since 1979, I will convey some of his early steps into the field, while simultaneously trying to tie up with my own algebraic interests in the period, pretty much till our last collaborations.
Exploring the Frontiers of Commutative Algebra: Optimization, Codes, and Graphs
Delio Jaramillo Velez
Universidad de la Laguna, Spain
Abstract: The main goal of this talk is to present and demystify part of the work of Rafael H. Villarreal. In doing so, I aim to show how his ideas have broadened the scope of inquiry in commutative algebra, creating fruitful connections with several other disciplines, including combinatorial optimization, coding theory, and graph theory. I will begin with some general comments on the impact of his research and the areas he has influenced. I will then focus on three main themes. First, we will discuss optimization, specifically the max-flow min-cut property of a clutter and the normality of its edge ideal. Second, we will explore the relationship between coding theory and commutative algebra, discussing how families of linear codes can be defined and how their basic parameters can be estimated using commutative algebraic techniques. Third, we will examine the Vasconcelos number, how it arises in coding theory, and its role in graph theory. Finally, I will conclude with some reflections on how Villarreal’s ideas have helped shape the way we think about algebraic objects.
3,247,490 vs 3,247,488 vs 3,247,482: why graphs on 11 or more vertices are interesting
Adam Van Tuyl
McMaster University, Canada
Abstract: Combinatorial commutative algebra has greatly benefited from Rafael Villareal's numerous contributions. In this talk, I will highlight some interesting examples on graphs on 11 or more vertices related to Rafael's work on Cohen-Macaulay edge ideals, and more recently, the v-number of an ideal. In particular, I will explain what the three numbers in the title are counting.
Regular sequences and Hilbert series for graphs
Joseph Brennan
University of Central Florida, US
Abstract: This talk will survey recent results on determining regular elements and regular sequences in graphs.
Recent results on the determination of the Hilbert series for graphs will also be presented.
Ehrhart functions, Group Rings, and Rafael
Jesús A. De Loera
University of California, US
Abstract: Let $P$ be a $d$-dimensional convex polytope, the Ehrhart function of $P$ is defined by $ehr(P;n) = |nP\cap \mathbb{Z}^d|$ for nonnegative integers $n$. Similarly, the Ehrhart series of $P$ is, as a formal power series in the variable $t$ with integer coefficients, given by
$$\mathcal E(P;t) = \sum_{n\geq 0} ehr(P;n)t^n=\sum_{n\geq 0}\mathcal |nP\cap \mathbb{Z}^d|t^n.$$
Ehrhart functions and their series have been the subject of a lot research for many years and in various versions and numerous applications in combinatorics and commutative algebra because they are Hilbert functions of monomial algebras.
Rafael has himself been very interested to put weights/gradings on the function.
We present a new generalization of Ehrhart counting functions that unifies several results in the existing theory, and in particular uses of weights the lattice points in high generality. Our method uses groups and rings as follows. Let $G$ be an abelian group, $\phi:\mathbb{Z}^d \to G$ be a homomorphism, and $R$ be a commutative ring with unity.
We now define the $\phi$-Ehrhart function of $P$ as
$$ ehr(P,\varphi;n) = \sum_{\alpha \in nP \cap \mathbb{Z}^d}\varphi(\alpha) \in R[G].$$
We also define the $\phi$-Ehrhart series of $P$
$$\mathcal E(P,\varphi;t) = \sum_{n\geq 0} ehr(P,\phi;n)t^n$$
as a formal power series in the variable $t$ with coefficients in the group ring $R[G]$.
I will review this subject quickly (with highlights of Rafael’s recent joint work) and then show consequences of our new group-ring definition that show it is the most general version so far.
Tuesday 9
Quasipolynomial growth in multigraded homological algebra
Ezra Miller
Duke University, US
Abstract: Families of modules or vector spaces indexed by an integer $n$, for instance arising from powers of ideals, often have numerical invariants that grow polynomially. These invariants might be lengths, Betti or Bass numbers, or regularity, or combinations of these with deeper homological constructions.
Sometimes the growth is only quasipolynomial: there are polynomials $P_1$,...,$P_r$ such that the numerical invariant takes the value $P(n) = P_i(n)$ whenever $n$ is congruent to $i$ (mod $r$).
What drives this kind of growth and periodicity? Joint work with Hailong Dao, Jonathan Montaño, Christopher O'Neill, and Kevin Woods provides a general answer in the multigraded setting -- that is, for families of monomial ideals and other finely graded modules over affine semigroup rings. The theory is clarified by thinking so generally that it yields for free the case of multiple parameters, when the families are indexed not merely by a single integer n but by many integers; in that context, a numerical function is called quasipolynomial when its values agree with one of several polynomials, one for each coset of a given integer lattice. The proofs and constructions rest on foundations from applied topology, specifically tame modules in persistent homology, combined with Presburger arithmetic. No familiarity with either of these theories is assumed.
Cartwright-Sturmfels property of generalized binomial edge ideals
Emanuela de Negri
University of Genova, Italy
Abstract: Let $X$ be an $m\times n$-matrix of indeterminates and let $G=([n],E)$ be a graph. The generalized binomial edge ideal associated to $G$ is the ideal $I_G$ generated by the $2$-minors of $X$ obtained by choosing two arbitrary rows and two columns $j,k$ such that $\{j, k\}\in E$.
In erlier joint work with A. Conca and E. Gorla, it was shown that $I_G$ is Cartwright-Sturmfels in the case $G=K_n$ and for arbitrary graphs $G$ when $m=2$.
We prove that the Cartwright-Sturmfels property holds for all $m$ and $G$, by establishing general results on ideal constructions that preserve this property.
We also provides classes of examples and counterexamples for higher size minors.
This is joint works with A. Conca and V. Welker.
Jets of monomial ideals
Federico Galetto
Cleveland State University, US
Abstract: The notion of jets comes from differential geometry and it captures higher order differential approximations of a space, generalizing the concept of tangent bundle. Jets have been adopted in algebraic geometry as a tool to study singularities. Although the equations of jets can be easily written down, many questions about jet spaces remain open, particularly their decomposition into irreducible components. Our work focuses on spaces defined by squarefree monomial equations, which allows us to employ techniques from algebra and combinatorics. We define jets of clutters and study their covers to address decompositions of the corresponding spaces. We also define principal jets of clutters and show how to compute their Hilbert series and Betti numbers.
A combinatorial description of when a self-associated set of points fails to be arithmetically Gorenstein
Diego Ruano Benito
University of Valladolid, Spain
Abstract: We prove that the set of points associated to a self-dual code with no proportional columns is arithmetically Gorenstein if and only if the code is indecomposable. This answers a question asked by Tohăneanu. We do so by providing a combinatorial way to compute the dimension of the Schur (component-wise) square of a self-dual code through a zero-one symmetrization of its generator matrix. Our approach also allows us to compute the Gorenstein defect. As a consequence, we obtain a combinatorial characterization of arithmetically Gorenstein self-associated sets of points over an algebraically closed field. This is a joint work with G. Rodríguez-Pajares and F. Salizzoni.
On critical ideals, sandpile groups and the inverse eigenvalue problem of a graph
Ralihe Raul Villagran Olivas
Worcester Polytechnic Institute, US
Abstract:Critical ideals were introduced as a generalization of sandpile groups which serves as a tool to solve problems in this framework. Moreover, significant connections have been drawn between critical ideals and the inverse eigenvalue problem for graphs, most notably linking them to the zero forcing number and the minimum rank. Beyond these applications, critical ideals possess rich algebraic and combinatorial properties that make them compelling objects of study in their own right. In this talk, we will highlight the utility of critical ideals through these connections and present recent results regarding their intrinsic properties. Finally, we will present some open problems.
Zeros of polynomials on weighted projective spaces
Rodrigo San Jose Rubio
Virginia Tech, US
Abstract: We study the maximum number of zeros that a homogeneous polynomial can have on a weighted projective space, when the first weight is equal to 1. This solves a conjecture by Aubry, Castryck, Ghorpade, Lachaud, O'Sullivan, and Ram. We showcase how to adapt techniques such as the footprint bound for this case. We also exemplify the difficulties that arise in the general case where all the weights are greater than 1. This is joint work with Jade Nardi.
Monomial Polarization and Depolarization of Simplicial Complexes
Eduardo Sáenz de Cabezón
University of La Rioja, Spain
Abstract: We translate the operations of polarization and depolarization from monomial ideals in a polynomial ring to abstract simplicial complexes. As a result, we explicitly describe the relation between the Koszul simplicial complex of a monomial ideal and that of its polarization. Using the simplicial translation of depolarization we propose a way to reduce a simplicial complex to a smaller one with the same homology, and, in fact, the same strong homotopy type. This type of reduction, that can be interpreted as non-elementary collapse, can be used as a pre-process step for algorithms on simplicial complexes. We apply this methodology to the efficient computation of the Alexander dual of abstract simplicial complexes.
Wednesday 10
Monomial downsets and applications to Wilf’s conjecture
Shalom Eliahou
Université du Littoral Côte d'Opale, France
Abstract: A monomial downset is a set of monomials in $n$ commuting variables $x_1,\dots,x_n$ which is stable under taking divisors. A divset is a finite monomial downset. To a divset $X$ we associate a graph $G(X)$ whose edges are all pairs $\{u,v\}$ in $X \setminus \{1\}$ such that $uv \in X$, and whose vertices are the extremities of the edges. Divsets may be used as abstract models of Apéry sets of numerical semigroups, i.e. of cofinite submonoids $S$ of $\mathbb{N}$. In this talk, we will show how properties of $G(X)$, in particular its vertex-maximal matching number, can be used to verify Wilf’s conjecture (1978) for a large class of numerical semigroups.
Using divisibility relations to study square-free monomial ideals and their powers
Liana Sega
University of Missouri Kansas City, US
Abstract:A divisibility relation between the generators of a square-free monomial ideal formally encodes the situation when one generator divides the least common multiple of some other generators. For a set of divisibility relations, we study all square-free monomials satisfying the given relations, and also the powers of these ideals. In particular, we obtain effective bounds on the betti numbers of the powers, that can be made precise for low powers.
Homological invariants of neural codes
Rebecca Rebhuhn-Glanz
George Mason University, US
Abstract: The neural ideal was introduced by Curto, Itskov, et al in 2013 to study the spatial firing patterns of a set of neurons, turning problems in neuroscience and coding theory into algebraic questions. In this talk I will show how homological invariants can be applied to study and classify neural ideals. This work is joint with Hugh Geller, Alexandra Seceleanu, and Nora Youngs.
On the regularity index of the minimum distance function in projective nested cartesian codes
Cicero Carvalho
Universidade Federal de Uberlândia, Brazil
Abstract: Projective nested cartesian codes are a class of codes introduced in 2017 by Carvalho, Neumann and López, obtained by the evaluation of homogeneous polynomials of degree $d$ in certain points of a projective space. In that paper the authors presented a formula for the minimum distance for a special case of those codes. In this talk I would like to present a formula for the least degree $d$ from which the minimum distance of the code is one, which works for the general case. I will also present an indicator function for every point in the evaluation set, and relate these results to the $v$-number of a certain ideal. This is a joint work with Maria Vaz Pinto and Rafael Villarreal.
Recent Advances in Asymptotic Properties of Ideals and Their Extensions to Filtrations
Jonathan Toledo Toledo
INFOTEC, Mexico
Abstract: In this talk, I will present some of the main recent results on persistence phenomena and asymptotic properties of ideals, with particular emphasis on the strong persistence property and its connections with associated primes and stabilization behavior. I will also discuss several generalizations of these properties from powers of ideals to more general filtrations, highlighting both theoretical advances and illustrative examples.
The presentation will include a selection of problems and questions that I have addressed during my doctoral studies and throughout my subsequent research career, reflecting the evolution of this line of work and its current directions.
This work is deeply inspired by questions originally suggested by Rafael H. Villarreal during my master’s studies, which have guided a significant part of my research program. As a tribute to his influence, I will share recent progress and ongoing directions stemming from those problems, including new perspectives on persistence indices and structural properties of filtrations.
Betti numbers and sumsets
Philippe Gimenez
IMUVa, Universidad de Valladolid, Spain
Abstract:In this talk we will focus on the interaction between commutative algebra and additive combinatorics. Associated with a finite subset A of tuples of non-negative integers (and an infinite field), we have a projective variety and a (standard) homogeneous toric ideal. On the other hand, for any non-negative integer s, we can consider the s-fold iterated sumset of A, sA, which is the set formed by all sums of s elements in A. We will show in some specific cases (monomial curves, simplicial varieties either smooth or with a single singular point) how the syzygies of the toric ideal associated with A, and in particular its Castelnuovo-Mumford regularity, are related to some properties of the sumsets. This bridge between commutative algebra and additive combinatorics can be used in both directions to solve problems in one field using results from the other. This talk is based on joint work with Mario González-Sánchez and Ignacio García-Marco.
Resurgence of pairs and convex geometry of graded families of ideals
Tai Huy Ha
Tulane University, US
Abstract: Resurgence and asymptotic resurgence numbers were introduced in the study of containment between symbolic and ordinary powers of ideals. We shall give a generalization of these notions to non-containment thresholds associated to a pair of graded families of ideals. We shall discuss how to understand these non-containment thresholds from convex regions associated to the given graded families. This talk grows out of our attempt to understand Villarreal's result which computes the asymptotic resurgence of a squarefree monomial ideal from its Newton and symbolic polyhedrons.
Thursday 11
The edge ideals of rooted products of graphs
Naoki Terai
Okayama University, Japan
Abstract: In this talk we introduce the notion of 2-Cohen-Macaulay property with respect to a specific vertex and use it to characterize Cohen-Macaulay property of the rooted products of graphs.
This is based on a joint work with Y. Muta.
Geometric vertex decomposition and $F$-splitting
Elisa Gorla
University of Neuchâtel, Swiss
Abstract: Geometric vertex decomposition was introduced by Knutson, Miller, and Yong while studying the Groebner geometry of matrix Schubert varieties. It extends the idea of vertex decomposition for simplicial complexes to the setting of algebraic varieties and polynomial ideals and provides an inductive strategy for reducing questions to smaller or simpler pieces. Frobenius splitting is a technique used to study singularities in positive characteristic. Knutson showed that Frobenius splittings descend through geometric vertex decomposition to the link and the deletion, the two pieces in which the original object is decomposed. In this talk, I define and discuss these ideas and their relations and I present a joint result with De Negri, Klein, Rajchgot, and Seccia, which establishes a partial converse of the result of Knutson.
Complementary Multigraded Betti Numbers
Sara Faridi
Dalhousie University, Canada
Abstract:In this talk we describe the notion of complementary multigraded betti numbers for a monomial ideal in a polynomial ring. We show how complementation in the lcm lattice can affect the betti table and the multigraded minimal free resolution of a monomial ideal. We consider instances where a nontrivial betti numbers leads to other nontrivial ones. This talk will be based on joint work with Mayada Shahada, as well as ongoing work with Dharm Veer and Volkmar Welker.
Thirty years of Tsfasman-Boguslavsky Conjecture
Sudhir R. Ghorpade
Indian Institute of Technology Bombay, India
Abstract: About thirty years ago, Tsfasman and Boguslavsky proposed a remarkable conjectural formula for the maximum number $e_r(d,m)$ of solutions in the $m$-dimensional projective space over the finite field with $q$ elements that a system of $r$ linearly independent homogeneous polynomials of degree $d$ can have.
The case $r=1$ of a projective hypersurface was considered much earlier where the corresponding question of Tsfasman was settled by Serre, and independently, Sørensen, in 1991. We now know that Tsfasman-Boguslavsky Conjecture holds in the affirmative if $r$ is at most $m+1$, but it can be false, in general.
Newer conjectures have been proposed and are known to hold in the affirmative in several cases. However, the general case remains open even though there have been some interesting results recently.
We will review these developments and outline several known results and the techniques used in proving them. Connections to coding theory would also be described. Much of this talk is based on joint works with Mrinmoy Datta and with Peter Beelen and Mrinmoy Datta.
Friday 12
From Code Equivalence to Polynomial Isomorphism
Martin Kreuzer
University of Passau, Germany
Abstract: For two linear $[n,k]_q$-codes $C,C'$, the Linear Code Equivalence (LCE) problem asks whether their generator matrices $G,G'$ satisfy $G' = A G D P$ for some matrix $A \in GL_k(\mathbb{F}_q)$, a diagonal matrix $D\in GL_n(\mathbb{F}_q)$, and a permutation matrix $P \in GL_n(\mathbb{F}_q)$.
In this talk we first reduce the LCE problem to the Point Set Equivalence (PSE) problem for finite point sets $X,X'$ in $\mathbb{P}^{k-1}$, i.e., to the question whether $X,X'$ differ only by a linear change of coordinates. Then we introduce the canonical module $\omega_R$ of the homogeneous coordinate ring of $X$, its canonical ideal $J_X$, and its doubling $D_X = R/J_X$. We show that the PSE problem is equivalent to an algebra isomorphism problem for $D_X, D_{X'}$.
Finally, we use the Macaulay inverse system of the Artinian Gorenstein algebra $D_X$ to reduce the problem to a Polynomial Isomorphism (PI) problem. This reduction is exponential in general, but polynomial time for "good" codes. Moreover, in the case of iso-dual codes, we obtain a polynomial time reduction of the LCE search problem to the PI search problem in degree 3.
Weight polynomials of codes and matroids
Trygve Johnsen
UiT – The Arctic University of Norway, Norway
Abstract: An interesting issue when working with linear codes $C$ is to find how many codewords there are of each weight (the weight distribution). A more ambitious goal is to determine the number $A_w^{(r)}$ of subcodes of dimension $r$ and support weight $w$, for each possible $w$ and $r$. This is called finding the higher weight spectra. As a tool for determining these spectra one may first study the extended codes $C_m=C\otimes_{\mathbf{F_q}}\mathbf{F_{q^m}}$. and finding the weight distribution of all these codes. A conversion formula will then give the higher weight spectra.
It turns out that the number of codewords of $C_m$ of weight $w$ is $P_w(q^m)$, where $P_w(Z)$ is a polynomial in degree at most the dimension of the code, and that determining these finitely many polynomials of bounded degree thus gives the weight distributions for all the infinitely many extension codes simultaneously. The $P_w(C)$ are called the generalized weight polynomials of the code $C$.
In this note we will describe the generalized weight polynomials in a more generalized setting, for matroids. In a mostly expository talk we will describe different aspects of, and approaches to, weight polynomials, and various ways of how to find them.
An algebraic approach to Coding Theory: Some contributions from professor Rafael H. Villarreal
Manuel González Sarabia
UPIITA-IPN, Mexico
Abstract: In this talk we analyze the main contributions from professor Rafael H. Villarreal in the study of some algebraic tools to solve some problems in Coding Theory.
Asymptotic Resurgence of Ideals arising from Matroids
Louiza Fouli
New Mexico State University, US
Abstract: In this talk, we consider the Stanley-Reisner ideal and the facet ideal of the independence complex of a matroid. We will discuss some general results about the asymptotic resurgence of these ideals and show that for certain classes of matroids we can obtain exact formulas. This is joint work with Michael DiPasquale and Arvind Kumar.
Complete Intersection Toric Ideals of Graphs and Oriented Graphs
Lourdes Cruz González
UAM-Azcapotzalco, Mexico
Abstract: In this talk, I will present a survey of graphs and oriented graphs whose associated toric ideals are complete intersections.
We focus on the sensitivity of the complete intersection property to edge orientation, specifically how the orientation of a single edge can preserve or destroy this condition. Motivated by this phenomenon, we discuss recent results and new perspectives for both simple and oriented graphs.
Distance ideals of graphs
Carlos Alejandro Alfaro
Banxico, Mexico
Abstract: Distance ideals are ideals over a polynomial ring defined from the minors of the distance matrix with variables in the diagonal. These ideals give us a new perspective to study subclasses of perfect graphs and new ways to extend the Graham-Lovasz-Pollak result that gives a formula of the determinant of the distance matrix for trees. We are going to talk about results that motivate their exploration and advances in some problems we are interested in.
Powers of Square-free Monomial Ideals and a New Approach to the Conforti-Cornuéjols Conjecture
Susan Morey
Texas State University, US
Abstract: For each positive integer $q$, there is an extremal ideal ${\mathcal E}_q$ with $q$ generators that provides information on properties and invariants of all square-free monomial ideals with $q$ generators. This talk will focus on how extremal ideals can be used to gain information about powers of square-free monomial ideals, including ordinary and symbolic powers, and integral closures of powers, and applications of these results. New sharp bounds for invariants related to powers, including symbolic defect, resurgence, and asymptotic resurgence, arise from extremal ideals and will be presented. In addition, the associated primes of extremal ideals map in a predictable way to the associated primes of a general square-free monomial ideal. This creates a new set of techniques that can be used to approach the conjecture of Conforti and Cornuéjols that the packing property of a clutter is equivalent to the max-flow-min-cut property, which based on work by Villarreal and his coauthors has been translated to a question about the equality of ordinary and symbolic powers.
This talk is based on joint work with T. Chau, A. Duval, S. Faridi, T. Hollenben, L. Şega.