Plenary talks - Charlas plenarias

- The Pythagoras numbers of projective varieties. Mauricio Velasco, Universidad de los Andes. In this talk we will study the "Pythagoras number" of a projective variety, that is the smallest number of squares needed to express any sum-of-squares on $X$. We will find upper and lower bounds for this quantity and discuss classification theorems of some situations where our bound match. These results are joint work with Greg Blekherman, Rainer Sinn and Greg G. Smith.

- Patterns in Standard Young Tableaux. Sara Billey, University of Washington. Standard Young tableaux are fundamental in combinatorics, representation theory, and geometry. The major index statistic originally defined for permutations has been extended to tableaux and can be interpreted in each of these settings. We consider the probability distribution of the major index on standard tableaux of fixed partition shape chosen uniformly along with the corresponding generating function. We give an explicit hook length formula for all of the cumulants of these distributions using recent work of Chen--Wang--Wang. The cumulant formula allows us to classify all possible limit laws for any sequence of shapes in terms of a simple auxiliary statistic, aft, generalizing earlier results of Canfield--Janson--Zeilberger, Chen--Wang--Wang, and others. We show that any such sequence of distributions with aft approaching infinity is asymptotically normal. This leads to a series of questions concerning locations of zero coefficients, unimodality, and asymptotic estimates for the major index generating functions over all standard tableaux of a fixed shape. We settle the first of these by identifying mutation rules in terms of tableaux patterns leading to both a strong and weak poset structure on tableaux ranked by the major index. The classification of zeros can be interpreted as determining which irreducible representations of the symmetric group exist in each homogeneous components of the corresponding coinvariant algebra. We give conjectured answers concerning unimodality and asymptotic estimates. This talk is based on joint work with Matjaz Konvalinka and Joshua Swanson.

Contributed talks - Charlas contribuídas

- Properties of Binomial Edge Ideals. Ayah Almousa, Cornell University. Binomial edge ideals were introduced independently in by Herzog, et al in 2010 and by Ohtani in 2011 as a generalization of the ideal of 2-minors of a (2 x n)-matrix of indeterminates, which is a well-studied object in commutative algebra and algebraic geometry. I present an overview of the work of myself and others in using combinatorial methods to study important algebraic invariants of these ideals.

- Defectivity of families of full-dimensional point configurations. Christopher Borger, Otto-von-Guericke-Universität Magdeburg. The mixed discriminant of a family of point configurations encodes (if it is non-trivial) the conditions under which an associated system of Laurent polynomials has a multiple root. It presents a generalization of the $A$-discriminant of a single point configuration. However, the mixed discriminant may also be trivial and in this case we call the family of point configurations defective. Using a combinatorial criterion by Furukawa and Ito we give a necessary condition for defectivity of a family of full-dimensional configurations. This implies the conjecture by Cattani, Cueto, Dickenstein, Di Rocco and Sturmfels that a family of $n$ full-dimensional configurations in $\mathbb{Z}^n$ is defective if and only if the mixed volume of the convex hulls of its elements is $1$. This is joint work with Benjamin Nill. https://arxiv.org/abs/1801.07467

- On f-vectors and h-vectors of relative simplicial complexes. Giulia Codenotti, Freie Universitaet Berlin. A relative simplicial complex is a collection of sets given as the set theoretic difference of two simplicial complexes. Relative complexes played key roles in recent advances in algebraic and geometric combinatorics, but many questions about their general combinatorial structure are unanswered. I will start from the basic definitions, and introduce some of the questions and theorems about simplicial complexes, such as the Kruskal-Katona theorem, whose analogues are interesting to investigate in the relative setting. https://arxiv.org/abs/1711.02729

- Orbit-Counting in Polyhedra, Fundamental Domains, and Order Cones. Steven Collazos, University of Minnesota - Twin Cities. Motivated by the problem of counting isomorphism types of combinatorial objects that can be modeled by integer lattice points in a polytope, we attempt to construct a fundamental domain for the orbits that is polyhedral. We employ an algebraic-combinatorial method for constructing forest posets which determine linear inequalities bounding such a fundamental domain. This generalizes earlier work (unpublished) for constructing a tree poset to bound a fundamental domain that arises in enumerating a certain family of combinatorial designs. This is joint work with Matthias Beck (San Francisco State University) and Felix Breuer. http://math.sfsu.edu/beck/teach/masters/steven.pdf

- Pieri rules for Jack polynomials in the superspace. Jessica Gatica, Pontificia Universidad Católica de Chile. The Jack polynomials in superspace are symmetric polynomials depending in a parameter $\alpha$ and involving commuting and anticommuting variables. These polynomials generalize the symmetric Jack polynomials. In this talk we will show how to get Pieri rules for the Jack polynomials in superspace and we will recover Pieri rules for the symmetric case. https://arxiv.org/abs/1712.02416

- Involution words-a survey. Zachary Hamaker, University of Michigan. The combinatorics of Coxeter groups has long been a rich area of study with important applications to representation theory and geometry. Many of the key ideas in this realm have natural analogues when we restrict our attention to involutions in Coxeter groups. Based on pioneering work of Richardson and Springer, we will survey many results translated through the lens of involution words, which are the natural analog of reduced words for involutions. Some highlights include a new insertion algorithm, an intuitive combinatorial interpretation of the Chinese monoid and applications to the geometry of spherical varieties. These results only scratch the surface, and many open problems remain! This is joint work with Eric Marberg and Brendan Pawlowski. Contains results from 4,6,8, 9-11 at http://www-personal.umich.edu/~hamaker/research.html

- Private Information Retrieval from Coded Databases with Colluding Servers. Olga Kuznetsova, Aalto University, Finland. In coded Private Information Retrieval (PIR), a user wants to download a file from a coded database without revealing the identity of the file. We consider the setting where certain subsets of servers collude to deduce the requested file. These subsets form an abstract simplicial complex called the collusion pattern. In this talk we will present a general scheme for PIR. Our scheme builds on the star (or Schur) product of the storage code and another linear code selected by the user, called the retrieval code. We study the combinatorics of the relationship between collusion patterns, storage codes, and feasible retrieval codes. In particular, we construct optimal retrieval codes in several special cases: full collusion, disjoint colluding sets, and no collusion. This is work-in-progress towards my master's thesis. The talk will also cover the larger context of the PIR project by Algebra, NT and Applications Research Group at Aalto University. https://arxiv.org/abs/1611.02062 and https://arxiv.org/abs/1701.07636

- Cohomology of combinatorial species via hyperplane arrangements and Salvetti's complex. Fernando Martin, Universidad de Buenos Aires. In their monograph on combinatorial Hopf algebras [1], Marcelo Aguiar and Swapneel Mahajan suggested a cohomology theory for combinatorial species; its construction was explicited in Pedro Tamaroff's bachelor's thesis [2] and its computation carried out in various concrete examples. While computing the cohomology of the species of linear orders as a comonoid over itself (a case not considered in [2]), we identified the complexes computing the first page of an associated spectral sequence introduced in [2] with those computing the cellular homology of a CW-complex X with the homotopy type of the complement of an arrangement of hyperplanes known as the braid arrangement. On the other hand, given a (complexified) arrangement of hyperplanes there is a general mechanism, due to Mario Salvetti, to construct a CW-structure on a space which has the homotopy type of the complement of said arrangement. Salvetti's construction for the braid arrangement is precisely the CW-complex X we mentioned previously. In [1], the authors assign a species L(A) to every hyperplane arrangement A, and in the case where A is the braid arrangement, L(A) is the species of linear orders. In this talk we plan to show how these topological, combinatorial and homological ideas fit together, to hopefully generalize them to any species of the form L(A). (This is joint work with my PhD advisor, Mariano Suárez-Álvarez). [1] http://www.math.cornell.edu/~maguiar/a.pdf [2] http://cms.dm.uba.ar/academico/carreras/licenciatura/tesis/2017/Tamaroff.pdf

- Singular Hodge theory of matroids. Jacob Matherne, University of Massachusetts Amherst. To any matroid, I will associate a certain ring that, when the matroid is realizable, is the cohomology ring of a certain variety called the semi-wonderful model. I will show how the Hodge theory of this ring can conjecturally be used to establish the "top-heavy conjecture" of Dowling and Wilson from 1974, as well as the positivity of the Kazhdan-Lusztig polynomials of Elias, Proudfoot, and Wakefield. This is joint work with Tom Braden, June Huh, Nick Proudfoot, and Botong Wang. In progress.

- The Erdős-Ko-Rado property on pure flag simplicial complexes. Jorge Alberto Olarte, Freie Universität Berlin. We say that a pure simplicial complex has the pure Erdős-Ko-Rado property when the largest families of pairwise intersecting facets have a common vertex. In this talk we will discuss this property and conjecture several families of simplicial complexes to be pure EKR. We show that the most general of these families, namely pure flag simplicial complexes without boundary, satisfy the pure EKR property for dimensions 3 or less. This talk will be completely self contained so that is easy to follow without any prior knowledge. Joint work with Francisco Santos, Jonathan Spreer and Christian Stump. https://arxiv.org/pdf/1710.02518.pdf

- Geometry of Unipotent Polytopes. Julian David Pulido Castelblanco, Universidad Nacional de Colombia. A polytope is a geometric object that generalizes the concept of polyhedron to any dimension, an example of a family of polytopes are the Unipotent Polytopes, these polytopes were recently discovered and appear as part of the study of representations of unipotent subgroups of matrices on a finite field. In this paper we focus on a particular family of Unipotent Polytopes and present some geometrical and combinatorial aspects of the polytope, we describe vertices, edges and maximum faces of the polytope by means of partitions of the set $\{1,\dots, n \}$. We also calculate the Ehrhart polynomial and the volume for a particular case.

- Uniform Ideals. Simon Soto, Universidad de los Andes. Tropical ideals are homogeneous ideals in the semiring of tropical polynomials in which every graded piece is matroidal. These ideals were introduced by Maclagan and Rincón in their paper "Tropical Ideals", where it was shown that they satisfy several properties analogous to classical ideals. In this thesis we study a particular class of tropical ideals that we call uniform tropical ideals, in which all the matroids are direct sums of uniform matroids. We use the language of weighted partitions to describe the structure of these ideals, and we characterize the uniform tropical ideals that are saturated. https://arxiv.org/abs/1609.03838

- Seeking algebraic properties of an edge ideal in the structure of a graph. Azucena Tochimani Tiro, Universidad de Valladolid. Our aim here is to relate algebraic properties of the edge ideal I(G) associated to a graph G to combinatorial properties of G or other graphs associated to G. One wants to obtain information on the Betti numbers of I(G) in terms of the structure of the graph.

- Slack ideals of polytopes. Amy Wiebe, University of Washington. In this talk we discuss a new tool for studying the realization spaces of polytopes, namely the slack ideal associated to the polytope. These ideals were first introduced to study PSD rank of polytopes, and their structure also encodes other important polytopal properties, gives us a new way to understand important concepts such as projective uniqueness and realizability, and suggests connections with the study of other algebraic and combinatorial objects (toric ideals and graphs, for example). https://arxiv.org/abs/1708.04739