Abstracts

Mini-courses

Luca Moci: "An introduction to matroids over a ring and their basis polytopes"

Matroids are combinatorial structures satisfying axioms that encapsulate the notion of linear dependence for a list of vectors. Since its origins in the 1930s, matroid theory has enjoyed close connections to graph theory, optimization, and computer science; in recent years, however, it proved to have a rich interplay also with algebraic geometry, tropical geometry, and commutative algebra. In this view, a generalization of the notion of a matroid is desirable. Matroids over a ring R are structures satisfying axioms that encapsulate the "linear dependence" between elements in an R-module. When R is a Dedekind (or more generally Prüfer) ring, a theory of matroids over R can be fruitfully developed. In this case, a matroid over R can be described in terms of its localizations, that are matroids over valuation rings. These can be characterized by combinatorial inequalities, that are indeed the tropicalization of Plücker relations for the Grassmannian. With every (classical) matroid, one can associate the so-called basis polytope, deﬁned as the convex hull of the indicator functions of the bases of the matroid. This polytope, which was introduced by Gelfand, Goresky, MacPherson and Serganova, has several interesting properties: for instance, when the matroid can be realized by a matrix, the closure of its torus orbit in the Grassmannian is the toric variety associated with the basis polytope. Moreover the edges of the basis polytope have prescribed directions, corresponding to the elements of a root system of type A; conversely, every polytope satisfying those conditions arises as the basis polytope for some matroid. After recalling the basic notions on matroids and basis polytopes, we will introduce matroids over a ring. Then we will show that an analogue of the basis polytope can be defined for matroids over a valuation ring; moreover, a similar characterization in term of edge directions can be proved. This series of lectures is based on joint work with Alex Fink.

Volkmar Welker: "Random Simplicial Complexes and Stanley-Reisner Ideals"

In this series of lectures we decribe models for random simplicial complexes and associated Stanley-Reisner ideals. We review results by Ermann and Yang on the limit behavior of the Betti table for random Stanley-Reisner ideals in their model and explain results obtained by Babson and myself in more detail. Both results are achieved using the Hochster formula for Betti numbers of Stanley Reisner ideals. We explain which methods are applied by Erman/Yang and in our work with Babson to take advantage of the formula in a probabilistic setting. We also mention work by de Loera, Hosten, Krone and Silverstein on random (possibly non-squarefree) monomial ideals. Most of the mentioned results are obtained over the field with two elements. We explain challenges and methods when using different fields and rings.

Invited talks

Riccardo Biagioli: "Fully commutative elements in affine Coxeter groups"

An element of a Coxeter group W is fully commutative if any two of its reduced decompositions are related by a series of transpositions of adjacent commuting generators. These elements were extensively studied by Stembridge, in the finite case. They index naturally a basis of the generalized Temperley–Lieb algebra. In this talk, we give explicit descriptions of fully commutative elements when W is an affine Coxeter group. Using our characterizations we then enumerate these elements according to their Coxeter length, and we show that the corresponding generating function is ultimately periodic in each type.

Mats Boij: "Waring rank and SLP for annihilators of symmetric forms"

In a joint work with Migliore, Miró-Roig and Nagel, we study Gorenstein algebras defined by the annihilators of symmetric forms. In particular we obtain a complete picture of what happens for symmetric cubics in terms of the Strong Lefschetz Property (SLP) and the Waring rank. We also prove that complete symmetric forms of any degree give algebras satisfying the SLP and we provide a surprisingly short power sum decomposition for them.

Francesco Brenti: "Odd length in Weyl groups and odd Schubert varieties"

We define a new statistic on any Weyl group, which we call the odd length, and which reduces, for Weyl groups of types A, B, and D, to the statistics by the same name that have already been defined and studied in [Trans. Amer. Math. Soc., 361(2009), 4405-4436], [Electronic J. Combin., 20(2013), Paper 50], [Amer. J. Math., 136(2014), 501-⁠550], [Trans. Amer. Math. Soc., 369(2017), 7531-7547], and [Discrete Math, 340(2017), 2822-2833]. We show that, by results of the authors and J. Stembridge, the signed (by length) generating function of the odd length always factors explicitly, and we obtain multivariate analogues of these factorizations in types B and D. We then give a geometric interpretation of the odd length in type A as the dimension of a complex projective variety which one can associate to any permutation. This leads to a new combinatorial concept which we call the odd diagram of a permutation. If time allows, we will also propose and briefly study a definition of odd length for any Coxeter system. This is joint work with A. Carnevale.

Fabrizio Caselli: "Superconformal Lie algebras and a duality in the representations of linearly compact Lie superalgebras"

Motivated by a conjecture in the representation theory of the exceptional Lie superalgebra of type E(5,10) I will describe the construction of superconformal Lie algebras and show how this algebraic structure can be used to prove a useful duality among generalized Verma modules associated.

Giulia Codenotti: "Unimodular covers of 3-dimensional parallelepipeds and Cayley sums"

We will give an introduction to unimodular triangulations, covers and the integer decomposition property of lattice polytopes, and briefly touch on motivations for their study. We then show that the following special classes of lattice polytopes have unimodular covers, in dimension three: the class of parallelepipeds and the class of Cayley sums of two polytopes where one is a weak Minkowski summand of the other.

Alessio D'Ali: "Symmetric edge polytopes in combinatorics and beyond"

Symmetric edge polytopes are certain lattice polytopes arising from the data of a finite simple graph. In the present talk we introduce some of the pleasant combinatorial properties of these objects and explore some surprising connections to the Kuramoto synchronization model in physics and to the theory of finite metric spaces. If time permits, we will focus on the algebraic-combinatorial tools used to investigate several invariants of interest of these polytopes, e.g. facets and triangulations. This is joint work with E. Delucchi and M. Michałek.

Emanuele Delucchi: "Stanley-Reisner rings of symmetric simplicial complexes"

A classical theme in algebraic combinatorics is the study of face rings of finite simplicial complexes (named after Stanley and Reisner, two of the pioneers of this field). In this talk I will examine the case where the simplicial complexes at hand carry a group action and are allowed to be infinite. I will present the foundations of this generalized theory with a special focus on simplicial complexes associated to (semi)matroids, where the Cohen-Macaulay property can be characterized via an arithmetic invariant of the group action. Motivation for our work comes from the theory of arrangements in abelian Lie groups (e.g., toric and elliptic arrangements), and in particular from the quest of understanding numerical properties of the coefficients of characteristic polynomials and h-polynomials of arithmetic matroids. (Joint work with Alessio D'Alì.)

Gunnar Fløystad: "Polarizations of powers of the maximal ideal in polynomial rings"

In a polynomial ring S = k[x_1, \ldots, x_n] let I be an artinian monomial ideal in S. A polarization of I is a squarefree monomial ideal J in a larger polynomial ring T, such that S/I is a quotient of T/J by a regular sequence consisting of variable differences (so one gets from T to S by setting various variables of T to be equal). Well-known is the standard polarization of any monomial ideal in a polynomial ring. Consider the maximal ideal m = (x_1, \ldots, x_n) generated by all the variables. We describe combinatorially all the polarizations of powers of the maximal ideal m^n. We describe the Alexander duals of these polarizations. We show that the polarizations define (shellable) simplicial balls when we have three variables, and conjecture this to be true for any n.

Christian Krattenthaler: "Robinson-Schensted-Knuth correspondence, jeu de taquin, and growth diagrams"

The Robinson-Schensted correspondence, which sets up a bijection between permutations and pairs of standard Young tableaux, and its extension to words and semistandard tableaux due to Knuth play an important role in several contexts, including of course combinatorics, but also representation theory and commutative algebra. I want to promote the "modern" way to deal with these correspondences through Fomin's growth diagrams. I shall review all these ideas and then put them into action by establishing a recent conjecture of Sophie Burrill on a certain bijective relation between standard tableaux and oscillating tableaux.

Martina Juhnke-Kubitzke: "Bounds for Betti numbers of balanced complexes"

I discuss upper bounds for the graded Betti numbers of Stanley-Reisner rings of balanced simplicial complexes in various settings, including general balanced simplicial complexes, balanced Cohen-Macaulay complexes such as balanced normal pseudomanifolds. Moreover, I provide explicit formulas for the graded Betti numbers of cross-polytopal stacked spheres, and show that they only depend on the dimension and the number of vertices, rather than also the combinatorial type. This extends previous work of Satoshi Murai as well as Jürgen Herzog and Takayuki Hibi. This is joint work with Lorenzo Venturello.

Martina Lanini: "Twisted quadratic foldings of root systems"

In this talk I will discuss joint work with Kirill Zainoulline about twisted foldings of root systems. This construction generalises usual involutive foldings corresponding to automorphisms of Dynkin diagrams. Our motivating example is the non-split folding of the root system of type E8 which gives rise to the root system of type H4, appeared already in the Eighties in work of Lusztig. Using moment graph techniques we show that twisted quadratic foldings can be applied to obtain information about equivariant cohomology.

Antonio Macchia: "Binomial edge ideals of bipartite graphs"

Binomial edge ideals are ideals generated by binomials corresponding to the edges of a graph, naturally generalizing the ideals of 2-minors of a generic matrix with two rows. We give a combinatorial classification of Cohen-Macaulay binomial edge ideals of bipartite graphs providing an explicit construction in graph-theoretical terms. In the proof we use the dual graph of an ideal, showing in our setting the converse of Hartshorne’s Connectedness theorem. As a consequence, we prove for these ideals a Hirsch-type conjecture of Benedetti-Varbaro. This is a joint work with Davide Bolognini and Francesco Strazzanti.