Abstracts

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Mini-courses

Petter Brändén (KTH, Stockholm)
Title: Totally nonnegative matrices and real-rooted polynomials in combinatorics

Abstract: Several important families of polynomials in combinatorics are known or conjectured to have only real zeros, or to exhibit other related positivity properties. We will survey some of these results through the lens of negative dependence, and present recent results that are derived using properties of totally nonnegative matrices. This is based on joint work with Leonardo Saud Maia Leite and Lorenzo Vecchi.

Francesco Brenti (Università di Roma Tor Vergata)
Title: Stable permutations, graphs, and groups

Abstract: In these talks I will study stable permutations using combinatorial and algebraic techniques. More precisely, I will define this class of permutations, point out their relationship with Cuntz algebra automorphisms and their reduced Weyl group, explain what Cuntz algebras are, and then survey the main results known about the construction, enumeration and characterization of these permutations, and about the construction of subgroups of the automorphism groups of Cuntz algebras. I will conclude with a number of open problems.

Invited talks

Bruno Benedetti (University of Miami)
Title: Skeleton Chordalities
Abstract: There are many possible ways to generalize the notion of chordality from graphs to simplicial complexes. We present a new one, called “Skeleton-E-chordality”, and try to argue it’s the best. This is joint work with Marta Pavelka (U Copenhagen).

Basile Coron (École Polytechnique, Paris)
Title: Uniformly shellable posets and real-rootedness
Abstract: In this talk we will discuss two parallel real-rootedness conjectures regarding two a priori unrelated polynomials: h-polynomials and Chow polynomials of posets. We will introduce a new class of posets, called UMEL-shellable posets, whose structure allow for a recursive resolution of the aforementioned conjectures, via interlacing methods. This class, although very restrictive, contains several famous families of posets, such as projective geometries, partition lattices of type A and B, and more generally all rank-uniform supersolvable geometric lattices. 

Sylvie Corteel (University of California, Berkeley)
Title: The extra slow Tamari lattice
Abstract: The Tamari lattice is a partially ordered set on Catalan structures. It was introduced in the 60s as a way to organize all the different ways of parenthesizing a product into a partially ordered structure. Since then this poset has appeared in many contexts in algebra, geometry and combinatorics. For example the Tamari lattice is the 1-skeleton of the associahedron and in the cluster algebra of finite type $A_{n}$, the clusters correspond to triangulations of a convex polygon and the exchange graph of these cluster mutations is the Tamari lattice. Here we define an extra slow Tamari lattice on algebraic objects called faithfully balanced modules and we show that this new poset contains the Tamari lattice as a sublattice and has similar properties. This is joint work with Jihyeug Jang (Geneva) and Baptiste Rognerud (Paris).

Alex Fink (Queen Mary University of London)
Title: The omega invariant of a matroid via h* vectors of section rings
Abstract: This talk is based on joint work with Chris Eur and Matt Larson, arXiv:2510.05207. Speyer's 2005 f-vector conjecture asserted the nonnegativity of the coefficients of a matroid invariant he defined, notably the leading coefficient omega(M).  When M is represented by a hyperplane arrangement, Larson identified omega(M) as the Euler characteristic of a certain anti-nef line bundle L^-1 on the wonderful compactification of the arrangement, up to a predictable sign.  Eur and Larson used this to show that omega(M) is the leading coefficient in the "h* polynomial" of L, i.e. the numerator of the Hilbert series of the total coordinate ring of the image of the map to projective space defined by L.  If the image is sufficiently nice (arithmetically Cohen-Macaulay), then the h* polynomial necessarily has nonnegative coefficients. The work I'll be talking about completes the argument, giving a proof that omega is positive for all matroids.  We show that wonderful varieties degenerate inside the permutahedral toric variety to a Cohen--Macaulay union of torus orbits controlled by a second matroid.  This union of torus orbits can be defined when M is not representable, and the needed Euler characteristic can be computed on the degeneration. 

Jacopo Gandini (Università di Bologna)
Title: On the partial order among the spherical orbital varieties
Abstract: Let g be a semisimple Lie algebra over the complex numbers, with a fixed Borel subalgebra with nilradical n. Given a nilpotent orbit in g, its intersection with n decomposes into several irreducible components, all having the same dimension. An orbital variety in n is by definition such an irreducible component, for some nilpotent orbit in g. Following Steinberg and Joseph, there is a surjective map from the Weyl group W of g to the set of the orbital varieties: the fibers of this map provide geometric analogues of the Kazhdan-Lusztig cells in W. Orbital varieties are naturally ordered by the inclusion of their closures. In the talk I will consider orbital varieties coming from the spherical nilpotent orbits, and I will discuss some questions and conjectures relating them to suitable fully commutative elements in the corresponding cells.

Hankyung Ko  (Uppsala Universitet)
Title: Coxeter combinatorics of double cosets
Abstract: I will introduce a theory of parabolic double cosets in a Coxeter group, partly based on a joint work with Ben Elias. In particular I will discuss double coset analogues of the reduced expressions, Coxeter-braid relations, and Bruhat orders. What has currently been studied is motivated largely by specific problems in categorical representation theory, and there appear to be a number of unexplored aspects natural from the combinatorial viewpoint. I aim to draw attention to that.

Shiyue Li (University of Michigan)
Title: TBA
Abstract: TBA

Lorenzo Vecchi (KTH, Stockholm)
Title: Existence of Kähler algebras with Chow polynomials as Hilbert series
Abstract: In recent years, the introduction of Chow rings of matroids has led to breakthroughs in matroid theory, including proofs of combinatorial inequalities such as the Heron--Rota--Welsh and Mason conjectures. Chow polynomials of posets generalize the Hilbert--Poincaré polynomial of the Chow ring of matroids. While no such ring-theoretic interpretation is known for general posets, several properties surprisingly still hold for the coefficients of this polynomial (including positivity, unimodality and palindromicity). This observation has led to the conjecture that Chow rings should also generalize to arbitrary posets, and (among other requirements) satisfy the Hard Lefschetz property. As a step towards this conjecture, we show that the Chow polynomial of any poset satisfies the numerical constraints imposed by the Hilbert series of Hard Lefschetz algebras. Consequently, there exists an algebra with the Hard Lefschetz property (and even the full Kähler package) that has the Chow polynomial of any poset as its Hilbert--Poincaré polynomial. Equivalently, this shows that the coefficients of the Chow polynomial are the h-vector of a simple polytope. This is a joint work with Adam Schweitzer.

Lorenzo Venturello (Università di Siena)
Title: The  h^*-polynomial of symmetric edge polytopes
Abstract: The symmetric edge polytope of a simple undirected graph is the convex hull of the subset of the type A roots corresponding to the edges of the graph. It is a lattice polytope whose combinatorial properties are completely determined from the combinatorics of the graph. A conjecture of Ohsugi and Tsuchiya predicts that the h^*-polynomial of any symmetric edge polytope, which is a palindromic polynomial with nonnegative coefficients, is gamma-nonnegative. After reviewing the state of the art of this conjecture, I will discuss new ideas which are part of a joint work with Giulia Codenotti and Roberto Riccardi. In particular, I will present a conjectural formula which would yield a decomposition of the h^*-polynomial of a symmetric edge polytope into a sum of palindromic polynomials with the same center. Moreover, using related ideas we obtain a sharp lower bound on the number of edges of a symmetric edge polytope in terms of simple graph invariants.