| Sun 7/12 | Mon 7/13 | Tue 7/14 | Wed 7/15 | Thu 7/16 | Fri 7/17 | |
|---|---|---|---|---|---|---|
| 8:45 | Announcements | |||||
| 9:00 | Invited talk | Invited talk | Invited talk | Invited talk | Invited talk | |
| 9:30 | ||||||
| 10:00 | Short talk | Short talk | Short talk | Short talk | Short talk | |
| 10:30 | Coffee Break | |||||
| 11:00 | Short talk | Short talk | Short talk | Short talk | Short talk | |
| 11:30 | Short talk | Short talk | Short talk | Short talk | Short talk | |
| 12:00 | Lunch | Lunch | Lunch | Lunch | Lunch | |
| 12:30 | ||||||
| 1:00 | ||||||
| 1:30 | ||||||
| 2:00 | Invited talk | Short talk | Excursions | Invited talk | Short talk | |
| 2:30 | Short talk | Short talk | ||||
| 3:00 | Short talk | Short talk | Short talk | Short talk | ||
| 3:30 | Coffee Break | Coffee Break | Coffee Break | |||
| 4:00 | Posters | Posters | Short talk | Short talk | ||
| 4:30 | Short talk | Invited talk | ||||
| 5:00 | Short talk | |||||
| 5:30 | Banquet | Closing Remarks | ||||
| 6:00 | Ice Cream Social | |||||
| 6:30 | Reception | |||||
| 7:00 | ||||||
| 7:30 | ||||||
| 8:00 | ||||||
Title: Crossing, or Not, in the Quasisymmetric World
Abstract: Stanley introduced quasisymmetric generating functions in 1972 in his work on $P$-partitions, as a refinement of Schur functions; and Gessel, in 1984, formalized the theory introducing the fundamental basis. In the early 2000s, with Aguiar and Sottile, we showed that quasisymmetric functions play a universal role as combinatorial invariant enumerators in the theory of combinatorial Hopf algebras, while with Aval and F. Bergeron, we studied the quasisymmetric coinvariant space.
In this talk I will describe recent joint work with Gagnon, Nadeau, Spink, and Tewari in which quasisymmetric polynomials arise naturally from noncrossing permutations and the geometry of the flag variety. In type A, the resulting spaces have cohomology given by the quasisymmetric coinvariant space, and Schubert-type polynomials given by forest polynomials. This picture is part of a broader story, extending to all Coxeter types, living between algebraic combinatorics, geometry, and Coxeter-Catalan combinatorics. For this talk, I will restrict my attention to type A and conclude with some open problems.
Title: Kostant’s Partition Function: Support, Structure, and Surprises
Abstract: Kostant’s partition function is a fundamental object at the crossroads of Lie theory and algebraic combinatorics. Arising in Kostant’s weight multiplicity formula, it counts the number of ways a weight can be expressed as a nonnegative integral combination of positive roots, and thus encodes structure in representations of semisimple Lie algebras. In this talk, we survey a program developing combinatorial and structural perspectives on this function and, in particular, on the support of Kostant’s multiplicity formula. We highlight explicit descriptions in adjoint representations and structural results showing that the support forms an order ideal in the weak Bruhat order. These developments reveal surprising connections to Fibonacci-type phenomena, lattice and polyhedral models, and to multiplex juggling, positioning Kostant’s partition function as a unifying object in algebraic combinatorics.
Title: Face enumeration meets Ehrhart theory
Abstract: We study the Ehrhart $h^\ast$-polynomial of (the boundary of) a lattice polytope via regular unimodular triangulations and Gröbner degenerations of toric ideals. This allows us to connect the boundary $h^\ast$-polynomial to the $h$-polynomial of any regular unimodular triangulations, in analogy to the classical Betke-McMullen Theorem.
Providing a direct link between Ehrhart theory and the face enumeration of simplicial complexes, we transfer structural results from the theory of simplicial polytopes to the setting of lattice polytopes. In particular, we derive general Dehn-Sommerville-type relations between $h^\ast(P)$ and $h^\ast(\partial P)$. Under the additional assumption of $\partial P$ admitting a regular unimodular triangulation, we recover old and prove new characterization results concerning symmetry or unimodality, as well as upper and lower bounds for coefficient-wise differences within $h^\ast(P)$. This is joint work with Steffen Schlie.
Title: Lusztig’s q-weight multiplicities and their refinements
Abstract: Kostka polynomials encode graded weight multiplicities in type A and admit rich combinatorial models via tableaux. Their refinements, known as Catalan functions, admit a recursive structure via catabolism, introduced by Alain Lascoux and further developed in recent work of Blasiak-Morse-Pun-Summers.
In this talk, we study $q$-weight multiplicities defined by George Lusztig, which generalize Kostka polynomials to all Lie types. We show that type C $q$-weight multiplicities, as well as type B $q$-weight multiplicities for spin weights, coincide with those arising from appropriate classical highest weights in Kirillov–Reshetikhin crystals; these results are joint work with HyunJae Choi and DongHyun Kim. We also introduce Catalan-type refinements of these multiplicities and conjecture their positivity, by investigating semistandard oscillating tableaux.
Title: Algebraic combinatorics in deformation cones
Abstract: We will explore the rich combinatorial structure of the deformation cones of the permutahedron and the associahedron. We will focus in particular on lattice properties and simplicity criteria for certain families of deformed permutahedra, which naturally gives rise to a variety of intriguing combinatorial objects and enumerative questions. The talk is based on several joint works with various coauthors (including arXiv:2007.01008, arXiv:2111.12387, arXiv:2305.08471, arXiv:2411.09832, arXiv:2503.15053), as well as some ongoing projects.
Title: Tropical ideals
Abstract: Tropical ideals are combinatorial objects that capture the behavior of collections of subsets of lattice points arising as the supports of all polynomials in an ideal. Their structure is governed by a sequence of compatible matroids and, even though most tropical ideals are not ‘realizable’ by a polynomial ideal, they nonetheless share and generalize many of the properties of usual ideals.
In this talk, I will introduce tropical ideals and survey some developments from the past decade concerning their algebraic structure and associated varieties. I will also present results concerning the matroids whose Bergman fans arise as varieties of tropical ideals, as well as recent developments toward a tropical analog of the Nullstellensatz.
Title: TBA
Abstract: TBA
Title: A memorial tribute to Adriano Garsia
Abstract: This memorial tribute honors the life and mathematical legacy of Adriano Garsia, who passed away in San Diego on October 6, 2024, at the age of 96. After beginning his career in ergodic theory and analysis, Garsia’s shift in the 1970s to algebraic combinatorics helped shape the modern development of the field, leaving a lasting influence across several areas of mathematics. Over the course of six decades, he supervised 36 Ph.D. students, weaving mentorship, collaboration, and personal connection into a vibrant community. This talk will reflect on some of his most influential results and highlight how mathematics, people, and shared meals formed an inseparable part of his world and left a truly lasting mark on all who knew him.