Sophie Rehberg (FU Berlin)
Combinatorial reciprocity theorems for pruned inside-out polytopes and their application to generalized permutahedra and hypergraphs
Abstract:: Generalized permutahedra are a class of polytopes with many interesting combinatorial subclasses. We introduce pruned inside-out polytopes, a generalization of inside-out polytopes introduced by Beck-Zaslavsky (2006),which have many applications such as recovering the famous reciprocity result for graph colorings by Stanley. We study the integer point count of pruned inside-out polytopes by applying classical Ehrhart polynomials and Ehrhart-Macdonald reciprocity. This yields a geometric perspective on and a generalization of a combinatorial reciprocity theorem for generalized permutahedra by Aguiar-Ardila (2017) and Billera-Jia-Reiner (2009). Applying this reciprocity theorem to hypergraphic polytopes allows us to give an arguably simpler proof of a recent combinatorial reciprocity theorem for hypergraph colorings by Aval-Karaboghossian-Tanasa (2020). Our proof relies, aside from the reciprocity for generalized permutahedra, only on elementary geometric and combinatorial properties of hypergraphs and their associated polytopes. In this talk we will focus on the application to hypergraphs and their polytopes.
What is... Schubert calculus?
Abstract: I will give an introduction to Schubert calculus on Grassmannians and flag varieties, assuming that the audience has never heard of any of these words before. I will use my algebraic combinatorics lecture notes (sections 11.4 and 11.5).
What is... Schubert calculus? (part 2)
Comment: At the end of the talk, I got asked the question, "So, what is Schubert calculus?" Here is answer: it is about using combinatorics (thr symmetric group, partitions, tableaux) to model and solving geometry and topology problems involving linear subspaces and configurations of linear subspaces.
Marge Bayer will be speaking at the Graduate Online Combinatorics Colloquium on Wednesday, March 24.
Kevin Marshall will be speaking at the Graduate Online Combinatorics Colloquium on Wednesday, April 7.
Bryan Gillespie (Colorado State University)
Convexity in Ordered Matroids and the Generalized External Order
Abstract: In this talk we will use generalized matroid activity to construct the "external order" on the independent sets of an ordered matroid. The poset simultaneously generalizes the active orders on matroid bases first studied by Björner, as well as the discrete convexity theory of Las Vergnas and Edelman for oriented matroids. We will relate the defining operator of the external order to the theory of anti-exchange closure functions and convex geometries, and we will discuss a dual view of the order from the perspective of the class of greedoids called antimatroids. Time permitting, we will characterize the lattices isomorphic to the external order.
Julianne Vega (Kennesaw State University)
Triangulations, Order Polytopes and Generalized Snake Posets
Abstract: In this talk, we will introduce generalized snake posets \(Q\) and investigate circuits, flips, and regular triangulations of the corresponding order polytopes \(O(Q)\). In particular, we will begin with an introduction and some basic properties of \(O(Q)\) for certain \(Q\). By the end of the talk we will find ourselves immersed in the secondary polytope of \(O(Q)\) and considering "twists" which extend to an action on regular triangulations. This work is joint with the Kentucky Crew: Matias von Bell, Benjamin Braun, Derek Henely, Khrystyna Serhiyenko, Andrés Vindas Meléndez, and Martha Yip.
Byeongsu Yu (Texas A&M)
When is the quotient of a semigroup ring by a monomial ideal Cohen-Macaulay?
Abstract: We give a new combinatorial criterion for quotients of affine semigroup rings by monomial ideals to be Cohen-Macaulay, by computing the homology of finitely many polyhedral complexes. This provides a common generalization of well-known criteria for affine semigroup rings and monomial ideals in polynomial rings. This is joint work with Laura Matusevich.
Laura Escobar (Washington University in St. Louis)
Which Schubert varieties are Hessenberg varieties?
Abstract: Schubert varieties are subvarieties of the flag variety parametrized by permutations; they induce an important basis for the cohomology of the flag variety. Hessenberg varieties are also subvarieties of the flag variety with connections to both algebraic combinatorics and representation theory. I will discuss joint work with Martha Precup and John Shareshian in which we investigate which Schubert varieties in the full flag variety are Hessenberg varieties.
For seminars from previous semesters, please see the KU Combinatorics Group page.
Last updated Sat 5/1/21