Hopf Monoids - A Refresher
Abstract: This is an intentionally off-the-cuff talk, to simulate the answer I would give over coffee at a conference to the question, "So what are these things called Hopf monoids and what do you do with them?"
Hopf Algebras - Examples
Abstract: We are going to explore examples coming from group algebras, Lie algebras, symmetric functions, and posets. The goal is to see how Hopf algebras allow us to give a common framework for different situations. Title TBA
Shira Zerbib (Iowa State)
Colorful phenomena in discrete geometry and combinatorics via topological methods
Abstract: We will discuss two recent topological results and their applications to several different problems in discrete geometry and combinatorics involving colorful settings.
The first result is a polytopal-colorful generalization of the topological KKMS theorem due to Shapley. We apply this theorem to prove a colorful extension of the d-interval theorem of Tardos and Kaiser, as well as to provide a new proof to the colorful Caratheodory theorem due to Barany. Our theorem can be also applied to questions regarding fair-division of goods (e.g., multiple cakes) among a set of players. This is a joint work with Florian Frick.
The second result is a new topological lemma that is reminiscent of Sperner's lemma: instead of restricting the labels that can appear on each face of the simplex, our lemma considers labelings that enjoy a certain symmetry on the boundary of the simplex. We apply this to prove that the well-known envy-free division theorem of a cake is true even if the players are not assumed to prefer non-empty pieces, whenever the number of players is prime or equal to 4. This is joint with Frederic Meunier.
Bennet Goeckner (University of Washington)
Partition extenders and Simon's conjecture
Abstract: If a pure simplicial complex is partitionable, then its h-vector has a combinatorial interpretation in terms of any partitioning of the complex. Such an interpretation does not exist for non-partitionable complexes. Given a non-partitionable complex, we will construct a relative complex---called a partition extender---that allows us to write the h-vector of a non-partitionable complex as the difference of two h-vectors of partitionable complexes in a natural way. We will show that all pure complexes have partition extenders.
A similar notion can be defined for Cohen--Macaulay and shellable complexes. We will show precisely which complexes have Cohen--Macaulay extenders, and we will discuss a connection to a conjecture of Simon on the extendable shellability of uniform matroids. This is joint work with Joseph Doolittle and Alexander Lazar.
TBA (we may cancel today)
No seminar (Spring Break)
Jose Bastidas (Cornell)
No seminar (get ready for GPCC 2020!)
Laura Escobar (Washington U. in St. Louis)
No seminar (Stop Day)
For seminars from previous semesters, please see the KU Combinatorics Group page.
Last updated Mon 2/10/20