Welcome to Combinatorics and Optimization

Title: Covers of the complete graph, equiangular lines, and the absolute bound

Abstract: We discuss equiangular lines and covers of the complete graph. The relationship between these objects dates to at least 1992, when Godsil and Hensel showed that any distance-regular antipodal cover of the complete graph (DRACKN) produces an ensemble of equi-isoclinic subspaces. In the case of a regular abelian DRACKN, this produces equiangular lines.
In the first part of the talk, we combine Godsil and Hensel's theorem with a 2017 observation of Waldron to explain why (with a single exception) there DO NOT exist regular abelian DRACKNs that would create d^2 equiangular lines in d-dimensional complex space, to achieve Gerzon's "absolute bound". This rules out a family of otherwise feasible DRACKN parameters that were enunciated in a 2016 paper of Coutinho, Godsil, Shirazi, and Zhan.
In the second part of the talk, we introduce "roux", a slight generalization of regular abelian DRACKNs. Roux are covers of the complete graph that produce equiangular lines. They come up naturally in the classification of doubly transitive lines, all of which arise from roux. Keeping hope alive for the present, we enunciate an infinite family of feasible roux parameters that would produce equiangular lines achieving Gerzon's absolute bound.
Based on joint work with Dustin Mixon.

Title: Sequential Contracts on Matroids

Abstract: First, I will talk about the well-known pandora’s box problem, and then I will introduce the generalization of pandora’s box to matroids. We call this problem “sequential contracts on matroids” and we will discuss some recent results about this problem. In particular, we will look at complexity results of the problem. This is joint work with Kanstantsin Pashkovich and Jacob Skitsko for my URA project in Spring 2024.

Title: Positivity of P-Recursive Sequences Satisfying Linear Recurrences

 Abstract: Whether it is decidable to determine when sequences satisfying linear recurrences with constant coefficients have all positive terms is a notorious problem in enumerative combinatorics that has essentially been open for around 90 years. Nevertheless, a "meta-principle" states that all such sequences arising from combinatorial counting problems belong to a special class where positivity (and more general asymptotic

behaviour) is decidable. Here we discuss new software for determining positivity for sequences satisfying linear recurrences with *polynomial* coefficients. Originally motivated by a novel approach to proving genus one solution uniqueness for the Canham model for biomembrane shapes, our algorithm combines rigorous numeric analytic continuation of functions satisfying linear ODEs with singularity analysis techniques from analytic combinatorics. The main talk will be presented using a live Sage Jupyter notebook, and audience members who have access to Sage with a recent version of the ore_algebra package installed (available at

https://github.com/mkauers/ore_algebra) will be able to follow along and play with the package during the talk.

There will be a pre-seminar presenting relevant background at the beginning graduate level starting at 1pm,