Future students

Title: Algebraic structure of the Hopf algebra of double posets

Abstract:

A Hopf algebra of double posets was introduced by Claudia Malvenuto and Christophe Reutenauer in 2011, motivated by the study of pictures of tableaux as defined by Zelevinsky. Starting from the correspondence between top-cones in the braid arrangement and partial orders, we investigate several properties of the Hopf algebra of double posets as the image of a Hopf monoid (via the Fock functor). In particular, we obtain a non-cancellative formula for the antipode. A description of the primitive space is also discussed.

Title: Feynman integrals as algebraic graph theory

Abstract:

I will overview how Feynman integrals should feel very familiar to algebraic graph theorists, and then say a few words about current directions of interest to me, particularly the c_2 invariant.

Title: A proof of the Erdős–Faber–Lovász conjecture

Title: A tale of two integrals

Abstract:

The Harish-Chandra/Itzykson-Zuber (HCIZ) and Brezin-Gross-Witten (BGW) integrals are a pair matrix integrals which play a prominent role in quantum field theory. Remarkably, these ubiquitous special functions are also significant from the perspective of algebraic combinatorics: they are generating functions for certain classes of Hurwitz numbers.

Title: Circuit QED Lattices: Synthetic Quantum Systems on Line Graphs

Abstract:

After two decades of development, superconducting circuits have emerged as a rich platform for quantum computation and simulation. Lattices of coplanar waveguide (CPW) resonators realize artificial photonic materials or photon-mediated spin models. Here I will highlight the special property that these lattice sites are deformable and allow for the implementation of devices with graph-like configurational flexibility. In particular, I will show that it is possible to create synthetic materials in which microwave photons experience negative curvature, which is impossible in conventional electronic materials [1].

Title: A combinatorial approach to Minkowski tensors of polytopes

Abstract:

Intrinsic volumes of a convex body provide scalar data (volume, surface area, Euler characteristic, etc. ) about the geometry of a convex body independent of the ambient space. Minkowski tensors are the tensor-valued generalization of intrinsic volumes. They provide more complex geometric information about a convex body, such as its shape, orientation, and more.

Title: Why are Hoffman's bounds for alpha and chi truly duals of each other?

Abstract:

Two of the most well known eigenvalue bounds for graph parameters look suspiciously related. Our goal in this talk is to confirm this suspicion by casting these bounds into a framework of semidefinite optimization that will give us almost for free a duality relation. As one should always expect in this context, we will see a connection to the Lovász theta function of a graph.

Title: Centrosymmetric Stochastic Matrices

Abstract:

We consider the convex subset of m by n stochastic matrices that are centrosymmetric: stochastic matrices that are symmetric under rotation by 180 degrees. We consider the extreme points and bases of this set, as well as several other parameters associated to such matrices. We provide examples illustrating the results throughout. This is joint work with Lei Cao (Nova Southeastern University) and Darian McLaren (University of Waterloo).

Title: The chromatic index of strongly regular graphs (joint work with Sebastian M. Cioaba and Krystal Guo)

Abstract:

It follows from Vizing's theorem that the chromatic index (edge chromatic number) of a k-regular graph equals k or k+1, and that it equals k+1 if the graph has odd order.

We investigate the chromatic index of strongly regular graphs (SRGs) of even order.

Title: The Poset Conjecture: results, counterexamples, and open problems

Abstract:

In 1978, Neggers conjectured that a certain transform of the order polynomial of a partially ordered set (poset) has only real roots.

In the late 1980s, Stanley gave this to me as a thesis project, generalized to labelled posets.  For my thesis I proved the conclusion for series-parallel labelled posets and a bit more.  Br\"and\'en, and later Stembridge, found counterexamples to the conjecture in general.