Events

Title: Scaffolds

Abstract:

Building on the work of various authors who have used tensors in the study of association schemes and spin models, I propose the term "scaffold" for certain tensors that have been represented by what are sometimes called "star-triangle diagrams" in the literature. The main goal of the talk is to introduce and motivate these objects which somewhat resemble partition functions as they appear in combinatorics. (The exact definition is too cumbersome to include here.)

Title: Projective Planes, Finite and Infinite

Abstract:

A projective plane is a point-line incidence structure in which every pair of distinct points has a unique joining line, and every pair of distinct lines meets in a unique point. Equivalently (as described by its incidence graph), it is a bipartite graph of diameter 3 and girth 6. 

Title: Further progress towards Hadwiger's conjecture

Abstract:

In 1943, Hadwiger conjectured that every graph with no $K_t$ minor is $(t-1)$-colorable for every $t\ge 1$. In the 1980s, Kostochka and Thomason independently proved that every graph with no $K_t$ minor has average degree $O(t\sqrt{\log t})$ and hence is $O(t\sqrt{\log t})$-colorable. 

Title: Extensions of the Erdős-Ko-Rado theorem to 2-intersecting perfect matchings and 2-intersecting permutations

Abstract:

The Erdős-Ko-Rado (EKR) theorem is a classical result in extremal combinatorics. It states that if n and k are such that $n\geq 2k$, then any intersecting family F of k-subsets of [n] = {1,2,...,n} has size at most $\binom{n-1}{k-1}$. Moreover, if n>2k, then equality holds if and only if F is a canonical intersecting family; that is, $\bigcap_{A\in F}A = \{i\}$, for some i in [n].

Title: Foundations of Matroids without Large Uniform Minors, Part 2

Abstract:

In this talk, we take a look under the hood of last week’s talk by Matt Baker: we inspect the foundation of a matroid.

The first desired properties follow readily from its definition: the foundation represents the rescaling classes of the matroid and shows a functorial behaviour with respect to minors and dualization.

Title: Edge Deletion-Contraction in the Chromatic and Tutte Symmetric Functions

Abstract:

We consider symmetric function analogues of the chromatic and Tutte polynomials on graphs whose vertices have positive integer weights. We show that in this setting these functions admit edge deletion-contraction relations akin to those of the corresponding polynomials, and we use these relations to give enumerative and/or inductive proofs of properties of these functions.

Title: Leonard pairs, spin models, and distance-regular graphs

Abstract:

A Leonard pair is an ordered pair of diagonalizable linear maps on a finite-dimensional vector space, that each act on an eigenbasis for the other one in an irreducible tridiagonal fashion. In this talk we consider a type of Leonard pair, said to have spin.

Title: Sign variations and descents

Abstract:

In this talk we consider a poset structure on projective sign vectors. We show that the order complex of this poset is partitionable and give an interpretation of the h-vector using type B descents of the type D Coxeter group.

Title: Strongly cospectral vertices, Cayley graphs and other things

Abstract:

In this talk we will look at a connection between the number of pairwise strongly cospectral vertices in a translation graph (a Cayley graph of an abelian group) and the multiplicities of its eigenvalues. We will use this connection to give an upper bound on the number of pairwise strongly cospectral vertices in cubelike graphs.

Title: Total Dual Integrality for Convex, Semidefinite and Extended Formulations

Abstract:

Within the context of characterizations of exactness of convex relaxations of 0,1 integer programming problems, we present a notion of total dual integrality for Semidefinite Optimization Problems (SDPs), convex optimization problems and extended formulations of convex sets.