Future students

Title: Coxeter combinatorics and spherical Schubert geometry

Abstract:

This talk will introduce spherical elements in a finite Coxeter system. These spherical elements are a generalization of Coxeter elements, that conjecturally, for Weyl groups, index Schubert varieties in the flag variety G/B that are spherical for the action of a Levi subgroup.

Title: Pretty Good State Transfer and Minimal Polynomials

Abstract:

We examine conditions for a pair of strongly cospectral vertices to have pretty good quantum state transfer in terms of minimal polynomials, and provide cases where pretty good state transfer can be ruled out.

Title: Factorization problems in complex reflection groups

Abstract:

The study of factorizations in the symmetric group is related to combinatorial objects like graphs embedded on surfaces and non-crossing partitions. We consider analogues for complex reflections groups of certain factorization problems of permutations first studied by Jackson, Schaeffer, Vassilieva and Bernardi.

Title: Efficient $(j,k)$-Domination

Abstract:

A function $f:V(G)\rightarrow\{0,\ldots,j\}$ is an efficient $(j,k)$-dominating function on $G$ if $\sum_{u\in N[v]}f(u)=k$ for all $v\in V(G)$ (here $N[v]=N(v)\cup\{v\}$ is the closed neighbourhood of $v$).

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.

Title: Filtering Grassmannian cohomology via k-Schur functions

Abstract:

This talk concerns the cohomology rings of complex Grassmannians. In 2003, Reiner and Tudose conjectured the form of the Hilbert series for certain subalgebras of these cohomology rings. We build on their work in two ways. First, we conjecture two natural bases for these subalgebras that would imply their conjecture using notions from the theory of k-Schur functions. Second we formulate an analogous conjecture for Lagrangian Grassmannians.

Joint work with Michael Feigen, Victor Reiner, and Ajmain Yamin.

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: 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.