Events

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: Rota's Basis Conjecture holds asymptotically

Abstract:

Rota's Basis Conjecture is a well known problem, that states that for any collection of n bases in a rank n matroid, it is possible to decompose all the elements into n disjoint rainbow bases. Here an asymptotic version of this is will be discussed - that it is possible to find n − o(n) disjoint rainbow independent sets of size n − o(n).

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.

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: 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: Generalization bounds for rational self-supervised learning algorithms

Abstract:

The generalization gap of a learning algorithm is the expected difference between its performance on the training data and its performance on fresh unseen test samples.Modern deep learning algorithms typically have large generalization gaps, as they use more parameters than the size of their training set. Moreover the best known rigorous bounds on their generalization gap are often vacuous.

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: The Hepp bound of a matroid: flags, volumes and integrals

Abstract:

Invariants of combinatorial structures can be very useful tools that capture some specific characteristics, and repackage them in a meaningful way. For example, the famous Tutte polynomial of a matroid or graph tracks the rank statistics of its submatroids, which has many applications, and relations like contraction-deletion establish a very close connection between the algebraic structure of the invariant (e.g. Tutte polynomials) and the actual matroid itself.