Events - September 2020

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: Hard Combinatorial Problems, Doubly Nonnegative Relaxations, Facial
and Symmetry Reduction, and Alternating Direction Method of Multipliers

Abstract:

Semi-definite programming, SDP, relaxations have proven to be extremely successful both in theory and practice for many hard combinatorial problems. This is particularly true for the Max-Cut problem, where problems of dimension in the thousands have been solved to optimality. In contrast, the quadratic assignment problem, QAP, is an NP-hard problem where dimensions bigger than $30$ are still considered hard. SDP and in particular, the doubly nonnegative, DNN, relaxation have been successful in providing strong upper and lower bounds, and even solving many instances to optimality. 

The profound impact of early discovery, experimentation, and disruption through research and invention

Researchers today build on the knowledge and discoveries made by those who have come before them. How can today’s researchers light the early pathways and curiosities for the research breakthroughs of the future? How can we demonstrate the impact and potential of the yet-to-be known? And, what if any, role does academia, industry, the Faculty of Mathematics, and Canada play in increasing the discovery journey to these new frontiers?

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: 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.