Monday, September 28, 2020 11:30 AM EDT

Title: Strongly cospectral vertices, Cayley graphs and other things

Speaker: Soffia Arnadottir
Affiliation: University of Waterloo
Zoom: Contact Soffia Arnadottir


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.

Friday, September 25, 2020 1:00 PM EDT

Title: Rota's Basis Conjecture holds asymptotically

Speaker: Alexey Pokrovskiy
Affiliation: Birkbeck, University of London
Zoom: Please email Emma Watson


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).

Thursday, September 24, 2020 1:00 PM EDT

Title: Sign variations and descents

Speaker: Aram Dermenjian
Affiliation: York University
Zoom: Contact Karen Yeats


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.

Monday, September 21, 2020 11:30 AM EDT

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

Speaker: Paul Terwilliger
Affiliation: University of Wisconsin
Zoom: Contact Soffia Arnadottir


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.

Friday, September 18, 2020 3:30 PM EDT

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

Speaker: Henry Wolkowicz
Affiliation: University of Waterloo
Zoom: Please email Emma Watson


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. 

Thursday, September 17, 2020 2:30 PM EDT

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

Speaker: Logan Crew
Affiliation: University of Waterloo
Zoom: Contact Karen Yeats


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.

Monday, September 14, 2020 4:00 PM EDT

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?

Monday, September 14, 2020 3:00 PM EDT

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

Speaker: Oliver Lorscheid
Affiliation: Instituto Nacional de Matemática Pura e Aplicada
Zoom: Contact Rose McCarty


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.

Monday, September 14, 2020 11:30 AM EDT

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

Speakers: Andriaherimanana Sarobidy Razafimahatratra & Mahsa Nasrollahi Shirazi
Affiliation: University of Regina
Zoom: Contact Soffia Arnadottir


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].

Friday, September 11, 2020 3:30 PM EDT

Title: Further progress towards Hadwiger's conjecture

Speaker: Luke Postle
Affiliation: University of Waterloo
Zoom: Please email Emma Watson.


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. 

Monday, September 7, 2020 11:30 AM EDT

Title: Projective Planes, Finite and Infinite

Speaker: Eric Moorhouse
Affiliation: University of Wyoming
Zoom: Contact Soffia Arnadottir


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. 

Friday, September 4, 2020 3:30 PM EDT

Title: Recent proximity results in integer linear programming

Speaker: Joseph Paat
Affiliation: UBC Sauder School of Business
Zoom: Please email Emma Watson.


We consider the proximity question in integer linear programming (ILP) --- Given a vector in a polyhedron, how close is the nearest integer vector? Proximity has been studied for decades with two influential results due to Cook et al. in 1986 and Eisenbrand and Weismantel in 2018. We derive new upper bounds on proximity using sparse integer solutions and mixed integer relaxations of the integer hull. When compared to previous bounds, these new bounds depend less on the dimensions of the constraint matrix and more on the data in the matrix.

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