Events

*Note different start time

Title: Real Chromatic Roots of Graphs

Abstract:

In February 1988, I arrived at C&O Waterloo for a postdoc with the late Ron Read. He handed me a paper by Beraha, Kahane and Weiss, and told me to apply it to determining the location of the complex roots of chromatic polynomials.  I’ve returned to the topic every few years since then, with varying degrees of success---some positive results, but still many open problems and conjectures remain.

Title: Chain decompositions for q,t-Catalan numbers

Abstract:

The q,t-Catalan numbers Cat_n(q,t) are polynomials in q and t that reduce to the ordinary Catalan numbers when q=t=1. These polynomials have important connections to representation theory, algebraic geometry, and symmetric functions. Work of Garsia, Haglund, and Haiman has given us combinatorial formulas for Cat_n(q,t) as sums of Dyck vectors weighted by area and dinv. This talk narrates our ongoing quest for a bijective proof of the notorious symmetry property Cat_n(q,t)=Cat_n(t,q).

Title: Flat Littlewood Polynomials Exist

Abstract:

In a Littlewood polynomial, all coefficients are either 1 or -1. Littlewood proved many beautiful theorems about these polynomials over his long life, and in his 1968 monograph he stated several influential conjectures about them. One of the most famous of these was inspired by a question of Erdos, who asked in 1957 whether there exist "flat" Littlewood polynomials of degree n, that is, with |P(z)| of order n^{1/2} for all (complex) z with |z| = 1. 

Title: Perfect Colorings of the Generalized Petersen Graphs

Abstract:

For a graph G and an integer m, a mapping T:V(G) -> {1,...,m} is called a perfect m-coloring with matrix A=(a_ij), i,j \in {1,...,m}, if it is surjective, and for all i,j, for every vertex of color i, the number of its neighbors of color j is equal to a_ij. There is another term for this concept in literature as "equitable partition". In this talk, we present some important results about enumerating parameter matrices of all perfect 2-colorings and perfect 3-colorings of generalized Petersen graphs GP(n,k).

Title: Graphs and combinatorics with a relationship to algebra, geometry and physics

Abstract:

Several algebraic and geometric structures are most naturally encoded via graphs. These include restrictions, such as trees, and decorations, such as planar graphs, ribbon graphs, bi-partite graphs (aka. hypergraphs), directed versions, etc. Particularly nice properties satisfy some kind of hereditary condition. This affords a dual perspective. Either as (nested) subsets and decomposition, or as composition, gluing locally. Both views relate to category theory, algebra, and combinatorics in terms of finite sets, cospans etc. We will give examples of these phenomena and provide a general background.

Title: The Erdos-Szekeres theorem 

Abstract: 

What lies at the intersection of combinatorial geometry, graph theory, order theory, analysis, and statistics? Why, only one of the most beautiful theorems you may have never heard of. Let me take you on a journey from early 20th century Budapest through to the heights of modern mathematics and show you why this classic result of Erdos and Szekeres is worth adding to your mathematical repertoire. Along the way we'll even see a proof so good it must come from The Book.

Title: The embezzlement of entanglement and its applications

Abstract:

Embezzlement of entanglement is the (impossible) task of producing an entangled state from a product state via a local change of basis, when a suitable *catalytic* entangled state is available.

The possibility to approximate this task was first observed by van Dam and Hayden in 2002.  Since then, the phenomenon is found to play crucial roles in many aspects of quantum information theory.  In this talk, we will discuss aspects of embezzlement and some applications (such as why quantum correlations do not form a closed set, and why there are nonlocal games that cannot be played optimally with a finite amount of entanglement, and why additive quantities cannot be more than asymptotically continuous).

Title: Lexicographic products, wreath products, and generalisations

Abstract:

I will present a history and overview of some of the work that has been done on the lexicographic product of graphs, and related generalisations. The focus of my talk will be on the automorphism groups of such graphs, and the relationship to the wreath product of permutation groups.

Title: Equivalences of Wilson loop diagrams

Abstract:

I will talk about Wilson loop diagrams, explain a bit about what they are, and some of the combinatorial questions that come out of them, with a focus on when they are equivalent.  This is joint work with Susama Agarwala and Zee Fryer.

Title: An approximate solution to a 2,079,471-point traveling salesman problem

Abstract:

Together with Keld Helsguan, we have found a TSP tour through the 3D positions of 2,079,471 stars. We discuss how linear programming allows us to prove the tour is at most a factor of 0.0000074 longer than an optimal solution. The talk will focus on the use of minimum cuts and GF(2) linear systems, to drive the cutting-plane method towards strong LP relaxations.