Events

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.

Title: The chromatic index of strongly regular graphs (joint work with Sebastian M. Cioaba and Krystal Guo)

Abstract:

It follows from Vizing's theorem that the chromatic index (edge chromatic number) of a k-regular graph equals k or k+1, and that it equals k+1 if the graph has odd order.

We investigate the chromatic index of strongly regular graphs (SRGs) of even order.

Title: The Poset Conjecture: results, counterexamples, and open problems

Abstract:

In 1978, Neggers conjectured that a certain transform of the order polynomial of a partially ordered set (poset) has only real roots.

In the late 1980s, Stanley gave this to me as a thesis project, generalized to labelled posets.  For my thesis I proved the conclusion for series-parallel labelled posets and a bit more.  Br\"and\'en, and later Stembridge, found counterexamples to the conjecture in general.