Combinatorics

Dave Morris (University of Lethbridge, Canada)

We will discuss graphs that have a unique hamiltonian cycle and are vertex-transitive, which means there is an automorphism that takes any vertex to any other vertex. Cycles are the only examples with finitely many vertices, but the situation is more interesting for infinite graphs. (Infinite graphs do not have "hamiltonian cycles," but there are natural analogues.) The case where the graph has only finitely many ends is not difficult, but we do not know whether there are examples with infinitely many ends. This is joint work in progress with Bobby Miraftab.

A classical question in polytope theory is whether an abstract polytope can be realized as a concrete convex object. Beyond dimension 3, there seems to be no concise answer to this question in general. In specific instances, answering the question in the negative is often done via “final polynomials” introduced by Bokowski and Sturmfels. This method involves finding a polynomial which, based on the structure of a polytope if realizable, must be simultaneously zero and positive, a clear contradiction. The search space for these polynomials is ideal of Grassmann-Plücker relations, which quickly becomes too large to efficiently search, and in most instances where this technique is used, additional assumptions on the structure of the desired polynomial are necessary.

In this talk, I will describe how by changing the search space, we are able to use linear programming to exhaustively search for similar polynomial certificates of non-realizability without any assumed structure. We will see that, perhaps surprisingly, this elementary strategy yields results that are competitive with more elaborate alternatives and allows us to prove non-realizability of several interesting polytopes.

I will give an overview of a few places where combinatorial structures have an interesting role to play in quantum field theory and which I have been involved in to varying degrees, from the Connes-Kreimer Hopf algebra and other renormalization Hopf algebras, to the combinatorics of Dyson-Schwinger equations and the graph theory of Feynman integrals.

For other events in this series see the quanTA events website.

An account of how a great Guy and his Brown coauthor created a 300-page book entitled "The Unity of Combinatorics" out of a 30-page paper from 1995 of the same name. The latter was an outline of a proposed lecture series, whose purpose was to feature the many connections within the vast area of combinatorics, thereby dispelling the then prevalent notion that combinatorics is just a bag of tricks. In writing the book, we took this notion and ran with it --- and how!

I'll talk about a number of these connections and some topics that seem almost magical, including Beatty sequences, Conway worms, games played with turtles instead of coins, and a way of viewing the non-negative integers as a field. The book begins with a child playing with colored blocks on his living-room rug and ends with a description of the Miracle Octad Generator. Finally, I'll talk about working with this gentlemanly giant of the world of numbers and sequences and patterns and games.

A triangulation of a surface is a way to divide it into a finite number of triangles. Let us pick a random triangulation uniformly among all those with a fixed size and genus. What can be said about the behaviour of these random geometric objects when the size gets large? We will investigate three different regimes: the planar case, the regime where the genus is not constrained, and the one where the genus is proportional to the size. Based on joint works with Baptiste Louf, Nicolas Curien and Bram Petri.