Schedule

A Latin square of order n is an n x n grid filled with n symbols such that each symbol appears exactly once in each row and column. A transversal in a Latin square of order n is a collection of n cells such that each row, column and symbol appears exactly once in the collection.

Latin squares were introduced by Euler in the 1700s and he was interested in the question of when a Latin square decomposes fully into transversals.

We'll discuss some of the history of this question, including some recent joint work with Richard Montgomery.

Critical groups are groups associated with graphs. They are well-established in combinatorics; closely related to the graph Laplacian and arising in several contexts such as chip firing and parking functions. The critical group of a graph is finite and Abelian, and its order is the number of spanning trees in the graph, a fact equivalent to Kirchhoff’s Matrix--Tree Theorem.

What happens if we want to define critical groups for graphs embedded in surfaces, rather than for graphs in the abstract? 

In this talk I'll offer an answer to this question. I'll describe an analogue of the critical group for an embedded graph. We'll see how it relates to the classical critical groups, as well as to Chumtov's partial-duals, Bouchet's delta-matroids, and a Matrix--quasi-Tree Theorem of Macris and Pule, and describe how it arises through a chip-firing process on graphs in surfaces.

This is joint work with Criel Merino and Steven D. Noble.

The normal covering number \gamma(G) of a finite group G is the minimal size of a collection of proper subgroups whose conjugates cover the group. This definition is motivated by number theory and related to the concept of intersective polynomials. For the symmetric and alternating groups we will see how these numbers are closely connected to some elementary (as in "relating to basic concepts", not "easy") problems in additive combinatorics, and we will use this connection to better understand the asymptotics of \gamma(S_n) and \gamma(A_n) as n tends to infinity.

A foundational result in Ramsey theory appears in a paper of Erdős and Szekeres from 1935: any sequence of n^2 +1 distinct real numbers contains either an increasing or decreasing subsequence of length n+1. This simple result was one of the starting seeds for the development of Ramsey theory. We discuss a generalisation of the Erdős-Szekeres theorem to monotone arrays. We will show how to obtain improvements on a theorem proved by Fishburn and Graham 30 years ago thus confirming a conjecture posed by Bucic, Sudakov, and Tran. More precisely, we will show that a doubly exponential upper bound holds in all dimensions. Finally, we will see how this is intimately connected to a generalisation of Ramsey Theorem on the cartesian product of cliques.

Joint work with Antonio Girao and Alex Scott.

A mathematical structure is homogeneous if every isomorphism between two of its  finitely generated substructures can be extended to an automorphism of the whole. Fraı̈ssé's theorem says that if a countably infinite class of finitely generated structures obeys certain properties, and so is an amalgamation class, then there is a homogeneous countably infinite structure, its Fraı̈ssé limit, whose finitely generated substructures are precisely the elements of the amalgamation class. For example, the Fraı̈ssé limit of the class of all finite graphs is the well-known Rado graph.

We consider a Steiner triple system as a Steiner quasigroup so the substructures in this context are the subsystems; that is, we consider a Steiner triple system as a functional structure. Graphs are relational structures, so, although there are some analogies between homogeneous graphs and homogeneous Steiner triple systems, there are also significant differences. For example, a mathematical structure M is omega-categorical if and only if its automorphism group has finitely many orbits in its action on Mn. A homogenous relational structure is necessarily omega-categorical, but this is not the case for homogeneous functional structures.

Unlike the case for graphs, it is unknown whether it is possible to completely classify all homogeneous Steiner triple systems, or even the omega-categorical homogeneous Steiner triple systems. In this talk we will look at homogeneous, omega-categorical and some other related countably infinite Steiner triple systems and will consider future avenues of research that may help shed light on these difficult classification problems.