Title: Algebraic hypergraph decompositions
Abstract:
We examine edge decompositions of complete uniform hypergraphs whose parts are permuted transitively by a permutation of the vertex set. We present an algebraicmethod for constructing such a hypergraph decomposition which is related tothe well known Paley graph construction. The construction is derived from apartition of the cosets of a group and a function from the power set of thevertex set into this group. Several examples will be constructed using different groups and we discuss the symmetry and other properties of the hypergraphs we obtain.

Title: Classifying Lattice Walks in the Quarter Plane
Abstract: The enumeration of walks in restricted regions is an elegant and active area in combinatorics. Recently, much work has been done attempting to classify these walks by analytic properties of their generating functions. We discuss this classication and its usefulness, with a focus on walks in the quarter plane using small steps. In particular, we recap previous results in this area and outline a method of classifying the final walks of this type not yet proven by other means.

Title: RamseyScript and Double Arithmetic Progressions
Abstract: Ramsey Theory is the study of very persistent structures: structures which
are monochromatic under any finite coloring of some set. For example, every
finite coloring of the naturals contains monochromatic solutions to the
equation x + y = z. Equivalently, there is a least integer N(r) such that
every r-coloring of {1, 2, ..., N(r)} has a monochromatic solution. There
are no good bounds for N(r) and exact values are only known for small r
through computational search.
We introduce the program RamseyScript, which does computational searches
for this and similar problems for integer colorings, sequences, words on
finite alphabets and permutations. It is operated through a simple script
language and obviates much single-purpose code.
We also discuss an open Ramsey problem: does every finite coloring of the
natural numbers contain double arithmetic progressions? We discuss the
history of the problem and present evidence toward a solution gathered
through RamseyScript.

Title: The Coxeter Group Project: A Progress Report
Abstract: If one forms a Cayley graph on the symmetric group using only transpositions
for the connection set, then the graph is bipartite. In 2006, Araki proved that if such a
Cayley graph is connected, then for any two vertices u and v in different parts, there is
a Hamilton path whose terminal vertices are u and v. Of course, the symmetric group is
a Coxeter group. The speaker has embarked on a project of extending Araki's Theorem
to Cayley graphs on other families of Coxeter groups. This talks is a summary of what
has been achieved so far.