- A symmetric design is a certain type of incidence structure with additional properties. As is often the case, the existence of groups which act "nicely" on these designs is of interest to us. This algebraic question turns out to be equivalent to a combinatorial one: the existence of difference sets. Using projective geometry, we can construct families of difference sets in groups of certain orders.

- This talk will focus on the combinatoric and geometric structures from which cluster algebras arise. After being introduced by Fomin and Zelevinsky in the early 2000's, cluster algebras quickly become a very active area of mathematics with diverse applications. I will give elementary examples of cluster algebras leading to the full abstract definition. A few applications, results, and conjectures will also be discussed.

- The product replacement graph (PRG) of a group G is the set of generating k-tuples of G, with edges corresponding to Nielsen moves. It is conjectured that PRGs of infinite groups are nonamenable. We verify that PRGs have exponential growth when G has polynomial growth or exponential growth, and show that this also holds for a group of intermediate growth: the Grigorchuk group. We also provide some sufficient conditions for nonamenability of the PRG, which cover elementary amenable groups, linear groups, and hyperbolic groups.

- Combinatorial Games are a class of games with perfect information and deterministic sequential play. This includes classics like Go and Chess as well as many others such as Nim and Dominering. The analysis of these games is extremely simple from a game theoretic standpoint, but the data analysis is extremely intense. Attempts to categorize these games has led to the construction of a proper class called the Surreal Numbers which have a group structure. The talk will go through this construction with the motivations tied to combinatorial games.

- The concept of a ``random graph'' is fundamental to hundreds of questions in combinatorics and theoretical computer science. Consequently, understanding the properties that characterize random graphs is an interesting field of study: what kinds of properties are ``essential'' to random graphs? Given a sequence of graphs, is there a sensible way in which we can say that the elements of this sequence ``act like'' random graphs? In this talk, we will motivate and introduce the idea of a *quasirandom* graph, a concept that gives us a rigorous way to deal with the questions above. Some examples of these quasirandom graphs will be presented, and a series of results ranging from 1986 to the present day will be mentioned. No prerequisites beyond a minimal knowledge of graph theory and probability should be necessary to follow this talk.

- A collection of points is equidistant if all pairs of points in the set are the same distance apart. In euclidean space the maximal equidistant set is roughly the dimension of the space and the same is true in hyperbolic space and in spheres, but surprises happen in projective space. In this talk we discuss these surprises, prove a general upper bound and discuss three special configurations which achieve this upper bound.

- Given a finite collection S of points in the plane, each point in S generates a region in the plane that contains all points closest to it in relation to all other points of S. This subdivides the plane into convex polygonal regions, and such a subdivision is called a Voronoi Diagram. Voronoi Diagrams have many applications ranging from Astronomy to Zoology (see what I did there?), but are very tedious to calculate straight from the definition. In this talk I will discuss Fortune's Method, which is a more efficient way of building the Diagram, some properties of Voronoi Diagrams, and their relationship with Delaunay Triangulation.

- The study of regular polytopes in real two and three dimensional space is classical. Their higher-dimensional and complex cousins, along with the discrete groups generated by their symmetries, were studied and classified by Schläfli, Stott, Coxeter, Shephard, Todd, and Popov. In both the real and complex case, one can separate these groups into two categories: the finite and the infinite. Corresponding to each reflection group W (Coxeter group) is an Artin group, arising from the complexified hyperplane arrangement of W. The goal of this talk is to give a sample of what is known about these groups after discussing the history which led up to them.