Discrete Geometry & Combinatorics Seminar: Fall 2013
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Difference Sets, Projective
Geometry, and Symmetric Designs
| October 2
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.
Cluster Algebras, Triangulations, and
| October 9
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.
Growth and Nonamenability in
Product Replacement Graphs
(UCLA) | October 16
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 and the Construction of Surreal
Kyle Chapman | October 23
- 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.
| October 30
- 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.
Equidistant Points and Equiangular Lines
| November 6
- 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
Voronoi Diagrams and Fortune's Method
Josh Pankau | November 20
- 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
Real and Complex Reflection Groups
- 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.