Spring 2018

Speaker: Duncan Levear
Title: Hyperplane Arrangements via Finite Fields
Abstract: Hyperplane arrangements is a thriving area of combinatorics. In this talk, I will explain the basic questions and answers for some famous cases. Amazingly, one can deduce facts about certain hyperplane arrangements by considering the same arrangement over a finite field, thereby turning an infinite problem into a finite one. Such an approach is known as the finite field method; I will explain how it works and demonstrate some applications. This talk is sort of a continuation of my talk about Mobius Inversion, which will feature heavily and beautifully.

Speaker: Charles Stine
Title: Kirby Calculus
Abstract: I will present the Kirby moves for 4-manifolds and state the important results about them. After presenting some instructive examples, I will briefly discuss the applications of the Kirby moves in the proof of the h-cobordism theorem for smooth simply connected 4-manifolds.

Speaker: Abhishek Gupta
Title: An overview of class field theory
Abstract: Class field theory is a branch of algebraic number theory that studies abelian extensions of local fields and global fields and various arithmetic properties of such abelian extensions. We will explain the statements of the main theorems of class field theory and their importance.

Speaker: Eric Hanson
Title: An Introduction to Differential Privacy
Abstract: When releasing information about a dataset, it is important to protect the privacy of the individuals whose data it contains. This is made precise through the definition of differential privacy. In this talk, I will present an overview of this definition and some basic examples of how it can be applied. Our main example will be to describe a differentially private framework developed in a recent paper by Qardaji, Yang, and Li. This framework, known as PriView, allows for the private release of a large number of marginal contingency tables without obliterating data utility. There are no prerequisites for this talk other than a non-rigorous understanding of probability.

Speaker: Josh Eike
Title: Automatic Groups and Their Algorithmic Properties
Abstract: A presentation of a group by generators and relations is useful, but they can still be difficult to work with. For example, such a presentation is not sufficient to determine whether a word in terms of the generators is equal to the identity. An automatic structure can be regarded as a particularly good presentation of a group. They allow the word problem to not only be solved but solved in polynomial time. We'll give examples of automatic groups and talk about some of their useful properties.

Speaker: Job Rock
Title: Continuous Cluster Categories
Abstract: We’ll start with the original definition of cluster category (the finite discrete case) and the motivation for that construction. Then we’ll examine a previous construction of a continuous cluster category by Igusa and Todorov followed by a new construction. Along the way we’ll highlight the similarities and differences of these three constructions. There will be a lot of pictures!

Speaker: Matt Garcia
Title: Integrability in Categories
Abstract: The category of smooth manifolds can be embedded (in a "nice" way) into several larger categories which contain many objects that are not classically smooth manifolds. This allows for the reformulation of many results of classical differential geometry in a categorical context in such a way that (1) proofs are simplified due to the presence of "infinitesimal smooth manifolds" and (2) the results apply to a broader class of spaces, not just smooth manifolds in the usual sense. After a brief introduction to such embeddings of the category of smooth manifolds, we will focus on a version of the Frobenius theorem, a result that gives a geometric meaning to the problem of finding a maximal set of independent solutions to a special class of systems of first-order partial differential equations (PDE's). The properties of the category we are embedded in allow for a relatively simple proof. Time permitting, we will indicate some generalizations of the Frobenius theorem that apply to arbitrary systems of PDE's which are conjectured to hold in a similar categorical construction for real analytic manifolds rather than smooth.

Speaker: Angelica Deibel
Title: Some ideas about thickness and relative hyperbolicity in Coxeter groups
Abstract: I'm getting kind of tired of talking about my results (and you might be tired of hearing about them), so instead I'm going to talk about some new things I've been thinking about even though I haven't proved anything yet! I'll talk about the dichotomy between thick and relatively hyperbolic Coxeter groups, and a little bit about what questions I'm asking and why these properties might be relevant.

Speaker: Abhishek Gupta
Title: Galois Cohomology
Abstract: The Galois group of a field extension Gal(L/K) has a natural action on many abelian groups, for example L under addition, L* under multiplication etc. The cohomology groups of this action are useful in proving many theorems in algebraic number theory, for example, the main theorems in class field theory. In this talk, we shall recall the basics of group cohomology and will then talk about some basic theorems and examples of Galois cohomology.