Date 
Fall 2015 Schedule 
September 30 
Speaker: Cindy Tsang Title: The Inverse Galois Problem over Q Abstract: Given a finite group G, is there a Galois extension K/Q with Gal(K/Q)\simeq G? This is known as the Inverse Galois Problem and is conjectured to have a positive answer for any finite group G. In this talk, I will discuss some classical techniques used in solving this problem. 
October 7 
Speaker: Casey Blacker Title: The Weyl Character Formula Abstract: Let G be a compact, connected, semisimple Lie group, let T be a maximal torus in G, and let t and g be their respective Lie algebras. The Weyl character formula establishes a correspondence between the irreducible representations of G and a certain collection of dual vectors over t related to the root space decomposition of g. We will begin with a review of basic representation theory, prove the character orthogonality relations, introduce the global and infinitesimal weights of a representation, and finish with the statement of the character formula. 
October 14 
Speaker: Nadir Hajouji Title: Tarski's Transfer Principle and Hilbert's 17th Problem Abstract: Hilbert's 17th Problem asks, "Is every multivariate real polynomial
that is positive semidefinite (meaning it only evaluates to nonnegative real numbers) a sum
of squares of rational functions?" The question was answered by Artin using a special case of
a technique in model theory that would eventually be called Tarski's Transfer Principle.

October 21 
Speaker: Glen Frost Title: Conics Abstract: This talk will begin with a classification of conics and discuss some of the tools that projective space provides in the study of curves. Then I will give a naive proof that five points determine a conic. Then I will discuss a classic example of a variety: the twisted cubic. This will generalize into the rational normal curve and prove a second proof that five points determine a conic through considering a basic example of a parameter space. 
October 28 
Speaker: Yingying Wang Title: Dirichlet Lfunctions and its padic Analogy Abstract: This talk will be an introduction to Dirichlet Lfunction and its analogy padic L function. We will discuss some backgrounds and their arithmetic properties. 
November 4 
Speaker: Ebrahim Ebrahim Title: Subobject Classifiers Abstract: We will explore some ideas from chapter 1 of the sheaf theory text by Maclane and Moerdijk. The talk will be informal and very category theoretical, with no particular objective besides a tour of the interesting ideas in that chapter. Should be fun! Roughly: One can extract "truth values" for a logic out of certain categories by looking at the structure of subobjects in the category. Consider the category of sets as an example. Here subobjects would be subsets, and in order to fully determine a subset of a set X we may label each element of X by "member" or "nonmember." The set {true, false} could then be referred to as a "subobject classifier" for the category of sets. What are subobject classifiers for some other categories? 
November 11 
No Seminar 
November 18 
Speaker: David Wen Title: Blow ups, Blow downs, but never Blown over Abstract: This talk with be a quick (and maybe gritty) introduction to blow ups of points on an algebraic surface and some interesting consequences as a result. The blow up of a point on a surface replaces the point with a line, essentially "blowing up the point" on the surface, but done in a way that still encodes "useful" information around the point. It is a very algebraic geometric technique which means that has very strong geometric influences but is done very algebraically. Blow ups are usually the first interesting birational map seen in a course in algebraic geometry and leads to the study of birational geometry. 
November 25 
No Seminar 
December 2 
Speaker: Justin Kelz Title: Quaternionic Structures/Abstract Witt Rings Abstract: Ernst Witt formalized the algebraic theory of quadratic forms in the early 20th century, giving rise to the socalled Witt Ring of a Field. We will explore how fields give rise to Quaternionic Structures, and define an axiomatic scheme for a Quaternionic Structure in general. Furthermore, we will use a result by Murray Marshall to elucidate the power of Quaternionic Structures in computing the Witt Ring of a field, and to define, motivate, and explore Abstract Witt Rings. 