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.

Tarski's Transfer Principle says that if we have a prenex definition defined over a ring A, and R, S are two real closures of A, then the prenex definition holds over R if and only if it holds over S.

My talk will aim to explain what all of the words in the previous sentence mean, and show how they lead to a very pleasing solution to Hilbert's problem.

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 L-functions and its p-adic Analogy

Abstract: This talk will be an introduction to Dirichlet L-function and its analogy p-adic 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 so-called 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.