Winter 2014 Schedule

January 24

Speaker: Cindy Tsang

Title: Introduction to Supercharacters and Superclasses

Abstract: Supercharacter theory is a generalization of the classical character theory that arises in representation theory. In this talk, I will give a brief review of the classical theory, and then explain how one can generalize it to supercharacter theory. I will prove some basic results that are analogous to those in classical character theory and describe a way to construct supercharacter theory on a given group G. This talk should be very accessible if you have a little bit of background in representation/character theory.

January 31

No seminar

February 7

Speaker: Nathan Saritzky

Title: Schemes

Abstract: Schemes are the fundamental object of study in modern algebraic geometry. They have a reputation for being challenging to understand; why would anyone define a topological space out of the prime ideals of a commutative ring? The aim of this talk will be to convince you that this is a very natural thing to do. It's a notion that provides an incredibly deep connection between geometry and algebra.

February 14

Speaker: David Wen

Title: What IS a curve in projective space?

Abstract: Projective curves are fundamental objects in Algebraic Geometry and are defined as closed projective varieties whose irreducible components are of dimension 1. Even with the definition the question still arises: "What is it?" or a better question, of the geometric flavor, would be "What does it look like?" The goal of this talk is to describe the shape of projective curves and bridge the gap between the algebraic and geometric definitions of projective curves and of projective space itself.

February 21

Speaker: Yingying Wang

Title: Introduction to Universal Algebra

Abstract: Universal algebra studies algebraic systems (e.g. groups, rings, lattices). The algebraic systems we are familiar with can be considered as specific examples of \Omega-algebra in the language of universal algebra and we can study the common properties they share by studying the properties of \Omega-algebra. In this talk, I will give an introduction on \Omega-algebras (and its isomorphism theorems, free algebras), varieties(the class of all \Omega-algebras satisfying the same identities), and ultraproducts. I might also talk about its applications and relation to category theory by the end of the talk if there is enough time.

February 28

No Seminar

March 7

Speaker: Jason Murphy

Title: Elliptic Curves and Triangles

Abstract: I will be talking about similar things that Jordan brought up in his talk. Specifically, at last week's colloquium he most likely spoke about a family of elliptic curves that are associated to triangles. What I will be talking about is a particular invariant (called k^2) associated to these elliptic curves which can be regarded as a function on the upper half plane which is modular with respect to the congruence subgroup Gamma_0(3). We will prove a few of these facts about k^2 and discuss some of the implications they have for the family of triangles. There will also be unicorns, rainbows, and dragons so I promise that it will be an entertaining talk!

March 14

Speaker: Matt Tucker-Simmons

Title: Pontrjagin and Tannaka-Krein Duality

Abstract: We've all heard the maxim that a good way to study a group is via its representation theory. For (locally compact) abelian groups it's good enough to consider only the irreducible representations (characters) - this is Pontrjagin duality. For non-abelian groups the story is a little more complicated - for instance, the quaternion group Q and the dihedral group D4 have the same character tables, but are not isomorphic. However, for compact groups the Tannaka-Krein Duality Theorem asserts that we can recover a group from its category of representations. In this talk I will explain what extra structure the category G-Rep possesses (beyond the character table) that allows us to recover G. This talk will be mainly example-driven.