Greenslopes Seminar

[Department of Mathematics]
Thursdays 11AM ENG E105
Co-Organizers: Dean Bisogno and Tanner Strunk


Schedule:

For the seminar attendance sheet, click here.
Abstracts appear below.

Date Speaker Title Advisor
Jan 25 Mark Blumstein Principal $G$-Bundles and Where to Find Them Dr. Jeanne Duflot
Feb 1
Feb 8
Feb 15
Feb 22
Mar 1
Mar 8
Mar 15 No Greenslopes - Spring Break
Mar 22
Mar 29
April 5
April 12
April 19
April 26
May 3

Abstracts:





01/25/2018 - Principal $G$-Bundles and Where to Find Them - Mark Blumstein
In this talk, we will consider situations where topology, algebraic/differential geometry, algebra, and group theory all get together to have a good time. Rather than a rigorous math talk, this talk will be more like a walking tour, where we walk to one stop, pause to try and get some sense of what's happening or maybe look at a few simple examples, and then keep moving. The topological and geometric objects we will consider are spaces with a group action. At first it might not be completely clear why one would care about abstract groups acting on abstract spaces, so I'll try to briefly motivate this by pointing out examples familiar to everyone, e.g. the general linear group (group of invertible matrices) acting on a vector space. I'll also mention an example from differential geometry known as a "Klein geometry." Throughout the talk, most of the applications we will consider come from the case where $G$ is a Lie group, or $G$ is discrete (e.g. finite.) The walking tour will then take a turn towards the homotopy theoretic, so weather permitting, a few stops we might make are: 1a) The classifying space of a group, and how its topology completely determines a purely algebraic construct called group cohomology (defined algebraically by the Ext functor on the coefficient module.) 1b) How $\mathbb{C}$-vector bundles (objects familiar from geometry) are classified by the universal principal bundle over the unitary group. 2) How representation theory fits into the picture above (Atiyah's work on topological K-theory started out as a study of representations of finite groups.) 3) How ordinary singular cohomology is a "representable functor" and the "universal object" is the classifying space of the coefficient group. Meaning that if you want to compute the singular cohomology of any topological space $X$ with coefficients in $\mathbb{Z}$, you can take a homotopy theoretic stance and consider just how $X$ relates to the classifying space of $\mathbb{Z}$...it's pretty cool that the cohomology of any topological space can be boiled down to how it compares to these classifying spaces (i.e. Eilenberg-Maclane spaces) Any one of these bullet points could take a whole semester to explain with any rigor, and I'm not an expert in all (or any?) of these points, so the intent is really just to bring awareness to some neat connections between different areas of math. I'll also mention that this talk is geared largely towards pure math, but maybe in a future talk I can talk about applications. e.g. Lie was initially interested in his transformation groups because he wanted an analogue for solutions of systems of differential equations to Galois' work on the group of symmetries for the roots of polynomial systems of equations (or so I read.)



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04/19/2018



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