## Departmental PhD Thesis Exam – Tyler Holden

Everyone is welcome to attend.  Refreshments will be served in the Math Lounge before the exam.

Wednesday, May 4, 2016
2:10 p.m.
BA6183

PhD Candidate:  Tyler Holden
Supervisor:  Lisa Jeffrey
Thesis title: Convexity and Cohomology of the Based Loop Group

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Abstract:

Let $K$ be a compact, connected, simply connected Lie group and define $\Omega K$ to be the loops on $K$. Let $\Omega_\text{alg}K$ be those loops which are the restriction of algebraic maps $\Bbb C^\times \to K_\Bbb C$.  Herein we establish two distinct but related results.  In the first, we demonstrate the module structure for various generalized abelian equivariant cohomology theories as applied to equivariantly stratified spaces. This result is applied to the algebraic based loop group for the cases of equivariant singular cohomology, $K$-theory, and complex cobordism cohomology.     Subsequently, we examine the image of the based loop group under the non-abelian moment map. We show that both the Kirwan and Duistermaat convexity theorems hold in this infinite dimensional setting.

A copy of the thesis can be found here: ThesisPreview