This dissertation consists of two distinct parts, the first concerning S^1-equivariant cohomology of loop spaces and the second concerning stability in partial differential equations.
In the first part of this dissertation, we study the existence of S^1-equivariant characteristic classes on certain natural infinite rank bundles over the loop space LM of a manifold M. We discuss the different S^1-equivariant cohomology theories in the literature and clarify their relationships. We attempt to use S^1-equivariant Chern-Weil techniques to construct S^1-equivariant characteristic classes. The main result is the construction of a sequence of S^1-equivariant characteristic classes on the total space of the bundles, but these classes do not descend to the base LM. In addition, we identify a class of bundles for which a single S^1-equivariant characteristic class does admit an S^1-equivariant Chern-Weil construction.
In the second part of this dissertation, we study the Maslov index as a tool to analyze stability of steady state solutions to a reaction-diffusion equation in one spatial dimension. We show that the path of unstable subspaces associated to this equation is governed by a matrix Riccati equation whose solution S develops singularities when changes in the Maslov index occur. Our main result proves that at these singularities the change in Maslov index equals the number of eigenvalues of S that increase to +∞ minus the number of eigenvalues that decrease to -∞.
| Identifer | oai:union.ndltd.org:bu.edu/oai:open.bu.edu:2144/19529 |
| Date | 07 November 2016 |
| Creators | McCauley, Thomas |
| Source Sets | Boston University |
| Language | en_US |
| Detected Language | English |
| Type | Thesis/Dissertation |
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