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August 2019
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Mr Richard Green

Doctor of Philosophy graduate

Honours graduate

 

Research seminars

TitleSeries
A Fourier-Mukai transform for invariant differential cohomologyDifferential Geometry Seminar

Doctoral thesis

A Fourier-Mukai transform for invariant differential cohomology

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Honours thesis

An introduction to equivariant Cohomology

This thesis aims to give an accessible introduction to equivariant cohomology, which, for our purposes, is an adaptation of the usual cohomology to the study of group actions on manifolds. From a purely mathematical perspective equivariant cohomology has dual uses, allowing the study of objects by first reducing to a quotient or set of fixed points, or by lifting a problem to a larger more convenient space, at times even allowing a move to a universal setting. The importance of symmetry, and hence group actions, has a long tradition in mathematical physics, from its use in classical mechanics in reducing a system of interest to a more manageable one, to the prominence of group theory in in quantum field theory, where, for example, particles are classified by their transformation properties. It is therefore not surprising that equiv- ariant cohomology is often a powerful tool when working at the interface of mathematics and physics. The theory of equivariant cohomology is intimately related to the theory of principal bundles and connections on them. We begin with a concise treatment of the relevant theory, including a crucial classification theorem for principal bundles. The following chapter constitutes an introduction to equivariant cohomology proper, where we construct the equivariant cohomology theory both from the topological and the algebraic perspective. In the final chapter we aim to indicate how equivariant cohomology can provide a bridge into a universal setting, with a brief discussion of characteristic classes, Chern-Weil theory, and the universal Thom form of Mathai and Quillen. We assume the reader knows basic differential geometry, for example that they know what a manifold is, and have seen a description of vector fields as sections of the tangent bundle. We also assume quite a strong familiarity with Lie groups, Lie algebras and group actions. All of this material is treated in a very accessible way in [15], which the author highly recommends, supplemented by [16] for additional details. A passing knowledge of algebraic topology is also necessary, however provided one has encountered de Rham cohomology and the notion of homotopy, the text should not be problematic.