Power operations and central maps in representation theory

The theme of this thesis is the novel application of techniques of algebraic topology (specifically, Steenrod's operations and Smith's localization theory) to representation theory (especially in the context of the geometric Satake equivalence). In Chapter 2, we use Steenrod's construction to prove that the quantum Coulomb branch is a Frobenius-constant quantization. We also demonstrate the corresponding result for the K-theoretic version of the quantum Coulomb branch. In Chapter 3, we develop the theory of parity sheaves with coefficients in the Tate spectrum, and use it to give a geometric construction of the Frobenius-contraction functor. In Chapter 4, we discuss some related results, including a geometric construction of the Frobenius twist functor, and also discuss future directions of research. The content of Chapter 3 is joint work with S. Leslie.

Department dc:contributor.department

Massachusetts Institute of Technology. Department of Mathematics.

Grantor dc:publisher

Massachusetts Institute of Technology

Year dc:date.issued

2018

Author dc:creator

• Lonergan, Gus (Gus C.)

Advisor dc:contributor.advisor

• Roman Bezrukavnikov.

• Mathematics.

Statement dc:rights

• MIT theses are protected by copyright. They may be viewed, downloaded, or printed from this source but further reproduction or distribution in any format is prohibited without written permission.

Licence dc:rights.uri

• http://dspace.mit.edu/handle/1721.1/7582

Language dc:language.iso

eng

Handle dc:identifier.uri

http://hdl.handle.net/1721.1/117873

OAI identifier oai:identifier

oai:dspace.mit.edu:1721.1/117873