Modular Character Sheaves on Lie Algebras

Summary

Developing a theory of character sheaves with modular coefficients on the Lie algebras of reductive groups.

Supervisor(s)

Professor Anthony Henderson

Research Location

School of Mathematics and Statistics

Program Type

PHD

Synopsis

Finite groups of Lie type are a very important class of groups, including practically all the finite simple groups. Their ordinary representation theory is reasonably well understood; a vital ingredient in the computation of the irreducible characters was Lusztig's theory of character sheaves on reductive algebraic groups. A major goal of modern research in this area is to find an analogous theory of character sheaves with modular coefficients, to answer basic unsolved questions about the modular representations of these groups. The aim of this project is to develop a theory of modular character sheaves on the Lie algebras of reductive groups. Here there are two possible constructions, by Fourier transform or horocycle transform, and the first step would be to prove that these give the same class of sheaves, generalizing the results of Lusztig and Mirkovic for ordinary character sheaves.

Additional Information

A good Honours degree (or equivalent) majoring in some algebraic area of pure mathematics is essential. Prior knowledge of group representation theory and algebraic geometry is highly desirable.

The School of Mathematics and Statistics has a large and active Pure Mathematics research group including (as of 2017) more than twenty ongoing academic and research staff members, more than ten fixed-term research staff members, and more than twenty HDR students. Seminar series in Algebra, Computational Algebra, Geometry and Topology, and Partial Differential Equations showcase the research of the group and its many visitors.

In addition to the academic requirements set out in the Science Postgraduate Handbook, you may be required to satisfy a number of inherent requirements to complete this degree. Example of inherent requirement may include:

  • Confidential disclosure and registration of a disability that may hinder your performance in your degree;
  • Confidential disclosure of a pre-existing or current medical condition that may hinder your performance in your degree (e.g. heart disease, pace-maker, significant immune suppression, diabetes, vertigo, etc.);
  • Ability to perform independently and/or with minimal supervision;
  • Ability to undertake certain physical tasks (e.g. heavy lifting);
  • Ability to undertake observatory, sensory and communication tasks;
  • Ability to spend time at remote sites (e.g. One Tree Island, Narrabri and Camden);
  • Ability to work in confined spaces or at heights;
  • Ability to operate heavy machinery (e.g. farming equipment);
  • Hold or acquire an Australian driver’s licence;
  • Hold a current scuba diving license;
  • Hold a current Working with Children Check;
  • Meet initial and ongoing immunisation requirements (e.g. Q-Fever, Vaccinia virus, Hepatitis, etc.)

You must consult with your nominated supervisor regarding any identified inherent requirements before completing your application.

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Keywords

Algebra; representation theory; algebraic groups; sheaves; geometry

Opportunity ID

The opportunity ID for this research opportunity is: 2272

Other opportunities with Professor Anthony Henderson