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Professor Tim Austin - High-Dimensional Probability in Ergodic Theory and Representation Theory

Students successfully applying to this grant will be part of the CDT and follow the normal CDT structure. The first year research project will be under the direction of the project supervisor, forming part of the project detailed below.

About / Project Description:

Supervisor:

Description:

This project spans some connected areas of research in the ergodic theory and representation theory of countable groups. In both of these fields, we can try to understand general objects (measure-preserving actions or actions on Hilbert spaces) by approximating them with large finitary or finite-dimensional versions of those objects. This project explores when this is possible, with particular emphasis on the following:
  1. approximation of measure-preserving group actions on a probability space using permutations on finite sets, and the related notion of sofic entropy;
  2. analogous ideas for unitary representations of groups or C*-algebras, particularly the new notion of `almost periodic entropy';
  3. related combinatorial modes of convergence for sparse random graphs such as `local-global convergence';
  4. analogous modes of convergence for tuples of random matrices such as `strong convergence'.
A PhD student working on this project is likely to focus on questions from one of these areas, but will be encouraged to develop research-level expertise in the others as well.

Eligibility:

Level of study: PhD

Mode of study: Full Time

Fee status: Home / Overseas

Department/School: Mathematics Department

Other eligibility: the ideal applicant need not have direct experience with any of the specific topics 1–4 above, but will have taken some advanced courses containing relevant background. Examples could include ergodic theory, operator algebras, high-dimensional probability, or a course about `limit objects' for combinatorial structures.

Application Process:

To be considered for these scholarships and the CDT please apply in the Admission Portal indicating your interest in this Research Project and the name of the supervisor in your application.

Key Dates:

Application deadline: Ongoing

Application results date: April 2027

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