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Categorification in Representation Theory (MIEMIETZV_U20SCIEP)

University of East Anglia - Mathematics

Qualification Type: PhD
Location: Norwich
Funding for: UK Students, EU Students
Funding amount: Funding is for 3.5 years and will be available to successful candidates who meets the UK Research Council eligibility criteria
Hours: Full Time
Placed On: 9th December 2019
Closes: 30th January 2020
Reference: MIEMIETZV_U20SCIEP

Supervisor: Dr Vanessa Miemietz

In the last 20 years, major progress in representation theory, low-dimensional topology and related areas has been made through the process of categorification. This refers to the process of considering higher categorical objects with extra layers of information, whose decategorified shadows (when forgetting the extra information) describe the original problem one is interested in solving.

In practice, this usually means that instead of considering a group or an algebra acting on a vector space by linear transformations, one studies a so-called 2-category (a gadget with objects, 1-morphisms and 2-morphisms – the 1-morphisms categorifying the elements in the group or algebra) acting on categories by functors.

In specific examples (e.g. 2-categories categorifying Lie algebra or Hecke algebras), major breakthroughs in long open problems have been achieved in this way, such as the computation of decomposition numbers for Hecke algebras, a proof of the Kazhdan-Lusztig conjectures for all Coxeter groups, and counterexamples to James’ conjecture. Inspired by this, there has been an ongoing effort to develop an abstract 2-representation theory that captures the successful examples and provides a framework for future ones.

In this project, the student will work on both questions from abstract 2-representation theory, and on applying those to examples relevant in classical representation theory.

Person Specification: 

Acceptable first degree in Mathematics.

The standard minimum entry requirement is 2:1.

Funding notes: 

This PhD project is in an EPSRC studentship competition within the Faculty of Science.  Funding is for 3.5 years and will be available to successful candidates who meets the UK Research Council eligibility criteria.  These requirements are detailed in the Research Council Training Grant Guide which can be found at https://www.ukri.org/files/legacy/publications/rcuk-training-grant-guide-pdf/ (see Annex 1 for Residential Guidelines for Research Council Studentships).  In most cases UK and EU nationals who have been ordinarily resident in the UK for 3 years prior to the start of the course are eligible for a full-award.  Other EU nationals may qualify for a fees only award.

   
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