Combinatorial and higher-categorical techniques in low dimensional topology and topological quantum field theory

University of Leeds

United Kingdom, Leeds

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University of Leeds

United Kingdom , Leeds

Key Facts

Program Level

PhD (Philosophy Doctorate)

Study Type

Full Time

Delivery

On Campus

Campuses

Main Site

Program Language

English

Start & Deadlines

Next Intake DeadlinesOctober-2026

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Next Intake October-2026

Combinatorial and higher-categorical techniques in low dimensional topology and topological quantum field theory

About

Summary

This PhD project addresses the construction of topological invariants in low dimensional geometric topology, arising from combinatorial representation theory, and (possibly) also their applications to physics, in particular to mathematical models of topological phases of matter and topological quantum computing.

Full description

Our framework for understanding, constructing and computing topological invariants is to decompose topological objects (e.g. knots, manifolds, knotted surfaces, etc.) into smaller "generating" pieces, and obtain global invariants by combining all of its local values. The latter local values cannot be arbitrarily given: typically, compatibility relations must be satisfied in order for the end-result of combining local values not to depend on the way objects were decomposed. Formulating topological invariants is here mainly inspected through determining how to map topological generators in order that appropriate relations hold between them: this is where both combinatorial representation theory as well as higher category theory do a great job for us.

Higher categories provide an effective framework for combining local values of invariants in order to obtain globally defined quantities. This is because higher categories have morphisms of several different dimensions, 0, 1, 2, etc, and moreover several different ways to combine those higher order morphisms, along different directions. This repertoire of different ways to compose higher order morphisms gives a way to book-keep the multitude of different ways chunks of topological objects can be combined along different directions, and possibly in different orders.

This PhD project can pursue a number of different paths for finding new topological invariants. From applying homotopy-theoretical techniques, to following a differential geometric framework, or even a purely combinatorial/algebraic flavor. There may also be opportunities to explore applications to modelling topological phases of matter and to the ensuing paradigms for topological quantum computing.

Requirements

Entry Requirements

Applicants to research degree programmes should normally have at least a first class or an upper second class British Bachelors Honours degree (or equivalent) in an appropriate discipline. The criteria for entry for some research degrees may be higher, for example, several faculties, also require a Masters degree. Applicants are advised to check with the relevant School prior to making an application. Applicants who are uncertain about the requirements for a particular research degree are advised to contact the School or Graduate School prior to making an application.

English Program Requirements

The minimum English language entry requirement for research postgraduate research study is an IELTS of 6.0 overall with at least 5.5 in each component (reading, writing, listening and speaking) or equivalent. The test must be dated within two years of the start date of the course in order to be valid. Some schools and faculties have a higher requirement.