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

This PhD project addresses the construction of topological invariants in low dimensional geometric topology, arising from combinatorial representational theory, and (possibly) also their applications to physics, in particular to mathematical models of topological phases of matter and topological quantum computing: Our framework for understanding, constructing and computing topological invariants is to decompose topological objects (e.g. knots, manifolds, etc.) into smaller "generating" pieces, and obtain global invariants by combining all of its local values.

The latter 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 so 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 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 ways chunks of topological objects can be combined along different directions, and possibly in different orders. This PhD project can pursue a number of 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 be opportunities to explore applications to modelling topological phases of matter and to the ensuing paradigms for topological quantum computing.