Depending on student interest and background I can supervise a variety of topics related to algebraic topology.

Arithmetic algebraic geometry

How many solutions does the equation x2+y2 = z2 have if x, y, z are taken from Z/pZ? How many square-free polynomials of degree n are there with coefficients in Z/pZ? Questions of this kind have deep and seemingly unexpected connections with the arithmetic and geometry of algebraic varieties.

Combinatorial and Geometric Group Theory

Homotopy groups are extremely difficult to compute, and even the homotopy groups of spheres are only understood through some small range of degrees.

Can one write a computer program to decide whether two given rings are isomorphic? Or whether a given group is the trivial group? Or whether a given CW complex is a sphere? These are questions on the inteface of logic, theoretical computer science, and mathematics with fascinating results and open problems.

Investigate the relation between group theory and repeating 'crystal' patterns in 1, 2, and 3 dimensions.

Clifford algebras and Clifford modules; Spin structures and Dirac operators, their geometric properties, and some examples, possibly including Witten's proof of the positive mass theorem.

Monoidal categories describe the "quantum symmetries" of 2-dimensional topological phases of matter. Recently, it's been realised that enriched monoidal categories provide a useful model for 2-dimensional phases at the boundary of a 3-dimension phase.

Higher arithmetic. The failure of unique factorization in generalized number systems. Integer and rational solutions to algebraic equations. Fermat's Last Theorem.

Explore the analogy between the Galois theory of fields and the theory of covering spaces in topology.

This project is to explore various new topological invariants using topological field theory and dualities arising from gauge theory and string theory

Why are the trace and the determinant of a matrix special, as opposed to some other random function of the matrix entries? Why should the discriminant $b^2-4ac$ play a fundamental role, and not, say $a^2+4bc$? One answer (among many) is that the trace, the determinant, and the discriminant are distinguished functions

Index theory for Lie groupoids

The moment when research mathematicians need to pay attention to progress in automated theorem proving is coming closer and closer!