Description

Topology studies what characteristics of a space are invariant under smooth deformations that do not introduce tears and rips into the space. For instance, a mug is topologically equivalent to a doughnut, but different from a spoon (which is topologically equivalent to a plate).

More generally, closed two-dimensional orientable surfaces are classified topologically by their genus: how many "holes" they have. The mug and the doghnut both have one hole, so they are equivalent to each other, while the spoon and the plate have none, so they are equivalent to each other but not to the mug and doughnut.

These examples are clear to us, from the vantage point of a three dimensional perspective, but it is worth asking how would an ant confined to live on the two-dimensional surface measure whether there is a hole in the surface or not. A sufficiently mathematically sophisticated ant could use the Gauss-Bonnet theorem: $$ \frac{1}{2\pi}\int K = 2-2g $$ where \(K\) is the curvature of the surface (which can be defined purely in two dimensional terms), and \(g\) stands for the number of holes.

In fact, we can generalize the idea of the Gauss-Bonnet theorem to arbitrary dimensions: we can construct generalizations \(\hat{K}\) of the integral of the curvature \(\frac{1}{2\pi}\int K\) such that \(\hat{K}\) measures useful topological properties of the surface (a important class of generalizations are known as Topological Quantum Field Theories, or TQFTs for short). It turns out that many of the most interesting TQFTs also have beautiful and deep applications in modern physics.

During this project we will explore the links of physics to topology via the study of some of these higher dimensional generalizations. (A particularly important example, but not the only one, is Chern-Simons theory.) During the second half of the project you could explore one of the following areas:

• There is a natural generalization of the \(2-2g\) term to arbitrary dimensions, known as the Euler characteristic. Which property of the space is it measuring? And is there an analog of the Gauss-Bonnet theorem that computes it in terms of locally measurable quantities?
• The circle is the boundary of a two-dimensional space (the disc). In general, how would we know if some \(d\)-dimensional space can be the boundary of some other \(d+1\)-dimensional space? This can be detected by computing topological invariants of the \(d\) dimensional space.
• How do you know if a knot in three dimensions is actually knotted? It turns out that there is a beautiful way of detecting this is by using Chern-Simons theory, due to Witten.
• Imagine that you study the behaviour of some material in the presence of a current and a magnetic field. Surprisingly, in some situations the electrons carrying the current effective "split" into fractionally charged constituents. This is the Fractional Quantum Hall Effect, and Chern-Simons theory plays a fundamental role in our understanding of it.
• There is a proposed architecture for quantum computers using topological quantum field theories. How does this architecture implement the basic operations in quantum computing?

Pre-requisites and co-requisites

For the first two topics it would be optimal (but not required) to have taken Geometric Topology II and Topology III and it is required to be taking either Riemannian Geometry IV or General Relativity IV for the fourth year. For the rest of topics, which involve Chern-Simons theory more directly, it is required to have taken Mathematical Physics II (or alternatively Theoretical Physics II and III) and attending either Riemannian Geometry IV or General Relativity IV in the fourth year. In addition, for the last topic (Topological Quantum Computation) you should have taken Quantum Information III.

If you don't fulfill these requisites but are still interested please talk to me.

Reading material

• The Wikipedia pages linked above.
• A good set of lecture notes on the Fractional Quantum Hall effect by David Tong.
• A non-technical introduction to topological computation.
• Witten's paper on the relation between Chern-Simons theory and knot theory.