I mostly study the topology of manifolds, in particular the topology of manifolds of dimension 3 and 4, which is often called low dimensional topology. Having said that I am very interested in problems in high dimensional manifold topology as well. An important subset of manifold topology is knot theory, the theory of embeddings of one manifold in another.
I often use techniques of surgery theory, which is the main method we know for classifying manifolds. Research into manifold topology can involve methods from algebraic topology, differential topology, and point set topology, as well as from algebra and analysis. To name a few: Morse theory, algebraic K and L theory, cobordism theory, homotopy theory, decomposition space theory, gauge theory, Floer homology, L2 homology, group theory, and number theory.
As a graduate student at Glasgow, you would be part of the topology research group. Check out my colleagues here. We work closely together and run weekly learning seminars on a topic of current interest in topology.
The start of a PhD involves a lot of background reading to bring you from master's level to the cutting edge in your chosen topic. You will also take SMSTC courses during your first year. Here is a flavour of some of the questions, in no particular order, that I think would be fascinating to study.
For a given 4-manifold, what is the homotopy type of the space of diffeomorphisms of that 4-manifold? More concretely, what is the mapping class group? This is a very active area of research, for example see here and here. There are many open questions. For example the smooth mapping class group of the 4-sphere is unknown. Possible techniques include 4-dimensional gauge theory, high dimensional surgery theory, embedding calculus, and configuration space integrals.
What about in the topological category? For a fixed 4-manifold, can we compute the mapping class group, i.e. can we classify the self-homeomorphisms up to isotopy? There is an intermediate notion called pseudo-isotopy. The method for high dimensional manifolds, which we are trying to replicate in dimension 4, breaks up into studying the pseudo-isotopy classes, and then detecting the difference between pseudo-isotopy and isotopy. So far, this has only been completed in the simply connected case: see here and here.
A sphere embedded in 4-space is called a 2-knot. One can also embed a surface in 4-space. Which classification results can one prove for 2-knots, and more generally for knotted surfaces? The papers here and here made some recent progress. It is unknown whether every smooth 2-knot, whose complement has infinite cyclic fundamental group, is unknotted, and there are analogous questions for other surfaces. Existence questions for embeddings is also interesting. Given a 4-manifold and a second homology class, what is the minimal genus of a locally flat surface embedded in the 4-manifold, representing that homology class?
There are some beautiful known results of the classification of 4-manifolds up to homeomorphism. See Chapter 11 of this article for some statements and further references. For the case of 4-manifolds with a fixed boundary, classification results are only known for trivial and infinite cyclic fundamental groups. For nonorientable 4-manifolds, classifications are only known for fundamental group infinite cyclic and order two. Extensions to more situations would be very interesting.
We say that two 4-manifolds are stably diffeomorphic if one can take connected sum with copies of S2 x S2 to either so that the manifolds become diffeomorphic. This can be determined by algebraic invariants in some cases: see for example here, here, and here. The classification of 4-manifolds up to homotopy equivalence is also an interesting and ongoing area of research: see here and here for some recent progress. In both situations, further classifications would be very interesting. Our general aim is to classify 4-manifolds up to homeomorphism, homotopy eqiuvalence, and stable diffeomorphism, in as many situations as possible.
First, you have to apply to the graduate school, and it's best to have had some contact with your preferred supervisor. I recommend applying by the end of December, but there is no hard deadline to apply. The next stage is an informal interview with two professors. It might be advisable to visit Glasgow at some point, to see if you would like to live here. If you aren't lucky enough to have external funding, then you probably need a scholarship. These are highly competitive, so good luck! Please bear in mind that it can take a long time to hear back, as scholarships slowly become available.