Below is a broad summary of the research area where I personally work, intended for curious undergraduate of Masters students.

If you are interested in doing a Senior Project, Master's Thesis or other research with me, drop me an email!
Generally I would like to work with you on:

  • Knots and links
  • Manifold topology
  • Any other kind of topology or homology theory
  • Groups, rings, modules and commutative algebra
  • If you have another topic you think might fit, ask me!

Low-dimensional geometric topology

My research is in a branch of pure/abstract mathematics called geometric topology. Topology is the study of which properties of a shape or space still remain when you remove the concept of distance from the setup. After this, the rules of the game roughly become that you are allowed to stretch/squash/expand/squeeze shapes and they are still considered the same shape as long as you don't rip them. I am interested in what it is still possible to say about the shape, given these rules.

This is a branch of pure mathematics, so there is no intended application from the outset. Pure mathematics has a long history of unintended, unexpected and very useful real-world applications. But this is not why we study it! We study the subject because it is fascinating and beautiful.

Another very appealing feature of geometric topology is the many different approaches one can take to study it. Almost all branches of pure mathematics have been brought to bear on my subject of interest: algebra, algebraic geometry, analysis, combinatorics, differential geometry, number theory... so picking your favourite tool is definitely an option in this arena.

Knots, links and concordance

A mathematical knot is more-or-less what you expect. You should imagine taking a piece of string and then tangling/tying it up into a knot - we then fuse the two ends of the string to make one continuous loop. A collection of several knots, possibly all linked and tangled up together, is called a link. Here is an excellent database of knots and their properties. Basic questions you can ask about a knot are: when you are handed two knots, how can you tell them apart? Conversely, is there some computation I can make to guarantee two knots are the same?

Why knots? Well they're pretty fun, but for me they are more of a means to an end. The basic objects of geometric topology are fairly well-behaved shapes called manifolds. Knots come into play for me as part of a general approach to studying manifolds by studying their submanifolds; i.e. the manifolds that live inside them. Particularly important are knots/links inside 3-dimensional manifolds (like the space we live in), and knotted surfaces inside 4-dimensional manifolds (like the space-time we live in). Ordinary knots are related to knotted surfaces by the concept of knot concordance, which is a 4-dimensional equivalence relation that can be put on the set of knots, and by the related notion of a slice knot. These interrelated ideas make knot-theory an essential tool in my toolkit.

You like dimension 4 - why is that?

For manifolds of dimension 5 and above (high-dimensions), remarkably successful classification results have been achieved using the systematic application of a technique called Surgery Theory. Contrasting this systematic approach, in low-dimensions techniques entirely specific to each dimension are often required. Even so, in the last few decades the solution of long-standing conjectures have led to enormous progress in dimension 3 in particular. This leaves dimension 4.

I am interested in this border between high- and low-dimensions. In the strange world of 4-dimensional topology even many basic classification problems still remain open, and I want to solve some of them! (For the experts: I tend to work in the category of topological manifolds, not smooth necessarily, using a combination of techniques from Surgery Theory imported from high-dimensions, together with techniques specific to dimension 4.)