Research Fellows Directory
Professor Chris Wendl
University College London
My research deals with a version of the following question: given a surface (e.g. imagine the surface of a sphere, or of a doughnut), is it the boundary of something? For all the examples that you can probably think up, the answer is yes: you can imagine for instance the sphere as a round container that you could treat as a bottle and fill with water, so that the sphere is the boundary of the region that the water fills. This is not true however for all surfaces, e.g. consider the "Klein bottle", which you can imagine as the result of taking an ordinary bottle and bending its opening around through the glass into the inside, then connecting the opening to the floor of the bottle by curving the floor upward. This surface is like a Moebius strip but closed in on itself: there is no distinction between inside and outside. It is therefore not a boundary of anything; indeed, if you try to imagine filling it up with water, you soon realize that it's not even clear what "filling it up" would mean.
My research considers questions like the above in a somewhat more elaborate context: instead of surfaces, I deal with "contact manifolds". These have more dimensions to them (a surface has only two!), so you can't picture them, but one can write down equations and rules that describe them. In this context, the question "is it a boundary?" can't be answered just by thinking about filling a bottle. The truly fascinating thing is that the answer is connected to questions which seem at first totally unrelated, e.g. questions from particle physics. In some sense, a contact manifold is a simplistic model of a system of particles, and the physical laws by which those particles interact determines whether that manifold can be a boundary or not. The most exciting thing to me in this subject is that no one truly understands that connection. One may hope that by understanding the abstract mathematics better, we may also gain deeper insights into the fundamental laws of nature.