The first of my Defining topology through interviews series is with Thomas Wasserman, a PhD student at Oxford University studying mathematics. In particular his work is in the realm of topological quantum field theory.
1. What do you say when trying to explain your work to non-mathematicians?
Suppose you have a bunch of circles. Now imagine some of those circles merging together to one, and some of them splitting themselves in two. This merging and splitting over time sweeps out a surface, an example is shown below.
We view this surface as a kind of machine that takes your bunch of circles and spits out another bunch of circles. Observe that given two such machines, we can apply one after the other by sticking the first one on top of the second, as long as we have the same number of circles coming out of the first one as going into the second one.
As you’ll see from the other posts, us topologists have a lot of trouble telling circles apart. This means that if we want to say something about machines like this we need to do something clever. Let’s give each circle an extra bit of information and think of it (by flawed analogy) as a colour, chosen from some range of possibilities. On top of merging and splitting circles our machine can now also change the information associated to each circle, for example mix the colours if two circles merge.
If this new feature of our machine sounds completely arbitrary to you, you’re on the right track. In order to use this feature to study the original surface, it needs to be compatible with applying several machines in a row: the big machine formed from smaller machines should do exactly the same to the information as just applying the small machines in succession. I study how one can construct these extra features subject to these requirements.
2. What would your own personal description of ”topology” be?
Topology is the study of shapes. To me, that means studying shapes that you can imagine encountering in nature. As mathematicians, we have developed a way of describing these shapes that allows us to check our intuition about them and study them in a lot of detail. This is what makes topology such an interesting subject; it very explicitly creates a bridge between the intuition we have for the world around us and very abstract mathematical ideas.
3. How does your work relate, if at all, to the Nobel prize work?
My work fits very well into the way of thinking that the Nobel prize work has inspired in the last decades. In condensed matter physics, the area of physics that the Nobel prize work was done in, people spend a lot of time thinking about a type of (composite) particles called anyons. These anyons live on very thin slices of material and have the strange property that the way they behave is described only by their relative movement. That is, the only thing that matters is whether or not they merge, split or braid around each other. For example the image below shows a braiding followed by a merging.
Comparing this kind of behaviour to what we discussed in question one, we can relate them as follows. Suppose that instead of a colour, we assigned a type of particle to each circle. The circles merging then corresponds to the particles merging, and the circles splitting to the particle splitting. The particles winding around each other is encoded by the surface winding around itself, as in the figure below, and this relates to my work on surfaces such as these!