The second of my interviews for the Defining topology through interviews series in with Carmen Rovi. Carmen was a PhD student in mathematics at Edinburgh whilst I did my undergrad, and is now a Postdoctoral Fellow at Indiana University in Bloomington. She works in the exciting world of algebraic surgery theory!
1. What would your own personal description of “topology” be?
Topology studies the geometric properties of spaces that are preserved by continuous deformations. For a topologist any stretching or bending will not change the topological properties of an object, since we concentrate on much more essential aspects. This is why many times we hear the comment that a topologist can’t distinguish her coffee cup from her doughnut! Manifolds are the main examples of topological spaces, with the local properties of Euclidean space in an arbitrary dimension n. They are the higher dimensional analogs of curves and surfaces. For example, a circle is a one-dimensional manifold. Balloons and doughnuts are examples of two dimensional manifolds. A balloon cannot be deformed continuously into a doughnut, so we see that there are essential topological differences between them. An invariant of a topological space is a number or an algebraic structure such that topologically equivalent spaces have the same invariant. For example the essential topological difference between the balloon and the doughnut is calculated by the Euler characteristic, which is 2 for a balloon and 0 for a doughnut.
2. What do you say when trying to explain your work to non-mathematicians?
When I talk to non-mathematicians and I am asked this question I usually say “I do topology, which is an area of mathematics”. Unfortunately the conversation usually ends there when the other person says “mathematics… that’s difficult. I had a hard time in school with that”. Sometimes with a bit of luck they will want to know more and then I can explain that my work lies in a field called “surgery theory”. Like real surgeons, a mathematician doing surgery also does a lot of cutting and sewing. Of course like in real surgery, there are very precise rules for doing this! In the following figure, we perform a surgery on a 2-dimensional sphere. The first step is to cut out two patches (discs) from the sphere. This leaves two holes in the shape of two circles. We can then sew on a handle along those two circles. If we are careful on how we sew things together, we obtain a torus:
This is an example of how surgery can be used to describe the classification of surfaces: sphere, torus, torus with two holes … Of course, surgery can also be used in much more sophisticated situations and it is a very powerful tool for investigating classification questions in topology. The aim of classifying complicated topological objects gives rise to the description of invariants which give you certain information about the objects you are trying to classify. As we have seen with the work of the physics laureates this year, such invariants can then be applied in “real life” contexts and produce stunning results.
3. How does your work relate, if at all, to the Nobel prize work?
One of the questions that the physics laureates from this year were trying to understand was the electrical conductance in a layer of condensed matter. Experiments done in extremely low temperatures and under very powerful magnetic fields showed that the electrical conductance of the material assumed very precise values, which is extremely rare in this kind of experiments. They also noticed that the conductance changed in steps, when the change in the magnetic field was significant. The strange behaviour of the conductance changing stepwise made them think of topological invariants. Of course, there is a wide gap between having such an intuition and being able to formalise exactly which topological invariant is relevant in this situation. The key invariant for them is called “Chern numbers”, which for me is very surprising as this is an invariant which features frequently in my work.