First of all I would like to thank the AMS blog on math blogs for their recent mention of this series. It is always nice for Anna and I to know that people are reading and enjoying this blog!
1. What do you say when trying to explain your work to non-mathematicians?
Consider the five Platonic solids: the tetrahedron, the cube, the octahedron, the dodecahedron and the icosahedron (Fig. 1).
It turns out that if you take the number of vertices of any one of them, subtract the number of edges and add the number of faces, the result is 2 for each shape. This number is called the Euler characteristic χ, and the reason it is the same for each of the Platonic solids is the fact that χ is a topological invariant. Each Platonic solid is topologically a sphere, with different decompositions. In fact, you don’t even need to take such a nice decomposition of the sphere to get the same number. You can also think of a sphere as a disc that is glued along its boundary to a single point (Fig. 2). χ = 1 point – 0 edges +1 face = 2 again.
However, if we decompose a shape that is topologically different, such as a torus (i.e. doughnut, Fig. 3), we get a different number for χ, in this case 0. A double torus (i.e. a surface with two holes) gives χ = – 2, and in general for a surface with g holes we have χ = 2 – 2 g.
We call a shape orientable if it has an inside and an outside, and all orientable surfaces are one of these spheres with g holes. You might notice that they all have an even Euler characteristic (thus in particular, they can only be cut up into an even number of bits). For surfaces that are not orientable, this is not necessarily true. Fig. 4 shows the real projective plane, a non-orientable surface with χ = 1.
In higher dimensions, surfaces generalise to (closed) manifolds: shapes that are smooth everywhere and bounded in their size (essentially just nice shapes). Both orientability and the Euler characteristic can be generalised to higher dimensional manifolds. Being orientable means that the manifold has an inside and an outside. To calculate χ, we again cut up our manifold into bits and take the number of points, subtract the number of edges, add the number of discs, subtract the number of three-dimensional balls, add the number of four-dimensional balls, et cetera.
One of the things I did in my research was to generalise the statement that orientable surfaces have an even Euler characteristic, for some notion of “k-orientable“, where 0-orientable means “not necessarily orientable” and 1-orientable just means orientable:
Theorem: A k-orientable manifold (i.e. nice shape) has an even Euler characteristic unless the dimension is a multiple of 2k+1.
2. What would your own personal description of ‘topology’ be?
Topology is the study of shapes and spaces, where we consider two shapes to be the same if they can be deformed into each other without punching any holes. There are many different areas of research in topology. An example of a research direction in topology that I’m interested in is trying to make a list of all manifolds of a given dimension, in the way that I said above that all orientable surfaces are topologically a sphere with g holes for some integer g. One way to approach this question is by asking how manifolds can be cut up along manifolds of one dimension less and, oppositely, what fundamental building blocks are needed to build all manifolds. For example, any orientable surface can be built from gluing the building blocks shown in Fig. 5 along circles. A torus for example can be glued as: cup – co-pants – pants – cap (Fig. 6).
Alternatively, we can consider the shapes in Fig. 5 from the point of view of the circles: the pair of pants (yes that’s what we call it!) is the surface swept out by two circles merging over time, as described in Thomas’s interview. We call this a cobordism: manifolds evolving over time. Any cobordism between circles can also be built from the building blocks in Fig. 5.
3. How does your work relate, if at all, to the Nobel Prize work?
My work relates to the Nobel Prize because it is related to Topological Quantum Field Theories (TQFT’s). These theories are the mathematical framework behind the physics that was awarded the Prize, but they are also very interesting tools in mathematics. TQFT’s link cobordisms of manifolds on the one hand to algebraic structures on the other hand. The pair of pants, for example, takes two circles and merges them to one. This can be related to multiplying two numbers: you start with two and you merge them into one!
It turns out that the entire structure of gluing cobordisms of circles can be encoded as an algebraic structure called a Frobenius algebra (more specifically, the circle is the algebra and the cobordisms are operations such as multiplying two elements). A TQFT (in two dimensions) is an assignment of a specific algebra to the circle.
I personally think mostly about the cobordism side of TQFT’s. How can we understand the ways in which manifolds can be constructed from the cobordism pieces? Which manifolds are cobordant, i.e. related to each other by a cobordism? (Two manifolds are cobordant precisely if we can build one from the other with surgeries, see Carmen’s interview.)
Linking back to the Euler characteristic, it turns out that the parity of χ (whether it is odd or even), is preserved by cobordisms. That means that whenever I can evolve one manifold into another over time, sweeping out a cobordism, the Euler characteristic is either odd or even for both. So in order for both odd and even Euler characteristics to appear for a given dimension of manifolds of a certain type (e.g. “k-orientable”), there need to be at least two manifolds of that type that have no cobordism between them.