My research focuses on moduli spaces. These are spaces that classify certain kinds of objects. For example, there is the moduli space of triangles, which classifies the triangles of high school geometry. This means that every point of this moduli space corresponds to exactly one triangle.
Gromov-Witten invariants are important in string theory, the physical theory according to which elementary particles are tiny strings, instead of points, as was believed since Newton. String theory is the best candidate for the so-called Theory of Everything, which unifies all physical theories, in particular Einstein's theory of gravity and quantum field theory. The theory of Gromov-Witten invariants is based on the moduli space of string world sheets, which classifies the differnent ways a string particle can move through spacetime.
My research focuses on some very subtle but fundamental aspects of moduli spaces. These spaces exhibit surprising phenomena when some of the objects they classify are more symmetric than others. Some triangles (like equilateral or isosceles ones) are more symmetric than most triangles and so the space of triangles has singularities at the corresponding points. The way to deal with such phenomena was discovered in the 1960s by Grothendieck, one of the most brilliant mathematicians of our time. For this purpose, Grothendieck introduced so-called stacks. Many questions about stacks are still open. My research addresses some of these.
Other anomalies can occur, if some of the objects to be classified are able to wiggle around in different ways than others. These anomalies do not occur for triangles, but they are a central difficulty in the study of the moduli space of string world sheets. Some of these world sheets might have many more ways to deform into others than the average world sheet. The way to deal with these phenomena, is the method of virtual fundamental classes. These have only been discovered within the last 15 years and my research was central in this effort.
Recently, it was discovered that Gromov-Witten invariants are related to so-called Donaldson-Thomas invariants. This connection is, in fact, still a conjecture, but it is very important, because Donaldson-Thomas invariants can explain many strange phenomena exhibited by Gromov-Witten invariants.
In my recent research I discovered something surprising about Donaldson-Thomas invariants. They behave like Euler characteristics. The Euler characteristic of a shape is a number which does not change, if the shape is deformed as if it was made of rubber. The surface of a sphere, for example, has Euler characteristic 2. The surface of a donut shape has Euler characteristic 0. The fact that Donaldson-Thomas invariants are certain kinds of Euler characteristics has important consequences, also for Gromov-Witten invariants and hence for string theory.
The goal for the future is to understand these Euler characteristics more deeply and make them a more flexible tool. They should not just be numbers, but some more complicated structure. Numbers are for counting things, but the things are lost in the process. I will discover the "things" which are "counted" by the Euler characteristics which give rise to Donaldson-Thomas invariants (categorification).
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