New Directions in the Theory of Moduli

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Gromov-Witten invariants are important in String Theory, the physical theory according to which elementary particles behave like 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 relativity and quantum field theory. Recently, it was discovered that Gromov-Witten invariants are related to Donaldson-Thomas invariants. This connection is still conjectural, but it is very important, because Donaldson-Thomas invariants can explain many of the 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 were 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. One goal of this research is to understand these Euler characteristics more deeply and make them a more flexible tool. They should not just be numbers, but numbers abstracted from some more complicated structure. Numbers are for counting things, but the things themselves are lost in the process of counting. The goal is to discover the things which are counted by the Euler characteristics which give rise to Donaldson-Thomas invariants. Moduli spaces are multidimensional geometric shapes, each of whose points corresponds to a geometric object. For example, there is a moduli space of triangles up to similarity. Each point of this moduli space represents one triangle shape. Many mathematical or physical objects can be sorted into moduli spaces. In fact, Gromov-Witten invariants and Donaldson-Thomas invariants are numbers associated to various moduli spaces. The moduli space reflects properties such as symmetries and deformation behaviour of the objects being classified. The main purpose of this research is to study the geometry of moduli spaces. This sheds light on the numerical invariants, and deepens our understanding of the mathematics underlying physical theories such as String Theory. A particular goal of this research is to apply what we have learned about moduli spaces to number theory. There is a compelling analogy between number theory and geometry, in which prime numbers correspond to knots, for example. Exploiting this analogy will illuminate subtle questions in number theory.