Curve counting via categorification

Counting curves inside a given space is the most classical problem in geometry. For example, one can ask a question such as 'How many lines pass through two given points in the plane?' Sometimes the set of all curves satisfying certain geometric properties is finite, but other times not. A modern approach to enumerative geometry is to assign numbers, called `virtual invariants' to the spaces parametrising all curves we want to count, whether or not they are finite sets. There are several different virtual invariants, and some of them were introduced by physicists in string theory.

The first aim of this project is to understand how different kind of invariants are related. Another aim is to refine these numerical counts. Namely, I will investigate certain vector spaces and certain categorical structures defined from the spaces parametrising curves, which recover the numerical invariants by taking their dimensions. These refined invariants will distinguish geometries more finely and lead to a deeper understanding of counting invariants in enumerative geometry and string theory.