Some keywords: Algebraic geometry, Gromov-Witten invariants, quantum K-theory, Mirror Symmetry, q-difference equations, Stokes phenomenon
I am interested in invariants in enumerative geometry. Gromov-Witten invariants count the number of curves in a given space satisfying some incidence conditions. An example of such invariant is the number of rational curves of degree d in the complex projective plane P² meeting 3d-1 points in general position. In general, these numbers are realised as an intersection degree on the moduli stack of stable maps. These invariants can be encoded in an algebraic structure called the quantum D-module, which is a vector bundle equipped with a flat connection.
K-theoretical Gromov-Witten invariants are new invariants built by replacing objects of cohomological nature in Gromov-Witten invariants by their K-theoretical analogue. One of my main questions is to understand the relation between these two invariants. One point of view to compare these two theories is to construct an analogue to the quantum D-module in the K-theoretical setting. This analogue features q-difference equations, which I aim to understand.