Andrea is a mathematical physicist with broad interests in Geometry (particularly Algebraic Geometry), Topology and Mathematical Physics. The primary objects of concern in his research are curve counting invariants: these record the number of curves of given genus and degree which meet a prescribed collection of cycles in a given algebraic variety (or symplectic manifold) X. Their systematic study has been the subject of an explosion of activity in the last two decades, as a startling range of fundamental questions, both within and outside geometry, receive a complete answer through their calculation:
- In algebraic geometry, they codify enumerative information about the geometry of X, and provide the solution to a wide range of classical and modern enumerative-geometric problems;
- In symplectic geometry and topology, they supply a sophisticated, infinite set of invariants of the symplectic isotopy class of X;
- In high energy mathematical physics, they capture exact information about symmetry-protected observables of an important class of quantum gauge and string theories;
- Generating functions of the invariants often satisfy an infinite-dimensional group of symmetry constraints given by the flows of a classical integrable hierarchy: a very special non-linear partial differential equation possessing infinitely many commuting conserved currents.
Within the polarised relationship between geometers and mathematical physicists of his research field-at-large, Andrea has consistently strived to keep a dual-loyalty profile: his research has straddled the border between the two subjects, with research outputs in both Algebraic Geometry and Mathematical Physics, and he devotes most of his time figuring out ways to create, transfer and convert ideas and methods from one field to the other. One big pay-off is that hard questions in one subject can sometimes be tackled with unexpected and powerful tools inspired by a priori distant areas of Mathematics. His work in particular has explored and revealed new connections between Gromov-Witten theory and integrable systems, open Gromov-Witten theory and birational geometry, and quantum topology, enumerative geometry, and matrix models.