Tropical methods for moduli spaces of maps & new techniques in DT theory

Short Summary:

For the past few decades, enumerative geometry has driven the research in various areas of mathematics including algebraic and arithmetic geometry, mathematical physics and represen- tation theory. The revolution of this classical subject started around 30 years ago, when ideas from modern physics inspired mathematicians to develop new and deep techniques to handle the enumerative questions originating from string theory and mirror symmetry predictions.

This proposal is about the study of the geometry of moduli spaces relevant to enumerative geometry questions.

There are three directions of research in this plan, with one common factor: employ recently developed technology to reinterpret long standing problems and exploit the innovative point of view to advance towards a solution. In a nutshell: we study moduli spaces of curves and maps using tropical and logarithmic techniques, and we investigate moduli spaces of sheaves via cohomological and p-adic techniques.