I work in the fields of algebraic geometry and representation theory. More specifically, my research is focused on problems in Donaldson-Thomas theory for resolutions of singularities, such as those appearing in the minimal model program. The goal of DT theory is to extract invariants from certain moduli spaces of objects in the derived category of a variety. These spaces are often complicated beasts, which need to be tamed before one can make any substantial progress. To do this, we I use tools from various adjacent areas:
The goal of my research is therefore to develop the parts of these areas that improve our understanding of some of the moduli spaces we encounter in the MMP.
We develop theoretical aspects of refined Donaldson-Thomas theory for threefold flopping contractions, and use these to determine all DT invariants for infinite families of length 2 flops. Our results show that a refined version of the strong-rationality conjecture of Pandharipande-Thomas holds in this setting, and also that refined DT invariants do not determine flops. Our main innovation is the application of tilting theory to better understand the stability conditions and cyclic A∞-deformation theory of these spaces. Where possible we work in the motivic setting, but we also compute intermediary refinements, such as mixed Hodge structures.