My greatest contribution so far is the development of the log-local principle in Local Gromov-Witten Invariants are Log Invariants, On the log-local principle for the toric boundary and in upcoming work with A. Brini and P. Bousseau for quanitized/refined invariants. Here is a presentation on that work:
Investigating the log-local principle at the level of BPS invariants of log K3 surfaces lead to the series of works Local BPS Invariants: Enumerative Aspects and Wall-Crossing, Log BPS numbers of log Calabi-Yau surfaces and Contributions of degenerate stable log maps.
Underlying this work is Gross-Siebert program, which gives an algebra-geometric realization of the (symplecto-geometric/string theoretic) SYZ conjecture in mirror symmetry. The Gross-Siebert program is a powerful and versatile construction that geometrically explains mirror symmetry.
I gave some two introductory lectures to the scattering diagram of (CP2,E) and Bousseau’s proof of N. Takahashi’s conjecture. Here are the slides:
A different direction of my research is concerned with rationality properties of varieties. Whether a variety is rational or not is a fundamental and important yet difficult question that currently is approached on a case by case basis. In my ongoing collaboration with M. Florence, we aim to systematically determine rationality for a large class of varieties given as certain (easy to describe) birational quotients. A first case was established in A constructive approach to a conjecture by Voskresenskii.
More recently, we defined prelog Chow rings in Prelog Chow rings and degenerations, with a view towards studying Voisin’s criterion of the existence of a decomposition of the diagonal in semistable families, with a first calculation for cubic threefolds in Prelog Chow groups of self-products of degenerations of cubic threefolds.