I had some role in the initiation of correspondences of different enumerative theories in different dimensions, relating log, open and local Gromov-Witten invariants, quiver Donaldson-Thomas invariants, as well as sheaf counting invariants of local Calabi-Yau fourfolds. Here are some slides of my work with Pierrick Bousseau and Andrea Brini:
Underlying some of this work is the 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 mirror construction that geometrically explains mirror symmetry.
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 work with Mathieu Florence, we 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, with Christian Böhning and Hans-Christian von Bothmer 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.