Research

My research mostly surrounds variational formulations of canonical metrics in Kähler geometry. The variational approach often yield a way of proving existence or non-existence of the related geometric PDE's.

In one theme of my research, a variational approach for Monge-Ampère metrics on singular affine manifolds is pursued, with applications to the metric SYZ conjecture originating in mirror symmetry in string theory, and the study of degenerations of Calabi-Yau manifolds. As it turns out, this is quite related to the classical theory of optimally transporting probability measure on Euclidean space to one another, with respect to a squared Euclidean cost. The close connection to classical optimal transport exists also in other corners of Kähler geometry, such as for the toric Kähler-Einstein problem and recently also in a problem related to complete Ricci-flat metrics on non-compact Calabi-Yau manifolds.

In another theme, arithmetic augmentations of the functionals used in the variational approach to Kähler-Einstein metrics on projective varieties are utilized to define a new invariants in Arakelov geometry, a field of geometry combining complex hermitian geometry with arithmetic geometry. These invariants is simply defined as the extremal value of a these functionals. Interestingly, even though the invariant seems to be of analytic nature - for example, a naive way to compute them involves integrating various curvature quantities of the Kähler-Einstein metrics - the arithmetic structure is needed in its definition. These invariants has been useful in extracting sharp bounds on the height of arithmetic Fano varieties. Moreover, they show up in the Kudla program for Shimura curves.

I am also interested in probabilistic/statistical mechanical constructions of Kähler-Einstein metrics, whose starting point is the formal similarities between the variational approach and the Gibbs variational principle in thermodynamics. The aim is to construct an intrinsically defined statistical mechanical system of interacting particles on the variety, whose large N limit is precisely given by the variational theory of Kähler-Einstein metrics.