Complex geometry and differential invariants of period maps

Algebraic geometry studies geometric properties of sets that arise as solutions to systems of polynomial equations. Many familiar sets arise in this way, including lines and planes, and circles and spheres/soap bubbles. Algebraic geometry also has many applications outside of mathematics, including statistics, robotics, error-correcting codes, phylogenetics, game theory and integer programming. “Moduli problems” is a sub-branch of algebraic geometry that studies these sets in families. For example, one might consider all the lines passing through a fixed point. These families, and the related questions, can be quite complicated. An important tool in their study is the “period map”, a function that is associated with the family. The period map allows one to bring to bear techniques from other areas of mathematics, including complex geometry and representation theory (a sophisticated extension of linear algebra). This award research into properties of period maps that are motived by applications to moduli problems. The PI maintains an active mentoring program involving graduate student research, and will continue current efforts in proof formalization using artificial intelligence, opening up a wide range of activities at the undergraduate and graduate level. Half the projects continue a program initiated by the PI to study period maps via their “Fubini–Hodge” forms, a class if differential invariants associated to a period map that includes Griffiths—Yukawa couplings. The program is inspired by a close analogy with the very successful Hwang–Mok program to study Fano manifolds via their varieties of minimal rational tangents. The remaining projects address the long-standing problem to develop analogs of the Ash—Mumford—Rapoport—Tai for arbitrary period maps. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.