CAREER: Moduli of Algebraic Varieties and K-stability

About This Grant

This project is on algebraic geometry, which is the study of shapes defined by polynomial equations. High degree polynomials and their corresponding shapes can be used to model various real world phenomena, and therefore play an important role in fields such as mathematical physics, cryptography, computer vision, and optics. By varying the coefficients of the polynomials, one varies the corresponding shapes and this leads to the problem of moduli in algebraic geometry, which concerns the construction and study of parameter spaces for algebraic shapes. This research will use new techniques inspired by differential geometry to construct parameter spaces for such shapes in higher dimensions. Additionally, the project includes educational activities aimed at training the next generation of mathematicians. More specifically, this project focuses on K-stability, which is an algebraic notion introduced by differential geometers to characterize the existence of certain canonical metrics on complex projective varieties. While the notion initially seemed difficult to understand algebraically, in the past decade algebraic geometers have made significant progress in understanding this notion in the case of Fano varieties. This progress has culminated in the development of a projective moduli theory for K-semistable Fano varieties over the complex numbers. This project aims to extend these ideas to develop new moduli theories for higher dimensional varieties. One direction involves extending the Fano K-moduli theory to work over the integers in low dimensions. Another direction focuses on using ideas from K-stability to construct moduli spaces of Calabi-Yau varieties. This direction will focus on both specific examples (e.g., K3 surfaces and abelian varieties) and general theory. The project also includes educational activities to train young mathematicians. These efforts include completing a book on K-stability, organizing an undergraduate summer mini-course on algebraic geometry, and hosting a weekend graduate student workshop on a topic in algebraic geometry. 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.