Moduli and Arithmetic of Higher Dimensional Varieties

Algebraic geometry is the study of geometric shapes which arise as solutions to polynomial equations. Some fundamental examples of these shapes are circles or hyperboloids, and as such, algebraic geometry is largely intertwined with many other fields of mathematics. One of the guiding research directions in algebraic geometry as well as the overall focus of this research project is understanding the classification of these shapes, a subfield known as the study of moduli spaces. Algebraic geometry as well as the study of moduli spaces have numerous applications, for example within cryptography as well as in understanding the structure underlying our universe via string theory and mathematical physics. This project includes training opportunities for both undergraduate and graduate students, as well as outreach efforts involving high school students from the local community. While we have an extensive understanding of moduli spaces of algebraic curves, we understand far less about moduli spaces of higher dimensional algebraic varieties (i.e. varieties of complex dimension at least two). In short, the main goal of this project is to further our understanding of higher dimensional moduli by leveraging many recent results in moduli theory and birational geometry, such as wall-crossing results that the PI has obtained in joint work with collaborators. The first project aims to use wall-crossing techniques to understand the underlying geometric structure and geometric properties of moduli spaces of higher dimensional algebraic varieties. Additionally, as most known results regarding moduli spaces of higher dimensional varieties focus on two cases, namely (log) general type pairs and (log) Fano pairs, the goal of the second project is to construct and study moduli spaces of (log) Calabi-Yau pairs using tools from the minimal model program and K-stability. Finally, the third project uses techniques from birational geometry, moduli spaces, and the minimal model program to understand various notions of hyperbolicity on varieties of (log) general type, including the distribution of rational and integral points. 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.