Higher Dimensional Algebraic Varieties: Geometry and Arithmetic

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Algebraic geometry focuses on the study of solutions of polynomial equations. Guiding questions in this fields are to understand the geometric objects, called algebraic varieties, arising from polynomial equations and to classify such shapes. This project also concerns applications to number theory, with hopes to understand what the geometry of these shapes can tell us about the rational number or integral solutions to these polynomial equations. These problems have applications to other fields of math (e.g. differential geometry) as well as physics (e.g. string theory). Moduli spaces of higher dimensional algebraic varieties are not as well understood as their curve counterparts. The first overarching question in this project is to the understand compactifications of moduli spaces of higher dimensional algebraic varieties. The approaches taken will be a combination of techniques arising from, e.g. the minimal model program (MMP), as well as K-stability. In particular, the PI plans to understand explicit compactifications in specific cases (e.g. K3 surfaces), and to study how to interpolate between various compactifications in a modular way. The second overarching question, is to investigate how geometry of higher dimensional varieties and their moduli influences arithmetic, studying the rich interplay between algebraic geometry, hyperbolicity, and arithmetic 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.