Degenerations and Moduli in Algebraic Geometry

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This award supports research in algebraic geometry, a central branch of mathematics. It aims to understand, both practically and conceptually, solutions of systems of polynomial equations in many variables. Algebraic geometry has important applications to other fields of mathematics, such as number theory, topology, and analysis, as well as to physics, biology, cryptography, and engineering. The particular questions under study in this research project involve moduli spaces, which are sets that parameterize solutions of geometric classification problems. Some of the topics under study originated in string theory in physics, and results of this work may in turn find application there. Graduate students are involved in the project. The investigator will work on a range of questions in algebraic geometry centering on degenerations, compact moduli spaces of stable varieties and pairs, and connections to the minimal model program. A central project is the study of functorial compactifications of moduli spaces of K3 surfaces and their explicit combinatorial description. The investigator will also study the birational geometry of abelian six-folds.