CAREER: Hodge Theory and Moduli

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Algebraic varieties are the spaces of solutions to polynomial equations, and algebraic geometry seeks to describe those solutions geometrically. Ubiquitous in mathematics, algebraic varieties are critical objects of study in complex geometry, number theory, topology, and representation theory. It is often particularly important to understand their moduli, that is, how they vary as the coefficients of the polynomial equations defining them are varied. Hodge theory offers one perspective on this problem: the moduli of algebraic varieties can be understood in terms of how integrals of differential forms vary. This research project will use new methods stemming from model theory, representation theory, and complex geometry to understand this connection. The project especially aims to promote interactions with closely-related areas of mathematics and to encourage the participation of students and young researchers. The project will address two main types of research problems. Hodge theory is a powerful tool in algebraic geometry but it is fundamentally transcendental in nature. Recent work of the PI and coauthors has demonstrated that techniques from model theory and complex geometry can be used to bridge this divide systematically, and the first goal of the project is to develop the application of these techniques more deeply. Secondly, the project will develop a more detailed understanding of the moduli of certain varieties which are particularly closely related to Hodge theory, including abelian varieties and hyperkahler varieties. Building on previous work using techniques from differential and complex geometry, the PI will further investigate the geometry and arithmetic of the moduli spaces of these varieties.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.