Rationality and Stable Rationality of Algebraic Varieties

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This project is concerned with the study of properties of solutions of systems of nonlinear algebraic equations in many variables. A major open problem in this field is to determine when these solution spaces can be parametrized by independent variables; it can be done for linear or quadratic equations, but is already unknown in degree three. Such algebraic questions have wide-ranging applications, e.g., in data science, optimization, or robotics. Translating this problem into geometry, by focusing on geometric invariants of shapes defined by the systems of equations, opens the door to a plethora of more intuitive techniques: the study of small deformations, or limit shapes. The project also provides research training opportunities for graduate students.The main goal of the research project is to understand this parametrization property in dimensions 2 and 3, i.e., to study rationality and stable rationality of Del Pezzo surfaces and Fano threefolds over nonclosed fields. Among concrete problems to be addressed are: Obstructions to rationality and stable rationality, such as unramified cohomology, integral decomposition of the diagonal, and Burnside groups; Variation of rationality in families of algebraic varieties, and specialization; Invariants in equivariant birational geometry, in particular, the equivariant Burnside group, as well as its applications to the study of the Cremona group.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.