Equivariant birational geometry

This award is focused on the study of systems of nonlinear algebraic equations in many variables. Of particular interest are problems concerning the existence of simple parametrizations of solutions and of hidden symmetries of such solution spaces. Apart from intrinsic importance to the field of algebraic geometry, this research has potential applications to theoretical computer science, and as a consequence to problems in cryptography, information processing, and management of large data structures. Furthermore, it is potentially applicable to theoretical physics, where nonlinear systems play an important role. Moreover, it stimulates the development of efficient algorithms for the computation of discrete invariants of such systems, and provides many concrete problems and examples for the next generation of geometers. The PI will continue to train graduate students and engage in outreach activities bringing mathematical awareness to a broad audience. Specifically, the proposed research would combine in novel ways arithmetic and geometric insights to significantly advance our understanding of rationality and stable rationality in small dimensions, as well as linearizability and stable linearizability of actions of finite groups on algebraic varieties. Rationality constructions often involve the study of fibrations and thus the study of rationality over the function field of the base, a nonclosed field. In turn, geometry over nonclosed fields is tightly linked to equivariant birational geometry, as there are strong parallels between the action of the absolute Galois group and the action of automorphisms. Exploring these connections between geometry, arithmetic, and group theory is a major thrust of this proposal. One of the long-term goals is to obtain a full classification of such actions on rational varieties in dimensions up to three. Another goal is to explore the range of applicability of recently discovered invariants in birational geometry, in presence of actions of finite groups, volume forms, and other structures. A third goal is to develop the theory of universal torsors in the equivariant and orbifold context, and to apply it to produce new examples of stable birationalities. 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.