Probabilistic approaches to Brauer groups and rationality problems

William Marsh Rice University, Houston TX

Investigators

Abstract

Diophantine geometry seeks to understand the set of integer or rational solutions to systems of polynomial equations in several variables. Every collection of polynomial equations with integer coefficients has a geometric avatar, called a variety. The geometry of this variety often governs the set of rational solutions to the original system of equations. This idea is summarized by the mantra "Geometry determines Arithmetic". The project focuses on developing theoretical tools to understand large classes of diphantine equations. Applications of understanding these equations, and their related shadows over finite number systems, abound, e.g., in cryptography and coding theory. The PI has experience working on the latter applications. The project will also fund mentoring and outreach efforts, with particular attention to increasing retention of students and researchers from underrepresented backgrounds in Mathematics, including leading small research project groups outside his primary institution, and organizing research conferences that specifically foster a sense of community and belonging. The PI currently advises five PhD students and two postdoctoral scholars, and expects to maintain a vigorous research and training group. He will also embark on a book project on the arithmetic of algebraic surfaces, to fill a gap in the literature, to help educate future generations of diophantine geometers. This project addresses foundational questions in the arithmetic of Diophantine equations whose geometric avatars are surfaces. The PI will use Bayesian inference to devise probabilistic algorithms that take as input a set of equations defining, e.g., a low-degree del Pezzo or K3 surface, and determine, with a prescribed degree of confidence, if these systems of equations have rational solutions. It is expected that these ideas will have wide application in other problems of arithmetic geometry around Galois groups. In a related direction, the PI will systematically study the behavior of rationality over number-theoretic bases, by leveraging the connection between certain kinds of fourfolds and surfaces twisted by Brauer classes. The project naturally leads to considering new cases of the Tate conjecture for divisors on surfaces over finitely generated fields. 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.