CAREER: Rational Points via Asymptotics and Geometry

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A wide range of systems of interest in science and engineering can be modeled by systems of polynomial equations. Often, particularly in computer science and cyber security, important quantities can be described by the set of solutions to a system of polynomial equations with rational coefficients, in other words by an algebraic variety. This project is concerned with determining when such a system has a solution where all coordinates are rational numbers. For an arbitrary system of equations, it can be quite difficult to determine whether there exist any such rational solutions. This research project develops a new perspective on this problem using geometry and asymptotic behavior. The research in this proposal is complemented by educational and outreach activities, including workshops for mid-to-late career graduate students focusing on presentation skills and how these skills can be leveraged to develop and broaden one's research program. The research projects fall in two main directions. The first focuses on the change in behavior of rational points and obstructions as the base field varies. In this direction, the project pursues a direct connection between the existence of 0-cycles and rational points on geometrically rational surfaces and explores new asymptotic questions that will shed light on Colliot-Thélène's conjecture on 0-cycles. The second focuses on leveraging dominant rational maps emanating from a variety X to obtain information about the Brauer group, rational points, and 0-cycles of X.