CAREER: Birational Geometry and Algebraic Dynamics

This project focuses on the role of symmetry in algebraic geometry, which is the study of the solutions of systems of polynomial equations in many variables, and methods of exploiting symmetries in these systems to better understand their structure. Solving such systems is often very difficult: in general, it is hard to learn much about the set of solutions unless one knows some peculiarities of the specific equations at hand. This research lies at the intersection of algebraic geometry and dynamical systems, which studies the behavior of repeated applications of functions on certain sets of points. A key component of this project is to bring the techniques from each of these disciplines to bear on questions from the other. The project includes educational components at every level. At the most advanced levels, the project focuses on introducing algebraic dynamics to new audiences, including graduate students in dynamics and researchers in algebraic geometry. The project will also involve several undergraduate researchers learning the basics of the field, as well as the development of educational materials for younger students hoping to understand the breadth of mathematics beyond calculus.

More technically, a primary goal of the project is to understand the growth of degrees of iterates of rational maps, from both computational and theoretical points of view. There has been considerable effort directed at understanding lower bounds on degree growth, and the investigators will use new methods from the Sarkisov program to study this question. The researchers will also investigate computational questions and improve the capacity of the open-source SageMath system to compute dynamical degrees. In other directions, the researchers will focus on questions related to the growth rates of spaces of sections of tensor powers of line bundles as well as higher-dimensional versions of the bounded negativity conjecture.

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.