Non-Archimedean Methods in Complex Analysis, Dynamics, and Geometry

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Analysis, Geometry and Dynamics are areas of Mathematics, which since their discovery have been used in order to understand the world around us. They are crucial to other scientific fields, such as engineering, biology and economics. For example, the mathematical modeling of any phenomenon that undergoes change over time (such as the population of bacteria in a body, the stock market, etc.) can be viewed as a dynamical system. Similarly, geometry is the basis for many current industrial applications such as 3D printing. This research project will enhance the tools available in the aforementioned branches of mathematics. The development of geometry goes back to the ancient Greeks, who laid down axioms, or basic assumptions, from which all other reasonable properties could be logically deduced. The main focus of this project is a detailed study of certain phenomena that occur when the Archimedean Axiom, attributed to Archimedes of Syracuse, is no longer valid. The resulting mathematics turns out to be useful even when the primary object of study is of the usual, Archimedean, kind.This mathematics research project will employ methods from non-Archimedean analysis and geometry in order to study a range of problems in analysis, dynamics and geometry. The principal investigator will work with Berkovich spaces, non-Archimedean analogues of real and complex manifolds. Among many specific projects, one involves a detailed study of the space of metrics on an ample line bundle on a compact complex manifold. Geodesic rays inside this space can be studied using non-Archimedean means. The principal investigator will also continue his work with Boucksom on the Kontsevich-Soibelman conjecture that originally arose in the study of mirror symmetry. In analysis and dynamics, the principal investigator will try to use non-Archimedean techniques to construct invariant currents for quite general two-dimensional complex dynamical systems and study the growth of arithmetic complexity along orbits of certain arithmetic dynamical systems.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.