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

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This project investigates open questions in the areas of analysis, geometry, and dynamical systems. These mathematical disciplines are crucial for a host of scientific fields, including engineering, biology, and economics. For example, the mathematical modeling of any physical phenomenon that undergoes change over time can be viewed as a dynamical system. Similarly, geometry is the basis for numerous current industrial applications such as 3D printing. This research project will enhance the tools available in the aforementioned mathematical areas. The axiomatic development of geometry, which postulates a set of basic assumptions from which all other reasonable conclusions are logically deduced, dates back to ancient Greece. The major 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 postulated. The resulting mathematics turns out to be useful even when the primary object of study is of the usual, Archimedean, kind. The project will also generate research opportunities at a variety of levels, suitable for work by graduate and undergraduate students.This project will employ non-Archimedean tools to study a range of problems in analysis, dynamics, and geometry. Essential to this endeavor is an understanding of Berkovich spaces, which are non-Archimedean analogues of real and complex manifolds. One component of the project involves a detailed study of the space of metrics on an ample line bundle on a compact complex manifold. Geodesic rays within this space can be studied via non-Archimedean methods. Another topic for consideration is the Kontsevich-Soibelman conjecture, which originally arose in the study of mirror symmetry. In the areas of analysis and dynamics, non-Archimedean techniques will be employed to construct invariant currents for two-dimensional complex dynamical systems, allowing for a study of the growth of arithmetic complexity along orbits of 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.