Berkovich Spaces, Tropical Geometry, Combinatorics, and Dynamics

Georgia Tech Research Corporation, Atlanta GA

Investigators

Abstract

This research project is primarily concerned with work in number theory, a branch of mathematics with important applications to cryptography and coding theory. The work also features connections to graph theory, a research area with applications to the study of large-scale networks such as the internet. The project will have a number of broader impacts, including support for high school, undergraduate, and graduate research. The PI is currently supervising two Ph.D. students from Georgia Tech and one Ph.D. student from UC Berkeley, in addition to one undergraduate research student and two high school students, each of whom is working on projects related to this project. The PI also has been involved in a number of mathematical outreach activities, including an online course on Number Theory and Cryptography for gifted high school students, and he writes a popular math blog. These activities will be continued in this project, integrating ideas from the current research whenever feasible. The project will also contribute to the dissemination of mathematical ideas through conference organization and publication of proceedings. The primary intellectual merit of the project is that it will increase our understanding of Berkovich spaces, tropical geometry, combinatorics, and complex dynamics, and unearth new relationships between these different areas of research. Berkovich spaces -- the modern incarnation of the pioneering early twentieth century work of Kurt Hensel on "p-adic numbers" -- have in recent years found applications to numerous areas of mathematical research, including algebraic geometry, number theory, and complex dynamics (where they can be used to study fractals such as the Mandelbrot set). Berkovich spaces are also intimately connected with tropical geometry, a relatively new and very active area of research with applications to number theory, algebraic geometry, statistics, biology, and physics. One can think of tropical geometry as a simplified model classical algebraic geometry in which the set of common solutions to a system of polynomial equations is replaced by the set of common solutions to a much simpler system of linear inequalities. Surprisingly -- and rather mysteriously -- the tropical simplification remembers much more information about the original solutions than one might originally expect. The investigator's previous work involved unexpected new applications of Berkovich spaces and tropical geometry, and this project will build on and significantly advance that work.