Non-Archimedean Methods in Analysis, Dynamics and Geometry

Regents Of The University Of Michigan - Ann Arbor, Ann Arbor MI

Investigators

Abstract

This mathematics research project by Mattias Jonsson will use techniques from non-Archimedean analysis and geometry in order to study a range of problems in analysis, dynamics and geometry. Jonsson will work with Berkovich spaces, non-Archimedean analogues of real and complex manifolds. Among many specific projects, one amounts to showing that four different notions of equisingularity for plurisubharmonic functions are equivalent. This is a problem in complex analysis but its solution involves doing analysis on a Berkovich space. Another project is in arithmetic dynamics. Given a discrete-time two-dimensional dynamical system defined by a pair of polynomials with rational coefficients, Jonsson will investigate how the arithmetic complexity grows along orbits of the dynamics. Here the problem is formulated using rather elementary number theory but Jonsson will approach it using a detailed study of an induced dynamical system on a Berkovich space. Jonsson will also study non-Archimedean analysis and dynamics outright. For example, he aims to extend the successful analysis obtained jointly with his collaborators Boucksom and Favre on the non-Archimedean Monge-Ampere equation. This mathematics research project by Mattias Jonsson is in the areas of Analysis, Geometry and Dynamics which, ever 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 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a detailed study of certain phenomena that occur when the Archimedean Axiom, attributed to Archimedes of Syracuse, is no longer valid.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.