Non-Archimedean Techniques in Analysis, Dynamics, and Geometry

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

Investigators

Abstract

This mathematics research project concerns the areas of analysis, geometry, and dynamics, which are central to mathematics and 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 these branches of mathematics. The development of geometry goes back to the ancient Greeks, who laid down axioms, or basic assumptions, from which 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a detailed study of certain phenomena that occur when the Archimedean axiom, attributed to Archimedes of Syracuse, is no longer valid.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 research project will use techniques from non-Archimedean analysis and geometry in order to study a range of problems in analysis, dynamics, and geometry. The project makes use of Berkovich spaces, non-Archimedean analogues of real and complex manifolds. One part of the project involves Calabi-Yau manifolds, geometric objects that play an important role in mathematical physics. The principal investigator and collaborator will use non-Archimedean techniques to study a conjecture concerning one-parameter families of complex Calabi-Yau manifolds that degenerate to a real Calabi-Yau manifold. Another part of the project is in arithmetic dynamics. Given a discrete-time two-dimensional dynamical system defined by a pair of polynomials with rational coefficients, the research investigates how the arithmetic complexity grows along orbits of the dynamics. Here the problem is formulated in terms of elementary number theory but approached using a detailed study of an induced dynamical system on a Berkovich space.