Non-Archimedean methods in complex analysis and geometry

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

Investigators

Abstract

The purpose of this project is to use innovative tools to investigate open questions in the areas of analysis and geometry. These mathematical disciplines are crucial for a host of scientific fields, including engineering, biology, and physics. For example, geometry is the basis for numerous current industrial applications such as 3D printing. The PI will explore advanced methods 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 and geometry. A through-line to the work is a deep understanding of Berkovich spaces, which are non-Archimedean analogues of real and complex manifolds. One component of the project involves an attack on the celebrated Yau--Tian--Donaldson conjecture, on an algebro-geometric criterion for the existence of constant scalar curvature metrics. In another project, the PI will use variational and non-Archimedean methods to study the existence of complete Calabi--Yau metrics on affine complex manifolds. and complex analytic geometry. A third project is centered around the Kontsevich--Soibelman conjecture arising in Mirror Symmetry, on the asymptotic geometry of degenerating families of compact complex Calabi--Yau manifolds. All these projects use Berkovich spaces in a crucial way. 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.