CAREER: Connections Between Tropical Geometry, Arithmetic Geometry, and Combinatorics

University Of Washington, Seattle WA

Investigators

Abstract

This award is at the intersection of algebraic geometry, arithmetic geometry, and combinatorics. In algebraic geometry, one studies the geometry of solutions to a collection of polynomial equations. In arithmetic geometry, one is concerned with the study of integers (whole numbers) and rational numbers (fractions). Combinatorics is an area of mathematics concerned with the study of finite structures. An example of such finite structures is the notion of graphs or networks. The aim is to make substantial progress on various foundational questions in these areas. The interplay between these disparate fields will continue to have significant consequences and will create and stimulate entirely new research problems. The PI will study fundamental questions at the intersection of non-archimedean and tropical geometry, arithmetic and Arakelov geometry, and combinatorics and convex geometry. The three main aspects of the research are: further study of the interplay between tropical methods and arithmetic geometry of abelian varieties, the study of various combinatorial and convex geometric questions arising from this interplay, and the study of invariants on degenerating families of curves and abelian varieties. The educational and outreach activities will complement the research. The centerpiece is a yearly extended research experience for graduate students on projects at the intersection of combinatorics, tropical geometry, and arithmetic and Arakelov geometry. 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.