Connections Between Number Theory, Algebraic Geometry, and Combinatorics

Georgia Tech Research Corporation, Atlanta GA

Investigators

Abstract

This proposal is concerned with the development of new connections between arithmetic geometry, tropical geometry, Berkovich spaces, and combinatorics. The intellectual merit of the proposal lies primarily in the cross-fertilization between these different areas, and in the concrete applications being proposed. For example, the PI proposes to use ideas coming from algebraic geometry to provide new insight into the graph isomorphism problem, one of the most famous unsolved problems in graph theory and computer science. The PI will also show that harmonic morphisms, which play a prominent role in differential geometry and potential theory, arise naturally in arithmetic geometry, tropical geometry, and combinatorics. Applications will be given to a diverse array of subjects including component groups of Neron models, tropical intersection theory, and graph theory. Finally, the PI plans to develop new connections between Berkovich's theory of analytic spaces and tropical geometry. This will enable further development of the foundations of tropical geometry and potential theory on Berkovich spaces, and will also provide a more conceptual understanding of some recent results concerning tropical elliptic curves. The broader impacts of the proposed work will include applications to problems in the physical sciences, interaction with mathematicians in different fields, and support for undergraduate and graduate research. For example, the PI's new ideas on the graph isomorphism problem could potentially have applications to chemistry, biology, and computer science. Accomplishing the various goals laid out in this proposal will require the PI to interact with leading experts in the fields of number theory, algebraic geometry, combinatorics, and dynamical systems. The PI, who is currently supervising two graduate students and has been intensely involved for many years with undergraduate research, plans to work with students at all levels on research projects related to this proposal.