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.
This project focused on the development of new connections between number theory, tropical geometry, dynamical systems, and combinatorics. The intellectual merit of the project lay primarily its cross-fertilization between all of these different areas, and in the concrete applications obtained. Number theory is one of the oldest branches of mathematics, and it focuses on properties of integers, prime numbers, etc. It has significant applications to cryptography, which is of crucial importance in modern e-commerce. Tropical geometry is one of the newest branches of mathematics, and is a very active area of current mathematical research. At its core, tropical geometry focuses on solving complicated non-linear problems by first solving a linear approximation and then proving that the answer to the complicated problem and the much simpler one are the same. Dynamical systems have to do with iterating a given algebraic operation over and over again, infinitely many times, and studying the dichotomy between regions where this process is predictable and regions where it is not. Combinatorics is the branch of mathematics which deals with ways to count or enumerate complicated quantities whose exact value is often hard to determine. On the face of it, all of the above subjects are quite different, and to a large extent the researchers in these subjects do not spend a lot of time working together or talking to one another. The PI's work on this project has contributed to an improvement in this situation, and has resulted in new interdisciplinary collaborations. For example, some important new connections between Berkovich’s theory of analytic spaces (which is a part of number theory) and tropical geometry were established in the PI's joint paper with Sam Payne and Joe Rabinoff. Such connections have been the source of multiple recent conferences and workshops, including the 2010 Bellairs Workshop in Number Theory (where the PI was the Principal Speaker) and two workshops for which the PI was a co-organizer: a 2013 Simons Symposium and a 2013 AMS Math Research Communities workshop. During the period of this project the PI also began a fruitful collaboration with Laura DeMarco, a complex dynamicist, which involved both number theory and dynamical systems, and which has now led to two publications in top journals. The broader impacts of the project included interaction with mathematicians in different fields and support for undergraduate and graduate research. The PI is currently supervising four Ph.D. students and one postdoc and the NSF funding for this project was a crucial ingredient in the success of some of the PI's students' recent research breakthroughs. The PI has also engaged in a number of outreach activities, including mathematical magic shows for local schools and math clubs and a mathematical presentation for the Board of Regents of the University System of Georgia.
