Analysis on Berkovich spaces and applications

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DMS-0600027Matthew BakerThis proposal is concerned with further development of the theory of Berkovich spaces, together with arithmetic applications. In broad terms, the PI plans to do the following: (1) Develop a theory of Jacobians of metrized graphs, with applications to studying the relationship between a Berkovich analytic curve and its Jacobian; (2) generalize the PI's and Rumely's potential theory on the Berkovich projective line to higher dimensions; and (3) use the theory of Berkovich spaces to make new progress on some open problems in arithmetic dynamics. The methods to be employed in the proposed research involve a mixture of techniques from arithmetic geometry, analysis, topology, graph theory, and dynamical systems. In broad terms, the proposal aims to address a key unifying principle within number theory, the idea that all completions of a global field should be treated in a symmetric way. This has been a central theme in number theory for almost a hundred years, as illustrated for example by Chevalley's idelic formulation of class field theory and Tate's development of harmonic analysis on adeles.The main intellectual merit of this work is that it will lead to new developments in the theory of Berkovich spaces, an exciting subject which has already found applications to number theory (including the local Langlands correspondence), mathematical physics (e.g. mirror symmetry), and other diverse fields. This project will also show how Berkovich spaces can be applied to a variety of interesting questions related to dynamical systems. In addition, the PI will bring new ideas to the theory of metrized graphs. Metrized graphs, also known in the literature as metric or quantum graphs, have applications to a diverse array of fields, including physics, mathematical biology, and number theory. The broader impacts of the proposed work will include dissemination of research and expository papers, interaction with experts in different fields, and support for undergraduate, graduate, and postdoctoral research.