p-adic Methods in Number Theory

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This award provides support for participation in the conference "p-adic Methods in Number Theory" held at the University of California, Berkeley on May 26-30, 2015. Since their conception by Kurt Hensel around 1900, p-adic numbers have played a central role in number theory; for example, they are used in a crucial way in the proof of Fermat's Last Theorem. To a number theorist, p-adic numbers are just as "real" -- and just as important -- as real numbers. Both are ways of "filling in the gaps" left by considering just rational numbers. In their book "Number Theory I: Fermat's Dream," Kato, Kurokawa, and Saito write poetically, "In the long history of mathematics a number meant a real number, and it is only relatively recently that we realized that there is a world of p-adic numbers. It is as if those who had seen the sky only during the day are marveling at the night sky. [ ] Just as we can see space objects better at night, we begin to see the profound mathematical universe through the p-adic numbers." This conference will bring together experts in the many different facets of p-adic numbers and their applications, will promote a cross-fertilization of ideas between number theorists of all stripes, will expose graduate students and postdocs to state-of-the-art techniques and results, and will promote participation by underrepresented minorities and women in high-level number theory research. A conference on p-adic methods in number theory is timely and important, as many spectacular recent number-theoretic advances have made use of deep p-adic methods. We mention, for example, recent work establishing special cases of the p-adic local Langlands correspondence; the proof that most hyperelliptic curves of odd degree have just one rational point; developments on non-abelian Coleman integration and integral points on curves; work on the fundamental curve of p-adic Hodge theory; and recent results on perfectoid spaces. More Information can be found at https://sites.google.com/site/padicmethods2015/.