The University of Arizona hosts "Arizona Winter School" (AWS), an annual week-long conference and workshop in which mathematics graduate students and undergraduates learn from and work under the guidance of leading experts on questions at the forefront of research in number theory and arithmetic geometry. This award supports the 2020, 2021, and 2022 meetings in the AWS series. The 2020 Arizona Winter School will be held from March 9-13 on the topic of "Non-Abelian Chabauty." Future topics will be based on important mathematical developments and the availability of key participants. AWS advances mathematics by catalyzing new research and generating a wealth of pedagogical materials, including detailed lecture notes, research project descriptions, problem session outlines, and high-quality video recordings of all the lectures. These resources from past AWS conferences, along with information about upcoming Winter School topics and application materials, are available through the AWS website: http://swc.math.arizona.edu/.
AWS 2020 will focus on the method of Chabauty and Coleman, a cornerstone of the arithmetic of curves and a key computational and theoretical tool. Non-abelian techniques subsequently gave a motivic proof of finiteness of solutions to the unit equation and laid the foundations of a program to push p-adic analytic techniques beyond the limitations of abelian integrals. The last decade has witnessed rapid progress (and applications), despite the technical depth of the subject. To effectively disseminate such a technical topic, the lecture series will be closely coordinated, and include classical Chabauty, computational aspects and applications, heights, arithmetic intersection theory, quadratic Chabauty, and a series of lectures about the conceptual and conjectural framework. Potential future topics for the AWS series include "Shimura Varieties," which would prepare students to be users of this ubiquitous theory; "Unlikely Intersections," which would survey recent advances on finiteness theorems for geometrically interesting intersections (e.g. the conjectures of Manin-Mumford and the Andre-Oort conjecture); "Complexity of Arithmetic Geometry Algorithms," which would focus on the theoretical analysis of algorithms in number theory and algebraic geometry; and "Automorphic Forms Beyond GL_2," which would introduce students to automorphic forms on groups beyond GL_2, emphasizing their applications to concrete number-theoretic problems.
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.