Arizona Winter School in Arithmetic Geometry

亚利桑那大学主办“亚利桑那冬季学校”（AWS），这是一个为期一周的年度会议和研讨会，数学研究生和本科生在数论和算术几何研究前沿问题的领先专家的指导下学习和工作。该奖项支持AWS系列中的2020年、2021年和2022年会议。2020年亚利桑那州冬季学校将于3月9日至13日举行，主题为“非阿贝尔夏博蒂”。未来的主题将基于重要的数学发展和关键参与者的可用性。AWS通过催化新的研究和生成丰富的教学材料，包括详细的课堂讲稿、研究项目描述、问题会话大纲和所有讲座的高质量视频记录，推动了数学的发展。这些来自过去AWS会议的资源，以及即将到来的冬季学校主题和申请材料的信息，可通过AWS网站：http://swc.math.arizona.edu/获得。AWS 2020将重点关注Chabauty和Coleman的方法，这是曲线算法的基石，也是关键的计算和理论工具。非阿贝尔技术随后给出了单位方程解的有限性的动力证明，并奠定了将p进解析技术推向超越阿贝尔积分限制的程序的基础。过去十年见证了快速的发展（和应用），尽管该主题的技术深度。为了有效地传播这样一个技术话题，系列讲座将紧密协调，包括经典Chabauty，计算方面和应用，高度，算术交理论，二次Chabauty，以及一系列关于概念和猜想框架的讲座。AWS系列的潜在未来主题包括“志村品种”，这将使学生成为这一无处不在的理论的用户；“不太可能的交叉点”，它将调查几何上有趣的交叉点的有限定理的最新进展（例如Manin-Mumford猜想和Andre-Oort猜想）；“算术几何算法的复杂性”，侧重于数论和代数几何中算法的理论分析；“GL_2以外的自同构形式”，向学生介绍GL_2以外群上的自同构形式，强调它们在具体数论问题中的应用。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.