Southwest Center for Arithmetic Geometry Winter School

在该奖项的支持下，西南算术几何中心将于2019年3月2日至6日在亚利桑那州图森市的亚利桑那大学继续举办一年一度的冬季学校系列活动。自1997年成立以来，西南中心的主要活动是亚利桑那州冬季学校(AWS)，这是一个年度会议，已成为全国知名活动，为研究生提供算术几何及相关领域的高水平培训和研究经验。AWS每年围绕一个不同的中心主题组织，由领先的和新兴的专家精心设计和提供一系列课程和配套的研究项目。结果是传统数学会议和密集研究工作坊的独特融合：演讲者组织四到五个讲座的课程，提前提供讲座笔记，并提出研究项目供研究生在会议期间工作。每晚关于这些项目和单独的问题集的工作会议由演讲者和博士后研究员主持。在最后一天，学生们向会议的参与者展示他们的发现。结果是，每个参与者都将进行为期五天的特别紧张和专注的数学活动。在AWS，同行之间形成了联系，学生和高级研究人员之间建立了指导关系。各级参与者之间的后续协作是常态。学生们朝着成为研究数学家的方向迈出了具体的步伐，博士后助理在他们的学术生涯中获得了宝贵的指导经验，教师们培养了新的兴趣，看到了新的联系，导致了重要的出版成果。西南中心网站共享冬季学校的可重复使用的内容，包括课堂讲稿、项目描述和讲座的音频和视频(包括现场直播和永久存档)。通过这些透彻的记录，AWS开始的对话扩展到更大的社区，AWS参与者的努力自由和无限期地向所有人提供。关于即将和过去的亚利桑那州冬季学校项目的更多信息可以在西南中心的网站上找到：http://swc.math.arizona.edu/The 2019年冬季学校将以拓扑和算术为主题，探索高等同伦理论和算术几何的边界。高阶范畴/同伦理论的发展和成熟是近十年来拓扑学中最重要的成就之一，最近在数论和算术几何中得到了显著的应用。最早的应用是拓扑量子场论、上同调和椭圆上同调；即使在这里，广义上同调理论和形式群之间的相互作用也产生了算术几何感兴趣的结果(特别是对形式群的Lubin-Tate和Gross-Hopkins模空间的几何的阐明)。更新的进展比比皆是。Gaitsgory-Lurie通过拓扑技术研究Tamagawa数(全局域上代数群的基本不变量)。Bhatt-Morrow-Scholze在拓扑Hochschild同调上建立了一个权滤子的存在性，它的分次块既与经典上同调理论(de Rham，晶体)有关，也与积分p-进Hodge理论有关。Wickelgren等人的工作发展了A1-同伦理论的算术方面，将一些计数问题从复数推广到其他领域。2019年冬季学校将专注于阐述算术几何的应用和联系，而不是发展一般理论。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

With support from this award, the Southwest Center for Arithmetic Geometry will continue its series of annual "Winter Schools" in 2019 taking place March 2-6, 2019 at the University of Arizona in Tucson, AZ. Since its founding in 1997, the primary activity of the Southwest Center is the Arizona Winter School (AWS), an annual meeting which has become a prominent national event and provides high-level training and research experience for graduate students in arithmetic geometry and related areas. The AWS is organized around a different central topic each year and features a set of courses and accompanying research projects carefully designed and delivered by leading and emerging experts. The result is a unique fusion of traditional mathematics conference and intensive research workshop: the speakers organize courses of four or five lectures, provide lecture notes in advance, and propose research projects for graduate students to work on during the meeting. Nightly working sessions on these projects and on separate problem sets are run by the speakers and postdoctoral fellows. On the last day, students present their findings to the participants of the meeting. The result is a particularly intense and focused five days of mathematical activity for everyone involved. At the AWS, connections among peers are formed, and mentoring relationships between students and senior researchers are developed. Subsequent collaborations between participants at all levels are the norm. Students make concrete strides toward becoming research mathematicians, post-doctoral assistants gain valuable mentoring experience in their academic careers, and faculty develop new interests and see new connections that lead to important published results. The Southwest Center website shares reusable content from the Winter Schools, including lecture notes, project descriptions, and audio and video of lectures (both live-streamed and permanently archived). Through these thorough records, the dialogues begun at the AWS are extended to the greater community, and the efforts of the AWS participants are made freely and indefinitely available to all. More information about the upcoming and past Arizona Winter School programs can be found at the Southwest Center's website: http://swc.math.arizona.edu/The 2019 AWS will be held on the topic of Topology and Arithmetic, exploring the boundary of higher homotopy theory and arithmetic geometry. The development and maturation of higher category/homotopy theory is one of the most significant achievements in topology in the last decade, and has recently found striking applications to number theory and arithmetic geometry. The earliest applications were to topological quantum field theories, cobordisms, and elliptic cohomology; even here, the interaction between generalized cohomology theories and formal groups yielded results of interest to arithmetic geometry (especially, the elucidation of the geometry of Lubin--Tate and Gross--Hopkins moduli spaces of formal groups). More recent progress abounds. Gaitsgory--Lurie study Tamagawa numbers (a fundamental invariant of algebraic groups over global fields) via topological techniques. Bhatt--Morrow--Scholze establish the existence of a weight filtration on topological Hochschild homology whose graded pieces are related both to classical cohomology theories (de Rham, crystalline) and to integral p-adic Hodge theory. Work of Wickelgren and others develop arithmetic aspects of A1-homotopy theory, extending some enumerative problems from the complex numbers to other fields. The 2019 Winter School will focus on exposition of applications and connections to arithmetic geometry, rather than the development of a general theory.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.