Southwest Conference on Arithmetic Geometry

在该奖项的支持下，西南算术几何中心将于2023年至2025年在亚利桑那州图森市的亚利桑那大学继续其年度“冬季学校”系列，第一次将于2023年3月4日至8日举行。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，已成为一个突出的全国性活动，并为算术几何和相关领域的研究生提供高水平的培训和研究经验。AWS每年围绕一个不同的中心主题组织，并提供一系列课程和由领先和新兴专家精心设计和交付的附带研究项目。其结果是传统的数学会议和密集的研究研讨会的独特融合：演讲者组织四个或五个讲座的课程，并提出研究生在会议期间工作的研究项目。关于这些项目和单独问题集的夜间工作会议由演讲者和博士后研究员主持。在最后一天，学生们向与会者介绍他们的发现。其结果是一个特别激烈和集中的五天的数学活动，每个人都参与。这个周期西南中心将扩大其方案，包括初步亚利桑那州冬季学校（PAWS）。 这个虚拟项目针对来自代表性不足群体的高级本科生和大三研究生，将在秋季学期围绕与AWS相同的主题举办两个为期六周的课程。参与者将参加由高级研究生举办的每周问题会议，并参加社区建设和导师活动。在AWS和PAWS，同行之间的联系形成，并在学生和高级研究人员之间建立指导关系。随后各级参与者之间的合作成为常态。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，教师们发展了新的兴趣，并看到了导致重要成果发表的新联系。西南中心网站共享AWS和PAWS的可重用内容，包括课堂讲稿、项目描述以及讲座的音频和视频。通过这些全面的记录，西南中心参与者的努力是免费提供给所有人。有关即将到来的和过去的亚利桑那州冬季学校计划的更多信息可以在西南中心的网站上找到：http://swc.math.arizona.edu/In 2022-2023，PAWS和AWS将在“不太可能的交叉点”。“这个主题涉及某些罕见的算术或几何有趣的交叉的期望有限性。例如，给定一条曲线和它周围空间中的一个可数的“特殊”点集合，人们期望曲线与这组点的交点是有限的，除非曲线也是“特殊的”。“最近在这个领域的主要结构和补充问题上取得了实质性的进展，特别是在André-Oort/Lang-Manin-Mumford/Zilber-Pink结构和算术动力学的类似物上。这项工作不仅汇集了代数和算术几何的经典工具，而且还包括逻辑，高度，解析数论，动力学和p进几何的工具。该计划将向学生介绍这个蓬勃发展的领域的核心问题，应用和独特的工具。该奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

With support from this award, the Southwest Center for Arithmetic Geometry will continue its series of annual "Winter Schools" from 2023 to 2025 at the University of Arizona in Tucson, AZ, the first taking place March 4-8, 2023. 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 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. This cycle the Southwest Center will expand its programming to include Preliminary Arizona Winter School (PAWS). This virtual program aimed at advanced undergraduates and junior graduate students from underrepresented groups will feature two six-week-long courses during the fall semester organized around the same topic as the AWS. Participants will engage in weekly problem sessions run by advanced graduate students and participate in community building and mentorship activities. At both AWS and PAWS, 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 both AWS and PAWS, including lecture notes, project descriptions, and audio and video of lectures. Through these thorough records, the efforts of the Southwest Center participants are made freely 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/In 2022-2023, PAWS and AWS will be on "Unlikely Intersections." This topic concerns the expected finiteness of certain rare arithmetically or geometrically interesting intersections. For example, given a curve and a countable collection of "special" points in its ambient space, one expects the intersection of the curve with this set of points to be finite unless the curve is also "special." There has been substantial recent progress on the main conjectures and complementary problems of this field, especially on the André-Oort/Lang-Manin-Mumford/Zilber-Pink conjectures and analogues from arithmetic dynamics. This work draws together not just classical tools from algebraic and arithmetic geometry, but also tools from logic, heights, analytic number theory, dynamics, and p-adic geometry. The program will introduce students to the central problems, applications, and unique tools of this flourishing field.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.