Arithmetic Topology Conference

该奖项支持美国参与者参加2019年6月10日至14日在温哥华举行的太平洋数学科学研究所（PIMS）举办的“算术拓扑”研讨会。 近年来在数论的交叉点上取得了惊人的进展，特别是问方程有多少解的问题，以及研究形状和空间的数学性质的拓扑学。虽然数学家几十年来已经理解了单个方程有一个相关的空间，其属性反映了方程，但许多自然方程（和许多自然空间）都是无限序列。 在过去的十年里，我们在理解自然方程序列的渐近行为（“算术统计”）如何支配自然空间序列的渐近行为（“同调稳定性”）方面取得了惊人的进展（部分导致Venkatesh的2018年菲尔兹奖，数学界的最高荣誉）。该研讨会旨在汇集一群在数论，代数几何和拓扑学领域工作的领先和新兴研究人员，以获得快速新兴和多学科领域的全球视野，培训参与者可用的方法范围，并生成一个强大的问题列表，可以指导该领域的活动未来5-10年。在过去的10年里，代数拓扑学、数论和代数几何学的交叉点上出现了一系列活跃的现象。这就产生了大量的新定理，如Ellenberg-Venkatesh-韦斯特兰对函数域的Cohen-Lenstra拓扑学的证明; 2）拓扑学中拓扑学的新来源，如Vakil-Wood对Grothendieck环的预言，或Farb-Wolfson-Wood中同调密度的概念和重合; 3）利用现代拓扑工具对经典的计数定理进行了改进，如Kass-Wickelgren对三次曲面上27条线的算术计数; 4）重新关注不稳定同源性，如Galatius-Kupers-Randal-Williams和Miller-Wilson。研讨会的组织者认为，这些结果只是算术拓扑学新兴领域的开始。他们正在组织一个为期5天的研讨会，汇集了来自这些领域的各种初级和高级研究人员，目标是：1）让学员对一个快速发展的多学科领域有一个全面的认识，2）让学员详细了解可用的方法范围，以及3）出现了一个强大的问题清单，可以帮助指导该领域未来5-10年的活动。更多信息可以在会议网站上获得：https://www.pims.math.ca/scientific-event/190610-pwat.This奖项反映了NSF的法定使命，并被认为是值得的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，

This award supports US participants to attend the workshop "Arithmetic Topology" at the Pacific Institute of Mathematical Sciences (PIMS) from June 10-14, 2019 in Vancouver. Recent years have seen spectacular advances at the intersection of number theory, specifically problems asking how many solutions an equations has, and topology which studies mathematical properties of shapes and spaces. While mathematicians have understood for several decades that individual equations have an associated space whose properties reflect the equation, many natural equations (and many natural spaces) come in infinite sequences. The last decade has seen spectacular advances (leading in part to Venkatesh's 2018 Fields Medal, highest honor in mathematics) in our understanding of how the asymptotic behavior in natural sequences of equations ("arithmetic statistics") governs and is governed by the asymptotic behavior in natural sequences of spaces ("homological stability"). This workshop aims to bring together a diverse group of leading and emerging researchers working in number theory, algebraic geometry and topology to obtain a global view of a fast emerging and multidisciplinary area, to train participants in the range of methods available, and to generate a robust problem list that can guide activity in the area for the next 5-10 years. The last 10 years have brought a burst of activity at the intersection of algebraic topology, number theory and algebraic geometry. This has led to a wealth of:1) new theorems, such as Ellenberg-Venkatesh-Westerland's proof of theCohen-Lenstra heuristics for function fields; 2) new sources of heuristics in topology, such as Vakil-Wood's predictions from the Grothendieck ring, or the notion and coincidences of homological densities as in Farb-Wolfson-Wood; 3) refinements of classical enumerative theorems using modern topological tools, such as Kass-Wickelgren's arithmetic count of the 27 lines on a cubic surface; and4) a renewed focus on unstable homology, as in Galatius-Kupers-Randal-Williams and Miller-Wilson. The organizers of the workshop believe that these results are just the beginning of the emerging area of arithmetic topology. They are organizing a 5 day workshop to bring together a diverse group of junior and senior researchers from across these areas with the goal of: 1) giving participants a global view of a fast emerging and multidisciplinary area,2) giving participants a detailed awareness on the range of methods available, and3) emerging with a robust problem list which can help guide activity in the area for the next 5-10 years.More information is available at the conference website:https://www.pims.math.ca/scientific-event/190610-pwat.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.