Arithmetic Topology Conference

该奖项支持美国参与者参加2019年6月10日至14日在温哥华太平洋数学科学研究所(PIMS)举行的“算术拓扑学”研讨会。近年来，数论和拓扑学在数论和拓扑学的交叉领域取得了令人瞩目的进展，特别是关于一个方程有多少解的问题，拓扑学研究形状和空间的数学性质。虽然数学家们几十年来一直明白，单个方程有一个相关的空间，其性质反映了方程的性质，但许多自然方程(以及许多自然空间)是无限序列的。在过去的十年里，我们在理解自然方程序列中的渐近行为(“算术统计”)如何支配和受控于自然空间序列中的渐近行为(“同调稳定性”)方面取得了惊人的进步(部分导致了文卡特什2018年的菲尔兹奖，这是数学上的最高荣誉)。这次研讨会的目的是将在数论、代数几何和拓扑学领域工作的一批领先的和新兴的研究人员聚集在一起，以获得一个快速新兴的多学科领域的全球视角，培训参与者现有的方法范围，并生成一个强大的问题清单，以指导未来5-10年在该领域的活动。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股热潮。这导致了许多新的定理，例如Ellenberg-Venkatesh-Westland对函数域的Cohen-Lenstra启发式的证明；2)拓扑学中的启发式的新来源，例如Vakil-Wood从Grothendieck环的预测，或同调密度的概念和重合，如Farb-Wolfson-Wood；3)使用现代拓扑工具对经典计数定理的改进，例如Kass-Wickelgren对三次曲面上27条直线的算术计数；以及4)重新关注不稳定的同调，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，将来自这些领域的不同的初级和高级研究人员聚集在一起，目标是：1)让与会者对一个快速崛起的多学科领域有一个全球视野，2)让与会者对可用的方法范围有一个详细的认识，3)提出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。会议上提供了更多信息，website:https://www.pims.math.ca/scientific-event/190610-pwat.This奖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。

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.