Collaborative Research: Conference: Trisections Workshops: Connections with Knotted Surfaces and Diffeomorphisms

这项提案将资助将于2024年6月24-28日在内布拉斯加州大学林肯分校举行的“三叉树研讨会：与结面的联系”，以及将于2025年夏天在德克萨斯大学奥斯汀分校举行的“三叉树研讨会：与差异同构的联系”。研讨会的参与者将包括知名专家、早期职业研究人员和学生，该计划将积极吸引所有参与者。每天上午将有专家的全体演讲和/或闪电演讲，突出初级研究人员的工作。下午的时间将专门用来分组研究开放问题。这一系列的定期研讨会对于在低维拓扑学中发展一个热情的研究人员社区至关重要，帮助这个新的和不断增长的领域获得动力，并促进在职业阶段和人口统计方面的大量合作。主办方为他们努力营造的同志情谊和欢迎氛围感到自豪，许多社区成员对这些活动深表重视和赞赏。三分割法将一个4维空间分成三个简单的部分。自从大约十年前提出以来，三分集已被证明是研究光滑4-流形的一种成功的新工具，在此期间撰写了大量文章来发展三分集理论的基础。三分理论的一个重要优点是它与低维拓扑学中的各种其他主题相结合的方式。这个界面提供了一个机会，通过一个新的镜头来探索4-流形拓扑的许多经典领域。这些领域包括，例如，研究4-空间中的纽结曲面，4-流形的微分同胚，奇异光滑结构，群作用和(分支)覆盖空间，以及辛结构。这些讲习班的主要目标是将来自多个领域的研究人员聚集在一起，提出开放问题并开展工作，特别注重纳入早期职业研究人员。讲习班之前举行了一系列介绍性虚拟讲习班前讲座，这些讲座有助于使新的研究人员掌握最新情况，促进小组工作的完成，并吸收更广泛的全球受众。2024年研讨会的网站可以在这里找到：https://sites.google.com/view/tw2024.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 proposal will fund the “Trisections Workshop: Connections with Knotted Surfaces,” which will take place at the University of Nebraska-Lincoln from June 24-28, 2024, and the “Trisections Workshop: Connections with Diffeomorphisms,” which will take place at the University of Texas at Austin during one week in the summer of 2025. Workshop attendees will include established experts, early-career researchers, and students, and the program will actively engage all participants. Each morning will feature plenary talks by experts and/or lightning talks highlighting the work of junior researchers. The afternoons will be devoted to working in groups on open problems. This series of regular workshops has been critical to the development of an enthusiastic community of researchers in low-dimensional topology, helping this new and growing area gain momentum and fostering numerous collaborations across career stages and demographics. The organizers take pride in the camaraderie and welcoming atmosphere they strive to create, and many in the community deeply value and appreciate these events. A trisection splits a 4-dimensional space into three simple pieces. Since their introduction roughly a decade ago, trisections have proven to be a successful new tool with which to study smooth 4-manifolds, with numerous articles written in the interim to develop the foundations for trisection theory. An important strength of the theory of trisections is the way it interfaces with a variety of other topics in low-dimensional topology. This interface provides an opportunity to explore many classical areas of 4-manifold topology through a new lens. Such areas include, for example, the study of knotted surfaces in 4-space, diffeomorphisms of 4-manifolds, exotic smooth structures, group actions and (branched) covering spaces, and symplectic structures. The main goal of these workshops is to bring together researchers from multiple areas to propose and to work on open problems, with a particular focus on the inclusion of early career researchers. The workshops are preceded by a series of introductory virtual pre-workshop talks, which serve to bring new researchers up to speed, to facilitate the work to be done in groups, and to incorporate a broader, worldwide audience. The website for the 2024 workshop can be found here: https://sites.google.com/view/tw2024.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.