Conference: 1, 2, 3: Curves, Surfaces, and 3-Manifolds

该NSF奖项为2023年5月7日至11日在以色列Nahsholim和理工学院举办的名为“1、2、3；曲线、曲面和3-流形”的会议的美国与会者提供部分支持。会议的目标是聚集所有职业生涯阶段的研究人员，讨论1维、2维和3维几何和拓扑学的交集。在过去的40年里，这些主题已经紧密地交织在一起，并为整个数学领域的许多发展提供了动力，例如几何群论。重要的是，会议还将包括研讨会式的演讲，旨在帮助弥合经验丰富的研究人员和新一代数学家之间的差距。三维流形的拓扑和几何的融合，以及映射类群的几何性质，在过去几十年中得到了深入的研究，在各种数学学科中具有核心意义。例如，双曲3-流形通常由其中包含的某些曲面所支配。这些曲面从包含到三维流形中继承了它们自己的双曲结构，并且这些结构由曲面上的曲线集合非常粗略地编码。奇迹般地，在关键情况下，双曲三维流形可以从这组曲线中重构出来--这是结束分层定理的基本内容。为研究用于证明这一定理的曲线而发展起来的数学同时为研究映射类曲面群提供了一个强大的工具包。这些工具的抽象现在为从几何角度研究非常庞大和健壮的一类群体提供了框架。更多细节可以在会议网站上找到：https://cms-math.net.technion.ac.il/123-curves-surfaces-and-3-manifolds/This奖项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 NSF award provides partial support for U.S. based participants of a conference titled "1, 2, 3; Curves, Surfaces, and 3-Manifolds" in Nahsholim Israel and the Technion Institute of Technology, May 7-11, 2023. The goal of the conference is to bring together researchers at all career stages to discuss intersections in geometry and topology in dimensions 1, 2, and 3. These topics have become intimately intertwined over the past 40 years, and have provided the impetus for the much of the development of entire fields of mathematics, such as geometric group theory. Importantly, the conference will also include workshop-style talks aimed at helping bridge the gap between seasoned researchers and the next generation of mathematicians.The confluence of topology and geometry of three dimensional manifolds, and the geometric properties of mapping class groups, has been intensively studied over the last several decades and is of central importance in a variety of mathematical disciplines. For example, hyperbolic 3-manifolds are often governed by certain surfaces contained in them. These surfaces inherit their own hyperbolic structures from their inclusion into the 3-manifold, and these structures are very coarsely encoded by collections of curves on the surface. Miraculously, in key situations, the hyperbolic 3-manifold can be reconstituted from this collection of curves--this is the essential content of the Ending Lamination Theorem. The mathematics developed to study curves used in the proof of this theorem simultaneously provide a powerful toolkit for studying mapping class groups of surfaces. Abstractions of these tools now provide the framework for studying a very large and robust class of groups from a geometric perspective. More details can be found on the conference website: https://cms-math.net.technion.ac.il/123-curves-surfaces-and-3-manifolds/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.