Conference: Low-Dimensional Manifolds, their Geometry and Topology, Representations and Actions of their Fundamental Groups and Connections with Physics

该奖项为美国的参与者提供支持，以参加将于2023年4月至7月在ICMAT马德里和西班牙El Barco de Avila举行的一系列研讨会。正如标题所证明的那样，“低维流形，它们的几何和拓扑，它们的基本群的表示和作用以及与物理学的联系”，主题广泛经典，同时涉及现代数学的中心研究领域。研讨会将包括由顶尖专家进行的介绍性讲座和研究讲座，并为早期职业研究人员提供专业发展机会，展示他们自己的工作，并形成新的国际合作。这些研讨会看似不同的主题之间的多重联系可以通过一个离散群体到一个更大群体的代表空间所发挥的核心作用来阐明：例如李群、自同构群或同胚群。这些都是强大的工具，在拆包的结构，原来的群体，特别是当它是基本组的一个紧凑的流形。曲面群表示的经典情况特别值得注意，它与泰希穆勒理论、三流形拓扑和最近的高等泰希穆勒理论有联系。在高等泰希穆勒理论的背景下，通过一个致密表面（具有复杂结构）上希格斯束的透镜，给出了一个非常不同的视角。 自从希钦在大约30年前提出这一观点以来，这一观点在几何学、拓扑学和理论物理学中产生了巨大的影响。这些和许多其他研究领域，其中一个离散群体的表征空间是无处不在的将是主题方案的核心。有关研讨会的更多信息可在www.example.com上获得https://sites.google.com/view/javier-aramayona/agol-lab/activity-period-2023.This奖项反映了NSF的法定使命，并被认为值得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来支持。

This award provides support for U.S. based participants to attend a sequence of workshops, to be held at ICMAT Madrid and El Barco de Ávila, Spain, in April-July 2023. As evidenced by the title, “Low-dimensional manifolds, their geometry and topology, representations and actions of their fundamental groups and connections with physics”, the themes are broadly classical, while relating to central research areas in modern mathematics. The workshops will include introductory lectures and research talks by leading experts as well as offer the early career researchers the professional development opportunity to present their own work and form new international collaborations.The multiple connections between the seemingly disparate themes of these workshops can be articulated by the central role played by spaces of representations of a discrete group into a larger group: for example a Lie group, groups of diffeomorphisms or groups of homeomorphisms. These are powerful tools in unpacking the structure of the original group, particularly when it is the fundamental group of a compact manifold. The classical case of surface group representations is particularly noteworthy, providing connections with Teichmuller Theory, three-manifold topology and more recently Higher Teichmuller theory. In the setting of Higher Teichmuller theory there is a very different perspective given through the lens of Higgs bundles over a compact surface (equipped with a complex structure). Since Hitchin's original insight around thirty years ago, this viewpoint has had a tremendous impact in geometry, topology and theoretical physics. These and many other research areas where spaces of representations of a discrete group are ubiquitous will be at the core of the thematic program. More information about the workshops is available at https://sites.google.com/view/javier-aramayona/agol-lab/activity-period-2023.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.