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

该奖项为在美国的参与者提供支持，以参加将于2023年4-7月在西班牙马德里ICMAT和El Barco deávila举行的一系列研讨会。正如题为“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”所证明的那样，这些主题广泛地是经典的，同时涉及现代数学的中心研究领域。研讨会将包括主要专家的介绍性讲座和研究演讲，并为早期职业研究人员提供展示他们自己的工作和形成新的国际合作的职业发展机会。这些研讨会看似不同的主题之间的多重联系可以通过离散群到更大组的表示空间所起的中心作用来阐明：例如，李群、微分同胚群或同胚群。这些都是解开原始群的结构的有力工具，特别是当它是紧致流形的基本群时。曲面群表示的经典情况尤其值得注意，它提供了与TeichMuller理论、三流形拓扑和最近更高的TeichMuller理论的联系。在高等泰希穆勒理论的背景下，通过希格斯丛的透镜在紧凑的表面(配备了复杂的结构)给出了非常不同的视角。自从希钦大约30年前提出最初的见解以来，这一观点在几何学、拓扑学和理论物理学中都产生了巨大的影响。这些和许多其他研究领域，其中一个离散群体的表示空间无处不在，将是主题方案的核心。有关研讨会的更多信息可在https://sites.google.com/view/javier-aramayona/agol-lab/activity-period-2023.This上获得，该奖项反映了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 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.