Conference: I.H.E.S. Workshop: Homogeneous Dynamics and Geometry in Higher-Rank Lie Groups

该奖项的资金将用于支持美国参加将于2023年6月19日至23日在法国伊维特河畔布尔斯的高级科学学院举办的“高阶李群中的均匀动力学和几何”研讨会。研讨会的目的是将最近在研究高阶半单李群的离散子群方面取得重大进展的两个知识界聚集在一起：齐次动力学社区和高阶TeichMuller理论社区。这一事件是及时的，因为来自齐次动力学的技术在这两个领域的李群的离散子群的研究中正变得越来越重要。每天上午将有两个小型课程讲座，旨在向研究人员介绍这两个领域的技术，而下午将专门用于研究讲座。将这两个小组结合在一起将导致进一步的互动，并将加快这些令人兴奋的领域的发展。组织者致力于资助一个不同的数学家群体。齐次动力学通过离散的子群研究李群的商上的流。在有限体积半单李群的齐次空间中，即当离散子群是格时，关于轨道闭合、测度分类、计数和均匀分布的研究已经发展得很成熟，现在已经是经典的研究。这些想法在看似不相关的领域有许多应用，例如，数论中的奥本海姆和利特尔伍德猜想的解，以及最近在泰希穆勒动力学中的应用。在过去的十年里，在将这一理论扩展到研究非格的一阶李群的离散子群方面取得了显著的进展。这项工作的基本工具是与离散子群在双曲空间几何边界上的作用有关的Patterson-Sullivan测度理论。将这些工作推广到具有无限余体积的高阶齐次空间的时机已经成熟。这一理论在阿诺索夫代表的背景下已经有了令人兴奋的初步发展。关于研讨会的更多细节，包括完整的演讲者名单，可以在会议网站上找到：https://indico.math.cnrs.fr/event/8759/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。

Funding for this award will support US participants at a workshop on ``Homogeneous Dynamics and Geometry in Higher-Rank Lie Group'' to be held June 19--23, 2023, at Institut des Hautes Etudes Scientifiques in Bures-sur-Yvette, France. The aim of the workshop is to bring together two intellectual communities that have recently made significant advances in the study of discrete subgroups of higher rank semi-simple Lie groups: the homogeneous dynamics community and the higher rank Teichmuller theory community. The event is timely, since techniques from homogenous dynamics are becoming increasingly prominent in the study of discrete subgroups of Lie groups in both fields. Each morning there will be two mini-course lectures, intended to introduce researchers to techniques from both fields, while the afternoon will be devoted to research lectures. Bringing together these two groups will lead to further interactions and will accelerate development in these exciting areas. The organizers are committed to funding a diverse group of mathematicians.Homogeneous dynamics deals with flows on the quotients of Lie groups by discrete subgroups. There is a well-developed, now classical, study of orbit closures, measure classifications, counting and equidistribution results in homogeneous spaces of semisimple Lie groups of finite volume, i.e. when the discrete subgroup is a lattice. These ideas have had many applications in seemingly unrelated areas, for example, the solutions of the Oppenheim and Littlewood conjectures in number theory, and more recently in Teichmuller dynamics. In the last decade, there has been significant progress in extending this theory to study discrete subgroups of rank one Lie groups which are not lattices. The fundamental tool in this work is the theory of Patterson-Sullivan measures associated to actions of discrete subgroups on the geometric boundaries of hyperbolic spaces. The time is ripe for the pursuit of generalizations of these works to higher rank homogeneous spaces of infinite co-volume. This theory has already seen exciting preliminary development in the context of Anosov representations.Further details about the workshop, including a full list of speakers, are available at the conference website: https://indico.math.cnrs.fr/event/8759/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.