Noncommutative Geometry and Analysis on Homogeneous Spaces

非交换几何与齐次空间分析会议将于2021年5月24日至28日在弗吉尼亚州威廉斯堡的威廉玛丽学院举行，2019冠状病毒病应急计划将于2022年1月举行。谐波分析是当今数学的一个中心领域，与从数论到物理学的其他科学领域有着深远的联系。会议将集中讨论调和分析的新发展，将非交换几何的方法和观点与表示理论的方法和观点结合起来。在过去的几十年里，两者都取得了惊人的进步，使它们成为广阔而充满活力的研究领域，但它们大多是分开的。会议将汇集这两个领域的专家，以促进交流，并在培训新一代谐波分析研究人员方面建立新的联系。为了让这两个领域的研究人员，特别是早期职业研究人员了解双方的观点，该项目将开设两门专门针对研究生和博士后研究人员的迷你课程，由著名数学家讲授。其他活动包括研究讲座、讨论环节和海报环节，为参与者提供许多有意义的互动机会。早期职业研究人员将有可能在特别有利于形成长期师徒关系的环境中与知名专家进行互动。组织者将努力鼓励多样化的观众参与，包括妇女和来自数学科学领域代表性不足的团体、组织和地理区域的个人。调和分析的基础是一个众所周知的原理，即可以通过函数空间来研究几何空间，并且通过考虑空间的对称性来简化这些函数的分析。周期函数的傅立叶级数分解是这一思想的基本表现，这一思想是Gelfand， Mackey， Dixmier和许多其他人自20世纪40年代以来开创性工作的核心。在这方面，表示理论和非交换几何都深深植根于谐波分析。事实上，在20世纪下半叶，对调质表征的研究产生了大量的活动，最终导致了harich - chandra的Plancherel定理，离散级数的实现以及Knapp和Zuckerman对调质不可约表征的分类。最近，继Vogan的代数方法之后，进一步的努力已经导致了对真实代数群的所有酉表示的几乎完整的计算机辅助描述。现在，分类问题已经给出了几乎完整的答案，不同性质的问题正在受到更多的关注。从酉表示的角度分析齐次空间上的函数会引出一些基本问题，需要新的概念工具。同时，非交换几何的最新结果清楚地表明，目前有关指数论的研究以及通过Connes-Kasparov同构和麦基-希格森双射的缓变对偶的拓扑方法为表示理论提供了新的见解和新的组织原则。会议将重点讨论通过两个社区专家的共同努力，将目前的方法从群体案例扩展到同质空间。有关会议的更多信息可在以下网址找到：https://sites.google.com/view/ncgahs20.This奖项反映了NSF的法定使命，并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，被认为值得支持。

The conference Noncommutative Geometry and Analysis on Homogeneous Spaces, to take place at the College of William & Mary in Williamsburg, VA has been planned for May 24 - 28, 2021, with COVID contingency plans for January, 2022. Harmonic Analysis is a central area in today's Mathematics, with far-reaching connections to other fields of Science, from Number Theory to Physics. The meeting will focus on new developments in Harmonic Analysis that bring together the methods and perspectives of Noncommutative Geometry with those of Representation Theory. Both have seen spectacular advances over the past decades, leading them to become vast and vibrant areas of research, yet they remain mostly separated. The meeting will bring together experts in both fields to foster exchanges and incorporate new connections in the training of a new generation of researchers in Harmonic Analysis. In order to acquaint researchers in the two areas, especially early career researchers, with the perspectives from both sides, it will feature two mini-courses specifically aimed at graduate students and postdoctoral researchers and delivered by prominent mathematicians. Other activities will include research talks, discussion sessions and a poster session, giving participants many opportunities for meaningful interactions. Early career researchers will have the possibility to interact with established experts in a context particularly conducive to the formation of long-lasting mentoring relations. The organizers will strive to encourage participation of a diverse audience, including women and individuals from groups, organizations, and geographic regions that are underrepresented in the mathematical sciences.At the foundation of Harmonic Analysis are the well-known principles that a geometric space may be studied through its space of functions, and that the analysis of these functions is simplified by taking into account the symmetries of the space. The decomposition of periodic functions into Fourier series is an elementary manifestation of this idea, which has been at the core of the pioneering work of Gelfand, Mackey, Dixmier and many others since the 1940s. In that regard, the Representation Theory and Noncommutative Geometry are both deeply rooted in Harmonic Analysis. Indeed, the study of tempered representations generated an immense amount of activity in the second half of the twentieth century, culminating in Harish-Chandra's Plancherel theorem, the realization of the discrete series and the classification of tempered irreducible representations by Knapp and Zuckerman. More recently, further efforts following Vogan's algebraic approach have led to a nearly complete computer-assisted description of all unitary representations of real algebraic groups. Now that classification questions have been given nearly complete answers, problems of different natures are receiving more attention. The analysis of functions on homogeneous spaces in terms of unitary representations leads to fundamental questions for which new conceptual tools are necessary. At the same time, recent results in Noncommutative Geometry clearly indicate that current investigations related to Index Theory and the topological approach to the tempered dual via the Connes-Kasparov isomorphism and the Mackey-Higson bijection provide original insight and new organizing principles in Representation Theory. The meeting will focus on extending the current methods beyond the group case to that of homogeneous spaces through joint effort of specialists from the two communities. Further information about the conference may be found at: https://sites.google.com/view/ncgahs20.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.