Noncommutative Geometry and Analysis on Homogeneous Spaces

非交换几何与齐次空间分析

• 批准号: 2054725
• 负责人: Pierre Clare
• 金额: $ 3.97万
• 依托单位: College of William and Mary
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-03-01 至 2023-02-28
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2054725&HistoricalAwards=false
• 关键词: Noncommutative; Geometry; Analysis; Homogeneous; Spaces

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.

会议非交换几何和齐性空间分析，将在威廉斯堡的威廉玛丽学院举行，弗吉尼亚州已计划于2021年5月24日至28日，2022年1月COVID应急计划。调和分析是当今数学的核心领域，与其他科学领域有着深远的联系，从数论到物理学。会议将集中在调和分析的新发展，将非交换几何的方法和观点与表示论的方法和观点结合起来。在过去的几十年里，这两个领域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，但它们大多仍然是分开的。会议将汇集这两个领域的专家，以促进交流，并将新的联系纳入新一代谐波分析研究人员的培训中。为了使这两个领域的研究人员，特别是早期的职业研究人员，了解双方的观点，它将专门针对研究生和博士后研究人员开设两门迷你课程，并由著名数学家授课。其他活动将包括研究讲座，讨论会和海报会议，为参与者提供许多有意义的互动机会。早期的职业研究人员将有可能在特别有利于形成长期指导关系的背景下与知名专家进行互动。组织者将努力鼓励不同观众的参与，包括妇女和来自数学科学代表性不足的团体，组织和地理区域的个人。调和分析的基础是众所周知的原则，即几何空间可以通过其函数空间进行研究，并且考虑到空间的对称性，这些函数的分析可以简化。将周期函数分解为傅立叶级数是这一思想的一个基本表现形式，自20世纪40年代以来，这一直是Gelfand，Mackey，Dixlane和许多其他人的开创性工作的核心。在这方面，表示论和非交换几何都深深植根于调和分析。事实上，对调和表象的研究在20世纪后半叶产生了大量的活动，最终导致了哈里什-钱德拉的Plancherel定理，离散级数的实现以及克纳普和朱克曼对调和不可约表象的分类。最近，进一步努力沃根的代数方法已经导致了一个几乎完整的计算机辅助描述所有酉表示的真实的代数群。既然分类问题已经得到了近乎完整的答案，不同性质的问题也就受到了更多的关注。分析功能齐性空间的酉表示导致的基本问题，新的概念工具是必要的。与此同时，最近的结果在非交换几何清楚地表明，目前的调查有关的指标理论和拓扑方法的回火对偶通过康纳斯-卡斯帕罗夫同构和麦基-希格森双射提供了原来的见解和新的组织原则表示论。会议的重点是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站找到：https://sites.google.com/view/ncgahs20.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。