Noncommutative Geometry and Analysis on Homogeneous Spaces

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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.