Spectral Analysis On Standard Locally Homogeneous Spaces
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Spectral Analysis on Standard Locally Homogeneous Spaces
A groundbreaking theory has emerged for spectral analysis of pseudo-Riemannian locally symmetric spaces, extending beyond the traditional Riemannian framework. The theory introduces innovative approaches to global analysis of locally symmetric spaces endowed with an indefinite metric. Breakthrough methods in this area are introduced through the development of the branching theory of infinite-dimensional representations of reductive groups, which is based on geometries with spherical hidden symmetries. The book elucidates the foundational principles of the new theory, incorporating previously inaccessible material in the literature. The book covers three major topics. (1) (Theory of Transferring Spectra) It presents a novel theory on transferring spectra along the natural fiber bundle structure of pseudo-Riemannian locally homogeneous spaces over Riemannian locally symmetric spaces. (2) (Spectral Theory) It explores spectral theory for pseudo-Riemannian locally symmetric spaces, including the proof of the essential self-adjointness of the pseudo-Riemannian Laplacian, spectral decomposition of compactly supported smooth functions, and the Plancherel-type formula. (3) (Analysis of the Pseudo-Riemannian Laplacian) It establishes the abundance of real analytic joint eigenfunctions and the existence of an infinite L2 spectrum under certain additional conditions.
Symmetry in Geometry and Analysis, Volume 1
Symmetry in Geometry and Analysis is a Festschrift honoring Toshiyuki Kobayashi. The three volumes feature 35 selected contributions from invited speakers of twin conferences held in June 2022 in Reims, France, and in September 2022 in Tokyo, Japan. These contributions highlight the profound impact of Prof. Kobayashi’s pioneering ideas, groundbreaking discoveries, and significant achievements in the development of analytic representation theory, noncommutative harmonic analysis, and the geometry of discontinuous groups beyond the Riemannian context, among other areas, over the past four decades. The first volume of the Festschrift includes a survey article on Kobayashi’s innovative contributions to Mathematics, emphasizing their influence and introducing new perspectives across various fields. Original articles contained in Volume 1 focus on differential geometry with symmetries as well as algebraic and geometric aspects of representation theory of reductive Lie groups and related topics. Contributions are by Velleda Baldoni, Dan Barbasch, Leticia Barchini, Sigiswald Barbier, Yves Benoist, Sam Claerebout, Michael Eastwood, Wee Teck Gan, William M. Goldman, Roger Howe, Kazuki Kannaka, Toshihisa Kubo, Hung Yean Loke, Jia-Jun Ma, Reiko Miyaoka, Kento Ogawa, Takayuki Okuda, Yoshiki Oshima, Paul-Émile Paradan, Annegret Paul, Michael Pevzner, Yiannis Sakellaridis, Atsumi Sasaki, Gordan Savin, Hideko Sekiguchi, Binyong Sun, Yuichiro Tanaka, Koichi Tojo, Peter Trapa, Michèle Vergne, Joseph A. Wolf, Kayue Daniel Wong, and Chen-Bo Zhu. The Mathematical Work of Toshiyuki Kobayashi is available open access under a Creative Commons Attribution 4.0 International License via link.springer.com.
Mathematics Going Forward
This volume is an original collection of articles by 44 leading mathematicians on the theme of the future of the discipline. The contributions range from musings on the future of specific fields, to analyses of the history of the discipline, to discussions of open problems and conjectures, including first solutions of unresolved problems. Interestingly, the topics do not cover all of mathematics, but only those deemed most worthy to reflect on for future generations. These topics encompass the most active parts of pure and applied mathematics, including algebraic geometry, probability, logic, optimization, finance, topology, partial differential equations, category theory, number theory, differential geometry, dynamical systems, artificial intelligence, theory of groups, mathematical physics and statistics.