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This book discusses the modern theory of Laplace eigenfunctions through the lens of a new tool called geodesic beams. The authors provide a brief introduction to the theory of Laplace eigenfunctions followed by an accessible treatment of geodesic beams and their applications to sup norm estimates, L^p estimates, averages, and Weyl laws. Geodesic beams have proven to be a valuable tool in the study of Laplace eigenfunctions, but their treatment is currently spread through a variety of rather technical papers. The authors present a treatment of these tools that is accessible to a wider audience of mathematicians. Readers will gain an introduction to geodesic beams and the modern theory of Laplace eigenfunctions, which will enable them to understand the cutting edge aspects of this theory.
List of contents
Introduction.- The Laplace operator.- Axiomatic introduction to semiclassical analysis.- Basic properties of eigenfunctions and eigenvalues.- The Koch-Tataru-Zworski approach to L estimates.- Geodesic Beam Tools.- Applications of the geodesic beam decomposition.- Dynamical ideas.
About the author
Yaiza Canzani, Ph.D., is Associate Professor in the Department of Mathematics at the University of North Carolina at Chapel Hill. She received her Ph.D. in Mathematics at McGill University. After graduating, Dr. Canzani held postdoctoral positions at Harvard University and the Institute for Advanced Study. Her work has been recognized with a Sloan Fellowship, an NSF CAREER grant, and the AWM Sadosky Prize in Analysis.
Jeffrey Galkowski, Ph.D., is Professor in the Department of Mathematics at University College London. He received his Ph.D. in Mathematics at University of California at Berkeley. Dr. Galkowski held a NSF Postdoctoral Fellowship at Stanford University and the CRM-ISM postdoctoral fellowship at McGill University. His work has been recognized with an EPSRC early career fellowship as well as the Adams Prize in Mathematics from the University of Cambridge.
Report
"The book is adequate for advanced graduate students familiarized with semiclassical analysis. It might be an excellent choice for a reading course." (Emilio A. Lauret, zbMATH 1545.58001, 2024)
