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AMS/IP Studies in Advanced Mathematics 2009; 482 pp; softcover Volume: 47 ISBN10: 0821893939 ISBN13: 9780821893937 List Price: US$119 Member Price: US$95.20 Order Code: AMSIP/47.S See also: Stochastic Analysis on Manifolds  Elton P Hsu The Ricci Flow: An Introduction  Bennett Chow and Dan Knopf Spectrum and Dynamics: Proceedings of the Workshop Held in Montréal, QC, April 711, 2008  Dmitry Jakobson, Stephane Nonnenmacher and Iosif Polterovich  The heat kernel has long been an essential tool in both classical and modern mathematics but has become especially important in geometric analysis as a result of major innovations beginning in the 1970s. The methods based on heat kernels have been used in areas as diverse as analysis, geometry, and probability, as well as in physics. This book is a comprehensive introduction to heat kernel techniques in the setting of Riemannian manifolds, which inevitably involves analysis of the LaplaceBeltrami operator and the associated heat equation. The first ten chapters cover the foundations of the subject, while later chapters deal with more advanced results involving the heat kernel in a variety of settings. The exposition starts with an elementary introduction to Riemannian geometry, proceeds with a thorough study of the spectraltheoretic, Markovian, and smoothness properties of the Laplace and heat equations on Riemannian manifolds, and concludes with Gaussian estimates of heat kernels. Grigor'yan has written this book with the student in mind, in particular by including over 400 exercises. The text will serve as a bridge between basic results and current research. Titles in this series are copublished with International Press, Cambridge, MA. Readership Graduate students and research mathematicians interested in geometric analysis; heat kernel methods in geometry and analysis. Reviews "And the book under review is indeed a wonderful source for this material. The author's approach is both broad and deep, and extremely thorough and rigorous. ... poised to be an important book in the field and a valuable pedagogical contribution. It is bound to be an important player on the scene for years to come."  MAA Reviews Table of Contents



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