Overview

Full Product Details

9781108499705

Table of Contents

Reviews

The topic of decoupling is now a major active area of research in both harmonic analysis and analytic number theory. There are a number of survey articles and lecture notes already on these topics, but this book - by one of the leading contributors to the field - is more comprehensive than any of these, being almost completely self-contained and detailing a number of older results as well as the more recent ones. It also has a number of exercises and insightful commentary. This text is an excellent resource, both for students and for existing researchers in the field. Terence Tao, UCLA I think that the book Fourier restriction, decoupling and applications, by Ciprian Demeter, will be a really valuable addition to the literature, and I endorse it strongly. Restriction theory is an exciting area of Fourier analysis. It is organized around some deep central questions, raised by Stein more than 50 years ago, about the connection between geometry and the Fourier transform. These central questions are still open and look very difficult, but there has been fundamental progress in our understanding over that time, and especially in the last decade. The questions are interesting in their own right, but they also have a number of applications. Early work on restriction theory led to the Strichartz estimates, which are a ubiquitous tool in dispersive PDE, and later developments have also had impact in PDE. - for instance giving the sharp Strichartz estimates for periodic solutions of the Schrodinger equation. Recently, ideas coming from restriction theory have had a significant impact in number theory: for instance, they led to a proof of Vinogradov's mean value conjecture from the 1930s, which in turn gives improved estimates for Waring's problem. Ideas from restriction theory also play a role in the best current estimates for the Lindelhof hypothesis and the Gauss circle problem. The author is one of the main contributors to the recent developments that we mentioned above. This book discusses the major recent developments in the field. He has worked hard to explain them clearly and reduce the technical details. For instance, he presents relatively simple cases of each development first, and then describes how to build to more general cases. He pauses to discuss examples and heuristics, or to look back at the end of a proof and reflect on the strategy. He is also careful and rigorous. And he provides plenty of exercises. I work in this area myself and I have a number of students learning the field. In the past, I have used lecture notes fr

'The topic of decoupling is now a major active area of research in both harmonic analysis and analytic number theory. There are a number of survey articles and lecture notes already on these topics, but this book - by one of the leading contributors to the field - is more comprehensive than any of these, being almost completely self-contained and detailing a number of older results as well as the more recent ones. It also has a number of exercises and insightful commentary. This text is an excellent resource, both for students and for existing researchers in the field.' Terence Tao, University of California, Los Angeles 'This book gives a self-contained introduction to some major recent developments in Fourier analysis. The restriction conjecture, raised by Stein in the 1960s, is still open and looks very difficult, but there has been fundamental progress in the area over the last decade, leading to striking applications in PDE and analytic number theory. This book is written by one of the main players in those developments. Demeter has worked hard to present the key points clearly and to minimize the technical issues involved by starting with simple cases of each new idea and pausing to give heuristics and examples.' Larry Guth, Massachuetts Institute of Technology 'Restriction theory, which has long occupied a central place in Euclidean harmonic analysis, has gained new urgency and impetus with the development of decoupling. Ciprian Demeter is one of the main contributors to this area as well as an excellent expositor of it. New researchers have already been clamouring for instructional materials on the subject. Experienced harmonic analysts will still want to read it for new insights from one of the main players in the field. Starting from classical results re-interpreted in light of the current understanding of the subject, the book then proceeds to the more recent topics culminating in the Bourgain-Demeter-Guth proof of Vinogradov's conjecture. I have been awaiting this book eagerly, and it did not disappoint. I expect it to be a constant presence on my desk, from graduate teaching to my own research, for many years to come.' Izabella Laba, University of British Columbia 'This book deals with the spectacular recent developments in modern Fourier analysis, with an emphasis on restriction theory and decoupling. Some of the results are new and many are just a few years old, notably the breakthrough theorems of Bourgain and Demeter on decoupling and their many applications. It is wonderful that this material is available in book form so soon, especially as the author has succeeded admirably in his goal of bringing forth the central ideas without obscuring them with too many technical details. Thus the presentation is accessible to non-experts and the book will be valuable for a wide readership, including graduate students.' Pertti Mattila, University of Helsinki

Author Information

Ciprian Demeter is Professor of Mathematics at Indiana University, Bloomington. He is one of the world's leading experts in Fourier restriction theory and its applications to number theory, which he teaches regularly at the graduate level. He received the Sloan fellowship in 2009 and was an invited speaker at the 2018 International Congress of Mathematicians in Rio de Janeiro.

Tab Content 6

Customer Reviews

Recent Reviews

Countries Available