*Brian Conrad and Gopal Prasad*

- Published in print:
- 2015
- Published Online:
- October 2017
- ISBN:
- 9780691167923
- eISBN:
- 9781400874026
- Item type:
- book

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691167923.001.0001
- Subject:
- Mathematics, Numerical Analysis

This book goes further than the exploration of the general structure of pseudo-reductive groups to study the classification over an arbitrary field. An Isomorphism Theorem proved here determines the ...
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This book goes further than the exploration of the general structure of pseudo-reductive groups to study the classification over an arbitrary field. An Isomorphism Theorem proved here determines the automorphism schemes of these groups. The book also gives a Tits-Witt type classification of isotropic groups and displays a cohomological obstruction to the existence of pseudo-split forms. Constructions based on regular degenerate quadratic forms and new techniques with central extensions provide insight into new phenomena in characteristic 2, which also leads to simplifications of the earlier work. A generalized standard construction is shown to account for all possibilities up to mild central extensions. The results and methods developed in this book will interest mathematicians and graduate students who work with algebraic groups in number theory and algebraic geometry in positive characteristic.Less

This book goes further than the exploration of the general structure of pseudo-reductive groups to study the classification over an arbitrary field. An Isomorphism Theorem proved here determines the automorphism schemes of these groups. The book also gives a Tits-Witt type classification of isotropic groups and displays a cohomological obstruction to the existence of pseudo-split forms. Constructions based on regular degenerate quadratic forms and new techniques with central extensions provide insight into new phenomena in characteristic 2, which also leads to simplifications of the earlier work. A generalized standard construction is shown to account for all possibilities up to mild central extensions. The results and methods developed in this book will interest mathematicians and graduate students who work with algebraic groups in number theory and algebraic geometry in positive characteristic.

*Christopher D. Sogge*

- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691160757
- eISBN:
- 9781400850549
- Item type:
- book

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691160757.001.0001
- Subject:
- Mathematics, Numerical Analysis

Based on lectures given at Zhejiang University in Hangzhou, China, and Johns Hopkins University, this book introduces eigenfunctions on Riemannian manifolds. The book gives a proof of the sharp Weyl ...
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Based on lectures given at Zhejiang University in Hangzhou, China, and Johns Hopkins University, this book introduces eigenfunctions on Riemannian manifolds. The book gives a proof of the sharp Weyl formula for the distribution of eigenvalues of Laplace–Beltrami operators, as well as an improved version of the Weyl formula, the Duistermaat-Guillemin theorem under natural assumptions on the geodesic flow. The book shows that there is quantum ergodicity of eigenfunctions if the geodesic flow is ergodic. It begins with a treatment of the Hadamard parametrix before proving the first main result, the sharp Weyl formula. The book avoids the use of Tauberian estimates and instead relies on sup-norm estimates for eigenfunctions. It also gives a rapid introduction to the stationary phase and the basics of the theory of pseudodifferential operators and microlocal analysis. These are used to prove the Duistermaat-Guillemin theorem. Turning to the related topic of quantum ergodicity, the book demonstrates that if the long-term geodesic flow is uniformly distributed, most eigenfunctions exhibit a similar behavior, in the sense that their mass becomes equidistributed as their frequencies go to infinity.Less

Based on lectures given at Zhejiang University in Hangzhou, China, and Johns Hopkins University, this book introduces eigenfunctions on Riemannian manifolds. The book gives a proof of the sharp Weyl formula for the distribution of eigenvalues of Laplace–Beltrami operators, as well as an improved version of the Weyl formula, the Duistermaat-Guillemin theorem under natural assumptions on the geodesic flow. The book shows that there is quantum ergodicity of eigenfunctions if the geodesic flow is ergodic. It begins with a treatment of the Hadamard parametrix before proving the first main result, the sharp Weyl formula. The book avoids the use of Tauberian estimates and instead relies on sup-norm estimates for eigenfunctions. It also gives a rapid introduction to the stationary phase and the basics of the theory of pseudodifferential operators and microlocal analysis. These are used to prove the Duistermaat-Guillemin theorem. Turning to the related topic of quantum ergodicity, the book demonstrates that if the long-term geodesic flow is uniformly distributed, most eigenfunctions exhibit a similar behavior, in the sense that their mass becomes equidistributed as their frequencies go to infinity.

*Charles Fefferman, Alexandru D. Ionescu, D.H. Phong, and Stephen Wainger*

- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691159416
- eISBN:
- 9781400848935
- Item type:
- book

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691159416.001.0001
- Subject:
- Mathematics, Numerical Analysis

Princeton University's Elias Stein was the first mathematician to see the profound interconnections that tie classical Fourier analysis to several complex variables and representation theory. His ...
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Princeton University's Elias Stein was the first mathematician to see the profound interconnections that tie classical Fourier analysis to several complex variables and representation theory. His fundamental contributions include the Kunze–Stein phenomenon, the construction of new representations, the Stein interpolation theorem, the idea of a restriction theorem for the Fourier transform, and the theory of Hp Spaces in several variables. Through his great discoveries, through books that have set the highest standard for mathematical exposition, and through his influence on his many collaborators and students, Stein has changed mathematics. Drawing inspiration from Stein's contributions to harmonic analysis and related topics, this book gathers papers from internationally renowned mathematicians, many of whom have been Stein's students. The book also includes expository papers on Stein's work and its influence.Less

Princeton University's Elias Stein was the first mathematician to see the profound interconnections that tie classical Fourier analysis to several complex variables and representation theory. His fundamental contributions include the Kunze–Stein phenomenon, the construction of new representations, the Stein interpolation theorem, the idea of a restriction theorem for the Fourier transform, and the theory of Hp Spaces in several variables. Through his great discoveries, through books that have set the highest standard for mathematical exposition, and through his influence on his many collaborators and students, Stein has changed mathematics. Drawing inspiration from Stein's contributions to harmonic analysis and related topics, this book gathers papers from internationally renowned mathematicians, many of whom have been Stein's students. The book also includes expository papers on Stein's work and its influence.