Paperback | April 2016 | ISBN: 9780691170299
Hardcover | April 2016 | ISBN: 9780691170282
616 pp. | 7 x 10
The p-adic Simpson correspondence, recently initiated by Gerd Faltings, aims at describing all p-adic representations of the fundamental group of a proper smooth variety over a p-adic field in terms of linear algebra?namely Higgs bundles. This book undertakes a systematic development of the theory following two new approaches, one by Ahmed Abbes and Michel Gros, the other by Takeshi Tsuji. The authors mainly focus on generalized representations of the fundamental group that are p-adically close to the trivial representation.
The first approach relies on a new family of period rings built from the torsor of deformations of the variety over a universal p-adic thickening defined by J. M. Fontaine. The second approach introduces a crystalline-type topos and replaces the notion of Higgs bundles with that of Higgs isocrystals. The authors show the compatibility of the two constructions and the compatibility of the correspondence with the natural cohomologies. The last part of the volume contains results of wider interest in p-adic Hodge theory. The reader will find a concise introduction to Faltings' theory of almost etale extensions and a chapter devoted to the Faltings topos. Though this topos is the general framework for Faltings' approach in p-adic Hodge theory, it remains relatively unexplored. The authors present a new approach based on a generalization of P. Deligne's covanishing topos.
Ahmed Abbes is director of research at the French National Center for Scientific Research (CNRS) and the Institute of Advanced Scientific Studies (IHES), France. Michel Gros is a researcher at the CNRS. Takeshi Tsuji is a professor in the Graduate School of Mathematical Sciences at the University of Tokyo.
Paperback | June 2016 | ISBN: 9780691170558
Hardcover | June 2016 | ISBN: 9780691170541
272 pp. | 6 x 9 | 7 line illus.
This is the first book to present a complete characterization of Stein-Tomas type Fourier restriction estimates for large classes of smooth hypersurfaces in three dimensions, including all real-analytic hypersurfaces. The range of Lebesgue spaces for which these estimates are valid is described in terms of Newton polyhedra associated to the given surface.
Isroil Ikromov and Detlef Muller begin with Elias M. Stein's concept of Fourier restriction and some relations between the decay of the Fourier transform of the surface measure and Stein-Tomas type restriction estimates. Varchenko's ideas relating Fourier decay to associated Newton polyhedra are briefly explained, particularly the concept of adapted coordinates and the notion of height. It turns out that these classical tools essentially suffice already to treat the case where there exist linear adapted coordinates, and thus Ikromov and Muller concentrate on the remaining case. Here the notion of r-height is introduced, which proves to be the right new concept. They then describe decomposition techniques and related stopping time algorithms that allow to partition the given surface into various pieces, which can eventually be handled by means of oscillatory integral estimates. Different interpolation techniques are presented and used, from complex to more recent real methods by Bak and Seeger.
Fourier restriction plays an important role in several fields, in particular in real and harmonic analysis, number theory, and PDEs. This book will interest graduate students and researchers working in such fields.
Isroil A. Ikromov is professor of mathematics at Samarkand State University in Uzbekistan. Detlef Muller is professor of mathematics at the University of Kiel in Germany.
ISBN 978-3-319-27484-3
Due: February 7, 2016
Gives a systematic, self-contained account of the topic
Presents recent results for the first time
Intended for researchers and graduate students with background in real and functional analysis
These lecture notes provide a self-contained introduction to regularity theory for elliptic equations and systems in divergence form. After a short review of some classical results on everywhere regularity for scalar-valued weak solutions, the presentation focuses on vector-valued weak solutions to a system of several coupled equations. In the vectorial case, weak solutions may have discontinuities and so are expected, in general, to be regular only outside of a set of measure zero. Several methods are presented concerning the proof of such partial regularity results, and optimal regularity is discussed. Finally, a short overview is given on the current state of the art concerning the size of the singular set on which discontinuities may occur.
The notes are intended for graduate and postgraduate students with a solid background in functional analysis and some familiarity with partial differential equations; they will also be of interest to researchers working on related topics.
ISBN 978-3-319-27664-9
Due: February 18, 2016
Features profound and recent results of the spectral theory of automorphic surfaces
Provides a self-contained proof of the so-called Jacquet-Langlands correspondence
Includes an introduction to Lindenstrauss's ergodic theoretic proof of quantum unique ergodicity for compact arithmetic surfaces, for which he was awarded a Fields medal in 2010
This text is an introduction to the spectral theory of the Laplacian on compact or finite area hyperbolic surfaces. For some of these surfaces, called garithmetic hyperbolic surfacesh, the eigenfunctions are of arithmetic nature, and one may use analytic tools as well as powerful methods in number theory to study them.
After an introduction to the hyperbolic geometry of surfaces, with a special emphasis on those of arithmetic type, and then an introduction to spectral analytic methods on the Laplace operator on these surfaces, the author develops the analogy between geometry (closed geodesics) and arithmetic (prime numbers) in proving the Selberg trace formula. Along with important number theoretic applications, the author exhibits applications of these tools to the spectral statistics of the Laplacian and the quantum unique ergodicity property. The latter refers to the arithmetic quantum unique ergodicity theorem, recently proved by Elon Lindenstrauss.
The fruit of several graduate level courses at Orsay and Jussieu, The Spectrum
of Hyperbolic Surfaces allows the reader to review an array of classical
results and then to be led towards very active areas in modern mathematics.
Nicolas Bergeron is a Professor at Universite Pierre et Marie Curie in Paris. His research interests are in geometry and automorphic forms, in particular the topology and spectral geometry of locally symmetric spaces.
ISBN 978-3-319-27815-5
Due: February 15, 2016
Contains exercises ranging from easy to difficult, with level of difficulty designated
Features an encyclopedic volume of exercises in five core topics of mathematical analysis
Prepares students well for qualifying exams and tests their depth of understanding of the material
This second of two Exercises in Analysis volumes covers problems in five core topics of mathematical analysis: Function Spaces, Nonlinear and Multivalued Maps, Smooth and Nonsmooth Calculus, Degree Theory and Fixed Point Theory, and Variational and Topological Methods. Each of five topics corresponds to a different chapter with inclusion of the basic theory and accompanying main definitions and results,
followed by suitable comments and remarks for better understanding of the material. Exercises/problems are presented for each topic, with solutions available at the end of each chapter. The entire collection of exercises offers a balanced and useful picture for the application surrounding each topic.
This nearly encyclopedic coverage of exercises in mathematical analysis is the first of its kind and is accessible to a wide readership. Graduate students will find the collection of problems valuable in preparation for their preliminary or qualifying exams as well as for testing their deeper understanding of the material. Exercises are denoted by degree of difficulty. Instructors teaching courses that include one or all of the above-mentioned topics will find the exercises of great help in course preparation. Researchers in analysis may find this Work useful as a summary of analytic theories published in one accessible volume.
Leszek Gasi?ksi is the Chair of Optimization and Control Theory in the Institute of Computer Science at Jagiellonian University in Krakow, Poland. He is the co-author, along with Nikolaos S. Papageorgiou, of "Nonlinear Analysis" (CRC 2005) and "Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems" (CRC 2006). Nikolaos S. Papageorgiou is a Professor of Mathematics in the School of Applied Mathematical and Physical Sciences at National Technical University in Athens, Greece. He is the co-author, along with Leszek Gasi?ksi, of "Nonlinear Analysis" (CRC 2005) and "Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems" (CRC 2006).