Rigid Analytic Geometry and Its Applications

Author: Jean Fresnel,Marius van der Put

Publisher: Springer Science & Business Media

ISBN: 1461200415

Category: Mathematics

Page: 299

View: 5701

Rigid (analytic) spaces were invented to describe degenerations, reductions, and moduli of algebraic curves and abelian varieties. This work, a revised and greatly expanded new English edition of an earlier French text by the same authors, presents important new developments and applications of the theory of rigid analytic spaces to abelian varieties, "points of rigid spaces," étale cohomology, Drinfeld modular curves, and Monsky-Washnitzer cohomology. The exposition is concise, self-contained, rich in examples and exercises, and will serve as an excellent graduate-level text for the classroom or for self-study.

Parabolic Quasilinear Equations Minimizing Linear Growth Functionals

Author: Fuensanta Andreu-Vaillo,Vicent Caselles,José M. Mazon,José M. Mazón

Publisher: Springer Science & Business Media

ISBN: 9783764366193

Category: Mathematics

Page: 340

View: 4901

This book contains a detailed mathematical analysis of the variational approach to image restoration based on the minimization of the total variation submitted to the constraints given by the image acquisition model. This model, initially introduced by Rudin, Osher, and Fatemi, had a strong influence in the development of variational methods for image denoising and restoration, and pioneered the use of the BV model in image processing. After a full analysis of the model, the minimizing total variation flow is studied under different boundary conditions, and its main qualitative properties are exhibited. In particular, several explicit solutions of the denoising problem are computed.

Rigid Geometry of Curves and Their Jacobians

Author: Werner Lütkebohmert

Publisher: Springer

ISBN: 331927371X

Category: Mathematics

Page: 386

View: 3298

This book presents some of the most important aspects of rigid geometry, namely its applications to the study of smooth algebraic curves, of their Jacobians, and of abelian varieties - all of them defined over a complete non-archimedean valued field. The text starts with a survey of the foundation of rigid geometry, and then focuses on a detailed treatment of the applications. In the case of curves with split rational reduction there is a complete analogue to the fascinating theory of Riemann surfaces. In the case of proper smooth group varieties the uniformization and the construction of abelian varieties are treated in detail. Rigid geometry was established by John Tate and was enriched by a formal algebraic approach launched by Michel Raynaud. It has proved as a means to illustrate the geometric ideas behind the abstract methods of formal algebraic geometry as used by Mumford and Faltings. This book should be of great use to students wishing to enter this field, as well as those already working in it.

Berkovich Spaces and Applications

Author: Antoine Ducros,Charles Favre,Johannes Nicaise

Publisher: Springer

ISBN: 3319110292

Category: Mathematics

Page: 413

View: 334

We present an introduction to Berkovich’s theory of non-archimedean analytic spaces that emphasizes its applications in various fields. The first part contains surveys of a foundational nature, including an introduction to Berkovich analytic spaces by M. Temkin, and to étale cohomology by A. Ducros, as well as a short note by C. Favre on the topology of some Berkovich spaces. The second part focuses on applications to geometry. A second text by A. Ducros contains a new proof of the fact that the higher direct images of a coherent sheaf under a proper map are coherent, and B. Rémy, A. Thuillier and A. Werner provide an overview of their work on the compactification of Bruhat-Tits buildings using Berkovich analytic geometry. The third and final part explores the relationship between non-archimedean geometry and dynamics. A contribution by M. Jonsson contains a thorough discussion of non-archimedean dynamical systems in dimension 1 and 2. Finally a survey by J.-P. Otal gives an account of Morgan-Shalen's theory of compactification of character varieties. This book will provide the reader with enough material on the basic concepts and constructions related to Berkovich spaces to move on to more advanced research articles on the subject. We also hope that the applications presented here will inspire the reader to discover new settings where these beautiful and intricate objects might arise.

Algebraic and Combinatorial Aspects of Tropical Geometry

Author: Erwan Brugalle,Maria Angelica Cueto,Alicia Dickenstein,Eva-Maria Feichtner,Ilia Itenberg

Publisher: American Mathematical Soc.

ISBN: 0821891464

Category: Mathematics

Page: 350

View: 7968

This volume contains the proceedings of the CIEM workshop on Tropical Geometry, held December 12-16, 2011, at the International Centre for Mathematical Meetings (CIEM), Castro Urdiales, Spain. Tropical geometry is a new and rapidly developing field of mat

P-adic Geometry

Lectures from the 2007 Arizona Winter School

Author: Matthew Baker,David Savitt,Dinesh S. Thakur

Publisher: American Mathematical Soc.

ISBN: 0821844687

Category: Mathematics

Page: 203

View: 3890

In recent decades, $p$-adic geometry and $p$-adic cohomology theories have become indispensable tools in number theory, algebraic geometry, and the theory of automorphic representations. The Arizona Winter School 2007, on which the current book is based, was a unique opportunity to introduce graduate students to this subject. Following invaluable introductions by John Tate and Vladimir Berkovich, two pioneers of non-archimedean geometry, Brian Conrad's chapter introduces the general theory of Tate's rigid analytic spaces, Raynaud's view of them as the generic fibers of formal schemes, and Berkovich spaces. Samit Dasgupta and Jeremy Teitelbaum discuss the $p$-adic upper half plane as an example of a rigid analytic space and give applications to number theory (modular forms and the $p$-adic Langlands program). Matthew Baker offers a detailed discussion of the Berkovich projective line and $p$-adic potential theory on that and more general Berkovich curves. Finally, Kiran Kedlaya discusses theoretical and computational aspects of $p$-adic cohomology and the zeta functions of varieties. This book will be a welcome addition to the library of any graduate student and researcher who is interested in learning about the techniques of $p$-adic geometry.

Lie Theory

Lie Algebras and Representations

Author: Jean-Philippe Anker,Bent Orsted

Publisher: Birkhauser

ISBN: N.A

Category: Mathematics

Page: 328

View: 4362

* First of three independent, self-contained volumes under the general title, "Lie Theory," featuring original results and survey work from renowned mathematicians. * Contains J. C. Jantzen's "Nilpotent Orbits in Representation Theory," and K.-H. Neeb's "Infinite Dimensional Groups and their Representations." * Comprehensive treatments of the relevant geometry of orbits in Lie algebras, or their duals, and the correspondence to representations. * Should benefit graduate students and researchers in mathematics and mathematical physics.

Mathematical Reviews

Author: N.A

Publisher: N.A

ISBN: N.A

Category: Mathematics

Page: N.A

View: 8717

Sūgaku Expositions

Author: N.A

Publisher: N.A

ISBN: N.A

Category: Mathematics

Page: N.A

View: 1598