Zurich Lectures in Advanced Mathematics
ISBN 978-3-03719-095-1
DOI 10.4171/095
September 2011, 258 pages, softcover, 17 x 24 cm.
The notion of an invariant manifold arises naturally in the asymptotic stability analysis of stationary or standing wave solutions of unstable dispersive Hamiltonian evolution equations such as the focusing semilinear Klein?Gordon and Schrodinger equations. This is due to the fact that the linearized operators about such special solutions typically exhibit negative eigenvalues (a single one for the ground state), which lead to exponential instability of the linearized flow and allows for ideas from hyperbolic dynamics to enter.
One of the main results proved here for energy subcritical equations is that the center-stable manifold associated with the ground state appears as a hyper-surface which separates a region of finite-time blowup in forward time from one which exhibits global existence and scattering to zero in forward time. Our entire analysis takes place in the energy topology, and the conserved energy can exceed the ground state energy only by a small amount.
This monograph is based on recent research by the authors and the proofs rely on an interplay between the variational structure of the ground states on the one hand, and the nonlinear hyperbolic dynamics near these states on the other hand. A key element in the proof is a virial-type argument excluding almost homoclinic orbits originating near the ground states, and returning to them, possibly after a long excursion.
These lectures are suitable for graduate students and researchers in partial differential equations and mathematical physics. For the cubic Klein?Gordon equation in three dimensions all details are provided, including the derivation of Strichartz estimates for the free equation and the concentration-compactness argument leading to scattering due to Kenig and Merle.
EMS Series of Congress Reports
ISBN 978-3-03719-101-9
DOI 10.4171/101
September 2011, 740 pages, hardcover, x cm.
This book is concerned with recent trends in the representation theory of algebras and its exciting interaction with geometry, topology, commutative algebra, Lie algebras, combinatorics, quantum algebras, and theoretical physics. The collection of articles, written by leading researchers in the field, is conceived as a sort of handbook providing easy access to the present state of knowledge and stimulating further development.
The topics under discussion include quivers, quivers with potential, bound quiver algebras, Jacobian algebras, cluster algebras and categories, Calabi?Yau algebras and categories, triangulated and derived categories, quantum loop algebras, Nakajima quiver varieties, Yang?Baxter equations, T-systems and Y-systems, dilogarithm and quantum dilogarithm identities, stable module categories, localizing and colocalizing subcategories, cohomologies of groups, support varieties, fusion systems, Hochschild cohomologies, weighted projective lines, coherent sheaves, Kleinian and Fuchsian singularities, stable categories of vector bundles, nilpotent operators, Artin?Schelter regular algebras, Fano algebras, deformations of algebras, module varieties, degenerations of modules, singularities of orbit closures, coalgebras and comodules, representation types of algebras and coalgebras, Tits and Euler forms of algebras, Galois coverings of algebras, tilting and cluster tilting theory, algebras of small homological dimensions, Auslander?Reiten theory.
The book consists of thirteen self-contained expository survey and research articles and is addressed to researchers and graduate students in algebra as well as a broader mathematical community. They contain a large number of examples and open problems and give new perspectives for research in the field.
Heritage of European Mathematics
ISBN 978-3-03719-104-0
DOI 10.4171/104
October 2011, 421 pages, hardcover, 17 x 24 cm.
Although Evariste Galois was only 20 years old when he died, shot in a mysterious early-morning duel in 1832, his ideas, when they were published 14 years later, changed the course of algebra. He invented what is now called Galois Theory, the modern form of what was classically the Theory of Equations. For that purpose, and in particular to formulate a precise condition for solubility of equations by radicals, he also invented groups and began investigating their theory. His main writings were published in French in 1846 and there have been a number of French editions culminating in the great work published by Bourgne & Azra in 1962 containing transcriptions of every page and fragment of the manuscripts that survive. Very few items have been available in English up to now.
The present work contains English translations of almost all the Galois material. They are presented alongside a new transcription of the original French, and are enhanced by three levels of commentary. An introduction explains the context of Galois' work, the various publications in which it appears, and the vagaries of his manuscripts. Then there is a chapter in which the five mathematical articles published in his lifetime are reprinted. After that come the Testamentary Letter and the First Memoir (in which Galois expounded the ideas now called Galois Theory), which are the most famous of the manuscripts. There follow the less well known manuscripts, namely the Second Memoir and the many fragments. A short epilogue devoted to myths and mysteries concludes the text.
The book is written as a contribution to the history of mathematics but with mathematicans as well as historians in mind. It makes available to a wide mathematical and historical readership some of the most exciting mathematics of the first half of the 19th century, presented in its original form. The primary aim is to establish a text of what Galois wrote. Exegesis would fill another book or books, and little of that is to be found here.
This work will be a resource for research in the history of mathematics, especially algebra, as well as a sourcebook for those many mathematicians who enliven their student lectures with reliable historical background.
Schemas en groupes, (SGA 3, Tome I )
(Proprietes generales des schemas en groupes)
Documents mathematiques 7 (2011), xviii + 638 pages
Ce volume est une edition recomposee et annotee du livre ≪ Schemas en groupes ≫ (SGA 3), tome I (Proprietes generales des schemas en groupes) , Lecture Notes in Mathematics, 151, Springer-Verlag, Berlin-Heidelberg-New York, 1970, par Michel Demazure, Alexander Grothendieck et al. Ce tome introduit le langage des foncteurs representables et des faisceaux et donne des resultats generaux sur les schemas en groupes (Exp.I a VII), et aussi sur les groupes formels (Exp.VII).
Mots-clefs : Schemas en groupes, Groupes reductifs, Foncteurs representables, Theorie de la descente, Topologies de Grothendieck, Torseurs, Faisceaux quotients, Extensions infinitesimales, Groupes formels, Tores, Groupes de type multiplicatif, Donnees radicielles, Sous-groupes paraboliques.
This volume is a new, updated edition of the book ``Schemas en groupes (SGA 3), tome I (Proprietes generales des schemas en groupes)'', Lecture Notes in Mathematics, 151, Springer-Verlag, Berlin-Heidelberg-New York, 1970, by Michel Demazure, Alexander Grothendieck et al. This tome introduces the language of representable functors and sheaves and proves general results about group schemes (Exp. I to VII), and formal groups (Exp. VII).
Keywords: Group schemes, Reductive groups, Representable functors, Descent theory, Grothendieck topologies, Torsors, Quotient sheaves, Infinitesimal extensions, Formal groups, Tori, Groups of multiplicative type, Root data, Parabolic subgroups.
ISBN : 978-2-85629-323-2
Schemas en groupes, (SGA 3, Tome III )
(Structure des schemas en groupes reductifs)
Documents mathematiques 8 (2011), lv +337 pages
Ce volume est une edition recomposee et annotee du livre ≪ Schemas en groupes (SGA 3), tome III (Structure des schemas en groupes reductifs) ≫, Lecture Notes in Mathematics, 153, Springer-Verlag, Berlin-Heidelberg-New York, 1970, par Michel Demazure, Alexander Grothendieck et al. Ce tome donne la structure, sur un schema de base S quelconque, des S-schemas en groupes reductifs (Exp. XIX a XXV) et de leurs sous-groupes paraboliques (Exp. XXVI).
Mots-clefs : Schemas en groupes, Groupes reductifs, Foncteurs representables, Theorie de la descente, Topologies de Grothendieck, Torseurs, Faisceaux quotients, Extensions infinitesimales, Groupes formels, Tores, Groupes de type multiplicatif, Donnees radicielles, Sous-groupes paraboliques.
This volume is a new, updated edition of the book ``Schemas en groupes (SGA 3), tome III (Structure des schemas en groupes reductifs) , Lecture Notes in Mathematics, 153, Springer-Verlag, Berlin-Heidelberg-New York, 1970, by Michel Demazure, Alexander Grothendieck et al. This tome gives the structure of reductive S-group schemes (ExpXIX to XXV), and their parabolic subgroups (ExpXXVI), over an arbitrary base scheme S.
Keywords: Group schemes, Reductive groups, Representable functors, Descent theory, Grothendieck topologies, Torsors, Quotient sheaves, Infinitesimal extensions, Formal groups, Tori, Groups of multiplicative type, Root data, Parabolic subgroups.
ISSN : 978-2-85629-324-9