AMS Chelsea Publishing
1968; 519 pp; hardcover
ISBN-10: 0-8218-4342-7
ISBN-13: 978-0-8218-4342-0
--Mathematical Reviews
The theory of finite simple groups enjoyed a period of spectacular activity in the 1950s and 1960s. The first edition of Gorenstein's book was published in 1968, at the time of some of the first major classification results. The second edition was published in 1980, when it was clear that the classification was understood and the proof was within reach. Gorenstein's treatment of the subject proved prescient, as many of the developments between the two editions could be seen as continuations of the material in the book. Even now, the book remains one of the best sources for an introduction to finite groups and the classification of the simple groups. Gorenstein's insight provides a guiding light through the many pages that have been dedicated to the proof.
Readership
Undergraduates, graduate students, and research mathematicians interested in finite groups and the classification of the simple groups.
Table of Contents
Part I: Methods
Preliminaries
Some basic topics
Representations of groups
Character theory
Groups of prime power order
Solvable and \pi-solvable groups
Fusion, transfer, and p-factor groups
p-constrained and p-stable groups
Groups of even order
Part II: Applications
Fixed-point-free automorphisms
The Hall-Higman theorem
Groups with generalized quaternion Sylow 2-subgroups
Zassenhaus groups
Groups in which centralizers are nilpotent
Groups with self-centralizing Sylow 2-subgroups of order 4
Part III: General Classification Problems
Simple groups of low rank
The known simple groups
Bibliography
List of symbols
Index
Colloquium Publications, Volume: 55
2008; approx. 813 pp; hardcover
ISBN-10: 0-8218-4210-2
ISBN-13: 978-0-8218-4210-2
The unifying theme of this book is the interplay among noncommutative geometry, physics, and number theory. The two main objects of investigation are spaces where both the noncommutative and the motivic aspects come to play a role: space-time, where the guiding principle is the problem of developing a quantum theory of gravity, and the space of primes, where one can regard the Riemann Hypothesis as a long-standing problem motivating the development of new geometric tools. The book stresses the relevance of noncommutative geometry in dealing with these two spaces.
The first part of the book deals with quantum field theory and the geometric structure of renormalization as a Riemann-Hilbert correspondence. It also presents a model of elementary particle physics based on noncommutative geometry. The main result is a complete derivation of the full Standard Model Lagrangian from a very simple mathematical input. Other topics covered in the first part of the book are a noncommutative geometry model of dimensional regularization and its role in anomaly computations, and a brief introduction to motives and their conjectural relation to quantum field theory.
The second part of the book gives an interpretation of the Weil explicit formula as a trace formula and a spectral realization of the zeros of the Riemann zeta function. This is based on the noncommutative geometry of the adele class space, which is also described as the space of commensurability classes of Q-lattices, and is dual to a noncommutative motive (endomotive) whose cyclic homology provides a general setting for spectral realizations of zeros of L-functions. The quantum statistical mechanics of the space of Q-lattices, in one and two dimensions, exhibits spontaneous symmetry breaking. In the low-temperature regime, the equilibrium states of the corresponding systems are related to points of classical moduli spaces and the symmetries to the class field theory of the field of rational numbers and of imaginary quadratic fields, as well as to the automorphisms of the field of modular functions.
The book ends with a set of analogies between the noncommutative geometries underlying the mathematical formulation of the Standard Model minimally coupled to gravity and the moduli spaces of Q-lattices used in the study of the zeta function.
Readership
Graduate and research mathematicians interested in noncommutative geometry, quantum field theory and particle physics, number theory, and arithmetic algebraic geometry.
Table of Contents
Quantum fields, noncommutative spaces, and motives
The Riemann zeta function and noncommutative geometry
Quantum statistical mechanics and Galois symmetries
Endomotives, thermodynamics, and the Weil explicit formula
Appendix
Bibliography
Index
Contemporary Mathematics, Volume: 446
2007; 495 pp; softcover
ISBN-10: 0-8218-4190-4
ISBN-13: 978-0-8218-4190-7
In celebration of Haim Brezis's 60th birthday, a conference was held at the Ecole Polytechnique in Paris, with a program testifying to Brezis's wide-ranging influence on nonlinear analysis and partial differential equations. The articles in this volume are primarily from that conference. They present a rare view of the state of the art of many aspects of nonlinear PDEs, as well as describe new directions that are being opened up in this field. The articles, written by mathematicians at the center of current developments, provide somewhat more personal views of the important developments and challenges.
In his own work, Brezis has been a seminal influence in many important areas: critical growth in semi-linear equations, variational analysis of functionals in borderline compactness cases, vector valued equations, the Ginzburg-Landau theory, as well as generalized degree theory and fine properties of Sobolev spaces. This same breadth is reflected in the mathematics in this collection.
Researchers in nonlinear partial differential equations will find much of interest in this volume.
Readership
Research mathematicians interested in nonlinear partial differential equations.
Table of Contents
A. Aftalion -- Vortex patterns in Bose Einstein condensates
A. Ambrosetti and A. Malchiodi -- Concentration phenomena for nonlinear Schrodinger equations: Recent results and new perspectives
L. Ambrosio, C. De Lellis, and J. Maly -- On the chain rule for the divergence of BV-like vector fields: Applications, partial results, open problems
A. Bahri -- Compactness
H. Berestycki and F. Hamel -- Generalized travelling waves for reaction-diffusion equations
F. Bethuel and D. Chiron -- Some questions related to the lifting problem in Sobolev spaces
J. Bourgain -- Normal forms and the nonlinear Schrodinger equation
X. Cabre -- Extremal solutions and instantaneous complete blow-up for elliptic and parabolic problems
L. A. Caffarelli and A. Mellet -- Capillary drops on an inhomogeneous surface
P. Constantin -- Diffusive Lagrangian transformations, Navier-Stokes equations and applications
J.-M. Coron -- Some open problems on the control of nonlinear partial differential equations
L. C. Evans -- The 1-Laplacian, the \infty-Laplacian and differential games
J.-F. Le Gall -- Probabilistic approach to a class of semilinear partial differential equations
A. Haddad and Y. Meyer -- Variational methods in image processing
S. Klainerman -- Null hypersurfaces with finite curvature flux and a breakdown criterion in general relativity
Y. Li -- Some Liouville theorems and applications
F. Lin and Y. Yang -- Analysis on Faddeev knots and Skyrme solitons: Recent progress and open problems
M. Marcus and L. Veron -- The precise boundary trace of positive solutions of the equation \Delta u=u^q in the supercritical case
H. Matano -- Blow-up in nonlinear heat equations with supercritical power nonlinearity
P. Mironescu -- Sobolev maps on manifolds: Degree, approximation, lifting
P. Pucci, B. Sciunzi, and J. Serrin -- Partial and full symmetry of solutions of quasilinear elliptic equations, via the comparison principle
P. H. Rabinowitz -- Single and multi-transition solutions of a family of pde's
S. Serfaty
Fields Institute Communications, Volume: 51
2007; 339 pp; hardcover
ISBN-10: 0-8218-4274-9
ISBN-13: 978-0-8218-4274-4
This volume collects a set of contributions by participants of the Workshop "Partially hyperbolic dynamics, laminations, and Teichmuller flow" held at the Fields Institute in Toronto in January 2006. The Workshop brought together several leading experts in two very active fields of contemporary dynamical systems theory: partially hyperbolic dynamics and Teichmuller dynamics. They are unified by ideas coming from the theory of laminations and foliations, dynamical hyperbolicity, and ergodic theory. These are the main themes of the current volume. The volume contains both surveys and research papers on non-uniform and partial hyperbolicity, on dominated splitting and beyond (in Part I), Teichmuller dynamics with applications to interval exchange transformations and on the topology of moduli spaces of quadratic differentials (in Part II), foliations and laminations and other miscellaneous papers (in Part III). Taken together these papers provide a snapshot of the state of the art in some of the most active topics at the crossroads between dynamical systems, smooth ergodic theory, geometry and topology, suitable for advanced graduate students and researchers. Non-specialists will find the extensive, in-depth surveys especially useful.
Readership
Graduate students and research mathematicians interested in dynamical systems theory.
Table of Contents
Part I - Partially Hyperbolic Dynamics
L. J. Diaz -- How to put a stop to domination once and for all (and what comes afterwards)
F. R. Hertz, M. A. R. Hertz, and R. Ures -- A survey of partially hyperbolic dynamics
E. R. Pujals -- From hyperbolicity to dominated splitting
F. R. Hertz, M. A. R. Hertz, and R. Ures -- Some results on the integrability of the center bundle for partially hyperbolic diffeomorphisms
Part II - Teichmuller Dynamics
K. Calta and J. Smillie -- The J-invariant, exceptional surfaces and notions of periodicity
J. Smillie and B. Weiss -- Finiteness results for flat surfaces: A survey and problem list
M. Viana -- Lyapunov exponents of Teichmuller flows
A. Avila and A. Bufetov -- Exponential decay of correlations for the Rauzy-Veech-Zorich induction map
Y. Cheung and A. Eskin -- Unique ergodicity of translation flows
H. Masur and A. Zorich -- Combinatorial geometry of generic degenerations of quadratic differentials
Part III-Miscellaneous
A. Bis -- Dynamics of foliated spaces in codimension greater than one
A. Glutsyuk -- A survey on minimality of horospheric laminations for rational functions
Y. Pesin and K. Zhang -- Thermodynamics of inducing schemes and liftability of measures
M. Gerber -- Pseudo-Anosov maps and Wojtkowski's cone methods
M. Pollicott