Michel Ledoux, Universite Paul-Sabatier, Toulouse, France

The Concentration of Measure Phenomenon

Expected publication date is October 26, 2001

Description
The observation of the concentration of measure phenomenon is inspired by isoperimetric inequalities. A familiar example is the way the uniform measure on the standard sphere $S^n$ becomes concentrated around the equator as the dimension gets large. This property may be interpreted in terms of functions on the sphere with small oscillations, an idea going back to Levy. The phenomenon also occurs in probability, as a version of the law of large numbers, due to Emil Borel. This book offers the basic techniques and examples of the concentration of measure phenomenon. The concentration of measure phenomenon was put forward in the early seventies by V. Milman in the asymptotic geometry of Banach spaces. It is of powerful interest in applications in various areas, such as geometry, functional analysis and infinite-dimensional integration, discrete mathematics and complexity theory, and probability theory. Particular emphasis is on geometric, functional, and probabilistic tools to reach and describe measure concentration in a number of settings.

The book presents concentration functions and inequalities, isoperimetric and functional examples, spectrum and topological applications, product measures, entropic and transportation methods, as well as aspects of M. Talagrand's deep investigation of concentration in product spaces and its application in discrete mathematics and probability theory, supremum of Gaussian and empirical processes, spin glass, random matrices, etc. Prerequisites are a basic background in measure theory, functional analysis, and probability theory.

Contents
Concentration functions and inequalities
Isoperimetric and functional examples
Concentration and geometry
Concentration in product spaces
Entropy and concentration
Transportation cost inequalities
Sharp bounds of Gaussian and empirical processes
Selected applications
References
Index

Details:
Series: Mathematical Surveys and Monographs,Volume: 89
Publication Year: 2001
ISBN: 0-8218-2864-9
Paging: 181 pp.
Binding: Hardcover

Laurent Manivel, University of Grenoble, Saint Martin d'Heres, France

Symmetric Functions, Schubert Polynomials and Degeneracy Loci

Expected publication date is October 20, 2001

From reviews of the French Edition:

"Well-written book ... all of the concepts are clearly defined and presented in an informal and pleasant way ... an attractive book which presents the interplay between many diverse topics of algebraic combinatorics and their geometric realizations in Schubert calculus. It will be of great use to anyone wishing a brief and well-organized treatment of this material and particularly good for graduate students."

-- Mathematical Reviews

Description
This text grew out of an advanced course taught by the author at the Fourier Institute (Grenoble, France). It serves as an introduction to the combinatorics of symmetric functions, more precisely to Schur and Schubert polynomials. Also studied is the geometry of Grassmannians, flag varieties, and especially, their Schubert varieties. This book examines profound connections that unite these two subjects.

The book is divided into three chapters. The first is devoted to symmetric functions and especially to Schur polynomials. These are polynomials with positive integer coefficients in which each of the monomials correspond to a Young tableau with the property of being "semistandard". The second chapter is devoted to Schubert polynomials, which were discovered by A. Lascoux and M.-P. Schutzenberger who deeply probed their combinatorial properties. It is shown, for example, that these polynomials support the subtle connections between problems of enumeration of reduced decompositions of permutations and the Littlewood-Richardson rule, a particularly efficacious version of which may be derived from these connections. The final chapter is geometric. It is devoted to Schubert varieties, subvarieties of Grassmannians, and flag varieties defined by certain incidence conditions with fixed subspaces.

This volume makes accessible a number of results, creating a solid stepping stone for scaling more ambitious heights in the area. The author's intent was to remain elementary: The first two chapters require no prior knowledge, the third chapter uses some rudimentary notions of topology and algebraic geometry. For this reason, a comprehensive appendix on the topology of algebraic varieties is provided. This book is the English translation of a text previously published in French.

Contents
Introduction
The ring of symmetric functions
Schubert polynomials
Schubert varieties
A brief introduction to singular homology
Bibliography
Index

Details:
Series: SMF/AMS Texts and Monographs, Volume: 6
Publication Year: 2001
ISBN: 0-8218-2154-7
Paging: approximately 176 pp.
Binding: Softcover

Yves Meyer, Ecole Normale Superieure de Cachan, France

Oscillating Patterns in Image Processing and Nonlinear Evolution Equations: The Fifteenth Dean Jacqueline B. Lewis Memorial Lectures

Expected publication date is September 30, 2001

Description
Image compression, the Navier-Stokes equations, and detection of gravitational waves are three seemingly unrelated scientific problems that, remarkably, can be studied from one perspective. The notion that unifies the three problems is that of "oscillating patterns", which are present in many natural images, help to explain nonlinear equations, and are pivotal in studying chirps and frequency-modulated signals.

The first chapter of this book considers image processing, more precisely algorithms of image compression and denoising. This research is motivated in particular by the new standard for compression of still images known as JPEG-2000. The second chapter has new results on the Navier-Stokes and other nonlinear evolution equations. Frequency-modulated signals and their use in the detection of gravitational waves are covered in the final chapter.

In the book, the author describes both what the oscillating patterns are and the mathematics necessary for their analysis. It turns out that this mathematics involves new properties of various Besov-type function spaces and leads to many deep results, including new generalizations of famous Gagliardo-Nirenberg and Poincare inequalities.

This book is based on the "Dean Jacqueline B. Lewis Memorial Lectures" given by the author at Rutgers University. It can be used either as a textbook in studying applications of wavelets to image processing or as a supplementary resource for studying nonlinear evolution equations or frequency-modulated signals. Most of the material in the book did not appear previously in monograph literature.

Contents
Still images compression
The role of oscillations in some nonlinear PDE's
Frequency-modulated signals, chirps and the Virgo program
Conclusion
References

Details:
Series: University Lecture Series, Volume: 22
Publication Year: 2001
ISBN: 0-8218-2920-3
Paging: approximately 136 pp.
Binding: Softcover

Edited by: Alejandro Adem, University of Wisconsin, Madison, WI,
and Gunnar Carlsson and Ralph Cohen, Stanford University, CA

Topology, Geometry, and Algebra: Interactions and New Directions

Expected publication date is October 7, 2001

Description
This volume presents the proceedings from the conference on "Topology, Geometry, and Algebra: Interactions and New Directions" held in honor of R. James Milgram at Stanford University in August 1999. The meeting brought together distinguished researchers from a variety of areas related to algebraic topology and its applications.

Papers in the book present a wide range of subjects, reflecting the nature of the conference. Topics include moduli spaces, configuration spaces, surgery theory, homotopy theory, knot theory, group actions, and more. Particular emphasis was given to the breadth of interaction between the different areas.

Contents
G. Carlsson -- On Jim Milgram's mathematical work
M. Ando and J. Morava -- A renormalized Riemann-Roch formula and the Thom isomorphism for the free loop space
M. Bendersky and D. M. Davis -- The 1-line of the $K$-theory Bousfield-Kan spectral sequence for $Spin(2n+1)$
W. Browder -- Homologically exotic free actions on products of $S^m$
S. E. Cappell, R. Lee, and E. Y. Miller -- Surgery formulae for analytical invariants of manifolds
F. R. Cohen -- On genus one mapping class groups, function spaces, and modular forms
B. Hanke -- Poincare duality and deformations of algebras
S. Kallel -- An analog of the May-Milgram model for configurations with multiplicities
S. Kallel -- Configuration spaces and the topology of curves in projective space
M. Karoubi -- Quantum methods in algebraic topology
K. Liu and Z. Weiping -- Adiabatic limits and foliations
K. Mohnke -- Legendrian links of topological unknots
A. Ranicki -- Algebraic Poincare cobordism

Details:
Series: Contemporary Mathematics, Volume: 279
Publication Year: 2001
ISBN: 0-8218-2063-X
Paging: approximately 264 pp.
Binding: Softcover

Edited by: Avinoam Mann, Hebrew University of Jerusalem, Israel, Amitai Regev, Weizmann Institute of Science, Rehovot, Israel, Louis Rowen, Bar-Ilan University, Ramat Gan, Israel, David Saltman, University of Texas, Austin, TX, and Lance Small, University of California, San Diego, La Jolla, CA

Selected Papers of S. A. Amitsur with Commentary

Description
A lead figure in twentieth century noncommutative algebra, S. A. Amitsur's contributions are wide-ranging and enduring. This volume collects almost all of his work. The papers are organized into broad topic areas: general ring theory, rings satisfying a polynomial identity, combinatorial polynomial identity theory, and division algebras. Included are essays by the editors on Amitsur's work in these four areas and a biography of Amitsur written by A. Mann. This volume makes a fine addition to any mathematics book collection.

Contents
General ring theory
A. Mann -- Rings satisfying a polynomial identity
General ring theory
A. Regev -- Rings satisfying a polynomial identity
General ring theory
L. Rowen -- Rings satisfying a polynomial identity
A. S. Amitsur -- A generalization of a theorem on linear differential equations
S. A. Amitsur -- A general theory of radicals. I. Radicals in complete lattices
S. A. Amitsur -- A general theory of radicals. II. Radicals in rings and bicategories
S. A. Amitsur -- A general theory of radicals. III. Applications
A. S. Amitsur -- Algebras over infinite fields
S. A. Amitsur -- Radicals of polynomial rings
S. A. Amitsur -- Invariant submodules of simple rings
S. A. Amitsur -- Derivations in simple rings
S. A. Amitsur -- The radical of field extensions
S. A. Amitsur -- Countably generated division algebras over nondenumerable fields
S. A. Amitsur -- Commutative linear differential operators
S. A. Amitsur -- Rings with a pivotal monomial
S. A. Amitsur -- On the semi-simplicity of group algebras
S. A. Amitsur -- Derived functors in abelian categories
S. A. Amitsur -- Remarks on principal ideals rings
S. A. Amitsur -- Generalized polynomial identities and pivotal monomials
S. A. Amitsur -- Rings with involution
S. A. Amitsur -- Rings of quotients and Morita contexts
S. A. Amitsur -- Nil radicals. Historical notes and some new results
S. A. Amitsur -- On rings of quotients
S. A. Amitsur and J. C. Robson -- Recognition of matrix rings II
S. A. Amitsur -- Nil PI-rings
A. S. Amitsur -- An embedding of PI-rings
S. A. Amitsur -- On rings with identities
S. A. Amitsur -- The $T$-ideals of the free ring
S. A. Amitsur -- A generalization of Hilbert's Nullstellensatz
S. A. Amitsur -- Groups with representations of bounded degree II
S. A. Amitsur and C. Procesi -- Jacobson-rings and Hilbert algebras with polynomial identities
S. A. Amitsur -- Nil semi-groups of rings with a polynomial identity
S. A. Amitsur -- Rational identities and applications to algebra and geometry
S. A. Amitsur -- Prime rings having polynomial identities with arbitrary coefficients
S. A. Amitsur -- Identities in rings with involutions
S. A. Amitsur -- A noncommutative Hilbert basis theorem and subrings of matrices
S. A. Amitsur -- Embeddings in matrix rings
S. A. Amitsur -- Some results on rings with polynomial identities
S. A. Amitsur -- A note on PI-rings
S. A. Amitsur -- On universal embeddings in matrix rings
S. A. Amitsur -- Polynomial identities and Azumaya algebras
S. A. Amitsur -- Polynomial identities
S. A. Amitsur -- Central embeddings in semi-simple rings
S. A. Amitsur and L. W. Small -- Polynomials over division rings
S. A. Amitsur and L. W. Small -- Prime ideals in PI-rings
S. A. Amitsur and L. W. Small -- Finite-dimensional representations of PI algebras
S. A. Amitsur and L. W. Small -- GK-dimensions of corners and ideals
S. A. Amitsur -- Contributions of PI theory to Azumaya algebras
S. A. Amitsur and L. W. Small -- Finite-dimensional representation of PI algebras, II
S. A. Amitsur and L. W. Small -- Algebras over infinite fields, revisited
Combinatoiral polynomial identity theory
A. Mann -- Division algebras
S. A. Amitsur -- Finite-dimensional subalgebras of division rings
Combinatoiral polynomial identity theory
A. Regev -- Division algebras
D. Baum -- Finite-dimensional subalgebras of division rings
Combinatoiral polynomial identity theory
L. Rowen -- Division algebras
A. Mann -- Shimshon Avraham Amitsur (1921-1994)
A. S. Amitsur -- Contributions to the theory of central simple algebras
M. S. Amitsur -- La representation dalgebres centrales simples
M. S. Amitsur -- Construction d'algebres centrales simples sur des corps de caracteristique zero
A. S. Amitsur -- Non-commutative cyclic fields
A. S. Amitsur -- Differential polynomials and division algebras
S. A. Amitsur -- Generic splitting fields of central simple algebras
S. A. Amitsur -- Finite subgroups of division rings
S. A. Amitsur -- Some results on central simple algebras
S. A. Amitsur -- On arithmetic functions
S. A. Amitsur -- Simple algebras and cohomology groups of arbitrary fields
S. A. Amitsur -- Some results on arithmetic functions
S. A. Amitsur -- Finite dimensional central divison algebras
S. A. Amitsur -- Homology groups and double complexes for arbitrary fields
S. A. Amitsur -- On a lemma in elementary proofs of the prime number theorem
S. A. Amitsur -- Complexes of rings
S. A. Amitsur -- On central division algebras
S. A. Amitsur -- The generic division rings
S. A. Amitsur and D. Saltman -- Generic abelian crossed products and $p$-algebras
S. A. Amitsur, L. H. Rowen, and J. P. Tignol -- Division algebras of degree 4 and 8 with involution
S. A. Amitsur -- On the characteristic polynomial of a sum of matrices
S. A. Amitsur -- Generic splitting fields
S. A. Amitsur -- Extension of derivations to central simple algebras
J.-P. Tignol and S. A. Amitsur -- Kummer subfields of Malcev-Neumann division algebras
J. P. Tignol and S. A. Amitsur -- Symplectic modules
J.-P. Tignol and S. A. Amitsur -- Totally ramified splitting fields of central simple algebras over Henselian fields
S. A. Amitsur -- Galois splitting fields of a universal division algebra
S. A. Amitsur and L. H. Rowen -- Elements of reduced trace 0
A. S. Amitsur and J. Levitzki -- Minimal identities for algebras
A. S. Amitsur and J. Levitzki -- Remarks on minimal identities for algebras
S. A. Amitsur -- The identities of PI-rings
S. A. Amitsur -- Identities and generators of matrix rings
S. A. Amitsur -- Identities and linear dependence
S. A. Amitsur -- On a central identity for matrix rings
S. A. Amitsur -- Alternating identities
A. Regev and S. A. Amitsur -- PI-algebras and their cocharacters
S. A. Amitsur -- The sequence of codimensions of PI-algebras

Details:
Series: Collected Works
Publication Year: 2001
ISBN: 0-8218-0688-2
Paging: approximately 1224 pp.
Binding: Hardcover