History of Mathematics, Volume: 32
2007; 336 pp; hardcover
ISBN-13: 978-0-8218-4343-7
Expected publication date is August 8, 2007.
Algebra, as a subdiscipline of mathematics, arguably has a history going back some 4000 years to ancient Mesopotamia. The history, however, of what is recognized today as high school algebra is much shorter, extending back to the sixteenth century, while the history of what practicing mathematicians call "modern algebra" is even shorter still.
The present volume provides a glimpse into the complicated and often convoluted history of this latter conception of algebra by juxtaposing twelve episodes in the evolution of modern algebra from the early nineteenth-century work of Charles Babbage on functional equations to Alexandre Grothendieck's mid-twentieth-century metaphor of a "rising sea" in his categorical approach to algebraic geometry. In addition to considering the technical development of various aspects of algebraic thought, the historians of modern algebra whose work is united in this volume explore such themes as the changing aims and organization of the subject as well as the often complex lines of mathematical communication within and across national boundaries. Among the specific algebraic ideas considered are the concept of divisibility and the introduction of non-commutative algebras into the study of number theory and the emergence of algebraic geometry in the twentieth century.
The resulting volume is essential reading for anyone interested in the history of modern mathematics in general and modern algebra in particular. It will be of particular interest to mathematicians and historians of mathematics.
Readership
Graduate students and research mathematicians interested in the history of mathematics and algebra.
Table of Contents
J. J. Gray and K. H. Parshall -- Acknowledgments
J. J. Gray and K. H. Parshall -- Introduction
E. L. Ortiz -- Babbage and French Ideologie: Functional equations, language, and the analytical method
S. E. Despeaux -- "Very full of symbols": Duncan F. Gregory, the calculus of operations, and the Cambridge Mathematical Journal
O. Neumann -- Divisibility theories in the early history of commutative algebra and the foundations of algebraic geometry
H. M. Edwards -- Kronecker's fundamental theorem of general arithmetic
G. Frei -- Developments in the theory of algebras over number fields: A new foundation for the Hasse norm residue symbol and new approaches to both the Artin reciprocity law and class field theory
J. Schwermer -- Minkowski, Hensel, and Hasse: On the beginnings of the local-global principle
D. D. Fenster -- Research in algebra at the University of Chicago: Leonard Eugene Dickson and A. Adrian Albert
C. W. Curtis -- Emmy Noether's 1932 ICM lecture on noncommutative methods in algebraic number theory
L. Corry -- From Algebra (1895) to Moderne Algebra (1930): Changing conceptions of a discipline--A guided tour using the Jahrbuch uber die Fortschritte der Mathematik
N. Schappacher -- A historical sketch of B. L. van der Waerden's work in algebraic geometry: 1926-1946
S. Slembek -- On the arithmetization of algebraic geometry
C. McLarty -- The rising sea: Grothendieck on simplicity and generality
CRM Proceedings & Lecture Notes, Volume: 42
2007; 475 pp; softcover
ISBN-13: 978-0-8218-4089-4
Expected publication date is August 15, 2007.
This volume is based on talks given at a conference celebrating Stanislav Molchanov's 65th birthday held in June of 2005 at the Centre de Recherches Mathematiques (Montreal, QC, Canada). The meeting brought together researchers working in an exceptionally wide range of topics reflecting the quality and breadth of Molchanov's past and present research accomplishments. This collection of survey and research papers gives a glance of the profound consequences of Molchanov's contributions in stochastic differential equations, spectral theory for deterministic and random operators, localization and intermittency, mathematical physics and optics, and other topics.
Readership
Graduate students and research mathematicians interested in Molchanov's work.
Table of Contents
G. B. Arous, S. Molchanov, and A. Ramirez -- Transition asymptotics for reaction-diffusion in random media
L. V. Bogachev -- Extreme value theory for random exponentials
J. Breuer -- Singular continuous and dense point spectrum for sparse trees with finite dimensions
J.-M. Combes, P. D. Hislop, and F. Klopp -- Some new estimates on the spectral shift function associated with random Schrodinger operators
M. Cranston and S. Molchanov -- On phase transitions and limit theorems for homopolymers
G. Derfel, P. J. Grabner, and F. Vogl -- Asymptotics of the Poincare functions
A. Figotin and J. Schenker -- Hamiltonian extension and eigenfunctions for a time dispersive dissipative string
F. Germinet, P. D. Hislop, and A. Klein -- Localization at low energies for attractive Poisson random Schrodinger operators
Y. A. Godin, S. Molchanov, and B. Vainberg -- On the influence of random perturbations on the propagation of waves described by a periodic Schrodinger operator
A. Gordon, J. L. Holt, and S. Molchanov -- Spectral theory of 1-D Schrodinger operators with unbounded potentials
A. Gordon, S. Molchanov, and J. Quinn -- Fermi-Dirac generators and tests for randomness
R. Grigorchuk, D. Savchuk, and Z. Sunic -- The spectral problem, substitutions and iterated monodromy
V. Imaykin, A. Komech, and B. Vainberg -- On scattering of solitons for wave equation coupled to a particle
U. Kaluzhny and Y. Last -- Purely absolutely continuous spectrum for some random Jacobi matrices
W. Konig -- The parabolic Anderson model and its universality classes
L. Koralov -- An inverse problem for Gibbs fields
E. Kritchevski -- Hierarchical Anderson model
P. Kuchment -- Integral representations of solutions of periodic elliptic equations
A. Laptev, R. Shterenberg, and V. Sukhanov -- Inverse spectral problems for Schrodinger operators with energy depending potentials
N. Minami -- Theory of point processes and some basic notions in energy level statistics
L. Pastur and V. Vasilchuk -- On the law of addition of random matrices: Covariance and the central limit theorem for traces of resolvent
P. Poulin -- Green's functions of generalized Lapalcians
B. Simon -- Orthogonal polynomials with exponentially decaying recursion coefficients
M. Stoiciu -- Poisson statistics for eigenvalues: From random Schrodinger operators to random CMV matrices
Contemporary Mathematics, Volume: 435
2007; 394 pp; softcover
ISBN-13: 978-0-8218-4061-0
Expected publication date is August 31, 2007.
This book consists of contributions by the participants of the Fifth Conference on Function Spaces, held at Southern Illinois University in May of 2006. The papers cover a broad range of topics, including spaces and algebras of analytic functions of one and of many variables (and operators on such spaces), $L^{p}$-spaces, spaces of Banach-valued functions, isometries of function spaces, geometry of Banach spaces, and other related subjects. The goal of the conference was to bring together mathematicians interested in various problems related to function spaces and to facilitate the exchange of ideas between people working on similar problems. Hence, the majority of papers in this book are accessible to non-experts. Some articles contain expositions of known results and discuss open problems, others contain new results.
Readership
Graduate students and research mathematicians interested in functional analysis.
Table of Contents
Graduate Studies in Mathematics, Volume: 85
2007; 221 pp; hardcover
ISBN-13: 978-0-8218-4234-8
Expected publication date is August 17, 2007.
Since at least the time of Poisson, mathematicians have pondered the notion of recurrence for differential equations. Solutions that exhibit recurrent behavior provide insight into the behavior of general solutions. In Recurrence and Topology, Alongi and Nelson provide a modern understanding of the subject, using the language and tools of dynamical systems and topology.
Recurrence and Topology develops increasingly more general topological modes of recurrence for dynamical systems beginning with fixed points and concluding with chain recurrent points. For each type of recurrence the text provides detailed examples arising from explicit systems of differential equations; it establishes the general topological properties of the set of recurrent points; and it investigates the possibility of partitioning the set of recurrent points into subsets which are dynamically irreducible. The text includes a discussion of real-valued functions that reflect the structure of the sets of recurrent points and concludes with a thorough treatment of the Fundamental Theorem of Dynamical Systems.
Recurrence and Topology is appropriate for mathematics graduate students, though a well-prepared undergraduate might read most of the text with great benefit.
Readership
Undergraduate and graduate students interested in dynamical systems and topology.
Table of Contents
Flows
Recurrent points
Irreducible sets
Test functions
Afterword
Appendix A. Discrete dynamical systems
Appendix B. Circle rotations
Appendix C. The Hausdorff metric
Bibliography
Index
Mathematical Surveys and Monographs, Volume: 142
2007; approx. 392 pp; hardcover
ISBN-13: 978-0-8218-4289-8
Expected publication date is September 7, 2007.
This book starts with the basic theory of topological groups, harmonic analysis, and unitary representations. It then concentrates on geometric structure, harmonic analysis, and unitary representation theory in commutative spaces. Those spaces form a simultaneous generalization of compact groups, locally compact abelian groups, and riemannian symmetric spaces. Their geometry and function theory is an increasingly active topic in mathematical research, and this book brings the reader up to the frontiers of that research area with the recent classifications of weakly symmetric spaces and of Gelfand pairs.
Part 1, "General Theory of Topological Groups", is an introduction with many examples, including all of the standard semisimple linear Lie groups and the Heisenberg groups. It presents the construction of Haar measure, the invariant integral, the convolution product, and the Lebesgue spaces.
Part 2, "Representation Theory and Compact Groups", provides background at a slightly higher level. Besides the basics, it contains the Mackey Little-Group method and its application to Heisenberg groups, the Peter-Weyl Theorem, Cartan's highest weight theory, the Borel-Weil Theorem, and invariant function algebras.
Part 3, "Introduction to Commutative Spaces", describes that area up to its recent resurgence. Spherical functions and associated unitary representations are developed and applied to harmonic analysis on $G/K$ and to uncertainty principles.
Part 4, "Structure and Analysis for Commutative Spaces", summarizes riemannian symmetric space theory as a role model, and with that orientation delves into recent research on commutative spaces. The results are explicit for spaces $G/K$ of nilpotent or reductive type, and the recent structure and classification theory depends on those cases.
Parts 1 and 2 are accessible to first-year graduate students. Part 3 takes a bit of analytic sophistication but generally is accessible to graduate students. Part 4 is intended for mathematicians beginning their research careers as well as mathematicians interested in seeing just how far one can go with this unified view of algebra, geometry, and analysis.
Readership
Graduate students and research mathematicians interested in lie groups and their representations.
Table of Contents
General theory of topological groups
Basic topological group theory
Some examples
Integration and convolution
Representation theory and compact groups
Basic representation theory
Representations of compact groups
Compact Lie groups and homogeneous spaces
Discrete co-compact subgroups
Introduction to commutative spaces
Basic theory of commutative spaces
Spherical transforms and Plancherel formulae
Special case: Commutative groups
Structure and analysis for commutative spaces
Riemannian symmetric spaces
Weakly symmetric and reductive commutative spaces
Structure of commutative nilmanifolds
Analysis on commutative nilmanifolds
Classification of commutative spaces
Bibliography
Subject index
Symbol index
Table index
Contemporary Mathematics, Volume: 436
2007; 334 pp; softcover
ISBN-13: 978-0-8218-3814-3
Expected publication date is September 1, 2007.
This book is based on talks presented at the Summer School on Interactions between Homotopy theory and Algebra held at the University of Chicago in the summer of 2004. The goal of this book is to create a resource for background and for current directions of research related to deep connections between homotopy theory and algebra, including algebraic geometry, commutative algebra, and representation theory. The articles in this book are aimed at the audience of beginning researchers with varied mathematical backgrounds and have been written with both the quality of exposition and the accessibility to novices in mind.
Readership
Graduate students and research mathematicians interested in algebraic geometry and homological algebra.
Table of Contents
Introductory lecture series
P. Goerss and K. Schemmerhorn -- Model categories and simplicial methods
C. Huneke and A. Taylor -- Lectures on local cohomology
H. Krause -- Derived categories, resolutions, and Brown representability
H. Krause -- Exercises on derived categories, resolutions, and Brown representability
Topics lecture series
J. P. C. Greenlees -- Spectra for commutative algebraists
K. Hess -- Rational homotopy theory: A brief introduction
S. Iyengar -- Andre-Quillen homology of commutative algebras
W. G. Dwyer -- Local cohomology in commutative algebra, homotopy theory, and group cohomology
J. P. C. Greenlees -- First steps in brave new commutative algebra
M. Hovey -- Cotorsion pairs and model categories
K. Bruning and I. Burban -- Coherent sheaves on an elliptic curve
A. Adem -- Lectures on the cohomology of finite groups