Michael E. Taylor, University of North Carolina, Chapel Hill, NC
Measure Theory and Integration
Graduate Studies in Mathematics Volume: 76
2006; 319 pp; hardcover
ISBN-10: 0-8218-4180-7
ISBN-13: 978-0-8218-4180-8
Expected publication date is September 8, 2006.
This self-contained treatment of measure and integration begins with a
brief review of the Riemann integral and proceeds to a construction of
Lebesgue measure on the real line. From there the reader is led to the
general notion of measure, to the construction of the Lebesgue integral
on a measure space, and to the major limit theorems, such as the Monotone
and Dominated Convergence Theorems. The treatment proceeds to L^p spaces,
normed linear spaces that are shown to be complete (i.e., Banach spaces)
due to the limit theorems. Particular attention is paid to L^2 spaces as
Hilbert spaces, with a useful geometrical structure.
Having gotten quickly to the heart of the matter, the text proceeds to
broaden its scope. There are further constructions of measures, including
Lebesgue measure on n-dimensional Euclidean space. There are also discussions
of surface measure, and more generally of Riemannian manifolds and the
measures they inherit, and an appendix on the integration of differential
forms. Further geometric aspects are explored in a chapter on Hausdorff
measure. The text also treats probabilistic concepts, in chapters on ergodic
theory, probability spaces and random variables, Wiener measure and Brownian
motion, and martingales.
This text will prepare graduate students for more advanced
studies in functional analysis, harmonic analysis, stochastic
analysis, and geometric measure theory.
Table of Contents
Edited by: Alejandro Adem, Jesus Gonzalez, and Guillermo Pastor
Recent Developments in Algebraic Topology
Contemporary Mathematics Volume: 407
2006; 191 pp; softcover
ISBN-10: 0-8218-3676-5
ISBN-13: 978-0-8218-3676-7
Expected publication date is August 16, 2006.
This book is an excellent illustration of the versatility of
Algebraic Topology interacting with other areas in Mathematics
and Physics. Topics discussed in this volume range from classical
Differential Topology and Homotopy Theory (Kervaire invariant one
problem) to more recent lines of research such as Topological
Quantum Field Theory (string theory). Likewise, alternative
viewpoints on classical problems in Global Analysis and Dynamical
Systems are developed (a spectral sequence approach to normal
form theory).
This collection of papers is based on talks at the conference on
the occasion of Sam Gitler's 70th birthday (December, 2003). The
variety of topics covered in this book reflects the many areas
where Sam Gitler's contributions have had an impact.
Readership
Graduate students and research mathematicians interested in
algebraic topology.
Table of Contents
D. M. Davis -- The mathematical work of Sam Gitler, 1960-2003
N. A. Baas, R. L. Cohen, and A. Rami rez -- The topology of the
category of open and closed strings
M. Bendersky and R. C. Churchill -- A spectral sequence approach
to normal forms
F. R. Cohen and I. Johnson -- On the degree 2 map for a sphere
C. L. Douglas -- On the fibrewise Poincare-Hopf theorem
T. de Fernex, E. Lupercio, T. Nevins, and B. Uribe -- A
localization principle for orbifold theories
D. Juan-Pineda and I. J. Leary -- On classifying spaces for the
family of virtually cyclic subgroups
S. Kallel and P. Salvatore -- Symmetric products of two
dimensional complexes
K. Y. Lam and D. Randall -- Block bundle obstruction to Kervaire
invariant one
P. Sankaran and P. Zvengrowski -- Upper bounds for the span of
projective Stiefel manifolds
M. A. Xicotencatl -- On mathbb{Z}_2-equivariant loop spaces
Edited by: Juan Luis Vazquez, Universidad Autonoma de Madrid, Spain, Xavier Cabre, Universitat Politecnica de Catalunya, Barcelona, Spain, and Jose Antonio Carrillo, Universitat Autonoma de Barcelona, Bellaterra, Spain
Recent Trends in Partial Differential Equations
Contemporary Mathematics Volume: 409
2006; 123 pp; softcover
ISBN-10: 0-8218-3891-1
ISBN-13: 978-0-8218-3891-4
Expected publication date is August 17, 2006.
This volume contains the research and expository articles for the
courses and talks given at the UIMP-RSME Lluis A. Santalo Summer
School, "Recent Trends in Partial Differential Equations".
The goal of the Summer School was to present some of the many
advances that are currently taking place in the interaction
between nonlinear partial differential equations and their
applications to other scientific disciplines. Oriented to young
post-docs and advanced doctoral students, the courses dealt with
topics of current interest.
Some of the tools presented are quite powerful and sophisticated.
These new methods are presented in an expository manner or
applied to a particular example to demonstrate the main ideas of
the method and to serve as a handy introduction to further study.
Young researchers in partial differential equations and
colleagues from neighboring fields will find these notes a good
addition to their libraries.
This is a joint publication of the Real Sociedad Matematica
Espanola and the American Mathematical Society.
Readership
Research mathematicians interested in partial differential
equations.
Table of Contents
L. Ambrosio -- Steepest descent flows and applications to spaces
of probability measures
L. Desvillettes -- Hypocoercivity: the example of linear
transport
H. Koch and E. Zuazua -- A hybrid system of PDE's arising in multi-structure
interaction: Coupling of wave equations in n and n-1 space dimensions
A. Aftalion -- Some rigorous results for vortex patterns in Bose-Einstein
condensates
M. Escobedo and S. Mischler -- Qualitative properties of some
Boltzmann like equations which do not fulfill a detailed balance
condition
Carlo Mazza, Rutgers University, Piscataway, NJ, Vladimir Voevodsky, Institute for Advanced Study,
Princeton, NJ, and Charles Weibel, Rutgers University, New Brunswick, NJ
Lecture Notes on Motivic Cohomology
Clay Mathematics Monographs Volume: 2
2006; approx. 210 pp; hardcover
ISBN-10: 0-8218-3847-4
ISBN-13: 978-0-8218-3847-1
Expected publication date is September 9, 2006.
The notion of a motive is an elusive one, like its namesake
"the motif" of Cezanne's impressionist method of
painting. Its existence was first suggested by Grothendieck in
1964 as the underlying structure behind the myriad cohomology
theories in Algebraic Geometry. We now know that there is a
triangulated theory of motives, discovered by Vladimir Voevodsky,
which suffices for the development of a satisfactory Motivic
Cohomology theory. However, the existence of motives themselves
remains conjectural.
This book provides an account of the triangulated theory of
motives. Its purpose is to introduce Motivic Cohomology, to
develop its main properties, and finally to relate it to other
known invariants of algebraic varieties and rings such as Milnor
K-theory, etale cohomology, and Chow groups. The book is divided
into lectures, grouped in six parts. The first part presents the
definition of Motivic Cohomology, based upon the notion of
presheaves with transfers. Some elementary comparison theorems
are given in this part. The theory of (etale, Nisnevich, and
Zariski) sheaves with transfers is developed in parts two, three,
and six, respectively. The theoretical core of the book is the
fourth part, presenting the triangulated category of motives.
Finally, the comparison with higher Chow groups is developed in
part five.
The lecture notes format is designed for the book to be read by
an advanced graduate student or an expert in a related field. The
lectures roughly correspond to one-hour lectures given by
Voevodsky during the course he gave at the Institute for Advanced
Study in Princeton on this subject in 1999-2000. In addition,
many of the original proofs have been simplified and improved so
that this book will also be a useful tool for research
mathematicians.
Table of Contents
Presheaves with transfers
The category of finite correspondences
Presheaves with transfers
Motivic cohomology
Weight one motivic cohomology
Relation to Milnor K-theory
Etale motivic theory
Etale sheaves with transfers
The relative Picard group and Suslin's rigidity theorem
Derived tensor products
mathbb{A}^1-weak equivalence
Etale motivic cohomology and algebraic singular homology
Nisnevich sheaves with transfers
Standard triples
Nisnevich sheaves
Nisnevich sheaves with transfers
The triangulated category of motives
The category of motives
The complex mathbb{Z}(n) and mathbb{P}^n
Equidimensional cycles
Higher Chow groups
Higher Chow groups
Higher Chow groups and equidimensional cycles
Motivic cohomology and higher Chow groups
Geometric motives
Zariski sheaves with transfers
Covering morphisms of triples
Zariski sheaves with transfers
Contractions
Homotopy invariance of cohomology
Bibliography
Glossary
Index
Bruce C. Berndt, University of Illinois, Urbana-Champaign, IL
Number Theory in the Spirit of Ramanujan
Student Mathematical Library Volume: 34
2006; approx. 198 pp; softcover
ISBN-10: 0-8218-4178-5
ISBN-13: 978-0-8218-4178-5
Expected publication date is September 13, 2006.
Ramanujan is recognized as one of the great number theorists of the twentieth
century. Here now is the first book to provide an introduction to his work
in number theory. Most of Ramanujan's work in number theory arose out of
q-series and theta functions. This book provides an introduction to these
two important subjects and to some of the topics in number theory that
are inextricably intertwined with them, including the theory of partitions,
sums of squares and triangular numbers, and the Ramanujan tau function.
The majority of the results discussed here are originally due to Ramanujan
or were rediscovered by him. Ramanujan did not leave us proofs of the thousands
of theorems he recorded in his notebooks, and so it cannot be claimed that
many of the proofs given in this book are those found by Ramanujan. However,
they are all in the spirit of his mathematics.
The subjects examined in this book have a rich history dating
back to Euler and Jacobi, and they continue to be focal points of
contemporary mathematical research. Therefore, at the end of each
of the seven chapters, Berndt discusses the results established
in the chapter and places them in both historical and
contemporary contexts. The book is suitable for advanced
undergraduates and beginning graduate students interested in
number theory.
Readership
Undergraduate and graduate students interested in number theory, including
q-series and theta functions.
Table of Contents
Introduction
Congruences for p(n) and tau(n)
Sums of squares and sums of triangular numbers
Eisenstein series
The connection between hypergeometric functions and theta
functions
Applications of the primary theorem of Chapter 5
The Rogers-Ramanujan continued fraction
Bibliography
William M. Singer, Fordham University, Bronx, NY
Steenrod Squares in Spectral Sequences
Mathematical Surveys and Monographs Volume: 129
2006; 155 pp; hardcover
ISBN-10: 0-8218-4141-6
ISBN-13: 978-0-8218-4141-9
Expected publication date is September 9, 2006.
This book develops a general theory of Steenrod operations in
spectral sequences. It gives special attention to the change-of-rings
spectral sequence for the cohomology of an extension of Hopf
algebras and to the Eilenberg-Moore spectral sequence for the
cohomology of classifying spaces and homotopy orbit spaces. In
treating the change-of-rings spectral sequence, the book develops
from scratch the necessary properties of extensions of Hopf
algebras and constructs the spectral sequence in a form
particularly suited to the introduction of Steenrod squares. The
resulting theory can be used effectively for the computation of
the cohomology rings of groups and Hopf algebras, and of the
Steenrod algebra in particular, and so should play a useful role
in stable homotopy theory. Similarly the book offers a self-contained
construction of the Eilenberg-Moore spectral sequence, in a form
suitable for the introduction of Steenrod operations. The
corresponding theory is an effective tool for the computation of
the cohomology rings of the classifying spaces of the exceptional
Lie groups, and it promises to be equally useful for the
computation of the cohomology rings of homotopy orbit spaces and
of the classifying spaces of loop groups.
Readership
Graduate students and research mathematicians interested in
algebraic topology.
Table of Contents
Conventions
The spectral sequence of a bisimplicial coalgebra
Bialgebra actions on the cohomology of algebras
Extensions of Hopf algebras
Steenrod operations in the change-of-rings spectral sequence
The Eilenberg-Moore spectral sequence
Steenrod operations in the Eilenberg-Moore spectral sequence
Bibliography
Index