University Lecture Series, Volume: 40
2006; 192 pp; softcover
ISBN-10: 0-8218-4137-8
ISBN-13: 978-0-8218-4137-2
The "infinite-dimensional groups" in the title refer to
unitary groups of Hilbert spaces, the infinite symmetric group,
groups of homeomorphisms of manifolds, groups of transformations
of measure spaces, etc. The book presents an approach to the
study of such groups based on ideas from geometric functional
analysis and from exploring the interplay between dynamical
properties of those groups, combinatorial Ramsey-type theorems,
and the phenomenon of concentration of measure.
The dynamics of infinite-dimensional groups is very much unlike
that of locally compact groups. For instance, every locally
compact group acts freely on a suitable compact space (Veech). By
contrast, a 1983 result by Gromov and Milman states that whenever
the unitary group of a separable Hilbert space continuously acts
on a compact space, it has a common fixed point.
In the book, this new fast-growing theory is built strictly from
well-understood examples up. The book has no close counterpart
and is based on recent research articles. At the same time, it is
organized so as to be reasonably self-contained. The topic is
essentially interdisciplinary and will be of interest to
mathematicians working in geometric functional analysis,
topological and ergodic dynamics, Ramsey theory, logic and
descriptive set theory, representation theory, topological
groups, and operator algebras.
Readership
Graduate students and research mathematicians interested in
representation theory, dynamical systems, geometric functional
analysis, Ramsey theory, and descriptive set theory.
Table of Contents
Graduate Studies in Mathematics, Volume: 77
2006; 608 pp; hardcover
ISBN-10: 0-8218-4231-5
ISBN-13: 978-0-8218-4231-7
Ricci flow is a powerful analytic method for studying the
geometry and topology of manifolds. This book is an introduction
to Ricci flow for graduate students and mathematicians interested
in working in the subject. To this end, the first chapter is a
review of the relevant basics of Riemannian geometry. For the
benefit of the student, the text includes a number of exercises
of varying difficulty.
The book also provides brief introductions to some general
methods of geometric analysis and other geometric flows.
Comparisons are made between the Ricci flow and the linear heat
equation, mean curvature flow, and other geometric evolution
equations whenever possible.
Several topics of Hamilton's program are covered, such as short
time existence, Harnack inequalities, Ricci solitons, Perelman's
no local collapsing theorem, singularity analysis, and ancient
solutions.
A major direction in Ricci flow, via Hamilton's and Perelman's
works, is the use of Ricci flow as an approach to solving the
Poincare conjecture and Thurston's geometrization conjecture.
Readership
Graduate students and research mathematicians interested in
geometric analysis, the Poincare conjecture, Thurston's
geometrization conjecture, and 3-manifolds.
Table of Contents
Mathematical Surveys and Monographs, Volume: 132
2006; 441 pp; hardcover
ISBN-10: 0-8218-3922-5
ISBN-13: 978-0-8218-3922-5
This book develops rigorous foundations for parametrized homotopy
theory, which is the algebraic topology of spaces and spectra
that are continuously parametrized by the points of a base space.
It also begins the systematic study of parametrized homology and
cohomology theories.
The parametrized world provides the natural home for many
classical notions and results, such as orientation theory, the
Thom isomorphism, Atiyah and Poincare duality, transfer maps, the
Adams and Wirthmuller isomorphisms, and the Serre and Eilenberg-Moore
spectral sequences. But in addition to providing a clearer
conceptual outlook on these classical notions, it also provides
powerful methods to study new phenomena, such as twisted K-theory,
and to make new constructions, such as iterated Thom spectra.
Duality theory in the parametrized setting is particularly
illuminating and comes in two flavors. One allows the
construction and analysis of transfer maps, and a quite different
one relates parametrized homology to parametrized cohomology. The
latter is based formally on a new theory of duality in symmetric
bicategories that is of considerable independent interest.
The text brings together many recent developments in homotopy
theory. It provides a highly structured theory of parametrized
spectra, and it extends parametrized homotopy theory to the
equivariant setting. The theory of topological model categories
is given a more thorough treatment than is available in the
literature. This is used, together with an interesting blend of
classical methods, to resolve basic foundational problems that
have no nonparametrized counterparts.
Readership
Research mathematicians interested in recent advances in
algebraic topology.
Table of Contents
Prologue
Point-set topology, change functors, and proper actions
Introduction to Part I
The point-set topology of parametrized spaces
Change functors and compatibility relations
Proper actions, equivariant bundles and fibrations
Model categories and parametrized spaces
Introduction to Part II
Topologically bicomplete model categories
Well-grounded topological model categories
The qf-model structure on mathcal{K}_B
Equivariant qf-type model structures
Ex-fibrations and ex-quasifibrations
The equivalence between HoGmathcal{K}_B and hGmathcal{W}_B
Parametrized equivariant stable homotopy theory
Introduction to Part III
Enriched categories and G-categories
The category of orthogonal G-spectra over B
Model structures for parametrized G-spectra
Adjunctions and compatibility relations
Module categories, change of universe, and change of groups
Parametrized duality theory
Introduction to Part IV
Fiberwise duality and transfer maps
Closed symmetric bicategories
The closed symmetric bicategory of parametrized spectra
Costenoble-Waner duality
Fiberwise Costenoble-Waner duality
Homology and cohomology, Thom spectra, and addenda
Introduction to Part V
Parametrized homology and cohomology theories
Equivariant parametrized homology and cohomology
Twisted theories and spectral sequences
Parametrized FSP's and generalized Thom spectra
Epilogue: Cellular philosophy and alternative approaches
Bibliography
Index
Index of notation
Mathematical Surveys and Monographs, Volume: 133
2006; 378 pp; hardcover
ISBN-10: 0-8218-3739-7
ISBN-13: 978-0-8218-3739-9
The Fourier coefficients of modular forms are of widespread
interest as an important source of arithmetic information. In
many cases, these coefficients can be recovered from explicit
knowledge of the traces of Hecke operators. The original trace
formula for Hecke operators was given by Selberg in 1956. Many
improvements were made in subsequent years, notably by Eichler
and Hijikata.
This book provides a comprehensive modern treatment of the
Eichler-Selberg/Hijikata trace formula for the traces of Hecke
operators on spaces of holomorphic cusp forms of weight
mathtt{k}>2 for congruence subgroups of operatorname{SL}_2(mathbf{Z}).
The first half of the text brings together the background from
number theory and representation theory required for the
computation. This includes detailed discussions of modular forms,
Hecke operators, adeles and ideles, structure theory for
operatorname{GL}_2(mathbf{A}), strong approximation, integration
on locally compact groups, the Poisson summation formula, adelic
zeta functions, basic representation theory for locally compact
groups, the unitary representations of operatorname{GL}_2(mathbf{R}),
and the connection between classical cusp forms and their adelic
counterparts on operatorname{GL}_2(mathbf{A}).
The second half begins with a full development of the geometric
side of the Arthur-Selberg trace formula for the group
operatorname{GL}_2(mathbf{A}). This leads to an expression for
the trace of a Hecke operator, which is then computed explicitly.
The exposition is virtually self-contained, with complete
references for the occasional use of auxiliary results. The book
concludes with several applications of the final formula.
Readership
Graduate students and research mathematicians interested in
number theory, particularly modular forms, Hecke operators, and
trace formulas.
Table of Contents
Contemporary Mathematics, Volume: 421
2007; 273 pp; softcover
ISBN-10: 0-8218-3985-3
ISBN-13: 978-0-8218-3985-0
This volume consists of contributions by participants and
speakers at two conferences. The first was entitled Combinatorial
Group Theory, Discrete Groups and Number Theory and was held at
Fairfield University, December 8-9, 2004. It was in honor of
Professor Gerhard Rosenberger's sixtieth birthday. The second was
the AMS Special Session on Infinite Group Theory held at Bard
College, October 8-9, 2005. The papers in this volume provide a
very interesting mix of combinatorial group theory, discrete
group theory and ring theory as well as contributions to
noncommutative algebraic cryptography.
Readership
Graduate students and research mathematicians interested in group
theory.
Table of Contents
P. Ackermann -- A description of the arithmetic Fuchsian groups
with signature (2;-)
R. B. J. T. Allenby, G. Kim, and C. Y. Tang -- Outer automorphism
groups of certain orientable Seifert 3-manifold groups
M. Anshel and A. M. Gaglione -- The search for origins of the
commutator calculus
G. Baumslag, B. Fine, A. M. Gaglione, and D. Spellman -- A note
on nondiscrimination of nilpotent groups and Mal'cev completions
G. Baumslag, B. Fine, and X. Xu -- A proposed public key
cryptosystem using the modular group
H. Bluhm and M. Kreuzer -- Grobner basis techniques in the
computation of two-sided syzygies
M. Conder and P. Dobcsanyi -- Normal subgroups of the modular
group and other Hecke groups
O. B. Cristo and C. P. Milies -- Commutativity of units in group
rings
M. J. Evans -- Presentations of groups involving more generators
than are necessary. II.
B. Fine, A. M. Gaglione, and D. Spellman -- Unions of varieties
and quasivarieties
B. Fine, A. M. Gaglione, and D. Spellman -- Finitely presented
infinite torsion groups and a question of V. H. Dyson
A. Fonseca and R. M. Thomas -- Context-free irreducible word
problems in groups
D. Garrison, L.-C. Kappe, and D. Yull -- Autocommutators and the
autocommutator subgroup
J. Gilman -- Informative words and discreteness
R. Goldstein -- An algorithm for potentially positive words in F_2
G. Kern-Isberner -- Using group theory for knowledge
representation and discovery
M. Kreuzer, A. Myasnikov, G. Rosenberger, and A. Ushakov --
Quotient tests and Grobner bases
L. A. Kurdachenko and I. Y. Subbotin -- Transitivity of normality
and pronormal subgroups
C. Maclachlan -- Torsion in maximal arithmetic Fuchsian groups
S. Majewicz -- Nilpotent mathbb{Q}[x]-powered groups
R. F. Morse -- On the Rosenberger monster
C. F. Rocca, Jr. -- Density of test elements in finite abelian
groups
R. Weidmann -- Adjoining a root does not decrease the rank