Edited by: Jane Gilman, Rutgers University, Newark, NJ, William W. Menasco, State University of New York, Buffalo, NY, and Xiao-Song Lin, University of California, CA
Knots, Braids, and Mapping Class Groups-
-Papers Dedicated to Joan S. Birman
Description
There are a number of specialties in low-dimensional topology that can find in their "family tree" a common ancestry in the theory of surface mappings. These include knot theory as studied through the use of braid representations and 3-manifolds as studied through the use of Heegaard splittings. The study of the surface mapping class group (the modular group) is of course a rich subject in its own right, with relations to many different fields of mathematics and theoretical physics. But its most direct and remarkable manifestation is probably in the vast area of low-dimensional topology. Although the scene of this area has been changed dramatically and experienced significant expansion since the original publication of Professor Joan Birman's seminal work, Braids, Links, and Mapping Class Groups (Princeton University Press), she brought together mathematicians whose research span many specialties, all of common lineage.
The topics covered are quite diverse. Yet they reflect well the aim and spirit of the conference: to explore how these various specialties in low-dimensional topology have diverged in the past 20-25 years, as well as to explore common threads and potential future directions of development. This volume is dedicated to Joan Birman by her colleagues with deep admiration and appreciation of her contribution to low-dimensional topology.
Contents
J. Cantarella, D. DeTurck, and H. Gluck -- Upper bounds for the writhing of knots and the helicity of vector fields
O. T. Dasbach and B. S. Mangum -- The automorphism group of a free group is not subgroup separable
R. Ghrist -- Configuration spaces and braid groups on graphs in robotics
J. Gilman -- Alternate discreteness tests
S. P. Humphries -- Intersection-number operators for curves on discs and Chebyshev polynomials
O. Kharlampovich and A. Myasnikov -- Implicit function theorem over free groups and genus problem
M. E. Kidwell and T. B. Stanford -- On the z-degree of the Kauffman polynomial of a tangle decomposition
W. Li -- Knot invariants from counting periodic points
X.-S. Lin and Z. Wang -- Random walk on knot diagrams, colored Jones polynomial and Ihara-Selberg zeta function
F. Luo -- Some applications of a multiplicative structure on simple loops in surfaces
W. W. Menasco -- Closed braids and Heegaard splittings
J. H. Przytycki -- Homotopy and q-homotopy skein modules of 3-manifolds: An example in algebra Situs
T. Stanford and R. Trapp -- On knot invariants which are not of finite type
Details:
Series: AMS/IP Studies in Advanced Mathematics,Volume: 24
Publication Year: 2002
ISBN: 0-8218-2966-1
Paging: 176 pp.
Binding: Softcover
Mara D. Neusel, University of Notre Dame, IN,
and Larry Smith, Mathemtatisches Institut, Gottingen, Germany
Invariant Theory of Finite Groups
Description
The questions that have been at the center of invariant theory since the 19th century have revolved around the following themes: finiteness, computation, and special classes of invariants. This book begins with a survey of many concrete examples chosen from these themes in the algebraic, homological, and combinatorial context. In further chapters, the authors pick one or the other of these questions as a departure point and present the known answers, open problems, and methods and tools needed to obtain these answers. Chapter 2 deals with algebraic finiteness. Chapter 3 deals with combinatorial finiteness. Chapter 4 presents Noetherian finiteness. Chapter 5 addresses homological finiteness. Chapter 6 presents special classes of invariants, which deal with modular invariant theory and its particular problems and features. Chapter 7 collects results for special classes of invariants and coinvariants such as (pseudo) reflection groups and representations of low degree. If the ground field is finite, additional problems appear and are compensated for in part by the emergence of new tools. One of these is the Steenrod algebra, which the authors introduce in Chapter 8 to solve the inverse invariant theory problem, around which the authors have organized the last three chapters.
The book contains numerous examples to illustrate the theory, often of more than passing interest, and an appendix on commutative graded algebra, which provides some of the required basic background. There is an extensive reference list to provide the reader with orientation to the vast literature.
Contents
Invariants, their relatives, and problems
Algebraic finiteness
Combinatorial finiteness
Noetherian finiteness
Homological finiteness
Modular invariant theory
Special classes of invariants
The Steenrod algebra and invariant theory
Invariant ideals
Lannes's T-functor and applications
Review of commutative algebra
References
Typography
Notation
Index
Details:
Series: Mathematical Surveys and Monographs, Volume: 94
Publication Year: 2002
ISBN: 0-8218-2916-5
Paging: 371 pp.
Binding: Hardcover
Bernadette Perrin-Riou, Universite Paris-Sud, Orsay, France
Theorie d'Iwasawa des Representations
p-Adiques Semi-Stables
Description
Let F be a finite unramified extension of \mathbb{Q}_p and V a p-adic galois semi-stable representation on F of dimension d. The author develops Iwasawa theory for V and the \mathbb{Z}_p-cyclotomic extension: she constructs a logarithm (regulator map) from the Iwasawa module associated to the Galois cohomology of V in a very explicit module on an algebra generated by analytic functions on the annulus \{p^{-1/(p-1)} < \vert x\vert < 1\} and \mathrm{log} x.
Contents
Introduction
Anneaux de fonctions
Modules d'Iwasawa associes a un (\phi,N)-module
Construction d'elements de \mathcal{D}_{\infty,\ast}(\mathcal{D})
Theoremes de structure des \mathcal{D}_{\infty,\ast}(\mathcal{D})
Exponentielle
Normes universelles
A. Digression: le polylogarithme
B. Etude de \mathcal{B}^{\psi=0}
Quelques formules
Bibliographie
Index
Details:
Publisher: Societe Mathematique de France
Series: Memoires de la Societe Mathematique de France, Number: 84
Publication Year: 2001
ISBN: 2-85629-106-6
Paging: 111 pp.
Binding: Softcover
Michael Harris, Universite Paris, France,
and Steven Zucker, Johns Hopkins University, Baltimore, Maryland
Boundary Cohomology of Shimura Varieties, III: Coherent Cohomology on Higher-Rank Boundary Strata and Applications to Hodge Theory
Description
In this book, the authors complete the verification of the following fact: The nerve spectral sequence for the cohomology of the Borel-Serre boundary of a Shimura variety \mathrm{Sh} is a spectral sequence of mixed Hodge-de Rham structures over the field of definition of its canonical model. To achieve that, they develop the machinery of automorphic vector bundles on mixed Shimura varieties, for the latter enter in the boundary of the toroidal compactifications of \mathrm{Sh}; and study the nerve spectral sequence for the automorphic vector bundles and the toroidal boundary. They also extend the technique of averting issues of base-change by taking cohomology with growth conditions. They give and apply formulas for the Hodge gradation of the cohomology of both \mathrm{Sh} and its Borel-Serre boundary.
Contents
Introduction
automorphic vector bundles on mixed Shimura varieties
Mixed growth conditions and coherent cohomology
The nerve spectral sequence for coherent cohomology
Hodge theoretic applications
On the comparison of Hodge structures
Bibliography
Details:
Publisher: Societe Mathematique de France
Series: Memoires de la Societe Mathematique de France,Number: 85
Publication Year: 2001
ISBN: 2-85629-107-4
Paging: 116 pp.
Binding: Softcover