James F. Davis and Paul Kirk, Indiana University, Bloomington, IN
Lecture Notes in Algebraic Topology
Expected publication date is August 4, 2001
Description
The amount of algebraic topology a graduate student specializing in topology must learn can be intimidating. Moreover, by their second year of graduate studies, students must make the transition from understanding simple proofs line-by-line to understanding the overall structure of proofs of difficult theorems.
To help students make this transition, the material in this book is presented in an increasingly sophisticated manner. It is intended to bridge the gap between algebraic and geometric topology, both by providing the algebraic tools that a geometric topologist needs and by concentrating on those areas of algebraic topology that are geometrically motivated.
Prerequisites for using this book include basic set-theoretic topology, the definition of CW-complexes, some knowledge of the fundamental group/covering space theory, and the construction of singular homology. Most of this material is briefly reviewed at the beginning of the book.
The topics discussed by the authors include typical material for first- and second-year graduate courses. The core of the exposition consists of chapters on homotopy groups and on spectral sequences. There is also material that would interest students of geometric topology (homology with local coefficients and obstruction theory) and algebraic topology (spectra and generalized homology), as well as preparation for more advanced topics such as algebraic $K$-theory and the s-cobordism theorem.
A unique feature of the book is the inclusion, at the end of each chapter, of several projects that require students to present proofs of substantial theorems and to write notes accompanying their explanations. Working on these projects allows students to grapple with the "big picture", teaches them how to give mathematical lectures, and prepares them for participating in research seminars.
The book is designed as a textbook for graduate students studying algebraic and geometric topology and homotopy theory. It will also be useful for students from other fields such as differential geometry, algebraic geometry, and homological algebra. The exposition in the text is clear; special cases are presented over complex general statements.
Contents
Chain complexes, homology, and cohomology
Homological algebra
Products
Fiber bundles
Homology with local coefficients
Fibrations, cofibrations and homotopy groups
Obstruction theory and Eilenberg-MacLane spaces
Bordism, spectra, and generalized homology
Spectral sequences
Further applications of spectral sequences
Simple-homotopy theory
Bibliography
Index
Details:
Series: Graduate Studies in Mathematics, Volume: 35
Publication Year: 2001
ISBN: 0-8218-2160-1
Paging: 367 pp.
Binding: Hardcover
Solomon Friedberg, Boston College, Chestnut Hill, MA
Teaching Mathematics in Colleges and Universities:
Case Studies for Today's Classroom: Faculty Edition
Expected publication date is June 23, 2001
Description
Progress in mathematics frequently occurs first by studying particular examples and then by generalizing the patterns that have been observed into far-reaching theorems. Similarly, in teaching mathematics one often employs examples to motivate a general principle or to illustrate its use. This volume uses the same idea in the context of learning how to teach: By analyzing particular teaching situations, one can develop broadly applicable teaching skills useful for the professional mathematician. These teaching situations are the Case Studies of the title.
Just as a good mathematician seeks both to understand the details of a particular problem and to put it in a broader context, the examples presented here are chosen to offer a serious set of detailed teaching issues and to afford analysis from a broad perspective.
Each case raises a variety of pedagogical and communication issues that may be explored either individually or in a group facilitated by a faculty member. Teaching notes for such a facilitator are included for each Case in the Faculty Edition.
The methodology of Case Studies is widely used in areas such as business and law. The consideration of the mathematics cases presented here will help readers to develop teaching skills for their own classrooms.
This series is published in cooperation with the Mathematical Association of America.
Contents
Introduction
Fourteen case studies
Changing sections
Emily's test
Fundamental problems part I
Making the Grade (College algebra version/Calculus I version/Multivariable calculus version)
Making waves
Order out of chaos
Pairing up
The quicksand of problem four
Salad days
Seeking points
Study habits
Studying the exam (College algebra questions/Calculus II questions/Multivariable calculus questions)
There's something about Ted part I
What were they thinking?
Supporting materials for faculty
Developing effective mathematics teaching assistants using case studies: An introduction for faculty
Using case studies in a TA-development program
Types of cases
Summaries of cases
How these cases were created
Changing sections, teaching guide
Emily's test, teaching guide
Fundamental problems part II (Fundamental problems, teaching guide)
Making the grade, teaching guide
Making waves part II (Making waves, teaching guide)
Order out of chaos, teaching guide
Pairing up, teaching guide
The quicksand of problem four, teaching guide
Salad days, teaching guide
Seeking points, teaching guide
Study habits, teaching guide
Studying the exam, teaching guide (College algebra version/Calculus II version/Multivariable calculus version)
There's something about Ted Part II/There's something about Ted, teaching guide
What were they thinking?, teaching guide
Details:
Series: CBMS Issues in Mathematics Education, Volume: 10
Publication Year: 2001
ISBN: 0-8218-2875-4
Paging: 158 pp.
Binding: Softcover
Edited by: J. C. Taylor, McGill University, Montreal, PQ, Canada
Topics in Probability and Lie Groups: Boundary Theory
Expected publication date is July 6, 2001
Description
This volume is comprised of two parts: the first contains articles by S. N. Evans, F. Ledrappier, and Figa`-Talomanaca. These articles arose from a Centre de Recherches de Mathe'matiques (CRM) seminar entitiled, "Topics in Probability on Lie Groups: Boundary Theory".
Evans gives a synthesis of his pre-1992 work on Gaussian measures on vector spaces over a local field. Ledrappier uses the freegroup on $d$ generators as a paradigm for results on the asymptotic properties of random walks and harmonic measures on the Martin boundary. These articles are followed by a case study by Figa`-Talamanca using Gelfand pairs to study a diffusion on a compact ultrametric space.
The second part of the book is an appendix to the book Compactifications of Symmetric Spaces (Birkhauser) by Y. Guivarc'h and J. C. Taylor. This appendix consists of an article by each author and presents the contents of this book in a more algebraic way. L. Ji and J.-P. Anker simplifies some of their results on the asymptotics of the Green function that were used to compute Martin boundaries. And Taylor gives a self-contained account of Martin boundary theory for manifolds using the theory of second order strictly elliptic partial differential operators.
Contents
J.-P. Anker and L. Ji -- Heat kernel and Green function estimates on noncompact symmetric spaces. II
S. N. Evans -- Local fields, Gaussian measures, and Brownian motions
A. Figa`-Talamanca -- An application of Gelfand pairs to a problem of diffusion in compact ultrametric spaces
Y. Guivarc'h, J. C. Taylor, and L. Ji -- Compactifications of symmetric spaces and positive eigenfunctions of the Laplacian
F. Ledrappier -- Some asymptotic properties of random walks on free groups
J. C. Taylor -- The Martin compactification associated with a second order strictly elliptic partial differential operator on a manifold $\textbf M$
Details:
Series: CRM Proceedings & Lecture Notes,Volume: 28
Publication Year: 2001
ISBN: 0-8218-0275-5
Paging: 202 pp.
Binding: Softcover
Shigeyuki Morita, University of Tokyo, Japan
Geometry of Differential Forms
Expected publication date is August 17, 2001
Iwanami Series in Modern Mathematics
Description
Since the times of Gauss, Riemann, and Poincare', one of the principal goals of the study of manifolds has been to relate local analytic properties of a manifold with its global topological properties. Among the high points on this route are the Gauss-Bonnet formula, the de Rham complex, and the Hodge theorem; these results show, in particular, that the central tool in reaching the main goal of global analysis is the theory of differential forms.
The book by Morita is a comprehensive introduction to differential forms. It begins with a quick introduction to the notion of differentiable manifolds and then develops basic properties of differential forms as well as fundamental results concerning them, such as the de Rham and Frobenius theorems. The second half of the book is devoted to more advanced material, including Laplacians and harmonic forms on manifolds, the concepts of vector bundles and fiber bundles, and the theory of characteristic classes. Among the less traditional topics treated is a detailed description of the Chern-Weil theory.
The book can serve as a textbook for undergraduate students and for graduate students in geometry.
Contents
Manifolds
Differential forms
de Rham theorem
Answers to exercises
Index
Laplacian and harmonic forms
Vector bundles and characteristic classes
Fiber bundles and characteristic classes
Perspectives
References
Solutions
Index
Details:
Series: Translations of Mathematical Monographs, Volume: 201
Subseries: Iwanami Series in Modern Mathematics
Publication Year: 2001
ISBN: 0-8218-1045-6
Paging: approximately 352 pp.
Binding: Softcover
Edited by: Emma Previato, Boston University, MA
Advances in Algebraic Geometry Motivated by Physics
Expected publication date is July 4, 2001
Description
Our knowledge of objects of algebraic geometry such as moduli of curves, (real) Schubert classes, fundamental groups of complements of hyperplane arrangements, toric varieties, and variation of Hodge structures, has been enhanced recently by ideas and constructions of quantum field theory, such as mirror symmetry, Gromov-Witten invariants, quantum cohomology, and gravitational descendants.
These are some of the themes of this refereed collection of papers, which grew out of the special session, "Enumerative Geometry in Physics," held at the AMS meeting in Lowell, MA, April 2000. This session brought together mathematicians and physicists who reported on the latest results and open questions; all the abstracts are included as an Appendix, and also included are papers by some who could not attend.
The collection provides an overview of state-of-the-art tools, links that connect classical and modern problems, and the latest knowledge available.
Contents
A. I. Suciu -- Fundamental groups of line arrangements: Enumerative aspects
Enumerative or reality problems
S. J. Kova'cs -- Number of automorphisms of principally polarized abelian varieties
F. Sottile -- Rational curves on Grassmannians: Systems theory, reality, and transversality
Variational and moduli problems
D. Abramovich and A. Bertram -- The formula $12 = 10 + 2\times 1$ and its generalizations: Counting rational curves on $\mathbf{F}_2$
D. Abramovich and F. Oort -- Stable maps and Hurwitz schemes in mixed characteristics
L. Caporaso -- On modular properties of odd theta-characteristics
E. Cattani and J. Fernandez -- Asymptotic Hodge theory and quantum products
H. Clemens -- On rational curves in $n$-space with given normal bundle
R. Vakil -- A tool for stable reduction of curves on surfaces
Mirror symmetry and Gromov-Witten invariants
D. A. Cox, S. Katz, and Y.-P. Lee -- Virtual fundamental classes of zero loci
T. J. Jarvis, T. Kimura, and A. Vaintrob -- Gravitational descendants and the moduli space of higher spin curves
B. Kreuler -- Homological mirror symmetry in dimension one
A. R. Mavlyutov -- The Hodge structure of semiample hypersurfaces and a generalization of the monomial-divisor mirror map
A. Polishchuk and A. Vaintrob -- Algebraic construction of Witten's top Chern class
A. Postnikov -- Symmetries of Gromov-Witten invariants
S. Rosenberg and M. Vajiac -- Gauge theory techniques in quantum cohomology
C. Woodward -- Gromov-Witten invariants of flag manifolds and products of conjugacy classes
Appendix
E. Previato -- The Lowell meeting
Details:
Series: Contemporary Mathematics, Volume: 276
Publication Year: 2001
ISBN: 0-8218-2810-X
Paging: 294 pp.
Binding: Softcover