Frederick J. Almgren, Jr.
Plateau's Problem: An Invitation to Varifold Geometry
Description
There have been many wonderful developments in the theory of minimal surfaces and geometric measure theory in the past 25 to 30 years. Many of the researchers who have produced these excellent results were inspired by this little book--or by Fred Almgren himself.
The book is indeed a delightful invitation to the world of variational geometry. A central topic is Plateau's Problem, which is concerned with surfaces that model the behavior of soap films. When trying to resolve the problem, however, one soon finds that smooth surfaces are insufficient: Varifolds are needed. With varifolds, one can obtain geometrically meaningful solutions without having to know in advance all their possible singularities. This new tool makes possible much exciting new analysis and many new results.
Plateau's problem and varifolds live in the world of geometric measure theory, where differential geometry and measure theory combine to solve problems which have variational aspects. The author's hope in writing this book was to encourage young mathematicians to study this fascinating subject further. Judging from the success of his students, it achieves this exceedingly well.
Contents
The phenomena of least area problems
Integration of differential forms over rectifiable sets
Varifolds
Variational problems involving varifolds
References
Index
Details:
Publisher: American Mathematical Society
Distributor: American Mathematical Society
Series: Student Mathematical Library
Publication Year: 2001
ISBN: 0-8218-2747-2
Paging: approximately 88 pp.
Binding: Softcover
Armand Borel, Institute for Advanced Study, Princeton, NJ
Essays in the History of Lie Groups and Algebraic Groups
Description
Algebraic groups and Lie groups are important in most major areas of mathematics, occurring in diverse roles such as the symmetries of differential equations and as central figures in the Langlands program for number theory. In this book, Professor Borel looks at the development of the theory of Lie groups and algebraic groups, highlighting the evolution from the almost purely local theory at the start to the global theory that we know today. As the starting point of this passage from local to global, the author takes Lie's theory of local analytic transformation groups and Lie algebras. He then follows the globalization of the process in its two most important frameworks: (transcendental) differential geometry and algebraic geometry. Chapters II to IV are devoted to the former, Chapters V to VIII, to the latter.
The essays in the first part of the book survey various proofs of the full reducibility of linear representations of $SL_2M$, the contributions H. Weyl to representation and invariant theory for Lie groups, and conclude with a chapter on E. Cartan's theory of symmetric spaces and Lie groups in the large.
The second part of the book starts with Chapter V describing the development of the theory of linear algebraic groups in the 19th century. Many of the main contributions here are due to E. Study, E. Cartan, and above all, to L. Maurer. After being abandoned for nearly 50 years, the theory was revived by Chevalley and Kolchin and then further developed by many others. This is the focus of Chapter VI. The book concludes with two chapters on various aspects of the works of Chevalley on Lie groups and algebraic groups and Kolchin on algebraic groups and the Galois theory of differential fields.
The author brings a unique perspective to this study. As an important developer of some of the modern elements of both the differential geometric and the algebraic geometric sides of the theory, he has a particularly deep appreciation of the underlying mathematics. His lifelong involvement and his historical research in the subject give him a special appreciation of the story of its development.
Copublished with the London Mathematical Society. Members of the LMS may order directly from the AMS at the AMS member price. The LMS is registered with the Charity Commissioners.
Contents
Overview
Full reducibility and invariants for $\mathbf{SL}_2(\mathbb C)$
Hermann Weyl and Lie groups
Élie Cartan, symmetric spaces and Lie groups
Linear algebraic groups in the 19th century
Linear algebraic groups in the 20th century
The work of Chevalley in Lie groups and algebraic groups
Algebraic groups and Galois theory in the work of Ellis R. Kolchin
Name index
Subject index
Photo section
Details:
Publisher: American Mathematical Society, London Mathematical Society
Distributor: American Mathematical Society
Series: History of Mathematics,
Publication Year: 2001
ISBN: 0-8218-0288-7
Paging: approximately 184 pp.
Binding: Hardcover
V. A. Vassiliev, Independent University of Moscow, Russia
Introduction to Topology
From a review for the Russian edition ...
"The book is based on a course given by the author in 1996 to first and second year students at Independent Moscow University ... the emphasis is on illustrating what is happening in topology, and the proofs (or their ideas) covered are those which either have important generalizations or are useful in explaining important concepts ... This is an excellent book and one can gain a great deal by reading it. The material, normally requiring several volumes, is covered in 123 pages, allowing the reader to appreciate the interaction between basic concepts of algebraic and differential topology without being buried in minutiae."-- Mathematical Reviews
Description
This English translation of a Russian book presents the basic notions of differential and algebraic topology, which are indispensable for specialists and useful for research mathematicians and theoretical physicists. In particular, ideas and results are introduced related to manifolds, cell spaces, coverings and fibrations, homotopy groups, homology and cohomology, intersection index, etc. The author notes, "The lecture note origins of the book left a significant imprint on its style. It contains very few detailed proofs: I tried to give as many illustrations as possible and to show what really occurs in topology, not always explaining why it occurs." He concludes, "As a rule, only those proofs (or sketches of proofs) that are interesting per se and have important generalizations are presented.
Contents
Topological spaces and operations with them
Homotopy groups and homotopy equivalence
Coverings
Cell spaces ($CW$-complexes)
Relative homotopy groups and the exact sequence of a pair
Fiber bundles
Smooth manifolds
The degree of a map
Homology: Basic definitions and examples
Main properties of singular homology groups and their computation
Homology of cell spaces
Morse theory
Cohomology and Poincaré duality
Some applications of homology theory
Multiplication in cohomology (and homology)
Index of notations
Subject index
Details:
Publisher: American Mathematical Society
Distributor: American Mathematical Society
Series: Student Mathematical Library, Volume: 14
Publication Year: 2001
ISBN: 0-8218-2162-8
Paging: 149 pp.
Edited by: V. Turaev, Université Louis Pasteur - CNRS, Strasbourg, France, and
A. Vershik, Steklov Mathematical Institute, St. Petersburg, Russia
Topology, Ergodic Theory, Real Algebraic Geometry: Rokhlin's Memorial
Description
This book is dedicated to the memory of the outstanding Russian mathematician, V. A. Rokhlin (1919-1984). It is a collection of research papers written by his former students and followers, who are now experts in their fields. The topics in this volume include topology (the Morse-Novikov theory, spin bordisms in dimension 6, and skein modules of links), real algebraic geometry (real algebraic curves, plane algebraic surfaces, algebraic links, and complex orientations), dynamics (ergodicity, amenability, and random bundle transformations), geometry of Riemannian manifolds, theory of Teichmüller spaces, measure theory, etc. The book also includes a biography of Rokhlin by Vershik and two articles of historical interest.
Contents
A. M. Vershik -- Vladimir Abramovich Rokhlin
A. M. Vershik -- V. A. Rokhlin and the modern theory of measurable partitions
J. E. Andersen and V. Turaev -- Higher skein modules, II
F. Bihan -- Betti numbers of real numerical quintic surfaces
A. Bufetov -- Markov averaging and ergodic theorems for several operators
Z. Coelho and W. Parry -- Ergodicity of $p$-adic multiplications and the distribution of Fibonacci numbers
A. Degtyarev -- On the Pontryagin-Viro form
M. Farber -- Counting zeros of closed 1-forms
S. Finashin -- Rokhlin's question and smooth quotients by complex conjugation of singular real algebraic surfaces
I. Itenberg -- On the number of even ovals of a nonsingular curve of even degree in $\mathbb{R}P^2$
N. V. Ivanov -- Isometries of Teichmüller spaces from the point of view of Mostow rigidity
V. A. Kaimanovich -- Equivalence relations with amenable leaves need not be amenable
V. Kharlamov, J.-J. Risler, and E. Shustin -- Maximal smoothings of real plane curve singular points
Y. Kifer -- On the topological pressure for random bundle transformations
S. Yu. Orevkov -- Complex orientations of $M$-curves of degree 7
M. Pollicott and R. Sharp -- Poincaré series and comparison theorems for variable negative curvature
O. Viro -- Encomplexing the writhe
B. Weiss -- Monotileable amenable groups
A. V. Zhubr -- Spin bordism of oriented manifolds and the Hauptvermutung for 6-manifolds
Details:
Publisher: American Mathematical Society
Distributor: American Mathematical Society
Series: American Mathematical Society Translations--Series 2,Volume: 202
Publication Year: 2001
ISBN: 0-8218-2740-5
Paging: 286 pp.
Binding: Hardcover
W. J. Kaczor and M. T. Nowak, Maria Curie-Sklodowska University, Lublin, Poland
Problems in Mathematical Analysis II: Continuity and Differentiation
Description
This is the sequel to Problems in Mathematical Analysis I: Real Numbers, Sequences and Series (Volume 4 in the AMS series, the Student Mathematical Library). As in the first volume, this book is divided into two parts. The first is a collection of exercises and problems, and the second contains their solutions. The book mainly deals with real functions of one real variable. Topics include: properties of continuous functions, intermediate value property, uniform continuity, mean value theorems, Taylor's formula, convex functions, sequences and series of functions.
The book is mainly geared toward students studying the basic principles of analysis. However, given its selection of problems, organization, and level, it would be an ideal choice for tutorial or problem-solving seminars, particularly those geared toward the Putnam exam. It is also suitable for self-study. The presentation of material is designed to help student comprehension, to encourage them to ask their own questions, and to start research.
The collection of problems in the book will also help teachers who wish to incorporate problems into their lectures. Solutions for most problems are provided.
Contents
Problems
Limits and continuity
Differentiation
Sequences and series of functions
Solutions
Limits and continuity
Differentiation
Sequences and series of functions
Bibliography
Index
Details:
Publisher: American Mathematical Society
Distributor: American Mathematical Society
Series: Student Mathematical Library, Volume: 12
Publication Year: 2001
ISBN: 0-8218-2051-6
Paging: 398 pp.
Binding: Softcover