Kuo-Tsai Chen (1923-1987) is best known to the mathematics community for his work on iterated integrals and power series connections in conjunction with his research on the cohomology of loop spaces. His work is intimately related to the theory of minimal models as developed by Dennis Sullivan, whose own work was in part inspired by the research of Chen. An outstanding and original mathematician, Chen's work falls naturally into three periods: his early work on group theory and links in the three sphere; his subsequent work on formal differential equations, which gradually developed into his most powerful and important work; and his work on iterated integrals and homotopy theory, which occupied him for the last twenty years of his life. The goal of Chen's iterated integrals program, which is a de Rham theory for path spaces, was to study the interaction of topology and analysis through path integration. The present volume is a comprehensive collection of Chen's mathematical publications preceded by an article, "The Life and Work of Kuo-Tsai Chen," placing his work and research interests into their proper context and demonstrating the power and scope of his influence.

The papers contained in this volume are an indication of the topics th discussed and the interests of the participants of The 9 International Conference on Probability in Banach Spaces, held at Sandjberg, Denmark, August 16-21, 1993. A glance at the table of contents indicates the broad range of topics covered at this conference. What defines research in this field is not so much the topics considered but the generality of the ques tions that are asked. The goal is to examine the behavior of large classes of stochastic processes and to describe it in terms of a few simple prop erties that the processes share. The reward of research like this is that occasionally one can gain deep insight, even about familiar processes, by stripping away details, that in hindsight turn out to be extraneous. A good understanding about the disciplines involved in this field can be obtained from the recent book, Probability in Banach Spaces, Springer-Verlag, by M. Ledoux and M. Thlagrand. On page 5, of this book, there is a list of previous conferences in probability in Banach spaces, including the other eight international conferences. One can see that research in this field over the last twenty years has contributed significantly to knowledge in probability and has had important applications in many other branches of mathematics, most notably in statistics and functional analysis."

In 1992 two successive symposia were held in Japan on algorithms, fractals and dynamical systems. The first one was Hayashibara Forum '92: International Symposium on New Bases for Engineering Science, Algorithms, Dynamics and Fractals held at Fujisaki Institute of Hayashibara Biochemical Laboratories, Inc. in Okayama during November 23-28 in which 49 mathematicians including 19 from abroad participated. They include both pure and applied mathematicians of diversified backgrounds and represented 11 coun tries. The organizing committee consisted of the following domestic members and Mike KEANE from Delft: Masayosi HATA, Shunji ITO, Yuji ITO, Teturo KAMAE (chairman), Hitoshi NAKADA, Satoshi TAKAHASHI, Yoichiro TAKAHASHI, Masaya YAMAGUTI The second one was held at the Research Institute for Mathematical Science at Kyoto University from November 30 to December 2 with emphasis on pure mathematical side in which more than 80 mathematicians participated. This volume is a partial record of the stimulating exchange of ideas and discussions which took place in these two symposia.

In the large and thriving field of compact transformation groups an important role has long been played by cohomological methods. This book aims to give a contemporary account of such methods, in particular the applications of ordinary cohomology theory and rational homotopy theory with principal emphasis on actions of tori and elementary abelian p-groups on finite-dimensional spaces. For example, spectral sequences are not used in Chapter 1, where the approach is by means of cochain complexes; and much of the basic theory of cochain complexes needed for this chapter is outlined in an appendix. For simplicity, emphasis is put on G-CW-complexes; the refinements needed to treat more general finite-dimensional (or finitistic) G-spaces are often discussed separately. Subsequent chapters give systematic treatments of the Localization Theorem, applications of rational homotopy theory, equivariant Tate cohomology and actions on Poincare duality spaces. Many shorter and more specialized topics are included also. Chapter 2 contains a summary of the main definitions and results from Sullivan's version of rational homotopy theory which are used in the book.

The fundamental theorem of algebra states that any complex polynomial must have a complex root. This book examines three pairs of proofs of the theorem from three different areas of mathematics: abstract algebra, complex analysis and topology. The first proof in each pair is fairly straightforward and depends only on what could be considered elementary mathematics. However, each of these first proofs leads to more general results from which the fundamental theorem can be deduced as a direct consequence. These general results constitute the second proof in each pair. To arrive at each of the proofs, enough of the general theory of each relevant area is developed to understand the proof. In addition to the proofs and techniques themselves, many applications such as the insolvability of the quintic and the transcendence of e and pi are presented. Finally, a series of appendices give six additional proofs including a version of Gauss'original first proof. The book is intended for junior/senior level undergraduate mathematics students or first year graduate students, and would make an ideal "capstone" course in mathematics.

Daniel Quillen's definition of the higher algebraic K-groups of a ring emphasized the importance of computing the homology of groups of matrices. This text traces the development of this theory from Quillen's fundamental calculation of the cohomology of GLn (Fq). The stability theorems and low-dimensional results of A. Suslin, W. van der Kallen and others are presented as well as recent results for rank one groups. A chapter on the Friedlander-Milnor-conjecture concerning the homology of algebraic groups made discrete is also included. This marks the first time that these results have been collected in a single volume. The book should prove useful to graduate students and researchers in K-theory, group cohomology, algebraic geometry and topology.

Inverse limits with set-valued functions are quickly becoming a popular topic of research due to their potential applications in dynamical systems and economics. This brief provides a concise introduction dedicated specifically to such inverse limits. The theory is presented along with detailed examples which form the distinguishing feature of this work. The major differences between the theory of inverse limits with mappings and the theory with set-valued functions are featured prominently in this book in a positive light. The reader is assumed to have taken a senior level course in analysis and a basic course in topology. Advanced undergraduate and graduate students, and researchers working in this area will find this brief useful.

gentle introduction to the subject, leading the reader to understand the notion of what is important in topology with regard to geometry. Divided into three sections - The line and the plane, Metric spaces and Topological spaces -, the book eases the move into higher levels of abstraction. Students are thereby informally assisted in learning new ideas while remaining on familiar territory. The authors do not assume previous knowledge of axiomatic approach or set theory. Similarly, they have restricted the mathematical vocabulary in the book so as to avoid overwhelming the reader, and the concept of convergence is employed to allow students to focus on a central theme while moving to a natural understanding of the notion of topology. The pace of the book is relaxed with gradual acceleration: the first nine sections form a balanced course in metric spaces for undergraduates while also containing ample material for a two-semester graduate course. Finally, the book illustrates the many connections between topology and other subjects, such as analysis and set theory, via the inclusion of "Extras" at the end of each chapter presenting a brief foray outside topology.

"Categorical Perspectives" consists of introductory surveys as well as articles containing original research and complete proofs devoted mainly to the theoretical and foundational developments of category theory and its applications to other fields. A number of articles in the areas of topology, algebra and computer science reflect the varied interests of George Strecker to whom this work is dedicated. Notable also are an exposition of the contributions and importance of George Strecker's research and a survey chapter on general category theory. This work is an excellent reference text for researchers and graduate students in category theory and related areas. Contributors: H.L. Bentley * G. Castellini * R. El Bashir * H. Herrlich * M. Husek * L. Janos * J. Koslowski * V.A. Lemin * A. Melton * G. Preua * Y.T. Rhineghost * B.S.W. Schroeder * L. Schr"der * G.E. Strecker * A. Zmrzlina

A complete and self-contained account of the dimension theory of general topological spaces, with particular emphasis on the dimensional properties of non-metrizable spaces. It makes the subject accessible to beginning graduate students and will also serve as a reference work for general topologists. Two introductory chapters summarize standard results in general topology, and cover material on paracompactness and metrization. The principal definitions of dimension follow and their general properties are deduced. Many examples are analysed to show some of the more surprising or pathological aspects of dimension theory. Wherever it is useful to do so, proofs are given in detail.

The present report on spaces of holomorphic mappings was prepared for the Sexto Coloquio Brasileiro de Matematica (Po os de Caldas, Minas Gerais, Brazil, July 1967). I also had the oppor tunity of giving a series of lectures on this material while I was a visiting member at the Center for Theoretical Studies of the University of Miami (Coral Gables, Florida, USA, February 1968). The preparation of this report was sponsored in part by the USA National Science Foundation through a grant to the University of Rochester. I am glad to thank Professors Paul R. Halmos and Peter J. Hilton for accepting my text as part of the series Ergebnisse der Mathematik und ihre Grenzgebiete. Rochester, New York 1968 Leopoldo Nachbin Contents 1. Introduction. 1 2. Notation and Terminology 4 3. Continuous Polynomials 6 4. Convergent Power Series 11 5. Holomorphic Mappings. 16 6. The Cauchy Integral 20 7. Convergence of Taylor Series. 26 8. Topology on the Space of all Holomorphic Mappings 31 9. Holomorphy Types. 34 10. Differentiation of Holomorphy Types . 38 II. Topology on Spaces of Holomorphic Mappings. 43 12. Bounded Subsets. 49 13. Relatively Compact Subsets . 54 14. The Current Holomorphy Type 59 15. Bibliographical References. 62 Subject Index 65 1."

1.1 Background The subject of this book is Morse homology as a combination of relative Morse theory and Conley's continuation principle. The latter will be useda s an instrument to express the homology encoded in a Morse complex associated to a fixed Morse function independent of this function. Originally, this type of Morse-theoretical tool was developed by Andreas Floer in order to find a proof of the famous Arnold conjecture, whereas classical Morse theory turned out to fail in the infinite-dimensional setting. In this framework, the homological variant of Morse theory is also known as Floer homology. This kind of homology theory is the central topic of this book. But first, it seems worthwhile to outline the standard Morse theory. 1.1.1 Classical Morse Theory The fact that Morse theory can be formulated in a homological way is by no means a new idea. The reader is referred to the excellent survey paper by Raoul Bott [Bol.

Reprint from GAFA, Vol. 5 (1995), No. 2. Enlarged by a short biography of Mikhail Gromov and a list of publications. In the last decades of the XX century tremendous progress has been achieved in geometry. The discovery of deep interrelations between geometry and other fields including algebra, analysis and topology has pushed it into the mainstream of modern mathematics. This Special Issue of Geometric And Functional Analysis (GAFA) in honour of Mikhail Gromov contains 14 papers which give a wide panorama of recent fundamental developments in modern geometry and its related subjects. CONTRIBUTORS: J. Bourgain, J. Cheeger, J. Cogdell, A. Connes, Y. Eliashberg, H. Hofer, F. Lalonde, W. Luo, G. Margulis, D. McDuff, H. Moscovici, G. Mostow, S. Novikov, G. Perelman, I. Piatetski-Shapiro, G. Pisier, X. Rong, Z. Rudnick, D. Salamon, P. Sarnak, R. Schoen, M. Shubin, K. Wysocki, and E. Zehnder. The book is a collection of important results and an enduring source of new ideas for researchers and students in a broad spectrum of directions related to all aspects of Geometry and its applications to Functional Analysis, PDE, Analytic Number Theory and Physics.

This book consists essentially of a collection of papers that were contributed to a national meeting held in Houston, Texas, in 1983. The papers contained herein cover a wide range of electro- chemical engineering topics and should serve as useful starting points in the design of electrochemical cells. The editor would like to thank the authors for their contribu- tions and patience and the typists, Mrs. Susan 'Firth and Mrs. Jeri Saulsbury, for their efforts. Also, the editor would like to thank Mr. T. Nguyen and Ms. M. A. Nictrolson for their help in preparing the index of this book. R. E. White v CONTENTS Design and Development of Electrochemical Chlor-Alkali Cells * * * * * * * * * 1 S. N. Chatterjee A Simple Model of a Diaphragm - Type Chlorine Cell 25 R. E. White, J. S. Beckerdite, and J. Van Zee Design Principles for Chlorine Membrane Cells 61 K. H. Simmrock Hydroxyl Ion Migration, Chemical Reactions, Water Transport and Other Effects as Optimizing Parameters in Cross-, Co- and Countercurrently Operated Membrane Cells for the Chlor/Alkali Electrolysis . *******. **** 89 K. H. Simmrock Hydraulic Modelling as an Aid to Electrochemical Cell Design. 115 I. Wardle Calculating Mechanical Component Voltage Drops in 123 Electrochemical Cells * * * * I. Wardle Electrolysis Cell Design for Ion Exchange Membrane Chlor-Alkali Process * * * * * * * * * * * * 135 M. Seko, A. Yomiyama, and S.

It is a privilege for me to write a foreword for this unusual book. The book is not primarily a reference work although many of the ideas and proofs are explained more clearly here than in any other source that I know. Nor is this a text of the customary sort. It is rather a record of a particular course and Gordon Whyburn's special method of teaching it. Perhaps the easiest way to describe the course and the method is to relate my own personal experience with a forerunner of this same course in the academic year 1937-1938. At that time, the course was offered every other year with a following course in algebraic topology on alternate years. There were five of us enrolled, and on the average we knew less mathematics than is now routinely given in a junior course in analysis. Whyburn's purpose, as we learned, was to prepare us in minimal time for research in the areas in which he was inter ested. His method was remarkable."

Like any books on a subject as vast as this, this book has to have a point-of-view to guide the selection of topics. Naber takes the view that the rekindled interest that mathematics and physics have shown in each other of late should be fostered, and that this is best accomplished by allowing them to cohabit. The book weaves together rudimentary notions from the classical gauge theory of physics with the topological and geometrical concepts that became the mathematical models of these notions. The reader is asked to join the author on some vague notion of what an electromagnetic field might be, to be willing to accept a few of the more elementary pronouncements of quantum mechanics, and to have a solid background in real analysis and linear algebra and some of the vocabulary of modern algebra. In return, the book offers an excursion that begins with the definition of a topological space and finds its way eventually to the moduli space of anti-self-dual SU(2) connections on S4 with instanton number -1.

This account of the History of General Topology has grown out of the special session on this topic at the American Mathematical Society meeting in San Anto- nio, Texas, 1993. It was there that the idea grew to publish a book on the historical development of General Topology. Moreover it was felt that it was important to undertake this project while topologists who knew some of the early researchers were still active. Since the first paper by Frechet, "Generalisation d'un theoreme de Weier- strass", C.R.Acad. Sci. 139, 1904, 848-849, and Hausdorff's classic book, "GrundZiige der Mengenlehre", Leipzig, 1914, there have been numerous devel- opments in a multitude of directions and there have been many interactions with a great number of other mathematical fields. We have tried to cover as many of these as possible. Most contributions concern either individual topologists, specific schools, specific periods, specific topics or a combination of these.

The main topic of this work is the study of general complexes of differential operators between sections of vector bundles. Although the global situation and the local one are often similar in content, the invariant language permits the simplification of the notation and more clearly reveals the algebraic structure of some questions. Recent developments in the theory of complexes of differential operators are dealt with to some degree: formal theory; existence theory; global solvability problem; overdetermined boundary problems; generalized Lefschetz theory of fixed points; and qualitative theory of solutions of overdetermined systems. Considerable attention is paid to the theory of functions of several complex variables. Examples and exercises are included.

For many, modern functional analysis dates back to Banach's book [Ba32]. Here, such powerful results as the Hahn-Banach theorem, the open-mapping theorem and the uniform boundedness principle were developed in the setting of complete normed and complete metrizable spaces. When analysts realized the power and applicability of these methods, they sought to generalize the concept of a metric space and to broaden the scope of these theorems. Topological methods had been generally available since the appearance of Hausdorff's book in 1914. So it is surprising that it took so long to recognize that they could provide the means for this generalization. Indeed, the theory of topo- logical vector spaces was developed systematically only after 1950 by a great many different people, induding Bourbaki, Dieudonne, Grothendieck, Kothe, Mackey, Schwartz and Treves. The resulting body of work produced a whole new area of mathematics and generalized Banach's results. One of the great successes here was the development of the theory of distributions. While the not ion of a convergent sequence is very old, that of a convergent fil- ter dates back only to Cartan [Ca]. And while sequential convergence structures date back to Frechet [Fr], filter convergence structures are much more recent: [Ch], [Ko] and [Fi]. Initially, convergence spaces and convergence vector spaces were used by [Ko], [Wl], [Ba], [Ke64], [Ke65], [Ke74], [FB] and in particular [Bz] for topology and analysis.

This account of the History of General Topology has grown out of the special session on this topic at the American Mathematical Society meeting in San Anto- nio, Texas, 1993. It was there that the idea grew to publish a book on the historical development of General Topology. Moreover it was felt that it was important to undertake this project while topologists who knew some of the early researchers were still active. Since the first paper by Frechet, "Generalisation d'un theoreme de Weier- strass", C.R. Acad. Sci. 139, 1904, 848-849, and Hausdorff's classic book, "Grundziige der Mengenlehre", Leipzig, 1914, there have been numerous de- velopments in a multitude of directions and there have been many interactions with a great number of other mathematical fields. We have tried to cover as many of these as possible. Most contributions concern either individual topologists, specific schools, specific periods, specific topics or a combination of these.

This book grew out of the work developed at the University of Warwick, under the supervision of Ian Stewart, which formed the core of my Ph.D. Thesis. Most of the results described were obtained in joint work with Ian; as usual under these circumstances, many have been published in research journals over the last two years. Part of Chapter 3 was also joint work with Peter Ashwin. I would like to stress that these were true collaborations. We worked together at all stages; it is meaningless to try to identify which idea originated from whom. While preparing this book, however, I felt that a mere description of the results would not be fitting. First of all, a book is aimed at a wider audience than papers in research journals. More importantly, the work should assume as little as possible, and it should be brought to a form which is pleasurable, not painful, to read.

This book is about an investigation of recent developments in the field of sympletic and contact structures on four- and three-dimensional manifolds from a topologist 's point of view. In it, two main issues are addressed: what kind of sympletic and contact structures we can construct via surgery theory and what kind of sympletic and contact structures are not allowed via gauge theory and the newly invented Heegaard-Floer theory.

This book is devoted to some results from the classical Point Set Theory and their applications to certain problems in mathematical analysis of the real line. Notice that various topics from this theory are presented in several books and surveys. From among the most important works devoted to Point Set Theory, let us first of all mention the excellent book by Oxtoby [83] in which a deep analogy between measure and category is discussed in detail. Further, an interesting general approach to problems concerning measure and category is developed in the well-known monograph by Morgan [79] where a fundamental concept of a category base is introduced and investigated. We also wish to mention that the monograph by Cichon, W";glorz and the author [19] has recently been published. In that book, certain classes of subsets of the real line are studied and various cardinal valued functions (characteristics) closely connected with those classes are investigated. Obviously, the IT-ideal of all Lebesgue measure zero subsets of the real line and the IT-ideal of all first category subsets of the same line are extensively studied in [19], and several relatively new results concerning this topic are presented. Finally, it is reasonable to notice here that some special sets of points, the so-called singular spaces, are considered in the classi

In the summer of 1991 the Department of Mathematics and Statistics of the Universite de Montreal was fortunate to host the NATO Advanced Study Institute "Algebras and Orders" as its 30th Seminaire de mathematiques superieures (SMS), a summer school with a long tradition and well-established reputation. This book contains the contributions of the invited speakers. Universal algebra- which established itself only in the 1930's- grew from traditional algebra (e.g., groups, modules, rings and lattices) and logic (e.g., propositional calculus, model theory and the theory of relations). It started by extending results from these fields but by now it is a well-established and dynamic discipline in its own right. One of the objectives of the ASI was to cover a broad spectrum of topics in this field, and to put in evidence the natural links to, and interactions with, boolean algebra, lattice theory, topology, graphs, relations, automata, theoretical computer science and (partial) orders. The theory of orders is a relatively young and vigorous discipline sharing certain topics as well as many researchers and meetings with universal algebra and lattice theory. W. Taylor surveyed the abstract clone theory which formalizes the process of compos ing operations (i.e., the formation of term operations) of an algebra as a special category with countably many objects, and leading naturally to the interpretation and equivalence of varieties."