Book Description

The book is concerned with one of the most interesting and important topological invariants of 3-dimensional manifolds based on an original idea of Kurt Reidemeister (1935). This invariant, called the maximal abelian torsion, was introduced by the author in 1976. The purpose of the book is to give a systematic exposition of the theory of maximal abelian torsions of 3-manifolds. Apart from publication in scientific journals, many results are recent and appear here for the first time.

Students and researchers with basic background in algebraic topology and low-dimensional topology will benefit from this monograph. Previous knowledge of the theory of torsions is not required. Numerous exercises and historical remarks as well as a collection of open problems complete the exposition. ... Read more

Book Description

This book introduces the important ideas of algebraic topology emphasizing the relation of these ideas with other areas of mathematics. Rather than choosing one point of view of modern topology (homotropy theory, axiomatic homology, or differential topology, say) the author concentrates on concrete problems in spaces with a few dimensions, introducing only as much algebraic machinery as necessary for the problems encountered. This makes it possible to see a wider variety of important features in the subject than is common in introductory texts; it is also in harmony with the historical development of the subject. The book is aimed at students who do not necessarily intend on specializing in algebraic topology.

The first part of the book emphasizes relations with calculus and uses these ideas to prove the Jordan curve theorem. The study of fundamental groups and covering spaces emphasizes group actions. A final section gives a taste of the generalization to higher dimensions. ... Read more

Reviews (3)

A book of ideas
This book is an introduction to algebraic topology that is written by a master expositor. Many books on algebraic topology are written much too formally, and this makes the subject difficult to learn for students or maybe physicists who need insight, and not just functorial constructions, in order to learn or apply the subject. Anyone learning mathematics, and especially algebraic topology, must of course be expected to put careful thought into the task of learning. However, it does help to have diagrams, pictures, and a certain degree of handwaving to more greatly appreciate this subject.

As a warm-up in Part 1, the author gives an overview of calculus in the plane, with the intent of eventually defining the local degree of a mapping from an open set in the plane to another. This is done in the second part of the book, where winding numbers are defined, and the important concept of homotopy is introduced. These concepts are shown to give the fundamental theorem of algebra and invariance of dimension for open sets in the plane. The delightful Ham-Sandwich theorem is discussed along with a proof of the Lusternik-Schnirelman-Borsuk theorem. I would like to see a constructive proof of this theorem, but I do not know of one.

Part 3 is the tour de force of algebraic topology, for it covers the concepts of cohomology and homology. The author pursues a non-traditional approach to these ideas, since he introduces cohomology first, via the De Rham cohomology groups, and these are used to proved the Jordan curve theorem. Homology is then effectively introduced via chains, which is a much better approach than to hit the reader with a HOM functor. Part 4 discusses vector fields and the discussion reads more like a textbook in differential topology with the emphasis on critical points, Hessians, and vector fields on spheres. This leads naturally to a proof of the Euler characteristic.

The Mayer-Vietoris theory follows in Part 5, for homology first and then for cohomology.

The fundamental group finally makes its appearance in Part 6 and 7, and related to the first homology group and covering spaces. The author motivates nicely the Van Kampen theorem. A most interesting discussion is in part 8, which introduces Cech cohomology. The author's treatment is the best I have seen in the literature at this level. This is followed by an elementary overview of orientation using Cech cocycles.

All of the constructions done so far in the plane are generalized to surfaces in Part 9. Compact oriented surfaces are classified and the second de Rham cohomology is defined, which allows the proof of the full Mayer-Vietoris theorem.

The most important part of the book is Part 10, which deals with Riemann surfaces. The author's treatment here is more advanced than the rest of the book, but it is still a very readable discussion. Algebraic curves are introduced as well as a short discussion of elliptic and hyperelliptic curves.

The level of abstraction increases greatly in the last part of the book, where the results are extended to higher dimensions. Homological algebra and its ubiquitous diagram chasing are finally brought in, but the treatment is still at a very understandable level.

For examples of the author's pedagogical ability, I recommend his book Toric Varieties, and his masterpiece Intersection Theory.

This is one of the great algebraic topology books!
This is a book for people who want to think about topology, not just learn a lot of fancy definitions and then mechanically compute things. Fulton has put the essence of Algebraic Topology into this book, much in the way Mike Artin has done with his "Algebra". In my opinion, he should win some sort of expository award for it.

Probably better as a 2nd (or 3rd) course rather than 1st
Most mathematicians, I suspect, can relate to the "colloquium experience": the first minutes of a lecture go easily, followed by twenty or thirty of real edification, concluded by ten to fifteen of feeling lost. I regret to say that this was pretty much my experience with the book. Fulton writes with unusual enthusiasm and the first two- thirds of the book is a joy to read, even while it is real work. I imagine that he must be a remarkable teacher in person. He has some threads such as winding numbers and the Mayer-Vietoris Sequence that continue throughout the book, bringing unity to a wide selection of topics. There are a number of applications of the subject to other areas, such as complex analysis (Riemann surfaces) and algebraic geometry (the Riemann-Roch Theorem), to name only two. There are particularly interesting illustrations of the Brouwer Fixed Point Theorem and related results. Unfortunately, there are two rather major reservations I have about the book. The first, already alluded to, is that it seemed to me to become precipitously difficult towards the end. The second is that this book would be excellent for a second or perhaps third course in the subject rather than a first. While the topics he covers are interesting in their own right, I still favor a more "standard" approach covering simplicial complexes, homology, CW complexes, and homotopy theory with higher homotopy groups, such as in the books by Maunder, Munkres, or Rotman (the last two of which I recommend unreservedly). It is true that Fulton has some coverage these topics, and a particularly extensive discussion of group actions and G-spaces, but he presupposes a background or ability that the novice to algebraic topology is unlikely to have. I would like to recommend this book, as I found it very edifying, but it seems better suited for one with some prior acquaintance to the subject. ... Read more

Book Description

This book provides a broad introduction to the subject of dynamical systems, suitable for a one or two-semester graduate course. In the first chapter, the authors introduce over a dozen examples, and then use these examples throughout the book to motivate and clarify the development of the theory.Topics include topological dynamics, symbolic dynamics, ergodic theory, hyperbolic dynamics, one-dimensional dynamics, complex dynamics, and measure-theoretic entropy.The authors top off the presentation with some beautiful and remarkable applications of dynamical systems to areas such as number theory, data storage, and internet search engines. ... Read more

Book Description

This book contains essential material that every graduate student must know. Written with Serge Lang's inimitable wit and clarity, the volume introduces the reader to manifolds, differential forms, Darboux's theorem, Frobenius, and all the central features of the foundations of differential geometry. Lang lays the basis for further study in geometric analysis, and provides a solid resource in the techniques of differential topology. The book will have a key position on my shelf.

-Steven Krantz, Washington University in St. Louis

This is an elementary, finite dimensional version of the author's classic monograph, Introduction to Differentiable Manifolds (1962), which served as the standard reference for infinite dimensional manifolds. It provides a firm foundation for a beginner's entry into geometry, topology, and global analysis. The exposition is unencumbered by unnecessary formalism, notational or otherwise, which is a pitfall few writers of introductory texts of the subject manage to avoid. The author's hallmark characteristics of directness, conciseness, and structural clarity are everywhere in evidence. A nice touch is the inclusion of more advanced topics at the end of the book, including the computation of the top cohomology group of a manifolds, a generalized divergence theorem of Gauss, and an elementary residue theorem of several complex variables. If getting to the main point of an argument or having the key ideas of a subject laid bare is important to you, then you would find the reading of this book a satisfying experience. ... Read more

Book Description

Acquaints the specialist in relativity theory with some global techniques for the treatment of space-times and will provide the pure mathematician with a way into the subject of general relativity. ... Read more

Reviews (1)

Still a useful overview
First published in 1972, it is remarkable that this book is still in print, and this fact attests to the current interest in singularity theorems in general relativity. The author of course is well-known for his contributions in this area, and he has written these series of lectures primarily for the mathematician whose speciality is differential topology, and who is curious about its applications to general relativity. The author thinks in pictures in this book, and so it is well-suited for the physicist reader also. Detailed proofs are omitted for the singularity theorems, but references are given. Much work and discussion has taken place since this book was published, but it can still serve as an introduction to modern developments.

Section 1 sets the mathematical definitions and conventions used in the later sections. Spacetime is defined as a real, four-dimensional connected smooth Hausdorff manifold on which is defined a global smooth nondegenerate Lorentzian metric. In addition, it is assumed that spacetime is time-orientable, which is not too big a restriction since as the author remarks, one can always find a time-orientable twofold covering of spacetime. Jacobi fields are introduced also, with the goal of eventually using them to study maximal geodesics. Known to physicists as the equation of geodesic deviation, the author derives the Jacobi equation, the solutions of which form an 8-dimensional vector space of Jacobi fields.

In section 2, the author gives definitions that allow one to discuss causality and time ordering for curves on spacetime. Special types of non-smooth curves, called trips, which (piecewise) are future-oriented timelike geodesics, are used to define a chronological ordering of points on spacetime. Causal trips are more restrictive, in that the curves are causal geodesics. The chronological ordering is shown to imply causal ordering, and both orderings are shown to be transitive. This allows the partitioning of spacetime into chronological future and past, and causal future and past. The topological properties of these sets are studied, and conditions are given in terms of null geodesics and timelike curves for causal and chronological ordering.

The next section considers the properties of future and past sets. A future (past) set is one that is equal to the chronological future (past) of some set in spacetime. In addition, subsets of spacetime that do not contain any points that are chronologically related, called achronal sets, and subsets that are boundaries are considered. It is shown that spacetime can be written as the disjoint union of an achronal boundary, and a unique past and future set. It is also shown that achronal boundaries are fairly well-behaved objects: they are 3-dimensional topological manifolds.

In order to rule out "pathological" spacetimes that contain closed trips or closed causal trips, the author studies global causality conditions in section 4. Thus the author defines a spacetime to be future (past)-distinguishing if for any two distinct points, their chronological future (past) sets are unequal. He then defines a spacetime to be strongly causal if every point in it has arbitrarily small causally convex neighborhoods (causally convex meaning that it does not intersect a trip in a disconnected set). The author offers examples to show that local violations of causal convexity can be avoided, and so violations of strong causality at a point are due to the global structure of the spacetime. He shows that a spacetime which is strongly causal at a point must be future and past distinguishing at the point. The converse is not true, and the author gives a counterexample. The Alexandrov topology for spacetime is defined in this section also. Given two points in spacetime, a basis for the open sets for this topology is given by the intersection of the chronological future set of one point with the chronological past set of the other. The author shows that spacetime is strongly causal iff the Alexandrov topology equals the manifold topology iff the Alexandrov topology is Hausdorff. Defining a vicious point to be one through which passes a closed trip, and concentrating attention on the set of all vicious points, the author gives five conditions for strong causality to fail at a point, these conditions involving the boundary of the set of vicious points. He concludes the section by showing that if spacetime is compact, it must contain a closed trip.

Motivated by the notion of an initial value set from physicial considerations, the author defines in the next section domains of dependence for achronal subsets of spacetime, along with the future, past, or total Cauchy horizon for closed achronal subsets. These are related to the familiar Cauchy hypersurfaces from the theory of partial differential equations. It is proven that spacetime is globally hyperbolic iff a Cauchy hypersurface exists for it.

The space of causal curves is defined in the next section, on which is defined the C0-topology. It is shown to be compact under certain conditions. The study of geodesics as curves of maximal length is taken up in section 7. This entails matters of a more purely differential geometric point of view. The important inequality involving the Ricci curvature and an element of volume (or area) on a hypersurface. The author discusses briefly the importance of this inequality in the singularity theorems.

The last section is (unfortunately) very brief, wherein the author discusses the applications of the preceeding sections in singularity theorems. Referring to S. Hawking for the full proof, he gives a general argument and discusses the conditions as to when spacetime will have a past-endless geodesic in M which has a finite length.He defines a future-trapped set as one where the "future horizon" of the set, defined as the difference between its causal and chronological future, is compact. He then outlines a proof of the result that no spacetime can have the property that it contain no closed trips, have endless causal geodesics containing a pair of conjugate points, and contain a future-trapped set. ... Read more

Book 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, 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." ... Read more

Book Description

This book provides a careful and detailed algebraic introduction to Grothendieck's local cohomology theory, and illustrates many applications for the theory in commutative algebra and in the geometry of quasi-affine and quasi-projective varieties. Topics covered include Castelnuovo-Mumford regularity, theFulton-Hansen connectedness theorem for projective varieties, and connections between local cohomology and both reductions of ideals and sheaf cohomology. It is designed for graduate students who have some experience of basic commutative algebra and homological algebra, and also for experts in commutative algebra and algebraic geometry. ... Read more

Book Description

Homology is a powerful tool used by mathematicians to study the properties of spaces and maps that are insensitive to small perturbations. This book uses a computer to develop a combinatorial computational approach to the subject. The core of the book deals with homology theory and its computation. Following this is a section containing extensions to further developments in algebraic topology, applications to computational dynamics, and applications to image processing. Included are exercises and software that can be used to compute homology groups and maps. The book will appeal to researchers and graduate students in mathematics, computer science, engineering, and nonlinear dynamics. ... Read more

Book Description

The purpose of this book is to provide an integrated development of modern analysis and topology through the integrating vehicle of uniform spaces. The reader should have taken an advanced calculus course and an introductory topology course. It is intended that a subset of the book could be used for an upper-level undergraduate course whereas much of the full text would be suitable for a one-year graduate class. An attempt has been made to document the history of all the central ideas and references and historical notes are embedded in the text. These can lead the interested reader to the foundational sources where these ideas emerged. ... Read more

Book Description

This is a long awaited book on rational homotopy theory which contains all the main theorems with complete proofs, and more elementary proofs for many results that were proved ten or fifteen years ago. The authors added a frist section on classical algebraic topology to make the book accessible to students with only little background in algebraic topology. ... Read more

Reviews (1)

An excellent, very understandable overview
This book follows up and greatly extends the work of the topologist Dennis Sullivan on the rationalization of topological spaces and continuous maps between these rationalizations. For n greater than or equal to 2, both the nth-homotopy group the nth homology group are abelian, and this lead Sullivan to introduce the concept of a "rationalized space". For such a space, one studies its nth homology group over the rational numbers, and the nth homotopy group of a rationalized space is the tensor product of the nth homotopy group with the rational numbers. Information of course is lost in such an approach, but it has the advantage of being amenable to calculation. The authors give a detailed overview of just what can be done for rationalized spaces and they do an excellent job of presenting it to those who are not experts in the theory. The book can definitely be read by graduate students who have finished courses in algebraic and geometric topology, and professional mathematicians who have some background in topology and who are curious about the subject.

As the authors explain eloquently, the (computational) power of rational homotopy theory comes from its algebraic formulation, which was first discussed by Sullivan and the mathematician Daniel Quillen, and involves the use of graded objects with both an algebraic structure and a "differential". What is fascinating about the role of the differential is its connection with homotopy theory, and not just in homology and cohomology theory as encountered in first-year graduate courses in algebraic topology. The authors deal with three different graded categories with a differential in the book, namely modules over a differential graded algebra (R, d), commutative cochain algebras, and differential graded Lie algebras. In analogy to the free resolution of an arbitrary module over a ring, associated with these three cases is a modeling construction that in the first case is a semi-free resolution of a module over (R, d), in the second case a "Sullivan model" which is a commutative cochain algebra which is free as a commutative graded algebra, and in the third case a free Lie model, which is free as a graded Lie algebra.

The first case arises topologically when considering continuous maps between spaces and the singular cochain algebras via the induced cochain map. When the map is a fibration, the authors compute the cohomology of the fiber using a semifree resolution. The first case also arises in considering the action of a topological monoid over a space and the singular chains. When the action is a principal G-fibration X-> Y, the authors compute the homology of Y using semifree resolutions. The authors then give a proof of the Whitehead-Serre theorem using this result. The proof of this follows their plan to avoid diagram-chasing techniques as much as possible: they do not use spectral sequences.

The second case involves a generalization of the classical commutative cochain algebra of smooth differential forms on a manifold. The authors construct a "Sullivan functor" from topological spaces to commutative cochain algebras, the Sullivan model, and the Sullivan "realization functor", the latter of which converts a Sullivan algebra into a rational topological space. The rational homotopy types of a space are then in bijective correspondence to isomorphism classes of "minimal" Sullivan algebras, and the homotopy classes of maps between rational spaces are in bijective correspondence to homotopy classes of maps between minimal Sullivan algebras. The characterization of a Sullivan algebra as being "minimal" comes from the fact that for such algebras there is a natural isomorphism between the vector space on which the Sullivan algebra is modeled and integral homomorphisms of the homotopy group to the ground field.

The third case involves the use of differential graded Lie algebras. The authors construct the "homotopy Lie algebra" of a simply connected topological space, which is the homotopy group of the loop space tensored with the ground field, and the homotopy Lie algebra of a minimal Sullivan algebra. The latter is interesting in that it involves using the quadratic part of the differential in order to obtain the Lie bracket. These two constructions of homotopy Lie algebras are the same for the Sullivan algebra over the space. In this context, the authors consider "free Lie models" for differential graded Lie algebras, which can be thought of as an assignment of generators to each single n-cells of a CW complex. The authors give many helpful examples of free Lie models that illustrate this, such as for the sphere, adjunction spaces, projective spaces, and homotopy fibers.

Since rational and ordinary homotopy are different in terms of their information content, it is perhaps not surprising that the Lusternik-Schnirelmann category makes its appearance in this book. The rational LS category is the LS category of a rational CW complex in the rational homotopy type of the space, and the authors calculate it in terms of Sullivan models, verifying that the rational case is much easier to compute than the general case. As further verification, the authors show that the Postnikov fibers in a Postnikov decomposition of a simply connected finite CW complex all have finite rational LS category, which is not true in the integral case. Even further, they show that the rational LS category of a product is the sum of the products, contrary to the ordinary LS category which is not well-behaved for products and fibrations.

The authors also discuss various applications at the end of the book, involving how to break up n-dimensional simply connected finite CW complexes into two groups: those whose rational homotopy groups vanish in degrees greater than or equal to 2n, and those where they grow exponentially. The former are called "rationally elliptic" and the latter "rationally hyperbolic". This classification can be determined, as they show, from a calculation of the "Betti numbers" of the loop group of the space over the rationals. A collection of unsolved problems for the ambitious reader ends the book. ... Read more

Reviews (1)

An extraordinary book!
This book describes how to tabulate knots with the help of computers. You could ask, so, what is special about that and why is this difficult? It is difficult because a knot or link is a topological object and hence, it is not clear how to represent such an object that it can be processed by an computer at all. Even if you know the Dowker-Thistlethwaite notation it is still an art to do it.
In my opinion this book is ahead of its time, because the field of computational topology/knot theory is still in its infacncy. ... Read more

Book Description

Hardbound. ...there are reasons enough to warrant a coherent treatment of the main body of differential topology in the realm of Banach manifolds, which is at the same time correct and complete. This book fills the gap: whenever possible the manifolds treated are Banach manifolds with corners. Corners add to the complications and the authors have carefully fathomed the validity of all main results at corners. Even in finite dimensions some results at corners are more complete and better thought out here than elsewhere in the literature. The proofs are correct and with all details. I see this book as a reliable monograph of a well-defined subject; the possibility to fall back to it adds to the feeling of security when climbing in the more dangerous realms of infinite dimensional differential geometry.

Peter W. Michor ... Read more

Book Description

Extensive development of a number of topics central to topology, including elementary combinatorial techniques, Sperner’s Lemma, the Brouwer Fixed Point Theorem, homotopy theory and the fundamental group, simplicial homology theory, the Hopf Trace Theorem, the Lefschetz Fixed Point Theorem, the Stone-Weierstrass Theorem, and Morse functions. Includes new section of solutions to selected problems.
... Read more

Reviews (1)

A good, low cost intro to topology
This is a great book, especially considering the price. If you have some real analysis and group theory under your belt, then this will provide an enjoyable introduction to topology. It moves through point-set, basic combinatorial techniques, homotopy theory, simplicial homology, and a brief peek at differential manifolds. The author has done a good job of explaining basic concepts. By putting this in a Euclidean setting, you do not get a completely general approach, but you can cover a lot of (conceptual) ground with some economy.
If you need to brush up on analysis first, I recommend Rosenlict's INTRO TO ANALYSIS. ... Read more

Book Description

Reviews (1)

A superb job...packed full of insights
This book could be loosely characterized as an attempt to generalize the theory of Riemann surfaces to that of Riemannian manifolds. The reader familiar with the theory of Riemann surfaces will perhaps find this book easier to read than one who has not. But the author has not assumed that the reader has had any prior exposure to Riemann surfaces, and so the reader without such background will find the reading straightforward. The paradigm in the book is the connection between the topology of Riemannian manifolds and their metric geometry. It is the metric structure of Riemannian manifolds that is responsible for their fame, due especially to their use in physics. Through the use of de Rham cohomology, Hodge theory, and other techniques from differential geometry, the author shows how to give an overview of the intrinsic ("coordinate-free") global differential geometry of Riemannian manifolds and how that geometry is connected to its topology.

Chapter 1 is a review of elementary differential geometry that is to be used in the rest of the book. Then in chapter 2 the author begins with a review of singular homology and de Rham cohomology. The key point, proved in an appendix, is the de Rham theorem which establishes an isomorphism between de Rham and singular cohomology. The pth Betti number is then the number of linearly independent closed differential forms of degree p modulo the exact forms of degree p. The rest of the chapter is devoted to showing how this result was extended by the mathematician W.V.D Hodge to a restricted class of forms, the famous "harmonic forms". Now called Hodge theory, it is a homology theory based on the Laplace-Beltrami operator, which generalizes, as expected, Laplace's equation.

Chapter 3 is devoted to finding an explicit expression for the Laplace-Beltrami operator in local coordinates. This expression is dependent on the Riemannian curvature of the Riemannian manifold, and so the homology of a compact and orientable manifold will depend on its curvature. The issue then is finding harmonic forms of a given degree. The obstruction to the existence of these is given by a particular quadratic form involving the curvature tensor. The absence of harmonic forms of degree p gives that the pth Betti number is zero. In particular the author shows that the Betti numbers of a compact, orientable, conformally flat Riemannian manifold of positive definite Ricci curvature are all zero. The author then applies these results to compact Lie groups in chapter 4. The harmonic forms on compact Lie groups are those differential forms that are invariant under both left and right translations of the group. The author shows that the first and second Betti numbers of compact Lie groups are zero and shows the existence of a harmonic 3-form, the latter proving that the third Betti number is greater than or equal to one.

The author turns his attention to complex manifolds in chapter 5. He approaches these objects from the standpoint of first defining complex structures on separable Hausdorff spaces. The complex structures then allow a definition of a Riemannian metric on these spaces. If the metric does have any torsion, then one can associate a particular 2-form with the metric and the complex structure that is closed. This 2-form is the famous "Kaehler metric", and the resulting space is called a "Kaehler manifold". The local geometry of Kaehler manifolds is referred to as "Hermitian geometry", and the author studies in detail this geometry in this chapter. Loosely speaking, a Kaehler metric can be viewed as a generalization of "flatness" in the usual Riemannian case, for the author shows that at each point of a Kaehler manifold there exists a system of local complex coordinates which is geodesic. He also introduces the important concept of a holomorphic p-form, and shows that on a Kaehler manifold these are harmonic.

In chapter 6, the author studies in detail how curvature and homology are related for the case of Kaehler manifolds. The results in this chapter could be viewed as a generalization of the classical results concerning compact Riemann surfaces, namely that the universal covering space of a complex n-dimensional compact Kaehler manifold of constant holomorphic curvature K is a projective space for K > 0, the interior of a unit sphere for k < 0, and the space of complex variables for K = 0. After defining the holomorphic curvature, the author shows that the pth Betti number of a compact Kahler manifold M with positive constant holomorphic curvature is zero if p is odd and 1 if p is even. In addition, he shows that any holomorphic form of degree p, for p > 0 and p less than or equal to n, on a compact Kaehler manifold with positive definite Ricci curvature is zero. The author also gives the reader a taste of sheaf theory, in which he discusses briefly the Kodaira vanishing theorems.

In the last chapter, the author generalizes what was done in chapter 3 regarding conformal transformations on Riemannian manifolds, namely that an infinitesimal holomorphic transformation of a compact Kaehler manifold can be viewed as the solution of a system of differential equations which involve the Ricci curvature. Conditions are given for making this transformation an isometry, and the author shows that for a compact Kaehler manifold of complex dimension greater than 1, an infinitesimal conformal transformation is holomorphic if and only if it is an infinitesimal isometry. This leads him to consider the groups of holomorphic transformations, and he gives conditions under which a compact complex manifold cannot admit a transitive Lie group of holomorphic transformations. The author also studies the most general class of Riemannian manifolds for which an infinitesimal conformal transformation is also an infinitesimal isometry. These are the famous "almost Kaehler" manifolds, and the author shows that an infinitesimal conformal transformation of a compact almost Kaehler manifold of dimension 2n for n > 1 is an infinitesimal isometry. ... Read more

Book Description

Learn to juggle numbers! This book is the first comprehensive account of the mathematical techniques and results used in the modelling of juggling patterns. This includes all known and many new results about juggling sequences and matrices, the mathematical skeletons of juggling patterns.

Many useful and entertaining tips and tricks spice up the mathematical menu presented in this book. There are detailed descriptions of jugglable and attractive juggling sequences, easy zero-gravity juggling, robot juggling, as well as fun juggling of words, anti-balls, and irrational numbers.

The book also includes novel, or at least not very well known connections with topics such as bell ringing, knot theory, and the many body problem. In fact, the chapter on mathematical bell ringing has been expanded into the most comprehensive survey in the literature of the mathematics used by bell ringers.

Accessible at all levels of mathematical sophistication, this is a book for mathematically wired jugglers, mathematical bell ringers, combinatorists, mathematics educators, and just about anybody interested in beautiful and unusual applications of mathematics. ... Read more

Book Description

"Essential Topology" brings the most exciting – and useful - aspects of modern topology within reach of the average second-year undergraduate student. It contains all the essentials. The first chapter provides a complete account of continuity beginning at a level that a high school student could understand. The algebraic notions are introduced slowly through the text, leading the reader to the celebrated Hairy Ball theorem, and on to homotopy and homology – the cornerstones of contemporary algebraic topology.Each topic is introduced with a thorough explanation of why it is being studied, and the focus throughout is on providing interesting examples that will motivate the student. Emphasis is placed on the basic objects that occur in research topology, and in its applications to other areas of mathematics. This book is designed to provide a "one-stop shop" for undergraduate topology, providing enough material for two semester-long courses, and leaving students motivated and prepared for postgraduate study. ... Read more

Book Description

Hardbound. The Handbook presents an overview of most aspects of modern Banach space theory and its applications. The up-to-date surveys, authored by leading research workers in the area, are written to be accessible to a wide audience.In addition to presenting the state of the art of Banach space theory, the surveys discuss the relation of the subject with such areas as harmonic analysis, complex analysis, classical convexity, probability theory, operator theory, combinatorics, logic, geometric measure theory, and partial differential equations.

The Handbook begins with a chapter on basic concepts in Banach space theory which contains all the background needed for reading any other chapter in the Handbook.Each of the twenty one articles in this volume after the basic concepts chapter is devoted to one specific direction of Banach space theory or its applications.Each article contains a motivated introduction as well as an exposition of the ma ... Read more