Book Description

The concept of a period of an elliptic integral goes back to the 18th century. Later Abel, Gauss, Jacobi, Legendre, Weierstrass and others made a systematic study of these integrals. Rephrased in modern terminology, these give a way to encode how the complex structure of a two-torus varies, thereby showing that certain families contain all elliptic curves. Generalizing to higher dimensions resulted in the formulation of the celebrated Hodge conjecture, and in an attempt to solve this, Griffiths generalized the classical notion of period matrix and introduced period maps and period domains which reflect how the complex structure for higher dimensional varieties varies. The basic theory as developed by Griffiths is explained in the first part of the book. Then, in the second part spectral sequences and Koszul complexes are introduced and are used to derive results about cycles on higher dimensional algebraic varieties such as the Noether-Lefschetz theorem and Nori's theorem. Finally, in the third part differential geometric methods are explained leading up to proofs of Arakelov-type theorems, the theorem of the fixed part, the rigidity theorem, and more. Higgs bundles and relations to harmonic maps are discussed, and this leads to striking results such as the fact that compact quotients of certain period domains can never admit a Kahler metric or that certain lattices in classical Lie groups can't occur as the fundamental group of a Kahler manifold. ... Read more

Book Description

In this volume the author gives a unified presentation of some ofthe basic tools and concepts in number theory, commutativealgebra, and algebraic geometry, and for the first time in abook at this level, brings out the deep analogies between them.The geometric viewpoint is stressed throughout the book.Extensive examples are given to illustrate each new concept, andmany interesting exercises are given at the end of each chapter.Most of the important results in the one-dimensional case areproved, including Bombieri's proof of the Riemann Hypothesis forcurves over a finite field. While the book is not intended to bean introduction to schemes, the author indicates how many of thegeometric notions introduced in the book relate to schemes whichwill aid the reader who goes to the next level of this richsubject. ... Read more

Book Description

Recent progress in research, teaching and communication has arisen from the use of new tools in visualization. To be fruitful, visualization needs precision and beauty. This book is a source of mathematical illustrations by mathematicians as well as artists. It offers examples in many basic mathematical fields including polyhedra theory, group theory, solving polynomial equations, dynamical systems and differential topology. For a long time, arts, architecture, music and painting have been the source of new developments in mathematics. And vice versa, artists have often found new techniques, themes and inspiration within mathematics. Here, while mathematicians provide mathematical tools for the analysis of musical creations, the contributions from sculptors emphasize the role of mathematics in their work. ... Read more

Book Description

This elegant book by distinguished mathematician John Milnor, provides a clear and succinctintroduction to one of the most important subjects in modern mathematics. Beginning with basic concepts such as diffeomorphisms and smooth manifolds, he goes on to examine tangent spaces, oriented manifolds, and vector fields. Key concepts such as homotopy, the index number of a map, and the Pontryagin construction are discussed. The author presents proofs of Sard's theorem and the Hopf theorem. ... Read more

Reviews (5)

Take full advantage of the clear, encompassing exposition:
Do the exercises. Many were Ph.D. dissertation-level problems in the 1960s; today, they're aptly described as "elementary"- because Milnor MADE them elementary.

This book forms part of the toolkit you will need to fully explore the more modern work in dynamics, complexity, and applications (e.g., economics, physics).

The clarity of the exposition also forms an ideal example of how to communicate mathematics powerfully and simply.

Compact, readable text on the topology of manifolds
This book is very short, which is quite an asset for a math book to have. It also requires little knowledge of math beyond advanced calculus and point-set topology. I found it extremely readable, and I greatly enjoyed it. I recommend it highly, and especially enjoyed the proof on page 8 of the fundamental theorem of algebra. How far we've come since Gauss' first proof...

A good place to start
This book is exceptionally well written and easy to read. Milnor proves a major result on almost every page. One learns a lot per unit time spent on this book. Despite being less than 80 pages, the book covers a significant amount of material in a clear concise manner.

Excellent starting point for differential topology
One of the best points of this little book is its brevity and clear exposition of the basic ideas. It makes a great reference guide because it's so short and well-organized. Written by a distinguished mathematician, it's no wonder that other graduate-level texts such as Guillemin & Pollacks "Differential Topology" highly recommend reading it alongside their book. Milnor's booklet is a classic, whose style and ideas surely pervade other texts.

An excellent introduction to topology of manifolds.
Perfect for a first-year graduate or advanced undergraduate course, Milnor takes us on a brief stroll through elementary differential topology. Elegant and self-contained, this book serves as an excellent first taste of the subject. Milnor is a master expositor, and is at his best in this book. ... Read more

Book Description

"Trudeau meets the challenge of reaching a broad audience in clever ways...(The book) is a good addition to our literature on non-Euclidean geometry and it is recommended for the undergraduate library." -CHOICE, February 1988"...the author, in this remarkable book, describes in an incomparable way the fascinating path taken by the geometry of the plane in its historical evolution from antiquity up to the discovery of non-Euclidean geometry. This 'non-Euclidean revolution', in all its aspects, is described very strikingly here...Many illustrations and some amusing sketches complement the very vividly written text." -MATHEMATICAL REVIEWS How unque and definitive is Euclidean geometry in describing the "real" space in which we live?Richard Trudeau confronts the fundamental question of truth and its representation through mathematical models in 'The Non-Euclidean Revolution.' First, the author analyzes geometry in its historical and philosophical setting; second, he examines a revolution every bit as significant as the Copernican revolution in astronomy and the Darwinian revolution in biology; third, on the most speculative level, he questions the possibility of absolute knowledge in the world.Trudeau writes in a lively, entertaining, and highly accessible style. His book provides one of the most stimulating and personal presentations of a struggle with the nature of truth in mathematics and the physical world.A portion of the book won the Pólya Prize, a distinguished award from the Mathematical Association of America. ... Read more

Reviews (1)

A nicely designed book, with accepted views of the subject
As indicated in my other reviews, my views of the subject differ from accepted ones, and I will try to explain them further in relation to this book.

The book assumes a somewhat condescending attitude, with imaginary dialogues between the author and presumably a student, possibly from the author's experience. The student asks supposedly "common sense" questions, and the author answers with lengthy explanations, sounding to me like excuses that make the teacher come out the loser.

Thus in a section about "points" (pp.23-30) the author defends the current concept of lines as composed of points, although a point is taken to have zero dimension, and zeros plus zeros are understood to add up to zero. The concept of points as the constituents of lines is indeed recent and questionable. The author like others overlooks Euclid's definition 3, "The extremities of a line are points", and points are in fact used to delimit lines, as do "breadthless" lines delimit areas, either usage not adding to dimension.

Another dialogue discusses "line" (p.170), which has caused similar confusion. Today the word is used for "straight line", while contrariwise the term is also applied to curves like great circles on a sphere. But the sticking point to me is the way it is justified to leave "primitive terms" undefined and then "interpret" them as desired (e.g. p.169).

The justification is roughly as follows. Basic logical principles are so general that one needn't specify what they are about, and then can apply them to particular cases. This is indeed true if a principle holds for anything whatsoever. But, for instance, Euclid's 5th postulate applies specifically to straight lines in a plane, which is why to reinterpret those terms as curvatures, and say the postulate then does not apply and is hence unprovable, commits the fallacy of equivocation.

What is disappointing is that undoubtedly good heads so carelessly perpetuate illogicalities while laying claim to increased rigor.
... Read more

Book Description

This book brings together a selection of the best lectures from many graduate workshops held at the Australian National Institute for Theoretical Physics in Adelaide. The lectures presented here describe subjects currently of great interest, generally at the interface between mathematics and physics, and also where suitable expositions did not previously exist at a level suitable for graduate students. Topics covered include quantum groups, the operator algebra approach to the integer quantum Hall effect, solvable lattice models and Hecke algebras, Yangevins, equivariant cohomology and symplectic geometry, and von Neumann invariants of covering spaces. ... Read more

Book Description

Over the last forty years, researchers have made great strides in elucidating the laws of image formation, processing, and understanding by animals, humans, and machines. This book describes the state of knowledge in one subarea of vision, the geometric laws that relate different views of a scene. Geometry, one of the oldest branches of mathematics, is the natural language for describing three-dimensional shapes and spatial relations. Projective geometry, the geometry that best models image formation, provides a unified framework for thinking about many geometric problems relevant to vision. The book formalizes and analyzes the relations between multiple views of a scene from the perspective of various types of geometries. A key feature is that it considers Euclidean and affine geometries as special cases of projective geometry.

Images play a prominent role in computer communications. Producers and users of images, in particular three-dimensional images, require a framework for stating and solving problems. The book offers a number of conceptual tools and theoretical results useful for the design of machine vision algorithms. It also illustrates these tools and results with many examples of real applications. ... Read more

Book Description

Multiplicative invariant theory, as a research area in its own right within the wider spectrum of invariant theory, is of relatively recent vintage. The present text offers a coherent account of the basic results achieved thus far..

Multiplicative invariant theory is intimately tied to integral representations of finite groups. Therefore, the field has a predominantly discrete, algebraic flavor. Geometry, specifically the theory of algebraic groups, enters through Weyl groups and their root lattices as well as via character lattices of algebraic tori.

Throughout the text, numerous explicit examples of multiplicative invariant algebras and fields are presented, including the complete list of all multiplicative invariant algebras for lattices of rank 2.

The book is intended for graduate and postgraduate students as well as researchers in integral representation theory, commutative algebra and, mostly, invariant theory.

Book Description

The aim of this book is to provide a complete history of pi from the dawn of mathematical time to the present.The story of pi reflects the most seminal, the most serious and sometimes the silliest aspects of mathematics, and a suprising amount of the most important mathematics and mathematicians have contributed to its unfolding. Pi is one of the few concepts in mathematics whose mention evokes a response of recognition and interest in those not concerned professionally with the subject.Yet, despite this, no source book on pi has been published.One of the beauties of the literature on pi is that it allows for the inclusion of very modern, yet still accessible, mathematics.Mathematicians and historians of mathematics will find this book indespensable.Teachers at every level from the seventh grade onward will find here ample resources for anything from special topic courses to individual talks and special student projects. The literature on pi included in this source book falls into three classes: first a selection of the mathematical literature of four millennia, second a variety of historial studies or writings on the cultural meaning and significance of the number, and third, a number of treatments on pi that are fanciful, satirical and/or whimsical. In addition to correcting a few minor infelicities, this new edition includes an Appendix in which articles 9 and 12 by Viete and Huygens respectively are translated into English. ... Read more

Book Description

Benoit Mandelbrot's pioneering research in fractal geometry has affected many areas of mathematics, physics, finance and other disciplines. The papers reprinted in this third volume of his Selected Works center on a detailed study of fractional Brownian functions, best known as the mathematical tools behind the celebrated fractal landscapes. Extensive introductory material preceding the reprints incorporates striking new observations and conjectures. This book explores the fractal themes of "self-affinity" and "globality." The ubiquity of "wild" temporal and spatial variability led Mandelbrot, in the early 1960's, to conclude that those phenomena lie beyond the usual statistical techniques and represent a new state of indeterminism. New mathematical tools are needed, and this book contributes to their development. ... Read more

Book Description

This introductory text explores the translation of geometric concepts into the language of numbers in order to define the position of a point in space (the orbit of a satellite, for example). The two-part treatment begins with discussions of the coordinates of points on a line, coordinates of points in a plane, and the coordinates of points in space. Part 2 examines geometry as an aid to calculation and the necessity and peculiarities of four-dimensional space. Written for systematic study, it features a helpful series of "road signs" in the margins, alerting students to passages requiring particular attention, and an abundance of ingenious problems--with solutions, answers, and hints--promote habits of independent work. 1967 edition.
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Reviews (1)

A fascinating introduction to coordinate geometry.
The authors of this slim volume demonstrate the power of coordinate geometry, which they describe as a means of translating geometric figures into algebraic formulas, through their lucid exposition, interesting examples, and well-chosen exercises.

The authors begin with the coordinate geometry of the real line. They discuss absolute value and define what distance means. Next the authors examine the coordinate geometry of the plane. They define distance in the plane, show how relations among the coordinates define geometric figures, and discuss different coordinate systems that can be used in the plane. Their examples illustrate how algebraic methods developed by Rene Descartes make it possible to solve geometric problems efficiently that would be quite difficult to solve using synthetic geometry. The authors then treat the coordinate geometry of three-dimensional space in a similar manner.

The second part of the book begins with a problem concerning lattice points in the plane. The authors use this example and its generalizations to justify exploring the coordinate geometry of four-dimensional space. They carefully treat the example of a four-dimensional unit hypercube, examining its properties by considering its analogues in lower dimensions: the segment [0, 1] of the real number line, the unit square in the coordinate plane, and the unit cube in space.

Since the book was initially written for a correspondence course for high school students in the Soviet Union, it is designed for self-study and accessible to students who have had high school courses in algebra and geometry. Since students in the Soviet Union were able to mail their solutions to the exercises to the authors when the authors were professors at the University of Moscow, answers to most of the exercises are not provided. The exercises are thought-provoking and some are quite challenging.

I also highly recommend that you explore the other volumes in the Gelfand School Outreach Program. They include Algebra, Functions and Graphs, and Trigonometry. ... Read more

Book Description

Polyhedra have cropped up in many different guises throughout recorded history. Recently, polyhedra and their symmetries have been cast in a new light by combinatorics and group theory. This unique text comprehensively documents the many and varied ways that polyhedra have come to the fore throughout the development of mathematics. The author strikes a balance between covering the historical development of the theory surrounding polyhedra and rigorous treatment of the mathematics involved. Attractively illustrated--including 16 color plates--Polyhedra elucidates ideas that have proven difficult to grasp. Mathematicians, as well as historians of mathematics, will find this book fascinating. ... Read more

Reviews (3)

Comprehensive masterpiece!
This is the best book about polyhedra! But it's not always easy to read. He has chosen to take a chronological approach. That means that sometimes you have to look around a bit.

I picked up the book wanting to understand two things.

1. What are the exact definition of the Platonic and Archimedian solids, i.e., how to destinguish the Platonic from the the Deltahedra and the 13 Archimedian from their isomeric forms and the pyramids.

3. What's the reason behind the names for the Kepler-Poinsot solids. Why is the great stellated dodecahedron called the great stellated dodecahedron?

Cromwell answers the first question beautifully in Chapter 2. The second question is first discussed in Chapter 4, but I was still confused. It was only in Chapter 7 that it started to make sense.

I believe the book will answer most of your questions, but you may have to look around for it.

The _Best_ Polyhedra Book
I've read many books on polyhedra, and this is the best I have seen. It covers the history and mathematics of many different polyhedra; the Platonic and Archimedean solids are just the beginning. Kepler's rhombic polyhedra, stellated polyhedra, Miller's solid, etc. -- it's all here. The diagrams are exceptional. I teach high school geometry, and have found this book to be an essential resource in class. The level of detail is quite high, making the book useful as a straight-through read (for someone who is really into math) or a book to flip around in (for those who find heavy math intimidating, but still like polyhedra). Includes helpful tips for model-making. Buy it!

You should buy this!
It's a wonderful book for learning history of polyhedra, but I think it has too little 'mathematics' in. All in all, it's a masterpiece in my mathbook collection. ... Read more

Book Description

Fibre bundles, now an integral part of differential geometry, are also of great importance in modern physics--such as in gauge theory.This book, a succinct introduction to the subject by renown mathematician Norman Steenrod, was the first to present the subject systematically.

It begins with a general introduction to bundles, including such topics as differentiable manifolds and covering spaces.The author then provides brief surveys of advanced topics, such as homotopy theory and cohomology theory, before using them to study further properties of fibre bundles. The result is a classic and timeless work of great utility that will appeal to serious mathematicians and theoretical physicists alike. ... Read more

Reviews (3)

This is where it started
For those individuals who want an in-depth, insightful, and solid understanding of fiber bundles this book must be read. In spite of its date of publication, it still is of considerable value in this regard. Modern treatments of fiber bundles are very formal and the underlying motivation gets swept away in the thirst for rigor. Fiber bundles are now ubiquitous in differential topology, algebraic topology, differential geometry, and algebraic geometry, and have also found a place in theoretical physics, thanks to the success of gauge field theories. Therefore a mastery of fiber bundles is essential for entering any of these fields. But fiber bundles are fascinating objects in and of themselves, and studying them for their own sake needs no apology.

The author does use some antiquated notation, but that is not really a hindrance to the study of the book. The reader will no doubt have some background in differential geometry and topology before attempting this book, so the appropriate translation to more modern notation should be straightforward. Once started, and with a little thought adjustment to the idiosyncracies of the author's writing style, the reader will find a plethora of neat examples and insights into the subject. In particular, part 3 on the cohomology theory of bundles is exceptionally valuable in that it gives the reader a detailed overview of the origin of what are not called Stiefel-Whitney classes. The theory of characteristic classes has of course advanced and matured extensively since this book first appeared, but all of the modern treatments are lacking in that they do not give the reader an appreciation of the fundamentals of the subject. Indeed, the construction of the obstruction to the construction of a cross-section to a bundle is the starting point for many of the ideas in obstruction theory that one finds in differential topology. And yes, the procedures the author uses can be "cleaned-up" and made more concise, but the price one pays in such an endeavor is the loss of an appreciation of the concepts behind the scene.

Since the book is a monograph, there are no exercises, and this is probably the only minus to the book. Also, some knowledge of the German language would be useful to a reader who has it, since the author makes references to papers written in German and much of the terminology in the book shows its roots in the German language. One good example of this is the Reidemeister theory of cohomology groups based on a bundle of coefficients, called Uberdeckung by Reidemeister.

There is no question as to why this book remains in print, and it will no doubt continue to be well into the 21st century. IT is a good example of the idea that something new may not be something better. After finishing it, the reader will be amply prepared to enter into the continually-evolving theory of fiber bundles and their applications, all of which are interesting and important.

Excellent introduction to fiber bundles
This book supplies a lot of intuition and background that more modern texts seem to assume of the reader. Steenrod's writing is meticulous and extremely clear. My opinion is that one can learn just as much out of this seemingly outdated text and probably even more than from the modern texts.

... True, more slick machinery has been developed since Steenrod's time, but those big machines are hardly transparent. Steenrod assumes very little of the reader; he even has a quick course in homotopy groups, although he assumes the reader knows the basics of homology/cohomology. Perhaps most importantly, since many of the ideas in the book were new at the time, he doesn't assume that the reader is already comfortable with those ideas. All together this makes a very accessible book indeed.

Still attractive.
A nostalgic but still attractive book on (homotoy theory of) fiber bundles. This book is not very accessible as it predates the development of modern machinery of algebraic topology, but is worth reading. ... Read more

Book Description

This first edition of this book quickly became an established text in this fast-developing branch of mathematics. This second edition has been significantly revised and expanded. It includes a section on new developments and an expanded discussion of Taubes' and Donaldson's recent results. ... Read more

Reviews (1)

A must for researchers new to the field
An authoritative and comprehensive reference...McDuff and Salamon have done an enormous service to the symplectic community: their book greatly enhances the accessibility of the subject to students and researchers alike.

The discussion begins with classic topology and cover a variety of final year undergraduate topics such as complex manifolds and inverse differential techniques before moving into the vastly complex world of Symplectic Topology.

A must for researchers new to the field ... Read more

Book Description

An undergraduate introduction to the fundamentals of topology — engagingly written, filled with helpful insights, complete with many stimulating and imaginative exercises to help students develop a solid grasp of the subject.
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Reviews (4)

Good Introduction to Metric Spaces and Topology
I was not a mathematics major, and only in recent years have I ventured into abstract mathematics. I was motivated to learn about topology as an aid to understanding a particular 3-D earth modeling application.

I read Introduction to Topology in three stages: as a review of set theory and metric spaces (chapters 1 and 2), then as an introduction to topology (chapter 3), and lastly as a detailed look at two important topological properties, connectedness (chapter 4) and compactness (chapter 5). I had previously read (and reviewed) another book titled Metric Spaces by Victor Bryant, but Mendelson's book was my first serious look at topology.

My reading of Mendelson's 200-page text required about 100 hours, substantially longer than the 40 to 60 hours estimated by an earlier reviewer. No solutions are provided for the section problems, which are generally of the form 'Prove that '.'.

The first chapter provides a concise overview of set theory and functions that is essential for Mendelson's subsequent set-theoretic analysis of metric spaces and topology.

The second chapter is a solid introduction to metric spaces with good discussions on continuity, open balls and neighborhoods, limits from a metric space perspective, open sets and closed sets, subspaces, and equivalence of metric spaces. Chapter 2 concludes with a brief introduction to Hilbert space in a section titled 'an infinite dimensional Euclidian space'.

The third chapter introduces topological spaces as a generalization of metric spaces, and many theorems are largely restatements of the metric space theorems derived in chapter 2. I was thankful for this approach.

Mendelson begins chapter 3 by demonstrating that 1) open sets and neighborhoods are preserved in passing from a metric space to its associated topological space and 2) the existence of a one to one correspondence between the collection of all topological spaces and the collection of all neighborhood spaces.

He then reminds us that in a metric space we can say that there are points of a subset A arbitrarily close to a point x if the metric d(x, A) = 0. In characterizing this notion of 'arbitrary closeness' in a topological space, Mendelson introduces the closure of A, the interior of A, and the boundary of A. Other topics included topological functions, continuity, homeomorphism (the equivalence relation), subspaces, and relative topology. The final sections in chapter 3 on products of topological spaces, identification topologies, and categories and functors were more difficult.

In chapter 4 the initial sections (connectedness on the real line, the intermediate value theorem, and fixed point theorems) were largely familiar. But thereafter I became bogged down with the discussions of path-connected topological spaces, especially with the longer proofs involving the concepts of homotopic paths, the fundamental group, and simple connectedness.

Chapter 5, titled Compactness, was even more abstract and difficult, with topics like coverings, finite coverings, subcoverings, compactness, compactness on the real line, products of compact spaces, compact metric spaces, the Lebesgue number, the Bolzano-Weierstrass property, and countability. I will definitely need to look at another text or two before I can handle more advanced topics.

I suspect that a reader familiar with analysis would have substantially less difficulty with the last two chapters.

In summary, Introduction to Topology quite useful for self-study. Mendelson's short text was intended for a one-semester undergraduate course, and it is thereby ideal for readers that either require a basic introduction to topology, or need a quick review of material previously studied. The last two chapters on connectedness and compactness are substantially more difficult, but are still accessible to the persistent reader.

Ideal for self-study
This book is ideal for self-study. If you have not had the luxury of taking a topology course during your undergraduate studies, but you need to know some topology and you have to study it by yourself, this is the book you need. It is very readable and it explains carefully every concept. However, it is just an introductory text and it contains only basic material. You don't have to invest a lot of time to study the material in this book: let's say 40-60 hours of study are enough to grasp everything. I reccomend it especially to those graduate students of applied mathematics, finance, statistics or economics, who need to use some basic result from topology in their work.

The best introduction to point-set topology
since, for some reason, my school didn't offer any topology course, I decided to study topology on my own. It was very fortunate that I found this book in the library. That was right after I took my first analysis course. But I could understand most of the book at that time. After reviewing basic set theory, the author discussed metric spaces, and then he motivates the definition of topological spaces. This is great, I think, becuase many of introductory topology books often give the definition of topological spaces with any motivation. However it is very important to motivate each concept in mathematics especially in introductory level. And this book does this. And as I did, this book is even good for indivisual study. However, you can get almost no geometricl flavor of topology from this book. For example, there is only one section in one chapter in which the author discusses the fumdamental group. Thus, after all this is the best introduction to "point-set topology". So if you don't know almost anything about topolosy, I strongly recommend this book. And one more thing. If you are still wondering if you should buy this one, just look at the price!

A great book, especially for the price
I know that some people don't like Dover, but I think Dover is great, and Mendelson's Introduction of Topology is an example of why.

Although the book is very short (around 150 pages), it covers the basics of topology very thoroughly and should prepare the reader for the considerably more abstruse Spanier's Algebraic Topology or other texts of such ilk.

If you are a recreational topologist, or are simply tryinging to figure out which way is up in your first topology course, this is for you. ... Read more

Book Description

In this book, the authors take the viewpoint that number theory and fractal geometry can be fruitfully combined. They study, in particular, the vibrations of fractal strings (one-dimensional drums with fractal boundary) and zeros of zeta-functions.

In earlier publications on fractal and spectral geometry, the Riemann Hypothesis was studied and this hinted at the notion of complex dimension as a means to describe certain geometric properties of a fractal, such as its fractal (Minkowski) dimension or the oscillations in the volume of its tubular neighborhoods. This notion of complex dimension is now precisely defined in this book.

A central problem in contemporary mathematics-often expressed as "Can one hear the shape of a drum?" -- consists in describing the relationship between the shape (geometry) of a drum and its sound (its spectrum). In the case of fractal strings, the complex dimensions provide a unified description of the oscillations in the geometry and the spectrum. This description is provided by an explicit formula -- an analytical tool, originally developed for the proof of the Prime Number Theorem, which is extended here to apply to the zeta-functions associated with fractals.

The context of vibrating fractal strings enables the authors to put the Riemann Hypothesis in a geometric setting. This famous conjecture states that the zeros r in the critical strip 0 {\leq} Re {\rho} {\leq} 1 of the Riemann zeta-function all lie on the critical line Re {\rho} = . Here, this conjecture becomes an inverse spectral problem, and its interpretation in the language of fractal strings, which have complex dimensions with real part between 0 and 1, is "One can hear if a fractal string is Minkowski measurable provided that its fractal dimension is not ".

In the more restricted context of fractal Cantor strings, the complex dimensions of which form an infinite vertical arithmetic progression, the inverse spectral problem gets an affirmative answer. The number-theoretical interpretation of this insight is that the Riemann zeta-function does not have an infinite vertical arithmetic progression of zeros. This result is generalized to apply to many other zeta-functions.

This highly original, self-contained monograph will appeal to geometers, fractalists, mathematical physicists, and number theorists, as well as to graduate students in these fields. ... Read more

Book Description

This is the softcover reprint of the English translation of 1972 (available from Springer since 1989) of the first 7 chapters of Bourbaki's 'Algbre commutative'. It provides a very complete treatment of commutative algebra, enabling the reader to go further and study algebraic or arithmetic geometry. The first 3 chapters treat in succession the concepts of flatness, localization and completions (in the general setting of graduations and filtrations). Chapter 4 studies associated prime ideals and the primary decomposition. Chapter 5 deals with integers, integral closures and finitely generated algebras over a field (including the Nullstellensatz). Chapter 6 studies valuation (of any rank), and the last chapter focuses on divisors (Krull, Dedekind, or factorial domains) with a final section on modules over integrally closed Noetherian domains, not usually found in textbooks. Useful exercises appear at the ends of the chapters. ... Read more

Book Description

This is the second of two volumes on the qualitative theory of foliations. For this volume, the authors have selected three special topics: analysis on foliated spaces, characteristic classes of foliations, and foliated manifolds. Each of these is an example of deep interaction between foliation theory and some other highly-developed area of mathematics. In all cases, the authors present useful, in-depth introductions, which lead to further study using the extensive available literature. ... Read more