Book Description
This thorough and detailed exposition is the result of an intensive month-long course sponsored by the Clay Mathematics Institute. It develops mirror symmetry from both mathematical and physical perspectives. The material will be particularly useful for those wishing to advance their understanding by exploring mirror symmetry at the interface of mathematics and physics.
This one-of-a-kind volume offers the first comprehensive exposition on this increasingly active area of study. It is carefully written by leading experts who explain the main concepts without assuming too much prerequisite knowledge. The book is an excellent resource for graduate students and research mathematicians interested in mathematical and theoretical physics.
Customer Reviews:
Detailed overview of the subject.......2005-05-16
Mirror symmetry has become an established branch of mathematics and mathematical physics, and research in the subject has resulted in brilliant developments. This sizable book contains essentially some (polished) lecture notes of a seminar series in mirror symmetry that was given in the spring of 2000. This reviewer only studied Part 5 of the book, entitled "Advanced Topics" and so only that part will be reviewed here. In addition, space constraints then dictate only a small portion of this part can be reviewed. Needless to say, any reader who intends to tackle this book will need a substantial background in modern mathematics and advanced physics, and a sizable commitment in time. The time spent is well worth it though, as both the mathematics and physics behind mirror symmetry has to rank as one of the most fascinating research topics in the last two decades.
In the chapter entitled "Topological Strings" the authors consider the functional integration of worldsheet geometries. This project involves essentially the integration over the complex structures of Riemann surfaces. Referring to this procedure as "quantum gravity", they do not address it in-depth, but instead focus on the coupling of topological sigma models to worldsheet gravity, which is called `topological string theory' in the literature. The authors first consider the case where the target is a Kahler manifold whose first Chern class is zero, since for this case the quantum cohomology ring is less easy to obtain, i.e. it can obtain contributions from holomorphic maps of any degree. Even for the case where there is no coupling to gravity, the degree 0 contribution is related to the classical intersection number. The contributions from higher degree result in the deformation of the classical cohomology ring into the quantum cohomology ring. The authors then ask whether there are any other correlators that will give nontrivial (non-zero) invariants in genus 0. Posing this question leads to the WDVV equation and the genus 0 topological string partition function. The n-point correlation functions of topological strings can then be defined as the nth partial derivatives of this function. For higher genus cases, the correlators are all zero, but the authors show the connection between the higher genus partition function and holomorphic anomalies. The case of three-dimensional Calabi-Yau manifolds is special, if one concentrates on the integration over the complex structures of the worldsheet. When the complex dimension of this moduli space is 3(g-1) then there are isolated points where holomorphic maps exist. Defining a topological string theory for Calabi-Yau threefolds is straightforward, as the author shows, and proceeds analogously to the case of topological field theory. A measure is defined on the moduli space of Riemann surfaces of genus g that cancels the axial charge anomaly. A genus g (>1) topological string amplitude, which is a section of a bundle over the moduli space of Calabi-Yau manifolds, is then obtained from this procedure. Modulo the presence of holomorphic anomalies, the authors show that the definition of topological string amplitudes is consistent with the topological symmetry. The origin of these holomorphic anomalies is discussed in fair detail by the authors, having their origin in the boundaries of the moduli space.
The rigorous mathematical formulation of mirror symmetry is of course of great interest to mathematicians. Because of its origin in string theory and quantum field theory, mirror symmetry has not yet received this kind of rigor. Chapters 37 and 38 of this book discuss some of the approaches that attempt to put mirror symmetry on a more rigorous foundation. One of these involves the use of `derived categories,' an approach that was recommended by the mathematician Maxim Kontsevich. The discussion in these chapters takes place in the context of D-branes, and Kontsevich conjectures that mirror symmetry is the equivalence of two categories: the derived category of coherent sheaves, and the category of Lagrangian submanifolds with flat U(1) connections. Specifically the equivalence entails the equivalence between the bounded derived category of coherent sheaves or `B-cycles' and the category of A-cycles with compositions defined in terms of holomorphic maps from disks. This latter category is derived from the Fukaya A-infinity category, as is shown by the authors. They discuss in detail this category, being essentially a generalization of a differential, graded algebra, especially how to obtain the compositions. In chapter 37, the authors give an explicit example of the equivalence of these categories for the case of the elliptic curve. The elliptic curve is interesting in this regard in that it is its own mirror, i.e. the complex parameter is mapped to the complexified Kahler parameter by the mirror map.
The derived category has sometimes been a stumbling block to those who want to understand the Kontsevich conjecture. The authors do not attempt to give the reader the needed insight into this kind of category, but merely take it to be a collection of all holomorphic bundles and coherent sheaves. Sheaves in this category can be subtracted from each other using a map between them. Physically, this subtraction corresponds to the annihilation of branes and anti-branes via a tachyon. Derived categories though are straightforward to think about if one views them from the standpoint of algebraic topology. Derived categories are rich enough to include notions of localization and triangulated objects (i.e. "complexes") and maps (i.e. morphisms) between these objects. This is a kind of "homology" but what is of main interest are homotopies between the morphisms. The class of homotopic morphisms between two complexes forms an abelian group and one can then obtain a category consisting of complexes as objects and classes of homotopic morphisms as morphisms. A cohomology functor can then be defined on this category, along with graded objects and differentials between them. The homotopic category can be given a "triangulation" and morphisms in this category that give rise to isomorphisms in cohomology are given special status, called `quasimorphisms.' The localization of this category with respect to quasimorphisms is called a derived category.
Average customer rating:
- Beautiful
- Great Intuitive Coverage of Complex Variables
- Pretty but not a substitute for traditional text.
- Exceptionally pretty complex analysis book
- essential
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Visual Complex Analysis
Tristan Needham
Manufacturer: Oxford University Press, USA
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The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities (Maa Problem Books Series.)
ASIN: 0198534477 |
Book Description
This radical approach to complex analysis replaces the standard calculational arguments with new geometric ones. With several hundred diagrams, and far fewer prerequisites than usual, this is the first visually intuitive introduction to complex analysis. Although designed for use by undergraduates in mathematics and science, the novelty of the approach will also interest professional mathematicians. PDF extracts of the book can be read at http://www.usfca.edu/vcal/.
Customer Reviews:
Beautiful.......2006-05-11
This is one of the best math books ever written. It is insightful, reader friendly,and has an excellent set of exercises. It also covers a broad range of topics including applications to physics.
Great Intuitive Coverage of Complex Variables.......2006-02-17
This is a great book for a Complex analysis course, especially when combined with a more traditional text like Ahlfors or Knopp's Theory of Functions. Needham's geometric approach builds 'feel' for the subject and manages to include a huge range of topics. Sometimes just trying to work out what you are seeing in a particular visualization can lead to new insights. All in all its an innovative and well written book.
Pretty but not a substitute for traditional text........2005-10-09
My slightly harsh rating is an antidote to all the gushing about
this book. It is a nice book with lots of pretty pictures and
genuine geometrical insights and is well worth reading as a
supplement to traditional complex analysis texts. The geometrical
topics are actually quite good. If you are a maths major then
this book will be of limited use because its coverage of the
traditional topics is simply too weak. The geometrical approach
quickly runs out of steam, in my opinion, once it gets into
complex integration. Homotopy does not even rate a mention in the
index. My pet dislike was the almost complete omission of the
calculus of residues. The author dimisses that topic as being
old-fashioned. True, the application to computing real integrals
is reduced since the advent of computers. But I think that a
maths major would need to be aware of Jordan's Lemma and other
techniques to estimate the asymptotic behaviour of integrals
along curves. I also found that the treatment of multi-valued
functions and branch cuts quite confusing, which is surprising
in a book which is supposed to have a strong geometrical focus.
Exceptionally pretty complex analysis book.......2005-04-09
This is a very exciting introduction to complex analysis. Its most striking feature is the many excellent illustrations; pictures are used to explain things whenever possible. Needham is always eager to explain, and also to show meaning. Thus Möbius transformations are not just charming quirks; instead Needham gives a self-contained introduction to non-Euclidean geometry to show them in action. And when one needs to understand analytic functions as flows, then Needham gives a self-contained introduction to the ideas of vector analysis. Because the book always spills over on other topics like this, I keep my copy within reach at all times, as a treasure mine of beautiful, visual explanations of topics even outside of complex analysis proper. This wide scope works very well most of the time but it should be said that there are probably too many minor side topics than are appropriate in a first textbook. Sometimes Needham seems to include results not because they fit in or add something important, but because he has thought of such a pretty proof. But we quickly forgive him, for the book is so extremely loveable, and it is still by far my first choice as a first textbook of complex analysis.
essential.......2005-01-13
It is possible to memorize definitions, master proofs and work endless exercises and still feel that you don't understand what's going on. This is especially true in complex analysis. This book emphasizes the visual/geometric aspect of analytic functions at the expense of some loss of rigor. These insights are priceless.
Needham employs a wider range of mathematical tools than other books aimed at the upper level undergraduate, e.g., Palka and Brown/Churchill. This would include simple group theory, linear algebra, vector calculus and obviously geometry. Often this works very well at the expense of some digression. At other times, the more traditional algebraic approach is better.
This book is unique and fills an important need.
Book Description
The study of homogeneous spaces provides excellent insights into both differential geometry and Lie groups. In geometry, for instance, general theorems and properties will also hold for homogeneous spaces, and will usually be easier to understand and to prove in this setting. For Lie groups, a significant amount of analysis either begins with or reduces to analysis on homogeneous spaces, frequently on symmetric spaces. For many years and for many mathematicians, Sigurdur Helgason's classic Differential Geometry, Lie Groups, and Symmetric Spaces has been--and continues to be--the standard source for this material.
Helgason begins with a concise, self-contained introduction to differential geometry. He then introduces Lie groups and Lie algebras, including important results on their structure. This sets the stage for the introduction and study of symmetric spaces, which form the central part of the book. The text concludes with the classification of symmetric spaces by means of the Killing-Cartan classification of simple Lie algebras over $\mathbf{C}$ and Cartan's classification of simple Lie algebras over $\mathbf{R}$.
The excellent exposition is supplemented by extensive collections of useful exercises at the end of each chapter. All the problems have either solutions or substantial hints, found at the back of the book.
For this latest edition, Helgason has made corrections and added helpful notes and useful references. The sequels to the present book are published in the AMS's Mathematical Surveys and Monographs Series: Groups and Geometric Analysis, Volume 83, and Geometric Analysis on Symmetric Spaces, Volume 39.
Sigurdur Helgason was awarded the Steele Prize for Differential Geometry, Lie Groups, and Symmetric Spaces and Groups and Geometric Analysis.
Customer Reviews:
Superb Treatise and Indispensible Reference .......2007-06-26
The mere thought or mention of the name Helgason inspires respect and awe. This book gets five stars all the way on its merit alone, regardless of who wrote it. Difficult as it is, the book starts from the fundamentals and works up in a coherent logical manner, there are no gaps in his presentation. The negative review below is completely unjustified. If anyone would like to at least see some of what this book is like go to ocw.mit.edu and download Helgason's notes which use excerpts in this book. Some of the topics in this book are covered in a more easy going way in "Lie Groups, Lie Algebras, and Some of Their Applications" by Robert Gilmore. (If I'm not mistaken Gilmore was a student of Helgason.) This book is mathematical exposition at it's absolute finest and I don't think but 1 in 1,000 people reading this page need me to tell them that much less need a review to persuade them. This book has quite a reputation.
Unsurpassed, but demanding.......2007-05-28
As I reviewed this book at Amazon, I found only one review, which I considered to be too harsh. You should understand that Helgason is writing a graduate textbook. Students will learn about "modules" in their graduate algebra course. They will learn De Rham's theorem in an introductory analysis course or sometimes even in a topology course (yes, it can happen). So, most of the language for which another reviewer criticized him would usually be covered in other graduate courses.
Helgason writes tersely but extremely precisely. I know of no other author who gives similar sophistication of point of view and quick, to the point, proofs. He is a "best of breed," and I suppose that is part of the reason he has been a core member of the faculty at M.I.T. for such a long time. A serious student cannot really avoid reading the entire progression of these texts, particularly the "Groups and Geometric Analysis" title, perhaps second in the Helgason manuscripts.
Semisimple( Simple)->Bad.......2007-05-13
I certainly hate being cheated.
This book is advance as a textbook for a course in Lie Algebra.
I can picture the man who wrote this book lecturing to the future great minds of MIT
and putting them to sleep.
The fellow is the worst sort of pedant.
On page one he mentions one of the more difficult theorems in modern Mathematics,
De Rham's theorem, then drops it like it was too hot to handle.
On page three he introduces Hausdorff's difficult separation axiom
without any explanation at all.
Throughout the book he beats you over the head with terms like "module"
without adequate definition or explanation of terms.
He literally expects you to have learned
what he is supposed to be teaching
before you take his course?
In short , anyone taking the course with this book as a text book
will be hunting for a good text on Lie AlgebraSemi-Simple Lie Algebras and Their Representations (Dover Books on Mathematics) Lie Groups, Lie Algebras, and Some of Their Applications
and differential geometry,
since this one is entirely unreadable,
even by those who know and love the subjects.
Book Description
Reciprocity laws of various kinds play a central role in number theory. In the easiest case, one obtains a transparent formulation by means of roots of unity, which are special values of exponential functions. A similar theory can be developed for special values of elliptic or elliptic modular functions, and is called complex multiplication of such functions. In 1900 Hilbert proposed the generalization of these as the twelfth of his famous problems. In this book, Goro Shimura provides the most comprehensive generalizations of this type by stating several reciprocity laws in terms of abelian varieties, theta functions, and modular functions of several variables, including Siegel modular functions.
This subject is closely connected with the zeta function of an abelian variety, which is also covered as a main theme in the book. The third topic explored by Shimura is the various algebraic relations among the periods of abelian integrals. The investigation of such algebraicity is relatively new, but has attracted the interest of increasingly many researchers. Many of the topics discussed in this book have not been covered before. In particular, this is the first book in which the topics of various algebraic relations among the periods of abelian integrals, as well as the special values of theta and Siegel modular functions, are treated extensively.
Customer Reviews:
An example of masterity from the master of modular forms.......1999-07-03
I regard the book as a priceless gate to the ideas by which "Fermat's last theorem" has been concluded. Hence for any mathematician who would like to master in algebraic geometry, number theory or any alike subject it is an indispensable resource of first glance.
Book Description
From the reviews of the second edition: "The new methods of complex manifold theory are very useful tools for investigations in algebraic geometry, complex function theory, differential operators and so on. The differential geometrical methods of this theory were developed essentially under the influence of Professor S.-S. Chern's works. The present book is a second edition... It can serve as an introduction to, and a survey of, this theory and is based on the author's lectures held at the University of California and at a summer seminar of the Canadian Mathematical Congress....
The text is illustrated by many examples... The book is warmly recommended to everyone interested in complex differential geometry." #Acta Scientiarum Mathematicarum, 41, 3-4#
Average customer rating:
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Moduli Spaces of Curves, Mapping Class Groups and Field Theory
Manufacturer: American Mathematical Society
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ASIN: 0821831674 |
Book Description
This is a collection of articles that grew out of a workshop organized to discuss deep links among various topics that were previously considered unrelated. Rather than a typical workshop, this gathering was unique as it was structured more like a course for advanced graduate students and research mathematicians.
In the book, the authors present applications of moduli spaces of Riemann surfaces in theoretical physics and number theory and on Grothendieck's dessins d'enfants and their generalizations. Chapter 1 gives an introduction to Teichmüller space that is more concise than the popular textbooks, yet contains full proofs of many useful results which are often difficult to find in the literature. This chapter also contains an introduction to moduli spaces of curves, with a detailed description of the genus zero case, and in particular of the part at infinity. Chapter 2 takes up the subject of the genus zero moduli spaces and gives a complete description of their fundamental groupoids, based at tangential base points neighboring the part at infinity; the description relies on an identification of the structure of these groupoids with that of certain canonical subgroupoids of a free braided tensor category. It concludes with a study of the canonical Galois action on the fundamental groupoids, computed using Grothendieck-Teichmüller theory. Finally, Chapter 3 studies strict ribbon categories, which are closely related to braided tensor categories: Here they are used to construct invariants of 3-manifolds which in turn give rise to quantum field theories. The material is suitable for advanced graduate students and researchers interested in algebra, algebraic geometry, number theory, and geometry and topology.
Book Description
This text covers Riemann surface theory from elementary aspects to the fontiers of current research. Open and closed surfaces are treated with emphasis on the compact case, while basic tools are developed to describe the analytic, geometric, and algebraic properties of Riemann surfaces and the associated Abelian varities. Topics covered include existence of meromorphic functions, the Riemann-Roch theorem, Abel's theorem, the Jacobi inversion problem, Noether's theorem, and the Riemann vanishing theorem. A complete treatment of the uniformization of Riemann sufaces via Fuchsian groups, including branched coverings, is presented, as are alternate proofs for the most important results, showing the diversity of approaches to the subject. Of interest not only to pure mathematicians, but also to physicists interested in string theory and related topics.
Book Description
Complex Analysis with Mathematica offers a new way of learning and teaching a subject that lies at the heart of many areas of pure and applied mathematics, physics, engineering and even art. This book offers teachers and students an opportunity to learn about complex numbers in a state-of-the-art computational environment. The innovative approach also offers insights into many areas too often neglected in a student treatment, including complex chaos and mathematical art. Thus readers can also use the book for self-study and for enrichment. The use of Mathematica enables the author to cover several topics that are often absent from a traditional treatment. Students are also led, optionally, into cubic or quartic equations, investigations of symmetric chaos, and advanced conformal mapping. A CD is included which contains a live version of the book, and the Mathematica code enables the user to run computer experiments.
Customer Reviews:
A true delight.......2007-02-14
What a wonderful and delightful book. The author has taken a well-worn topic and infused it with insight and energy. Best of all, the author communicates simply and clearly. He has brought graduate complex analysis to the masses; I wish that I had a book like this as an undergraduate. It is also a great read: the kind of pick you could pick up and go cover to cover with. However, you'll probably want to be at your keyboard trying out his many examples. The unique thing about this tome is that it is well-written, it is mathematically ambitious and it is an invaluable reference for how to use Mathematica. Also, as we have come to expect from Cambridge, this book has excellent production quality.
Book Description
Spatial data analysis is a fast growing area and Voronoi diagrams provide a means of naturally partitioning space into subregions to facilitate spatial data manipulation, modelling of spatial structures, pattern recognition and locational optimization. With such versatility, the Voronoi diagram and its relative, the Delaunay triangulation, provide valuable tools for the analysis of spatial data. This is a rapidly growing research area and in this fully updated second edition the authors provide an up-to-date and comprehensive unification of all the previous literature on the subject of Voronoi diagrams.
Features:
* Expands on the highly acclaimed first edition
* Provides an up-to-date and comprehensive survey of the existing literature on Voronoi diagrams
* Includes a useful compendium of applications
* Contains an extensive bibliography
A wide range of applications is discussed, enabling this book to serve as an important reference volume on this topic. The text will appeal to students and researchers studying spatial data in a number of areas, in particular, applied probability, computational geometry, and Geographic Information Science (GIS). This book will appeal equally to those whose interests in Voronoi diagrams are theoretical, practical or both.
Customer Reviews:
Most complete reference work on Voronoi tessellations.......2005-07-30
The focus of this book is the theory and application of the Voronoi diagram in all its various forms. It is a complete and exhaustive reference work and is the de facto authoritative compendium on all things Voronoi. The book is well-organized, clearly written, and a must-have reference for spatial decomposition and computational problem-solving. Although many concepts are advanced, Dr. Okabe provides excellent examples and complete explanations throughout. The book begins with a history of the the Voronoi diagram and progresses from the simplest two-dimensional applications to more advanced models. There are numerous algorithms, pseudocode examples, and diagrams which illustrate the concepts and make them easy to understand. There are 70 pages of references which are reason enough to purchase the book.
Book Description
This is a modern introduction to the analytic techniques used in the investigation of zeta-function. Riemann introduced this function in connection with his study of prime numbers, and from this has developed the subject of analytic number theory. Since then, many other classes of "zeta-function" have been introduced and they are now some of the most intensively studied objects in number theory. Professor Patterson has emphasized central ideas of broad application, avoiding technical results and the customary function-theoretic approach.
Customer Reviews:
Analytic Number Theory through Zeta-Function.......2007-01-29
This introductory textbook gives a very good insight of analytic number theory through the special topic of Riemann Zeta-Function. It also includes of course an excellent exposition of its relationship with the Prime Number Theorem and with Riemann and Lindelöf Hypotheses. Moreover, seven appendices provide analytic technical complements needed in the core of the text. Many well-chosen exercises and problems accompany each chapter. I use this textbook in my course on Zeta-Function with great success.
An excellent resource for those interested in Riemann's.......2003-04-12
Zeta function. This book contains a lot of application, theory, and, to my surprise, several practice problems at the end of each section to maximize the learning experience. The chapters are concise and the mathematics is relatively easy follow for those with some experience in special functions.
That, however, is the major thing to note: one needs some experience with special functions in order to find this material accessible. (Obviously, right? Otherwise one wouldn't be buying this book! However, much of this material is beyond the grasp of the average mathematics student that stopped at a bachelor's degree.)
Although this book is called an introduction, I don't think that view is entirely appropriate. The material is quite extensive, and the historocity of the zeta function and its development were kept to a minimum. The precursors to the zeta function and its development by Euler and Riemann (especially Euler's original proof to the Basel Problem) are fantastic. If you're interested in Euler's role in the development, I would look to Dunham's Euler: The Master of Us All, and for Riemann, one should turn to Edwads' Riemann's Zeta Function to read Riemann's original paper.
If you're looking for depth, conciseness, and a broad view of Riemann's zeta function, this book should suit your purposes. If you want a more historical view, I would suggest either of the other books I've mentioned, and not this one.
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- MP for Convective Heat & Mass Transfer
- Nonlinear Control Systems (Communications and Control Engineering)
- Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Applied Mathematical Sciences Vol. 42)
- Numerical Solution of Partial Differential Equations: An Introduction
- Optimal Control Theory: An Introduction
- Optimal Control Theory: An Introduction
- Optimization by Vector Space Methods (Series in Decision and Control)
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