Texts and Readings in Mathematics – serie

According to a remark attributed to Mark Kac 'Probability Theory is a measure theory with a soul'. This book with its choice of proofs, remarks, examples and exercises has been prepared taking both these aesthetic and practical aspects into account.

This book offers a self-contained elementary introduction to the fundamental concepts and techniques of Algebraic Geometry, leading to some gems of the subject like Bezout’s Theorem, the Fundamental Theorem of Projective Geometry, and Zariski’s Main Theorem.The book contains a detailed treatment of algebraic plane curves with a special emphasis on elliptic curves and their birational classification. The role played by elliptic curves in modern theory of cryptology is illustrated.A novel feature of the book is a discussion of the state of the art on the Jacobian Problem and its relation to the Epimorphism Theorem. The recently introduced Tame Transformation Method of Cryptosystems, is sketched.Prerequisities are limited to a knowledge of finite Galois Theory, and of commutative Noetherian rings. All the Commutative Algebra needed is presented in Chapter 1, and could form the basis for a mini course on the subject. The exposition retains classroom flavour. About 300 exercises are included, often with adequate hints.

This book arose out of courses taught by the author. It covers the traditional topics of differential manifolds, tensor fields, Lie groups, integration on manifolds and basic differential and Riemannian geometry. The author emphasizes geometric concepts, giving the reader a working knowledge of the topic. Motivations are given, exercises are included, and illuminating nontrivial examples are discussed.Important features include the following: Geometric and conceptual treatment of differential calculus with a wealth of nontrivial examples. A thorough discussion of the much-used result on the existence, uniqueness, and smooth dependence of solutions of ODEs. Careful introduction of the concept of tangent spaces to a manifold. Early and simultaneous treatment of Lie groups and related concepts. A motivated and highly geometric proof of the Frobenius theorem. A constant reconciliation with the classical treatment and the modern approach. Simple proofs of the hairy-ball theorem and Brouwer's fixed point theorem. Construction of manifolds of constant curvature à la Chern. This text would be suitable for use as a graduate-level introduction to basic differential and Riemannian geometry.

A collection of problems from a competition for college students organised by the Iranian Mathematical Society. It compiles problems from these competitions between 1973 and 2007 and provides solutions to most of them. It is suitable for students of mathematics preparing for competitions and for advanced studies.

This book serves as an advanced text for a graduate course on stochastic algorithms for the students of probability and statistics, engineering, economics and machine learning. This second edition gives a comprehensive treatment of stochastic approximation algorithms based on the ordinary differential equation (ODE) approach which analyses the algorithm in terms of a limiting ODE. It has a streamlined treatment of the classical convergence analysis and includes several recent developments such as concentration bounds, avoidance of traps, stability tests, distributed and asynchronous schemes, multiple time scales, general noise models, etc., and a category-wise exposition of many important applications. It is also a useful reference for researchers and practitioners in the field.

This is a textbook meant to be used at the advanced undergraduate or graduate level. It is an introduction to the theory of Lie groups and Lie algebras. The book treats real and p-adic groups in a unified manner. The first chapter outlines preliminary material that is used in the rest of the book. The second chapter is on analytic functions and is of an elementary nature; this material is included to cater to students who may not be familiar with p-adic fields. The third chapter introduces analytic manifolds and contains standard material; the only notable feature being that it covers both real and p-adic analytic manifolds. Chapters 4 and 5 are on Lie groups. All the standard results on Lie groups are proved here. Some of the proofs are different from those in the earlier literature. The last two chapters are on Lie algebras and cover their structure theory as found in the first of the Bourbaki volumes on the subject. Some proofs here are new.

This second edition has been enlarged and considerably rewritten. Among the new topics are infinite product spaces with applications to probability, disintegration of measures on product spaces, positive definite functions on the line, and additional information about Weyl's theorems on equidistribution. Topics that have continued from the first edition include Minkowski's theorem, measures with bounded powers, idempotent measures, spectral sets of bounded functions and a theorem of Szego, and the Wiener Tauberian theorem. Readers of the book should have studied the Lebesgue integral, the elementary theory of analytic and harmonic functions, and the basic theory of Banach spaces. The treatment is classical and as simple as possible. This is an instructional book, not a treatise. Mathematics students interested in analysis will find here what they need to know about Fourier analysis. Physicists and others can use the book as a reference for more advanced topics.

This book, taking a holistic view of geometry, introduces the reader to axiomatic, algebraic, analytic and differential geometry. Starting with an informal introduction to non-Euclidean plane geometries, the book develops the theory to put them on a rigorous footing. It may be considered as an explication of the Kleinian view of geometry a la Erlangen Programme. The treatment in the book, however, goes beyond the Kleinian view of geometry.Some noteworthy topics presented include:various results about triangles (including results on areas of geodesic triangles) in Euclidean, hyperbolic, and spherical planesaffine and projective classification of conicstwopoint homogeneity of the three planes and the fact that the set of distance preserving maps (isometries) are essentially the same as the set of lengths preserving maps of these planes.Geometric intuition is emphasized throughout the book. Figures are included wherever needed. The book has several exercises varying from computational problems to investigative or explorative open questions.

This book on integration theory is based on the lecture notes for courses that the author gave at the Tata Institute of Fundamental Research, Mumbai, and at ETH, Zürich. The subject matter is classical. The goal of the notes is to provide a concise, clear, and accurate treatment of the basic ideas of the subject.

This book contains the author's notes for a course that he taught at ETH, Zürich. The aim is to lead the reader to a proof of the Peter-Weyl theorem, the basic theorem in the representation theory of compact topological groups. The topological, analytical, and algebraic groundwork needed for the proof is provided as part of the course.

This book presents the previously unpublished notes from a series of lectures given by the author at the Tata Institute of Fundamental Research in 1961. Basic material on affine connections and on locally or globally Riemannian and Hermitian symmetric spaces is covered. The final chapter proves the basic theorems on maximal compact subgroups of Lie groups. Readers should be familiar with differential manifolds and the elementary theory of Lie groups and Lie algebras.

The newly developed field of Seiberg-Witten gauge theory has become a well-established part of the differential topology of four-manifolds and three-manifolds. This book offers an introduction and an up-to-date review of the state of current research.

This book is an introductory graduate-level text on game theory that grew out of courses in mathematics and economics and operations research. Game theory deals with mathematical models of conflict and cooperation. In the first nine chapters, the author covers noncooperative games in extensive and strategic form and some economic applications. Relations with the theory of linear programing and the theory of linear complementarity are indicated. The last chapter treats different types of cooperative games and solution concepts. Discussed are economic applications and applications in OR-situations with multiple agents. The book includes a rich collection of exercises, partly with solutions.

This is an introductory book on Ergodic Theory. The presentation has a slow pace and the book can be read by any person with a background in basic measure theory and metric topology. A new feature of the book is that the basic topics of Ergodic Theory such as the Poincaré recurrence lemma, induced automorphisms and Kakutani towers, compressibility and E. Hopf's theorem, the theorem of Ambrose on representation of flows are treated at the descriptive set-theoretic level before their measure-theoretic or topological versions are presented. In addition, topics around the Glimm-Effros theorem are discussed. In the third edition a chapter entitled `Additional Topics' has been added. It gives Liouville's Theorem on the existence of invariant measure, entropy theory leading up to Kolmogorov-Sinai Theorem, and the topological dynamics proof of van der Waerden's theorem on arithmetical progressions.

Like the first volume, this is a special collection of articles describing the work of some of the best-known mathematicians from India. It contains eight articles written by experts, each of whom has chosen one major research contribution by an Indian mathematician and explained its context, significance, and impact. This is done in a way that makes the main ideas accessible to someone whose own research interests might be in a different area. Included here are commentaries on important works by:R. C. Bose, S. S. Shrikhande and E. T. ParkerH. Cramer and C. R. RaoV. B. Mehta and A. RamanathanR. NarasimhaK. R. Parthasarathy, R. Ranga Rao, and V. S. VaradarajanR. ParthasarathyD. N. VermaN. Wiener and P. R. Masani

The aim of this book is to give an introduction to what has come to be known as Standard Monomial Theory (SMT). SMT deals with the construction of nice bases of finite dimensional irreducible representations of semi-simple algebraic groups or, in geometric terms, nice bases of coordinate rings of flag varieties (and their Schubert subvarieties) associated to these groups. Besides its intrinsic interest, SMT has applications to the study of the geometry of Schubert varieties. SMT has its origin in the work of Hodge, giving bases of the coordinate rings of the Grassmannian and its Schubert subvarieties by ""standard monomials"". In its modern form, SMT was developed by the author in a series of papers written in collaboration with V. Lakshmibai and C. Musili.This book is a reproduction of a course of lectures given by the author in 1983-84 which appeared in the Brandeis Lecture Notes series. The aim of this course was to give an introduction to the series of papers by concentrating on the case of the full linear group. In recent years, there has been great progress in SMT due to the work of Peter Littelmann. Seshadri's course of lectures (reproduced in this book) remains an excellent introduction to SMT.In the second edition, Conjectures of a Standard Monomial Theory (SMT) for a general semi-simple (simply-connected) algebraic group, due to Lakshmibai, have been added as Appendix C. Many typographical errors have been corrected, and the bibliography has been revised.

The material presented in this book is suited for a first course in Functional Analysis which can be followed by masters students. While covering all the standard material expected of such a course, efforts have been made to illustrate the use of various theorems via examples taken from differential equations and the calculus of variations, either through brief sections or through exercises. In fact, this book will be particularly useful for students who would like to pursue a research career in the applications of mathematics.The book includes a chapter on weak and weak*topologies and their applications to the notions of reflexivity, separability and uniform convexity. The chapter on the Lebesgue spaces also presents the theory of one of the simplest classes of Sobolev spaces. The book includes a chapter on compact operators and the spectral theory for compact self-adjoint operators on a Hilbert space.Each chapter has large collection of exercises at the end. These illustrate the results of the text, show the optimality of the hypotheses of various theorems via examples or counterexamples, or develop simple versions of theories not elaborated upon in the text.

These notes are a record of a one semester course on Functional Analysis given by the author to second year Master of Statistics students at the Indian Statistical Institute, New Delhi. Students taking this course have a strong background in real analysis, linear algebra, measure theory and probability, and the course proceeds rapidly from the definition of a normed linear space to the spectral theorem for bounded selfadjoint operators in a Hilbert space.The book is organised as twenty six lectures, each corresponding to a ninety minute class session. This may be helpful to teachers planning a course on this topic. Well prepared students can read it on their own.

Several generations of students of algebraic geometry have learned the subject from David Mumford's fabled "Red Book" containing notes of his lectures at Harvard University. Their genesis and evolution are described in the preface as:Initially notes to the course were mimeographed and bound and sold by the Harvard math department with a red cover. These old notes were picked up by Springer and are now sold as the "Red book of Varieties and Schemes". However, every time I taught the course, the content changed and grew. I had aimed to eventually publish more polished notes in three volumes...This book contains what Mumford had then intended to be Volume II. It covers the material in the "Red Book" in more depth with several more topics added. The notes have been brought to the present form in collaboration with T. Oda.

This is a basic text on combinatorics that deals with all the three aspects of the discipline: tricks, techniques and theory, and attempts to blend them. The book has several distinctive features. Probability and random variables with their interconnections to permutations are discussed. The theme of parity has been specially included and it covers applications ranging from solving the Nim game to the quadratic reciprocity law. Chapters related to geometry include triangulations and Sperner's theorem, classsification of regular polytopes, tilings and an introduction to the Eulcidean Ramsey theory. Material on group actions covers Sylow theory, automorphism groups and a classification of finite subgroups for orthogonal groups. All chapters have a large number of exercises with varying degrees of difficulty, ranging from material suitable for Mathematical Olympiads to research.

This is the second (revised and enlarged) edition of the book originally published in 2003. It introduces the first concepts of Algebraic Topology like general simplicial complexes, simplicial homology theory, fundamental groups, covering spaces and singular homology theory in detail. The text has been designed for undergraduate and beginning graduate students of Mathematics. It assumes a minimal background of linear algebra, group theory and topological spaces. The author has dealt with the basic concepts and ideas in a very lucid manner giving suitable motivations and illustrations. As an application of the tools developed in this book, some classical theorems like Brouwer’s fixed point theorem, the Lefschetz fixed point theorem, the Borsuk-Ulam theorem, Brouwer’s separation theorem and the theorem on invariance of domain have been proved and illustrated. Most of the exercises are elementary but some are more challenging and will help the readers in their understanding of the subject.

This book is an introduction to basic topics in inequalities and their applications. These include the arithmetic mean-geometric mean inequality, Cauchy-Schwarz inequality, Chebyshev inequality, rearrangement inequality, convex and concave functions and Muirhead's theorem. More than 400 problems are included in the book and their solutions are explained. A chapter on geometric inequalities is a special feature of this book. Most of these problems are from International Mathematical Olympiads and from many national mathematical olympiads.The book is intended to help students who are preparing for various mathematical competitions. It is also a good source book for graduate students in consolidating their knowledge of inequalities and their applications. The author has been involved in training Indian students for mathematical olympiads for more than 25 years.

This book deals with topics usually studied in a masters or graduate level course on the theory of measure and integration. It starts with the Riemann integral and points out some of its shortcomings which motivate the theory of measure and the Lebesgue integral.Starting with abstract measures and outermeasures, the Lebesgue measure is constructed and its important properties are highlighted. Measurable functions, different notions of convergence, the Lebesgue integral, the fundamental theorem of calculus, product spaces, and signed measures are studied. There is a separate chapter on the change of variable formula and one on Lp- spaces. Most of the material in this book can be covered in a one semester course. The prerequisite for following this book is familiarity with basic real analysis and elementary topological notions, with special emphasis on the topology of the N- dimensional euclidean space. Each chapter is provided with a variety of exercises.

This book treats some basic topics in the spectral theory of dynamical systems. The treatment is at a general level, but two more advanced theorems, one by H. Helson and W. Parry and the other by B. Host, are presented. Moreover, Ornstein's family of mixing rank one automorphisms is described with construction and proof. Systems of imprimitivity and their relevance to ergodic theory are discussed, and Baire category theorems of ergodic theory, scattered in the literature, are derived in a unified way. Riesz products are considered and they are used to describe the spectral types and eigenvalues of rank one automorphisms.The major change in this edition is that a new chapter titled Calculus of Generalized Riesz Products has been added. This is based on some recent work of the author with El Houcein El Abdalaoui and supplements the material presented elsewhere in the book.

This is an introductory text on a broad class of statistical estimators that are minimizers of convex functions. It covers the basics of U-statistics and Mm-estimators and develops their asymptotic properties. It also provides an elementary introduction to resampling, particularly in the context of these estimators. The last chapter is on practical implementation of the methods presented in other chapters, using the free software R.

This is an introductory text on a broad class of statistical estimators that are minimizers of convex functions. It covers the basics of U-statistics and Mm-estimators and develops their asymptotic properties. It also provides an elementary introduction to resampling, particularly in the context of these estimators. The last chapter is on practical implementation of the methods presented in other chapters, using the free software R.

This book discusses basic topics in the spectral theory of dynamical systems. It also includes two advanced theorems, one by H. Helson and W. Parry, and another by B. Host.  Moreover, Ornstein’s family of mixing rank-one automorphisms is given with construction and proof. Systems of imprimitivity and their relevance to ergodic theory are also examined. Baire category theorems of ergodic theory, scattered in literature, are discussed in a unified way in the book. Riesz products are introduced and applied to describe the spectral types and eigenvalues of rank-one automorphisms. Lastly, the second edition includes a new chapter “Calculus of Generalized Riesz Products”, which discusses the recent work connecting generalized Riesz products, Hardy classes,  Banach's problem of simple Lebesgue spectrum in ergodic theory and flat polynomials.