Fields Institute Monographs, Volume: 14
2000; 146 pp; softcover
ISBN-13: 978-0-8218-4435-9
This volume offers an introduction to large deviations. It is divided into two parts: theory and applications. Basic large deviation theorems are presented for i.i.d. sequences, Markov sequences, and sequences with moderate dependence. The rate function is computed explicitly. The theory is explained without too much emphasis on technicalities. Also included is an outline of general definitions and theorems. The goal is to expose the unified theme that gives large deviation theory its overall structure, which can be made to work in many concrete cases. The section on applications focuses on recent work in statistical physics and random media.
This book contains 60 exercises (with solutions) that should elucidate the content and engage the reader. Prerequisites for the book are a strong background in probability and analysis and some knowledge of statistical physics. It would make an excellent textbook for a special topics course in large deviations.
Theory
Large deviations for i.i.d. sequences: Part 1
Large deviations for i.i.d. sequences: Part 2
General theory
Large deviations for Markov sequences
Large deviations for dependent sequences
Applications
Statistical hypothesis testing
Random walk in random environment
Heat conduction with random sources and sinks
Polymer chains
Interacting diffusions
Solutions to the exercises
Bibliography
Index
Glossary of symbols
Errata
2009; 248 pp; hardcover
ISBN-13: 978-81-85931-89-0
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 organized 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.
Graduate students and research mathematicians interested in functional analysis.
Banach spaces
Dimensionality
New Banach Spaces from old
The Hahn-Banach theorem
The uniform boundedness principle
The open mapping theorem
Dual spaces
Some applications
The weak topology
The second dual and the weak* topology
Hilbert spaces
Orthonormal bases
Linear operators
Adjoint operators
Some special operators in Hilbert space
The resolvent and the spectrum
Subdivision of the spectrum
Spectra of normal operators
Square roots and the polar decomposition
Compact operators
The spectrum of a compact operator
Compact operators and invariant subspaces
Trace ideals
The spectral theorem-I
The spectral theorem-II
The spectral theorem-III
Index
2009; 288 pp; hardcover
ISBN-13: 978-81-85931-92-0
Flag varieties are important geometric objects and their study involves an interplay of geometry, combinatorics, and representation theory. This book is a detailed account of this interplay. In the area of representation theory, the book discusses complex semisimple Lie algebras and semisimple algebraic groups; in addition, the representation theory of symmetric groups is discussed. In the area of algebraic geometry, the book explains in detail Grassmannian varieties, flag varieties, and their Schubert subvarieties.
Because of the connections with root systems, many of the geometric results admit elegant combinatorial description, a typical example being the description of the singular locus of a Schubert variety. This is shown to be a consequence of standard monomial theory (abbreviated SMT). Thus the book includes SMT and some important applications--singular loci of Schubert varieties, toric degenerations of Schubert varieties, and the relationship between Schubert varieties and classical invariant theory.
Graduate students and research mathematicians interested in flag varieties.
Introduction
Preliminaries
Structure theory of semisimple rings
Representation theory of finite groups
Representation theory of the symmetric group
Symmetric polynomials
Schur-Weyl duality and the relationship between representations of $Sd$ and $GL_n (C)$
Structure theory of complex semisimple Lie algebras
Representation theory of complex semisimple Lie algebras
Generalities on algebraic groups
Structure theory of reductive groups
Representation theory of semisimple algebraic groups
Geometry of the grassmannian, flag and their Schubert varieties via standard monomial theory
Singular locus of a Schubert variety in the flag variety $SL_n/B$
Applications
Appendix: Chevalley groups
Bibliography
List of symbols
Index
2009; 400 pp; hardcover
ISBN-13: 978-81-85931-88-3
This book is an introduction to the study of fundamental inequalities such as the arithmetic mean-geometric mean inequality, the Cauchy-Schwarz inequality, the Chebyshev inequality, the rearrangement inequality, and the inequalities for convex and concave functions. The emphasis is on the use of these inequalities for solving problems. The book's special feature is a chapter on the geometrical inequalities that studies relations between various geometrical measures. It contains more than 300 problems, many of which are applications of inequalities. A large number of problems are taken from the International Mathematical Olympiads (IMO) and many national olympiads from countries across the world.
The book should be very useful for students participating in mathematical contests. It should also help graduate students consolidate their knowledge of inequalities by way of applications.
Undergraduate and graduate students interested in analysis.
Some basic inequalities
Techniques for proving inequalities
Geometric inequalities
Applications involving inequalities
Problems on inequalities
Solutions to problems
2009; 210 pp; softcover
ISBN-13: 978-81-85931-90-6
This book is an elaboration of a series of lectures given at the Harish-Chandra Research Institute. The reader will be taken through a journey on the arithmetical sides of the large sieve inequality which, when applied to the Farey dissection, will reveal connections between this inequality, the Selberg sieve and other less used notions such as pseudo-characters and the $\Lambda_Q$-function, as well as extend these theories.
One of the leading themes of these notes is the notion of so-called local models that throws a unifying light on the subject. As examples and applications, the authors present, among other things, an extension of the Brun-Tichmarsh Theorem, a new proof of Linnik's Theorem on quadratic residues, and an equally novel one of the Vinogradov's Three Primes Theorem; the authors also consider the problem of small prime gaps, of sums of two squarefree numbers and several other ones, some of them new, like a sharp upper bound for the number of twin primes $p$ that are such that $p+1$ is squarefree. In the end the problem of equality in the large sieve inequality is considered, and several results in this area are also proved.
Graduate students and research mathematicians interested in number theory.
Introduction
The large sieve inequality
An extension of the classical arithmetical theory of the large sieve
Some general remarks on arithmetical functions
A geometric interpretation
Further arithmetical applications
The Siegel zero effect
A weighted hermitian inequality
A first use of local models
Twin primes and local models
The three primes theorem
The Selberg sieve
Fourier expansion of sieve weights
The Selberg sieve for sequences
An overview
Some weighted sequences
Small gaps between primes
Approximating by a local model
Selecting other sets of moduli
Sums of two squarefree numbers
On a large sieve equality
Appendix
Notations
References
Index
2009; 281 pp; hardcover
ISBN-13: 978-81-85931-87-6
The material presented in this book is suited for a first course in Functional Analysis which can be followed by master's students. While all the standard material expected of such a course is covered, 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 on in the text.
Graduate students interested in functional analysis.
Preliminaries
Normed linear spaces
Hahn-Banach theorems
Baire's theorem and applications
Weak and weak* topologies
$L^p$ spaces
Hilbert spaces
Compact operators
Bibliography
Index