Graduate Texts in Mathematics
Treats both classical and recent results in ergodic theory from a modern analytic perspective
Assumes no background in ergodic theory, while providing a review of basic results in functional analysis
Provides a foundation for understanding recent applications of ergodic theory to combinatorics and number theory
Stunning recent results by Host?Kra, Green?Tao, and others, highlight the timeliness of this systematic introduction to classical ergodic theory using the tools of operator theory. Assuming no prior exposure to ergodic theory, this book provides a modern foundation for introductory courses on ergodic theory, especially for students or researchers with an interest in functional analysis. While basic analytic notions and results are reviewed in several appendices, more advanced operator theoretic topics are developed in detail, even beyond their immediate connection with ergodic theory. As a consequence, the book is also suitable for advanced or special-topic courses on functional analysis with applications to ergodic theory.
an intuitive introduction to ergodic theory
an introduction to the basic notions, constructions, and standard examples of topological dynamical systems
Koopman operators, Banach lattices, lattice and algebra homomorphisms, and the Gelfand-Naimark theorem
measure-preserving dynamical systems
von Neumann’s Mean Ergodic Theorem and Birkhoff’s Pointwise Ergodic Theorem
strongly and weakly mixing systems
an examination of notions of isomorphism for measure-preserving systems
Markov operators, and the related concept of a factor of a measure-preserving system
compact groups and semigroups, and a powerful tool in their study, the Jacobs-de Leeuw-Glicksberg decomposition
an introduction to the spectral theory of dynamical systems
the theorems of Furstenberg and Weiss on multiple recurrence
applications of dynamical systems to combinatorics (theorems of van der Waerden, Gallai, and Hindman, Furstenberg’s Correspondence Principle, theorems of Roth and Furstenberg-Sarkozy)
Beyond its use in the classroom, Operator Theoretic Aspects of Ergodic Theory can serve as a valuable foundation for doing research at the intersection of ergodic theory and operator theory.
Lecture Notes in Mathematics
Explores the non-standard geometric view of the Wentzell-Freidlin theory of rare transition events
The general geometric framework may spawn applications outside of probability theory
Key results and their explanations are well-separated from the necessary technical proofs, making it easy to quickly use the proven existence criteria in practice
Includes many intuitive examples with color illustrations
Only a knowledge of graduate level analysis is required; all non-standard concepts are introduced as needed
Provides detailed complete proofs that do not require any additional work by the reader to fill the gaps
Presenting a study of geometric action functionals (i.e., non-negative functionals on the space of unparameterized oriented rectifiable curves), this monograph focuses on the subclass of those functionals whose local action is a degenerate type of Finsler metric that may vanish in certain directions, allowing for curves with positive Euclidean length but with zero action. For such functionals, criteria are developed under which there exists a minimum action curve leading from one given set to another. Then the properties of this curve are studied, and the non-existence of minimizers is established in some settings.
Applied to a geometric reformulation of the quasipotential of Wentzell-Freidlin theory (a subfield of large deviation theory), these results can yield the existence and properties of maximum likelihood transition curves between two metastable states in a stochastic process with small noise.
The book assumes only standard knowledge in graduate-level analysis; all higher-level mathematical concepts are introduced along the way.
Lecture Notes in Mathematics
Presents sheaves with a clear connection to the set-theoretic foundations
Strives for a maximum of rigor (concerning proofs, statements, definitions, and notation)
Overcomes the “canonical isomorphism” paradigm; all morphisms are given/constructed explicitly
Introduces a Gaus-Manin connection for families of possibly non-compact manifolds
The text consists in three parts. The first part develops a theory of period mappings of “Hodge-de Rham type” for families of complex manifolds, the essential argument being that the manifolds in question need neither be compact nor Kahler. For compact Kahler manifolds we fall back on Griffiths’ classical theory of period mappings. The second part investigates the degeneration behavior of the relative Frolicher spectral sequence associated to a submersive morphism of complex manifolds, the main point again being that the fibers of the morphism in question need neither be compact nor Kahler. In turn, the third part focuses on applications of the previous parts to the study of irreducible symplectic complex spaces. The latter notion generalizes the idea of an irreducible symplectic manifold, dubbed a hyperkahler manifold in differential geometry, to possibly singular spaces. The three parts of the work offer valuable resources, both individually and as a combined whole.
Lecture Notes in Mathematics
Touches upon a huge number of hot topics in probability
The methods presented have become indispensable working tools for researchers in probability theory and mathematical physics
The lectures are written in an informal style, focusing on the essentials rather than technicalities
This volume presents five different methods recently developed to tackle the large scale behavior of highly correlated random systems, such as spin glasses, random polymers, local times and loop soups and random matrices. These methods, presented in a series of lectures delivered within the Jean-Morlet initiative (Spring 2013), play a fundamental role in the current development of probability theory and statistical mechanics. The lectures were: Random Polymers by E. Bolthausen, Spontaneous Replica Symmetry Breaking and Interpolation Methods by F. Guerra, Derrida's Random Energy Models by N. Kistler, Isomorphism Theorems by J. Rosen and Spectral Properties of Wigner Matrices by B. Schlein.
This book is the first in a co-edition between the Jean-Morlet Chair at CIRM and the Springer Lecture Notes in Mathematics which aims to collect together courses and lectures on cutting-edge subjects given during the term of the Jean-Morlet Chair, as well as new material produced in its wake. It is targeted at researchers, in particular PhD students and postdocs, working in probability theory and statistical physics.
Lecture Notes in Mathematics
Presents the latest developments on commutants and reducing subspaces of multiplication operators on the Bergman space
Reveals von Neumann algebras generated by multiplication operators on the Bergman space
Establishes a fascinating connection between operator theory and von Neumann algebra
About this book
This book deals with various aspects of commutants and reducing subspaces of multiplication operators on the Bergman space, along with relevant von Neumann algebras generated by these operators, which have been the focus of considerable attention from the authors and other experts in recent years. The book reviews past developments and offers insights into cutting-edge developments in the study of multiplication operators. It also provides commentary and comparisons to stimulate research in this area.
2016 300 pages
Series: Chapman & Hall/CRC Monographs on Statistics & Applied Probability
Hardback:
978-1-46-656840-2
March 14th 2016
Multi-state models describe stochastic processes that consist of transitions between states over time, such as the three-state illness-death model. Interval-censored data is extremely common as the exact time of transition from one state to another is unknown?only an interval of time is known. This book shows how to study this type of data. Many examples using real data from the medical and social science literature illustrate the methods. R and WinBUGS code is also provided to implement the methods, with the code and data available for download online.
Introduction. Modeling Survival Data. Three-State Illness-Death Models. General Multi-State Survival Model. Bayesian Inference for Multi-State Survival Models. Frailty Models. Residual Life Expectancy. Additional Topics. Appendix: Software. References. Index.
Nankai Tracts in Mathematics: Volume 14
Etale cohomology is an important branch in arithmetic geometry. This book covers the main materials in SGA 1, SGA 4, SGA 4 1/2 and SGA 5 on etale cohomology theory, which includes decent theory, etale fundamental groups, Galois cohomology, etale cohomology, derived categories, base change theorems, duality, and ?-adic cohomology. The prerequisites for reading this book are basic algebraic geometry and advanced commutative algebra.
Descent Theory
Etale Morphisms and Smooth Morphisms
Etale Fundamental Groups
Group Cohomology and Galois Cohomology
Etale Cohomology
Derived Categories and Derived Functors
Base Change Theorems
Duality
Finiteness Theorems
-Adic Cohomology
Readership: Graduate students and researchers in pure mathematics.
This book provides a systematic treatment of algebraic and topological properties of convex sets (possibly non-closed or unbounded) in the n-dimensional Euclidean space. Topics under consideration include general properties of convex sets and convex hulls, cones and conic hulls, polyhedral sets, the extreme structure, support and separation properties of convex sets.
Lectures on Convex Sets is self-contained and unified in presentation. The book grew up out of various courses on geometry and convexity, taught by the author for more than a decade. It can be used as a textbook for graduate students and even ambitious undergraduates in mathematics, optimization, and operations research. It may also be viewed as a supplementary book for a course on convex geometry or convex analysis, or as a source for independent study of the subject, suitable for non-geometers.
The Affine Structure of ?n
Convex Sets
Convex Hulls
Convex Cones and Conic Hulls
Recession and Normal Directions
Support and Separation Properties
The Extreme Structure of Convex Sets
The Exposed Structure of Convex Sets
Polyhedra
Readership: Graduate students in mathematics, optimization and operations research.