DATE PUBLISHED: April 2020
FORMAT: HardbackISBN: 9780521193573
Part of Cambridge Monographs on Mathematical Physics
The two pillars of modern physics are general relativity and quantum field theory, the former describes the large scale
structure and dynamics of space-time, the latter, the microscopic constituents of matter. Combining the two yields quantum
field theory in curved space-time, which is needed to understand quantum field processes in the early universe and black
holes, such as the well-known Hawking effect. This book examines the effects of quantum field processes back-reacting on the
background space-time which become important near the Planck time (10-43 sec). It explores the self-consistent description of
both space-time and matter via the semiclassical Einstein equation of semiclassical gravity theory, exemplified by the
inflationary cosmology, and fluctuations of quantum fields which underpin stochastic gravity, necessary for the description
of metric fluctuations (space-time foams). Covering over four decades of thematic development, this book is a valuable
resource for researchers interested in quantum field theory, gravitation and cosmology.
Introduces mathematical formalisms and methodology by explicitly showing how they are implemented using field theory examples
Includes examples of advanced field theory techniques used to solve problems in the early universe and black holes, preparing
readers for future research
Contains information on new concepts and techniques, and provides links to new research problems in non-equilibrium quantum
processes, gravitational quantum physics and quantum gravity
Preface
1. Overview: main themes. Key issues. Reader's guide
Part I. Effective Action and Regularization, Stress Tensor and Fluctuations:
2. `In-out' effective action. Dimensional regularization
3. `In-in' effective action. Stress tensor. Thermal fields
4. Stress-energy tensor and correlators: zeta-function method
5. Stress-energy tensor and correlation. Point separation
Part II. Infrared Behavior, 2Pi, 1/n Backreaction and Semiclassical Gravity:
6. Infrared behavior of interacting quantum fields
7. Advanced field theory topics
8. Backreaction of early universe quantum processes
Part III. Stochastic Gravity:
9. Metric correlations at one-loop: in-in and large N
10. The Einstein-Langevin equation
11. Metric fluctuations in Minkowski spacetime
Part IV. Cosmological and Black Hole Backreaction with Fluctuations:
12. Cosmological backreaction with fluctuations
13. Structure formation in the early universe
14. Black hole backreaction and fluctuations
Part V. Quantum Curvature Fluctuations in De Sitter Spacetime:
15. Stress-energy tensor fluctuations in de Sitter space
16. Two-point metric perturbations in de Sitter
17. Riemann tensor correlator in de Sitter
18. Epilogue: linkage with quantum gravity
References
Index.
Part of Lecture Notes in Logic
PUBLICATION PLANNED FOR: June 2020
FORMAT: HardbackISBN: 9781108478984
The last two decades have seen a wave of exciting new developments in the theory of algorithmic randomness and its
applications to other areas of mathematics. This volume surveys much of the recent work that has not been included in
published volumes until now. It contains a range of articles on algorithmic randomness and its interactions with closely
related topics such as computability theory and computational complexity, as well as wider applications in areas of
mathematics including analysis, probability, and ergodic theory. In addition to being an indispensable reference for
researchers in algorithmic randomness, the unified view of the theory presented here makes this an excellent entry point for
graduate students and other newcomers to the field.
Synthesizes a range of results previously scattered across various journal articles
Includes an introductory survey targeted at newcomers to the field
Details the interaction between algorithmic randomness and various areas of classical mathematics as well as computable
analysis and areas of theoretical computer science
1. Key developments in algorithmic randomness Johanna N. Y. Franklin and Christopher P. Porter
2. Algorithmic randomness in ergodic theory Henry Towsner
3. Algorithmic randomness and constructive/computable measure theory Jason Rute
4. Algorithmic randomness and layerwise computability Mathieu Hoyrup
5. Relativization in randomness Johanna N. Y. Franklin
6. Aspects of Chaitin's Omega George Barmpalias
7. Biased algorithmic randomness Christopher P. Porter
8. Higher randomness Benoit Monin
9. Resource bounded randomness and its applications Donald M. Stull
Index.
Part of Cambridge Studies in Advanced Mathematics
PUBLICATION PLANNED FOR: January 2021
FORMAT: HardbackISBN: 9781108495790
Elementary treatments of Markov chains, especially those devoted to discrete-time and finite state-space theory, leave the
impression that everything is smooth and easy to understand. This exposition of the works of Kolmogorov, Feller, Chung, Kato
and other mathematical luminaries focuses on time-continuous chains but is not so far from being elementary itself. It
reminds us once again that the first impression is false: an infinite, but denumerable state-space is where the fun begins.
If you have not heard of Blackwell's example (in which all states are instantaneous), do not understand what the minimal
process is, or do not know what happens after explosion, dive right in. But beware lest you are enchanted: 'There are more
spells than your commonplace magicians ever dreamed of.'
Takes a much simpler approach than the existing literature
Encourages the reader to discover the facts for themselves by examining examples before learning the theorem
Contains unusual, fascinating examples of Markov chains, gathered from the works of Blackwell, Feller, Kolmogorov, Kendall,
Levy and Reuter
A non-technical introduction
1. A guided tour through the land of operator semigroups
2. Generators versus intensity matrices
3. Boundary theory: core results
4. Boundary theory continued
5. The dual perspective
Solutions and hints to selected exercises
Commonly used notations
References
Index.
Part of Encyclopedia of Mathematics and its Applications
PUBLICATION PLANNED FOR: November 2020
FORMAT: HardbackISBN: 9781107074682
This is a companion book to Asymptotic Analysis of Random Walks: Heavy-Tailed Distributions by A.A. Borovkov and K.A.
Borovkov. Its self-contained systematic exposition provides a highly useful resource for academic researchers and
professionals interested in applications of probability in statistics, ruin theory, and queuing theory. The large deviation
principle for random walks was first established by the author in 1967, under the restrictive condition that the distribution
tails decay faster than exponentially. (A close assertion was proved by S.R.S. Varadhan in 1966, but only in a rather special
case.) Since then, the principle has always been treated in the literature only under this condition. Recently, the author
jointly with A.A. Mogul'skii removed this restriction, finding a natural metric for which the large deviation principle for
random walks holds without any conditions. This new version is presented in the book, as well as a new approach to studying
large deviations in boundary crossing problems. Many results presented in the book, obtained by the author himself or jointly
with co-authors, are appearing in a monograph for the first time.
The first unified systematic exposition of large deviation theory for light-tailed random walks
A leading specialist details the current state of affairs in this important research area
Offers a logical complement to Asymptotic Analysis of Random Walks: Heavy-Tailed Distributions
1. Preliminaries
2. Distribution approximations for sums of random variables
3 Boundary problems for random walks
4. Large deviation principles for trajectories of random walks
5. Moderately large deviation principles for trajectories of random walks and processes with independent increments
6. Applications to mathematical statistics.
Part of Cambridge Tracts in Mathematics
PUBLICATION PLANNED FOR: February 2021
FORMAT: HardbackISBN: 9781107142596
In the middle of the last century, after hearing a talk of Mostow on one of his rigidity theorems, Borel conjectured in a
letter to Serre a purely topological version of rigidity for aspherical manifolds (i.e. manifolds with contractible universal
covers). The Borel conjecture is now one of the central problems of topology with many implications for manifolds that need
not be aspherical. Since then, the theory of rigidity has vastly expanded in both precision and scope. This book rethinks the
implications of accepting his heuristic as a source of ideas. Doing so leads to many variants of the original conjecture -
some true, some false, and some that remain conjectural. The author explores this collection of ideas, following them where
they lead whether into rigidity theory in its differential geometric and representation theoretic forms, or geometric group
theory, metric geometry, global analysis, algebraic geometry, K-theory, or controlled topology.
Introduces tools from a variety of fields, useful to students and researchers in topology, geometry, operator theory, and
geometric group theory
Uses both true and false variations on the conjecture, to gain a deeper understanding of it
Makes much more concrete an area where recent work has been expressed very abstractly
1. Introduction
2. Examples of aspherical manifolds
3. First contact ? The proper category
4. How can it be true?
5. Playing the Novikov game
6. Equivariant Borel conjecture
7. Existential problems
8. Epilogue ? A survey of some techniques
References
Index.