Part of Mathematical Sciences Research Institute Publications
This collection of survey and research articles focuses on recent developments concerning various quantitative aspects of “thin groups.” There are discrete subgroups of semisimple Lie groups that are both big (i.e., Zariski dense) and small (i.e., of infinite co-volume). This dual nature leads to many intricate questions. Over the past few years, many new ideas and techniques, arising in particular from arithmetic combinatorics, have been involved in the study of such groups, leading, for instance, to far-reaching generalizations of the strong approximation theorem in which congruence quotients are shown to exhibit a spectral gap, referred to as superstrong approximation. This book provides a broad panorama of a very active field of mathematics at the boundary between geometry, dynamical systems, number theory, and combinatorics. It is suitable for professional mathematicians and graduate students in mathematics interested in this fascinating area of research.
This is a very timely book describing the most recent advances in an exciting subject that has developed very rapidly in recent years Gathers a collection of well-written surveys by the best experts in the field Includes an overview of the field by Peter Sarnak, IAS Princeton
Product details
Publication planned for: January 2014
format: Hardback
isbn: 9781107036857
dimensions: 234 x 156 mm
available from January 2014
1. Some Diophantine applications of the theory of group expansion Jean Bourgain
2. A brief introduction to approximate groups Emmanuel Breuillard
3. Superstrong approximation for monodromy groups Jordan S. Ellenberg
4. The ubiquity of thin groups Elena Fuchs
5. The orbital circle method Alex V. Kontorovich
6. Sieve in discrete groups, especially sparse Emmanuel Kowalski
7. How random are word maps? Michael Larsen
8. Constructing thin groups Darren Long and Alan W. Reid
9. On ergodic properties of the Burger-Roblin measure Amir Mohammadi
10. Harmonic analysis, ergodic theory and counting for thin groups Hee Oh
11. Generic elements in Zariski-dense subgroups and isospectral locally symmetric spaces Gopal Prasad and Andrei Rapinchuk
12. Growth in linear groups Laszlo Pyber and Endre Szabo
13. On strong approximation for algebraic groups Andrei Rapinchuk
14. Generic phenomena in groups: some answers and many questions Igor Rivin
15. Affine sieve and expanders Alireza Salehi Golsefidy
16. Growth in linear groups Peter Sarnak.
Part of London Mathematical Society Lecture Note Series
Publication planned for: July 2014
available from July 2014
format: Paperback
isbn: 9781107689497
The theory of optimal transportation has its origins in the eighteenth century when the problem of transporting resources at a minimal cost was first formalised. Through subsequent developments, particularly in recent decades, it has become a powerful modern theory. This book contains the proceedings of the summer school 'Optimal Transportation: Theory and Applications' held at the Fourier Institute in Grenoble. The event brought together mathematicians from pure and applied mathematics, astrophysics, economics and computer science. Part I of this book is devoted to introductory lecture notes accessible to graduate students, while Part II contains research papers. Together, they represent a valuable resource on both fundamental and advanced aspects of optimal transportation, its applications, and its interactions with analysis, geometry, PDE and probability, urban planning and economics. Topics covered include Ricci flow, the Euler equations, functional inequalities, curvature-dimension conditions, and traffic congestion.
Part I. Short Courses:
1. Introduction to optimal transport theory Filippo Santambroggio
2. Models and applications of optimal transport in economics, traffic and urban planning Filippo Santambroggio
3. Logarithmic Sobolev inequality for diffusions and curvature-dimension condition Ivan Gentil
4. Lecture notes on variational methods for incompressible Euler equations Luigi Ambrosio and Alessio Figalli
5. Ricci flow: the foundations via optimal transportation Peter Topping
6. Lecture notes on gradient flows and optimal transport Sara Danieri and Guiseppe Savare
7. Ricci curvature, entropy, and optimal transport Shin-Ishi Ohta
Part II. Survey and Research Papers:
8. Computing the time-continuous optimal mass transport without Lagrangian techniques Olivier Besson, Martine Picq and Jerome Poussin
9. On the duality theory for the Monge?Kantorovich transport problem Mathias Beiglbock, Chrsitian Leonard and Walter Schachermayer
10. Optimal coupling for mean field limits Francois Bolley
11. Functional inequalities via Lyapunov conditions Patrick Cattiaux and Arnaud Guillin
12. Size of the medial axis and stability of Federer's curvature measures Quentin Merigot
Part of Australian Mathematical Society Lecture Series
Publication planned for: June 2014
available from June 2014
format: Paperback
isbn: 9780521186490
Despite their classical nature, continued fractions are a neverending research area, with a body of results accessible enough to suit a wide audience, from researchers to students and even amateur enthusiasts. Neverending Fractions brings these results together, offering fresh perspectives on a mature subject. Beginning with a standard introduction to continued fractions, the book covers a diverse range of topics, from elementary and metric properties, to quadratic irrationals, to more exotic topics such as folded continued fractions and Somos sequences. Along the way, the authors reveal some amazing applications of the theory to seemingly unrelated problems in number theory. Previously scattered throughout the literature, these applications are brought together in this volume for the first time. A wide variety of exercises guide readers through the material, which will be especially helpful to readers using the book for self-study, and the authors also provide many pointers to the literature.
Offers fresh perspectives on a classical subject Derived from the authors' many years of experience teaching the subject at different levels Includes unpublished work of the late Alf van der Poorten
Preface
1. Some preliminaries from number theory
2. Continued fractions, as they are
3. Metric theory of continued fractions
4. Quadratic irrationals through a magnifier
5. Hyperelliptic curves and Somos sequences
6. From folding to Fibonacci
7. The integer part of qα + β
8. The Erd?s-Moser equation
9. Irregular continued fractions
Appendix. Selected continued fractions
Bibliography
Index.
Part of London Mathematical Society Lecture Note Series
Publication planned for: June 2014
available from June 2014
format: Paperback
isbn: 9781107633223
Focusing on the role that automorphisms and equivalence relations play in the algebraic theory of minimal sets provides an original treatment of some key aspects of abstract topological dynamics. Such an approach is presented in this lucid and self-contained book, leading to simpler proofs of classical results, as well as providing motivation for further study. Minimal flows on compact Hausdorff spaces are studied as icers on the universal minimal flow M. The group of the icer representing a minimal flow is defined as a subgroup of the automorphism group G of M, and icers are constructed explicitly as relative products using subgroups of G. Many classical results are then obtained by examining the structure of the icers on M, including a proof of the Furstenberg structure theorem for distal extensions. This book is designed as both a guide for graduate students, and a source of interesting new ideas for researchers.
Part I. Universal Constructions:
1. The Stone?Cech compactification βT
Appendix to Chapter 1. Ultrafilters and the construction of βT
2. Flows and their enveloping semigroups
3. Minimal sets and minimal right ideals
4. Fundamental notions
5. Quasi-factors and the circle operator
Appendix to Chapter 5. The Vietoris topology on 2^X
Part II. Equivalence Relations and Automorphisms:
6. Quotient spaces and relative products
7. Icers on M and automorphisms of M
8. Regular flows
9. The quasi-relative product
Part III. The τ-Topology:
10. The τ-topology on Aut(X)
11. The derived group
12. Quasi-factors and the τ-topology
Part IV. Subgroups of G and the Dynamics of Minimal Flows:
13. The proximal relation and the group P
14. Distal flows and the group D
15. Equicontinuous flows and the group E
Appendix to Chapter 15. Equicontinuity and the enveloping semigroup
16. The regionally proximal relation
Part V. Extensions of Minimal Flows:
17. Open and highly proximal extensions
Appendix. Extremely disconnected flows
18. Distal extensions of minimal flows
19. Almost periodic extensions
20. A tale of four theorems.
Publication planned for: June 2014
available from June 2014
format: Hardback
isbn: 9781107035904
From the basics to the forefront of modern research, this book presents all aspects of probability theory, statistics and data analysis from a Bayesian perspective for physicists and engineers. The book presents the roots, applications and numerical implementation of probability theory, and covers advanced topics such as maximum entropy distributions, stochastic processes, parameter estimation, model selection, hypothesis testing and experimental design. In addition, it explores state-of-the art numerical techniques required to solve demanding real-world problems. The book is ideal for students and researchers in physical sciences and engineering.
Preface
Part I. Introduction:
2. The meaning of probability
3. Basic definitions
4. Bayesian inference
5. Combinatrics
6. Random walks
7. Limit theorems
8. Continuous distributions
9. The central limit theorem
10. Poisson processes and waiting times
Part II. Assigning Probabilities:
11. Transformation invariance
12. Maximum entropy
13. Qualified maximum entropy
14. Global smoothness
Part III. Parameter Estimation:
15. Bayesian parameter estimation
16. Frequentist parameter estimation
17. The Cramer-Rao inequality
Part IV. Testing Hypotheses:
18. The Bayesian way
19. The frequentist way
20. Sampling distributions
21. Bayesian vs frequentist hypothesis tests
Part V. Real World Applications:
22. Regression
23. Inconsistent data
24. Unrecognized signal contributions
25. Change point problems
26. Function estimation
27. Integral equations
28. Model selection
29. Bayesian experimental design
Part VI. Probabilistic Numerical Techniques:
30. Numerical integration
31. Monte Carlo methods
Nested sampling
Appendixes
References
Index.
Part of Encyclopedia of Mathematics and its Applications
Publication planned for: July 2014
available from July 2014
format: Hardback
isbn: 9781107043060
This first systematic account of the basic theory of normed algebras, without assuming associativity, includes many new and unpublished results and is sure to become a central resource for researchers and graduate students in the field. This first volume focuses on the non-associative generalizations of (associative) C*-algebras provided by the so-called non-associative Gelfand?Naimark and Vidav?Palmer theorems, which give rise to alternative C*-algebras and non-commutative JB*-algebras, respectively. The relationship between non-commutative JB*-algebras and JB*-triples is also fully discussed. The second volume covers Zel'manov's celebrated work in Jordan theory to derive classification theorems for non-commutative JB*-algebras and JB*-triples, as well as other topics. The book interweaves pure algebra, geometry of normed spaces, and complex analysis, and includes a wealth of historical comments, background material, examples and exercises. The authors also provide an extensive bibliography.
Preface
1. Foundations
2. Beginning the proof of the non-associative Vidav?Palmer theorem
3. Concluding the proof of the non-associative Vidav?Palmer theorem
4. Jordan spectral theory
References
Symbol index
Subject index.