Hardback (ISBN-13: 9780521514378)
Most scholars think of David Hilbert's program as the most demanding and ideologically motivated attempt to provide a foundation for mathematics, and because they see technical obstacles in the way of realizing the program's goals, they regard it as a failure. Against this view, Curtis Franks argues that Hilbert's deepest and most central insight was that mathematical techniques and practices do not need grounding in any philosophical principles. He weaves together an original historical account, philosophical analysis, and his own development of the meta-mathematics of weak systems of arithmetic to show that the true philosophical significance of Hilbert's program is that it makes the autonomy of mathematics evident. The result is a vision of the early history of modern logic that highlights the rich interaction between its conceptual problems and technical development.
* Argues forcefully against the ‘received view’ of Hilbert’s program
presenting it in a new light * Will appeal to historians, mathematicians
and philosophers * Presents complicated mathematics in an accessible way
Preface; 1. A new science; 2. David Hilbert's naturalism; 3. Arithmetization; 4. Intensionality; 5. Interpreting G2 for Q; 6. Autonomy in context; Bibliography; Index.
‘Franks' book is remarkable for the clarity of the prose and the originality of the arguments. It is a beautiful introduction to Hilbert's thought on the nature of mathematics and a natural extension of his work.’ Jeffrey Barrett, University of California, Irvine
Series: London Mathematical Society Lecture Note Series (No. 358)
Paperback (ISBN-13: 9780521757249)
Geometric group theory is a vibrant subject at the heart of modern mathematics. It is currently enjoying a period of rapid growth and great influence marked by a deepening of its fertile interactions with logic, analysis and large-scale geometry, and striking progress has been made on classical problems at the heart of cohomological group theory. This volume provides the reader with a tour through a selection of the most important trends in the field, including limit groups, quasi-isometric rigidity, non-positive curvature in group theory, and L2-methods in geometry, topology and group theory. Major survey articles exploring recent developments in the field are supported by shorter research papers, which are written in a style that readers approaching the field for the first time will find inviting.
* An ideal introduction to the field for non-experts * Survey articles
capture the most recent research in this vibrant area of modern mathematics
* Designed for graduate students and researchers in pure mathematics, particularly
geometry, topology and group theory
Preface; List of participants; 1. Notes on Sela's work: limit groups and Makanin-Razborov diagrams M. Bestvina and M. Feighn; 2. Solutions to Bestvina & Feighn's exercises on limit groups H. Wilton; 3. L2-Invariants from the algebraic point of view W. Luck; 4. Constructing non-positively curved spaces and groups J. McCammond; 5. Homology and dynamics in quasi-isometric rigidity of once-punctured mapping class groups L. Mosher; 6. Hattori-Stallings trace and Euler characteristics for groups I. Chatterji and G. Mislin; 7. Groups of small homological dimension and the Atiyah conjecture P. H. Kropholler, P. Linnell and W. Luck; 8. Logarithms and assembly maps on Kn(Zl[G]) V. P. Snaith; 9. On complete resolutions O. Talelli; 10. Structure theory for branch groups J. S. Wilson.
Hardback (ISBN-13: 9780521896191)
This up-to-date account of algebraic statistics and information geometry explores the emerging connections between the two disciplines, demonstrating how they can be used in design of experiments and how they benefit our understanding of statistical models, in particular, exponential models. This book presents a new way of approaching classical statistical problems and raises scientific questions that would never have been considered without the interaction of these two disciplines. Beginning with a brief introduction to each area, using simple illustrative examples, the book then proceeds with a collection of reviews and some new results written by leading researchers in their respective fields. Part III dwells in both classical and quantum information geometry, containing surveys of key results and new material. Finally, Part IV provides examples of the interplay between algebraic statistics and information geometry. Computer code and proofs are also available online, where key examples are developed in further detail.
* Chapters written by leading researchers in the field * Includes introductory
and review chapters, and a glossary of terms from algebraic geometry *
Online material develops in detail some key examples, and provides computer
code, technical material and detailed proofs
List of contributors; Frequently used notations and symbols; Preface; 1.
Algebraic and geometric methods in statistics P. Gibilisco, E. Riccomagno,
M. P. Rogantin and H. P. Wynn; Part I. Contingency Tables: 2. Maximum likelihood
estimation in latent class models S. E. Fienberg, P. Hersh, A. Rinaldo
and Y. Zhou; 3. Algebraic geometry of 2 x 2 contingency tables A. Slavkovic
and S. E. Fienberg; 4. Model selection for contingency tables with algebraic
statistics A. Krampe and S. Kuhnt; 5. Markov chains, quotient ideals, and
connectivity Y. Chen, I. Dinwoodie and R. Yoshida; 6. Algebraic category
distinguishability E. Carlini and F. Rapallo; 7. Algebraic complexity of
MLE for bivariate missing data S. Ho*ten and S. Sullivant; 8. The generalized
shuttle algorithm A. Dobra and S. E. Fienberg; Part II. Designed Experiments:
9. Generalised design H. Maruri-Aguilar and H. P. Wynn; 10. Design of experiments
and biochemical network inference R. Laubenbacher and B. Stigler; 11. Replicated
measurements and algebraic statistics R. Notari and E. Riccomagno; 12.
Indicator function and sudoku designs R. Fontana and M. P. Rogantin; 13.
Markov basis for design of experiments and three-level factors S. Aoki
and A. Takemura; Part III. Information Geometry: 14. Non-parametric estimation
R. F. Streater; 15. Banach manifold of quantum states R. F. Streater; 16.
On quantum information manifolds A. Jen*ova; 17. Axiomatic geometries for
text documents G. Lebanon; 18. Exponential manifold by reproducing kernel
Hilbert spaces K. Fukumizu; 19. Extended exponential models D. Imparato
and B. Trivellato; 20. Quantum statistics and measures of quantum information
F. Hansen; Part IV. Information Geometry and Algebraic Statistics: 21.
Algebraic varieties vs differentiable manifolds G. Pistone; Part V. On-Line
Supplements: Coloured Figures for Chapter 2; 22. Maximum likelihood estimation
in latent class models Y. Zhou; 23. The generalized shuttle algorithm A.
Dobra and S. E. Fienberg; 24. Indicator function and sudoku designs R.
Fontana and M. P. Rogantin; 25. Replicated measurements and algebraic statistics
R. Notari and E. Riccomagno; 26. Extended exponential models D. Imparato
and B. Trivellato.
Series: Cambridge Studies in Advanced Mathematics (No. 118)
Hardback (ISBN-13: 9780521194525)
The theory of random matrices plays an important role in many areas of pure mathematics and employs a variety of sophisticated mathematical tools (analytical, probabilistic and combinatorial). This diverse array of tools, while attesting to the vitality of the field, presents several formidable obstacles to the newcomer, and even the expert probabilist. This rigorous introduction to the basic theory is sufficiently self-contained to be accessible to graduate students in mathematics or related sciences, who have mastered probability theory at the graduate level, but have not necessarily been exposed to advanced notions of functional analysis, algebra or geometry. Useful background material is collected in the appendices and exercises are also included throughout to test the reader’s understanding. Enumerative techniques, stochastic analysis, large deviations, concentration inequalities, disintegration and Lie algebras all are introduced in the text, which will enable readers to approach the research literature with confidence.
* A rigorous yet accessible introduction for beginners at the graduate
level * Contains over 70 exercises so the reader can test their understanding
of the material * A valuable text not only for mathematicians, but also
physicists and electrical engineers
Preface; 1. Introduction; 2. Real and complex Wigner matrices; 3. Hermite polynomials, spacings, and limit distributions for the Gaussian ensembles; 4. Some generalities; 5. Free probability; Appendices; Bibliography; General conventions; Glossary; Index.
Series: Cambridge Tracts in Mathematics (No. 180)
Hardback (ISBN-13: 9780521116732)
This elegant book is sure to become the standard introduction to synthetic differential geometry. It deals with some classical spaces in differential geometry, namely ‘prolongation spaces’ or neighbourhoods of the diagonal. These spaces enable a natural description of some of the basic constructions in local differential geometry and, in fact, form an inviting gateway to differential geometry, and also to some differential-geometric notions that exist in algebraic geometry. The presentation conveys the real strength of this approach to differential geometry. Concepts are clarified, proofs are streamlined, and the focus on infinitesimal spaces motivates the discussion well. Some of the specific differential-geometric theories dealt with are connection theory (notably affine connections), geometric distributions, differential forms, jet bundles, differentiable groupoids, differential operators, Riemannian metrics, and harmonic maps. Ideal for graduate students and researchers wishing to familiarize themselves with the field.
* Authored by one of the originators of synthetic geometry * A research
monograph that can also be used as an invitation to differential geometry
* Suitable for any mathematician interested in differential geometry
Preface; 1. Calculus and linear algebra; 2. Geometry of the neighbour relation; 3. Combinatorial differential forms; 4. The tangent bundle; 5. Groupoids; 6. Lie theory; non-abelian covariant derivative; 7. Jets and differential operators; 8. Metric notions; Appendix; Bibliography