Stevo Todorcevic - Böcker

Ramsey theory is a fast-growing area of combinatorics with deep connections to other fields of mathematics such as topological dynamics, ergodic theory, mathematical logic, and algebra. The area of Ramsey theory dealing with Ramsey-type phenomena in higher dimensions is particularly useful. Introduction to Ramsey Spaces presents in a systematic way a method for building higher-dimensional Ramsey spaces from basic one-dimensional principles. It is the first book-length treatment of this area of Ramsey theory, and emphasizes applications for related and surrounding fields of mathematics, such as set theory, combinatorics, real and functional analysis, and topology. In order to facilitate accessibility, the book gives the method in its axiomatic form with examples that cover many important parts of Ramsey theory both finite and infinite. An exciting new direction for combinatorics, this book will interest graduate students and researchers working in mathematical subdisciplines requiring the mastery and practice of high-dimensional Ramsey theory.

This compilation of papers presented at the 2000 European Summer Meeting of the Association for Symbolic Logic marks the centenial anniversery of Hilbert's famous lecture. Held in the same hall at La Sorbonne where Hilbert first presented his famous problems, this meeting carries special significance to the Mathematics and Logic communities. The presentations include tutorials and research articles from some of the world's preeminent logicians. Three long articles are based on tutorials given at the meeting, and present accessible expositions of devloping research in three active areas of logic: model theory, computability, and set theory. The eleven subsequent articles cover seperate research topics in all areas of mathematical logic, including: aspects in Computer Science, Proof Theory, Set Theory, Model Theory, Computability Theory, and aspects of Philosophy.

This compilation of papers presented at the 2000 European Summer Meeting of the Association for Symbolic Logic marks the centenial anniversery of Hilbert's famous lecture. Held in the same hall at La Sorbonne where Hilbert first presented his famous problems, this meeting carries special significance to the Mathematics and Logic communities. The presentations include tutorials and research articles from some of the world's preeminent logicians. Three long articles are based on tutorials given at the meeting, and present accessible expositions of devloping research in three active areas of logic: model theory, computability, and set theory. The eleven subsequent articles cover seperate research topics in all areas of mathematical logic, including: aspects in Computer Science, Proof Theory, Set Theory, Model Theory, Computability Theory, and aspects of Philosophy.

Describing some interactions of topology with other areas of mathematics, this text assumes only a basic knowledge of the subject. The first chapter deals with the topology of pointwise convergence and proves results of Bourgain, Fremlin, Talagrand and Rosenthal on compact sets of Baire class-1 functions. In the second chapter some topological dynamics of beta-N and its applications to combinatorial number theory are presented. The third chapter gives a proof of the Ivanovskii-Kuzminov-Vilenkin theorem that compact groups are dyadic. The last chapter presents Marjanovic's classification of hyper-spaces of compact metric zerodimensional spaces.

The solution of the distortion problem for the Hilbert space, the unconditional basic sequence problem for Banach spaces, and the Banach homogeneous space problem are samples of the most important recent advances where Ramsey methods are used. The main goal of this book is to expose the general principles and methods that lie hidden behind and are most likely useful for further developments.

This is a collection of articles on set theory written by some of the participants in theResearchProgrammeonSetTheoryanditsApplicationsthattookplaceatthe Centre de Recerca Matem' atica (CRM) in Bellaterra (Barcelona). The Programme run from September 2003 to July 2004 and included an international conference on set theory in September 2003, an advanced course on Ramsey methods in ? analysis in January 2004, and a joint CRM-ICREA workshop on the foundations of set theory in June 2004, the latter held in Barcelona. A total of 33 short and long term visitors from 15 countries participated in the Programme. This volume consists of two parts, the ?rst containing survey papers on some of the mainstream areas of set theory, and the second containing original research papers. All of them are authored by visitors who took part in the set theory Programme or by participants in the Programme's activities. The survey papers cover topics as Omega-logic, applications of set theory to lattice theory and Boolean algebras, real-valued measurable cardinals, complexity of sets and relations in continuum theory, weak subsystems of axiomatic set t- ory, de?nable versions of large cardinals, and selection theory for open covers of topological spaces.As for the research papers, they range from topics such as the number of near-coherence classes of ultra?lters, the consistency strength of bounded forcing axioms,P (?) combinatorics,someapplicationsof morasses,subgroupsofAbelian ? Polish groups, adding club subsets of ? with ?nite conditions, the consistency 2 strength of mutual stationarity, and new axioms of set theory.

The analysis of the characteristics of walks on ordinals is a powerful new technique for building mathematical structures, developed by the author over the last twenty years. This is the first book-length exposition of this method. Particular emphasis is placed on applications which are presented in a unified and comprehensive manner and which stretch across several areas of mathematics such as set theory, combinatorics, general topology, functional analysis, and general algebra. The intended audience for this book are graduate students and researchers working in these areas interested in mastering and applying these methods.