Foreword to Part 1 This part of the book emphasizes the use of analytic tools for extracting geometric information out of foliated spaces. The first chapter describes the construction, due to A. Connes, of the C∗-algebra of a foliated space. Also introduced here is the fundamentally important concept of the graph of a foliated space. Our treatment of these topics is only introductory, being intended as an aid to readers who would like to begin studying some of the vast literature on this subject. The second chapter is devoted to the harmonic measures of L. Garnett. These measures provide a universally applicable tool for studying ergodic- theoretic problems on foliated spaces. In the third chapter, we treat topological and geometric applications, featuring a beautiful theorem of E. Ghys (Theorem 3.1.4) concerning the ends of almost all leaves. The first two chapters require significant analytic background, namely C∗-algebras, diffusion on manifolds and Brownian motion. While there are numerous texts treating these topics, the authors feel that the reader will be well served by a concise survey of each of the theories in the form of three appendices. These appendices are intended to provide a guided tour of these theories and are no substitute for the extensive literature to which they refer. 3
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