Igor Nikolaev Poradie kníh

This book covers the basics of noncommutative geometry (NCG) and its applications in topology, algebraic geometry, and number theory. The author takes up the practical side of NCG and its value for other areas of mathematics. A brief survey of the main parts of NCG with historical remarks, bibliography, and a list of exercises is included. The presentation is intended for graduate students and researchers with interests in NCG, but will also serve nonexperts in the field. ContentsPart I: BasicsModel examplesCategories and functorsC∗-algebrasPart II: Noncommutative invariantsTopologyAlgebraic geometryNumber theoryPart III: Brief survey of NCGFinite geometriesContinuous geometriesConnes geometriesIndex theoryJones polynomialsQuantum groupsNoncommutative algebraic geometryTrends in noncommutative geometry

Focusing on the intersection of geometric topology and operator algebras, this monograph introduces innovative functors that connect topological spaces to operator algebras, aiding in the resolution of complex problems in topology. The first chapter presents new invariants for three-dimensional manifolds derived from operator algebras. The second chapter explores the relationship between homology invariants and K-theory of Cuntz-Krieger algebras. Finally, the third chapter addresses the Harvey conjecture for the mapping class group through noncommutative geometry techniques.

This book presents a comprehensive, encyclopedic approach to the subject of foliations, one of the major concepts of modern geometry and topology. It addresses graduate students and researchers and serves as a reference book for experts in the field.