Introduction to Noncommutative Spaces and Their Geometries

These notes arose from a series of introductory seminars on noncommutative geometry I gave at the University of Trieste in September 1995 during the X WorkshoponDi?erentialGeometricMethodsinClassicalMechanics.Itwas Beppe Marmo s suggestion that I wrote notes for the lectures. The notes are mainly an introduction to Connes noncommutative geo- try. They could serve as a ?rst aid kit before one ventures into the beautiful but bewildering landscape of Connes theory. The main di?erence from other available introductions to Connes work, is the emphasis on noncommutative spaces seen as concrete spaces. Important examples of noncommutative spaces are provided by nonc- mutative lattices. The latter are the subject of intense work I am doing in collaboration with A.P. Balachandran, Giuseppe Bimonte, Elisa Ercolessi, Fedele Lizzi, Gianni Sparano and Paulo Teotonio-Sobrinho. These notes are also meant to be an introduction to this research. There is still a lot of work in progress and by no means can these notes be considered as a review of - erything we have achieved so far. Rather, I hope they will show the relevance and potentiality for physical theories of noncommutative lattices. Cambridge, October 1997. Preface to the Online Edition Since this monograph ?rst appeared in October 1997, there has been a host of important developments in the subject. My personal taste would select the following three lines of research but there is much more which I am not mentioning here for lack of space(-time). 1. Applications of noncommutative geometry to string theories.