1 Introduction 1
2 Differential Geometry 7
2.1 Differential manifolds 7
2.2 Metrics and connections 23
2.3 Cohomology 38
3 Matrix Geometry 45
3.1 Differential forms Ⅰ 46
3.2 Differential forms Ⅱ 65
3.3 Tensor products 74
3.4 Metrics 77
3.5 Yang-Mills connections 87
3.6 Linear connections 97
3.7 Curvature 109
4 Noncommutative Geometry 118
4.1 General algebras 118
4.2 Poisson structures 149
4.3 Topological algebras 155
4.4 Quantum groups 191
5 Vector Bundles 213
5.1 K-theory 213
5.2 A matrix analogue 228
5.3 Fredholm modules 231
6 Cyclic Homology 241
6.1 The universal calculus 241
6.2 Cyclic homology 250
6.3 Morita equivalence 255
6.4 The Loday-Quillen theorem 257
7 Modiflcations of Space-Time 260
7.1 Noncommutative space-time 261
7.2 A finite model 270
7.3 Fuzzy physics 280
8 Extensions of Space-Time 313
8.1 The spinning particle 313
8.2 Noncommutative electrodynamics 317
8.3 Modified Kaluza-Klein theory 325