Notes on Noncommutative Geometry

Noncommutative geometry has roots in and is a synthesis of a number of diverse areas of mathematics, including: • Hilbert space and single operator theory; • Operator algebras (C*-algebras and von Neumann algebras); • Spin geometry – Dirac operators – index theory; • Algebraic topology – homological algebra. It has certainly also been inspired by quantum mechanics, and, besides feedback to the above areas that it comes from, noncommutative geometry has applications to (at least): 1. Foliation theory; 2. Number theory – arithmetic algebraic geometry; 3. Deformation theory – quantization theory; 4. Quantum field theory – renormalization; 5. Elementary particle physics – Standard Model; 6. Solid state physics – Quantum Hall effect; In this sense, as a general mathematical formalism with such a wide range of deep applications to both mathematics and physics, noncommutative geometry may be compared with Newton's calculus. The interaction between the above areas plays an important role in noncommutative geometry, especially the unexpected use of tools from algebraic topology (like K-theory) and homological algebra (like Hochschild (co)homology) in the context of operator algebras and more general complex associative algebras. But the reverse direction, where operator techniques are e.g. used to redevelop and generalize spin geometry and index theory, is at least as fruitful and is arguably even more unexpected. The history of noncommutative geometry goes back to John von Neumann's work on the mathematical structure of quantum physics, as presented in his book Mathematische Grundlagen der Quantenmechanik (Springer, 1932), and his subsequent invention of the theory of operator algebras (written down is a series of papers published between 1936 and 1949, partly with his assistant F.J. Murray). Other events of great importance to noncommutative geometry were the definition of C*-algebras and the first results in this area by Gelfand and Naimark in 1943, and the development of index theory by Atiyah and Singer from 1968 onwards. Connes himself brought the " introverted " period in the history of operator algebras to a close with his magnificent classification of injective factors in 1976, and subsequently opened up the field by relating it to foliated manifolds and index theory. This led to a series of papers by Connes in the period 1979-1985 that launched noncommutative geometry as a new area of mathematics. An important feature of this area was and is the interplay between abstract theory and examples; what makes it difficult to enter the field is that …