Survey of noncommutative geometry
The origin of Noncommutative Geometry is twofold. On the one hand there is a wealth of examples of spaces whose coordinate algebra is no longer commutative but which have obvious geometric meaning. The first examples came from phase space in quantum mechanics but there are many others, such as the leaf spaces of foliations, duals of nonabelian discrete groups, the space of Penrose tilings, the noncommutative torus which plays a role in M-theory compactification, and finally the space of Q-lattices which is a natural geometric space carrying an action of the analogue of the Frobenius for global fields of zero characteristic.
On the other hand the stretching of geometric thinking imposed by passing to noncommutative spaces forces one to rethink about most of our familiar notions. The difficulty is not to add arbitrarily the adjective quantum behind our familiar geometric language but to develop far reaching extensions of classical concepts. This has been achieved a long time ago by operator algebraists as far as measure theory is concerned. The theory of nonabelian von-Neumann algebras is indeed a far reaching extension of measure theory, whose main surprise is that such an algebra inherits from its noncommutativity a god-given time evolution.
The development of the topological ideas was prompted by the Novikov conjecture on homotopy invariance of higher signatures of ordinary manifolds as well as by the Atiyah-Singer Index Theorem. It has led to the recognition that not only the Atiyah-Hirzebruch K-theory but more importantly the dual K-homology admit Noncommutative Geometry as their natural framework. The cycles in K-homology are given by Fredholm representations of the C*-algebra A of continuous functions. A basic example is the group ring of a discrete group and restricting oneself to commutative algebras is an obviously undesirable assumption.
The development of differential geometric ideas, including de Rham homology, connections and curvature of vector bundles, took place during the eighties thanks to cyclic cohomology which led for instance to the proof of the Novikov conjecture for hyperbolic groups but got many other applications. Basically, by extending the Chern-Weil characteristic classes to the general framework it allows us for many concrete computations on noncommutative spaces.
The very notion of Noncommutative Geometry comes from the identification of the two basic concepts in Riemann's formulation of Geometry, namely those of manifold and of infinitesimal line element. It was recognized at an early stage that the formalism of quantum mechanics gives a natural place both to infinitesimals (the compact operators in Hilbert space) and to the integral (the logarithmic divergence in an operator trace). It was also recognized long ago by geometers that the main quality of the homotopy type of a manifold, (besides being defined by a cooking recipee) is to satisfy Poincare duality not only in ordinary homology but in K-homology.
In the general framework of Noncommutative Geometry the confluence of the two notions of metric and fundamental class for a manifold led very naturally to the equality ds=1/D which expresses the infinitesimal line element ds as the inverse of the Dirac operator D, hence under suitable boundary conditions as a propagator. The significance of D is two-fold. On the one hand it defines the metric by the above equation, on the other hand its homotopy class represents the K-homology fundamental class of the space under consideration.
We shall discuss three of the recent developments of Noncommutative Geometry. The first is the understanding of the noncommutative nature of spacetime from the symmetries of the Lagrangian of gravity coupled with matter. The starting point is that the natural symmetry group G of this Lagrangian is isomorphic to the group of diffeomorphisms of a space X, provided one stretches one's geometrical notions to allow slightly noncommutative spaces. The spectral action principle allows to recover the Lagrangian of gravity coupled with matter from the spectrum of the line element ds.
The second has to do with various appearances of Hopf algebras relevant to Quantum Field Theory which originated from my joint work with D.Kreimer and led recently in joint work with M.Marcolli to the discovery of the relation between renormalization and one of the most elaborate forms of Galois theory given in the Riemann-Hilbert correspondence and the theory of motives. A tantalising unexplained bare fact is the appearance in the universal singular frame eliminating the divergence of QFT of the same numerical coefficients as in the local index formula. The latter is the corner stone of the definition of curvature in noncommutative geometry.
The third is the spectral interpretation of the zeros of the Riemann zeta function from the action of the idele class group on the space of Q-lattices and of the explicit formulas of number theory as a trace formula of Lefschetz type.