We continue on: ---The index problem for elliptic PDE's, characteristic classes via Chern-Weil theory, ---Miraculous cancellations, Getzler's supersymmetric proof of the Atiyah-Singer index theorem, special cases: Gauss-Bonnet-Chern, Hirzebruch signature theorem, and Riemann-Roch, --- Approach via path integrals and quantum mechanics.

The talk is concerned with regularity of weak solutions to second order infinitely degenerate elliptic equations. It is known that regularity of weak solutions can be studied by studying properties of certain metric spaces associated to the operator, namely, subunit metric spaces. The problem arising in the infinitely degenerate case is that the measures of subunit balls are non doubling. As a consequence many classical tools such as Sobolev-type inequalities become unavailable. We show that in certain cases a weaker version of Sobolev inequality can be established which allows to perform Moser iterations to obtain boundedness and continuity of weak solutions.

I shall briefly present robotics, with images. I shall in particular explain the notions of serial architecture and parallel architecture. Then I shall introduce the central topic of my talk: singularities of parallel robots. I intend to explain why roboticians studying the kinematics of parallel robots are specially interested in cusps. I shall also consider some examples: asymptotic singularities of planar parallel robots, rationality of the set of singular configurations of a Stewart platform.