The Local Index Theorem

This article is based on author's talk at the International Conference "Alexandroff Reading", Moscow 21 - 25 May, 2012. The material presented in article is a programme intended to organise the ingredients of the index formula. The first results results obtained in this project were announced at the International Conference on Non-commutative Geometry, Trieste, November 2007. Progress obtained along the path of the project was reported at different conferences in Crakovia (June 2011), "K-Theory, C*-Algebras and Index Theory International Conference", Goettingen (November 2010) and Iasi (September 2011). The unifying idea behind our program is to localise K-theory and the non-commutative geometry basic tools (Hochschild, cyclic homology and co-homology, Connes-Karoubi Chern character) along the lines of Alexander-Spanier co-homology and homology. The motivation for the realisation of this programme is four-fold: -1) the classical Atiyah-Singer type index formula is a global statement with local controle, -2) the non-localised existing tools are not fine enough to capture sufficient information in the case of Banach algebras, -3) one wants to make so that the Alexander-Spanier (co)-homology becomes a natural ingredient of non-commutative geometry and -4) the Alexander-Spanier co-homology, with respect to the existing non-commutative geometry tools, has the advantage that it does need extra regularity beyond the ordinary topology. Author's last reference publications represent parts of this programme