We shall explain how to give substance to an old dream of Tits, to invent exotic new zeta functions, and discover the skeleton of algebraic varieties (toric manifolds and tropical geometry).
We shall explain how to give substance to an old dream of Tits, to invent exotic new zeta functions, and discover the skeleton of algebraic varieties (toric manifolds and tropical geometry).
We shall report on the use of algebraic geometry for the calculation of Feynman amplitudes (work of Bloch, Brown, Esnault and Kreimer). Or how to combine Grothendieck's motives with high energy physics in an unexpected way, radically distinct from string theory.
We construct spaces of manifolds of various dimensions following
Vassiliev's approach to the theory of knots. These are infinite-dimensional
spaces with hypersurface, corresponding to manifolds with Morse singularities.
Connected components of the complement to this discriminant are homotopy
equivalent to the covering spaces of BDiff(M). These spaces appear to be a
natural base over which one can consider parametrised versions of Floer and
Seiberg-Witten theories.