| Abstract. I will describe briefly how to do some geometry on commutative differential graded algebras, on simplicial algebras and on commutative ring spectra. Here the geometry has to take the intrinsic homotopical flavor of such objects into account. Then I will describe some applications of this 'homotopical geometry' to problems raised outside its framework such as the problem of constructing a universal elliptic cohomology theory and the so-called geometric Langlands correspondence. At the end I will sketchily discuss some open problems and current directions in this field. |