Deformations and cohomology in non-commutative derived geometry.

My research project is at the crossroads of non-commutative geometry (in the sense of Kontsevich, Van den Bergh, . . . ) and homotopical derived geometry (in the sense of Toën, . . . ). An important inspiration is the fact [6] that a smooth proper scheme is equivalent in the derived sense to a differential graded (dg) algebra [28], and smoothness and properness boil down to properties of this dg algebra. Hence, dg algebras become models of "noncommutative schemes" [37], [60]. This approach has proven useful in topics ranging from deformation quantization to homological mirror symmetry. In this spirit, we study dg algebras [28], their twins, A1-algebras [27], stacks, and in particular deformations and Hochschild cohomology of these objects.

Keywords:COHOMOLOGY, GEOMETRY

Disciplines:Algebra, Geometry