Derived algebraic geometry over [special characters omitted]-rings

We develop a theory of less commutative algebraic geometry where the role of commutative rings is assumed by [special characters omitted]-rings, that is, rings with multiplication parametrized by configuration spaces of points in [special characters omitted]. As n increases, these theories converge to the derived algebraic geometry of Toën-Vezzosi and Lurie. The class of spaces obtained by gluing [special characters omitted]-rings form a geometric counterpart to [special characters omitted]-categories, which are higher topological variants of braided monoidal categories. These spaces further provide a geometric language for the deformation theory of general [special characters omitted] structures. A version of the cotangent complex governs such deformation theories, and we relate its values to [special characters omitted]-Hochschild cohomology. In the affine case, this establishes a claim made by Kontsevich. Other applications include a geometric description of higher Drinfeld centers of [special characters omitted]-categories, explored in work with Ben-Zvi and Nadler. (Copies available exclusively from MIT Libraries, Rm. 14-0551, Cambridge, MA 02139-4307. Ph. 617-253-5668; Fax 617-253-1690.)