"Brave New" algebraic geometry and global derived moduli spaces of ring spectra

Contributo su invito. Risultati originali non pubblicati precedentemente. ABSTRACT- We develop homotopical algebraic geometry ([To-Ve 1, To-Ve 2]) in the special context where the base symmetric monoidal model category is that of spectra S, i.e. what might be called, after Waldhausen, brave new algebraic geometry. We discuss various model topologies on the model category of commutative algebras in S, and their associated theories of geometric S-stacks (a geometric S-stack being an analog of Artin notion of algebraic stack in Algebraic Geometry). Two examples of geometric S-stacks are given: a global moduli space of associative ring spectrum structures, and the stack of elliptic curves endowed with the sheaf of topological modular forms.