Abstract:
This is a report on recent progress concerning the interactions between derived algebraic geometry and deformation quantization. We present the notion of derived algebraic stacks, of shifted symplectic and Poisson structures, as well as the construction of deformation quantization of shifted Poisson structures. As an application we propose a general construction of the quantization of the moduli space of $G$-bundles on an oriented space of arbitrary dimension.

Abstract:
This is the integral text of my thesis. The first part is an expanded version of "Riemann-Roch theorems for Deligne-Mumford stacks", where I deal with Artin stacks over general bases. In the second part, I prove some Riemann-Roch statment for D-modules on Deligne-Mumford stacks, and I also consider the problem of algebraization of analytic stacks.

Abstract:
For a (semi-)model category M, we define a notion of a ''homotopy'' Grothendieck topology on M, as well as its associated model category of stacks. We use this to define a notion of geometric stack over a symmetric monoidal base model category; geometric stacks are the fundamental objects to "do algebraic geometry over model categories". We give two examples of applications of this formalism. The first one is the interpretation of DG-schemes as geometric stacks over the model category of complexes and the second one is a definition of etale K-theory of E_{\infty}-ring spectra. This first version is very preliminary and might be considered as a detailed research announcement. Some proofs, more details and more examples will be added in a forthcoming version.

Abstract:
This is the first of a series of papers devoted to lay the foundations of Algebraic Geometry in homotopical and higher categorical contexts (for part II, see math.AG/0404373). In this first part we investigate a notion of higher topos. For this, we use S-categories (i.e. simplicially enriched categories) as models for certain kind of \infty-categories, and we develop the notions of S-topologies, S-sites and stacks over them. We prove in particular, that for an S-category T endowed with an S-topology, there exists a model category of stacks over T, generalizing the model category structure on simplicial presheaves over a Grothendieck site of A. Joyal and R. Jardine. We also prove some analogs of the relations between topologies and localizing subcategories of the categories of presheaves, by proving that there exists a one-to-one correspondence between S-topologies on an S-category T, and certain left exact Bousfield localizations of the model category of pre-stacks on T. Based on the above results, we study the notion of model topos introduced by C. Rezk, and we relate it to our model categories of stacks over S-sites. In the second part of the paper, we present a parallel theory where S-categories, S-topologies and S-sites are replaced by model categories, model topologies and model sites. We prove that a canonical way to pass from the theory of stacks over model sites to the theory of stacks over S-sites is provided by the simplicial localization construction of Dwyer and Kan. We also prove a Giraud's style theorem characterizing model topoi internally. As an example of application, we propose a definition of etale K-theory of ring spectra, extending the etale K-theory of commutative rings.

Abstract:
These are expanded notes from some talks given during the fall 2002, about ``homotopical algebraic geometry'' (HAG) with special emphasis on its applications to ``derived algebraic geometry'' (DAG) and ``derived deformation theory''. We use the general framework developed in Toen, Vezzosi, ``Homotopical Algebraic Geometry I: Topos theory'', and in particular the notions of model topology, model sites and stacks over them, in order to define various derived moduli functors and study their geometric properties. We start by defining the model category of D-stacks, with respect to an extension of the etale topology to the category of non-positively graded commutative differential algebras, and we show that its homotopy category contains interesting objects, such as schemes, algebraic stacks, higher algebraic stacks, dg-schemes ... . We define the notion of ``geometric D-stack'' and present some related geometric constructions ($\mathcal{O}$-modules, perfect complexes, K-theory, derived tangent stacks, cotangent complexes, various notion of smoothness ... .). Finally, we define and study the derived moduli problems classifying local systems on a topological space, vector bundles on a smooth projective variety, and $A_{\infty}$-categorical structures. We state geometricity and smoothness results for all of these examples.

Abstract:
It is now well known that the K-theory of a Waldhausen category depends on more than just its (triangulated) homotopy category (see [Schlichting]). The purpose of this note is to show that the K-theory spectrum of a (good) Waldhausen category is completely determined by its Dwyer-Kan simplicial localization, without any additional structure. As the simplicial localization is a refined version of the homotopy category which also determines the triangulated structure, our result is a possible answer to the general question: ``To which extent $K$-theory is not an invariant of triangulated derived categories ?''

Abstract:
In math.AG/0207028 we began the study of higher sheaf theory (i.e. stacks theory) on higher categories endowed with a suitable notion of topology: precisely, we defined the notions of S-site and of model site, and the associated categories of stacks on them. This led us to study a notion of \textit{model topos} (orginally due to C. Rezk), a model category version of the notion of Grothendieck topos. In this paper we treat the analogous theory starting from (1-)Segal categories in place of S-categories and model categories. We introduce notions of Segal topologies, Segal sites and stacks over them. We define an abstract notion of Segal topos and relate it with Segal categories of stacks over Segal sites. We compare the notions of Segal topoi and of model topoi, showing that the two theories are equivalent in some sense. However, the existence of a nice Segal category of morphisms between Segal categories allows us to improve the treatment of topoi in this context. In particular we construct the 2-Segal category of Segal topoi and geometric morphisms, and we provide a Giraud-like statement characterizing Segal topoi among Segal categories. As an example of applications, we show how to reconstruct a topological space up to homotopy from the Segal topos of locally constant stacks on it, thus extending the main theorem of Toen, "Vers une interpretation Galoisienne de la theorie de l'homotopie" (to appear in Cahiers de top. et geom. diff. cat.) to the case of un-based spaces. We also give some hints of how to define homotopy types of Segal sites: this approach gives a new point of view and some improvements on the \'etale homotopy theory of schemes, and more generally on the theory of homotopy types of Grothendieck sites as defined by Artin and Mazur.

Abstract:
We develop homotopical algebraic geometry (see math.AG/0207028) in the special context where the base symmetric monoidal model category is the category S of spectra, 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, the associated theories of geometric S-stacks (a geometric S-stack being an analog of Artin notion of algebraic stack in Algebraic Geometry), and finally show how to define global moduli spaces of associative ring spectra structures and a moduli space related to topological modular forms as geometric S-stacks.

Abstract:
We use techniques from relative algebraic geometry and homotopical algebraic geometry in order to construct several categories of schemes defined "under Spec Z". We define this way the categories of N-schemes, F_1-schemes, S-schemes, S_+-schemes, and S_1-schemes, where from a very intuitive point of view N is the semi-ring of natural numbers, F_1 is the field with one element, S is the sphere ring spectrum, S_+ is the semi-ring spectrum of natural numbers and S_1 is the ring spectrum with one element. These categories of schemes are related by several base change functors, and they all possess a base change functor to Z-schemes (in the usual sense). Finally, we show how the linear group Gl_n and toric varieties can be defined as objects in certain of these categories.

Abstract:
This is the second part of a series of papers devoted to develop Homotopical Algebraic Geometry. We start by defining and studying generalizations of standard notions of linear and commutative algebra in an abstract monoidal model category, such as derivations, etale and smooth maps, flat and projective modules, etc. We then use the theory of stacks over model categories introduced in \cite{hagI} in order to define a general notion of geometric stack over a base symmetric monoidal model category C, and prove that this notion satisfies the expected properties. The rest of the paper consists in specializing C to several different contexts. First of all, when C=k-Mod is the category of modules over a ring k, with the trivial model structure, we show that our notion gives back the algebraic n-stacks of C. Simpson. Then we set C=sk-Mod, the model category of simplicial k-modules, and obtain this way a notion of geometric derived stacks which are the main geometric objects of Derived Algebraic Geometry. We give several examples of derived version of classical moduli stacks, as for example the derived stack of local systems on a space, of algebra structures over an operad, of flat bundles on a projective complex manifold, etc. Finally, we present the cases where C=(k) is the model category of unbounded complexes of modules over a char 0 ring k, and C=Sp^{\Sigma} the model category of symmetric spectra. In these two contexts, called respectively Complicial and Brave New Algebraic Geometry, we give some examples of geometric stacks such as the stack of associative dg-algebras, the stack of dg-categories, and a geometric stack constructed using topological modular forms.