Categorification in algebraic geometry – CatAG

Derived algebraic geometry goes back to intersection theory and particularly to the famous Serre's intersection formula introduced in the 50'. This formula express an intersection number as an alternating sum of dimensions of the higher Tor's of the structure sheaves of two algebraic sub-varieties. In the early 90', Kontsevich has pushed the story one step further by introducing the notion of quasi-manifolds and virtual fundamental classes in his treatment of the moduli space of stable maps for then purpose of enumerative geometry of curves. The theory of quasi-manifolds and virtual classes have evolved independently in the late 90'. On the one side Kapranov and Ciocan-Fontanine introduced the notion of dg-schemes as a formalization of the notion of
quasi-manifold. On the other side virtual classes has been defined in great generality by Behrend and Fantechi based on the notion of obstruction theories. Both of these notions have a serious drawback: the lack of functoriality, which in practice implies un-necessary technical complications as well as un- reachable constructions. This has led several authors to develop new foundations for the whole subject. The modern foundations of what is now called "derived algebraic geometry" have been developed during the last decade by Toën-Vezzosi (HAG1, HAG2), and later on by Lurie. They introduced the notion of derived Artin n-stacks, a far reaching generalization of the notion of algebraic stacks in the sense of Artin, higher algebraic stacks in the sense of Simpson, and of dg-schemes in the sense of Kapranov and Ciocan-Fontanine. These foundations are based on techniques from homotopical algebra and higher category theory making the subject extremely flexible and therefore extremely rich in examples. Derived algebraic geometry has been developing fast during the last few years, with the works of various mathematicians: Pantev, Toën, Vaquié, Vezzosi, Lurie, Francis, Gaitsgory, Rozenblyum, Preygel, Ben-Zvi, Nadler, Brav, Bussi, Joyce, Costello, Ginot, Calaque, Bhatt, Schuerg. Thanks to these recent developments, derived algebraic geometry is today a central subject, with very solid foundations as well as rich interactions with various subjects in mathematics, as for instance: singularity theory (matrix factorizations), symplectic geometry (shifted or derived symplectic structures, moduli of objects in Fukaya categories), quantization by deformation (formality and higher formality theorems via derived algebraic geometry), moduli theory (moduli space of objects in a Calabi-Yau 3-fold), enumerative geometry (Donaldson-Thomas and Gromov-Witten invariants), \etc

The main objective of the present proposal is to bring together mathematicians with international recognition whose research domains are related to categorification's problems. We divide them in two main themes : Gromov-Witten theory and quantization. In Gromov-Witten theory, we plan to refund the subject using derived algebraic geometry to find hidden functorialty properties and also to define Gromov-Witten invariants in non-archimedian geometry. On the quantization side, we plan to categorify Donaldson-Thomas invariants and study deformation quantization.