Categorification in Algebraic GeometryANR-17-CE40-0014 |
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 technical complications as well as un- reachable constructions. This has led several authors to develop new foundations for the whole subject. These foundations are based on techniques from homotopical algebra and higher category theory making the subject extremely flexible and therefore extremely rich in examples. In this project, we plan to use derived algebraic geometry to find hidden functoriality properties and also to define Gromov-Witten invariants in non-archimedian geometry. Moreover we plan to categorify Donaldson-Thomas invariants and study deformation quantization. |