This is the first of a series of papers about quantization in the context of derived algebraic
geometry. In this first part, we introduce the notion of n-shifted symplectic structures (n …

We study the interaction between geometric operations on stacks and algebraic operations
on their categories of sheaves. We work in the general setting of derived algebraic …

Derived algebraic geometry Page 1 EMS Surv. Math. Sci. 1 (2014), 153–240 DOI 10.4171/EMSS/4
EMS Surveys in Mathematical Sciences c European Mathematical Society Derived algebraic …

We define and study a sheaf-theoretic cohomological Hall algebra for suitably geometric
Abelian categories $\mathcal {A} $ of homological dimension at most two, and a sheaf …

We characterize two objects by universal property: the derived de Rham complex and
Hochschild homology together with its Hochschild-Kostant-Rosenberg filtration. This …

We show that a Calabi–Yau structure of dimension d on a smooth dg category CC induces a
symplectic form of degree 2-d 2-d on 'the moduli space of objects' M _ C MC. We show …

We introduce and survey a Betti form of the geometric Langlands conjecture, parallel to the
de Rham form developed by Beilinson-Drinfeld and Arinkin-Gaitsgory, and the Dolbeault …

We prove a Darboux theorem for derived schemes with symplectic forms of degree $ k< 0$,
in the sense of Pantev, Toën, Vaquié, and Vezzosi. More precisely, we show that a derived …

For various 2-Calabi-Yau categories $\mathscr {C} $ for which the stack of objects
$\mathfrak {M} $ has a good moduli space $ p\colon\mathfrak {M}\rightarrow\mathcal {M} …

In this work we study the failure of the HKR theorem over rings of positive and mixed
characteristic. For this we construct a filtered circle interpolating between the usual …