**Title:** Cartan calculus in derived and noncommutative geometry. III

**Speaker:** Nick Rozenblyum

**Speaker Info:** University of Chicago

**Brief Description:**

**Special Note**:

**Abstract:**

Classical Cartan calculus concerns the action of vector fields on differential forms of a smooth manifold via Lie derivative and contraction. The key result is the Cartan magic formula which expresses the relation between the two actions.In the algebro-geometric setting, I will describe an interpretation of the Cartan calculus as Griffiths transversality for the Gauss-Manin connection on the universal deformation space. This naturally generalizes to the setting of derived algebraic geometry and also to Tamarkin-Tsygan calculus in noncommutative geometry which involves the action of Hoschschild cohomology on Hochschild homology of a DG category.

I will explain a very general theorem relating derived loop spaces and connections, which is used to give a precise relationship between the two. Moreover, I will describe some applications of these results to shifted symplectic structures on moduli spaces in topology and representation theory.

This is joint work with Christopher Brav.

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