Abstract：
Classical differential geometry is underpinned by the Cartan calculus. The structural essence of this calculus transcends the setting of manifolds, vector fields and differential forms.
This talk will describe the formalization of this structure as a Tamarkin–Tsygan calculus. We will begin with a brief review of the classical Cartan relations before transitioning to the abstract definition, which identifies a calculus as a module over a Gerstenhaber algebra equipped with a compatible differential.
We will then explore a primary motivation for this abstraction: the Hochschild theory of associative algebras. We will demonstrate how the Hochschild cohomology (acting as the "fields") and Hochschild homology (acting as the "forms") naturally assemble into a Tamarkin–Tsygan calculus.
To conclude, we will briefly discuss the deep connection between the classical Cartan calculus and the algebraic Hochschild calculus through the lens of Deformation Quantization. We will review the landmark formality theorems of Kontsevich (for cochains) and Tsygan-Shoikhet (for chains), which together establish that for any smooth manifold, the classical Cartan calculus and the Hochschild calculus are isomorphic.
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