A Proof of Tsygan's Formality Conjecture for an Arbitrary Smooth   Manifold
  Proofs of Tsygan's formality conjectures for chains would unlock important algebraic tools which might lead to new generalizations of the Atiyah-Patodi-Singer index theorem and the Riemann-Roch-Hirzebruch theorem. Despite this pivotal role in the traditional investigations and the efforts of various people the most general version of Tsygan's formality conjecture has not yet been proven. In my thesis I propose Fedosov resolutions for the Hochschild cohomological and homological complexes of the algebra of functions on an arbitrary smooth manifold. Using these resolutions together with Kontsevich's formality quasi-isomorphism for Hochschild cochains of R[[y_1, >..., y_d]] and Shoikhet's formality quasi-isomorphism for Hochschild chains of R[[y_1,..., y_d]] I prove Tsygan's formality conjecture for Hochschild chains of the algebra of functions on an arbitrary smooth manifold. The construction of the formality quasi-isomorphism for Hochschild chains is manifestly functorial for isomorphisms of the pairs (M,\nabla), where M is the manifold and \nabla is an affine connection on the tangent bundle. In my thesis I apply these results to equivariant quantization, computation of Hochschild homology of quantum algebras and description of traces in deformation quantization. 
Vasiliy A. Dolgushev