In the first half of the talk I shall present a new algebraic proof of a result of Deligne-Illusie about the degeneration of the Hodge-de Rham spectral sequence. The idea is to reduce the main technical point of their proof to a question about the formality of a derived intersection in an Azumaya space.

In the second half of the talk I shall discuss the main technical difficulty that arises when trying to extend these techniques to obtain a proof of a famous claim of Barannikov-Kontsevich. This claim, which was first proved by analytic methods by Sabbah, expresses the hypercohomology of the twisted de Rham complex in terms of computations with coherent sheaves. It is conceptually the analogue of the Hodge-de Rham degeneration statement for dg categories of matrix factorizations.

This is joint work with Dima Arinkin and Marton Hablicsek.