Coline Emprin: Formality transmission

Abstract: Here are seemingly unrelated problems:

• Koszul duality for the category of a reductive group in representation theory,
• the existence of a K-contact non-Sasakian manifold in differential geometry,
• splitting Drinfeld space’s de Rham complex in the p-adic Langlands program,
• deformation quantization of Poisson manifolds in mathematical physics.

And yet, all of them boil down to the same question: formality. A differential graded algebraic structure A (e.g. an associative algebra, a Lie algebra, a pre-Calabi-Yau algebra, etc.) is formal if it is related to its homology H(A) by a zig-zag of quasi-isomorphisms preserving the algebraic structure.

The examples mentioned above rely on criteria that guarantee formality. These include formality descent, formality in families, intrinsic formality, domination techniques, the behavior of formality in fibrations, and many others. In this talk, I will prove that all these criteria arise as special cases of a single theorem: formality transmission. On the one hand, this theorem unifies the aforementioned results into a single theory; on the other hand, it generalizes these criteria in diverse directions, in particular over any coefficient ring and for algebraic structures with several outputs: algebras encoded by properads.