Higher structures and deformations in representation theory

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The research proposal is focusing on higher algebraic structures arsing from Hochschild, and related, cohomology and on higher categorical structures such as A-infinity categories. I aim to develop new and advanced techniques applicable to a wide range of problems in algebra, representation theory, topology, and geometry. Over the past few years, I have been investigating higher algebraic structures arising from singular Hochschild cohomology. I have shown that singular Hochschild cohomology is endowed with the same rich structure as classical Hochschild cohomology: a Gerstenhaber algebra in cohomology and a B-infinity algebra at the complex level. Suggested by this result, Keller proved that singular Hochschild cohomology is isomorphic to the Hochschild cohomology of the dg singularity category. He also conjectured that this isomorphism lifts to a quasi-isomorphism of B-infinity algebras. The first objective in my research proposal is to prove Keller's conjecture stated above. To achieve this, jointly with X. -W. Chen we introduced an explicit dg enhancement of singularities categories. In this proposal, we plan to construct a natural action of the B-infinity structure on singular Hochschild cohomology on this dg enhancement, which will yield a B-infinity quasi-isomorphism lifting Keller's isomorphism. The second objective is to explore further applications of singular Hochschild cohomology in topology and geometry. Jointly with M. Rivera we have shown that there is a rich higher algebraic structure on the singular Hochschild cohomology of any dg Frobenius algebra. In this proposal, we plan to show that this higher structure actually comes from a natural action of the Sullivan dg PROP, originally used to model operations in string topology. This proposal also concerns (A-infinity) deformations of (graded) algebras and their applications to representation theory. Recently jointly with S. Barmeier we have developed a combinatorial method to study the deformation theory of a path algebra of any finite quiver with relations, which allows us to easily compute particular classes of examples. The third objective aims to understand the behaviour of singularity categories under deformation quantisation, motivated by Keller's conjecture. Particularly, we expect an equivalence between singularity categories of a cyclic quotient singularity and of the deformation quantisation of a natural Poisson structure on the singularity. The fourth objective is to describe the A-infinity deformations of graded gentle algebras up to derived equivalence, in terms of their surface models. In particular, we expect that some A-infinity deformations correspond to doing surgery on the surface models.The fifth objective is to use our combinatorial method to study and verify Stroppel's conjecture for extended Khovanov arc algebras, which yields the intrinsic formality of these algebras.