DefAlgS · Deformation theory of algebraic structures

Deformation theory of algebraic structures

"The research project leading this action is divided in three goals, each one based on the development of new methods to solve important open problems lying at the crossroads of topology, geometry and mathematical physics. A first goal is to understand the structures underlying Poincaré duality on the cochains of manifolds by their geometric and homotopical properties. The deformation theory and classification up to a suitable notion of equivalence of such structures can be formalized by using an appopriate notion of moduli space of homotopy algebraic structures. The idea is to build such a moduli space by considering geometrically meaningful constructions of this cochain-level Poincaré duality, and to understand the homotopy type and the tangential properties of these moduli spaces. A main application is to solve a longstanding open problem of Sullivan about the classification of known geometric invariants up to the underlying homotopy type of the manifold. A second goal is to use new methods based on previous works of the researcher, combined with derived geometry and factorization homology/higher Hochschild cohomology, to solve longstanding open problems relating deformation theory of E-n algebras and quantum group theory. In particular, two important and related conjectures are one of Gerstenhaber-Schack about the kind of structure controlling the deformation theory of quantum groups (beginning of the 90s), and one of Kontsevich (in his work on deformation quantization of Poisson manifolds) about the deeper formality theorem underlying Etingof-Kazdhan's deformation quantization of Lie bialgebras. A third goal is to use the recent developments of deformation quantization in derived geometry to get new families of invariants for spaces of links and higher dimensional manifolds, by deformation quantization of the appropriate moduli spaces of algebraic structures. This will give a deformation quantization ""in families"" of structures naturally appearing at the chain level in string topology, and consequently quantum invariants of higher dimensional manifolds generalizing the known invariants of 3-manifolds coming from Turaev's work on TQFTs and algebras of loops on surfaces. Each of this three goals has important short-term and long-term outcomes in Mathematics, first by solving longstanding open problems in the concerned fields (topology, geometry, mathematical physics), second by proposing each time new methods that could be applied later to other related problems. During the period covered by the action (01 Mar 2016 - 01 Sep 2016), the researcher focused on the second goal in a work in collaboration with Gregory Ginot (now available as a preprint arXiv:1606.01504) . Briefly, they proved an equivalence between the appropriate homotopy theories of E-n algebras and bialgebras (the kind of structure underlying quantum groups) and solved the Gerstenhaber-Schack conjecture, stating that the deformation complex of a bialgebra is an E-3 algebra. In the case of the symmetric bialgebra, this complex is formal as an E-3 algebra, which on the one hand solves Kontsevich's formality conjecture about such complexes, and on the other hand allows to prove a generalized version of Etingof-Kazdhan deformation quantization of Lie bialgebras (including, in particular, a new proof of the original theorem). Moreover, the new methods used in this work provide along the way a conceptual explanation of some (until now) mysterious aspects of algebraic deformation theory."

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