## Periodic Reporting for period 1 - DefAlgS (Deformation theory of algebraic structures)

Reporting period: 2016-03-01 to 2017-10-31

"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."

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."

During the period covered by the report (01 Mar 2016 - 01 Sep 2016), the researcher wrote a preprint with Gregory Ginot which is available on arxiv and has been submitted for publication:

https://arxiv.org/abs/1606.01504

This preprint not only completely solve the tasks associated to the second goal of the project, proving longstanding (more than 20 years old) conjectures raised especially by Kontsevich in deformation quantization, but solved them at a greater level of generality than initially expected, and also provides a new conceptual framework to completely understand some (until now) mysterious aspects of algebraic deformation theory.

Briefly, we precisely relate the homotopy theory of bialgebras to the one of algebras over the little two-disks operad and prove that the deformation theory of a bialgebra is equivalent to the E-2 deformation theory of an appropriate cobar construction on it (in the sense of Lurie's approach to formal moduli problems). Using the higher Deligne conjecture, this equips the Gerstenhaber-Schack complex of a bialgebra with an E_3-algebra structure and solves a longstanding conjecture of Gerstenhaber-Schack (1990). In the case of a differential graded symmetric bialgebra, we prove that this complex is E_3-formal, solving a longstanding conjecture of Kontsevich (2000). Finally we apply these results to give a new proof of Etingof-Kazdhan deformation quantization of Lie bialgebras which works in a more general setting. Moreover, these important theorems are based on results of independent interest about formal moduli problems of algebraic structures giving a conceptual and concrete framework to understand their variants.

These results have been exposed both by the researcher and his coauthor in various places for seminars or conferences.

https://arxiv.org/abs/1606.01504

This preprint not only completely solve the tasks associated to the second goal of the project, proving longstanding (more than 20 years old) conjectures raised especially by Kontsevich in deformation quantization, but solved them at a greater level of generality than initially expected, and also provides a new conceptual framework to completely understand some (until now) mysterious aspects of algebraic deformation theory.

Briefly, we precisely relate the homotopy theory of bialgebras to the one of algebras over the little two-disks operad and prove that the deformation theory of a bialgebra is equivalent to the E-2 deformation theory of an appropriate cobar construction on it (in the sense of Lurie's approach to formal moduli problems). Using the higher Deligne conjecture, this equips the Gerstenhaber-Schack complex of a bialgebra with an E_3-algebra structure and solves a longstanding conjecture of Gerstenhaber-Schack (1990). In the case of a differential graded symmetric bialgebra, we prove that this complex is E_3-formal, solving a longstanding conjecture of Kontsevich (2000). Finally we apply these results to give a new proof of Etingof-Kazdhan deformation quantization of Lie bialgebras which works in a more general setting. Moreover, these important theorems are based on results of independent interest about formal moduli problems of algebraic structures giving a conceptual and concrete framework to understand their variants.

These results have been exposed both by the researcher and his coauthor in various places for seminars or conferences.

"Before the aforementionned preprint came out, the state of the art in the related topics was the following:

-Equivalence between deformation theories of bialgebras and E-n algebras unknown;

-precise relation between variants of deformation complexes of algebraic structures in the litterature and corresponding formal moduli problems unknown;

-Gerstenhaber-Schack conjecture unsolved;

-Kontsevich E-3 formality conjecture unsolved.

The work performed by the researcher and his coauthor answer all of these questions, taking a step far beyond the state of art in these topics.

The paper ""Deformation theory of bialgebras, higher Hochschild cohomology and formality"" (available on arXiv as arXiv:1606.01504) contains all these new results. Moreover, the researcher gave the following talks about these results:

-May 2017: same title, conference Lens Topology and geometry 2017, Lens.

-March 2017: Deformation theory of bialgebras via higher Hochschild cohomology, formality and quantization, Institut de Mathématiques de Toulouse.

-June 2016: same title, conference Higher structures in Geometry and Physics, MATRIX institute, Melbourne.

-March 2016: same title, LAGA, Paris 13.

-March 2016: same title, LAREMA, Angers.

-February 2016: same title, Centre for Symmetry and Deformation, Copenhagen University.

-December 2015: Moduli spaces of bialgebras, higher Hochschild cohomology and Formality, Université Paris-Diderot.

There will be also a Proceedings book edited by Springer (part of a new MATRIX book series) and covering the topics of the conference held at MATRIX institute, in which a survey article by the researcher about this work will appear. This survey ""Moduli spaces of (bi)algebra structures in topology and geometry"" is also available on arXiv as arXiv:1611.03662.

Moreover, the new methods developped in this work have other potential outcomes to more general problems in deformation quantization, factorization homology and derived algebraic geometry. The researcher and his coauthor are currently working on various projects based on such methods and aimed to solve several open problems in these fields."

-Equivalence between deformation theories of bialgebras and E-n algebras unknown;

-precise relation between variants of deformation complexes of algebraic structures in the litterature and corresponding formal moduli problems unknown;

-Gerstenhaber-Schack conjecture unsolved;

-Kontsevich E-3 formality conjecture unsolved.

The work performed by the researcher and his coauthor answer all of these questions, taking a step far beyond the state of art in these topics.

The paper ""Deformation theory of bialgebras, higher Hochschild cohomology and formality"" (available on arXiv as arXiv:1606.01504) contains all these new results. Moreover, the researcher gave the following talks about these results:

-May 2017: same title, conference Lens Topology and geometry 2017, Lens.

-March 2017: Deformation theory of bialgebras via higher Hochschild cohomology, formality and quantization, Institut de Mathématiques de Toulouse.

-June 2016: same title, conference Higher structures in Geometry and Physics, MATRIX institute, Melbourne.

-March 2016: same title, LAGA, Paris 13.

-March 2016: same title, LAREMA, Angers.

-February 2016: same title, Centre for Symmetry and Deformation, Copenhagen University.

-December 2015: Moduli spaces of bialgebras, higher Hochschild cohomology and Formality, Université Paris-Diderot.

There will be also a Proceedings book edited by Springer (part of a new MATRIX book series) and covering the topics of the conference held at MATRIX institute, in which a survey article by the researcher about this work will appear. This survey ""Moduli spaces of (bi)algebra structures in topology and geometry"" is also available on arXiv as arXiv:1611.03662.

Moreover, the new methods developped in this work have other potential outcomes to more general problems in deformation quantization, factorization homology and derived algebraic geometry. The researcher and his coauthor are currently working on various projects based on such methods and aimed to solve several open problems in these fields."