### Speakers/Abstracts

### Seminars

- Serguei Barannikov : Batalin-Vilkovisky formalism on cyclic words and EA-matrix integrals

Quantum master equation on cyclic cochains (QMECC) appears naturally in various counting problems, like counting of higher genus curves with boundaries on lagrangian submanifolds. In the lowest degree it defines cyclic A-infinity algebras. Solutions to the QMECC equation define equivariantly closed differential forms on matrix spaces. The asymptotic expansions of the integrals of these forms in the Batalin-Vilkovisky formalism define families of cohomology classes of compactified moduli spaces of curves. In the simplest example of the associative algebra of one element, this family is the generating function of all products of psi-classes in the total cohomology of compactified moduli spaces of curves. Relevant papers: arxiv:1803.11549 / hal-00429963(2009) arxiv:0912.5484 / hal-00102085(2006) - Pierre Bieliavsky: Symmetric spaces, midpoints and strict deformation quantization

I shall explain an old conjecture by Alan Weinstein about the specific form of an invariant star-product on a Hermitian symmetric space in terms of the geometry of its geodesic triangles. I shall present a proof of this conjecture in the case of the hyperbolic plane. If time permits, I shall show how the solution of this conjecture leads to a notion of noncommutative Riemann surfaces that generalises the one of noncommutative torus in the context of operator algebras. - Alberto Cattaneo: Poisson sigma model by cut and paste

In this talk I will start recalling the classical theory of the Poisson sigma model (PSM) and its relation to the integration of Poisson manifolds. I will then describe its quantum theory and its relation to deformation quantization. Next I will describe the globalization issue and describe how it can be incorporated into the PSM via the BV formalism. Finally, I will present an application of the BV-BFV formalism: the quantization of the relational symplectic groupoid and the lift (up to homotopy) of deformation quantization to the path space. - Janusz Grabowski : Remarks on contact geometry

We present an approach to contact (and Jacobi) geometry that makes many facts, presented in the literature in an overcomplicated way, much more natural and clear. The key role is played by homogeneous symplectic (and Poisson) manifolds. The difference with the existing literature is that the homogeneity of the Poisson structure is related to a principal GL(1;R)-bundle structure on the manifold and not just to a vector field. This allows for working with nontrivial line bundles that drastically simplifies the picture. Contact manifolds of degree 2 and contact analogs of Courant algebroids are studied as well. Based on a joint work with A. J. Bruce and K. Grabowska. - Simone Gutt: $L_\infty$-formality question for the universal algebra of semisimple Lie algebras

We show that the Hochschild complex of the universal enveloping algebra of a non abelian reductive Lie algebra is not formal. In the case of $\mathfrak{so}(3)$, we show that adding one higher bracket of order $3$ restores a $L-\infty$-quasi-isomorphism. This is joint work with Martin Bordemann, Olivier Elchinger and Abdenacer Makhlouf. - Donatella Iacono: DG-Lie algebroids and deformations of log varieties

For every smooth algebraic manifold X and every effective divisor D on X, we associate a DG-Lie algebroid such that the DG-Lie algebra of derived global sections controls the deformation theory of the pair (X,D). Joint work in progress with Marco Manetti. - Niels Kowalzig: Higher brackets on cyclic and negative cyclic (co)homology

In this talk, we will embed the string topology bracket developed by Chas-Sullivan and Menichi on negative cyclic cohomology groups as well as the dual bracket found by de Thanhoffer de Voelcsey-Van den Bergh on negative cyclic homology groups into the global picture of a noncommutative differential (or Cartan) calculus up to homotopy on the (co)cyclic bicomplex in general, in case a certain Poincare' duality is given. For negative cyclic cohomology, this in particular leads to a Batalin-Vilkovisky algebra structure on the underlying Hochschild cohomology. In the special case in which this BV bracket vanishes, one obtains an e_3-algebra structure on Hochschild cohomology. The results are given in the general and unifying setting of (opposite) cyclic modules over (cyclic) operads. (joint work with D. Fiorenza) - Camille Laurent-Gengoux: Quasi-Poisson Lie groupoids and shifted+1 Poisson structure

Joint work with F. Bonechi, N. Ciccoli, P. Xu. We show that the notion of quasi-Poisson structure on a Lie groupoid goes through Morita equivalence. We suggest it as a definition of shifted +1-Poisson structure on a differentiable stacks. - Ioan Marcut: A local model around Poisson submanifolds

The first jet of a Poisson tensor at a fixed-point encodes precisely the structure of a Lie algebra (i.e. the isotropy Lie algebra). The corresponding linear Poisson structure on the dual of the isotropy Lie algebra (i.e. the Kirillov-Kostant-Souriau Poisson structure) represents the first jet approximation of the Poisson tensor. Much more intricate is the semi-global version of this construction due to Yuri Vorobiev, which provides a first order approximation for a Poisson structure around a symplectic leaf, depending only on the first order jet at the leaf (encoded by a transitive Lie algebroid). In this talk I will explain a similar model for first order approximations of Poisson structures around Poisson submanifolds. This model generalizes Vorobiev's construction, it depends only on the first jet of the Poisson structure, it is unique up to isomorphisms, but does not always exist. I will also discuss an existence criterion. This is joint work with Rui Loja Fernandes. - Eva Miranda: Desingularizing Poisson and Jacobi structures

I will present a desingularization technique (joint work with Victor Guillemin and Jonathan Weitsman) for $b^m$-Poisson structures which fulfill good tranversality properties and describe some applications to the study of their geometry and Dynamics. This desingularization can be also extended to $b^m$-contact manifolds (joint work with Cédric Oms) which are odd dimensional Jacobi manifolds satisfying some transversality assumptions. This technique can be used to prove existence of $b^m$-contact structures and the Weinstein conjecture in this realm. - Ryszard Nest: On a Γ-equivariant index theorem

We will formulate and sketch a proof of a Γ-equivariant version of the algebraic index theorem, where Γ is a discrete group of automorphisms of a formal deformation of a symplectic manifold. The particular cases of this result are the algebraic version of the transversal index theorem related to the theorem of A. Connes and H. Moscovici for hypoelliptic operators and the index theorem for the extension of the algebra of pseudodifferential operators by a group of diffeomorphisms of the underlying manifold due to A. Savin, B. Sternin, E. Schrohe and D. Perrot. While this lecture concentrates on the algebraic version, these results can be used to prove the corresponding theorenms in the analytic context. This is joint work with Alexander Gorokhovsky and Niek de Kleijn. - Cristian Ortiz: Shifted symplectic structures on differentiable stacks

In this talk I will explain the notion of n-shifted symplectic structure on a stack presented by a Lie groupoid. After discussing the main examples, we will see how Dirac Geometry appears in connection with the description of certain Lagrangian structures in 1-shifted symplectic stacks. This is part of ongoing work joint with G. Ginot and D. Stefani. - Volodya Roubtsov: Lie algebroid cohomology and Lie algebroid extensions

We consider the extension problem for Lie algebroids over schemes over a field. Given a locally free Lie algebroid Q over a scheme (X,O), and a sheaf of finitely generated Lie O-algebras L, we determine the obstruction to the existence of extensions 0 --> L --> E --> Q --> 0, and classify the extensions in terms of a suitable Lie algebroid hypercohomology group. Based on joint work with Ettore Aldrovandi and Ugo Bruzzo (arXiv arXiv:1711.05156, to appear in J. of Alg.) - Pavol Severa: Quantization of Poisson-Lie groups and a little bit beyond

There is a rather simple description of (all) Hopf algebras in terms of braids. It is closely related to the idea of Hopf algebra valued holonomies on surfaces. A similar description works also for Hopf algebroids with a commutative base. As an application we get an explicit quantization of Poisson Lie groups (or of Poisson Hopf algebras) to Hopf algebras, and also a quantization of Poisson groupoids with a commutative base to Hopf algebroids. The construction works also for quantization of suitable Poisson structures on higher Lie groupoids, but the full understanding of these Poisson structures and of the corresponding quantum objects is lacking. - Georgy Sharygin: Deformation quantization of Lie algebra actions and commutative families

As one knows, every Poisson manifold M admits a deformation quantization, i.e. an associative *-product on the space of formal power series in a variable \hbar with coefficients in the algebra of smooth functions on M, coherent with the Poisson bracket on M. In my talk I will address the question, whether for a Poisson commutative subalgebra S in C^\infty(M) there exists an extension of S to a commutative subalgebra with respect to the *-product. This question (sometimes referred to as the quantum integrability) is closely related with the problem of finding an extension of a Lie algebra action on M to its action (by derivatives) on the deformed algebra. Both questions are in a great measure still open and depend on the structure of the singularities of the corresponding foliation. In my talk I will try to describe the few known facts about these problems, and give some concrete examples of these constructions. - Jan Slovák: BGG calculus on symplectic manifolds

As observed in several examples, symplectic manifolds enjoy certain complexes of invariant differential operators reminding rather the de Rham complexes from higher dimensions. Following our work with Michael Eastwood, I will explain how is this phenomenon linked to conformally Fedosov structures and their suitable contactizations. This brings the analogy of the algebraic construction of the Bernstein-Gelfand-Gelfand complexes on partial flag manifolds into the play. I shall also comment on further development, in particular the recent series of papers by A. Cap and T. Salac. (joint work with M.G. Eastwood). - Stefan Waldmann: Coisotropic Triples and Reduction

In this talk I will report on a joint project with Chiara Esposito and Marvin Dippell on an algebraic and noncommutative version of coisotropic reduction: the main example comes from deformation quantization of coisotropic submanifolds and their (quantum) reduction. The framework allows for a conceptual discussion of Morita equivalences before and after reduction and has a good classical limit. - Ping Xu: Formality theorem for dg manifolds

The Atiyah class of a dg manifold (M,Q) is the obstruction to the existence of an affine connection that is compatible with the homological vector field Q. The Todd class of dg manifolds extends both the classical Todd class of complex manifolds and the Duflo element of Lie theory. We establish a formality theorem for smooth dg manifolds: given any finite-dimensional dg manifold (M,Q), there exists an L_oo quasi-isomorphism of the dgla of polyvector fields on (M,Q) to the dgla of polydifferential operators on (M,Q), whose first Taylor coefficient is the composition of the action of the square root of the Todd class with the Hochschild-Kostant-Rosenberg map. We also we prove the Kontsevich-Shoikhet conjecture: a Kontsevich-Duflo type theorem holds for all finite-dimensional smooth dg manifolds. This is a joint work with Hsuan-Yi Liao and Mathieu Stienon. - Marco Zambon: Singular subalgebroids

We introduce singular subalgebroids of a Lie algebroid A. Two extreme examples are Lie subalgebroids (this is the constant rank case), and singular foliations (the case where A is the tangent bundle). Upon fixing a Lie groupoid G integrating the Lie algebroid A, one can "integrate" a singular subalgebroid canonically to a topological groupoid, which comes witha morphism to G and which in a sense is "minimal". This is in analogy with - and encompasses - the integration of Lie subalgebroids by Moerdijk-Mrcun as well as the holonomy groupoids of Androulidakis-Skandalis. Explicit examples are given, and we discuss the functoriality of the construction.