Speakers/Abstracts

Minicourses

• Alberto Cattaneo: Perturbative quantum gauge theories on manifolds with boundary
  
  Classical and quantum field theories may be thought of as appropriate functors from (some version of) the cobordism category. In these lectures (based on joint work with Mnev and Reshetikhin) after a short introduction to the BV formalism for quantum field theories, I will introduce the BV-BFV formalism for theories on manifolds with boundary and its quantum version. I will then apply it to the example of BF theories. In particular, I will show how the procedure is compatible with gluing along boundary components. An outlook to the general case will be presented, time permitting.
• Marco Manetti: Homotopy types of DG-Lie algebras and deformation theory
  
  Lecture 1. Homotopy theory in the category of DG-Lie algebras, homotopy fibers and totalization, transfer of homotopic abelianity, formal DG-Lie algebras, Massey powers and formality criteria (without proofs). Examples. Lecture 2. Maurer-Cartan and deformation functors associated to a DG-Lie algebra, tangent spaces and obstructions, gauge vs homotopy equivalence, homotopy invariance of deformation functors. Examples. Lecture 3. L-infinity algebras, homotopy transfer, minimal models, fibrations and weak equivalences, Maurer-Cartan and deformation functors, formality criteria. Examples.

Seminars

• Ruggero Bandiera: Higher Deligne groupoids
  
  We define the higher Deligne groupoid of a pronilpotent L_∞ algebra L as the simplicial set of Maurer-Cartan cochains on the standard cosimplicial simplex with coefficients in L, and show that this is equivalent to a previous construction by Getzler. We introduce a left adjoint functor from simplicial sets to pronilpotent DG Lie algebras and discuss some potential applications to rational homotopy theory. Finally, we show that the higher Deligne groupoid functor commutes with homotopy limits and recover as a consequence Hinich's theorem on descent of Deligne groupoids.
• Francesco Bonechi: Multiplicative integrable models from Poisson-Nijenhuis structures
  
  The symplectic groupoid is a canonical symplectic manifold that can be associated to every integrable Poisson manifold. We discuss in this talk integrable models on the symplectic groupoid that are compatible with groupoid multiplication. The main motivation of their definition lies in the problem of quantizing the underlying Poisson manifold. After reviewing definition and motivation, we present a procedure to define such models out of a maximal rank Poisson Nijenhuis structure on the Poisson manifold. We will discuss in detail the case of the Gelfand-Cetlin system on the Grassmannians as a tool for quantizing a pencil of SU(n) covariant Poisson structures.
• Gennaro Di Brino: Model Categories and DG D-algebras
  
  In this talk, we summarize the main points of two recent papers in the framework of an ongoing joint project with D. Pištalo and N. Poncin, eventually aiming at a derived approach for the BRST-BV formalism in the algebro-geometric setting. In particular, we will outline how to induce a projective model structure on dgDa, the commutative monoid objects in the category ch_≥0D of non-negatively graded chain complexes of D-modules. Next, we will show how the description of a functorial cofibrant replacement functor for such a structure yields a model categorical version of the classical Koszul-Tate resolution.
• Chiara Esposito: Quantization of Poisson-Hamiltonian systems
  
  In this talk we will introduce the concept of Hamiltonian system in the canonical and Poisson settings and we discuss a new reduction procedure for the Poisson setting. We will discuss the quantization of the Hamiltonian systems in the Poisson context, using formal deformation quantization and propose a non-formal approach of the Poisson-Hamiltonian spaces for triangular Poisson Lie groups.
• Donatella Iacono: Deformations of pairs (X,D)
  
  We study infinitesimal deformations of pairs (X,D), where D is a smooth divisor in a smooth projective variety X, using differential graded Lie algebras, the Cartan homotopy construction and the differential Batalin-Vilkovisky algebras. In particular, we are able to prove unobstructedness of deformations of the pair (X,D) in some cases.
• Niels Kowalzig: Higher Structures on Modules over Operads
  
  We show under what additional ingredients a comp (or opposite) module over an operad with multiplication can be given the structure of a cyclic module and how the underlying simplicial homology gives rise to a Batalin-Vilkovisky module over the cohomology of the operad. In particular, one obtains a generalised Lie derivative and a generalised (cyclic) cap product that obey a Cartan-Rinehart homotopy formula, and hence yield the structure of a noncommutative differential calculus in the sense of Nest, Tamarkin, Tsygan, and others. Examples include the calculi known for the Hochschild theory of associative algebras, for Poisson structures, but above all the calculus for general Hopf algebroids with respect to general coefficients (in which the classical calculus of vector fields and differential forms is contained); that is, a calculus structure on the pair of Ext- and Tor-groups over generalised bialgebras over noncommutative rings. Time permitting, we will also discuss Batalin-Vilkovisky algebras and their relationship to Cotor-groups.
• Elena Martinengo: Singularities of moduli spaces of sheaves on K3 surfaces
  
  In the eighties Mukai proved that the singularities of the moduli space of sheaves on a K3 surface are contained in the locus of strictly semistable sheaves, that is not empty just if the polarization is non generic or if the Mukai vector is non primitive. In the first case, Kaledin, Lehn and Sorger conjectured that the dg-algebra that controls deformations of sheaves on a K3 is formal. This would give a complete description of the singularities of the moduli space. The conjecture was proved in some cases by Kaledin-Lehn and Zhang. The tecniques they used are similar and they consist in pulling back the sheaves on the K3 to the twistor family and to apply Kaledin's theorem of formality in families. In a work in progress with Manfred Lehn we aim to complete the proof of the conjecture. We proved the conjecture of the remaining case and we are trying to extend our ad hoc construction to a general proof.
• Pavel Mnev: Cellular BV-BFV-BF theory
  
  We will introduce the cellular version of BF theory and explain how it fits into symplectic cohomological ("BF-BFV") quantization programme. Partition functions are given by finite-dimensional integrals, satisfy Segal-like gluing property, are invariant with respect to cellular aggregations (which play the role of Wilson's renormalization flow) and satisfy BV quantum master equation modified by a boundary term. Partition functions can be expressed in terms of torsions and the data of rational homotopy type; they also contain a mod 16 phase - a model for the eta invariant appearing in the phase of Chern-Simons partition function. This is a report on joint work with A. S. Cattaneo and N. Reshetikhin.
• Ryszard Nest: Perturbation vectors and torsion
  
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• Vincent Schlegel: Manifolds in infinitesimal homotopy theory
  
  I will discuss manifolds in the context of Urs Schreiber's differential cohesion for infinity-topoi, a powerful abstract framework combining the techniques of synthetic differential geometry and homotopy theory. I will discuss a recent result on gluing manifolds with corners in this context, and interpret it in terms of classical field theories presented by moduli stacks.
• Paul Stapor: Convergence of the Gutt star product
  
  Every Poisson structure on a finite-dimensional space admits formal deformations. In particular, if the Poisson tensor is linear, such system is a Lie algebra. We can endow such a Poisson algebra with the Gutt star product and the so deformed Poisson algebra is known to be isomorphic to the universal enveloping algebra of the deformed Lie algebra. Therefore, it carries a natural Hopf structure. In this work, we start from the more general case of a locally convex Lie algebra and get again a deformed Poisson Hopf algebra with the Gutt star product. This paper gives a sufficient condition on the Lie bracket to construct an explicit locally convex topology on it such that the Gutt star product becomes convergent. We also show that it depends holomorphically on the formal parameter and that the locally convex Hopf structure is continuous.
• Mathieu Stienon: Infinite jets of exponential maps
  
  Exponential maps arise naturally in the contexts of Lie theory and connections on smooth manifolds. The infinite jets of these classical exponential maps are related to the Poincaré-Birkhoff-Witt isomorphism and the complete symbol of differential operators. We will explain how these maps can be extended to differential graded manifolds and how this problem leads naturally to an interesting class of L_∞-algebras and to Fedosov-type resolutions.
• Alessandro Valentino: Central extensions of mapping class groups from characteristic classes
  
  I will discuss a functorial construction of extensions of mapping class groups of smooth manifolds which are induced by extensions of (higher) diffeomorphisms groups via the group stack of automorphisms of manifolds equipped with higher degree topological structures. The problem of constructing such extensions arises naturally in the study of topological quantum field theories, in particular in 3d Chern-Simons theory. Joint work with Domenico Fiorenza and Urs Schreiber.
• Yannick Voglaire: Invariant connections and PBW theorem for Lie groupoid pairs
  
  Given a closed wide Lie subgroupoid A of a Lie groupoid L, i.e. a Lie groupoid pair, I will introduce an associated Atiyah class and explain how it is the obstruction to the existence of fibrewise affine connections on the homogeneous space L/A. For Lie groupoid pairs with vanishing Atiyah class, I will show that the left A-action on the quotient space L/A can be linearized, I will outline an alternative proof of a result of Calaque about the Poincaré-Birkhoff-Witt map for such objects, and explain an interpretation of the Molino class of foliations suggested by this approach. This is joint work with Camille Laurent-Gengoux.
• Sinan Yalin: Derived stacks of algebraic structures
  
  I will start by explaining how various bialgebra structures can be parametrised by props, which generalize operads and their algebras. Resolutions of such props define algebraic structures up to homotopy, which occur in various contexts in topology and geometry. A highly non trivial result shows that such a definition does not depend, up to homotopy, on the choice of a resolution. A relevant idea to understand the behaviour of these structures on a given object is to organize them as a moduli space. I will thus define a notion of simplicial moduli space of algebraic structures, then explain how these moduli spaces fit in the setting of derived algebraic geometry in the sense of Toen-Vezzosi.
• Aissa Wade: On generalized contact bundles
  
  Generalized contact bundles are natural extensions of contact structures. They also can be viewed as odd-dimensional analogues of generalized complex structures. One of their apparent advantages is that they incorporate both coorientable and non-coorientable contact structures. A fundamental fact is that there is always a Jacobi structure underlying any given generalized contact bundle. After describing multiplicative Atiyah forms on Lie groupoids, I will give a complete characterization of generalized contact bundles having a non-degenerate associated Jacobi structure. I will then explain aspects of generalized contact bundle from the point of view of Lie algebroids and the Lie groupoids. This is a joint work with Luca Vitagliano.