• Morita equivalences, moduli spaces and flag varieties. Double Bruhat cells in a complex semisimple Lie group G emerged as a crucial concept in the work of S. Fomin and A. Zelevinsky on total positivity and cluster algebras. Double Bruhat cells are special instances of a broader class of cluster varieties known as generalized double Bruhat cells. These can be studied collectively as Poisson subvarieties of a family of Poisson groupoids which is are products of G and a power of the flag variety of G. 

In this work, we describe such Poisson groupoids as decorated moduli spaces of flat G-bundles over a disc. As a consequence, we integrate them, we show that these integrations are Morita equivalent and we relate our integration with the work of P. Boalch on meromorphic connections. 

• Symplectic double groupoids and the generalized Kähler potential (with Marco Gualtieri and Yucong Jiang).  A description of the fundamental degrees of freedom underlying a generalized Kähler manifold, which separates its holomorphic moduli from the space of compatible metrics in a similar way to the Kähler case, has been sought since its discovery in 1984.  In this paper, we describe a full solution to this problem for arbitrary generalized Kähler manifolds, which involves the new concept of a holomorphic symplectic Morita 2-equivalence between double symplectic groupoids,  equipped with a Lagrangian bisection of its real symplectic core.  We demonstrate the theory by constructing explicitly the above Morita 2-equivalence and Lagrangian bisection for the well-known generalized Kähler structures on compact even-dimensional semisimple Lie groups, which have until now escaped such analysis. We construct the required holomorphic symplectic manifolds by expressing them as moduli spaces of flat connections on surfaces with decorated boundary, through a quasi-Hamiltonian reduction.

• Homological vector fields over differentiable stacks (with Miquel Cueca). In this work we solve the problem of providing a Morita invariant definition of Lie and Courant algebroids over Lie groupoids. By relying on supergeometry, we view these structures as instances of vector fields on graded groupoids which are homological up to homotopy. We describe such vector fields in general from two complementary viewpoints: firstly, as Maurer-Cartan elements in a differential graded Lie algebra of multivector fields and, secondly, we also view them from a categorical approach, in terms of functors and natural transformations. Thereby, we obtain a unifying conceptual framework for studying many examples which include flat 2-connections on 2-bundles and cotangents of quasi-Poisson groupoids.

• Transitive Courant algebroids and double symplectic groupoids [Int. Math. Res. Not. IMRN  (2024), no. 9, 7526-7551] We classify exact Courant algebroids over Lie groupoids and we use that result to study transitive Courant algebroids and Dirac structures inside them. As an application we extend the Lu-Weinstein construction of double symplectic groupoids to any Lie bialgebroid whose associated Courant algebroid is transitive and with integrable underlying Atiyah algebroid.

• Poisson groupoids and moduli spaces of flat bundles over surfaces [Adv. Math. 440 (2024), Paper no. 109523] Let G be a Lie group equipped with a bi-invariant pseudo-Riemannian metric. We show that the decorated moduli space of flat G-bundles over a compact and oriented surface with nonempty boundary possesses a compatible Poisson groupoid structure, provided that the decoration is symmetric with respect to a decomposition of the surface as the double of another surface with nonempty boundary.

• Reduction of symplectic groupoids and quotients of quasi-Poisson manifolds [Transform. Groups, Vol. 28, no. 4, 1357-1374, (2023)] In this work we study the integrability of quotients of quasi-Poisson manifolds. Our approach allows us to put several classical results about the integrability of Poisson quotients in a common framework. By categorifying one of the already known methods of reducing symplectic groupoids we also describe double symplectic groupoids which integrate the recently introduced Poisson groupoid structures on gauge groupoids.

• Complete Lie algebroid actions and the integrability of Lie algebroids [Proc. Amer. Math. Soc., 149 (2021), 4923-4930] We give a new proof of the equivalence between the existence of a complete action of a Lie algebroid on a surjective submersion and its integrability. The main tools in our approach are double Lie groupoids and multiplicative foliations, our proof relies on a simple characterization of those vacant double Lie groupoids which induce a Lie groupoid structure on their orbit spaces.

• Integrability of quotients in Poisson and Dirac geometry [Pacific J. Math., 311-1 (2021), pp. 1-32] In this work we show that several results about the integrability of quotient Poisson (and Dirac) structures can be deduced from an elementary characterization of Morita fibrations. As new applications of this observation, we integrate (1) Dirac homogeneous spaces of Dirac-Lie groups and (2) a class of Poisson homogeneous spaces of symplectic groupoids which are in duality with respect to the classical Poisson homogeneous spaces of Drinfeld.

• Leaves of stacky Lie algebroids [C. R. Math. Acad. Sci. Paris, Volume 358 (2020) no. 2, pp. 217-226] In this paper we show that the leaves of an LA-groupoid which pass through the unit manifold are -up to a connectedness issue- Lie groupoids themselves. This implies, in particular, that the symplectic leaves of a Poisson groupoid provide whole families of symplectic groupoids. For instance, the coadjoint orbits in the dual of a Lie 2-algebra are symplectic groupoids with their KKS symplectic forms.

• Integration of action Courant algebroids (with Marco Gualtieri)

• Shifted lagrangian structures in Poisson geometry (with Henrique Bursztyn and Miquel Cueca)

• Poisson manifolds of complex reductive type (with Maarten Mol)