Infinite-dimensional differential geometry is often viewed as a fairly arcane subject with little connection to geometric questions arising in (finite-dimensional) applications. The aim of this talk is to show that this impression could not be further from the truth. We will take a scenic tour to a multitude of examples, connecting finite, infinite-dimensional and higher geometry. While some of these are well known classics such as Euler-Arnold theory for partial differential equations, also new results with surprising applications (such as in rough path integration theory) will be presented. As this talk is intended as a gentle introduction to these topics, no prior knowledge of infinite-dimensional geometry will be necessary.

• Thematic program: Fellows of the Institut CNRS Pauli (fellows 2021/2022)
• Event: Workshop on "A finite and infinite-dimensional meeting on Lie groupoids, Poisson geometry and integrability" (2021)

Diffeological groupoids appear in many areas of mathematics, such as infinite-dimensional Lie theory, classical field theory, deformation theory, and moduli spaces. The category of diffeological spaces, however, is too general and does not have a good differential calculus, which would be needed for a Lie theory of diffeological groupoids. I will introduce the notion of elastic diffeological spaces and show that these form a subcategory with an abstract tangent structure in the sense of Rosicky. The tangent structure yields a Cartan calculus consisting of vector fields, differential forms, the de Rham differential, inner derivatives, and Lie derivatives, satisfying the usual relations. Surprisingly, all diffeological groups are elastic. I then introduce the notion of diffeological Lie algebroids and show that the invariant vector fields of an elastic diffeological groupoid form a diffeological Lie algebroid. As application, I will revisit a diffeological groupoid that arises in lorentzian geometry whose diffeological Lie algebroid encodes the Poisson brackets of the Gauss-Codazzi constraint functions.

• Thematic program: Fellows of the Institut CNRS Pauli (fellows 2021/2022)
• Event: Workshop on "A finite and infinite-dimensional meeting on Lie groupoids, Poisson geometry and integrability" (2021)

A Jacobi structure is a Lie bracket on the sections of a line bundle. These brackets encode time-dependent mechanics in the same way Poisson brackets encode mechanics. Contact groupoids are finite-dimensional models for the "integrations" of these infinite-dimensional Lie algebras. In this talk, we explain how, under a certain compactness hypothesis, one can adapt the argument of Gray-Moser to these multiplicative contact structures and point out some applications.

• Thematic program: Fellows of the Institut CNRS Pauli (fellows 2021/2022)
• Event: Workshop on "A finite and infinite-dimensional meeting on Lie groupoids, Poisson geometry and integrability" (2021)

In this talk we outline how one can use the Nash-Moser method to prove Poisson linearization results of compact semisimple Lie algebras. We use Conn's idea to prove a more general linearization result.

• Thematic program: Fellows of the Institut CNRS Pauli (fellows 2021/2022)
• Event: Workshop on "A finite and infinite-dimensional meeting on Lie groupoids, Poisson geometry and integrability" (2021)

Local normal form theorems in differential geometry are often the manifestation of rigidity of the structure in normal form. For example, the existence of local Darboux coordinates in symplectic geometry follows from the fact that, locally, the standard symplectic structure has no deformations. After introducing closed pseudogroups and their associated sheaf of Lie algebras, I will discuss a general local rigidity result for solutions to PDE’s under the action of a closed pseudogroup of symmetries. The result is of the form: “infinitesimal tame rigidity” implies “tame rigidity”; it is in the smooth setting, and the proof uses the Nash-Moser fast convergence method. Several classical theorems fit in our setting: e.g. the Newlander-Nirenberg theorem in complex geometry, Conn’s theorem in Poisson geometry. This is a joint work with Roy Wang.

• Thematic program: Fellows of the Institut CNRS Pauli (fellows 2021/2022)
• Event: Workshop on "A finite and infinite-dimensional meeting on Lie groupoids, Poisson geometry and integrability" (2021)

see external webpage

• Thematic program: Fellows of the Institut CNRS Pauli (fellows 2021/2022)
• Event: Workshop on "A finite and infinite-dimensional meeting on Lie groupoids, Poisson geometry and integrability" (2021)

We plan to discuss certain genuine Poisson geometrical structures that arise in the theory of operator algebras on Hilbert spaces. Lecture 1 should be a gentle introduction to the basic notions on operator algebras that are needed later, with emphasis on the so-called standard form of von Neumann algebras that goes back to the PhD thesis of of U. Haagerup (1973). In Lecture 2, the focus is on the Poisson bracket carried by the predual of any von Neumann algebra, which turns out to admit smooth symplectic leaves, just as in the case of finite-dimensional Poisson manifolds. This lecture is partly based on joint work with T.S. Ratiu (2005). Finally, in Lecture 3, the geometric structures underlying the standard representations are pointed out, thereby presenting infinite-dimensional versions of presymplectic groupoids. This lecture is based on joint work with A. Odzijewicz (2019).

• Thematic program: Fellows of the Institut CNRS Pauli (fellows 2021/2022)
• Event: Workshop on "A finite and infinite-dimensional meeting on Lie groupoids, Poisson geometry and integrability" (2021)

see external homepage

• Thematic program: Fellows of the Institut CNRS Pauli (fellows 2021/2022)
• Event: Workshop on "A finite and infinite-dimensional meeting on Lie groupoids, Poisson geometry and integrability" (2021)

Analytic structure on a manifold (adapted to a specific analytic atlas) is a special type of G-structure of infinite order. I will report on work in progress that aims to answer the following questions: What is an almost analytic structure? What are obstructions to integrability? Does formal integrability imply integrability? What natural geometric objects define corresponding analytic structures?

• Thematic program: Fellows of the Institut CNRS Pauli (fellows 2021/2022)
• Event: Workshop on "A finite and infinite-dimensional meeting on Lie groupoids, Poisson geometry and integrability" (2021)

Quantizations of infinitesimal gauge symmetries are classified in terms of the continuous Lie algebra cohomology group of gauge algebras in degree 2. For gauge bundles with semisimple fibers, this space was calculated by Janssens-Wockel (2013), their method relying heavily on the low degree of the cohomology group. In this talk, we extend these results to homology in higher degree. To this end, we review some homological algebra for topological chain complexes and use it to lift the well-known Loday-Quillen-Tsygan-Theorem (1983, 1984) from a statement in algebraic Lie algebra homology to one that takes topological data into account. For globally trivial gauge algebras whose fibres are classical Lie algebras, this calculates a certain stable part of continuous homology. A similar description was given by Feigin (1988), but lacking a detailed proof. Finally, we use the results for trivial bundles to construct a Gelfand Fuks-like local-to-global spectral sequence from which homological information about nontrivial gauge algebras can be extracted. If time permits, we discuss obstructions to a full understanding of this spectral sequence. This talk is based on joint work with Bas Janssens.

• Thematic program: Fellows of the Institut CNRS Pauli (fellows 2021/2022)
• Event: Workshop on "A finite and infinite-dimensional meeting on Lie groupoids, Poisson geometry and integrability" (2021)

see external webpage

• Thematic program: Fellows of the Institut CNRS Pauli (fellows 2021/2022)
• Event: Workshop on "A finite and infinite-dimensional meeting on Lie groupoids, Poisson geometry and integrability" (2021)

see external webpage

• Thematic program: Fellows of the Institut CNRS Pauli (fellows 2021/2022)
• Event: Workshop on "A finite and infinite-dimensional meeting on Lie groupoids, Poisson geometry and integrability" (2021)

I will discuss different aspects of infinite-dimensional symplectic geometry. Why is it interesting and what are important applications? What are the common technical issues in the infinite-dimensional setting and how to overcome them? In particular, I will explain how the Marle-Guillemin-Sternberg local normal form and symplectic reduction work in infinite dimensions.

• Thematic program: Fellows of the Institut CNRS Pauli (fellows 2021/2022)
• Event: Workshop on "A finite and infinite-dimensional meeting on Lie groupoids, Poisson geometry and integrability" (2021)

We plan to discuss certain genuine Poisson geometrical structures that arise in the theory of operator algebras on Hilbert spaces. Lecture 1 should be a gentle introduction to the basic notions on operator algebras that are needed later, with emphasis on the so-called standard form of von Neumann algebras that goes back to the PhD thesis of of U. Haagerup (1973). In Lecture 2, the focus is on the Poisson bracket carried by the predual of any von Neumann algebra, which turns out to admit smooth symplectic leaves, just as in the case of finite-dimensional Poisson manifolds. This lecture is partly based on joint work with T.S. Ratiu (2005). Finally, in Lecture 3, the geometric structures underlying the standard representations are pointed out, thereby presenting infinite-dimensional versions of presymplectic groupoids. This lecture is based on joint work with A. Odzijewicz (2019).

• Thematic program: Fellows of the Institut CNRS Pauli (fellows 2021/2022)
• Event: Workshop on "A finite and infinite-dimensional meeting on Lie groupoids, Poisson geometry and integrability" (2021)

Diffeological groupoids appear in many areas of mathematics, such as infinite-dimensional Lie theory, classical field theory, deformation theory, and moduli spaces. The category of diffeological spaces, however, is too general and does not have a good differential calculus, which would be needed for a Lie theory of diffeological groupoids. I will introduce the notion of elastic diffeological spaces and show that these form a subcategory with an abstract tangent structure in the sense of Rosicky. The tangent structure yields a Cartan calculus consisting of vector fields, differential forms, the de Rham differential, inner derivatives, and Lie derivatives, satisfying the usual relations. Surprisingly, all diffeological groups are elastic. I then introduce the notion of diffeological Lie algebroids and show that the invariant vector fields of an elastic diffeological groupoid form a diffeological Lie algebroid. As application, I will revisit a diffeological groupoid that arises in lorentzian geometry whose diffeological Lie algebroid encodes the Poisson brackets of the Gauss-Codazzi constraint functions.

• Thematic program: Fellows of the Institut CNRS Pauli (fellows 2021/2022)
• Event: Workshop on "A finite and infinite-dimensional meeting on Lie groupoids, Poisson geometry and integrability" (2021)

see external webpage

• Thematic program: Fellows of the Institut CNRS Pauli (fellows 2021/2022)
• Event: Workshop on "A finite and infinite-dimensional meeting on Lie groupoids, Poisson geometry and integrability" (2021)

Diffeological groupoids appear in many areas of mathematics, such as infinite-dimensional Lie theory, classical field theory, deformation theory, and moduli spaces. The category of diffeological spaces, however, is too general and does not have a good differential calculus, which would be needed for a Lie theory of diffeological groupoids. I will introduce the notion of elastic diffeological spaces and show that these form a subcategory with an abstract tangent structure in the sense of Rosicky. The tangent structure yields a Cartan calculus consisting of vector fields, differential forms, the de Rham differential, inner derivatives, and Lie derivatives, satisfying the usual relations. Surprisingly, all diffeological groups are elastic. I then introduce the notion of diffeological Lie algebroids and show that the invariant vector fields of an elastic diffeological groupoid form a diffeological Lie algebroid. As application, I will revisit a diffeological groupoid that arises in lorentzian geometry whose diffeological Lie algebroid encodes the Poisson brackets of the Gauss-Codazzi constraint functions.

• Thematic program: Fellows of the Institut CNRS Pauli (fellows 2021/2022)
• Event: Workshop on "A finite and infinite-dimensional meeting on Lie groupoids, Poisson geometry and integrability" (2021)

see external webpage

• Thematic program: Fellows of the Institut CNRS Pauli (fellows 2021/2022)
• Event: Workshop on "A finite and infinite-dimensional meeting on Lie groupoids, Poisson geometry and integrability" (2021)

In this talk I will present the diffeological space of paths along a singular foliation and its groupoid structure. I will also show how to construct the fundamental groupoid of a singular foliation from its diffeological space of paths. This is a presentation of the joint work with Joel Villatoro entitled "Integration of singular foliations via paths" and to be published on IMRN.

• Thematic program: Fellows of the Institut CNRS Pauli (fellows 2021/2022)
• Event: Workshop on "A finite and infinite-dimensional meeting on Lie groupoids, Poisson geometry and integrability" (2021)

In this talk I will discuss how one can construct an infinite dimensional space of paths associated to a sheaf of Lie-Rinehart algebras. We will briefly examine some of the topological properties of this path space and how it can be used to construct a diffeological groupoid which appears to integrate the underlying sheaf. We will also take a look at some motivating examples for studying sheaves of Lie-Rinehart algebras over manifolds.

• Thematic program: Fellows of the Institut CNRS Pauli (fellows 2021/2022)
• Event: Workshop on "A finite and infinite-dimensional meeting on Lie groupoids, Poisson geometry and integrability" (2021)

see external webpage

• Thematic program: Fellows of the Institut CNRS Pauli (fellows 2021/2022)
• Event: Workshop on "A finite and infinite-dimensional meeting on Lie groupoids, Poisson geometry and integrability" (2021)

see external webpage

• Thematic program: Fellows of the Institut CNRS Pauli (fellows 2021/2022)
• Event: Workshop on "A finite and infinite-dimensional meeting on Lie groupoids, Poisson geometry and integrability" (2021)

see external webpage

• Thematic program: Fellows of the Institut CNRS Pauli (fellows 2021/2022)
• Event: Workshop on "A finite and infinite-dimensional meeting on Lie groupoids, Poisson geometry and integrability" (2021)

We plan to discuss certain genuine Poisson geometrical structures that arise in the theory of operator algebras on Hilbert spaces. Lecture 1 should be a gentle introduction to the basic notions on operator algebras that are needed later, with emphasis on the so-called standard form of von Neumann algebras that goes back to the PhD thesis of of U. Haagerup (1973). In Lecture 2, the focus is on the Poisson bracket carried by the predual of any von Neumann algebra, which turns out to admit smooth symplectic leaves, just as in the case of finite-dimensional Poisson manifolds. This lecture is partly based on joint work with T.S. Ratiu (2005). Finally, in Lecture 3, the geometric structures underlying the standard representations are pointed out, thereby presenting infinite-dimensional versions of presymplectic groupoids. This lecture is based on joint work with A. Odzijewicz (2019).

• Thematic program: Fellows of the Institut CNRS Pauli (fellows 2021/2022)
• Event: Workshop on "A finite and infinite-dimensional meeting on Lie groupoids, Poisson geometry and integrability" (2021)

see external webpage

• Thematic program: Fellows of the Institut CNRS Pauli (fellows 2021/2022)
• Event: Workshop on "A finite and infinite-dimensional meeting on Lie groupoids, Poisson geometry and integrability" (2021)

Exponential maps arise naturally in Lie theory and in the context of smooth manifolds endowed with affine connections. The Poincaré--Birkhoff--Witt isomorphism and the complete symbols of differential operators are related to these classical exponential maps through their infinite-order jets. The construction of (jets of) exponential maps can be extended to differential graded (dg) manifolds. As a consequence, the space of vector fields of any dg manifold inherits an L-infinity algebra structure, which is related to the Atiyah class of the dg manifold. Specializing this construction to the dg manifold arising from a foliation of a smooth manifold, one obtains an L-infinity structure on the de Rham complex of the foliation. In particular, a complex manifold can be regarded as a sort of `complexified' foliation. It turns out that the induced L-infinity structure is quasi-isomorphic to the L-infinity structure associated to the Atiyah class of the holomorphic tangent bundle on the Dolbeault complex first discovered by Kapranov. This is a joint work with Mathieu Stiénon and Ping Xu.

• Thematic program: Fellows of the Institut CNRS Pauli (fellows 2021/2022)
• Event: Workshop on "A finite and infinite-dimensional meeting on Lie groupoids, Poisson geometry and integrability" (2021)

We define the notion of Morita equivalence for singular Riemannian foliations (SRFs) such that the underlying singular foliations are Hausdorff-Morita equivalent as recently introduced by Garmendia and Zambon. We then define a functor from SRFs to the category of I-Poisson manifolds, where the objects are Poisson manifolds together with appropriate ideals and morphisms are defined as a particular relaxation of Poisson maps. We show that Morita equivalent SRFs are mapped to I-Poisson manifolds with isomorphic Poisson algebra of smooth functions on the symplectically reduced spaces. This is joint work in progress with T. Strobl.

• Thematic program: Fellows of the Institut CNRS Pauli (fellows 2021/2022)
• Event: Workshop on "A finite and infinite-dimensional meeting on Lie groupoids, Poisson geometry and integrability" (2021)

see external webpage

• Thematic program: Fellows of the Institut CNRS Pauli (fellows 2021/2022)
• Event: Workshop on "A finite and infinite-dimensional meeting on Lie groupoids, Poisson geometry and integrability" (2021)

The talk deals with the restricted Grassmannian which is a Hilbert manifold and related Banach Lie-Poisson spaces. One of the integrable systems related to this setup is of course the KdV equation. Using Magri method it is also possible to define another infinite hierarchy of differential equations on a certain central extension of a Banach Lie-Poisson space. Using integral of motions it is possible to write down solutions in particular cases.

• Thematic program: Fellows of the Institut CNRS Pauli (fellows 2021/2022)
• Event: Workshop on "A finite and infinite-dimensional meeting on Lie groupoids, Poisson geometry and integrability" (2021)