Schedule and abstracts

Click here to download the booklet with all the abstracts and further information and here for the Youtube playlist with all the videos.

• Daniel Beltiţă, Poisson geometrical aspects of von Neumann algebras (minicourse) - video lecture 1, lecture 2, lecture 3

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).

• Christian Blohmann, Diffeological groupoids (minicourse) - lecture notes - video lecture 1, lecture 2, lecture 3

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.

• Stefan Haller and Cornelia Vizman, Infinite dimensional Grassmannians and flag manifolds - video lecture 1, lecture 2, lecture 3, lecture 4

Nonlinear Grassmannians are Fréchet manifolds Gr_S(M) of submanifolds of a manifold M, having the diffeomorphism type of a closed manifold S. By allowing each submanifold to have an extra structure of a given type, we get decorated nonlinear Grassmannians. The decoration types used in this lecture are: weights, closed 1-forms, curves, and foliations. A nonlinear flag is a finite sequence N₁ ⊆ N₂ ⊆ ... ⊆ N_r ⊆ M of nested closed submanifolds. They build Fréchet manifolds and can also be endowed with extra structures.

The first lecture is dedicated to coadjoint orbits of the Hamiltonian group Ham(M) that are modeled on nonlinear Grassmannians of symplectic submanifolds resp. weighted isotropic submanifolds of the symplectic manifold M. In the second lecture we present coadjoint orbits of the group of contact diffeomorphisms that are modeled on nonlinear Grassmannians of weighted isotropic submanifolds of the contact manifold. Via the projectivized cotangent bundle, this leads to a geometric description of some coadjoint orbits of the full diffeomorphism group. In the third lecture we discuss certain coadjoint orbits of Ham(M) that can be modeled with symplectic nonlinear flags (nested sets of symplectic submanifolds of M) resp. weighted isotropic nonlinear flags (nested sets of isotropic submanifolds of M, each submanifold endowed with a volume density). The fourth lecture deals with coadjoint orbits of the group of volume preserving diffeomorphisms that can be modeled with (decorated) nonlinear Grassmannians, casting singular vorticities.

• Alexander Schmeding, Connecting finite, infinite-dimensional and higher differential geometry (opening talk) - slides - video talk

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.

• Camilo Angulo, Gray stability for contact groupoids - slides - video talk

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.

• Tobias Diez, A journey through the infinite lands of symplectic geometry - slides - video talk

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.

• Alfonso Garmendia, Path Integration: The fundamental groupoid of a singular foliation - video talk

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.

• Tomasz Goliński, Restricted Grassmannian and integrable systems around it - slides - video talk

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.

• Bas Janssens, Localization for positive energy representations of gauge groups - video talk

Let M be a manifold with a fixed point free time evolution ℝ x M → M, and let P → M be a principal fibre bundle with a lift of this action. Then a projective unitary representation of the compactly supported gauge group Γ_c(Ad(P)) is of `positive energy' if it is compatible with a 1-parameter unitary group of time translations whose generator (the Hamilton operator) has spectrum bounded from below. We show that this positive energy condition provides one with a surprising degree of analytic control. In particular, we prove a localization theorem which essentially reduces the positive energy representation theory to the case of 1-dimensional base manifolds. (Joint work with K-H. Neeb)

• Lory Kadiyan, The Lie algebroids of diffeological groupoids - slides - video talk

A diffeological groupoid is a groupoid object internal to the category of diffeological spaces. In 2006, Severa has shown that the infinitesimal counterpart of a higher Lie groupoid, namely its L_∞ algebroid, is given by the inner hom in the category of simplicial supermanifolds from the pair groupoid of ℝ^0|1 to the nerve of the groupoid at hand. Using the language of (co)ends, we give a categorical generalization of Severa’s idea for the case of diffeological groupoids. If time permits, I will discuss possible applications to geometric deformation theory. This is joint work with Christian Blohmann.

• Igor Khavkine, The geometry of analytic structures - slides - video talk

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?

• Gabriel Larotonda, Hamiltonian actions of compact Lie groups and their induced geometry - slides - video talk

If a compact Lie group K has a Hamiltonian action on a symplectic manifold (M,ω), we can pull-back Hofer's metric from the group of Hamiltonian Symplectomorphis Ham(M,ω) to the Lie group K. This metric is highly non-Riemannian, in the sense that it does not come from an inner product; moreover its unit sphere in Lie(K) has vertices and faces: it is the convex closure of the unitary orbit in Lie(K) of Kirwan's momentum polytope for the group action. The main example of this setting is obtained by taking M as the coadjoint orbit O_λ⊂Lie(K) of a regular element in the Lie algebra. It is well-known that this is a symplectic manifold when given the canonical KKS form ω(x,y)=<λ,[x,y]>, where the pairing is given by the Killing form of K. Then the coadjoint action of K in the orbit is Hamiltonian, and the polytope P is simply the intersection of the orbit with a positive Weyl chamber in Lie(K).

In this talk we will discuss the relation among short (i.e. distance minimizing for the metric) in K, and short paths in the group of Hamiltonian Symplectomorphisms Ham(M,ω) with its natural Hofer metric. We will specialize certain (previously obtained) results for bi-invariant distances on Lie groups to this setting, to characterize these short paths by means of the Weyl chambers, and the properties of the extremal points of the polytope.

• Ioan Mărcuț, Rigidity of solutions to PDEs with symmetry - video talk

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.

• Lukas Miaskiwskyi, Continuous Lie Algebra Homology of Gauge Algebras - slides - video talk

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.

• Hadi Nahari, Morita equivalence of singular Riemannian foliations and I-Poisson geometry - slides - video talk

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.

• Stephen Preston, Breakdown of the mu-Camassa-Holm equation - video talk

The family of PDEs m_t + u m_x + b u_x m = 0, with m = μ –u_xx, where μ is the average value of u on the circle, is a variation of the Camassa-Holm equation proposed in two papers by Khesin, Lenells, Misiolek, and Tiglay. In case b=2 it is called the μ-Camassa-Holm equation, and if b=3 it is the Degasperis-Procesi equation; both represent geodesics on the diffeomorphism group of the circle. Using a new geometric interpretation, I will show how to prove the breakdown result for C² initial data: the solution exists for all time if and only if the initial momentum m(0,x) does not change sign on the circle.

• Leonid Ryvkin, L_∞ Extensions for the Poisson algebra of a symplectic manifold - slides - video talk

Let (M, ω) be a non-compact symplectic manifold. The universal central extension of the Lie algebra C^∞(M) can be realized as the quotient Ω¹(M)/δΩ²(M), where δ is the differential of the canonical homology complex. We will show how the Lie algebra Ω¹(M)/δΩ²(M) extends to an L_∞-algebra structure on the canonical homology complex. The extension procedure is inspired by a similar approach for general multisyplectic manifolds developed by Christopher Rogers, but is combinatorially more intricate and relies on results from symplectic Hodge theory. (j.w. Bas Janssens and Cornelia Vizman)

• Seokbong Seol, Formal exponential map of differential graded manifolds - slides - video talk

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.

• Aneta Sliżewska, Fibre-wise linear Poisson structures related to W^*-algebras - slides - video talk

We investigate the fiber-wise linear Poisson structures as well as the Lie groupoid and Lie algebroid structures which are defined in the canonical way by the structure of a W^*-algebra (von Neumann algebra) M. The main role in this theory is played by the complex Banach-Lie groupoid of partially invertible elements of M over the lattice of orthogonal projections of this algebra. The Atiyah sequence and the predual Atiyah sequence corresponding to this groupoid are investigated from the point of view of Banach Poisson geometry. In particular we show that the predual Atiyah sequence fits in a short exact sequence of complex Banach sub-Poisson VB-groupoids with groupoid of partially invertible elements as the side groupoid.

• Joel Villatoro, Paths in Lie-Rinehart algebras - slides - video talk

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.

• Marco Zambon, Singular subalgebroids and their integrations - slides - video talk

We introduce singular subalgebroids of an integrable Lie algebroid, extending the notion of Lie subalgebroid by dropping the constant rank requirement. We review how one can associate canonically to them a topological groupoid, called holonomy groupoid, adapting the procedure of Androulidakis-Skandalis for singular foliations. In the special case of Lie subalgebroids this extends work of Moerdijk-Mrčun.

From the holonomy groupoid and its natural diffeological structure, one can recover the singular subalgebroid. We single out a class of diffeological groupoids for which this differentiation procedure yields a singular subalgebroid. This allows to introduce the notion of integration of a singular subalgebroid, for which we present several examples. This is based on joint work with Iakovos Androulidakis.

• Florian Zeiser, Poisson linearization using the Nash-Moser method - slides - video talk

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.