Lie Pseudogroups: Old and New

This is a workshop on Lie pseudogroups, nearby fields and applications (geometric structures, Lie groupoids, PDEs, Poisson geometry, etc.), supported by Universiteit Utrecht, which will be held on October 2-4 2017 in Driebergen (The Netherlands).  

Speakers:

• Francesco Cattafi (Universiteit Utrecht)
• Marius Crainic (Universiteit Utrecht)
• Bas Janssens (Technische Universiteit Delft)
• Ioan Mărcuț (Radboud Universiteit Nijmegen)
• Antonio de Nicola (Università degli Studi di Salerno)
• Hessel Posthuma (Universiteit van Amsterdam)
• Ivan Struchiner (Universidade de São Paulo)
• Luca Vitagliano (Università degli Studi di Salerno)
• Ori Yudilevich (Katholieke Universiteit Leuven)
• Marco Zambon (Katholieke Universiteit Leuven)

 

Other participants:

• Luca Accornero (Universiteit Utrecht)
• Davide Alboresi (Universiteit Utrecht)
• Álvaro del Pino Gómez (Universiteit Utrecht)
• Alfonso Garmendia (Katholieke Universiteit Leuven)
• Ralph Klaasse (Université libre de Bruxelles)
• Maarten Mol (Universiteit Utrecht)
• Lauran Toussaint (Universiteit Utrecht)
• Kirsten Wang (Universiteit van Amsterdam)
• Aldo Witte (Universiteit Utrecht)
• Florian Zeiser (Radboud Universiteit Nijmegen)

   

Schedule:

Each speaker will talk for one hour and a half, divided in two blocks of 45 minutes each.    

Titles and abstracts:

• Introduction to Lie pseudogroups (Crainic)
  Abstract: TBA
• On Cartan’s Structure Theory for Lie Pseudogroups (Yudilevich)
  In this talk, I will give an overview of Élie Cartan’s structure theory for Lie pseudogroups, told from a modern perspective. The starting point of Cartan’s theory is the Second Fundamental Theorem for Lie pseudogroups, which states that any Lie pseudogroup is equivalent to a pseudogroup in normal form, and, hence, that the study of Lie pseudogroups (up to equivalence) amounts to the study of pseudogroups in normal form. In turn, Cartan’s notion of normal form is encoded in an infinitesimal geometric structure we call a Cartan algebroid and in its so-called realizations. After introducing these structures, I will discuss some of their interesting features, and, time permitting, I will explain the proof of the Second Fundamental Theorem. This talk is based on joint work with Marius Crainic.
• Pfaffian Morita equivalence (Cattafi)
  Morita equivalence is a powerful tool which allows us to “transport” objects and properties from one Lie groupoid to another; in this talk, using jets of Lie pseudogroups as motivating example, we extend this setting to Pfaffian groupoids and explain which of the relevant structures are preserved and under which conditions. This is joint work with Marius Crainic.
• Morita equivalence of singular foliations (Zambon)
  Singular foliations in the sense of Androulidakis-Skandalis carry more information than the underlying partition into leaves, and in particular allow to associated to them a topological groupoid called holonomy groupoid. We will discuss the notion of Morita equivalence for singular foliations, which essentially preserves the “geometry transverse to the leaves”. In particular, we related to the notion of Morita equivalence of the associated holonomy groupoids. The talk is based on work in progress by my doctoral student Alfonso Garmendia and myself.
• Homotopy algebras and PDEs (Vitagliano)
  A PDE can be described geometrically within jet spaces. An interesting consequence of this is that there is a rich cohomology theory attached to a PDE: the so-called horizontal cohomology. To some extent, horizontal cohomologies can be interpreted as vector fields, differential forms etc., on the space of solutions of the PDE. This interpretation is supported by the fact that horizontal cohomologies are canonically equipped with the same algebraic structures as standard vector fields, differential forms, etc. The aim of the talk is twofold: 1) reviewing horizontal cohomologies and their interpretation; 2) showing that the algebraic structures on horizontal cohomologies do actually come from homotopy algebras at the level of cochains.
• Lie Theory and Classification Problems for G-structures (Struchiner)
  The infinitesimal data attached to a (finite type) class of G-structures with connections are its structure equations. Such structure equations give rise to Lie algebroids endowed with extra geometric information following from the fact that they came from G-structures. On the other hand, given such a Lie algebroid (called a G-algebroid) a natural question is that of the existence of a G-structure with connection which corresponds to the given G-algebroid through differentiation. This integration problem is called Cartan’s Realization Problem for G-structures.In this talk I will describe a way of solving the realization problem by integrating the G-algebroid. I will focus on the case of Riemannian metrics (G=O(n)). The talk is based on joint work with Rui Loja Fernandes.
• Rigidity of solutions to PDEs with symmetry (Mărcuț)
  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.
• Almost formality of Vaisman manifolds with applications to nilmanifolds (de Nicola)
  We provide models that are as close as possible to being formal for a large class of compact manifolds that admit a transversely Kähler structure, including Vaisman and quasi-Sasakian manifolds. As an application we are able to classify the corresponding nilmanifolds.
• Lie algebra cohomology for Lie-Rinehart algebras (Janssens)
  For a certain class of Lie-Rinehart algebras (algebraic versions of Lie algebroids), we show that every central Lie algebra extension can be realised by a differential operator of order at most 5. This holds in particular for the Lie algebra of sections of every Lie algebroid whose anchor is either identically zero or everywhere nonvanishing (but not necessarily of constant rank!).
• Hopf algebroids and Lie pseudogroups (Posthuma)
  Abstract: TBA

 

Venue:

All activities will take place at Bergse Bossen Conference Centre, Traaij 299, 3971 GM Driebergen-Rijsenburg. We will provide you accommodation (single room) and full board (breakfast x2, lunch x3, dinner x2) at the conference centre for the entire duration of the workshop, unless you asked differently. Breakfast and lunch are buffet style and in the evening there is a sit-down 3-course dinner; please tell us in advance if you have any food allergies and/or other dietary requirements. There is free wifi on location. The lecture room (school style) will be equipped with blackboards and a beamer.  

How to get there:

From the International Schiphol Airport take an NS train to Driebergen-Zeist (depending on the time of the day, there are direct trains or you may have to change at Utrecht Centraal). You need either a single-use chipkaart or an OV-chipkaart: both are available from the yellow ticket machines, which are located near the platforms at Schiphol Plaza. Tickets are also available from the ticket- and service desks, which are situated close to the red/white-checked cube at Schiphol Plaza. You need to check in and out with your chipkaart at the train station before and after your journey. UPDATE: because of railway works, on Sunday 1 October there will be no train connections between Utrecht Centraal and Driebergen-Zeist; in place of that, every half an hour (until past midnight) there will be a “snelbus” in direction Ede-Wageningen, which will bring you to Driebergen-Zeist in 22 minutes. On Monday morning trains should ride as usual. From Driebergen-Zeist station you can take Syntus bus 381 (direction “Austerlitz via Zeist”) and get off at the stop “Bergse Bossen” (8 minutes journey), which is one minute walk from the conference centre. Unfortunately, this works only during the week; the people coming on Sunday have to take Syntus bus 56 (direction “Wijk bij Duurstede via Zeist/Doorn”) and get off at the stop “Welgelegenlaan” (5 minutes journey) and then walk for 11 minutes to the conference centre. Tickets can be bought on the bus (at a higher price), or you can use your OV-chipkaart. You can also take the environmentally friendly shuttle service and taxi transport directly to the conference centre. The shuttle bus has a special parking/departing area (notice signs “Groen centraal”), costs € 6,40 per person but can be reserved only at specific times (and not during the weekend); the taxi costs instead € 14,00 (up to 4 people) or € 27,00 (up to 8 people) and can be reserved at any time (including the weekend). For reservations or information please check www.ikreisgroen.nl. Other alternatives from the station include walking (40 minutes) or biking (15-20 minutes); there is an OV-fiets service (bikes for rent, using the OV-chipkaart) available at the station, open from Monday to Friday, from 7.00 to 20.00. An invaluable source for travel information within the Netherlands is the route planner.     For any further questions, please contact f.cattafi[at]uu.nl