Geometric Structures in Utrecht
Utrecht University - Mathematics Department

Higher Geometric Structures along the Lower Rhine VI

September 24-26, 2015 - Utrecht University

Organizers: Marius Crainic, Ieke Moerdijk, Christian Blohmann


This is the sixth in a series of  short workshops jointly organized by the Geometry/Topology groups in Bonn, Nijmegen, and Utrecht, all situated along the Lower Rhine. The focus lies on the development and application of new structures in geometry and topology such as Lie groupoids, di fferentiable stacks, Lie algebroids, generalized complex geometry, topological quantum fi eld theories, higher categories, homotopy algebraic structures, higher operads, derived categories, and related topics.

Web pages of the previous workshops: Bonn (Jan 2012), Nijmegen (Dec 2012), Utrecht (Oct 2013), Bonn (Feb 2014), Nijmegen (Jun 2014)


Matija Basic - University of Zagreb

Yael Fregier - Artois University in Lens and MPIM in Bonn

Benoit Fresse - Université Lille 1

Vladimir Hinich - University of Haifa

João Nuno Mestre - Utrecht University

Steffen Sagave - University of Bonn

Cornelia Vizman - West University of Timisoara

Yannick Voglaire - Université du Luxembourg

Marco Zambon - KU Leuven


The workshop will take place in:

Van der Valk Hotel De Bilt

De Holle Bilt 1, 3732 HM De Bilt

+31 30 220 58 11

This is a hotel and conference center in the town of “De Bilt” neighboring Utrecht. We encourage participants to stay overnight at the hotel. The stay includes all meals. Please specify in the registration form your dates of arrival and departure and whether you will stay overnight at the hotel.

Registration - Deadline: September 14th, 2015

Click here to register. For any inquiries, please send an email to

Schedule Titles and Abstracts

Thursday - September 24



12:30 - 14:00


14:00 - 15:00

Steffen Sagave - Rigidification of homotopy coherent commutative multiplications

In various contexts, the use of functors defined on the category I of finite sets and injections makes it possible to replace E-infinity objects by strictly commutative ones. For example, E-infinity spaces can be replaced up to weak equivalence by strictly commutative monoids in I-diagrams of spaces, and E-infinity quasi-categories can be replaced up to weak equivalence by strictly commutative monoids in I-diagrams of simplicial sets. Building on the latter result, we show that every presentably symmetric monoidal infinity-category is represented by a symmetric monoidal model category. This can be viewed as a rigidification result about multiplicative homotopy theories.

(This is report on joint work with Christian Schlichtkrull, with Dimitar Kodjabachev, and with Thomas Nikolaus.)

15:30 - 16:30

Yannick Voglaire - Invariant connections and PBW theorem for homogeneous spaces of Lie groupoids

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 L-invariant 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.

17:00 - 18:00

Vladimir Hinich - Enriched infinity categories and enriched infinity-operads

We suggest a definition of enriched category and enriched colored opeard in the infinity-categorical setting. The definition of enriched category is based on a (more or less obvious) notion of enriched quiver.



Friday - September 25



10:00 - 11:00

Yael Fregier - Homotopy moment map

The symmetries of a Hamiltonian systems on a symplectic manifold can be, in the good cases, described in terms of a moment map. On the other hand, there exists a generalization of Hamiltonian mechanics for closed forms of degree greater or equal to two. This appears for example when one considers symplectic structures on loop spaces. The role of Poisson brackets is then played by an $L_\infty$ algebra. It is therefore natural to seek for the analogue of the notion of moment map in this setting. We have introduced, in a commun work with Martin Callies, Chris Rogers and Marco Zambon, the notion of moment map up to homotopy as an $L_\infty$ morphism between the Lie algebra encoding the symmetries and the „Poisson" $L_\infty$ algebra. In this new framework, the correspondence due to Atiyah and Bott between cocycles in equivariant cohomology and couples (moment map, symplectic form) can be extended.

11:30 - 12:30

Cornelia Vizman - Integrability of central extensions of the Poisson Lie algebra

We present a geometric construction of some central S1-extensions of the quantomorphism group of a compact symplectic prequantizable manifold. On the way we obtain central S1-extensions of the group of exact strict contact transformations, and of the universal cover of the group of Hamiltonian diffeomorphisms.

12:30 - 13:30


14:30 - 15:30

Benoit Fresse - Rational homotopy and intrinsic formality of $E_n$-operads

The theory of E_n-operads has considerably developed since a decade. Let us mention, among other applications, the second generation of proofs of the Kontsevich formality theorem, based on the formality of $E_2$-operads, which has hinted the existence of an action of the Grothendieck-Teichmüller group on moduli spaces of deformation-quantization of Poisson algebras, and the description of the Goodwillie-Weiss approximations of embedding spaces in terms of functions spaces on structures associated to $E_n$-operads.

The goal of this talk is to explain an intrinsic formality statement which gives a rational homotopy characterization of $E_n$-operads in topological spaces in terms of their homology when $n\geq 3$. Recall that the homology of an $E_n$-operad is identified with the operad governing Poisson algebra structures of degree $n-1$. I will show that any operad in topological spaces which has the same rational homology as an $E_n$-operad for some $n\geq 3$, and is additionally equipped with an involutive isomorphism that mimicks the action of a hyperplane reflection in the case $n|4$, is rationally weakly-equivalent to an operad in topological spaces which is determined by this homology operad of Poisson algebras of degree $n-1$.

This is a joint work with Thomas Willwacher."

16:00 - 17:00

João Nuno Mestre - Measures and volumes for differentiable stacks

We explain how an extension of Haefliger's approach to transverse measures for foliations allows us to define and study measures and geometric measures (densities) on differentiable stacks.

The abstract theory works for any differentiable stack, but it becomes very concrete for proper stacks - for example, when computing the volume associated with a density, we recover the formulas that are taken as definition by Weinstein. This talk is based on joint work with Marius Crainic.



Saturday - September 26



10:00 - 11:00

Matija Basic - Stable homotopy theory of dendroidal sets

Dendroidal sets were introduced as combinatorial models for topological operads. After recalling the definition and history of dendroidal sets, we will explain how dendroidal sets also model connective spectra. We will introduce homology theory of dendroidal sets and show how to use it to identify spectra that are associated to some dendroidal sets. This is joint work with Thomas Nikolaus.

11:30 - 12:30

Marco Zambon - L-infinity actions

An infinitesimal action of a Lie algebra on a manifold can be encoded by a Lie algebroid, called transformation Lie algebroid, which in turn gives rise to a graded manifold endowed with a homological vector field (i.e., a Q-manifold).

We describe how a similar phenomenon occurs for  actions of L-infinity algebras on graded manifolds, and provide some examples. The same holds even ion the more general context of actions of an L-infinity algebroid A. Many examples occur when A is a Lie algebroid, one of them being representations up to homotopy.

This talk is based on work with R. Metha and on work in progress with O. Brahic.