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Abstracts for Higher Geometric Structures along the Lower Rhine

Alternatively have a look at the program.

Homotopy 'groups' of Lie algebroids and obstruction of integration

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Speaker: 
Chenchang Zhu
Affiliation: 
Göttingen
Date: 
Thu, 2012-01-12 13:45 - 14:45
Location: 
MPIM Lecture Hall

Lie algebroids are certain degree 1 super manifolds. We propose a definition of fibration in the category of Lie algebroids, and prove the long exact sequence of homotopy groups associated to a fibration. On the other hand, certain Lie algebroids, unlike (finite dimensional) Lie algebras which can always integrate to Lie groups, can not integrate to Lie groupoids. The obstructions are classified by Crainic and Fernandes. It turns out that the long exact sequence of homotopy groups associated to a suitable fibration encodes the obstruction of integration.

Higher Segal spaces

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Speaker: 
Mikhail Kapranov
Affiliation: 
Yale
Date: 
Thu, 2012-01-12 15:00 - 16:00
Location: 
MPIM Lecture Hall

A Segal space is a simplicial topological space $X = (X_n)$ with a certain condition expressing $X_n$ as an $n$-fold homotopy fiber product of $X_1$ over $X_0$. This concept can be seen as encoding a weak (higher) categorical structure.

Simplicial approaches to derived $C^\infty$-geometry

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Speaker: 
Dennis Borisov
Affiliation: 
MPIM
Date: 
Thu, 2012-01-12 16:30 - 17:30
Location: 
MPIM Lecture Hall

I will prove that derived $C^\infty$-geometry, defined by gluing simplicial $C^\infty$-spaces up to homotopy, can be faithfully expressed through the usual category of simplicial $C^\infty$-rings. In particular, I will show equivalence to D.Spivak's construction. (Joint work with J.Noel) 

Global stable homotopy theory

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Speaker: 
Stefan Schwede
Affiliation: 
Universität Bonn
Date: 
Fri, 2012-01-13 09:30 - 10:30
Location: 
MPIM Lecture Hall

I will present some basic notions and results of "global" homotopy theory, where objects are spectra (in the sense of stable homotopy theory) equipped with actions of all compact Lie groups, in a way compatible with transfers and restriction along group homomorphisms.

The precise implementation proceeds via a new model structure on the category of orthogonal spectra, with "global equivalences" as weak equivalences.

Sheaf Theory for Étale Stacks

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Speaker: 
David Carchedi
Affiliation: 
MPIM
Date: 
Fri, 2012-01-13 11:00 - 12:00
Location: 
MPIM Lecture Hall

Étale differentiable stacks are a generalization of manifolds which include such examples as orbifolds and leaf spaces of foliated manifolds. In this talk we will discuss certain aspects of sheaf theory for étale differentiable stacks and emphasize the role of  "the universal étale stack".

Topos theory and C$^*$-algebras

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Speaker: 
Klaas Landsman
Affiliation: 
Nijmegen
Date: 
Fri, 2012-01-13 14:00 - 15:00
Location: 
MPIM Lecture Hall

Toposes and C$^*$-algebras both represent generalized concepts of space. The utmost generality seems achieved by studying C$^*$-algebras in (Grothendieck) toposes, as first done by Banaschewski and Mulvey. In the commutative case, Gelfand duality holds and one gets interesting examples of internal locales.

Morita equivalence in geometry and algebra

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Speaker: 
Henrique Bursztyn
Date: 
Fri, 2012-01-13 15:30 - 16:30
Location: 
MPIM Lecture Hall

Morita equivalence is an important equivalence relation for algebras, based on comparing categories of representations. There is also a geometric version of Morita equivalence in the realm of Poisson geometry, based on the notion of "dual pairs". In this talk, we will discuss how deformation quantization provides a concrete link between Morita equivalence of Poisson structures and their quantum algebras.

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