Aug 03, 2015 to Sep 03, 2015

WITH GREAT REGRETS THIS CONFERENCE HAS BEEN CANCELLED DUE TO THE SECURITY SITUATION FOLLOWING THE SOUSSE ATTACK.

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MIMS and the Laboratoire LATAO (Faculté des Sciences de Tunis) organize a conference in Topology once every two years.

This year's edition will bring together a dozen of the finest topologists from around the world. The conference will present recent advances in the field with applications to other branches of geometry, including configuration spaces, moduli spaces and operads.

**Organizing Commitee:**

Sadok Kallel, Gael Collinet, Ines Saihi, Ali Baklouti

**Scientific Commitee:**

Gael Collinet, Sadok Kallel

**Sponsors:**

MIMS

**Preliminary list :**

- Natalia Castellana (Universitat Autònoma de Barcelona)
- Benoit Fresse (University of Lille1, France)
- Martin Palmer (Westfälische Wilhelms-Universität Münster, Germany)
- Petar Pavešić (University of Ljubljana)
- Paolo Salvatore (University of Roma, Tor Vergata, Italy)
- Jerome Scherer (Universite de Lausanne, Switzerland)
- Paul Arnaud Songhafouo (University of Louvain, Belgium)
- Pierre Vogel (University of Paris VII)
- Matthias Wendt (University of Duisburg-Essen, Germany)

**Titles and Abstracts :**

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**Benoit Fresse**

*Rational homotopy and intrinsinc formality of E_n-operads*

In the first part of my talks, I will briefly recall the definition of an E_n-operad, as an operad which is weakly-equivalent to the operad of little n-discs, and I will survey some applications of these objects. I will also recall the Kontsevich formality theorem which asserts that E_n-operads are weakly-equivalent to a model determined by the homology of these operads. In the case n=2, this formality result can also be deduced from the existence of Drinfeld's associators. The main purpose of my talks is to explain that E_n-operads satisfy a stronger intrinsic formality statement for n>=3. This intrinsic formality result implies that E_n-operads are detected by their homology (up to rational weak-equivalence of operads), at least when n>=3.

(Based on joint works with Victor Turchin and Thomas Willwacher.)

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**Martin Palmer**

*Configuration spaces, homological stability and closed manifolds*

Homological stability is the (surprisingly common) phenomenon that, given a sequence of spaces X_n, their homology "stabilises" as n goes to infinity, meaning that it is eventually independent of n in each degree. It was proved by McDuff and Segal in the 1970s that this phenomenon holds for unordered configuration spaces C_n(M) of n points in a manifold M, as long as M is non-compact and connected. I will talk about two generalisations of this result, concerning closed (i.e. compact without boundary) manifolds. One is about configurations in a closed manifold M - in this case, the situation is more subtle, since homological stability in its strongest form is false, but there are nevertheless several stability phenomena if one looks more carefully (for example a homological periodicity result for homology with field coefficients). This part of the talk represents joint work with Federico Cantero. The second generalisation of McDuff-Segal's result is where the ambient manifold M is again non-compact, but we consider configurations of disjoint closed submanifolds instead of points. In this case, the statement of homological stability generalises directly, as long as the submanifolds have sufficiently small dimension relative to M. As a corollary, one obtains homological stability results for various diffeomorphism groups of manifolds.

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**Paul Arnaud Songhafouo**

*Graph-complexes computing the rational homology and homotopy of high dimensional string links*

__Abstract__: The goal of this talk is to explicitly describe graph-complexes computing the rational homology and homotopy of high dimensional string links.

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**Jerome Scherer**

*Nilpotency for groups and loop spaces*

__Abstract__: Classically the notion of nilpotency has been defined for groups in terms of commutators or central extensions. These ideas have been taken to the homotopy category in the sixties and led to the study for example of homotopy commutativity for loop spaces or more generally H-spaces. From the point of view of homotopy theory however this is not quite the right analogue of the group theoretical concept. I will explain how Goodwillie calculus has allowed Biedermann and Dwyer to define the correct notion of homotopy nilpotency for loop spaces. I will also present recent results with Costoya and Viruel about a numerical approximation of homotopy nilpotency by what we call ``extension by principal fibrations length'', or epfl for short. A generalization of Hubbuck's Torus Theorem follows. Whereas the original statement is about homotopy commutative finite H-spaces, ours deals with finite homotopy nilpotent groups.

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**Matthias Wendt**

*Homology of linear groups over elliptic curves*

__Abstract__:In the talk I will explain an ongoing project concerning computations of homology of general linear groups $GL_n(k[E])$ over function rings of affine elliptic curves. These computations are based on the study of the action of such general linear groups on the associated Bruhat-Tits buildings. The identification of orbits of vertices in terms of vector bundles on the complete elliptic curve is well-known; I will explain how this can be generalized to an explicit description of the complete equivariant cell structure of the building in terms of diagrams of moduli of vector bundles (at least in the small ranks $n\leq 4$). The first application of explicit equivariant cell structures for Bruhat-Tits buildings are computations of cohomology of $GL_3$ of function rings of elliptic curves over finite fields; this provides new information on the function field analogue of Quillen's conjecture.

The second application is a new construction of a complex which computes the weight 2 part of the K-theory of an elliptic curve; this provides an integral refinement of the elliptic motivic weight 2 complex constructed by Goncharov and Levin.

Aug 03, 2015 to Sep 03, 2015

No participants

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