** FLOER LECTURES 2017 **

** JUNE 29-30, BOCHUM**

We are delighted to have **Michael Weiss** (Münster) and** Thomas Willwacher** (ETH Zürich) as our two speakers. They will each give two talks.

This is a workshop hosted by the Floer Center of Geometry and funded by the Bochum-Cologne SFB collaboration program **CRC/TRR 191 "Symplectic Structures in Geometry, Algebra and Dynamics”**.

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**Schedule:**

**Thursday**

16:15-17:15 M . Weiss -* **"Rational Pontryagin classes of euclidean fiber bundles"*

17:15-17:45 Coffee break

17:45-18:45 T. Willwacher - *"Graph complexes in topology"** *

**Friday**

15:15-16:15 T. Willwacher - *"Graph complexes in topology"
*

16:15-16:45 Coffee break

16:45-17:45 M . Weiss -* **"Rational Pontryagin classes of euclidean fiber bundles"*

**Talks** will take place in room NA 01/99, which is on the ground floor of the Mathematics Department (building NA).

**Coffee breaks** will take place in the Friedrich-Sommer Raum NA 1/58, which is on the first floor up of the same building.

**Abstracts:**

T. Willwacher -* **"Graph complexes in topology"*

Graph complexes are vector spaces of formal linear combinations of combinatorial graphs, with a

differential, and often further algebraic structure.

Following seminal work of Kontsevich and others it has been found in recent years that many

questions of topological interest can be reduced to questions about graph complexes and can hence to a certain degree be combinatorialized.

I will give a general overview about graph complexes, the state of the art, and their appearance in algebra and topology.

Furthermore, I will report on recent work with B. Fresse and V. Turchin in which we (mostly) settle the rational homotopy theory of the little n-disks operads, by showing that they are intrinsically formal for n>=3, and by computing the rational homotopy type of the function spaces between these operads.

As an application we obtain completely combinatorial (graphical) descriptions of the spaces of long embeddings of R^m into R^n in codimension >=3.

M . Weiss -* **"**Rational Pontryagin classes of euclidean fiber bundles"*

A euclidean fiber bundle is a fiber bundle with fiber Rn for some n. It may be helpful to think of euclidean vector bundles as something like real vector bundles, but without the linear structure. Real vector bundles have certain characteristic classes, the Pontryagin classes; the Pontryagin classes of a real vector bundle are essentially the Chern classes of the complexified vector bundle. In the late 1950s and early 1960s it emerged that the Pontryagin classes of a real vector bundle do not depend on the linear structure, as long as we use cohomology with rational coefficients. In other words, euclidean fibers bundles have (rational) Pontryagin classes, too. Nowadays we can state this rather easily by saying that the classifying space for stable euclidean fiber bundles, BTOP, has the same rational cohomology as the classifying space for stable vector bundles, BO. But this does not do justice to the process of discovery.

The old definition of Pontryagin classes for real vector bundles in terms of complexification and Chern classes implies some vanishing relations. Specifically, the Chern class ck (in 2k-dimensional cohomology) of an n-dimensional complex vector bundle vanishes if k > n, so that the Pontryagin classes in cohomological dimension > 2n of an n-dimensional real vector bundle are also zero. But the definition of rational Pontryagin classes for euclidean fiber bundles does not use complexification. (Complexification of euclidean fiber bundles does not seem to make any sense!) Instead it is inspired by the Hirzebruch signature theorem which expresses the signature of a smooth closed manifold in terms of the Pontryagin classes of its tangent bundle (and this can also be read the other way round). Therefore the question arises: do the rational Pontryagin classes of euclidean fiber bundles satisfy the same vanishing relations, i.e., do they always vanish in cohomology of degree > 2n if the fiber dimension is n? It turns out that this is not the case. The counterexamples available so far are more statistical in nature than explicit, and they are not claimed to be best or worst possible, but in any case one can go nearly as far cohomological dimension equal to 9/2 times the fiber dimension n and still see nonzero rational Pontryagin classes, provided n is large enough.

The construction of these (counter)examples takes us back to the Hirzebruch signature theorem. Apart from that it exploits recent advances in the homotopical theory of diffeomorphism groups of highly connected manifolds (especially the work of Galatius and Randal-Williams), and advances in the homotopical theory of spaces of smooth embeddings which in turn lead to the construction of interesting families of homeomorphisms.

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Anyone is welcome to attend!

There is no formal **registration**, but please send an email to Frau Dzwigoll (ursula.dzwigoll@rub.de) so that we can estimate the number of participants. Everyone is welcome to join us for a **dinner** on the Thursday evening.

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**Hotels**

Here are some of the hotels in Bochum, please contact Frau Dzwigoll if you would like assistance with booking a room.

**Park Inn** - conveniently located next to central station, large rooms, but not cheap (ca 85 Euro incl. breakfast)).

**Ibis City** und **Ibis Zentrum** - next to central station, small rooms, good price but nothing special - Ibis Zentrum is probably a bit more quiet (ca 65 incl. breakfast).

**Art Hotel Tucholsky **in the "Bermuda triangle" (party mile), that is, can be noisy at night, but if you reserve a room to the back yard it's very fine, art hotel with funny accessoires - great breakfast, probably the most interesting hotel in the list (ca. 75 Euros incl. breakfast)

**Jugendherberge (youth hostel) Bochum **again in the "Bermuda triangle", it is supposed to be very good and has differnet kinds of rooms (ca. 62 Euros for single rooms incl. breakfast)

General directions including a map of the campus can be found here.

For assistance or questions please contact Ursula Dzwigoll.

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*We hope to see you there!*