**Takes place on Thursdays following the following
schedule**

**Place:
Big room of CRM**

**Morning session
1130-1230**

**Afternoon session 1500-1600**

(from 11 on, there will be cookies and coffee in front
of the conference room)

**May 9, Friday (this week there is this conference at CRM that overlaps
interests with our seminar that is why the seminar takes place on Friday
afternoon when the conference will be over)**

15:00 p.m Dieter Kotschick (Ludwig-Maximilians-Universität Munchen), Hyperkähler and hypersymplectic structures

In this lecture I shall discuss hyperkähler structures from a purely symplectic point of view. This point of view yields a strict generalization of hyperkähler geometry. It also leads to the consideration of hypersymplectic and other structures, whose definitions are completely analogous to the symplectic version of hyperkähler geometry. (Joint work with G. Bande.)

**May 15, Thursday**

11:30 am Mark Hamilton (CRM-MFO, Oberwolfach),

Geometric quantization of singular reductionOne question that has received a great deal of attention over the years is the "quantization commutes with reduction," first proposed by Guillemin and Sternberg in 1980, and proved by them under suitable hypotheses of regularity. Attempts to generalize "quantization commutes with reduction" have the problem of suitably defining the reduction in cases when the reduced space is not necesssarily a smooth manifold. Two particular methods of reduction for singular spaces, called algebraic and singular reduction, take a differential space approach to the reduced space, studying it by looking at its ring of smooth functions. In the case that the action of is free and proper, both coincide with the usual symplectic reduction.

Sniatycki has defined the quantization of the algebraic reduction of a symplectic manifold. In this talk, I will describe a similar construction for the quantization of the singular reduction of a manifold equipped with a proper action of a Lie group G. I will compare the quantization of the two types of reduction, and say something about when they agree. If time, I will give an example where the reductions differ but the resulting quantizations are the same.

This is joint work with J. Sniatycki, R. Cushman, and L. Bates.

**15:00 pm:
Jaume Amoros (UPC),
Homotopy Properties of Aspherical
Symplectic Four-manifolds**

The interrelation of homotopical properties such as formality, Hard Lefschetz or Massey products is discussed in closed, aspherical four-manifolds. A construction using mapping tori for surface diffeomorphisms in the Torelli group shows the flexibility of this class of manifolds, compared to their algebraic analogues.

**May 21-13 GESTA
2008**

Minicourses by## · Marius Crainic (University of Utrecht)

## · Yakob Eliashberg (Stanford University)

## · Rui Loja Fernandes (Instituto Superior Técnico, Lisboa): Stability of symplectic leaves

## · Pol Vanhaecke (Université de Poitiers): Algebraic integrability

## ·

Jean-Yves Welschinger (CNRS, ENS de Lyon)Additional lecture by Gang Tian

Look at the

webpagefor the schedule.

**
May 28 Wednesday,
Special Talk by John Morgan **

15:00 pm (big room of CRM) John Morgan (Columbia University)Structure of 3-manifolds with (locally) collapsed volume.

(This talk is a presentation of joint work with Gang Tian.)

The final step in the classification of 3-manifolds (Thurston's geometrization conjecture) using Ricci flow with surgery is a purely geometric/topological study of 3-manifolds that are locally volume-collapsed on a scale where the curvature is bounded below. The result that must be established is that these manifolds are graph manifolds, i.e. connected sums of manifolds which themselves decompose along incompressible tori into Seifert fibrations. We present a proof of these result along the lines suggested by Perelman, namely using the theory of Alexandrov spaces.

**
May 29 Thursday **

11:30 am Camille Laurent-Gengoux (Université de Poitiers),Action-angle in the Poisson context

We prove the action-angel theorem in the general and most natural context of (possibly non-commutative)integrable systems on Poisson manifolds, thereby generalizing several results which are well-known

in the symplectic context.

15:00 pm: Carlos Currás-Bosch (UB) Singular cotangent model

The semilocal classification of completely integrable systems, with non degenerate singularities, is still an open problem. Some approaches to solve this question are known.

As the local description of non degenerate singularities is given in terms of products of elliptic, hyperbolic and focus-focus components; the number of these elements at each point of the level will play an essential role in the classification problem. Furthermore, the affine structure with singularities on the level $L_0$, furnished by the Hamiltonian vector fields associated to the moment map, gives strong restrictions on the set of invariants that we know.

Our proposal in this talk is to show that this affine structure on the level allows us to the construction of a " natural "completely integrable system in a neighborhood of the level. The cotangent bundle to the desingularisation of $L_0$ is used to find this model. This construction looks like the simplest one with the given affine structure on the level.

**
May 29 Thursday,
Special Talk by Gang Tian at Cosmocaixa**

19:00 pm, Auditori, Cosmocaixa, Gang Tian, Princeton University La conjectura de Poincaré i la geometria

La conjectura de Poincaré, presentada l'any 1904, dóna una caracterització topològica de l'espai tridimensional més simple (és a dir, la 3-esfera) i ha estat la qüestió central de la topologia, una branca molt important de les matemàtiques. La conjectura, des que es va formular ara fa més de cent anys, ha estat temptejada repetidament amb tota mena de mètodes topològics, però no hi ha hagut manera de resoldre-la. L'objectiu d'aquesta conferència és proporcionar d'una manera intuïtiva una breu panoràmica de la conjectura de Poincaré i la geometria relacionada.