We give a brief survey on the theory of symplectic connections over a symplectic manifold $(M,\omega)$, ie a torsion free linear connection such that $\nabla \omega=0$. A result of P. Tondeur assures that these connections always exist. Some unsolved problems still arise in the case of $(\mathbb{R}^{2n},\omega)$ and thus we look carefully on a new result for translation invariant connections.

Next we introduce the bundle of all compatible complex structures on the tangent spaces of $M$, which is called a Twistor Space if it is seen together with a canonical almost complex structure $J^\nabla$ induced by a symplectic connection on $M$. The integrability equation of $J^\nabla$ appears in terms of the curvature: it agrees with the vanishing of the Weyl curvature tensor. With this theorem, which in a final correct version is due to J. Rawnsley, we may draw a bridge between the study of symplectic connections and that of twistor spaces. We proceed with the presentation of a certain biholomorphism of the twistor space induced by some symplectomorphism on the base manifold. This is used in understanding farther the action of $\operatorname{Symp}(M,\omega)$ on the space of symplectic connections. We give examples. Finally and having time, we show some complex geometrical properties of the twistor space, namely a kählerian structure if and only if $\nabla$ is flat, the degree of holomorphic completeness and the vanishing of the Penrose transform in the symplectic setting.

Notice that the referred result on the Weyl tensor strongly resembles the analogue of twistor theory in riemannian geometry. Hence we take the opportunity of this seminar to make a commemoration of the 25 years of that landmark in twistors which is Self-duality in four-dimensional riemannian geometry from M. Atiyah, N. Hitchin and I. Singer.

We state the existence of Levi decompositions for (formal, analytical and smooth) Poisson structures and give applications of this result on some linearization problems.

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Amphitheatre of Interdisciplinary Complex at the University of Lisbon

Segundo o teorema de Decomposição de Weinstein (1983), por cada ponto de uma variedade de Poisson passam: a) uma folha simplética e b) uma variedade de Poisson a ela transversa. Estas estruturas (simpléctica e transversa) são locais e únicas (a menos de difeomorfismo de Poisson local). Neste seminário consideraremos uma classe especial de variedades de Poisson: o dual de uma álgebra de Lie com a sua estrutura de Lie-Poisson. A folha simpléctica por um ponto não é mais do que a órbita coadjunta por esse ponto. Usando uma escolha adequada da variedade transversa (Roberts & Cushman - 2002) deduziremos uma fórmula simples para o cálculo da estrutura de Poisson transversa à órbita coadjunta.

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Room P3.10, Mathematics Building

Arnaud Beauville, Laboratoire J. A. Dieudonné, Université de Nice

Moduli spaces and their Picard group. The determinant bundle. The case of vector bundles: the linear system $|\mathcal{L}|$ and its geometry, the $\theta$ divisor of a vector bundle.

Given a compact complex algebraic surface $S$, the geometric genus $p_g$ of $S$ is the dimension over $\mathbb C$ of the space of holomorphic (or regular) $2$-forms and the irregularity $q$ of $S$ is the dimension of the space of holomorphic (or regular) $1$-forms. The complex projective plane $\mathbb{P}^2$ satisfies $p_g=q=0$, but there are surfaces satisfying $p_g=q=0$ which are not bimeromorphically equivalent to $\mathbb{P}^2$. Although many examples of such surfaces of general type are known, few general results are known.

In this seminar, after a brief historical introduction, some recent general results related to the bicanonical map of surfaces of general type with $p_g=q=0$ will be presented.

We shall describe in this seminar the initial value formulation of Einstein's equations, which yields an evolution problem for a Riemannian metric on a 3-manifold. This formulation will then be used to construct “warp drive” type solutions, which allow arbitrarily high effective speeds. Properties of these solutions will be analysed.

In joint work with Sílvia Anjos we continue the analysis by Abreu, MacDuff and Anjos of the topology of the group of symplectomorphisms of $S^2\times S^2$ when the ratio of the areas of the spheres lie in the interval $]1,2]$. We express the group of symplectomorphisms up to homotopy as the pushout (or amalgam) of certain compact Lie subgroups. We use this to give a homotopy decomposition of its classifying space and compute the corresponding ring of characteristic classes for symplectic fibrations.

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Room P3.10, Mathematics Building

Nguyen Tien Zung, Université Paul Sabatier - Toulouse

I will discuss the convexity properties of momentum maps in symplectic geometry, including Atiyah - Guillemin -Sternbarg - Kirwan theorems and "nonlinear" convexity theorems, from an intrinsic point of view. The "Poisson convexity conjecture" of Weinstein will be formulated and a proof of it indicated. The local structure of (symplectic) proper groupoids plays a key role in this conjecture.

K-Theory and its twisted versions: definitions and properties

Definition of K-theory of a space, using vector bundles and using families of bounded (Fredholm) operators. Group actions and equivariant K-theory. The Chern character. Twistings for K-theory and the twisted Chern character.

References

Bouwknegt et. al: Twisted K-theory and K-theory of bundle gerbes. Comm. Math. Phys. 228 (2002).

A refresher on loop groups and their positive-energy representations, the fusion product and the Verlinde algebra. Computation of the twisted \(K_G(G)\) in simple cases (\(S^1\), \(SU(2)\), \(SO(3)\)). Gradings and graded representations.

References

Pressley, Segal: Loop Groups. Oxford University Press.

Freed: The Verlinde algebra is twisted equivariant K-theory. Turkish J. Math. 25 (2001).

Kostant's "cubic" Dirac operator. The Dirac operator on a loop group. Construction of the twisted K-class for a positive energy representation of a loop group, by coupling the Dirac operator to a connection.

References

Landweber: Multiplets of representations and Kostant's Dirac operator for equal rank loop groups. Duke Math. J. 110 (2001).

Mickelsson: Gerbes, (twisted) K-theory, and the supersymmetric WZW model. hep-th/0206139.

In this talk I will make use of two instanton numbers called the height and the width of the instanton, whose sum gives the topological charge. I will define these numbers. Then I will give two applications: first I will show that these numbers provide good stratifications of moduli of bundles on a blown up surface and second I will use these numbers to distinguish curve singularitites.

Twisted K-theory and the moduli of holomorphic G-bundles on a Riemann surface

The Frobenius algebra structure and relation to the index theory for the moduli of G-bundles on Riemann surfaces. The moduli space of flat G-bundles and the stack of all holomorphic G-bundles. (*Time permitting: higher twistings and general index formulas).

O-minimal structures are a generalization of real algebraic and analytic geometry and are considered to be the realization of Grothendieck program of "topologie moderee". In this talk we will give a brief introduction to o-minimal geometry and compare the o-minimal singular cohomology with the real etale cohomology.

Given a principal G bundle E® B (where G is a compact Lie group and E and B are closed manifolds), one would like to know if the isometry group of B can be mada to act on E via bundle maps. This question was asked in the 60's by Palais and Stewart and they answered it when G is a torus. There were various suibsequent proofs of this result. We give another proof based on Yang-Mills theory. We also formulate the question for the case of G being a general compact Lie group and discuss related issues.

I am going to prove that in many cases the rational cohomology ring of groups of symplectomorphisms is infinitely generated (as a ring!). The first part of the talk will be concerned with algebraic topology methods (rational homotopy, Gottlieb groups etc). The main result will appear as the (not so direct) application. Details can be found in my recent paper Evaluation fibrations and topology of symplectomorphisms (math.AT/0305325). To understand the talk it is enough to know basic algebraic topology.

With the use of Atiyah-Bott-Berline-Vergne formula in equivariant cohomology, we count lattice points in the moment polytope of a toric manifold with isolated fixed points. Lattice points are counted with weights depending on the codimension of the smallest dimensional face in the polytope that contains them. We derive a way of computing the equivariant Hirzebruch characteristic of a line bundle over a toric manifold and get, as a particular case, a weighted version of quantization commutes with reduction principle for toric manifolds. More generally, we obtain a polar decomposition of a simple polytope, whose proof is purely combinatorial. The talk is based on my latest work on the subject.

Let X be a Calabi-Yau 3-fold. GW theory concerns maps of curves to X. Holomorphic Chern-Simons theory, defined by S. Donaldson and R. Thomas in the mid 90's, concerns the moduli ideal sheaves on X. I will describe a conjectural GW/DT correspondence which equates these two theories via a simple change of variables (joint work with D. Maulik, N. Nekrasov, and A. Okounkov).

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Room P3.10, Mathematics Building

Oana Dragulete, École Polytechnique Fédérale de Lausanne

We extend the theorems concerning the equivariant symplectic reduction of the cotangent bundle to contact geometry. The role of the cotangent bundle is taken by the cosphere bundle. We use Albert's method for the reduction at zero and Willett's method for non-zero reduction.