1999 seminars


Amphitheatre of Interdisciplinary Complex at the University of Lisbon

Rui Albuquerque, Universidade de Évora

On Twistor Spaces

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.


Room P3.10, Mathematics Building

Philippe Monnier, Instituto Superior Técnico

Levi decompositions of Poisson structures

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


Amphitheatre of Interdisciplinary Complex at the University of Lisbon

Inês Cruz, Universidade do Porto

Estrutura de Poisson transversa a uma folha simpléctica

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.


Room P3.10, Mathematics Building

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

Vector bundles on curves and generalized theta functions. Abelian theta functions.

Definition, interpretation as sections of line bundles. The associated maps into projective space. The heat equation. The Heisenberg group.

References

Any good book on abelian varieties and theta functions, like:

O. Debarre. Tores et variétés abéliennes complexes. Cours Spécialisés 6. Société Mathématique de France, EDP Sciences (1999).


Room P3.10, Mathematics Building

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

Vector bundles on curves and generalized theta functions. Vector bundles and principal bundles on curves.

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.

References

A. Beauville. Vector Bundles on Curves and Generalized Theta Functions: Recent Results and Open Problems. Current topics in complex algebraic geometry, MSRI Publications 28, 17-33; Cambridge University Press (1995).


Room P3.10, Mathematics Building

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

Vector bundles on curves and generalized theta functions. Generalized $\theta$ functions and conformal blocks

The moduli space as a double coset space of an infinite dimensional group. Applications: the Picard group; the space of conformal blocks.

References

Ch. Sorger. La formule de Verlinde. Exp. 794 du Séminaire Bourbaki, Astérisque 237 (1996), 87–114.


Room P3.10, Mathematics Building

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

Vector bundles on curves and generalized theta functions. Generalized theta functions and conformal field theory.

Conformal field theories and conformal blocks; the Verlinde formula.

References

A. Beauville. Vector bundles on Riemann surfaces and Conformal Field Theory. Algebraic and Geometric Methods in Mathematical Physics, 145-166; Kluwer (1996).


Room P3.10, Mathematics Building

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

Vector bundles on curves and generalized theta functions. Generalized theta functions and topological field theory.

Topological field theories. The Chern-Simons action. The heat equation for generalized theta functions.

References

E. Witten. Quantum field theory and the Jones polynomial. Comm. Math. Phys. 121 (1989), 351–399.

M. Atiyah. The geometry and physics of knots. Lezioni Lincee, CUP, Cambridge,1990.

N. Hitchin. Flat connections and geometric quantization. Comm. Math. Phys. 131 (1990), 347–380.


Room P3.10, Mathematics Building

Margarida Mendes Lopes, CMAF, Universidade de Lisboa

Surfaceswith no global 1-forms and 2-forms

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.


Room P3.10, Mathematics Building

José Natário, Instituto Superior Técnico

Warp Drive

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.


Room P3.10, Mathematics Building

Gustavo Granja, Instituto Superior Técnico

Ahomotopy decomposition for a group of symplectomorphisms of $S^2\times S^2$

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.


Room P3.10, Mathematics Building

Nguyen Tien Zung, Université Paul Sabatier - Toulouse

Proper groupoids and intrinsic convexity of momentum maps

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.


Room P3.10, Mathematics Building

Constantin Teleman, Cambridge University

Twisted K-Theory and Applications I

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).
  • Freed: ICM Proceedings 2002.
  • Freed, Hopkins, Teleman: math.AT/0206257.


Room P3.10, Mathematics Building

Constantin Teleman, Cambridge University

Twisted K-Theory and Applications II

The Verlinde algebra as twisted K-theory

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).
  • Freed, Hopkins, Teleman: math.AT/0206257.


Room P3.10, Mathematics Building

Constantin Teleman, Cambridge University

Twisted K-Theory and Applications III

The Dirac-Ramond operator for a loop group

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.


Room P3.10, Mathematics Building

Elizabeth Gasparim, New Mexico State University

Two applications of instanton numbers

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.


Room P3.10, Mathematics Building

Constantin Teleman, Cambridge University

Twisted K-Theory and Applications IV

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).

References

  • Beaville, Laszlo: Conformal blocks and generalized theta functions. Comm. Math. Phys. 164 (1994).
  • Teleman: Borel-Weil-Bott theory on the moduli stack of G-bundles over a curve. Invent. Math. 134 (1998).


Amphitheatre of Interdisciplinary Complex at the University of Lisbon

Mário Edmundo, CMAF, Universidade de Lisboa

O-minimal singular (co)homology and real etale cohomology

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.


Room P3.10, Mathematics Building

Nitya Kitchloo, Johns Hopkins University

On lifting isometries

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.


Room P3.10, Mathematics Building

Peter Newstead, University of Liverpool

Coherent Systems on Algebraic Curves

We will discuss the latest developments on coherent systems on algebraic curves and implications for Brill-Noether theory.


Room P3.10, Mathematics Building

Jaroslaw Kedra, University of Szczecin

Groups of symplectomorphisms have complicated topology

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.


Room P3.10, Mathematics Building

José Agapito, U. C. Santa Cruz

A weighted version of quantization commutes with reduction principle for a toric manifold

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.


Room P3.10, Mathematics Building

Rahul Pandharipande, Princeton University

Gromov-Witten theory and Donaldson-Thomas theory

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).


Room P3.10, Mathematics Building

Oana Dragulete, École Polytechnique Fédérale de Lausanne

Cosphere bundle reduction in contact geometry

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.