Programa

El programa incluye:

• Dos mini-cursos.
  

  Fernando Barbero (IMAFF, CSIC): "Introduction to General Relativity"
  
  The purpose of this minicourse is to provide a bridge from the mathematics of differential geometry to the gravitational physics of general relativity. After discussing the basic geometric framework I will pay attention to the problem of extracting physical information from Lorentzian metrics and the physics of curvature. After this I will introduce the Einstein field equations and discuss, in particular, the mechanism by which matter curves the geometry of spacetime. Finally I will devote some time to a review of current research topics on classical general relativity covering both geometrical aspects and physical applications.
  
  Juan Pablo Ortega (CNRS, Niza): "Momentum maps and Hamiltonian Reduction"
  
  The different techniques to associate conservation laws to the symmetries of a hamiltonian dynamical system is collectively known as momentum map theory and has been
  a major subject of research during the last forty years. Momentum maps are used to simplify
  the study of the dynamical features of symmetric systems via a procedure generically referred
  to as reduction. Reduction techniques in the context of symplectic and Poisson geometry are the
  main subject of this course in which we will mainly focus in the relation of these topics with
  Lie theory, normal forms, stratifications, singularities, groupoid actions, and Hamiltonian distributions.
  

• Conferencias invitadas de 45 minutos.
  
  Ana Bela Cruzeiro (Grupo de Fisica Matematica, Univ. de Lisboa, and and IST, Univ. Técnica de Lisboa): "Riemannian path space geometry"
  
  I shall revew the basic definitions of a Riemannian geometry on the space of paths of a manifold, which is endowed with the probability measure of the Brownian motion of the underlying manifold. This geometry was defined in A.B. Cruzeiro & P.Malliavin, J. Funct. Anal. 139 (1996).
  
  More recentely, in collaboration with X.Zhang, we reconstructed this geometry via finite-dimensional approximations. This approach allows to reconstruct the Ornstein-Ulhenbeck semigroups on the path space and prove some (infinite-dimensional) inequalities, as those of Harnack or Littlewood-Paley type.
  
  Rui Loja Fernandes (Instituto Superior Técnic, Universidade Técnica de Lisboa): "Rigidity vs softness in Poisson Geometry"
  
  Rigidity and flexibility phenomenon are well-known in symplectic geometry. In this talk I will argue that questions about rigidity and flexibility are harder to study in Poisson geometry, and I will give some new results in this direction.
  
  Eric Loubeau (Département de Mathématiques, U. de Bretagne Occidentale, Brest, Francia): "An apercu of harmonic maps and morphisms"
  
  I will start with an overview of the theory of harmonic maps. Choosing the standpoint of the Calculus of Variations, the energy of a map between Riemannian manifolds is a functional, its critical points being the harmonic maps. I will review their best-known properties and most appealing applications, in particular, (non-)existence, minimal submanifolds, stability problems.
  Then, specialising to geometrical aspects, harmonic morphisms will be brought in, along with their numerous features: semi-conformality, curvature of the fibres, stability. The last part of the talk will be a showcase of frameworks where the above concepts could be of interest to mathematical physicists:
  i) sections of bundles: Choosing the target space to be the total space of a vector bundle over the domain, one can investigate sections which are harmonic maps. It is well-known that for the tangent bundle of a Riemannian manifold equipped with the Sasaki metric, the answer is parallel vector fields. Extra conditions must be imposed to obtain interesting problems.
  ii) Einstein manifolds: The curvature conditions on the fibres of a harmonic morphism make the case of Einstein manifolds particularly rich. Harmonic morphisms with one-dimensional fibres from an Einstein manifold have been classified by Pantilie as : a) Induced by Killing vector fields; b) warped-products; c), when the domain has dimension four, corresponding to solutions of the Beltrami field equations.
  iii) Weyl geometry: The geometric flavour of harmonic morphisms, makes conformal geometry a natural framework, where Twistor Theory invites itself in.
  iv) semi-Riemannian manifolds and gravitation: Both harmonic maps and morphisms can be extended to semi-Riemannian geometry. Recently, Mustafa has shown how harmonic morphisms can be applied to analyse a coupled gravity system with a geometrical meaning, and obtain spontaneous compactification and splitting.
  v) biharmonic maps: There are two natural generalisations of harmonic maps: $p$-harmonic maps and biharmonic maps. Opting for the latter, which models elasticity, we will describe the new challenges of this burgeoning field.
  
  Juan Carlos Marrero (U La Laguna): "Subvariedades lagrangianas en algebroides de Lie y su aplicacion en la descripcion de la dinamica"
  
  En 2001 E. Martinez probo que un algebroide de Lie puede ser prolongado de tal manera que sobre el nuevo espacio se pueden definir todos los objetos que son importantes para la construccion de sistemas lagrangianos. Usando estos objetos se derivan las ecuaciones de Euler-Lagrange (tipo algebroide de Lie) de forma intrinseca. Recientemente, completando algunas investigaciones ya iniciadas tambien por Martinez, hemos probado que es posible obtener las ecuaciones de Hamilton sobre un algebroide de Lie E, usando un formalismo simplectico y sin necesidad de acudir a la estructura de Poisson lineal sobre el fibrado dual E^*, la cual tiene la desventaja de ser singular.
  
  El próposito de esta charla es mostrar que las soluciones de las ecuaciones de Euler-Lagrange y de Hamilton asociadas a un algebroide de Lie son curvas distinguidas en cierto tipo de subvariedades de algebroides de Lie simplecticos. Tales subvariedades han sido denominadas lagrangianas. Como una primera aplicacion de esta construccion uno recupera algunos resultados probados por Tulczyjew en los 70 sin mas que tomar como algebroide de Lie de partida el trivial, esto es, el fibrado tangente del espacio de configuracion. Otra aplicacion no trivial y que motiva el estudio general realizado consiste en tomar como algebroide de Lie de partida el algebroide de Atiyah asociado a un fibrado principal con grupo de estructura G. De hecho, haciendo esta eleccion uno obtiene interpretaciones elegantes e interesantes de las ecuaciones de Lagrange-Poincare y de Hamilton-Poincare para un sistema mecanico que admite como grupo de Lie de simetrias a G.
  
  Elena Medina Reus: "Nuevas soluciones del modelo de Nizhnik-Novikov-Veseñov mediante la teoris de transformaciones de simetria"
  
  El modelo de Nizhnik-Novikov-Veselov es un conocido modelo integrable, propuesto como una extensión de la ecuación de KdV en dimensión 2+1, que a diferencia de la ecuación de KP, tiene simetría respecto al intercambio de las variables espaciales x,y. Se han determinado algunas soluciones de este modelo, por ejemplo soluciones tipo breathers, y procesos de interacción entre ellos, o algunos tipos de estructuras coherentes.
  
  Haciendo uso de la teoría de transformaciones de simetría en EDPs, construimos nuevas soluciones. Entre ellas, soluciones que describen procesos de explosión, soluciones en las que ciertas estructuras emergen de fuentes instantáneas, procesos de difusión o creación de estructuras, o soluciones tipo fuente o sumidero.
  
  Stefano Montaldo (Dipartimento di Matematica. Universita di Cagliari): "Invariant surfaces of a three-dimensional manifold with constant Gauss curvature"
  
  We give a reduction procedure to determine (locally) the surfaces with constant Gauss curvature in a three-dimensional manifold which are invariant under the action of a one parameter subgroup of the isometry group of the ambient space. We apply this procedure to classify all invariant surfaces with constant Gauss curvature in the Heisenberg space and in the product of the Hyperbolic space with the real line.
  
  Roger Picken (Departamento de Matematica Instituto Superior Tecnico): "Classical and quantum geometry of moduli spaces arising in 3-dimensional gravity"
  
  I will present some results obtained together with J. Nelson (Turin) concerning a description of the phase space of 3-dimensional Einstein gravity when space is a torus and with negative cosmological constant. The approach uses the holonomy matrices of flat SL(2,R) connections on the torus to parametrise the geometry. After quantization, matrices with non-commuting entries appear, with interesting properties, in particular q-commutation relations between them. Based on Phys.Lett. B471 (2000) 367-372, Lett.Math.Phys. 52 (2000) 277-290, Lett.Math.Phys. 59 (2002) 215-226 and an article in preparation.
  
  Estelita Vaz (Universidade do Minho, Guimaraes): "Symmetries in warped and duoble warped space-times"
  
  The importance of studying symmetries of space-times is well known. In particular, Killing vectors have been quite widely investigated. Here we will focus attention in decomposable metrics as well as in metrics which are conformal to decomposable ones, the conformal factor being separable on the variables. The attention will be focused on Killing vector fields (i.e. vector fields along which the metric remains unchanged) and on curvature collineations (i.e. vector fields preserving the curvature tensor). Some examples of spacetimes admitting those symmetries will also be presented.
  
  Jean-Claude Zambrini (Grupo de Fisica-Matematica, Univ. de Lisboa): "Brownian motion and symplectic geometry"
  
  After a summary of what is known about Brownian Motion on a Riemannian manifold (the theory is almost as old as the Brownian motion itself) I will explain why there are no such results for Symplectic manifolds. Then I will show a new way to approach this problem and various examples illustrating the insight this approach provides into many fundamental properties of the Brownian motion itself. In the second part, I'll show how the same approach gives a more general definition of symmetries in Quantum Mechanics. Most of the new quantum symmetries obtained in this way are directly inspired by their probabilistic counterparts.
  

• Comunicaciones cortas de 25 minutos.
  

  Antonio Jesus Cañete Martin: "Least-perimeter partitions of the disk"
  
  In the Calculus of Variations, problems related to isoperimetric partitions have multiple applications in physical sciences.They can properly model multitude of natural phenomena, such as the shape of a cellular tissue or the interfaces separating several fluids.

  In this talk we will consider the isoperimetric problem of partitioning a planar disk into $n$ regions of given areas with the least possible perimeter; we will discuss specially the case of three regions, whose unique solution is the standard configuration, consisting of three circular arcs or segments meeting orthogonally the boundary of the disk, and meeting in threes at $120$ degrees in an interior vertex.

  Alberto Enciso Carrasco, Daniel Peralta Salas: "A spectral theory approach to quantum
  integrability"
  
  We prove that any $n$-dimensional Hamiltonian operator with pure point spectrum is completely integrable via self-adjoint first integrals [*]. This theorem is a consequence of the following result [A]: Let $C$ be a sequence of real numbers. Then there exists an integrable $n$-dimensional Hamiltonian $H$ with pure point spectrum, whose $n$ commuting first integrals can be chosen to be self-adjoint, which realizes the sequence $C$ as its point spectrum.
  
  We provide a constructive proof for this proposition. The definition of pure point spectrum we use is that of [**]. Another consequence of Proposition A is that given any closed set $\Sigma\subset\RR$ there exists an integrable $n$-dimensional Hamiltonian which realizes it as its spectrum. Finally, we develop some nontrivial applications of our integrability criterion. For instance, a remarkable result is that: For almost all Hamiltonians with pure point spectrum, its point spectrum is uniformly distributed.
  
  As discussed in [*], this proposition ensures that Berry's conjecture, as studied by physicists, holds for the class of Hamiltonians with pure point spectrum. Although extensive numerical research has been carried out, this is, to the best of our knowledge, the widest class of Hamiltonians for which this conjecture has managed to be rigourously proved.

  References
  [*] Enciso, A. and Peralta-Salas, D.: Integrability of Hamiltonians with pure point spectrum and Berry's conjecture. Preprint.
  [**] Reed, M. and Simon, B.: {\it Functional Analysis}. Academic Press. New York (1972).

  Isabel Fernández Delgado (con Francisco López): "Maximal surfaces with conelike type singularities in complete flat three space-times"
  
  We deal with the geometry and the conformal structure of complete embedded maximal surfaces with isolated singularities in complete flat Lorentzian 3-manifolds. We also prove that a complete flat three dimensional space-time containing a surface of this kind must be the quotient of the Lorentz-Minkowski space under a translational group of rank less than or equal to two. At this point it has been crucial a result by Mess about the non existence a of certain type of Margulis space-times.
  
  Benito Hernandez Bermejo: "Nuevas soluciones de las ecuaciones de Jacobi
  para sistemas Hamiltonianos generalizados"
  
  Los sistemas dinámicos de Poisson son una generalización de los sistemas Hamiltonianos clásicos. Tienen la forma $\dot{x} = J \cdot \nabla H$, donde H es el Hamiltoniano y $J$ es la denominada matriz de estructura, que está caracterizada por dos propiedades: (1) es antisimétrica; y (2) cumple las identidades de Jacobi, que son el siguiente conjunto de $n$ ecuaciones diferenciales parciales no lineales acopladas: $ \sum_{l=1}^n ( J_{li} \partial_l J_{jk} + J_{lj} \partial_l J_{ki} + J_{lk} \partial_l J_{ij} ) = 0 $, donde las $J_{ij}$ son las entradas de dicha matriz. La gran potencia de los sistemas de Poisson está en que suponen una generalización muy amplia de los sistemas Hamiltonianos clásicos (dado que por ejemplo permiten sistemas de dimensión impar, y aun en el caso de dimensión par engloban una variedad de sistemas mucho mayor) mientras que el carácter Hamiltoniano de la dinámica se preserva (Teorema de Darboux). Los sistemas de Poisson también presentan propiedades muy generales en otros aspectos, ya que por ejemplo todo cambio diferenciable de las coordenadas es una transformación canónica (es decir preserva el formato de sistema de Poisson). Entre los problemas no resueltos relativos a los sistemas de Poisson destaca el de identificar y escribir un sistema dado como estructura de Poisson. La dificultad principal es encontrar una matriz de estructura apropiada, ya que para ello es necesario encontrar una solución adecuada de las ecuaciones de Jacobi. Lo más habitual en la práctica viene siendo proceder mediante ansatzs sencillos y comprobar si llevan a posibilidades válidas, a falta de un conocimiento adecuado de familias suficientemente amplias de soluciones de dichas ecuaciones. Este trabajo presenta un enfoque alternativo, que considera a las ecuaciones de Jacobi como un problema per se. De esta forma se han obtenido familias de soluciones muy generales que engloban ansatzs anteriores, de manera que sistemas de Poisson ya conocidos, y hasta ahora aparentemente inconexos, resultan ser casos particulares de estructuras mucho más generales. A menudo esto ha permitid o, además, desarrollar métodos comunes para la caracterización global de propiedades independientes del Hamiltoniano, tales como la estructura simpléctica y la forma canónica de Darboux.
  
  Antonio Lopez Almorox: "Cohomology of Lie groups associated to a set of compatrible linear 1-cocycles and new Lie group structures on its cotangent bundle"
  
  It is well known that the cotangent bundle $T^*G$ of a Lie group $G$ has a Lie group structure with the product $\omega_g \cdot {\omega'}_{g'} = { (L_{g^{-1}})}^T_{*,\, g {g'}} [{\omega'}_{ g'}] + {({R}_{{g'} } )}^T_{*, \, g {g'}} [w_g]$. As the vector bundle $T^*G $ is trivial, one can define non standard cotangent lifts of the left and the right actions of $G$ using $ Aut\, {\mathcal G}^*$-valued 1-cocycles. Given a set of compatible linear 1-cocycles $( {\cal L},{\tilde{\cal R}}) $ for $G$, one can define a cohomoly $H^\cdot_{({\cal L},{\tilde{\cal R}})}(G, {\mathcal G}^*)$ associated to them. In this talk, we will construct new Lie groups structures on $T^*G \simeq G \times {\mathcal G}^*$ using sets of compatible linear 1-cocycles $( {\cal L},{\tilde{\cal R}}) $ for $G$ and an element $\xi \in Z^2_{({\cal L},{\tilde{\cal R}})}(G, {\mathcal G}^*)$. In general, these Lie group structures for $T^*G$ are not isomorphic to the standard one and they are (non central) Lie group extensions of the Lie group $G$ by ${\mathcal G}^*$. We will also analyze when these group structures on $T^*G$ are semidirect products of $G$ by ${\mathcal G}^*$ but this depends of the triviality of the class $[\xi] \in H^2_{({\cal L},{\tilde{\cal R}})}(G, {\mathcal G}^*)$. These news products eill be be used in order to construct new $T^*G $-invariant symplectic forms or $T^*G $-invariant (almost) complex structures on $T^*G$.
  

  Josep LLosa Carrasco (con Daniel Soler): "Riemannian metrics as deformations of a constant curvature metric"
  
  It is known, since an old result by Riemann, that a n-dimensional metric has f = n(n-1)/2 degrees of freedom, i.e. it is locally equivalent to the giving of f functions. As this feature is related to some particular choices of local charts it seems to be generically a not covariant property.

  According to it, a two-dimensional metric has f=1 degrees of freedom. In this case, however, a stronger result holds, as it is well known, namely: any two-dimensional metric g is locally conformally flat, g=\phi\eta, \phi being the conformal deformation factor and \eta the flat metric.

  Contrarily to what the above Riemann's general result suggests, the two-dimensional case is intrinsic and covariant, as it only needs the knowledge of the metric g and only involves tensor quantities, specifically, the sole degree of freedom is represented by a scalar, the conformal deformation factor \phi.

  The question thus arises of, whether or not, for n>2 there exist similar intrinsic and covariant local relations between an arbitrary metric g, the corresponding flat one \eta and a covariant set of f quantities. We here proof that this conjecture is true, at least locally.

  Jose M. Martin Senovilla: "Trapped submanifolds in Lorentzian geometry"
  
  In Lorentzian geometry, the concept of trapped submanifold will be introduced by means of the properties of the mean curvature vector. Trapped submanifolds are generalizations of the standard maximal hypersurfaces and minimal surfaces, of geodesics, and also of the trapped surfaces introduced by Penrose. Examples and selected applications to gravitational theories will be presented.
  
  Gil Salgado Gonzalez, A. O. Sánchez-Valenzuela: "Superalgebras de Lie basadas en gl(n)"
  
  Clasificamos superálgebras de Lie tales que su ${\Bbb Z}_2$ graduación esta dada por $\frak{gl}_n \oplus \frak{gl}_n$ y la acción del $\frak{gl}_n$ par en el $\frak{gl}_n$ impar es por medio de la representación adjunta. Demostramos que para $n \geq 3$ hay una familia (a un parámetro) de superálgebras de Lie no isomorfas, más un conjunto finito de diferentes clases de isomorf\'{\i}a. Para $n=2$ hay 10 diferentes clases de isomorf\'{\i}a si el campo considerado es $\Bbb R$, mientras que hay 8 si el campo es $\Bbb R$. Para $n=1$ hay dos diferentes clases de isomorf\'{\i}a independientemente del campo considerado. Damos representantes de cada clase de isomorf\'{\i}a y determinamos sus respectivos grupos de automorfismos. También respondemos la pregunta de cuales clases de isomorf\'{\i}a admiten estructuras geométricas $\operatorname{ad}$-invariantes ${\Bbb Z}_2$ graduadas (de tipo ortogonal o simpléctico).
  
  
• Sesiones de posters
  

  Angel Ballesteros Castañeda (joint work with F.J. Herranz and O. Ragnisco): "q-Poisson coalgebras and integrable dynamics on spaces with curvature"
  
  A direct connection between $q$-deformation and integrability on spaces with curvature is presented. The non-standard deformation of a $sl(2)$ Poisson coalgebra is used to introduce an integrable Hamiltonian that describes the geodesic motion on a two-dimensional space with negative and non-constant curvature. Another Hamiltonian defined on the same deformed coalgebra is also shown to generate integrable motions on the two-dimensional Cayley-Klein spaces. In this case, the constant curvature of the space is just the deformation parameter $z=\ln q$.
  
  Manuel Calixto Molina: "Higher-spin representations of U(p,q), coherent states and Kähler structures on flag manifolds"
  
  Using Berezin and Geometric Quantization techniques, we develop the representation theory of (pseudo-)unitay groups U(p,q), calculate arbitrary-spin coherent states and derive K\"ahler structures on flag manifolds. They are essential ingredients to define operator symbols, their star-commutators and to discuss the classical limit.
  
  Fernando Etayo, Rafael Santamaría, Ujué R. Trías: "Triple structures on a manifold"

  Many geometric concepts can be defined by a suitable algebraic formalism. This point of view has interest because one can compare different geometric structures having similar algebraic expressions. In the present paper we study manifolds endowed with three (1,1)-tensor fields $F$, $P$ and $J$ satisfying $$F^2=\pm 1\, ,\, P^2=\pm 1\, ,\, J=PF\, ,\, PF\pm FP=0.$$

  We analyse the geometries arising from the above algebraic conditions. According to the choosen signs there exist eight different geometries. We show that, in fact, there are only four. In particular, hypercomplex manifolds and manifolds endowed with a 3-web fit to this construction. We study geometric objects associated to these manifolds (such as metrics, connections, etc.), restrictions on dimensions, etc., and we show significative examples.

  Raúl Manuel Falcón Ganfornina and Juan Núñez Valdés: "Basic concepts on Lorentz symmetry and Minkowsky isogeometry by using the MCIM isotopic model"
  
  At 1978, R.M. Santilli proposed to generalize the conventional Lie theory by using isotopies. To do it, he considered that the basic unit of any mathematical structure can depend on external factors (such as position, speed, acceleration, time, temperature or density): $\wh{I}=\wh{I}(x,\stackrel{\bullet }{x},\stackrel{\bullet \bullet }{x},...,\mu ,\tau ,...)$. In this way, he obtains a more general Lie theory, based on nonassociative, nonlinear and nonlocal-integral systems, which allows him to identify Euclidean, Riemannian and Minkowskian spaces, when working with extended particles or high energies into unusual physical conditions in exterior or interior dynamic systems. Since the 80's, several mathematicians and physicists have investigated in Lie-Santilli's isotheory. Particularly, Lorentz symmetry and Minkowsky isogeometry have been studied by Santilli and A.K. Aringazin in the 90's. We propose to review some of these concepts by using the MCIM isotopic model studied at 2001, that generalize Santilli's one. So, we will need to generalize the construction of isovectorspaces and the isodifferential calculus.
  
  Olga Gil-Medrano y Ana Hurtado: "Volume, energy and generalized energy of unit vector fields on Berger's spheres. Stability of Hopf vector fields"
  
  We study to what extent the known results concerning the behaviour of Hopf vector fields, with respect to volume, energy and generalized energy functionals, on the round sphere are still valid for the metrics obtained by performing the canonical variation of the Hopf fibration.
  
  Xavier Gracia Sabate, Ruben Martin Grillo: "Vector hulls of affine spaces and affine bundles"
  
  For every affine space $A$ there exists a canonical immersion $A \to \hat A$ as a hyperplane in a vector space, which is called the vector hull of $A$. This fact is greatly clarifying, both for affine geometry and for its applications. In particular, the construction of the vector hull can be extended to affine bundles.

  In this contribution, we will study some aspects of this construction and we will show that the vector hull of some interesting affine bundles can be identified with well-known vector bundles. For example, given a bundle $M \to {\bf R}$, the tangent bundle $\mathrm{T} M$ is a model for the vector hull of the jet bundle ${\rm J}^1 M$.

  These results can be applied to the study of differential equations. In particular, given a time-dependent differential equation of a certain class, we show a method to obtain an equivalent autonomous one.

  Julio Guerrero: "Quantum Dynamics of a particle in de Sitter universe"
  
  The dynamics of a quantum test particle in a 1+1D de Sitter background is considered. The difficulties and inconsistencies that arise are analysed and an interpretation in terms of the Continuous Series of representations of $SL(2,{\mathbb R})$ (a double covering of the symmetry group $SO(2,1)$) is given.
  
  David Iglesias, Juan Carlos Marrero, Diana Sosa, Edith Padrón: "Compatibility between affine Jacobi and Lie affgebroid structures on an affine bundle"
  
  On the first jet manifold of a fibration with total space a Poisson manifold and base space the real line, we can define an affine Jacobi structure and a Lie affgebroid structure which are compatible in a certain sense. This fact suggests us to introduce the compatibility notion between these types of structures defined over an arbitrary affine bundle. In this note we will study several aspects related with this concept of compatibility and we will present some examples which illustrate this notion.
  
  Miguel Ángel Javaloyes Victoria: "Geometrical models of relativistic particles: a special case"

  The Lagrangian whose density depends on the curvatures of the curve is a good model for relativistic particles with spin. The idea is to describe the freedom degrees with the intrinsical geometry of the curve instead of adding more dimensions. In a former work we study this kind of Lagrangian and use two Killing vector fields along the curve to obtain the critical curves. For that goal, we need one of the two Killing vector fields not to be constant. In this work we analyse what happens when all the two Killing vector fields are constant.

  M. de León, J. Marín-Solano, J. C. Marrero, M. C. Muñoz-Lecanda, N. Román-Roy: "Constraint algorithm for singular field theories"
  
  It is well known that for systems of ODE's describing singular dynamical systems, the existence and uniqueness of solutions are not assured. This happens in the Lagrangian and Hamiltonian formalisms of system described by non-regular Lagrangians, and for certain systems of ODE's arising from some technical applications. In all these cases, there are geometrical constraint algorithms that, in the most favourable cases, give a maximal submanifold of the phase space of the system, where consistent solutions exist.
  
  The same problems arise when considering system of PDE's associated with field theories described by singular Lagrangians (both in the Lagrangian and Hamiltonian formalisms), as well as in some other applications related with optimal control theories. Working in the framework of the multisymplectic description for this kind of theories, we present a geometric algorithm for finding the maximal submanifold where there are consistent solutions of the system.

  Eva Miranda: "Singularidades en sistemas integrables"
  
  Las singularidades estan presentes en muchos sistemas integrables. El objetivo de esta conferencia es presentar diversos resultados de estructura semi-local à la Arnold-Liouville para singularidades de sistemas integrables en variedades simplécticas y de contacto. Algunos de dichos resultados estan contenidos en trabajos conjuntos con Nguyen Tien Zung y Carlos Currás-Bosch.
  
  Daniel Peralta-Salas: "On the geometry and topology of the equilibrium shapes on
  manifolds"

  Let $(P1)$ be certain elliptic free-boundary problem on a Riemannian manifold $(M,g)$. In this work [*] we study the restrictions on the topology and geometry of the fibres $f^{-1}(t)$, $t \in f(M)$, of the solutions $f$ to $(P1)$. We study these restrictions with geometrical rather than with analytical techniques. The form of $(P1)$ is a generalization of the equations modeling static self-gravitating fluids. We only study the class of solutions of $(P1)$ satisfying certain regularity condition of the fibres (the analytic representation hypothesis). This assumption is of geometrical nature and is motivated by physical reasons. We prove [*] that the partition of $(M,g)$ induced by these solutions is an equilibrium partition (An analytic function $I$ is an equilibrium function if $I$, $\|\nabla I \|^2$ and $\Delta I$ agree fibrewise. The partition induced by an equilibrium function is called an equilibrium partition.) For general Riemannian manifolds we prove [*] the following theorem: Any equilibrium partition of a Riemannian manifold possesses a fibre bundle local structure, each one of its fibres has constant mean curvature and locally the neighbour fibres are geodesically parallel.
  
  We also give \cite{ref} further properties of the equilibrium partitions in more specific spaces, i. e. locally symmetric, conformally flat and constant curvature Riemannian manifolds. We apply \cite{ref} these results to the classical problem in physics of classifying the equilibrium shapes of both Newtonian and relativistic self-gravitating fluids. Our techniques are completely different to the previous ones appeared in the literature of this topic. We recover and generalize the classical theorems of Lichtenstein, Lindblom, ... and lay down the foundations of a general theory of equilibrium shapes in general spaces \cite{ref}.
  
  [*] Pelayo, A. and Peralta-Salas, D.: Topological and geometrical restrictions, free-boundary problems and self-gravitating fluids. Preprint available at math-ph/0305038.

  Miguel Rodriguez Olmos: "Symmetry reduction of cotangent-lifted abelian actions"

  We will expose recent results dealing with the symplectic reduction program applied to symplectic manifolds which are cotangent bundles and the symmetry is given by a (possibly non free) group action by cotangent lifts. We will show how the general theory of Hamiltonian reduction particularizes to this case, and explain the particularities in the stratification of the involved reduced spaces. Currently this scheme is developped for reduction at momentum values which are trivial coadjoint orbits, allowing us then to provide a full description of reduction for abelian actions. The dynamic properties of symmetric mechanical systems defined on these spaces will be briefly explored.

  Manual César Rosales Lombardo: "Stable constant mean curvature hypersurfaces inside convex domains"
  
  Constant mean curvature (CMC) surfaces are geometric objects that model multitude of physical phenomena; they appear, for example, as the interfaces between two inmiscible fluids or between two gases at different pressures. In the context of the Calculus of Variations, CMC surfaces arise as critical points of the area for volume preserving variations.

  In this talk we study stable CMC hypersurfaces inside a domain $D$: by definition, these are local minima of the area among the hypersurfaces in $D$ separating a given amount of volume. By using variational and geometric arguments, we give a complete description of compact, stable CMC hypersurfaces inside certain convex domains $C$ which are invariant under the action of a group of dilations (convex cones, for example). As a consequence, we also characterize the isoperimetric hypersurfaces in $C$ -global minima of the area inside $C$ for fixed volume-.

  E. Torrente-Lujan, G. G. Volkov: "Root systems from Toric Calabi-Yau Geometry. Towards new algebraic structures and symmetries in physics?"
  
  The algebraic approach to the construction of the reflexive polyhedra that yield Calabi-Yau spaces in three or more complex dimensions with K3 fibres reveals graphs that include and generalize the Dynkin diagrams associated with gauge symmetries. In this presentation, after a general introduction, we present some recent results concerning the study of the structure of graphs obtained from $CY_3$ reflexive polyhedra. We show how some particularly defined integral matrices can be assigned to these diagrams. This family of matrices and its associated graphs may be obtained by relaxing the restrictions on the individual entries of the generalized Cartan matrices associated with the Dynkin diagrams that characterize Cartan-Lie and affine Kac-Moody algebras. These graphs keep however the affine structure, as it was in Kac-Moody Dynkin diagrams. We presented a possible root structure for some simple cases. We conjecture that these generalized graphs and associated link matrices may characterize generalizations of these algebras.

  Luis Ugarte Vilumbrales: "Strong K\"ahler with torsion structures and balanced metrics in six dimensions"
  
  For any Hermitian manifold $(M,J,g)$, there is a unique connection for which $g$ and $J$ are parallel and the torsion $g(\cdot,T(\cdot,\cdot))$ is totally skew-symmetric. This connection was introduced by Bismut, and the resulting 3-form can be identified with $J\, d\Omega$, where $\Omega$ is the fundamental form. Metric connections with such torsion have applications in string theory, supersymmetric $\sigma$-models and the geometry of black hole moduli spaces.

  If $J\, d\Omega$ is closed but non-zero (which excludes the K\"ahler case) then the Hermitian structure is called {\it strong K\"ahler with torsion} (SKT for short). Recently, Fino, Parton and Salamon have classified 6-dimensional nilmanifolds admitting invariant SKT structures, and they have described the space of such structures on the Iwasawa manifold. We show an alternative description of the space of SKT structures on any 6-dimensional nilmanifold. In particular, in our description SKT structures on the Iwasawa manifold are parametrized by the points inside an ovaloid in the 3-dimensional Euclidean space.

  A Hermitian metric is said to be {\it balanced} if its associated Lee form vanishes identically.
  Fino and Grantcharov have recently shown the existence of invariant complex structures on the Iwasawa manifold which do not admit a Hermitian structure whose Bismut connection has restricted holonomy in SU(3), because they have no compatible balanced metrics.
  This provides counter-examples to a conjecture of Gutowski, Ivanov and Papadopoulos. We classify 6-dimensional nilmanifolds having such property.
  
  Eduardo J. S. Villaseñor (conjunto con J. Fernando Barbero, Guillermo A. Mena Marugan): "Perturbative and non-perturbative quantum Einstein-Rosen waves"

  We discuss the connection between the Fock space introduced by Ashtekar and Pierri for the quantization of the Einstein-Rosen waves and its perturbative counterpart, based on the concept of particle that arises in linearized gravity with a de Donder gauge. We show that, in the Ashtekar and Pierri approach, the perturbative vacuum is not accesible as a normalizable state analytic in the interaction constant, and interpret this fact as an indication that the two Fock quantizations are unitarily inequivalent.

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