Resumen

Usando cirugía simpléctica a lo largo de subvariedades simplécticas construimos una clase particular de variedades simplécticas de dimensión 4. La característica fundamental es que dado un entero positivo k, una de estas variedades simplécticas tendrá grupo de Flux isomorfo a Z^k.

Los resultados que presentaremos son parte de una colaboración con I. Hasse.

Resumen

I will begin by discussing the Segal-Bargmann transform for the unitary group U(N), which maps an L^2 space of function on U(N) to a space of holomorphic L^2 function on the general linear group GL(N;C). The idea is to replace the Gaussian functions in the classical transforms of Segal and Bargmann by heat kernels on U(N) or GL(N;C). I will then discuss what happens when the size N of the matrices goes to infinity. This “large-N limit” has a surprising application to random matrix theory.

Resumen

In this talk, we will present mechanical systems subjected to external forces in the framework of symplectic geometry. We obtain Noether’s theorem for Lagrangian systems with external forces, among other results regarding symmetries and conserved quantities, as well as the reduction procedure when the system is invariant under the action of a Lie group. We also introduce a Hamilton-Jacobi theory and a discrete version of mechanical systems with external forces.

Resumen

We study the propagation of semi-classical localized solutions of Schroedinger or wave equations (Gaussian beams) on a certain class of singular spaces. These spaces are obtained by connecting a number of smooth manifolds by several segments. Laplacians on such spaces are defined with the help of extension theory and depend on boundary conditions in the points of gluing. Statistics of a number of Gaussian packets is governed by the behavior of geodesics on manifolds and is connected with certain problems of analytic number theory, in particular, with the problem of distribution of abstract primes.

Resumen

Las teselaciones de Voronoi surgen de manera natural en las aplicaciones: pigmentación de animales, patrones en reacciones químicas, crecimiento urbano y de colonias de bacterias, etc. En el contexto de las ecuaciones diferenciales parciales, las teselaciones de Voronoi surgen cuando se considera la interacción de frentes a partir de fuentes puntuales. En esta plática discutimos varios ejemplos de EDPs que incluyen:

• Un juego de colonización estocástica.

• Interacción de frentes difusivos.

• El principio de Huygens en óptica.

• Un problema de transporte óptimo de masas y la ecuación de Monge-Ampère.

Resumen

Slow-fast Hamiltonian systems are characterized by a separation of the phase space into slow and fast parts typically identified by a small (or slow) parameter. This kind of Hamiltonian system is not integrable, in general; even though in one degree of freedom. In this lecture, we study an integrable model associated with a slow-fast Hamiltonian system of two degrees of freedom. Thinking of the slow parameter as a perturbative one, making suitable symmetry assumptions, and using normal form theory, we show that a slow-fast Hamiltonian system in two degrees of freedom is close to an integrable Hamiltonian model. What we gain with this model is the possibility to associate a family of Lagrangian 2-tori which is almost invariant with respect to the original slow-fast Hamiltonian system. As an important application, this family of almost invariant Lagrangian 2-tori can be used to compute approximations to the spectrum of the quantum model associated with the slow-fast Hamiltonian systems.

Resumen

The canonical Maslov operator is one of the powerful tools for constructing semiclassical asymptotics. Classical objects arising in the canonical operator are Lagrangian manifolds in phase space, the features of their projection onto the configuration (physical) space are caustics, focal points, and other Lagrangian singularities. In their neighborhood, the asymptotics determined by the canonical operator is represented as an integral over the impulse variables (or part of them). The representation recently proposed by S.Y. Dobrokhotov, V.E. Nazaikinsky and A.I. Shafarevich relies on integration in the corresponding domains (maps) directly by some coordinates on a Lagrangian manifold. This representation significantly simplifies the construction of the canonical operator and allows us to expand its scope to frequently encountered problems with non-smooth Lagrangian manifolds. In addition, these representations for a number of problems allow us to write asymptotic solutions uniformly in the form of special functions of a complex argument in a wide neighborhood of caustics. As applications, we consider examples from the theory of wave beams and the linear theory of waves on water. This talk is based on joint work with V.E. Nazaikinskii, A.I. Shafarevich, A. Yu. Anikin, D. S. Minenkov, A. A. Tolchennikov and A. V. Tsvetkova.

Resumen

We presented a symplectic realization of the dynamic structure of two interacting spin-two fields in three dimensions. A significant simplification refers to the treatment of constraints: instead of performing a Hamiltonian analysis á la Dirac, we worked out a method that only uses properties of the pre-symplectic two-form matrix and its corresponding zero-modes to investigate the nature of constraints and the gauge structure of the theory. For instance, we demonstrate that the contraction of the zero-modes with the potential gradient, yields explicit expressions for the whole set of constraints on the dynamics of the theory. In the case of gauge structure, the transformation laws for the entire set of dynamical variables are more straightforwardly derived from the structure of the remaining zero-modes; in this sense, the zero-modes must be viewed as the generators of the corresponding gauge transformations. Thereafter, we use an appropriate gauge-fixing procedure, the time gauge, to compute both the quantization brackets and the functional measure on the path integral associated with our model. Finally, we confirm that three-dimensional bi-gravity has two physical degrees of freedom per space point. With the above, we provide a new perspective for a better understanding of the dynamical structure of theories of interacting spin-two fields, which does not require the constraints to be catalogued as first- and second-class ones as in the case of Dirac's method.

Resumen

Se presenta una descripción y algunas propiedades de una clase de estructuras de Poisson en variedades fibradas llamadas de casi-acoplamiento. Término que refiere a cierta compatibilidad que se puede definir entre estructuras de Poisson y estructuras fibradas. Además, se muestra cómo las estructuras de Poisson de casi-acoplamiento ayudan al estudio de la geometría semilocal e infinitesimal de las llamadas subvariedades de Poisson. Lo que en particular da lugar a un formalismo algebraico que se relaciona con los denominados Módulos de Poisson.

Resumen

Presentamos una manera de construir estructuras de Poisson que nos permiten estudiar en un contexto puramente algebraico las llamadas álgebras de Poisson infinitesimales que surgen en el estudio de la geometría semilocal de subvariedades de Poisson. Dicha construcción se lleva a cabo en extensiones triviales de álgebras y es basada en la noción de derivada contravariante. En particular, estudiamos un tipo de estructuras asociadas a derivadas contravariantes planas, llamadas Módulos de Poisson. Además, caracterizamos los elementos de Casimir, las derivaciones de Poisson y las derivaciones Hamiltonianas de tales estructuras e introducimos su primera cohomología de Poisson reducida.