Grupo de Análisis Geométrico y sus Aplicaciones

Departamento de Matemáticas

Universidad de los Andes

IV Taller

Geometría Sub-Riemanniana

Operadores de Dirac en espacio-tiempo curvo

24 - 28 de Septiembre 2007

Entrada libre, pero requiere una inscripción. Se entregará certificado de asistencia a los que lo soliciten sin costo alguno.

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Programa

Durante el taller se llevarán a cabo cursos cortos sobre Geometría Sub-Riemanniana, Operadores de Dirac y sus aplicaciones a la física teórica, así como charlas sobre temas afines por parte de los participantes. El evento está dirigido a estudiantes de maestría o doctorado en matemáticas y física, y a estudiantes avanzados de pregrado.

• Cursillos:

  • "Block operator matrices and an application to Dirac Operators", (Course in English),
    Monika Winklmeier, Universität Bern, Switzerland

  http://www.math.unibe.ch/lenya/math_stat/live/ueberuns/dozassist/winklmeier.html

  • Lecture 1:This lecture will be concerned with 2x2 block operator matrices, i.e., with 2x2 matrices whose entries are operators. For example, the free Dirac operator in flat space time has a realisation as a block operator matrix.  We will see that the Dirac equation in the Kerr-Newman spacetime can be decoupled into a system of two ordinary differential equations: the radial and the angular equations. Details for the decoupling process will be covered in the course by Prof. Batic. Both equations have a realisation as eigenvalue equations for block operator matrices.
    The remainder of this course will focus on the operator associated with the angular equation of the Dirac equation. We will discuss some of its properties and its spectrum qualitatively.
  • Lecture 2:In this lecture, the spectrum of the angular operator is discussed in more details. We will derive analytic bounds for its eigenvalues and compare them with numerical values found in the literature.
  • Lecture 3: Using the results on the angular operator, we will prove that the Dirac operator in the Kerr-Newman metric has a selfadjoint realisation. Moreover, we will discuss its spectrum.
    
    
  • "Dirac Operators and applications", (Course in English),
    Davide Batic, Universidad de los Andes, Bogotá D.C., Colombia
    [dbatic@uniandes.edu.co]
    • Lecture 1: Introduction to the Generalized Heun Equation: I will speak about some basic properties of  this Fuchsian ordinary differential equation and emphasize its relationship with the Heun equation.
    • Lecture 2: In the second lecture I will introduce the Dirac equation in the Kerr-Newman metric, i.e. in the geometry of a charged rotating black hole. I will show how to separate such an equation into a radial and an angular part by means of the Chandrasekhar ansatz. Finally, I will show that the separated equations can be reduced to Generalized Heun Equations after a suitable transformation. This was a joint work with Monika Winklmeier (University of Bern, Math. Dpt., Switzerland) (Batic, Schmid, Winklmeier; J.Phys. A39 (2006)12559- 12564).
    • Lecture 3: In the last lecture I will present a rigorous derivation of the Dirac propagator in the Kerr-Newman metric. I will first show how to prove the completeness of the Chandrasekhar ansatz for the Dirac equation in the above mentioned metric. Based on this, I derive an integral representation for smooth compactly supported functions which in turn I use to obtain an integral representation for the propagator of solutions of the Cauchy problem with initial data in the above class of functions.
      As a by-product, I also obtain the propagator for the Dirac equation in the Minkowski space-time in oblate spheroidal coordinates. This was a joint work with Harald Schmid (University of Regensburg, Germany) (Batic and Schmid; Prog.Theor.Phys. 116 (2006) 517-544). Finally, I will mention a relevant application of our result for the construction of wave operators Dollard modified asymptotically at infinity and classical (i.e. without modification) at the event horizon of a spinning charged black hole for massive Dirac particles. (Batic; J. Math. Phys. 48, (2007) 022502-022527).
  • "Sub-Riemannian Geometry and applications", (Course in English),
    Mikhail malakhaltsev, Kazan State University, Russia
    [Mikhail.Malakhaltsev@ksu.ru]
    • Lecture 1: Geometry of nonholonomic surface in three-dimensional Euclidean
      space. Principal curvatures of first and second type. Total and mean
      curvature of first and second type of nonholonomic surfaces. Geodesics of
      nonholonomic surfaces.
    • Lecture 2: Geometry of metric spaces. Carnot-Caratheodory distance on Lie
      groups. Geodesics. Pansu differentiability. Basic example: Heisenberg
      group.
    • Lecture 3: Frames and charts adopted to sub-Riemannian structure.
      Deformations of sub-Riemannian manifolds. Curvature of sub-Riemannian
      manifold.
      
      
      
• Conferencias:
  • "El Operador de Dirac en Geometría", Alexander Cardona G., Universidad de los Andes, Bogotá D.C., Colombia.
  • "El Problema de Yamabe", Jean carlos Cortissoz, Universidad de los Andes, Bogotá D.C., Colombia.
  • "El problema de Dido", José Ricardo Arteaga B., Universidad de los Andes, Bogotá D.C., Colombia.

 

• Materiales
  • Mikhail Malakhaltsev, "Geometría Sub-Riemanniana". (2007)
  • Mikhail Malakhaltsev, "Open problems" (2007)

Mayor información:

• Alexander CARDONA acardona@uniandes.edu.co

• José Ricardo ARTEAGA jarteaga@uniandes.edu.co

Fotos

• Finca el Noviciado, clausura