Kollokvium: Boundary value problems of complex analysis on complementary regions

  • Datum:
  • Plats: Ångströmlaboratoriet
  • Föreläsare: Eric Schippers (http://server.math.umanitoba.ca/~schippers/)
  • Kontaktperson: Jordi-Lluís Figueras
  • Seminarium

Abstract:

Given a Jordan curve in the Riemann sphere, let A and B be the bounded and unbounded components of the complement.  Consider the following two problems. (1) The transmission problem: given a complex harmonic function on A, find a corresponding harmonic function on B with the same boundary values.  (2) The jump problem: given a function on the curve, find holomorphic functions g and h on A and B respectively such that u=g-h.    When the functions in question are of bounded Dirichlet energy, these questions are solvable precisely for quasicircles, which are very rough non-rectifiable curves arising in Teichmüller theory and complex dynamics.  Similar results hold for curves dividing Riemann surfaces.

These two problems are closely related to global approximability of holomorphic functions by polynomials, and the Faber and Grunsky operators of complex function theory.  I will give a non-technical account of this circle of ideas, and shed new light on these classical objects.  If time allows I will discuss applications to conformal field theory and Teichmüller theory.