Tobias Kaiser (University of Regensburg, Germany)
Hilbert 16, the Riemann Mapping Theorem, the Dirichlet problem, and o-minimality
Abstract: We investigate the connection of o-minimality and the following three concepts from analysis. 1. Hilbert's 16th Problem: What can be said about the number of limit cycles of a polynomial vector field in the plane? 2. The Riemann Mapping Theorem: A proper simply connected domain in the plane can be mapped biholomorphically onto the unit ball. 3. The Dirichlet problem: The following PDE with boundary value problem is considered. Given a domain U and a continuous boundary function h the Dirichlet solution for h is harmonic in U and can be continuously extended by h to the boundary of U. Important tools in the work on Hilbert 16 are Poincare return maps resp. transition maps at singular boundary points of the vector field. In the context of the Riemann Mapping Theorem we are interested in the case that the domain is globally semianalytic. In the context of the Dirichlet problem we are interested in the case that the domain is in the plane and that both domain and boundary function are globally semianalytic. The main result is the following. There is an o-minimal structure such that transition maps at non-resonant hyperbolic singular points, the Riemann map and the Dirichlet solution (under an additional condition on the angles at singular boundary points in both cases) are definable in this structure. In the first part of the talk we introduce the three concepts from analysis. A key step in Ilyashenko's work on Hilbert 16 is the introduction of a certain quasianalytic class in which transition maps at hyperbolic singularities live. We show that both the Riemann map and the Dirichlet solution (with semianalytic raw data) can also be realized in this quasianalytic class. In the second part we show how we obtain an o-minimal structure from this quasianalytic class. The main difficulty is the right definition of the quasianalytic classes in several variables. In the third part we discuss further developments in this direction. (This talk is sequel to the one in the Logic seminar on Tuesday Sept. 23rd.) |
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