Working Seminar on Mathematical Physicsof HSE University and Skoltech Center for Advanced Studies. Speaker: Artem Posadskij (MIPT)
Conformal mapping onto a polygon with several cuts
I will talk about the generalization of the Kufarev's method for finding accessory parameters in the Christoffel-Schwartz integral. It is based on the use of Löwner's differential equation, which describes one-parameter families of conformal maps of the canonical domain onto polygons with a straight cut. Using the operation of cutting, it is possible to reduce the problem of finding parameters to solving several Cauchy problems for ODE systems.
We propose a modification of Kufarev's method for the case of several cuts growing together. The work considers some questions that arise when constructing a generalized method, derives the corresponding Löwner equation and a system of ODEs describing the dynamics of accessory parameters. Numerical calculations were also made to demonstrate the operability of this method