Vorträge in der Woche 01.08.2016 bis 07.08.2016

Dienstag, 02.08.2016: Analytic torsion and dynamical zeta functions of locally symmetric spaces

Polyxeni Spilioti

Donnerstag, 04.08.2016: Discretization of the Schwarzian derivative

Toshiaki Itoh (Doshisha University, Japan)

The Schwarzian derivative is old topics, but it appears often in many research areas even now. For example, the interest of discretization of analytically integrable or geometrical system sometimes leads us to the Schwarzian derivative.There were many works about the Schwarzian derivative already. Therefore at the beginning in this study, we review the correspondence between the Schwarzian derivative and the Cross-ratio, then consider their application. As an example, we apply discretized Schwarzian derivative to conformal mapping. It is well known that the Schwarzian derivative gives conformal mapping from upper half plane of complex space to inside of multi-arc polygon. In this application, since discrete counterpart of the Schwarzian derivative is the Cross-ratio actually, we apply Cross-ratio to conformal mapping. The problems of what we meet on such mapping with Cross-ratio are treatment of singularities of the mapping at the vertices of polygon and evaluation of accessary parameters. It found we can solve this by coming and going between the Schwarzian derivative and Cross-ratio. Moreover this approach leads us to the entrance of discrete integrable system of special functions from the classical treatment of 2nd order ODE and difference equations. Such discrete integrable system is composed of simultaneous equation of the two Möbius transformations or discrete Riccati's equations. Acknowledgements. I would like to express my gratitude to Numerical Analysis group of Mathematisches Institut, Universität Tübingen in Germany for many assistance of my visiting and research.