# Algebra and Geometry Seminar:Jenn Qian, From the hypergeometric differential equation to conformal maps of curvilinear polygons

### Event Description:

title: from the hypergeometric differential equation to conformal maps of curvilinear polygons

abstract: The Schwarzian differential operator plays an important role in conformal maps due to its invariant property. We will show how to use the Schwarzian to construct a differential equation for the conformal map of any curvilinear polygon, a domain that is bounded by a finite number of circular arcs. We will also mention the special case of the hypergeometric equation and the hypergeometric series, which is the only case that has been solved for arbitrary angles. In general, such conformal maps remain unknown since the so-called accessory parameters in the differential equation are difficult to determine.