AMS Graduate Chapter at Ohio University is delighted to feature Doctoral Candidate Khaled Qaraman as our speaker on WEDNESDAY, February 7, 2024, from 04: 10 PM to 5: 05 PM at MORTON HALL, 219. Qaraman will talk about Inversion in Higher Dimensions.

Abstract: My work aims to contribute to the study of the inversion in the curves on the 2D surfaces, specifically the curves on the Riemann Sphere. Tools required for this study include Schwarz function, Quaternions and Stereographic projection.

A basis for this study requires investigating the relations between these tools and the conformal mappings. One important Theorem in that regard includes the Invariance property of symmetric points under biholomorphic transformation. After that we are going to study the inversion in a circle on the Riemann Sphere in depth and explain the relation between this inversion and the reflection with respect to a circle or a line in the complex plane.

Next, we are going to study the inversion in a spherical ellipse, then we consider more complicated spherical curves and explain the general method to study the inversion in a spherical curve and its relationship with the conformal mapping from the unit circle to the stereographic projection of this spherical curve on the complex plane.