Dissertation defence (Mathematics): MSc Rahim Kargar


26.6.2024 at 12.00 - 16.00
MSc Rahim Kargar defends the dissertation in Mathematics titled “Metrics and Quasiconformal Maps” at the University of Turku on 26 June 2024 at 12.00 (University of Turku, Agora, XX lecture hall, Turku).

The audience can participate in the defence by remote access: https://utu.zoom.us/j/61578190536

Opponent: Professor Swadesh Kumar Sahoo (Indian Institute of Technology Indore, India)
Custos: Professor Matti Vuorinen (University of Turku)

Doctoral Dissertation at UTUPub: https://urn.fi/URN:ISBN:978-951-29-9765-7


Summary of the Doctoral Dissertation:

In my dissertation, I studied the geometric function theory, which is a subfield of classical analysis. The subject of research is conformal and quasi-conformal mappings, as well as harmonic mappings and their properties. One key topic is to study the distortion of the distances of pairs of points under the mappings with the help of special functions.

The importance of the research field is based on connections and applications, e.g. physics, technology, and other fields of mathematics, e.g. dynamical systems, potential theory, and number theory.

In my research, I examine various methods of determining the distances between pairs of points within subdomains of Euclidean space with the help of metrics and the transformation of the distances measured in this way under the mappings. The most suitable metrics include hyperbolic, quasihyperbolic, and visual angle metrics.

The most important observations are the following:
• comparison of the hyperbolic metric with intrinsic metrics within a convex
polygonal domain;
• an efficient algorithm for calculating special functions that occur;
• new formulas for the visual angle metric of the unit disk and with the help of these,
a new version of Schwarzs lemma was stated;
• considerations related to Harnacks inequality;
• presentation of topics suitable for further research.

My research brings new ideas to the theory of this research area and illustrates the accuracy of the achieved results with examples.
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