Ru, Min2019-09-142019-09-14May 20192019-05May 2019Portions of this document appear in: Ru, Min, and Richard Walden. "Uniqueness Results for Holomorphic Mappings on the Disc." Acta Mathematica Vietnamica: 1-11.https://hdl.handle.net/10657/4657This dissertation gives a brief exposition of the history of Value Distribution Theory, often times referred to as Nevanlinna theory, and studies the case for Nevanlinna theory for holomorphic maps where the source is a disc developed by Ru-Sibony [16]. We start with a motivation into the subject and lay out some classical formulations with a focus on applications to the shared value problems. These problems are also referred to as uniqueness theorems. It is known that if two complex polynomials P and Q share two values without counting multiplicities, then they are the same. Such problem is called the shared value problem. In this dissertation, we focus on the study of the shared value problem for holomorphic maps where the source is a disc. There are derivations for several new unicity results for a class of holomorphic mappings from the disc into compact Riemann surfaces and n-dimensional complex projective space.application/pdfengThe author of this work is the copyright owner. UH Libraries and the Texas Digital Library have their permission to store and provide access to this work. UH Libraries has secured permission to reproduce any and all previously published materials contained in the work. Further transmission, reproduction, or presentation of this work is prohibited except with permission of the author(s).Nevanlinna theoryComplex geometry2019-09-14Thesisborn digital