Milivojević, Aleksandar (2015) Postojanje meromorfnih funkcija na Riemannovim plohama. Diploma thesis, Faculty of Science > Department of Mathematics.

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Abstract
This thesis is an introduction to the theory of Riemann surfaces. We define Riemann surfaces and holomorphic maps between them, and prove some basic properties. We then apply complex analytic techniques to address the question of existence of meromorphic functions on an arbitrary Riemann surface. We study differential 1forms and decompose the Hilbert space of squareintegrable 1forms. Using this decomposition, we prove that there exist meromorphic 1forms with prescribed singularities, whence we obtain the existence of meromorphic functions as well. Our focus then narrows to compact Riemann surfaces. We define algebraic curves, and then briefly expound the necessary theory to state the RiemannRoch theorem, which links the degree of a divisor to the dimension of a related meromorphic function space on an algebraic curve. Applying this theorem, we prove that every algebraic curve can be embedded into projective space. We then comment on the fact that the previously proven result on the existence of meromorphic functions on Riemann surfaces implies that all Riemann surfaces are algebraic curves.
Item Type:  Thesis (Diploma thesis) 

Supervisor:  Muić, Goran 
Date:  2015 
Number of Pages:  81 
Subjects:  NATURAL SCIENCES > Mathematics 
Divisions:  Faculty of Science > Department of Mathematics 
Depositing User:  Iva Prah 
Date Deposited:  23 Oct 2015 08:49 
Last Modified:  23 Oct 2015 08:49 
URI:  http://digre.pmf.unizg.hr/id/eprint/4173 
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