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master's thesis
Postojanje meromorfnih funkcija na Riemannovim plohama
Zagreb: University of Zagreb, Faculty of Science, 2015. urn:nbn:hr:217:680399
Milivojević, Aleksandar
University of Zagreb
Faculty of Science
Department of Mathematics

Institutional repository: Repository of the Faculty of Science

Cite this document

Milivojević, A. (2015). Postojanje meromorfnih funkcija na Riemannovim plohama (Master's thesis). Zagreb: University of Zagreb, Faculty of Science. Retrieved from https://urn.nsk.hr/urn:nbn:hr:217:680399

Milivojević, Aleksandar. "Postojanje meromorfnih funkcija na Riemannovim plohama." Master's thesis, University of Zagreb, Faculty of Science, 2015. https://urn.nsk.hr/urn:nbn:hr:217:680399

Milivojević, Aleksandar. "Postojanje meromorfnih funkcija na Riemannovim plohama." Master's thesis, University of Zagreb, Faculty of Science, 2015. https://urn.nsk.hr/urn:nbn:hr:217:680399

Milivojević, A. (2015). 'Postojanje meromorfnih funkcija na Riemannovim plohama', Master's thesis, University of Zagreb, Faculty of Science, accessed 23 December 2024, https://urn.nsk.hr/urn:nbn:hr:217:680399

Milivojević A. Postojanje meromorfnih funkcija na Riemannovim plohama [Master's thesis]. Zagreb: University of Zagreb, Faculty of Science; 2015 [cited 2024 December 23] Available at: https://urn.nsk.hr/urn:nbn:hr:217:680399

A. Milivojević, "Postojanje meromorfnih funkcija na Riemannovim plohama", Master's thesis, University of Zagreb, Faculty of Science, Zagreb, 2015. Available at: https://urn.nsk.hr/urn:nbn:hr:217:680399

Metadata
TitlePostojanje meromorfnih funkcija na Riemannovim plohama
AuthorAleksandar Milivojević
MentorGoran Muić (mentor)
Committee memberGoran Muić (predsjednik povjerenstva)
Committee memberBoris Širola (član povjerenstva)
Committee memberŽeljka Milin-Šipuš (član povjerenstva)
Committee memberTina Bosner (član povjerenstva)
GranterUniversity of Zagreb
Faculty of Science
(Department of Mathematics)
Zagreb
Defense date and country2015-07-10, Croatia
Scientific / art field,
discipline and subdiscipline		NATURAL SCIENCES
Mathematics
Abstract
Ovaj rad je uvod u teoriju Riemannovih ploha. Definiramo Riemannove plohe i holomorfna preslikavanja medu njima i dokazujemo neka njihova osnovna svojstva. Zatim tehnikama iz kompleksne analize pristupamo pitanju postojanja meromorfnih funkcija na proizvoljnoj Riemannovoj plohi. Prouˇcavamo diferencijalne 1-forme i činimo dekompoziciju Hilbertovog prostora kvadratno integrabilnih 1-formi. Pomoću te dekompozicije dokazujemo da postoje meromorfne 1-forme s unaprijed zadanim singularitetima, odakle dobivamo i meromorfne funkcije. Zatim se usredotočimo na kompaktne Riemannove plohe. Definiramo algebarske krivulje, te ukratko izlažemo teoriju potrebnu da bi se iskazao Riemann-Rochov teorem, koji povezuje stupanj divizora i dimenziju pripadnog prostora meromorfnih funkcija na algebarskoj krivulji. Pomoću tog teorema dokazujemo da se svaka algebarska krivulja može uložiti u projektivan prostor, te komentiramo da prethodno dokazan rezultat o postojanju meromorfnih funkcija na Riemannovoj plohi povlači da je svaka kompaktna Riemannova ploha algebarska krivulja.
Abstract (english)
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 1-forms and decompose the Hilbert space of square-integrable 1-forms. Using this decomposition, we prove that there exist meromorphic 1-forms 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 Riemann-Roch 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.
Keywords
Keywords (english)
Languagecroatian
URN:NBNurn:nbn:hr:217:680399
Study programmeTitle: Pure Mathematics
Study programme type: university
Study level: graduate
Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
Type of resourceText
File originBorn digital
Access conditionsOpen access
Terms of useLicence In copyright icon
Created on2019-01-24 12:48:46