250067 VO Riemann surfaces (2018S)
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Details
max. 25 Teilnehmer*innen
Sprache: Englisch
Prüfungstermine
Donnerstag
28.06.2018
Montag
09.07.2018
Dienstag
10.07.2018
Montag
17.09.2018
Donnerstag
11.10.2018
Mittwoch
19.12.2018
Mittwoch
20.11.2019
Dienstag
22.09.2020
Lehrende
Termine (iCal) - nächster Termin ist mit N markiert
Dienstag
06.03.
09:45 - 12:15
Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
Dienstag
13.03.
09:45 - 12:15
Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
Dienstag
20.03.
09:45 - 12:15
Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
Dienstag
10.04.
09:45 - 12:15
Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
Dienstag
17.04.
09:45 - 12:15
Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
Dienstag
24.04.
09:45 - 12:15
Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
Dienstag
08.05.
09:45 - 12:15
Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
Dienstag
15.05.
09:45 - 12:15
Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
Dienstag
29.05.
09:45 - 12:15
Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
Dienstag
05.06.
09:45 - 12:15
Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
Dienstag
12.06.
09:45 - 12:15
Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
Dienstag
19.06.
09:45 - 12:15
Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
Dienstag
26.06.
09:45 - 12:15
Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
Information
Ziele, Inhalte und Methode der Lehrveranstaltung
Art der Leistungskontrolle und erlaubte Hilfsmittel
Mündliche Prüfung
Mindestanforderungen und Beurteilungsmaßstab
Der Student muss die Inhalte der Vorlesung verstanden haben und sie wiedergeben können.
Prüfungsstoff
In der Vorlesung besprochene Themen.
Literatur
Ein Vorlesungsskriptum wird zur Verfügung gestellt:
http://www.mat.univie.ac.at/~armin/lect/Riemann_surfaces.pdf
Die Vorlesung basiert auf dem folgenden Buch.O. Forster, Lectures on Riemann surfaces, Graduate Texts in Mathematics, vol. 81, Springer- Verlag, New York, 1991.
http://www.mat.univie.ac.at/~armin/lect/Riemann_surfaces.pdf
Die Vorlesung basiert auf dem folgenden Buch.O. Forster, Lectures on Riemann surfaces, Graduate Texts in Mathematics, vol. 81, Springer- Verlag, New York, 1991.
Zuordnung im Vorlesungsverzeichnis
MANV, MGEV
Letzte Änderung: Mi 23.09.2020 00:28
plane to get a single valued function on the covering space.Abstract Riemann surfaces are by definition connected complex one-dimensional manifolds. They are the natural domains of definitions of holomorphic functions in one variable.First we will introduce Riemann surfaces and discuss basic properties. We develop the fundamentals of the theory of topological covering spaces including the fundamental group, the universal covering, and deck transformations. It will turn out that non-constant holomorphic maps between Riemann surfaces are covering maps, possibly with branch points.Then we get acquainted with the language of sheaves. It proves very useful in the construction of Riemann surfaces which arise from the analytic continuation of germs of holomorphic functions. Some attention is devoted to the Riemann surfaces of algebraic functions, i.e., functions which satisfy a polynomial equation with meromorphic
coefficients.For the further study of Riemann surfaces we need the calculus of differential forms. We also briefly discuss periods and summands of automorphy.Another important tool for the investigation of the geometry of Riemann surfaces is Cech cohomology. We will develop the basics of this theory. We shall only need the cohomology groups of zeroth and first order. The long exact cohomology sequence will prove useful for the computation of cohomology groups. On Riemann surfaces we
prove versions of Dolbeault's and deRham's theorem.Next we will focus on compact Riemann surfaces. We present and prove the main classical results, like the Riemann--Roch theorem, Abel's theorem, and the Jacobi inversion problem. Following Serre, all the main theorems are derived from the fact that the first cohomology group with coefficients in the sheaf of holomorphic functions is a finite dimensional complex vector space. The proof of this fact
is based on a functional-analytic result due to Schwartz. Its dimension is the genus of the Riemann surface. By means of the Serre duality theorem we will see that the genus equals the maximal number of linearly independent holomorphic one-forms on the compact Riemann surface. Eventually, it will turn out that the genus is a topological invariant. Much of this part is concerned with the existence of meromorphic functions on compact Riemann
surfaces with prescribed principal parts or divisors.Non-compact Riemann surfaces are studied next. The function theory of non-compact Riemann surfaces has many similarities with the one
on regions in the complex plane. In contrast to compact Riemann surfaces, there are analogues of Runge's theorem, the Mittag--Leffler theorem, and the Weierstrass' theorem. The solution of the Dirichlet problem, based on Perron's method, will provide a further existence theorem. It will lead to a proof of Rado's theorem that every Riemann surface has a countable topology. We shall also prove the uniformization theorem for Riemann surfaces: any simply connected Riemann surface is isomorphic to one of three normal forms, i.e,
the Riemann sphere, the complex plane, or the unit disk. Evidently, this is a generalization of the Riemann mapping theorem. As a consequence we get the classification of Riemann surfaces: every Riemann surface is isomorphic to the quotient of one of the three normal forms by a group of Möbius transformations isomorphic to the fundamental group of the Riemann surface which acts discretely and fixed point freely.Apart from some familiarity with basic complex analysis, general topology, and basic algebra no other prerequisites are demanded. All necessary tools will be developed when needed.