Seminar: Riemann surfaces, Wintersemester 2021/22

Dozent: Prof. Dr. Vladimir Lazić
Sprechstunde: Nach Vereinbarung per Email.

Assistenten: Dr. Nikolaos Tsakanikas, Dr. Zhixin Xie
Sprechstunde: Nach Vereinbarung per Email.

NEWS (July 2021)

The seminar begins on Tuesday, 19 October 2021. The first meeting will take place online via Zoom.

The seminar will be in English.

To register for the Seminar: please send an email to Prof. Dr. Lazić lazic [add @math.uni-sb.de]. Please mention 2 or 3 topics of your preference. This email should include also the following data: Name, Surname, Matrikelnummer and your Uni Saar email address.

Dates: Tuesdays 14-16

Topics

The topic of this seminar are Riemann surfaces. These objects generalise the concept of complex curves, that is, the zero loci in C2 of complex polynomials in two variables; they can also be viewed as real algebraic objects of real dimension two, hence the name surfaces. They were first introduced and extensively studied by Riemann in the 19th century and have a very rich geometry which has inspired much of modern mathematics. We will study their algebraic, analytic, geometric and topological properties in this seminar.

Prerequisites: Linear Algebra I and II, Analysis I and II, as well as Complex Analysis (Funktionentheorie) are necessary. Some knowledge of Algebra is recommended, some knowledge of Algebraic Geometry and Topology is useful, but not necessary.

Important: The talks will be distributed on the first come/first served principle. The content of each talk should be discussed with Dr. Tsakanikas and Dr. Xie a week or two before the date of the talk.

Exam Criteria

  • Regular participation at the Seminar.
  • Successful presentation of a topic (90 minutes).
  • Studients of the new Studienordnung, who need 7 CP, have to write a short essay (around 5 pages).

Distribution of talks

Each talk follows the corresponding part of Kirwan's book (see Literature below).

Date

TopicSpeaker
19.10.2021

Introduction to complex curves and Riemann surfaces (Chapter 1)

Nikolaos Tsakanikas, Zhixin Xie
26.10.2021Topological preliminaries I: topological spaces, continuous maps, compactness, Haudorff spaces, connectedness, quotient topologiesNikolaos Tsakanikas, Zhixin Xie
02.11.2021Topological preliminaries II: covering projections (Section C.1)Nikolaos Tsakanikas, Zhixin Xie
09.11.2021Complex curves and the projective spaces (Sections 2.1 and 2.2)Igor Schlegel
16.11.2021Affine and projective curves and the weak Bézout's theorem (Sections 2.3, 2.4 and 3.1 until 3.9, including possibly only sketches of proofs of 3.3, 3.4 and 3.6)Leonard DeSimone
23.11.2021Strong Bézout's theorem (Section 3.1 between 3.18 and 3.22 with sketches of some applications)TBA
30.11.2021Cubic curves and branched covers of P(Sections 3.2 and 4.2)TBA
07.12.2021The degree-genus formula (Section 4.3)TBA
14.12.2021

The Weierstrass p-function (Section 5.1)

Maximilian Tornes
04.01.2022Riemann surfaces (Section 5.2)TBA
11.01.2022Holomorphic differentials and the statement of Abel's theorem (Sections 6.1 and 6.2 until 6.24, possibly without 6.16)TBA
18.01.2022Proof of Abel's theorem and divisors of meromorphic differentials (Section 6.2 from 6.23 and Section 6.3 until 6.36)TBA
25.01.2022The Riemann-Roch theorem (Section 6.3 from 6.37 with possibly only sketches of proofs of 6.29 and 6.45)TBA
01.02.2022  
08.02.2022  

 

Literature

Zu dieser Veranstaltung wurde von der Campus-Bibliothek für Informatik und Mathematik einen Semesterapparat eingerichtet. Eine Übersicht befindet sich hier.

Address

Fachrichtung Mathematik
Campus, Gebäude E2 4
Universität des Saarlandes
66123 Saarbrücken
Germany