Course description: Riemann surfaces are one-dimensional complex manifolds, obtained by gluing together pieces of the complex plane by holomorphic maps. This course will be an introduction to the the theory of Riemann surfaces, with an emphasis on analytical and topological aspects. After describing examples and constructions of Riemann surfaces, the topics covered would include branched coverings and the Riemann-Hurwitz formula, holomorphic 1-forms and periods, the Weyl's Lemma and existence theorems, the Hodge decomposition theorem, Riemann's bilinear relations, Divisors, the Riemann-Roch theorem, theorems of Abel and Jacobi, the Uniformization theorem, Fuchsian groups and hyperbolic surfaces.
Prerequisites: Topology (MA 231), Complex Analysis (MA 224), Introduction to Algebraic Topology (MA 232) or equivalent courses.
Recommended books: H.M. Farkas and I. Kra, Riemann surfaces, Springer GTM 1992.
R. Miranda, Algebraic Curves and Riemann Surfaces, AMS Graduate Studies in Mathematics, 1995.
W. Schlag, A Course in Complex Analysis and Riemann surfaces, AMS Graduate Studies in Mathematics, 2014.
Grading policy: 50% homework and quizzes , 50% Final project.
There will be no class on January 5th (Tuesday).
There will be no class on December 17th (Thursday).
There will be no class on November 19th (Thursday). Also, no lectures would be held in the midterm week of Nov 23 -27.
A new Teams code has been emailed to the mailing list.
There will be weekly homework also.
There will be no lecture on Tuesday, October 6th.
Lectures will be held on Tuesdays and Thursdays, 3:30-5pm.
Our first meeting will be held on Thursday October 1st , 3:30-5pm.
Classes will be online, at the beginning over Microsoft Teams. The first meeting link will be put up here, and on the IISc Intranet Bulletin board, before the meeting time.
A set of problems will be put up here each week (Thursday/Friday), to be submitted (via uploading to Teams) before the Thursday class.
Homework 1 (due on Oct 15th)
Homework 2 (due on Oct 22nd)
Homework 3 (due on Oct 29th)
Homework 4 (due on Nov 5th)
Homework 5 (due on Nov 12th)
Homework 6 (due on Nov 19th)
Midterm Homework (due on Nov 27th)
Homework 8 (due on Dec 10th)
Homework 9 (due on Dec 18th)
Homework 10 (due on Jan 1st)
The recorded videos will be available on the Teams channel.
Lecture 1 Definitions, examples (eg. smooth affine curves).
Lecture 2 More examples -- surfaces with a Riemannian metric, polyhedral surfaces, smooth projective algebraic curves.
Lecture 3 Quotients, complex tori, Holomorphic functions -- definition and examples
Lecture 4 More on holomorphic functions, meromorphic functions on the Riemann sphere, properties of holomorphic maps
Lecture 5 Local normal form of a holomorphic map, definitions of multiplicity, ramification and branch points, branched coverings and degree.
Lecture 6 Riemann-Hurwitz formula and applications
Lecture 7 Discussion of homework problems, another example applying the R-H formula, finite group actions
Lecture 8 Hyperelliptic curves revisited, finished the discussion on finite group actions, started with elliptic functions
Lecture 9 Elliptic functions
Lecture 10 Finishing discussion of elliptic functions, Riemann surface of a holomorphic germ
Lecture 11 Discussion of homework problems, Riemann surface of an algebraic germ
Lecture 12 Ramified Riemann surfaces and algebraic curves
Lecture 13 Quotients of the unit disk, review of differential forms on surfaces, holomorphic differentials.
Lecture 14 Harmonic forms, Hodge decomposition theorem
Lectures 15 &16 Existence of meromorphic differentials and meromorphic functions
Lecture 17 Discussion of a homework problem, Intersection number, Holomorphic differentials on a compact Riemann surface
Lecture 18 Divisors, the Riemann-Roch theorem
Lecture 19 Proof of Riemann-Roch, some applications
Lecture 20 More applications of Riemann-Roch, Riemann's bilinear relations
Lecture 21 Finishing Riemann's bilinear relations, the Jacobian variety, the Abel-Jacobi map
Lecture 22 Proof of Abel's theorem
Lecture 23 Discussion of the last homework, Jacobi's theorem
Lecture 24 Proof of the Uniformization Theorem
Lecture 25 Finishing proof of the Uniformization Theorem, Riemann's moduli space
A possible list of topics will be discussed, and students will choose their topic sometime after mid-semester.
A typical topic will be something related to the material covered in class, and the student will be responsible for choosing the material to read and learn. The evaluation will be based on a report, exercises/problems solved, and on a telephonic or online viva. For a sampling of the kind of reports expected, check out the submitted mini-project reports for last semester's MA 338 course here.
Update: submitted reports below.
The Riemann-Roch theorem for line-bundles
ODEs with meromorphic coefficients on the complex plane
Riemann-Theta functions: divisor of zeroes and applications