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Introduction to Riemann Surfaces, Spring 2016

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Course Information

• Syllabus
• Information
• Ceiba (link retired)
• Lectures: Tuesday, 13:45 ~ 15:00 and Thursday 13:45 ~ 13:00 at Astro-Math 302
• Office hours: Monday and Tuesday 15:10 ~ 16:00, or by appointment, at Astro-Math 458
• Grading scheme:
  • Homework 30%
    You have three jokers: the lowest three grades will be discarded.
  • Midterm 30%
    The midterm will be a written exam. It will be held during the class of April 21 or April 28.
  • Final 40%
    The format of the final will be decided later.
• References:
  • [FK] Hershel M. Farkas and Irwin Kra, Riemann surfaces. Second edition. MR
  • [W] Hermann Weyl, The concept of a Riemann surface. MR
  • [S] George Springer, Introduction to Riemann surfaces. MR

Lecture summaries and references

• (Week 1) basic examples and properties of Riemann surfaces, maps, differential forms. Reference: [FK, §I.1 and I.3], note.
• (Week 2) more about the differential forms on a Riemann surface, Weyl's lemma. Reference: [FK, §I.3, I.4, II.1 and II.2], note and note.
• (Week 3) harmonic differential, Hodge decomposition, harmonic function with singularities. Reference: [FK, §II.3 and II.4], note.
• (Week 4) meromorphic differential, and meromorphic function. topology of compact Riemann surfaces. Reference: [FK, §II.4, II.5 and I.2], note and note.
• (Week 5) topology of compact Riemann surfaces (continued), harmonic and holomorphic differentials, bilinear relation. Reference: [FK, §I.2, III.1 and III.2], note and note.
• (Week 6) bilinear relation (continued), periods of meromorphic differentials, simplest case of Riemann-Roch. Reference: [FK, §III.3 and III.4], note.
• (Week 7) divisors and the Riemann-Roch theorem. Reference: [FK, §III.4], note.
• (Week 8) some applications of the Riemann-Roch theorem, Weierstrass points. Reference: [FK, §III.4 and III.5], note and note.
• (Week 9) Abel's theorem and Jacobi inversion problem. Reference: [FK, §III.6], note.
• (Week 10) Midterm. Elliptic integral, elliptic function and Weierstrass ℘-function. solution and note.
• (Week 11) Restoration: no class this week.
• (Week 12) Uniformization theorem: Perron's method, Green's function. Reference: [Gamelin, §XVI], note.
• (Week 13) Uniformization theorem: bipolar Green's function and the uniformization theorem. Reference: [Gamelin, §XVI], note.
• (Week 14) Torelli theorem: more on the Jacobian varieties, symmetric product. Reference: [FK, §III.11], note.
• (Week 15) Torelli theorem: translation properties of W_r, sketch of the proof of Torelli theorem. Reference: [FK, §III.11], note.
• (Week 16) (Jacobi) theta function. Reference: [FK, §VI], note.

Homework

• Homework 01: due Tuesday, March 8.
• Homework 02: due Tuesday, March 15.
• Homework 03: due Tuesday, March 22.
• Homework 04: due Tuesday, March 29.
• Homework 05: due Thursday, April 7.
• Homework 06: due Tuesday, April 12.
• Homework 07: due Tuesday, April 19.
• Homework 08: due Tuesday, May 10.
• Homework 09: due Tuesday, May 17.
• Homework 10: due Tuesday, May 24.
• Homework 11: due Tuesday, May 31.