Last update on 11/18/99.

MAT 543 Fall 1999 Complex Analysis II

Course Instructor: Irwin Kra 4-111, 2-8273, irwin@math.sunysb.edu. Course meets: Tu, Th 2:20-3:40pm in Harriman Hall 205. The first meeting is on Thursday 9/2. IK office hours: W 3-4, Th 1-2, F 11-12 (in UG office) and by appointment.

The course will will cover:

1. Elliptic function theory (an introduction to Riemann surfaces via θ-functions).

2. Analytic continuation (the Monodromy theorem).

3. The Dirichlet problem and uniformization of multiply connected domains.

4. Normal families; theorems of Picard, Montel, Marty.

5. The Ahlfors version of the Schwarz-Pick lemma with applications.

6. (If time permits) An introduction to combinatorial number theory (via complex analysis).

There will be no formal homework assignments nor any examinations in the course. Each student will be expected to present a 15 minute talk on a subject related to the material of the lectures. These talks should be relatively self contained (using mathematical ideas, terminology and results from first year graduate courses only) presenting a proof of a single theorem or outlining the state of current knowledge in a narrow area of complex analysis.

If you have any condition, such as a physical or mental disability, that will make it difficult to complete the course, please notify me as soon as possible so that appropriate arrangements can be made.

There are many fine books on the subject. Although I will not follow any one completely, reading one will greatly increase your understanding. Some references on elliptic functions are [1], [2] (notes will be handed out for this), [6]; for analytic continuation, [1]; for normal families, [4]; for the Schwarz-Pick lemma, [5].

References

[1] L.V. Ahlfors, Complex Analysis (third edition), MaGraw-Hill, 1979.

[2] H.M. Farkas and I. Kra, Theta Constants, Riemann Surfaces and the Modular Group, monograph in preparation.

[3] , Riemann Surfaces (second edition), Graduate Texts in Mathematics, vol. 71, Springer-Verlag, 1992.

[4] E. Hille, Analytic Function Theory, Volume II, Blaisdell, 1962.

[5] S. Krantz, Complex Analysis, the Geometric Viewpoint, vol. 1990, Mathematical Association of America.

[6] H. McKean and V. Moll, Elliptic Curves; Function Theory, Geometry, Arithmetic, Cambridge University Press, 1997.

[7] L. Zalcman, A heuristic principle in complex function theory, Amer. Math. Monthly 82 (1975), 813–817.

These books have been placed on reserve in the library.

This announcement and its updates will be posted on the web. There will be a pointer to it in my home page (www:math.sunysb.edu/˜ irwin).

Lectures (including reference to material covered and student assignments) L1: 9/2/99. The family of tori, [2]. L2: 9/7. The algebraic curve associated to a torus, [2]. L3: 9/9. The differential equation satisfied by the ℘-function, [2]. L4: 9/14. Algebraic invariants for tori. Plumbing, [2]. 9/16. School closed because of “Floyd.” L5: 9/21. Projective embeddings of tori. Theta functions, [2]. L6: 9/23. Applications of theta function to elliptic function theory, [2]. L7: 9/28. Applications of theta function to elliptic function theory, [2] (continuation). L8: 9/30. Applications of theta function to elliptic function theory, [2] (conclusion). Analytic continuation, [1]. L9: 10/5.Analytic continuation, [1] (continuation). Addition formulae for the ℘-function, [1] (Teo). L10: 10/7. Monodromy, [1]. Branch points, [1]. L11: 10/12. Algebraic functions, [1], [3]. L12: 10/14. Analytic configurations and algebraic functions. The Riemann surface of w^{2} = P (z), [1], [3]. L13: 10/19. The Riemann surface of w^{3} = z(z − 1)(z − 2)(z − 3), [3] (Milea). Ordinary differential equations – ordinary points for linear equations, [1]. L14: 10/21. Ordinary differential equations – singular points for linear equations, [1]. L15: 10/26. Lacunary values and Picards’s theorem, [1] (Kim). Hypergeometric functions, [1]. L16: 10/28. Riemann surfaces for some simple algebraic equations, [3] (Tanaka). Normal families , [1]. L17: 11/2. Zalcman’s heuristic principle,[7]. L18: 11/4. Koebe’s distortion without bounds (Rempe). Harnack’s principle, [1]. L19: 11/9. Perron’s Theorem; solution to the Dirichlet problem, [1].

L20: 11/11. Reflection principle, conformal mappings, [1]. L21: 11/16. Green’s function, [1]. L22: 11/18. Schwarzian derivatives and univalent functions. L23: 11/23. Properties of the Schwarzian. The Poincaré metric. L24: 11/30. Univalence criteria. L25: 12/2. The Ahlfors-Scwarz-Pick lemma. L26: 12/7. Roots of the discriminant. L27: 12/9. Partitions and divisors.