| Week | Monday | Wednesday | Assignments, suggested reading |
|---|---|---|---|
| 1/23 | Introduction and some history Derivatives |
Complex analytic functions, Complex integration | Zakeri sections 1.1-1.3 Notes Mon 1/23, Wed 1/25 Homework hw #1, due Wednesday 2/1 |
| 1/30 | Cauchy's theorem in a disk, Goursat's theorem | Morera's theorem, Liouville's theorem, Cauchy integral formula | Zakeri section 1.3, 1.4 Notes Mon 1/30, Wed 2/1 Homework hw #2, due Wednesday 2/8 |
| 2/6 | Mapping properties of holomorphic functions |
Local normal form, Open mapping theorem, Homotopy | Zakeri sections 1.5, 2.1 Notes Mon 2/6, Wed 2/8 Homework hw #3, due Wednesday 2/15 |
| 2/13 | Covering properties of the exponential | More on covering by the exponential; Winding numbers and the Jump Principle | Zakeri sections 2.2-2.3 Notes Mon 2/13, Wed 2/15 Homework hw #4, due Wednesday 2/22 |
| 2/20 | Homology and Cauchy's theorem | Singularities, The Riemann sphere | Zakeri sections 2.4,2.5, 3.1, 3.2 Notes Mon 2/20, Wed 2/22 Homework hw #5, due Wednesday 3/1 |
| 2/27 | Laurent series, Residues | The residue theorem; index of fixed points | Zakeri sections 3.3,3.4 Notes Mon 2/27, Wed 3/1 Homework hw #6, due Wednesday 3/8 |
| 3/6 | The argument principle (see this animation), Rouché's theorem | The Möbius group | Zakeri sections 3.5, 4.1 Notes Mon 3/6, Wed 3/8 Homework hw #7, due Wednesday 3/22 |
| 3/13 | Spring Break (probably not like this) |
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| 3/20 | Automorphisms of the Riemann sphere, complex plane, unit disk, and upper half-plane; Schwarz lemma; dynamics of Möbius maps | midterm Here is the midterm and the solutions. Grades are on Brightspace |
Zakeri sections 4.2, 4.3 Notes Mon 3/20 Homework none this week |
| 3/27 | Conformal metrics | The hyperbolic metric | Zakeri sections 4.4,4.5 Notes Mon 3/27, Wed 3/29 Homework hw #8, due Wednesday 4/5 |
| 4/3 | Compact convergence; Arzela-Ascoli theorem | Convergence of holomorphic functions; Hurwitz's theorem and Montel's theorem | Zakeri sections 5.1-5.2 Notes Mon 4/3, Wed 4/5 Homework hw #9, due Wednesday 4/12 |
| 4/10 | Normal families of meromorphic maps; Marty's theorem | The Riemann mapping theorem; Schlict functions. See also this illustration of the square-root trick. | Zakeri sections 5.3, 6.1, 6.2 Notes Mon 4/10, Wed 4/12 Homework hw #10, due Wednesday 4/19 |
| 4/17 | Bieberbach-deBranges Theorem, Koebe 1/4-Theorem | Boundary behavior of Riemann maps: Caratheodory's theorem, Schwarz Reflection Principle. | Zakeri sections 6.2, 6.3 Notes Mon 4/17, Wed 4/19 Homework hw #11, due Wednesday 4/26 |
| 4/24 | Properties of harmonic functions; the mean-value property and the maximum principle for harmonic functions. | Poisson integral formula, the Poisson kernel. | Zakeri sections 7.1, 7.2, 7.3 Notes Mon 4/24, Wed 4/26 Homework hw #12, due Wednesday 5/3 |
| 5/1 | Poisson kernel, harmonic and holomorphic Schwarz Reflection Principle, Schwarz-Christoffel Theorem | Covering spaces, holomorphic coverings, inverse branches | Zakeri sections 10.3, 12.1, 12.2, 12.3 Notes Mon 5/1, Wed 5/3 Homework HW is impractical this week |
| 5/8 | Final Cumulative Friday, May 12, 11:15-1:45, Physics P-127 |