Topics & Pacing
The following is a rough list of arguments covered in class. It is (or will be) updated on a weekly basis.
| Date | Topic |
|---|---|
| Week 1 (Jan 12 & 14) | Proof of the Riemann mapping theorem (From Stein & Shakarchi) |
| Week 2 (Jan 19 & 21) | Fractional linear (or Möbius) transformations of the extended complex plane (Jones & Singerman, Chap. 2; see also Stein & Shakarchi, Chap. 8) |
| Week 3 (Jan 26 & 28) | Cross ratio. Permutations of three and four points |
| Week 4 (Feb 4; no class on Feb 2) | Fixed points and classification of fractional linear transformations. Multipliers, traces. |
| Week 5 (Feb 9 & 11) |
Conjugacy classes and classification of fractional linear
transformations (Jones and Singerman Chap. 2 up to sect. 11). Harmonic functions (Lang, Chap. VIII, sect. 1, 2.) |
| Week 6 (Feb 16 & 18) | Harmonic functions: mean value, maximum modulus, harmonic functions on an annulus. (Lang, Chap. VIII, sect. 2, 3.) |
| Week 7 (Feb 23 & 25) |
Harmonic functions: harmonic functions on punctured
sets. (Lang, Chap VIII, sect 3, Theorem 3.9.) Poisson formula, Poisson kernel (Lang, Chap VIII, sect. 4). |
| Week 8 (Mar 2 & 4) |
Harmonic functions: construction of harmonic functions
via the Poisson kernel (Lang, Chap VIII, sect. 5) First midterm exam. |
| Week 9 (Mar 9 & 11) |
Discussion of midterm problems Analytic continuation (Lang, chap XI; Jones & Singerman, chap 4, sect. 1): Examples: Logarithm and Gamma functions. |
| Weeks 10 & 11 (Mar 16 & 18, 23 & 25) | Analytic continuation (Lang, chap XI; Jones & Singerman, chap 4, sect. 1): monodromy theorem, algebraic functions. Example of a function defined on the unit disc that cannot be continued. Dilogarithm. |