| Date | Topics covered | Notes |
| Monday August 24 |
Introduction: Complex numbers, polar coordinates, Riemann sphere, complex numbers as matrices, conformal linear maps. |
|
| Wednesday August 26 |
Topology 1: Topological spaces, continuous maps, metric spaces. |
For review, you may want to read Chapter 6 in Mathematical
analysis. An introduction, Andrew Browder,
Undergraduate Texts in Mathematics. Springer-Verlag, New
York, 1996. (Electronic
version available from the library.) |
| Friday August 28 |
Topology 2: Constructing topological spaces. |
|
| Monday
August 31 |
Topology 3: Sequences in metric spaces, compact spaces. |
|
| Wednesday
September 2 |
Topology 4: Connectedness, path connectedness. |
|
| Friday
September 4 |
Differentiation 1: Differentiable maps, basic properties, and complex differentiable maps. |
Homework 1 For review, you may want to read Chapter 8 in Mathematical analysis. An introduction, Andrew Browder, Undergraduate Texts in Mathematics. Springer-Verlag, New York, 1996. (Electronic version available from the library.) |
| Monday
September 7 |
NO CLASS |
LABOR DAY |
| Wednesday September 9 |
Differentiation 2: Derivations, tangent spaces, and differentials. |
|
| Friday September 11 |
Differentiation 3: Inverse and implicit function theorems. |
Homework 2 |
| Monday
September 14 |
Differentiation 4: Complex differentials revisited. |
|
| Wednesday
September 16 |
Analytic functions 1: Review of uniform convergence. |
|
| Friday
September 18 |
Analytic functions 2: Definitions and basic properties of real and complex analytic functions. |
Homework 3 For review, you may want to read Chapters 2-3 in Mathematical analysis. An introduction, Andrew Browder, Undergraduate Texts in Mathematics. Springer-Verlag, New York, 1996. (Electronic version available from the library.) |
| Monday September 21 |
Analytic functions 3: Real and complex analytic functions, continued. |
|
| Wednesday September 23 |
Integration 1: Overview of integration, manifolds with boundary, and differential forms. |
|
| Friday
September 25 |
Integration 2: Review of path integrals, and path integrals for vector valued functions. |
Homework 4 We will be covering topics out of Chapters 11-14 in Mathematical analysis. An introduction, Andrew Browder, Undergraduate Texts in Mathematics. Springer-Verlag, New York, 1996. (Electronic version available from the library.) |
| Monday
September 28 |
Integration 3: Complex path integrals. |
|
| Wednesday
September 30 |
Integration 4: Cauchy integral formula and first applications. |
|
| Friday
October 2 |
Manifolds 1: Manifolds, manifolds with boundary. |
Homework 5 We will be covering topics out of Chapters 11-14 in Mathematical analysis. An introduction, Andrew Browder, Undergraduate Texts in Mathematics. Springer-Verlag, New York, 1996. (Electronic version available from the library.) |
| Monday October 5 |
Manifolds 2: Tangent bundles to manifolds. |
|
| Wednesday October 7 |
Manifolds 3: Tangent bundles to manifolds, continued. |
|
| Friday October 9 |
Manifolds 4: Linear algebra. |
Homework 6 We will be covering topics out of Chapters 11-14 in Mathematical analysis. An introduction, Andrew Browder, Undergraduate Texts in Mathematics. Springer-Verlag, New York, 1996. (Electronic version available from the library.) |
| Monday
October 12 |
Manifolds 5: Differential forms and de Rham's theorem. |
|
| Wednesday
October 14 |
Manifolds 6: Integration on manifolds. |
|
| Friday
October 16 |
Manifolds 7: Stokes' theorem. |
Homework 7 |
| Monday October 19 |
Applications of Stokes' theorem 1: Integral formulas in the complex plane. |
|
| Wednesday October 21 |
Applications of Stokes' theorem 2: Integral formulas continued. |
|
| Friday October 23 |
Consequences of Cauchy's integral formula 1: Zeros of holomorphic functions, and singularities of holomorphic functions. |
Homework 8 |
| Monday
October 26 |
Consequences of Cauchy's
integral formula 2: Local structure of holomorphic maps, the open mapping theorem, and the maximum modulus principle. |
|
| Wednesday
October 28 |
Consequences of Cauchy's
integral formula 3: Uniform convergence on compact sets. |
|
| Friday October 30 |
Laurent series 1: Meromorphic functions, and introduction to Laurent series. |
Homework 9 |
| Monday November 2 |
Residue theory 1: Residue theorem, and applications. |
|
| Wednesday November 4 |
Residue theory 2: Residue theorem, and applications. |
|
| Friday November 6 | Meromorphic functions: Definitions, and basic properties. |
Homework 10 |
| Monday
November 9 |
Series and products 1: Motivation, and the Mittag-Lefler problem. |
|
| Wednesday
November 11 |
Series and products 2: Mittag-Lefler continued. |
|
| Friday
November 13 |
Series and products 3: Canonical products |
Homework 11 |
| Monday November 16 |
Series and products 4: Canonical products continued, the Weierstrass Factorization Theorem. |
|
| Wednesday November 18 |
Series and products 5: Weierstrass Factorization Theorem continued. |
|
| Friday November 20 |
Series and products 6: The interpolation problem. |
Homework 12 |
| Monday November 23 - Friday
November 27 |
NO CLASS | THANKSGIVING BREAK |
| Monday
November 30 |
Introduction to the gamma
function |
|
| Wednesday
December 2 |
The gamma function continued |
|
| Friday
December 4 |
Homework 13 |
|
| Monday December 6 |
||
| Wednesday December 9 |
||
| Friday December 11 |
Homework 14 |
|
| Sunday December 13 |
FINAL
EXAM 4:30-7:00 PM MUEN
E432 |
FINAL EXAM |