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. SpringerVerlag, 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. SpringerVerlag, 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 23 in Mathematical analysis. An introduction, Andrew Browder, Undergraduate Texts in Mathematics. SpringerVerlag, 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 1114 in Mathematical analysis. An introduction, Andrew Browder, Undergraduate Texts in Mathematics. SpringerVerlag, 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 1114 in Mathematical analysis. An introduction, Andrew Browder, Undergraduate Texts in Mathematics. SpringerVerlag, 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 1114 in Mathematical analysis. An introduction, Andrew Browder, Undergraduate Texts in Mathematics. SpringerVerlag, 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 MittagLefler problem. 

Wednesday
November 11 
Series and products 2: MittagLefler 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 4:307:00 PM MUEN
E432 
FINAL EXAM 