UGA List of topics

Scheduling

• Week 1 (May 19): Preliminary Review
  • \({\mathbb{C}}, {\mathbb{D}}, {\mathbb{H}}, {\mathbb{CP}}^1\)
  • Complex arithmetic
  • Arguments, branch cuts, complex log
  • Elementary geometry, conic sections
  • Analysis Review
    • Uniform (continuity, differentiability, convergence)
    • Heine-Cantor
  • Calculus review
    • Implicit function theorem
    • Inverse function theorem
    • Green's theorem
    • Stokes' theorem
• Week 2 (May 26): Differentiation and integration
  • Calculus preliminaries
  • The Laplace equation,
  • |Cauchy-Riemann,
  • Harmonic functions
  • Contours, parameterization, primitives
  • Cauchy's integral formula
  • Cauchy's inequality
• Week 3 (June 2): Analytic Functions and singularities
  • Power series and convergence
  • Laurent expansions
  • Isolated singularities
    • Poles, meromorphic functions, essential singularities
  • Riemann's removable singularity theorem
• Week 4 (June 9): Integration I
  • Cauchy's theorem and Morera's theorem
  • Cauchy's Integral Formula
  • Maximum Modulus Principle]
  • Schwarz reflection principle
  • Mean value theorem
• Week 5 (June 16): Integration II
  • Residues
  • Techniques in contour integration
• Week 6 (June 23): Entire functions
  • Liouville's theorem
  • The fundamental theorem of algebra
• Week 7 (June 30): Roots
  • Argument principle
  • Rouche's theorem
• Week 8 (July 7): Conformal Maps
  • Conformal maps
  • Mobius transformations and linear fractional transformations
  • The big 9 standard maps
  • Automorphisms of the disc and plane,
• Week 9 (July 14): Schwarz lemma
  • Blaschke Factors
  • Schwarz lemma
  • Open Mapping Theorem
• Week 10 (July 21): Omitted values
  • Casorati-Weierstrass
  • Picard's theorem
• Week 11 (July 28): Montel
  • The identity principle
  • Normal families
  • Montel's theorem
  • Riemann mapping theorem and unformization
• Other misc topics:
  • \({\operatorname{SL}}_2({\mathbb{R}})\), modularity and elliptic curves
  • The hyperbolic metric
  • Riemann surfaces
  • Special functions:
    • The Gamma function, zeta functions and elliptic functions
  • Canonical products
  • Dirichlet’s problem
  • Hurwitz’s theorem