Lectures for Mathematics 311, Spring 2007-08
- Monday 1/28: Introduction, preview
- Tuesday 1/29: More preview
- Wednesday 1/30: Complex numbers
- Friday 2/1: Point set topology
- Monday 2/4: [Assignment #1 due]
Uniform continuity, uniform convergence
- Tuesday 2/5: [Add/section-change deadline]
Complex differentiability, Cauchy-Riemann equations
- Wednesday 2/6: More Cauchy-Riemann, harmonic functions
- Friday 2/8: Complex functions as mappings: exp, log, powers
- Monday 2/11: Complex functions as mappings: (z+1/z)/2
- Tuesday 2/12: [Assignment #2 due]
Complex integration
- Wednesday 2/13: More integration
- Friday 2/15: Cauchy's Theorem
- Monday 2/18: [Assignment #3 due]
Finish Cauchy's Theorem, Cauchy's formula, higher derivatives
and power series representation modulo technicalities
- Tuesday 2/19: Technicalities of higher derivatives and power series representation
- Wednesday 2/20: Homology and homotopy, Morera's theorem, recap
- Friday 2/22: [In-class quiz]
- Monday 2/25: Uniform limit of continuous functions, limsup
- Tuesday 2/26: [Assignment #4 due]
Power series radius of convergence,
differentiability of power series
- Wednesday 2/27: Differentiability of limits more generally
- Friday 2/29: Uniqueness theorem
- Monday 3/3: [Assignment #5 due, no-W drop
deadline]
Maximum principle, Fundamental Theorem of Algebra, state
Casorati-Weierstrass
- Tuesday 3/4: Prove Casorati-Weierstrass, Laurent expansion, examples
- Wednesday 3/5: Classification of singularities, Residue Theorem,
Argument Principle
- Friday 3/7: Rouche's Theorem, behavior at infinity,
Local Mapping Theorem, Open Mapping Theorem
- Monday 3/10: [Assignment #6 due,
midterm out - due Saturday 7 p.m.]
General Casorati-Weierstrass Theorem, Riemann's Theorem,
the Hurwitz Theorem
- Tuesday 3/11: Examples of the residue theorem
- Wednesday 3/12: More examples of the residue theorem: zeta(k) for even k,
Fourier transform of the Gaussian
- Friday 3/14: Miscellany on cotangent and zeta
- Saturday 3/15: [midterm due 7 p.m.]
- Spring break week
- Monday 3/24: Conformality, stereographic projection is conformal
- Tuesday 3/25: Fractional linear transformations, circles and lines
- Wednesday 3/26: Associativity
- Friday 3/28: Triple transitivity, reflection in a circle, cross ratio,
conformal mappings examples
- Saturday 3/29: [Junior Qual]
- Monday 3/31: More conformal mapping examples, dynamics
- Tuesday 4/1: [Assignment #7 due] Automorphisms
dynamics
- Wednesday 4/2: Automorphisms of the plane and of the sphere,
rotations of the sphere
- Friday 4/4: Automorphisms of H and D, Schwarz's lemma
- Monday 4/7: [Assignment #8 due,
withdraw/leave deadline]
Arzela-Ascoli Theorem (student lecture)
- Tuesday 4/8: Start Riemann Mapping Theorem
- Wednesday 4/9: Finish Riemann Mapping Theorem,
start Dirichlet's Problem on the disk
- Friday 4/11: finish Dirichlet's Problem on the disk
- Monday 4/14: [Assignment #9 due,
final out - due 5/14 5 p.m.]
Topology of compact Riemann surfaces and their universal covers
- Tuesday 4/15: Complex-isomorphism classes of tori
- Wednesday 4/16: Escher's "Print Gallery," isogenies of tori
- Friday 4/18: Necessary conditions, Weierstrass P and P'
- Monday 4/21: Complex tori as elliptic curves
- Tuesday 4/22: Sufficient conditions (student lecture), elliptic integrals
- Wednesday 4/23: Duplication law for P, Klein's j-function,
- Friday 4/25: Elliptic curve isomorphisms, rational j => rational curve,
RS structure of X(1)
- Monday 4/28: Continuation of zeta (student lecture)
- Tuesday 4/29: RS structure, modular curves, Modularity Theorem
- Wednesday 4/30: Picard's Theorem, course evaluations
- Friday 5/2: (No meeting)
- Wednesday 5/14: [Final due 5 p.m.]
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