Lectures for Mathematics 311, Spring 2020-21
- Monday 1/25: Introduction, preview
- Tuesday 1/26: More preview
- Wednesday 1/27: Complex numbers
- Friday 1/29: Point set topology
- Monday 2/1: [Assignment 1 due]
Uniform continuity, uniform convergence
- Tuesday 2/2: Complex differentiability, Cauchy-Riemann equations
- Wednesday 2/3: More Cauchy-Riemann, harmonic functions, complex
functions as mappings: exp
- Friday 2/5: [Add/section-change
deadline]
Complex functions as mappings: log, powers, (z+1/z)/2
- Monday 2/8: Complex integration
- Tuesday 2/9: [Assignment 2 due]
More integration
- Wednesday 2/10: More integration
- Friday 2/12: Cauchy's Theorem, complex fundamental theorem of calculus
- Monday 2/15: [Assignment 3 due]
Finish Cauchy's Theorem, Cauchy's formula, higher derivatives
and power series representation modulo technicalities
- Tuesday 2/16: Technicalities of higher derivatives and power series representation
- Wednesday 2/17: Homology and homotopy, Morera's theorem, recap
- Friday 2/19: Uniform limit of continuous functions, limsup
- Monday 2/22: Power series radius of convergence,
differentiability of power series
- Tuesday 2/23: [Assignment 4 due]
Differentiability of limits more generally
- Wednesday 2/24: Uniqueness theorem
- Friday 2/26: Maximum principle, Fundamental Theorem of Algebra, state
Casorati-Weierstrass
- Monday 3/1: [Assignment 5 due]
Prove Casorati-Weierstrass, Laurent expansion, examples
- Tuesday 3/2: Classification of singularities, Residue Theorem,
Argument Principle
- Wednesday 3/3: Rouche's Theorem, behavior at infinity,
Local Mapping Theorem, Open Mapping Theorem
- Friday 3/5: General Casorati-Weierstrass Theorem, Riemann's Theorem,
the Hurwitz Theorem
- Monday 3/8: [Assignment 6 due]
Examples of the residue theorem
- Tuesday 3/9: More examples of the residue theorem: zeta(k) for even k,
Fourier transform of the Gaussian
- Wednesday 3/10: No meeting
- Friday 3/12: Miscellany on cotangent and zeta
- Monday 3/15: [Midterm due]
Conformality, stereographic projection is conformal
- Tuesday 3/16: Fractional linear transformations, circles and lines
- Wednesday 3/17: Associativity
- Friday 3/19: Triple transitivity, reflection in a circle, cross ratio,
conformal mappings examples
- Monday 3/22: More conformal mapping examples, dynamics
- Tuesday 3/23: [Assignment 7 due] Automorphisms
- Wednesday 3/24: Automorphisms of the plane and of the sphere,
rotations of the sphere
- Friday 3/26: Automorphisms of H and D, Schwarz's lemma
- Monday 3/29: [Assignment 8 due,
no-W drop deadline,
withdraw/leave deadline]
Arzela-Ascoli Theorem
- Tuesday 3/30: Start Riemann Mapping Theorem
- Wednesday 3/31: Finish Riemann Mapping Theorem,
start Dirichlet's Problem on the disk
- Friday 4/2: finish Dirichlet's Problem on the disk
- Monday 4/5: [Assignment 9 due,
final out - due in electronic form
5/12 5 p.m.]
Topology of compact Riemann surfaces and their universal covers
- Tuesday 4/6: Complex-isomorphism classes of tori
- Wednesday 4/7: Escher's "Print Gallery," isogenies of tori
- Friday 4/9: Necessary conditions, Weierstrass P and P'
- Spring break week
- Monday 4/19: Complex tori as elliptic curves
- Tuesday 4/20: Sufficient conditions, elliptic integrals
- Wednesday 4/21: Duplication law for P, Klein's j-function,
- Friday 4/23: Elliptic curve isomorphisms, rational j => rational curve,
RS structure of X(1)
- Monday 4/26: Continuation of zeta
- Tuesday 4/27: RS structure, modular curves, Modularity Theorem
- Wednesday 4/28: Picard's Theorem, course evaluations
- Friday 4/30: (No meeting)
- Wednesday 5/12: [Final due in electronic form 5
p.m.]
Assignments
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Solutions
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Writeups
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