Lectures for Mathematics 311, Spring 2024-25
- Monday 1/27: Introduction, preview
- Tuesday 1/28: More preview
- Wednesday 1/29: Complex numbers
- Friday 1/31: Point set topology
- Monday 2/3: [Assignment 1 due]
Uniform continuity, uniform convergence
- Tuesday 2/4: Complex differentiability, Cauchy-Riemann equations
- Wednesday 2/5: More Cauchy-Riemann, harmonic functions, complex
functions as mappings: exp
- Friday 2/7: [Add/section-change
deadline]
Complex functions as mappings: log, powers, (z+1/z)/2
- Monday 2/10: Complex integration
- Tuesday 2/11: [Assignment 2 due]
More integration
- Wednesday 2/12: More integration
- Friday 2/14: Cauchy's Theorem, complex fundamental theorem of calculus
- Monday 2/17: [Assignment 3 due]
Finish Cauchy's Theorem, Cauchy's formula, higher derivatives
and power series representation modulo technicalities
- Tuesday 2/18: Technicalities of higher derivatives and power series representation
- Wednesday 2/19: Homology and homotopy, Morera's theorem, recap
- Friday 2/21: Uniform limit of continuous functions, limsup
- Monday 2/24: Power series radius of convergence,
differentiability of power series
- Tuesday 2/25: [Assignment 4 due]
Differentiability of limits more generally
- Wednesday 2/26: Uniqueness theorem
- Friday 2/28: Maximum principle, Fundamental Theorem of Algebra, state
Casorati-Weierstrass
- Monday 3/3: [Assignment 5 due]
Prove Casorati-Weierstrass, Laurent expansion, examples
- Tuesday 3/4: Classification of singularities, Residue Theorem,
Argument Principle
- Wednesday 3/5: Rouche's Theorem, behavior at infinity,
Local Mapping Theorem, Open Mapping Theorem
- Friday 3/7: General Casorati-Weierstrass Theorem, Riemann's Theorem,
the Hurwitz Theorem
- Monday 3/10: [Assignment 6 due]
Examples of the residue theorem
- Tuesday 3/11: More examples of the residue theorem: zeta(k) for even k,
Fourier transform of the Gaussian
- Wednesday 3/12: No meeting
- Friday 3/14: Miscellany on cotangent and zeta
- Monday 3/17: [Midterm due]
Conformality, stereographic projection is conformal
- Tuesday 3/18: Fractional linear transformations, circles and lines
- Wednesday 3/19: Associativity
- Friday 3/21: Triple transitivity, reflection in a circle, cross ratio,
conformal mappings examples
- Spring break week
- Monday 3/31: More conformal mapping examples, dynamics
- Tuesday 4/1: [Assignment 7 due] Automorphisms
- 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,
no-W drop deadline,
withdraw/leave deadline]
Arzela-Ascoli Theorem
- 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 in electronic form
5/12 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, 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
- 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 in electronic form 5
p.m.]
Assignments
|
Solutions
(at a Google drive)
|
Writeups
|
Back to my home page
|