Jen Paulhus, Ph.D.

Senior Seminar: Riemann Surfaces

email: paulhus@math.grinnell.edu     pgp key
Office Hours: Tuesday/Thursday 2:15 PM - 3:15 PM or by appointment
Class Meetings: 1:00-3:00 PM Monday, Wednesday, Friday
WebEx/Zoom Links: See PWeb

Syllabus
Homework

Daily Topics

Class Date Book Pages/Sections Topics
Oct. 30 Bak and Newman Ch. 1 and 2 : pp. 1-32 introductions, complex differentiation
Nov. 2 Bak and Newman Ch. 3 and 4: pp. 35-56 analytic functions, complex line integrals
Nov. 4 Bak and Newman Ch. 5 and 6: pp. 58-66, 77-88 analytic functions as power series, Cauchy integral formula, uniqueness theorem, minimum modulus theorem, open mapping theorem
Nov. 6 Bak and Newman Ch. 7 and 8: pp. 93-115 analytic functions as power series, Cauchy integral formula, uniqueness theorem, minimum modulus theorem, open mapping theorem
Nov. 9 Bak and Newman Ch. 9 and 10: pp. 117-140 singularities, Laurent expansions, winding numbers, Cauchy residue theorem, meroorphic functions
Nov. 11 Manetti, Lee Ch. 1: pp. 1-19, Ch. 1: pp. 1-17 finish up complex analysis discussion, big picture of topology/manifolds
Nov. 13 Manetti, Lee Ch. 3 pp. 39-57, Ch. 2,3: pp. 19-31, 33-35 49-55 topological spaces, homeomorphisms, metric spaces, subspace
Nov. 16 Manetti, Lee Ch. 3,4: pp. 58-61, 63-75, Ch. 2,3,4: 31-32 pp. 60-63, 64-65, 85-100 product space, Hausdorff, connectedness, compact spaces
Nov. 18* Miranda, Manetti, Lee Ch. 3: pp. 75, Ch. 4: pp. 79-82, Ch. 3: pp. 77-81 group actions, topological groups
Nov. 20 Manetti, Lee Ch. 5: pp. 87-97, Ch. 3: pp. 65-73 identifications, quotient topology, projective space
Nov. 23 Manetti, Lee, Cavalieri & Miles Ch. 6: pp. 105-108, Ch. 2: pp. 36-45, Ch.2: 14-31 second countable, manifolds
Nov. 25 Miranda, Cavalieri & Miles Ch. I.1, I.2: pp. 1-4, 7-9, Ch. 1.4, 3.1, 3.2: 9-13, 32-35, 38-41 riemann surfaces definition, first examples: projective line, complex tori, and kth roots, inverse function theorem
Nov. 30 Miranda, Cavalieri & Miles Ch. I.2, I.3: pp. 10-18, Ch. 3: pp. 35-37, 41-46 graphs, affine plane curves, projective curves
Dec. 2 Miranda Ch. II.1, II.2: pp. 21-38 holomorphic and meromorphis functions on Riemann surfaces, singularities
Dec. 4 Miranda, Cavalieri & Miles Ch II.3: pp. 38-42,, Ch. 4: pp. 47-54 maps between Riemann surfaces
Dec. 7 Miranda, Cavalieri & Miles Ch. II.4, III.1: pp. 44-53 and 57-65, Ch. 4: pp 54-61 degree, (Riemann-)Hurwitz formula, hyperelliptic Riemann surfaces, maps between complex tori
Dec. 9 Miranda Ch. III.3: pp. 75-83 quotient Riemann surfaces, ramification, Hurwitz's theorem on automorphisms
Dec. 11 Cavalieri & Miles, Miranda Ch. 5: pp. 63-79, Ch. III.4: pp. 84-86 homotopy, fundamental groups, coverings
Dec. 14 Miranda, Cavalieri & Miles Ch. III.4: pp. 86-93, Ch. 6, 7: pp. 80-96 monodromy, Riemann Existence Theorem
Dec. 16 tie up all the loose ends
* also see Mileti notes on Group Actions on Pweb.

Homework Assignments

HW Due Date Problems
# PS 1 Nov. 13 Complex Analysis
# PS 2 Nov. 25 Topology
# PS 3 Dec. 9 Riemann Surfaces #1
# PS 4 Dec. 18 Riemann Surfaces #2