Lectures: Tuesday 13:20 ~ 15:10 and Thursday 13:20 ~ 15:10 at Astro-Math 101.
Grading scheme:
Homework 40%
Students are asked to do the homework on the blackboard at 14:20 ~ 15:10 on Thursdays.
Final Exam 60%
Main Topics:
This is a continuation of Complex Analysis (I). The main topic is the theory of Riemann surfaces, including Riemann--Roch theorem, uniformization, Abel-Jacobi theorem, etc.
References:
Simon Donaldson, Riemann surfaces.
Farkas and Kra, Riemann surfaces. Second edition.
Lecture summaries and references
(week 1)
holomorphic function
implicit function theorem via Cauchy integral
analytic continuation and complex ODE
classification of surface: Klein bottle ≌ RP2 # RP2, Klein bottle # RP2 ≌ T2 # RP2
reference: Donaldson ch.1 and ch.2.
(week 2)
definition of Riemann surface
basic examples
projective space and algebraic curve
quotients
reference: Donaldson ch.3.
(week 3)
holomorphic map between Riemann surfaces
covering spaces and fundamental groups
reference: Donaldson ch.4.
(week 4)
monodromy representation of holomorphic map
Riemann's existence theorem
basic theory of differential forms on oriented surfaces
reference: Donaldson ch.4 and ch.5.
(week 5)
Poincare duality
decomposition of d on Riemann surfaces
the Laplace operator and Dirichlet energy
Euler characteristic and Poincare-Hopf theorem
reference: Donaldson ch.5 and ch.6.
(week 6)
zero and poles of meromorphic 1-forms
Riemann-Hurwitz formula
degree-genus formula
cohomology defined by \bar{\partial}
reference: Donaldson ch.6 and ch.8.
(week 7)
genus 0 Riemann surfaces
genus 1 Riemann surfaces
Riemann-Roch theorem (restricted version)
reference: Donaldson ch.8.
(week 8)
completion with respect to the Dirichlet energy and the bounded linear functional
Poincare type inequality in 2D
Weyl's lemma
reference: Donaldson ch.9.
(week 9)
solving the Laplace on non-compact surfaces
behavior at infinity
uniformization theorem
reference: Donaldson ch.10.
(week 10)
uniformization theorem (continued)
Cech cohomology
sheaf and sheaf cohomology
relation to the existence of meromorphic functions