Differential Geometry (2023 Spring)

Course Description

The material that we would like to cover in the course includes (but not limiting to) the following:

De Rham cohomology, Poincare Lemma, chain homotopy
Differential complex, zig-zag, Mayer-Vietoris, cohomology group of Riemann surfaces
Poincare duality: Laplace operator on functions and on differential forms, harmonic forms
Representing cohomology class by Harmonic forms, Hodge decomposition, Hodge theory
Compactly supported cohomology, Mayer-Vietoris for compatly supported cohomology
General form of Poincare duality using compactly supported cohomology
Other geometries: almost complex, symplectic, complex, kaehler

Hodge theory etc:
Do Carmo, Riemannian Geometry.
J. Jost, Riemannian Geometry and Geometric analysis.
R. Bott and L. Tu, Differential forms in algebraic topology
General theory:
I.M. Singer and J.A. Thorpe, Lecture notes on Elementary Topology and Geometry, UTM.
S. Kobayashi, Foundations of Differential Geometry I and II.
M.Spivak, A Comprehensive Introduction to Differential Geometry I-V.
Other geometries:
A. Silva, Lectures on Symplectic Geometry
D. McDuff, D. Salamon, Introduction to Symplectic Topology
D. Huybrechts, Complex geometry, an introduction.

4/10: Representing cohomology class by Harmonic forms, Poincare duality, Hodge decomposition, Hodge theory
4/17: Compactly supported cohomology
4/24: Compactly supported cohomology
5/1: Symplectic manifolds
5/8: Almost complex manifolds (補課在6/12)
5/15: Almost complex manifolds
5/22: Midterm II up to compactly supported cohomology
5/29: Complex manifolds and Dolbeault cohomology
6/5*: Kaehler manifolds
6/12*: Hodge theory on Kaehler manifolds

Last Updated: May 29, 2023
URL: http://www.math.nthu.edu.tw/~nankuo/DG2023S.html