The lecture notes were prepared by Jonathan Campbell, a student in the class. They are available as a single file (PDF - 1.4 MB) or mapped to the lecture topics below. The notes for lectures 16, 17, and 18 are from the Supplementary Notes on Elliptic Operators.
Course notes.
Lec # |
Topics |
Complex Variable Theory on Open Subsets of C^{n} |
1 |
Functions of one Complex Variable, Cauchy Integral Formula, Taylor Series, Analytic Continuation (PDF) |
2 |
Cauchy Integral Formula (cont.), Inhomogeneous C.R. Equation, Riemann Equation in One Variable, Functions of Several Complex Variables (PDF) |
3 |
The Inhomogeneous Cauchy-Riemann Equation in Several Variables, Hartog's Theorem (PDF) |
4 |
Applying Hartog's Theorem, The Dolbeault Complex, Exactness of the Dolbeault Complex on Polydisks (PDF) |
5 |
The Holomorphic Version of the Poincare Lemma (PDF) |
6 |
The Inverse Function Theorem and the Implicit Function Theorem for Holomorphic Mappings (PDF) |
Theory of Complex Manifolds, Kaehler Manifolds |
7 |
Complex Manifolds: Affine and Projective Varieties (PDF) |
8 |
Complex Manifolds: Affine and Projective Varieties (cont.) (PDF) |
9 |
Sheaf Theory and Sheaf Cohomology (PDF) |
10 |
The DeRham Theorem for Acyclic Covers (PDF) |
11 |
Identification of Cech Cohomology Groups with the Cohomology Groups of the Dolbeault Complex (PDF) |
12 |
Linear Aspects of Symplectic and Kaehler Geometry (PDF) |
13 |
The Local Geometry of Kaehler Manifolds, Strictly Pluri-subharmonic Functions and Pseudoconvexity (PDF) |
14 |
The Ricci Form and the Kaehler Einstein Equation (PDF) |
15 |
The Fubini Study Metric on CP^{n} (PDF) |
Elliptic Operators and Pseudo-differential Operators |
16 |
Differential Operators on R^{n} and Manifolds (PDF) |
17 |
Smoothing Operators, Fourier Analysis on the n-torus (PDF) |
18 |
Pseudodifferential Operators on T^{n} and Open Subsets of T^{n}, Elliptic Operators on Compact Manifolds (PDF) |
Hodge Theory on Kaehler Manifolds |
19 |
Systems of Elliptic Operators and Elliptic Operators on Vector Bundles (PDF) |
20 |
Elliptic Complexes and Examples (PDF) |
21 |
Hodge Theory, the *-operator (PDF) |
22 |
Computing the *-operator (PDF) |
23 |
The *-operator in Kaehler Geometry (PDF) |
24 |
The *-operator in Kaehler Geometry (cont.) (PDF) |
25 |
The Symplectic Version of the Hodge Theory (PDF) |
26 |
The Symplectic Version of the Hodge Theory (cont.) (PDF) |
27 |
The Brylinski Conjecture and the Hard Lefchetz Theorem, Hodge Theory on Riemannian Manifolds (PDF) |
28 |
Basic Facts About Representations of SL(2,R), SL(2,R) Modules of Finite H-type (PDF) |
29 |
Hodge Theory on Kaehler Manifolds (PDF) |
30 |
Hodge Theory on Kaehler Manifolds (cont.) (PDF) |
Geometric Invariant Theory |
31 |
Actions of Lie Groups on Manifolds, Hamiltonian G Actions on Symplectic Manifolds (PDF) |
32 |
Symplectic Reduction (PDF) |
33 |
Kaehler Reduction and GIT Theory (PDF) |
34 |
Toric Varieties (PDF) |
35 |
The Cohomology Groups of Toric Varieties (PDF) |
36 |
Stanley's Proof of the McMullen Conjecture (PDF) |