# Lecture Notes

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 Cn
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 CPn (PDF)
Elliptic Operators and Pseudo-differential Operators
16 | Differential Operators on Rn and Manifolds (PDF)
17 | Smoothing Operators, Fourier Analysis on the n-torus (PDF)
18 | Pseudodifferential Operators on Tn and Open Subsets of Tn, 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)