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) |