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.

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) |