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