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Topics in Several Complex Variables >> Content Detail



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Lec #topics
Complex Variable Theory on Open Subsets of Cn
1Functions of one Complex Variable, Cauchy Integral Formula, Taylor Series, Analytic Continuation
2Cauchy Integral Formula (cont.), Inhomogeneous C.R. Equation, Riemann Equation in One Variable, Functions of Several Complex Variables
3The Inhomogeneous Cauchy-Riemann Equation in Several Variables, Hartog's Theorem
4Applying Hartog's Theorem, The Dolbeault Complex, Exactness of the Dolbeault Complex on Polydisks
5The Holomorphic Version of the Poincare Lemma
6The Inverse Function Theorem and the Implicit Function Theorem for Holomorphic Mappings
Theory of Complex Manifolds, Kaehler Manifolds
7Complex Manifolds: Affine and Projective Varieties
8Complex Manifolds: Affine and Projective Varieties (cont.)
9Sheaf Theory and Sheaf Cohomology
10The DeRham Theorem for Acyclic Covers
11

Identification of Cech Cohomology Groups with the Cohomology Groups of the Dolbeault Complex

12Linear Aspects of Symplectic and Kaehler Geometry
13The Local Geometry of Kaehler Manifolds, Strictly Pluri-subharmonic Functions and Pseudoconvexity
14The Ricci Form and the Kaehler Einstein Equation
15The Fubini Study Metric on CPn
Elliptic Operators and Pseudo-differential Operators
16Differential Operators on Rn and Manifolds
17Smoothing Operators, Fourier Analysis on the n-torus
18Pseudodifferential Operators on Tn and Open Subsets of Tn, Elliptic Operators on Compact Manifolds
Hodge Theory on Kaehler Manifolds
19Systems of Elliptic Operators and Elliptic Operators on Vector Bundles
20Elliptic Complexes and Examples
21Hodge Theory, the *-operator
22Computing the *-operator
23The *-operator in Kaehler Geometry
24The *-operator in Kaehler Geometry (cont.)
25The Symplectic Version of the Hodge Theory
26The Symplectic Version of the Hodge Theory (cont.)
27The Brylinski Conjecture and the Hard Lefchetz Theorem, Hodge Theory on Riemannian Manifolds
28Basic Facts About Representations of SL(2,R), SL(2,R) Modules of Finite H-type
29Hodge Theory on Kaehler Manifolds 
30Hodge Theory on Kaehler Manifolds (cont.)
Geometric Invariant Theory
31Actions of Lie Groups on Manifolds, Hamiltonian G Actions on Symplectic Manifolds
32Symplectic Reduction
33Kaehler Reduction and GIT Theory
34Toric Varieties
35The Cohomology Groups of Toric Varieties
36Stanley's Proof of the McMullen Conjecture

 








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