Teaching

I have taught the graduate course "Microlocal analysis and complex geometry" at Mathematics Institute of the Universitat zu Köln on 2012 (winter semester). 

The goal of this lecture is to give an introduction to Microlocal Analysis with applications in Complex Geometry. We will emphasize the application of microlocal analysis to the description of the Bergman and Szegö kernels. The singularities of these kernels explain many phenomena of the theory of several complex variables and have many applications to Complex Geometry .

Here is the Syllabus of this lecture:

• I. Distribution Theory and Schwartz kernel Theorem

• II. Fourier transform and the method of stationary phase formula

• III. Symbols and Oscillatory integral and local theory of Fourier integral operators

• IV. Pseudodifferential operators

• V. Application to Elliptic operators and L2 continuity

• VI. Compact complex manifolds, holomorphic vector bundles and Chern classes of vector bundles

• VII. Pseudodifferential operators on complex manifolds, Harmonic Theory on compact complex manifolds and Riemann-Roch-Hirzebruch Theorem

• VIII. Wave front set of a distribution

• IX. Symplectic Geometry

• X. Global theory of Fourier integral operators

• XI. The Malgrange Preparation Theorem and the method of complex stationary phase formula

• XII. Global Theory of complex Fourier integral operators

• XIII. CR manifolds, Kohn Laplacian, L2 estimates of Kohn-Hörmander and understand Boutet-Sjöstrand's Theorem

• XIV. Apply L2 estimates of Kohn-Hörmander and Boutet-Sjöstrand's Theorem to compact complex manifolds. We will deduce: the existence of the Bergman kernel asymptotic expansion for a high power of positive line bundle, Kodaira vanishing Theorem and Kodaira embedding Theorerm

• XV. Prove Boutet-Sjöstrand's Theorem