Seminar on Complex Analysis and Complex Geometry - Fall 2010

Wednesday, September 22, 2010, 1:30pm in PGH 646A

Title: The Degenerated Second Main Theorem and Schmidt's Subspace Theorem

Speaker: Min Ru, UH

Abstract: We establish a Second Main Theorem for algebraically degenerated holomorphic mapping $f:{\mathbb{C}}\ra {\mathbb{P}}^n({\mathbb{C}})$ intersecting hypersurfaces in general position. The related Diophantine problems are also considered. This is a joint work with Zhihua Chen and Qiming Yan. The work was done while the speaker was visiting Tongji University this summer.

Wednesday, November 3, 2010, 3:00pm in SEC 204 (Colloquium)

Title: Minimal desingularizations of planes in space

Speaker: Michael Wolf, Rice University

Abstract: We prove that there is only one way to 'desingularize' the intersection of two planes in space and to obtain a periodic minimal surface as a result. The proof is mostly an exercise in, and an introduction to, basic Teichmueller theory: we translate the geometry of minimal surface in space into a statement about a moduli space of flat structures on Riemann surfaces, and then study deformation theory and degenerations in this moduli space to prove the result. We remark on the general (non-periodic) case.

CANCELLED--Wednesday, November 10, 2010, 1:30pm in PGH 646A

Title: Indices and Relative Indices in CR-geometry

Speaker: Charles Epstein, University of Pennsylvania

Abstract: It is well known that most CR-structures on strictly pseudoconvex 3d-CR manifolds cannot be obtained as the boundary of a Stein space. We discuss an approach to studying the set of embeddable perturbations of an embeddable structure that uses a global invariant, called the relative index. We explain how to define and compute this index and its application to show that the set of embeddable perturbations is closed in certain cases.

CANCELLED--Wednesday, November 10, 2010, 3:00pm in SEC 204 (Colloquium)

Title: A new method for the numerical solution of Maxwell's equations

Speaker: Charles Epstein, University of Pennsylvania

Abstract: I will discuss recent joint work with Leslie Greengard on a new representation of solutions to the time harmonic Maxwell Equations. Using this representation we reduce the solution of the classical problems of scattering off of a smooth bounded interface to the solution of Fredholm integral equations of second kind on the interface. What distinguishes our representation is that it does not have any spurious "interior" resonances, or suffer from low frequency breakdown. It also reveals some interesting topological features of the time harmonic equations at non-zero frequencies.

Monday (!!!), November 15, 2010, 1:30pm in PGH 646A

Title: Syzygies of compact complex hyperbolic manifolds

Speaker: Jun-Muk Hwang, Korea Institute for Advanced Study

Abstract: By Kodaira's embedding theorem, compact complex hyperbolic manifolds are projective algebraic varieties. However, very little is known about their defining equations. In a joint work with Wing-Keung To, we study the syzygies of the compact complex hyperbolic manifolds, via a volume inequalty in a tubular neighborhood of the diagonal.

Monday (!!!), November 15, 2010, 4:00pm in PGH 646A (Colloquium)

Title: Injectivity radius and gonality of a compact Riemann surface

Speaker: Jun-Muk Hwang, Korea Institute for Advanced Study

Abstract: A compact Riemann surface of genus at least 2 is a hyperbolic manifold and at the same time an algebraic curve. We discuss how these two different geometric aspects interact via the concepts of injectivity radius and gonality. The key ingredient is a certain volume inequality in a tubular neighborhood of the diagonal. This is a joint work with Wing-Keung To.