SCV Seminar

# Several Complex Variables Seminar

This semester (Fall 2007) the seminar meets Fridays 3:00pm-3:50pm, in Milner 216.

Sep. 7:
Emil Straube, Texas A & M
A sufficient condition for global regularity of the $\overline{\partial}$-Neumann operator, I
Abstract: A theory of global regularity of the $\overline{\partial}$-Neumann operator is developed which unifies the two principal approaches to date, namely the one via compactness due to Kohn-Nirenberg and Catlin and the one via plurisubharmonic defining functions and/or vector fields that commute approximately with $\overline{\partial}$ due to Boas and the author.

Sep. 14:

Emil Straube, Texas A & M
A sufficient condition for global regularity of the $\overline{\partial}$-Neumann operator, II
Abstract: A theory of global regularity of the $\overline{\partial}$-Neumann operator is developed which unifies the two principal approaches to date, namely the one via compactness due to Kohn-Nirenberg and Catlin and the one via plurisubharmonic defining functions and/or vector fields that commute approximately with $\overline{\partial}$ due to Boas and the author.

Sep. 21: (Joint with RAA!)

Scott Zrebiec, Texas A & M
Computing the expected behavior of random holomorphic functions
Abstract: This is a talk based on a paper of Bleher, Shiffman and Zelditch. During this talk we will define random holomorphic functions and random sections, discuss how the expected behavior maybe computed and discuss how this behavior may be shown to be universal in a very general setting.

Sep. 28: (Joint with RAA!)

Scott Zrebiec, Texas A & M
The decay of the hole probability for holomorphic functions
Abstract: This talk will focus on efforts to compute how often no zeros can be found in a domain where many are expected for a random function. In particular, a good deal is known in the codimension 1 case, and this will be presented or summarized.

Oct. 12: (Joint with RAA!)

Scott Zrebiec, Texas A & M
Correlations between zeros of random Bargman-Fock functions
Abstract: In this talk we shall illustrate how the correlation between zeros of a random function may be computed by solving them for a model system. This talk will be based on the work of Bleher, Shiffman and Zelditch, who have solved this system. The techniques used will be quite general, and have been used successfully by the previously mentioned authors to solve more general systems.

Oct. 19:

Michael Fulkerson, Texas A & M
Title: M-harmonic functions
Abstract: M-harmonic functions on the ball are functions which locally satisfy an "invariant" mean-value property. We will discuss the basic properties of M-harmonic funtions and the invariant Laplacian. The talk will be mostly introductory, however an open question will be discussed at the end.

Oct. 26:

Debraj Chakrabarti, The University of Western Ontario
Holomorphic extension of CR functions from non-smooth hypersurfaces
Abstract: We show that the analog of Trepreau's Theorem (local holomorphic extension of all CR functions to one side of a real hypersurface not containing a complex hypersurface) does not hold for hypersurfaces with singularities. We formulate a geometric condition (two-sided support) which gives a necessary condition for the simplest types of singularities, the quadratic cones.

Nov. 2:

Roman Dwilewicz, U. Missouri-Rolla
Global Holomorphic Extensions of Cauchy-Riemann Functions
Abstract: After a short introduction to the Cauchy-Riemann (CR) theory, results about solution of d-bar problem in complex fiber bundles will be presented. Then some consequences will be given about the global holomorphic extension problem for CR functions. As an application, for vector bundles over complex tori, a possibility of holomorphic extension of CR functions can be characterized in terms of theta functions.

Nov. 9:

Zbigniew Slodkowski, University of Illinois at Chicago
Positive closed currents on pseudoconcave sets with thin fibers
Abstract: We consider closed, pseudoconcave subsets K of DxC with totally disconnected compact fibers over points of the unit disk D. Such sets arise (for example) as K = h(X)X, where "h" stands for the polynomially convex hull of X and X is a compact subset of bDxC, with all fibers of X over points of bD (the boundary of D) having zero logarithmic capacity. We study the question (opened in general) under what additional condition the set K supports a positive, closed 1,1 current on DxC. The projection map of K onto D can be viewed upon as a generalized branched covering map with with highly complicated branching structure. We show, that under certain topological restrictions on the character of branching, and under assumption that the image of the branching set under the projection map has logarithmic capacity zero, the pseudoconcave set K supports a nontrivial positive closed current.

Nov. 16:

Albert Boggess, Texas A&M University
CR Functions and their Approximation by Entire Functions
Abstract: This will be an introductory talk on the subject of CR functions. Background will be presented along with an overview of some of the important questions concerning CR function theory. The latter half of the talk will devoted to the subject of approximating CR functions by entire functions. The local theory will be surveyed and some recent results on global approximation will be presented. Few proofs will be given in this introductory talk. If there is interest, some of the details of proofs will be given in a later talk.

Nov. 30:

Roman Dwilewicz, U. Missouri-Rolla
Relations Between Values of the Riemann Zeta Function
Abstract: In this very elementary talk, a short review of known results will be given about values of the Riemann zeta function at integers. Then a result about various relations between the values will be presented in terms of holomorphic functions of two variables.