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.