SCV Seminar
# Several Complex Variables Seminar

This semester (Fall 2006) the seminar meets Fridays 3:00pm-3:50pm, in Milner
216.

Sept. 22: Dr. Harold Boas, Texas A & M

Everything you know is wrong (about convergence of power series)

Abstract:
I will illustrate some of the subtleties and surprises
in the theory of convergence of power series via
examples from the past, the present, and the future.
Sept. 29:

Mehmet Celik, Texas A&M

Invariance of compactness and subelliptic estimates for smooth metrics
Abstract

Oct. 6: Mehmet Celik, Texas A&M

Invariance of compactness and subelliptic
estimates for smooth metrics. (Part II)

Abstract:
I will continue talking on the proofs of
invariance of the subelliptic and compactness
estimates for a smooth metric. Moreover, I will
discuss different results according to the different
properties of the smooth metric.
Oct. 13:

Mehmet Celik, Texas A&M

Ideal of Compactness Multipliers

Abstract: In $1979$ J. J. Kohn developed a theory of subelliptic
multipliers. He invented an interesting algorithmic
procedure for computing certain ideals; these ideals
at least in the real analytic case, control both
whether there is a complex analytic variety in the
boundary and whether there is a subelliptic estimate.

Analogously to the definition of the subelliptic
multipliers I define the compactness multipliers
associated to the compactness estimate. The set of
compactness multipliers comes as a (radical) ideal. An
obstruction for the compactness of the
$\overline{\partial}$-Neumann operator will be shown
and then characterized for some classes of
pseudoconvex domains in $\C^{n}$.
Oct. 20:

Michael Fulkerson, Texas A&M

Radial Limits of Holomorphic Functions
Abstract

Oct. 27: Dr. Fanny Dos Reis, Texas A&M

On the regularity of elliptic cycles

Abstract: If $D$ is the unit complex disk and $M$ is a 4-manifold, there
is an elliptic structure associated to every first order elliptic partial
differential equation on $C^\infty(D,M)$. An elliptic structure $E$ is
the set of all admissible tangent planes, and
elliptic structures generalize
the notion of an almost complex structure. M. Gromov introduced
elliptic structures in his article on pseudoholomorphic curves.

I consider the following question: Is a
closed rectifiable elliptic current an $E$-curve? The answer is yes if $E$
is an almost complex structure.

After introducing the problem and necessary background material, I will
state partial results and sketch some proofs.
Nov. 10:

Dr. Andy Raich, Texas A&M

Pointwise heat kernel estimates with applications to complex analysis

Abstract: I discuss a number of problems in several complex variables
whose analysis reduces to the study of a
one-parameter family of heat equations in RxC. I solve
the heat equations and find pointwise estimates on the heat kernels and their derivatives.
Dec. 1: (Joint with
Geometry)

** Special Time: 4-5pm **

Dr. Andrea Nicoara, Harvard

Equivalence of Types on Smooth Domains

Abstract: In 1979, Joseph J. Kohn defined the first multiplier ideal sheaf while investigating the subellipticity of the $\bar\partial$-Neumann problem. He designed an algorithm that generates an increasing chain of ideals, whose termination implies subellipticity. This termination condition is called Kohn finite ideal type. In that same paper, Kohn proved that for a domain in $C^n$ with real-analytic boundary, subellipticity of the $\bar\partial$-Neumann problem on the domain for (p,q) forms is equivalent to Kohn finite ideal type and also equivalent to the property that all holomorphic varieties of complex dimension q have finite order of contact with the boundary of the domain, known as finite D'Angelo type. The equivalence of these two notions of finite type for domains with smooth boundary is known as the Kohn Conjecture. I will present my very recent proof of the Kohn Conjecture and perhaps explain a little bit how this equivalence works on domains with Denjoy-Carleman quasianalytic boundary.