SCV Seminar

Several Complex Variables Seminar

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

Sept. 22: 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:

Oct. 6: 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:

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:

Oct. 27: 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:

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

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.