Lecturer: 
Michael Christ
Title:
The d-bar Neumann problem, magnetic Schrodinger operators,
and the Aharonov-Bohm phenomenon

Abstract:
The d-bar Neumann problem is a boundary value problem for Laplace's equation 
on domains in several complex dimensions.  The boundary condition, and thereby 
the regularity of solutions, depend on the complex geometry of the boundary, 
in a manner which has been extensively studied but is still only partially understood. 

One fundamental problem is to characterize compactness 
of the solution operator in complex geometric terms, that is, to 
decide when solutions are necessarily 
slightly smoother than data, as solutions of a differential equation 
ought to be. We discuss this problem for the special class of domains 
possessing a one-dimensional symmetry group. 

There is a well-known but not entirely sufficient condition, known as (P). 
For domains with one-dimensional symmetry, both (P) and compactness 
are equivalent to certain properties of quantum particles in electric 
and magnetic fields, asymptotically as Planck's constant tends to zero, 
related to diamagnetism and paramagnetism. 

We describe a class of nonsmooth domains for which compactness arises
 through an effect not captured by condition (P). 
This is based on an extreme form of the Aharonov-Bohm 
effect, in which a quantum particle experiences a magnetic field 
even though it is confined to a region in which the field vanishes identically.  

However, our main result paradoxically asserts that for smooth domains, 
(P) is equivalent to compactness.  This equivalence comes about 
in a surprising way, via an exceptional sequence of Fourier modes 
which must exist but can be quite sparse.

This is joint work with Siqi Fu.