Department of Mathematics
Texas A & M - Commerce
Student/Faculty Colloquium
Tuesday, March 27, 2007
Henderson 304, 2:00pm
Mehmet Celik,
Texas A & M University math department,
will speak on:
Solving the CR equations using the del-bar Neumann problem
Abstract: Solving del-bar v = u is a fundamental
problem in
several complex variables. One of the ways to obtain the solution is
solving the del-bar Neumann problem. I will start with a question about
a
version of the Wirtinger inequality for continuous functions on the
unit
interval [0,1]. Then, I will give an example of how the fundamental
problem comes to the scene. Moreover, I will show that a solution
of del-bar Neumann problem solves del-bar v = u. The way I am going to
solve the
Wirtenger inequality gives a rough idea of how the question for the
fundamental problem in several complex variables is solved. Finally, I
will mention some interesting ideas related with the regularity of the
del-bar Neumann problem. During the talk there will be only one sketchy
proof
(for a version of the Wirtinger inequality) and all the other proofs
will
be tautologies, so the talk will be easy to follow for everybody.
[By the way: If z = x + i y, then "dz" is dx + i dy, so "d zbar" is
dx
- i dy, and ∂ f is
the operator
giving (∂ f/∂ x - i ∂ f/∂ y) dz. Del-bar is the
complex conjugate of ∂ and so del-bar f is (∂ f/∂ x + i ∂ f/∂ y) d
zbar.
The
del-bar operator thus yields a concise way to express the
Cauchy-Riemann
equations in one variable: f is analytic iff del-bar f = 0.]
All students and faculty are welcome to attend!