(Tuesday, May 21, 2002)
Philip D. Loewen
Gradients of smooth bumps in Banach spaces
A bump is a real-valued function whose support is
nonempty and bounded; continuously differentiable
bumps are called smooth.
In this talk I will outline the results of recent
investigations (joint with J. Borwein, I. Kortezov,
and M. Fabian) into the kinds of sets obtainable as
the range of the gradient for a smooth bump. I will
show how to build a smooth bump on the plane whose
gradient range is not simply connected. Passing to
infinite dimensional Banach spaces, I will explain
how to use a given smooth bump to build another one
whose gradient range exactly reproduces the closure
of a preassigned convex neighbourhood of the origin;
the same procedure covers other reasonable shapes.
Some tantalizing open problems that motivated this
research remain unsolved: I'll mention one of these
and its current status.
The speaker is a candidate for the position of Head, Department of Mathematics.