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.