School of Mathematical Sciences

School Seminar

Wednesday, April 26, 2006

3:35 – 4:25 p.m.

Ross 2238 (Conference Room)

Speaker: Steve Leth

“Infinitesimals and Fixed Points”

Nonstandard analysis is a technique for proving theorems by exploiting the existence of larger mathematical “universes” in which the real numbers have “infinitesimal” elements. In this talk I will provide an introduction to the use of nonstandard methods, and how they might provide insight into a classic open problem about the fixed point property in the plane.

The Brouwer Fixed Point Theorem says that any continuous function from the disk (a circle with its interior) into itself must have a fixed point, i.e. a point such that . The classic open question referred to above is whether or not any closed, bounded connected subset of the plane whose complement is also connected must have this same property. Thus, the question is, if is a continuous function from such a set into itself, must there be some point such that ?