THE TWELFTH COLLOQUIUMFEST
Speaker: Franz-Viktor Kuhlmann (University of Saskatchewan, Canada)
Title: A common generalization of metric, ultrametric and topological
fixed point theorems
Abstract:
We present a general fixed point theorem which can be seen as the
quintessence of the principles of proof for Banach's Fixed Point
Theorem, ultrametric and topological fixed point theorems. It works in a
minimal setting, not involving any metrics or topology, only based on
the notion of ``ball'' and the condition that certain descending chains
of balls have nonempty intersection. We demonstrate its applications to
the ultrametric case and discuss how such fixed point theorems can be
used to prove Hensel's Lemma. For ordered abelian groups and fields, we
discuss the possible choices for the
balls: order balls (induced by the ordering), ultrametric balls (induced
by the natural valuation), and combinations of both of them.
This is joint work with Katarzyna Kuhlmann.