Franz-Viktor Kuhlmann (University of Silesia at Katowice)

Title: Ball spaces - theory and applications

Abstract:
A theorem of considerable importance in valuation theory is the ultrametric version of Banach's Fixed Point Theorem (FPT)
 which was first proved by S. Priess-Crampe. This version requires an analogue of completeness which is called "spherical completeness".
 These and related ultrametric theorems constitute underlying principles for important results and tools in valuation theory, reducing 
 their proofs to the essential core. Driving this abstraction even further, one can ask for the common denominator of metric and 
 ultrametric FPTs and the connections to topological methods. We present an abstract but simple and efficient approach to such
 a unification of methods in Fixed Point Theory and beyond. Inspired by the ultrametric world, we base this approach on the notion
 of spherical completeness. The key idea is to let it refer to any collection of distinguished subsets of an arbitrary set, instead
 of just the ultrametric balls in an ultrametric space. It turns out that important completeness properties can be formulated in terms
 of spherical completeness. For example, a topological space is compact if and only if it is spherically complete with respect to its 
 nonempty closed sets. Areas of applications include metric, ultrametric and topological spaces, ordered abelian groups and fields,
 posets and lattices.

The approach allows us to prove FPTs and Coincidence Point Theorems in many settings that work with some sort of contracting functions.
 Having a common denominator for the various applications also allows us to transfer known results from one application to another
 where they had not been previously observed. Examples for such transfers are the Knaster-Tarski Theorem from the theory of complete
 lattices and the Tychonoff Theorem for products of compact topological spaces. Moreover, ball spaces pose completely new questions such as:
- how to define the notion of "continuity" of functions on ball spaces,
- how to generate new spherically complete ball spaces from given ones,
- how to classify the various levels of "strength" of spherically complete ball spaces.

We will cover joint work with Katarzyna Kuhlmann, Wieslaw Kubis and René Bartsch.