Professor Phil Rossignol

Entomology Department, Oregon State University

Emeritus Professor Andrzej Granas

Mathematics Department, University of Montreal

On the Leray-Schauder principle in the ANR spaces

The topic of the talk lies on the border-line of geometric topology (theory of ANR-s) and nonlinear analysis. In the first part of the talk we shall recall the definition and basic properties of the ANR spaces (these spaces include,for example,polyhedra, Banach manifolds,finite unions of closed convex sets in Banach spaces). Then we address the question of compact extendability of a compact map into an ANR space; we show that this problem depends only on the compact homotopy class of a given compact map.Precisely,in the context of the theory of compact maps, we establish an analogue of the classical Homotopy Extension Theorem of Borsuk. In the second part of the talk, the above extended result is applied to obtain a Generalized Schauder fixed point theorem and the Leray-Schauder principle in arbitrary ANR-s will be established.The classical Schauder theorem and the classical Leray-Schauder principle in Banach spaces are special cases of the abovetheorems. The methods used in the proofs of the above results are elementary and no use is made of the fixed point index for ANR-s.

JOINT OSU/U of O COLLOQUIUM

Speaker: TBA