Event Type:

Department Colloquium

Date/Time:

Monday, February 28, 2005 - 08:00

Location:

Covell 221 <b>(NOTE: unusual day, time, and location)</b>

Guest Speaker:

Andrzej Granas

Institution:

Department of Mathematics, University of Montreal and University of Olsztyn, Poland

Abstract:

This talk is suitable for a general audience. In the article, Uber eine Klasse von lokal zusammenhangende Raumen, published in 1932 in Fundamenta Mathematicae Karol Borsuk introduced, in the setting of separable metric spaces, the class of spaces now called ANRs (Absolute Neighborhood Retracts). One of the corollaries of his main results was the following:THEOREM. Let X be a compact ANR and f be a continuous -homotopic map of X into itself (i.e. f is homotopic to a constant map ). Then f has a fixed point. After giving the proof, Borsuk raised the following question in the article: Under what conditions is the theorem valid without the assumption that X is compact? In this talk (using essentially the mathematical tools and means available before the Second World War), I will show that the answer to the above question leads to results that imply the so-called Leray-Schauder Alternative. As is well known today, this alternative is the foundation for numerous existence results and is one of the central principles of modern Nonlinear Analysis. One historical comment is in order: the Leray-Schauder alternative was originally established as one of the main conclusions of the theory of degree for the so-called compact vector fields in Banach spaces. This theory was developed by J. Leray and J. Schauder in 1933, one year after theorem of Borsuk mentioned above.