Set covering problems: from combinatorial geometry to the geometry of numbers

Abstract
In our talk, we will speak about two, apparently quite different, types of problems. On the one hand, we shall discuss some classical questions of combinatorial geometry such as Borsuk's partition problem and Nelson--Hadwiger's problem on coloring metric spaces. On the other hand, we shall treat of a problem in the geometry of numbers. A bridge between the two types of problems will be built by one more problem in pure combinatorics --- the set covering problem.