A family F of infinite subsets of N is called almost disjoint whenever A ∩ B is finite for all distinct A,B ∈ F. A separation of an almost disjoint family F is a set C ⊆ ℕ such that F = FC ∪ Fℕ\C, where

FC = {A ∈ F : A \ C is finite}, Fℕ\C = {A ∈ F : A ∩ C is finite}:

In 1977 S. Mrówka has constructed a maximal almost disjoint family M where one of the sets MC or Mℕ\C must be finite for any separation C of M. Although the constructed family is far from being Borel, the main trick of the proof relies on the properties of the Borel sets.

Mrówka's family and the method of its construction found many applications in diverse parts of mathematics like compactifications of locally compact separable spaces, projections and complemented subspaces in Banach spaces, the multiplier and corona algebras and the stability of C*-algebras. We will describe some of them and will formulate a couple of open problems.

Refreshments will be served in N620 Ross Building at 3:30p.m.