The group algebra RG is finite dimensional (as a vector space) only if G is a finite group. Thus dimension counting arguments are not available if G is infinite. Matters change when one passes to the Hilbert space completion of the group algebra. In that setting we have a dimension notion that behaves very much like the familiar vector space dimension. In my talk I will discuss how one can apply Hilbert space techniques to questions in low dimensional topology, in particular Whitehead’s asphericity conjecture. This talk will be aimed at students with some background in linear algebra and topology.