For any real number p greater than or equal to one, the Banach space l_p induces an equivalence relation on the space of real sequences: one declares two sequences to be equivalent provided they differ by an element in l­_p. In 1999, Dougherty and Hjorth considered these equivalence relations and showed that the l­_p equivalence relation Borel reduces to the l_q equivalence relation if and only if p is less than or equal to q. This shows that if we restrict ourselves to just the F sigma equivalence relations (the simplest beyond smooth in terms of the Borel hierarchy) the poset of Borel cardinalities is still quite rich. In fact, there is a continuum size chain of inequivalent elements.

In this two part talk, we will present Dougherty and Hjorth's proof. The first talk is devoted to an introduction to invariant descriptive set theory and the proof of the reduction. The reduction map is defined via functions whose graphs are fractals. Thus, the proof is interesting since it connects ideas from descriptive set theory, Banach space theory, and fractal geometry. In the second part of the talk, we will give the proof of nonreduction.