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
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.