Manlio Valenti on the computational strength of a Hausdorff oracle

This Thursday Manlio will give a talk "On the computational strength of a Hausdorff oracle" as a part of our seminar series.

Abstract: Hausdorff dimension is one of the most important notions of dimension in geometric measure theory. It provides a means to describe the "size" of a set in terms of "how tightly" it can be covered with open sets. Recent work has highlighted an interesting connection between the Hausdorff dimension of a set and the computability-theoretical properties of its points. In particular, the effective (Hausdorff) dimension of a point can be defined in terms of its (in)compressibility, so that incompressible (aka random) points have maximum effective dimension. The Point-to-Set Principle states that the Hausdorff dimension of a set can be obtained as the supremum of the effective dimension of its points relative to a fixed parameter (also called oracle). This raises a natural question: how hard is it to compute such an oracle and what is its computational strength? In this talk, we will discuss some recent results in this direction. This is joint work with Ben Koch, Elvira Mayordomo, Arno Pauly, and Cecilia Pradic.