Analysis and randomness in Auckland Dec 12 and 13, 2011

This meeting is dedicated to the rapidly growing interaction between computable analysis and algorithmic randomness.
It is a satellite meeting of the ALC 2011 in Wellington, which takes place Dec 15-20.

Speakers

• Andre Nies.
  TITLE: K-triviality in computable metric spaces . With A. Melnikov.
  A point $x$ in a computable metric space is called $K$-trivial if for each distance a positive rational $\delta$, there is an approximation $p$ at distance at most $\delta$ from $x$ such that the pair $p, \delta$ is highly compressible in the sense that $K(p, \delta) \le K(\delta) + O(1)$. We show that this local definition is equivalent to the point having a rapid Cauchy name that is $K$-trivial when viewed as a function from $\NN$ to $\NN$. We use this to transfer known results on $K$-triviality for functions to the more general setting of metric spaces. For instance, we show that each computable Polish space without isolated points contains an incomputable $K$-trivial point.

Title:Randomness and separation axioms Abstract Algorithmic randomness is usually studied on Cantor space and is also generalized to a computable metric space recently. Using the framework of TTE, randomness on a computable topological space can be considered. However some natural properties do not hold in general. Separation axioms in general topology are effectivized by Weihrauch and SCT_3 is a sufficient condition for that the space can be embedded to a computable metric space with the same topology. I propose SCT_3 is a natural condition for that ML-randomness is a natural randomness notion and SCT_2 is not sufficient.