Computability, Randomness, and Analysis

We study the computability theory in three different contexts. Firstly, we study the relationship between PA degrees and Martin-Löf randomness. In particular, we focus on the nonuniformity of the PA degrees computed from optimal c.e. supermartingales. Secondly, we investigate the Rademacher series, by considering the summation \sum_n x_n a_n , where x_n\in\{-1,1\}^\infty and (a_n) is a sequence of square-summable reals. On the one hand, to ensure that the summation converges for any computable (a_n), we ask how random x should be. On the other hand, we study the class of x which makes the summation converge, and compare it with other randomness notions. Lastly, we explore the computablity theory in analysis, especially for the complexity of the collection of Banach spaces that have the local basis structure.