Higher randomness theory investigates the notions of effective
randomness one obtains when replacing "computable" by
"hyperarithmetic" and "c.e." by "Pi^1_1" in the usual definitions
(Martin-Löf randomness, Schnorr randomness, computable randomness,
etc). After recalling the basics of the theory, I will present some
recent work in collaboration with Noam Greenberg and Benoit Monin.
The main question we will address is the following: do the (very
rich) interactions between randomness and Turing degrees have a
counterpart in the higher computability? We will argue that this is
indeed the case, provided one correctly translates the notion of
Turing reduction in the higher setting. We will thus introduce the
notion of higher Turing reduction and show that a significant part
of the classical theory translates accordingly. However, we will
also see that the two landscapes (classical and "higher") differ
dramatically on some key aspects, such as the existence of a
uniform oracle tests and measures. If time permits, I will discuss
the impact this has on the study of lowness and triviality, and
will ask some open questions.