This talk is part of a two-part series.
In part 1 (today), we will discuss some results about Turing-invariant functions from reals into \(\omega_1\) under the Axiom of Determinacy. In part 2, we'll see how these results can be used to prove Martin's Conjecture for order-preserving functions up to the double hyperjump (and some other related things, time permitting).