Martin's conjecture is an attempt to classify the behavior of all definable functions on the Turing degrees under strong set theoretic hypotheses. Very roughly it says that every such function is either eventually constant, eventually equal to the identity function or eventually equal to a transfinite iterate of the Turing jump. It is typically divided into two parts: the first part states that every function is either eventually constant or eventually above the identity function and the second part states that every function which is above the identity is eventually equal to a transfinite iterate of the jump. If true, it would provide an explanation for the unique role of the Turing jump in computability theory and rule out many types of constructions on the Turing degrees.

In this thesis we will introduce a few tools which we use to prove several cases of Martin's conjecture. It turns out that both these tools and these results on Martin's conjecture have some interesting consequences both for Martin's conjecture and for a few related topics.

The main tool that we introduce is a basis theorem for perfect sets, improving a theorem due to Groszek and Slaman. We also introduce a general framework for proving certain special cases of Martin's conjecture which unifies a few pre-existing proofs. We will use these tools to prove three main results about Martin's conjecture: that it holds for regressive functions on the hyperarithmetic degrees (answering a question of Slaman and Steel), that part 1 holds for order preserving functions on the Turing degrees, and that part 1 holds for a class of functions that we introduce, called measure preserving functions.

This last result has several interesting consequences for the study of Martin's conjecture. In particular, it shows that part 1 of Martin's conjecture is equivalent to a statement about the Rudin-Keisler order on ultrafilters on the Turing degrees. This suggests several possible strategies for working on part 1 of Martin's conjecture, which we will discuss.

The basis theorem that we use to prove these results also has some applications outside of Martin's conjecture. We will use it to prove a few theorems related to Sacks' question about whether it is provable in ZFC that every locally countable partial order of size continuum embeds into the Turing degrees. We will show that in a certain extension of ZF (which is incompatible with ZFC), this holds for all partial orders of height two, but not for all partial orders of height three. We will also present an obstacle to embedding height three partial orders into the Turing degrees in ZFC which shows that one of the most natural ways of trying to do so cannot work.

We will end the thesis with a list of open questions related to Martin's conjecture, which we hope will stimulate further research.