Axiom of determinancy in action I was wondering if anybody could give examples of actual works which utilized the axiom of determinancy. My issue is that I have heard of it, and read the Wikipedia, but have had trouble actually finding anything that uses it. That is, choice-based proofs tend to have certain general themes (e.g. make this big family of sets and use Zorn to find one set, or well-order this set and do certain things with that well-order), but I have absolutely no flavor for how AD is used. If you can give any reference that actually does something with determinancy, I'd really appreciate it.
Thanks.
 A: There is a fairly extense literature detailing uses of determinacy in a variety of situations. A good place to start is Akihiro Kanamori's The higher infinite. The last part of the book is devoted to determinacy. Eventually, Aki concentrates on the question of the consistency of determinacy from large cardinals, but before getting there, he provides many examples of the sort you are asking for. Once you are past Kanamori's book, you may want to look at the Cabal Volumes, probably the reissues rather than the original series (There will be 4 reissue volumes in total, three are published or in press so far, but all volumes have been written). There is also Hugh Woodin's book, and many papers in journals and conference proceedings.  
Some examples: Determinacy implies regularity properties of sets of reals. For each of these properties (Lebesgue measurability, the perfect set property, the property of Baire), the argument is a fairly direct appeal to determinacy: Since the times of the Scottish book, games have been identified that in a direct way isolate the key features of the regularity property under investigation. Determinacy gives us a winner, and this translates into a dichotomy (two cases, depending on which player won), from which we can conclude the regularity property. For instance, given a set $A\subseteq\mathbb 2^{\mathbb N}$, the perfect set game for $A$, due to Morton Davis, has the feature that if the first player has a winning strategy, then $A$ contains a perfect subset, while if it is the second player who has a winning strategy, then $A$ is countable.
There is a further regularity property that is a notorious exception, namely the Ramsey property, studied by Adrian Mathias (you can find his thesis here). We do not have a direct game-theoretic argument here. The Ramsey property of sets of reals can be established from $\mathsf{AD}^+$, a strengthening of determinacy first considered by Woodin (what we use is that sets of reals are $\infty$-Borel. This allows us to implement Solovay's argument from his paper on Lebesgue measurability in $L(\mathbb R)^{\mathrm{Col}(\omega,<\kappa)}$ if $\kappa$ is inaccessible). Whether we can prove the Ramsey property for sets of reals directly from determinacy is still open.
One of the areas where determinacy has been most successful is in the study of scale properties of sets of reals. Many papers in the first of the (reissued) Cabal volumes are devoted to scales. The definition of a scale is somewhat technical, but what matters is that they act as surrogates, giving us a way to circumvent the lack of choice in the universe. They have proved essential to carry out core model inductions, the tool we use to extract large cardinal strength from combinatorial properties. This is perhaps the key modern application of determinacy: From determinacy, we can conclude that there are Woodin cardinals in certain inner models (the Koellner-Woodin chapter of the Handbook of set theory is devoted to this topic). The core model induction proceeds by showing that the combinatorial property we are studying gives us enough determinacy ("locally") to carry out this conclusion. The scales are used to improve on the amount of determinacy we have (if we know that all sets of reals in certain pointclass are determined, and we know that either this pointclass, or related ones, have the scale property, we can maneuver this to conclude that all sets of reals in a larger pointclass are determined). More determinacy gives us more large cardinals. This all means that determinacy is essential to this study of lower bounds in consistency strength. For many combinatorial properties, we do not know how to derive large cardinal strength without the detour through determinacy. Descriptive inner model theory is the outcome of this realization (that last link is to a .ps file), and the passage through determinacy is no longer considered a detour, but instead a natural step in this process.
Another area of research that makes use of determinacy is the topic of Woodin's book linked to above: Assuming determinacy, Woodin identified a family of forcing notions ($\mathbb P_{max}$ being the best known). Forcing with these posets gives us back models of choice, with additional combinatorial structure that we typically do not know how to force by traditional means. Woodin's book has many examples of applications of this technique. For a different collection of examples, to which I contributed myself, see here. Let me emphasize that determinacy is essential in the study of these posets. A good exposition of the technique and of the relevance of determinacy, besides Woodin's monograph, is Paul Larson's chapter in the Handbook of set theory.
Let me close with a couple of additional miscellaneous examples. One I like very much is the study of partition calculus. The relevance of partition relations in determinacy was identified very early, through work of Martin and Kunen. A modern extension of some of these results is here. Determinacy, ostensibly a property of sets of reals, is used to establish combinatorial properties of large families of (well-ordered) cardinals. The paper I linked to centers on the Jónsson property; measurability of regular cardinals is another example.
Finally, in joint work with Richard Ketchersid, I established a trichotomy theorem for uncountable sets under determinacy. We proved that, in natural models of determinacy, any uncountable set (not just a set of reals, or a set of sets of reals) is either well-orderable, or else it contains a copy of $\mathbb R$. If the latter, then either the set is linearly orderable, or else it contains a copy of $\mathbb R/E_0$, the quotient of $\mathbb R$ by the Vitali equivalence relation (this set is not linearly orderable in the presence of determinacy. The argument for this goes back to Sierpiński). We have further extensions of these results to well known descriptive set theoretic dichotomies. 
The last few examples make use of $\mathsf{AD}^+$. My paper with Ketchersid should also serve as an introduction to this theory.
