**Question 1:** A good example is Woodin's extender algebra. One reference describing the discovery of the extender algebra is the introduction to Neeman's book *The Determinacy of Long Games*. I am basically repeating Neeman's account.

A soft consequence of ${\bf \Delta}^1_2$-determinacy is that the theory of $(\text{HOD}^{L[x]},\omega_2^{L[x]})$ is stable on a Turing cone. Woodin guessed that this has the following inner model theoretic explanation: $ \text{HOD}^{L[x]}$ is an iterated ultrapower (via an iteration tree) of the minimal canonical inner model $M_1$ with a Woodin cardinal, and $\omega^{L[x]}_2$ is the image of $M_1$'s Woodin cardinal. (**EDIT:** We add that a lot of interesting descriptive set theory, not all due to Woodin, went into this guess: for example, it was conjectured by Kechris-Martin-Solovay in *Introduction to Q-Theory* that the set $$Q_3 = \{x\in \omega^\omega : x\text{ is }\Delta^1_3\text{ in a countable ordinal}\}$$ is the set of reals in the ultimate inner model with a $\Delta^1_3$ wellorder of the reals, which at that time was believed to have large cardinals at the level of $I_3$ (!). The same paper includes a proof of the following theorem of Martin: assuming ${\bf \Delta}^1_2$-determinacy, for a Turing cone of $x$, $Q_3 = \omega^\omega \cap \text{HOD}^{L[x]}$. Then Woodin, in his work with Shelah, realized that the large cardinal level of this ultimate inner model should actually be exactly one Woodin cardinal, since any significantly stronger hypothesis implies there is no $\Delta^1_3$ wellorder of the reals.)

By Vopenka's theorem, for any real $x$, $x$ is generic over $\text{HOD}^{L[x]}$ for a partial order $\mathbb P \subseteq\omega^{L[x]}_2$. If the guess that on a cone of $x$, $\text{HOD}^{L[x]}$ is an iterate of $M_1$ is correct, then this predicts every real is generic over some iterate of $M_1$. But how could you iterate $M_1$ to make a real generic? Perhaps by imitating the process of comparison by least disagreement from inner model theory, instead constructing an iteration tree that inductively eliminates extenders that "disagree" with the real $x$ being generic.

This motivated Woodin's definition of the extender algebra, which turns out to work in the coarse context as well: roughly, if $M$ is any iterable model with a Woodin cardinal $\delta$ (fine structural or not), there is a ${<}\delta$-cc partial order $\mathbb Q\subseteq V_\delta^M$, with $\mathbb Q\in M$ such that for any real $x$, there is an iteration tree on $M$ with branch embedding $i : M\to N$ such that $x$ is $N$-generic for $i(\mathbb Q)$.

(The fine-structural identity of $\text{HOD}^{L[x]}$ on a cone is still unknown, but the smaller model $\text{HOD}^{L[x][G]}$ where $G\subseteq \text{Col}(\omega,{<}\kappa)$ is $L[x]$-generic and $\kappa$ is the least inaccessible of $L[x]$ was analyzed by Woodin, building on work of Steel, and is a fine-structural one Woodin model, although it is *not* an iterate of $M_1$. See Steel-Woodin's paper *HOD as a Core Model*.)

**Question 2:** An example in cardinal arithmetic is the following theorem of Shelah, a sketch of which appears in James Cummings's EFI paper, *Some Challenges for the Philosophy of Set Theory*: if $\kappa$ is the $\omega_1$-th $\aleph$-fixed point then $\kappa^{\aleph_1}$ is less than the $(2^{2^{\aleph_1}})^+$th $\aleph$-fixed point. The proof is by dichotomy, conditioned on whether there is some $A\subseteq (2^{2^{\aleph_1}})^+$ such that there is no inner model $M$ such that $A\in M$ and $M$ has a Ramsey cardinal. If there is some such $A$, one uses covering over $K(A)$. If there is no such $A$, Shelah uses a determinacy result coming from the existence of inner models for Ramsey cardinals over all $A\subseteq (2^{2^{\aleph_1}})^+$.

There are many applications of core model theory at the level of Woodin cardinals in determinacy theory. One is Steel's theorem that PFA implies $\text{AD}^{L(\mathbb R)}$ (see his paper *PFA implies $\text{AD}^{L(\mathbb R)}$*). This is proved by the core model induction, another discovery of Woodin's. Another example is Sargsyan's result (see his thesis *A Tale of Hybrid Mice*) that if there is a Woodin limit of Woodin cardinals, then there is a pointclass $\Gamma\subseteq P(\mathbb R)$ such that $L(\mathbb R,\Gamma)$ satisfies $\text{AD}_\mathbb R + \Theta\text{ is regular}$. Before Sargsyan's work, this theory was conjectured to be extremely strong, at least at the level of a supercompact cardinal. There are also Steel's theorems that assuming AD and $V = L(\mathbb R)$, (1) every regular cardinal below $\Theta$ is measurable and (2) $\text{HOD}$ is a model of GCH. The proofs of both theorems involve analyzing HOD as a fine structure model; $\text{HOD}$ is closely related to (but distinct from) the minimal inner model $M_\omega$ with $\omega$ Woodin cardinals, for example, $V_\Theta^\text{HOD}$ is an iterate of $M_\omega|\delta$ where $\delta$ is the least Woodin of $M_\omega$. Again see *HOD as a Core Model.*