On Dec 1st, 2017, at 10:00am I gave a talk in the CUNY Set Theory Seminar.

*Abstract:* An essential question regarding the theory
of inner models is the analysis of the class of all hereditarily
ordinal definable sets $\operatorname{HOD}$ inside various inner
models $M$ of the set theoretic universe $V$ under appropriate
determinacy hypotheses. Examples for such inner models $M$ are
$L(\mathbb{R})$, $L[x]$ and $M_n(x)$. Woodin showed that under
determinacy hypotheses these models of the form $\operatorname{HOD}^M$
contain large cardinals, which motivates the question whether they are
fine-structural as for example the models $L(\mathbb{R})$, $L[x]$ and
$M_n(x)$ are. A positive answer to this question would yield that they
are models of $\operatorname{CH}, \Diamond$, and other combinatorial
principles.

The first model which was analyzed in this sense was $\operatorname{HOD}^{L(\mathbb{R})}$ under the assumption that every set of reals in $L(\mathbb{R})$ is determined. In the 1990’s Steel and Woodin were able to show that $\operatorname{HOD}^{L(\mathbb{R})} = L[M_\infty, \Lambda]$, where $M_\infty$ is a direct limit of iterates of the canonical mouse $M_\omega$ and $\Lambda$ is a partial iteration strategy for $M_\infty$. Moreover Woodin obtained a similar result for the model $\operatorname{HOD}^{L[x,G]}$ assuming $\Delta^1_2$ determinacy, where $x$ is a real of sufficiently high Turing degree, $G$ is $\operatorname{Col}(\omega, {<}\kappa_x)$-generic over $L[x]$ and $\kappa_x$ is the least inaccessible cardinal in $L[x]$.

In this talk I will give an overview of these results (including some background on inner model theory) and outline how they can be extended to the model $\operatorname{HOD}^{M_n(x,g)}$ assuming $\boldsymbol\Pi^1_{n+2}$ determinacy, where $x$ again is a real of sufficiently high Turing degree, $g$ is $\operatorname{Col}(\omega, {<}\kappa_x)$-generic over $M_n(x)$ and $\kappa_x$ is the least inaccessible cardinal in $M_n(x)$.

This is joint work with Grigor Sargsyan.