# Talk: The hereditarily ordinal definable sets in inner models with finitely many Woodin cardinals

On August 14th, 2017 I gave a talk in the special session on set theory at the Logic Colloquium 2017 (August 14-20, 2017).

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 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 cutpoint in $M_n(x)$ which is a limit of cutpoints in $M_n(x)$.

This is joint work with Grigor Sargsyan.

This abstract will be published in the Bulletin of Symbolic Logic (BSL).
My slides can be found here. A preprint containing these results will be uploaded on my webpage soon.

# Talk: HOD in inner models with Woodin cardinals.

On July 25th I gave a talk at the 4th Münster conference on inner model theory.

Abstract: Abstract: We analyze $\operatorname{HOD}$ in the inner model $M_n(x,g)$ for reals $x$ of sufficiently high Turing degree and suitable generics $g$. Our analysis generalizes to other canonical minimal mice with Woodin and strong cardinals. This is joint work with Grigor Sargsyan.

Notes taken by Ralf Schindler during my talk can be found here. These notes include a sketch of the proof of our main result, the corresponding preprint will be uploaded on my webpage soon.

# Talk: Combinatorial Variants of Lebesgue’s Density Theorem

On July 3rd 2017 I gave a contributed talk at the 6th European Set Theory Conference in Budapest.

Abstract: We introduce alternative definitions of density points in Cantor space (or Baire space) which coincide with the usual definition of density points for the uniform measure on ${}^{\omega}2$ up to a set of measure $0$, and which depend only on the ideal of measure $0$ sets but not on the measure itself. This allows us to define the density property for the ideals associated to tree forcings analogous to the Lebesgue density theorem for the uniform measure on ${}^{\omega}2$. The main results show that among the ideals associated to well-known tree forcings, the density property holds for all such ccc forcings and fails for the remaining forcings. In fact we introduce the notion of being stem-linked and show that every stem-linked tree forcing has the density property.

This is joint work with Philipp Schlicht, David Schrittesser and Thilo Weinert.

# Talk: HOD in $M_n(x,g)$

On January 26th 2017 I gave a talk at the Arctic Set Theory Workshop 3 in Kilpisjärvi, Finland, about $\operatorname{HOD}$ in $M_n(x,g)$. Here are my (very sketchy!) slides.

The following pictures are taken by Andrés Villaveces. Thank you Andrés!

# Talk: $\operatorname{HOD}^{M_n(x,g)}$ is a core model

On December 1st 2016 I gave a talk in the KGRC Research Seminar.

Abstract: Let $x$ be a real of sufficiently high Turing degree, let $\kappa_x$ be the least inaccessible cardinal in $L[x]$ and let $G$ be $Col(\omega, {<}\kappa_x)$-generic over $L[x]$. Then Woodin has shown that $\operatorname{HOD}^{L[x,G]}$ is a core model, together with a fragment of its own iteration strategy.

Our plan is to extend this result to mice which have finitely many Woodin cardinals. We will introduce a direct limit system of mice due to Grigor Sargsyan and sketch a scenario to show the following result. Let $n \geq 1$ and let $x$ again be a real of sufficiently high Turing degree. Let $\kappa_x$ be the least inaccessible strong cutpoint cardinal of $M_n(x)$ such that $\kappa_x$ is a limit of strong cutpoint cardinals in $M_n(x)$ and let $g$ be $Col(\omega, {<}\kappa_x)$-generic over $M_n(x)$. Then $\operatorname{HOD}^{M_n(x,g)}$ is again a core model, together with a fragment of its own iteration strategy.

This is joint work in progress with Grigor Sargsyan.

Many thanks to Richard again for the great pictures!

# Talk at the 1st IRVINE CONFERENCE on DESCRIPTIVE INNER MODEL THEORY and HOD MICE

On July 19th and 21st I gave talks at the 1st IRVINE CONFERENCE on DESCRIPTIVE INNER MODEL THEORY and HOD MICE.

Producing $M_n^\#(x)$ from optimal determinacy hypotheses.

Abstract: In this talk we will outline a proof of Woodin’s result that boldface $\boldsymbol\Sigma^1_{n+1}$ determinacy yields the existence and $\omega_1$-iterability of the premouse $M_n^\#(x)$ for all reals $x$. This involves first generalizing a result of Kechris and Solovay concerning OD determinacy in $L[x]$ for a cone of reals $x$ to the context of mice with finitely many Woodin cardinals. We will focus on using this result to prove the existence and $\omega_1$-iterability of $M_n^\#$ from a suitable hypothesis. Note that this argument is different for the even and odd levels of the projective hierarchy. This is joint work with Ralf Schindler and W. Hugh Woodin.

You can find notes taken by Martin Zeman here and here.

More pictures and notes for the other talks can be found on the conference webpage.

# Talk at the YSTW 2016 in Kopenhagen

From 13th to 17th June 2016 the Young Set Theory Workshop will be held in Kopenhagen. For more information see the webpage of the YSTW 2016. The title of my talk will be the following:

A Journey Through the World of Mice and Games – Projective and Beyond.

Abstract: This talk will be an introduction to inner model theory the at the
level of the projective hierarchy and the $L(\mathbb{R})$-hierarchy. It will
focus on results connecting inner model theory to the determinacy of
certain games.

Mice are sufficiently iterable models of set theory. Martin and Steel
showed in 1989 that the existence of finitely many Woodin cardinals
with a measurable cardinal above them implies that projective
determinacy holds. Neeman and Woodin proved a level-by-level
connection between mice and projective determinacy. They showed that
boldface $\boldsymbol\Pi^1_{n+1}$ determinacy is equivalent to the fact that the
mouse $M_n^\#(x)$ exists and is $\omega_1$-iterable for all reals $x$.

Following this, we will consider pointclasses in the $L(\mathbb{R})$-hierarchy
and show that determinacy for them implies the existence and
$\omega_1$-iterability of certain hybrid mice with finitely many
Woodin cardinals, which we call $M_k^{\Sigma, \#}$. These hybrid mice
are like ordinary mice, but equipped with an iteration strategy for a
mouse they are containing, which enables them to capture certain sets
of reals. We will discuss what it means for a mouse to capture a set
of reals and outline why hybrid mice fulfill this task.

Slides.

# Talk in the KGRC Research Seminar

On June 9th 2016 I gave a talk in the KGRC Research Seminar in Vienna.

Hybrid Mice and Determinacy in the $L(\mathbb{R})$-hierarchy.

Abstract: This talk will be an introduction to inner model theory the at the
level of the $L(\mathbb{R})$-hierarchy. It will
focus on results connecting inner model theory to the determinacy of
certain games.

Mice are sufficiently iterable models of set theory. Martin and Steel
showed in 1989 that the existence of finitely many Woodin cardinals
with a measurable cardinal above them implies that projective
determinacy holds. Neeman and Woodin proved a level-by-level
connection between mice and projective determinacy. They showed that
boldface $\boldsymbol\Pi^1_{n+1}$ determinacy is equivalent to the fact that the
mouse $M_n^\#(x)$ exists and is $\omega_1$-iterable for all reals $x$.

Following this, we will consider pointclasses in the $L(\mathbb{R})$-hierarchy
and show that determinacy for them implies the existence and
$\omega_1$-iterability of certain hybrid mice with finitely many
Woodin cardinals, which we call $M_k^{\Sigma, \#}$. These hybrid mice
are like ordinary mice, but equipped with an iteration strategy for a
mouse they are containing, which enables them to capture certain sets
of reals. We will discuss what it means for a mouse to capture a set
of reals and outline why hybrid mice fulfill this task. If time allows we will sketch a proof that determinacy for sets of reals in the $L(\mathbb{R})$-hierarchy implies the existence of hybrid mice.

Many thanks to Richard for the pictures!