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!

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!

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!

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.

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.

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!