# My latest posts

• 09 Jun 2018 » UMI-SIMAI-PTM, Wroclaw - Large Cardinals in the Stable Core

In September 2018 I am invited to give a talk in the Thematic Session in Set Theory and Topology at the joint meeting of the Italian Mathematical Union, the Italian Society of Industrial and Applied Mathematics and the Polish Mathematical Society (UMI-SIMAI-PTM) in Wrocław.

Large Cardinals in the Stable Core

Abstract: The Stable Core $\mathbb{S}$, introduced by Sy Friedman in 2012, is a proper class model of the form $(L[S],S)$ for a simply definable predicate $S$. He showed that $V$ is generic over the Stable Core (for $\mathbb{S}$-definable dense classes) and that the Stable Core can be properly contained in HOD. These remarkable results motivate the study of the Stable Core itself. In the light of other canonical inner models the questions whether the Stable Core satisfies GCH or whether large cardinals is $V$ imply their existence in the Stable Core naturally arise. We answer these questions and show that GCH can fail at all regular cardinals in the Stable Core. Moreover, we show that measurable cardinals in general need not be downward absolute to the Stable Core, but in the special case where $V = L[\mu]$ is the canonical inner model for one measurable cardinal, the Stable Core is in fact equal to $L[\mu]$.

This is joint work with Sy Friedman and Victoria Gitman.

• 08 Jun 2018 » KNAW Academy Colloquium on Generalised Baire Spaces - Lebesgue's Density Theorem for tree forcing ideals

On August 24th, 2018 I will give an short talk at the KNAW Academy Colloquium on Generalised Baire Spaces taking place August 23rd and 24th in Amsterdam, The Netherlands.

Abstract: Lebesgue introduced a notion of density point of a set of reals and proved that any Borel set of reals has the density property, i.e. it is equal to the set of its density points up to a null set. 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.

• 07 Jun 2018 » 1st Girona conference on inner model theory - Long games and Woodin cardinals

I am invited to give a talk at the 1st Girona conference on inner model theory which takes place July 16–27, 2018 in Girona.

Long games and Woodin cardinals

Abstract: Itay Neeman proved the determinacy of analytic games of length $\omega \cdot \theta$ for countable $\theta > \omega$ from a sharp for $\theta$ Woodin cardinals. We aim for a converse at successor ordinals and show how to obtain $\omega+1$ Woodin cardinals from the determinacy of analytic games of length $\omega \cdot (\omega+1)$.

This is joint work with Juan P. Aguilera.

• 01 Jun 2018 » (with R. Carroy and A. Medini) Every zero-dimensional homogeneous space is strongly homogeneous under determinacy

Submitted. PDF. arXiv. Bibtex.

• 22 Mar 2018 » Oberseminar Konstanz - Large cardinals from the determinacy of games

On July 9th, 2018 I am invited to give a 90min talk in the Oberseminar Mathematical Logic at the University of Konstanz, Germany.

Large cardinals from the determinacy of games

Abstract: We will study infinite two player games and the large cardinal strength corresponding to their determinacy. In particular, we will consider mice, which are sufficiently iterable models of set theory, and outline how they play an important role in measuring the exact strength of determinacy hypotheses. After summarizing the situation within the projective hierarchy for games of length $\omega$, we will go beyond that and consider the determinacy of even longer games. In particular, we will sketch the argument that determinacy of analytic games of length $\omega \cdot (\omega+1)$ implies the consistency of $\omega+1$ Woodin cardinals. This part is joint work with Juan P. Aguilera.

• 15 Mar 2018 » Turin - Combinatorial Variants of Lebesgue's Density Theorem

In the first week of April 2018 I will visited the Mathematical Logic group of Turin and gave a 2h talk on April 6.

Abstract: Lebesgue introduced a notion of density point of a set of reals and proved that any Borel set of reals has the density property, i.e. it is equal to the set of its density points up to a null set. 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.

• 10 Mar 2018 » (with P. Schlicht, D. Schrittesser and T. Weinert) Combinatorial variants of Lebesgue's density theorem
• 06 Mar 2018 » GDMV Jahrestagung Paderborn - Projective homogeneous spaces and the Wadge hierarchy

On March 06, 2018 I gave an invited talk in the Section in Logic at the annual meeting GDMV of the German Mathematical Society (DMV) joint with GDM taking place March 5th to 9th in Paderborn, Germany.

Abstract: In his PhD thesis Wadge characterized the notion of continuous reducibility on the Baire space ${}^\omega\omega$ in form of a game and analyzed it in a systematic way. He defined a refinement of the Borel hierarchy, called the Wadge hierarchy, showed that it is well-founded, and (assuming determinacy for Borel sets) proved that every Borel pointclass appears in this classification. Later Louveau found a description of all levels in the Borel Wadge hierarchy using Boolean operations on sets. Fons van Engelen used this description to analyze Borel homogeneous spaces.

In this talk, we will discuss the basics behind these results and show the first steps towards generalizing them to the projective hierarchy, assuming projective determinacy (PD). In particular, we will outline that under PD every homogeneous projective space is in fact strongly homogeneous.

This is joint work with Raphaël Carroy and Andrea Medini.

• 26 Feb 2018 » (with J. Aguilera) Mice with infinitely many Woodin cardinals from the determinacy of long games
• 26 Feb 2018 » (with J. Aguilera and P. Schlicht) Determinacy Transfer for sigma-Projective Sets of Reals
• 10 Jan 2018 » (with G. Sargsyan) HOD in inner models with Woodin cardinals

Submitted. PDF. Bibtex.

• 01 Dec 2017 » CUNY Set Theory Seminar - Canonical inner models and their HODs

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.

• 14 Aug 2017 » Logic Colloquium Stockholm - 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.

• 25 Jul 2017 » Münster conference on inner model theory - HOD in inner models with Woodin cardinals

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

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.

• 03 Jul 2017 » European Set Theory Conference - 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.

• 26 Jan 2017 » Arctic Set Theory Workshop - 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!

• 01 Dec 2016 » KGRC Research Seminar - $HOD^{M_n(x,g)}$ 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!

• 21 Oct 2016 » (with R. Schindler and W. H. Woodin) Mice with Finitely many Woodin Cardinals from Optimal Determinacy Hypotheses

Submitted. PDF. Bibtex.

• 21 Oct 2016 » Graduation from the University of Münster

Last week I defended my PhD thesis and finally graduated from the University of Münster. Many thanks to everybody who was there to celebrate with me and especially to Anna, Dorothea, Fabiana and Svenja for the amazing graduation hat.

• 20 Oct 2016 » Pure and Hybrid Mice with Finitely Many Woodin Cardinals from Levels of Determinacy

Dissertation. PDF. Bibtex.

• 19 Jul 2016 » 1st Irvine Conference on Descriptive Inner Model Theory and HOD Mice - Producing M_n^#(x) from optimal determinacy hypotheses

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

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 $M_n^\sharp(x)$ 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 $M_n^\sharp$ 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.

• 13 Jun 2016 » YSTW 2016 Copenhagen - A Journey Through the World of Mice and Games - Projective and Beyond

On June 13th, 2016 I gave a talk at the Young Set Theory Workshop in Copenhagen. For more information see the webpage of the YSTW 2016.

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 $M_n^\sharp(x)$ 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 $M_k^{\Sigma, \sharp}$. 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.

• 09 Jun 2016 » KGRC Research Seminar - Hybrid Mice and Determinacy in the L(IR)-hierarchy

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

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 $M_n^\sharp(x)$ 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 $M_k^{\Sigma, \sharp}$. 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!

• 22 May 2012 » Writeup of an account of Woodin’s HOD conjecture

Masterthesis (in German). PDF