# Vorträge im Sommersemester 2021

Das Seminar findet dieses Semester über Zoom statt. Um teilzunehmen kontaktieren Sie bitte rds [at] wwu.de.
• #### April 16th 10:00am - Liang Yu (Nanjing University). A basis theorem for $$\Pi^1_1$$-sets.

Abstract: It was claimed by Harrington, but never published, that every non-thin $$\Pi^1_1$$-set ranges over an upper cone of hyperarithmetic degrees. We shall give a proof via a full approximation argument.

# Vorträge im Wintersemester 2020/21

Das Seminar findet dieses Semester über Zoom statt. Um teilzunehmen kontaktieren Sie bitte rds [at] wwu.de.
• #### Novemver 4th - Grigor Sargsyan (Gdansk). Determinacy, forcing axioms and inner models.

Abstract: We will exposit some recent results of the speaker and others
that connect determinacy axioms, forcing axioms and inner models. A
culmination of this work is a recent proof that the most liberally
backgrounded construction of a model build from an extender sequence
cannot be shown to converge in ZFC alone. In this construction, which
is a type of $$K^c$$ construction, one uses extenders that are certified by
a Mostowski collapse. This result challenges common perceptions of the
role of the model $$K^c$$ in the inner model program.

We will mention a specific consistency result showing that the failure
of $$\square_{\omega_3}$$ and $$\square(\omega_3)$$ with
$$2^\omega=2^{\omega_1}=\omega_2$$ and $$2^{\omega_2}=\omega_3$$ is weaker than a
Woodin cardinal that is a limit of Woodin cardinals.

Many people have been involved in this project. The work is heavily
based on the efforts of Steel, Jensen, Woodin, Schindler, Mitchell,
Schimmerling, Trang, Larson, Neeman, Zeman, Schlutzenberg, the speaker
and many others.

• #### Novemver 11th - Grigor Sargsyan (Gdansk). Determinacy, forcing axioms and inner models.

Continuation

• #### Novemver 18th - Grigor Sargsyan (Gdansk). Determinacy, forcing axioms and inner models.

Continuation

• #### November 26th - Grigor Sargsyan (Gdansk). Determinacy, forcing axioms and inner models.

Update: This talk has been moved to thursday 4:15 pm!

Continuation

• #### December 2nd -  Paul Larson (Miami Univ., Oxford OH). Square principles in Pmax extensions of Chang models.

Abstract: We show that the statements $$\square(\omega_{3})$$ and $$\square(\omega_{4})$$ both fail in the $$\mathbb P_{\mathrm{max}}$$ extension of a variation of the Chang model introduced by Sargsyan. This is joint work with Grigor Sargsyan.

Continuation
• #### December 16th - Paul Larson (Miami Univ., Oxford OH). Square principles in Pmax extensions of Chang models.

Continuation

• #### January 13th - Ralf Schindler. The Stable Core.

Abstract: S. Friedman introduced the stable core, $$L[S]$$, and showed that $$V$$ is class generic over $$L[S]$$. He also showed that $$L[S]$$ is a
definable inner model of an iterate of a mouse which has a measurable
definable Woodin cardinal. We will produce versions of these results.

#### Update: This talk will start at 3:30 pm.

Abstract: We continue our investigation on the forcability of regularity properties implied by projective determinacy in showing that one can force the $$\Pi^1_3$$ Reduction Property over the constructible universe $$L$$.

Continuation

#### Update: This talk is postponed to 5:00 pm on Monday, February the 8th.

Abstract: We will describe some aspects of full normalization for
stacks of normal iteration trees. More precisely (but somewhat
approximately), given a normal iteration strategy $$\Sigma$$ for a fine
structural premouse $$M$$, such that $$\Sigma$$ and $$M$$ both satisfy
appropriate condensation properties, we construct a strategy
$$\Sigma^*$$ for transfinite stacks of iteration trees on $$M$$, such that
the iterates via $$\Sigma^*$$ are in fact iterates via $$\Sigma$$. This is
joint work with John Steel.

• #### February 15th - Farmer Schlutzenberg. Full normalization.

This talk starts at 5:00 pm.

Continuation.

• #### February 23rd - Farmer Schlutzenberg. Full normalization.

This talk starts at 5:00 pm.

Continuation.

• #### March 2nd - Farmer Schlutzenberg. Full normalization.

This talk starts at 5:00 pm.

Continuation.

# Vorträge im Sommersemester 2020

Das Seminar findet dieses Semester über Zoom statt. Um teilzunehmen kontaktieren Sie bitte rds [at] wwu.de.
• #### April 21st - Farmer Schlutzenberg. Non-definability of embeddings $$j:V_\lambda\rightarrow V_\lambda$$

Abstract: Assume $$ZF$$. We show that there is no limit ordinal $$\lambda$$ and
$$\Sigma_1$$-elementary $$j:V_\lambda\to V_\lambda$$ which is definable
from parameters over $$V_\lambda$$.
• #### April 28th - Matteo Viale (Turin). Tameness for set theory

Abstract: We show that (assuming large cardinals) set theory is a tractable (and we dare to say tame) first order theory when formalized in a first order signature with natural predicate symbols for the basic definable concepts of second and third order arithmetic, and appealing to the model-theoretic notions of model completeness and model companionship. Specifically we develop a general framework linking generic absoluteness results to model companionship and show that (with the required care in details) a $$\Pi_2$$-property formalized in an appropriate language for second or third order number theory is forcible from some T extending ZFC + large cardinals if and only if it is consistent with the universal fragment of T if and only if it is realized in the model companion of T.
Part (but not all) of our results require Schindler and Asperó that Woodin’s axiom $$(\ast)$$ can be forced by a stationary set preserving forcing.
• #### May 5th - Andreas Lietz. How to force $$(\ast)$$ from less than a supercompact

Abstract:  Asperó-Schindler have shown that Woodin's axiom $$(\ast)$$ is a consequence of $$\operatorname{MM}^{++}$$ and the latter is known to be forceable from a supercompact cardinal. $$(\ast)$$ however has consistency strength of $$\omega$$-many Woodin cardinals, so it should be possible to force it from a much weaker assumption. We present a construction that does so from strictly less than a $$\kappa^{+++}$$-supercompact cardinal $$\kappa$$ (+GCH). The strategy will be to iterate the forcing from the proof of $$\operatorname{MM}^{++}\Rightarrow(\ast)$$. Two main difficulties arise: Whenever we want to use that forcing we will have to make sure that it is semiproper and that $$\operatorname{NS}_{\omega_1}$$ is saturated. We hope that the large cardinal assumption can be lowered to around the region of an inaccessible limit of Woodin cardinals. This is joint work with Ralf Schindler.
• #### May 12th - Farmer Schlutzenberg. $$j:V_\delta\to V_\delta$$ in $$L(V_\delta)$$

Abstract: Assuming $$\mathrm{ZF}+V=L(V_\delta)$$ where $$\delta$$ is a
limit ordinal of uncountable cofinality, we show there is no
non-trivial $$\Sigma_1$$-elementary $$j:V_\delta\to V_\delta$$. Reference:
Section 8 of Reinhardt cardinals and non-definability'', arXiv
2002.01215.
• #### May 19th - Stefan Hoffelner. Forcing the $$\bf{\Sigma^1_3}$$-separation property

Abstract: The separation property, introduced in the 1920s, is a classical notion in descriptive set theory. It is well-known due to Moschovakis, that $$\bf{\Delta^1_2}$$-determinacy implies the $$\bf{\Sigma^1_3}$$-separation property; yet $$\bf{\Delta^1_2}$$-determinacy implies an inner model with a Woodin cardinal. The question whether the $$\bf{\Sigma^1_3}$$-separation property is consistent relative to just ZFC remained open however since Mathias’ „Surrealist Landscape“-paper from 1968. We show that one can force it over L.

#### UPDATE: This talk starts at 10:15 am

Abstract: I will discuss a proof of the joint consistency of​ BPFA and $$\Delta_1$$-definablity of $$NS_{\omega_1}$$. Joint work with Stefan Hoffelner and Ralf Schindler.
• #### June 9th - Farmer Schlutzenberg. Remarks on rank-into-rank embeddings

Abstract: Recall that Woodin's large cardinal axiom $$I_0$$ gives an ordinal $$\lambda$$ and an elementary embedding $$j:L(V_{\lambda+1})\to L(V_{\lambda+1})$$ with critical point $$<\lambda$$. Using methods due to Woodin, we show that if $$ZFC+I_0$$ is consistent then so is $$ZF+DC_\lambda \ +$$ "there is an ordinal $$\lambda$$ and an elementary $$j:V_{\lambda+2}\to V_{\lambda+2}$$''. (A version with the added assumption that $$V_{\lambda+1}^\#$$ exists is due to the author, and Goldberg observed that the appeal to $$V_{\lambda+1}^\#$$ could actually be replaced by some further calculations of Woodin's.)
Reference: https://arxiv.org/abs/2006.01077, "On the consistency of  $$ZF$$ with an elementary embedding from $$V_{\lambda+2}$$ into $$V_{\lambda+2}$$".

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• #### July 14th - Ralf Schindler. $$\mathrm{MM}$$ and $$(\ast)^{++}$$

Abstract: The axiom $$(\ast)^{++}$$ is a strengthening of $$(\ast)$$ which was
also introduced by Woodin. $$(\ast)^{++}$$ says that the set of sets of reals,
or equivalently, $$H_{c^+}$$, is contained in a $$\mathbb P_{\mathrm{max}}$$ extension of a
determinacy model. Woodin showed that $$(\ast)^{++}$$ is false in all the known
models of Martin's Maximum. We will give a proof of this result. It
remains open if $$\mathrm{MM}^{++}$$ refutes $$(\ast)^{++}$$.

# Vorträge im Wintersemester 2019

• #### October 8th - Farmer Schlutzenberg.  Strategy extensions of $$M_1$$

Abstract: We discuss the following two questions:
(1) Let $$x$$ be a real of high complexity. Given a cardinal $$\eta$$ of $$L[x]$$,
what is the $$\eta$$-mantle of $$L[x]$$? Does it model ZFC?
(2) Recall that $$M_1$$ is the minimal proper class iterable mouse with a Woodin cardinal $\delta$. Woodin showed that one can add a significant fragment
of $$M_1$$'s iteration strategy to $$M_1$$ without destroying the Woodinness of $$\delta$$. How much exactly can be added?

Continuation
• #### October 22nd - Ralf Schindler. A universe where $$NS_{\omega_1}$$ is saturated, $$\Delta_1(\omega_1)$$-definable and $$MA_{\omega_1}$$ holds.

Abstract: The definability of the nonstationary ideal can serve as a test question for the 'good' behaviour of subsets of $$\omega_1$$. Recent results of Larson, Schindler and Wu and Aspero-Schindler indicate that strong forcing axioms imply that $$NS_{\omega_1}$$ behaves well, i.e. can not be $$\Delta_1(\omega_1)$$-definable. We show that on the other hand Martin’s Axiom is consistent with the non stationary ideal being $$\Delta_1(\omega_1)$$-definable and saturated, assuming the existence of a Woodin cardinal. This work builds heavily on a coding technique which forces first a suitable ground model and in a second step does the coding over the generically created universe. Goal of the talk will be to highlight some of its properties.

Continuation

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• #### December 3rd - Rahman Mohammadpour (Paris). Some Combinatorial Properties Concerning Guessing Models

Abstract: C. Weiß formulated some combinatorial properties that capture the essence of some large cardinal properties, mostly at level of supercompactness, but can hold at small cardinals. He and M. Viale could then use them to show that any standard forcing construction of a model of PFA requires at least a strongly compact cardinal. In this talk, I will introduce $$GM+(\omega_3,\omega_1)$$ a combinatorial principle strengthening those by Weiß, and will discuss also the consequences of it. This is joint work with B. Velickovic.

Continued
• #### January 21st - Farmer Schlutzenberg. Reinhardt embeddings and non-definability

Abstract: Reinhardt embeddings and non-definability: A Reinhardt cardinal is the
critical point of a Reinhardt embedding, that is, a non-trivial
elementary embedding $$j:V\rightarrow V$$. Kunen showed that there is no Reinhardt
embedding - assuming ZFC. They are, however, not known to be
inconsistent with ZF alone - at least when formulated correctly. In
the last few years, Woodin, Koellner, Bagaria and Cutolo have
investigated Reinhardt and stronger larger cardinals (all incompatible
with AC). We will discuss these large cardinal principles, and prove
some non-definability consequences of them, including that if there is
a Reinhardt cardinal then $$V$$ is not of the form $$HOD(X)$$ for a set X. We
will also discuss the question as to whether V_delta can support a
"local" Reinhardt embedding $$j:V_\delta\rightarrow V_\delta$$ with $$j$$ in $$L(V_\delta)$$
-- while I do not know the answer to this question, I will show that
such a $$j$$ cannot be constructed extremely soon above $$V_\delta$$.

# Vorträge im Sommersemester 2019

• #### April 9th - Farmer Schlutzenberg. The core model in mice

Abstract: We will discuss a proof showing that if $$M$$ is a mouse below a superstrong which has a least Woodin cardinal $$\delta_0$$ and $$M|\delta_0$$ satisfies I am fully iterable'', then $$M|\delta_0$$ satsifies $$V=K$$''. As a corollary, (i) for any $$M$$-generic $$G$$ for a poset $$P\in V_{\delta_0}^M$$, we have $$M|\delta_0 \subseteq HOD^{M[G]}$$, and (ii) $$M$$ has no proper ground via a forcing in $$V_{\delta_0}^M$$.
*These results may already be known to others, but I am not aware of a written account up until now.

Continuation
• #### April 23rd - Ralf Schindler. $$\mathrm{MM}^{++} \implies (*)$$

where $$(*)$$ is the statement: $$\mathrm{AD}^{L(\mathbb{R})}$$ holds and there is some $$g \in V$$ which is $$\mathbb{P}_{\mathrm{max}}$$-generic over $$L(\mathbb{R})$$ such that $$H_{\omega_2}^V \subseteq L(\mathbb{R})[g]$$.

#### Update: We meet 16:00 in lecutre hall M4.

Abstract: In this talk we shall investigate the bi-embeddability relation on countable periodic groups with techniques from (descriptive) set theory. First we shall show that bi-embeddability is not a Borel equivalence relation in the case of primary groups. Next we shall use forcing and the theory of pinned names to show that the isomorphism and bi-embeddability relations on countable periodic groups are incomparable up to Borel reducibility. To contrast this result, we shall discuss how bi-embeddability is strictly simpler than isomorphism under $$\Delta^1_2$$ reducibility and some mild large cardinal assumptions. This is joint work with Simon Thomas.
• #### May 8th - Dima Sinapova (UIC). Singularizing cardinals

UPDATE: WE MEET 16:00 IN LECUTRE HALL M5.

Abstract: It is well known that if $$\kappa$$ is inaccessible in $$V$$ and $$W$$ is an outer model of $$V$$ such that $$(\kappa^+)^V = (\kappa^+)^W$$ and $$\mathrm{cf}^W(\kappa) = \omega$$, then $$\square_{\kappa,\omega}$$ holds in $$W$$. Many strengthenings of this theorem have been obtained as well. We prove that this theorem does not generalize to uncountable cofinalities. Using Magidor's forcing, we show that we can singularize $$\kappa$$ in a cardinal preserving way, and have that in the outer model $$\square_{\kappa,\tau}$$ fails for all $$\tau<\kappa$$.
• This is joint work with Maxwell Levine

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# Vorträge im Wintersemester 2018

• #### 28.08. - Liuzhen Wu (Chinese Academy of Sciences). Lifting argument for Neeman's forcing with side condition

Abstract: Neeman introduces the generalized side condition forcing using two type of models. In this talk, we will describe a lifting argument for a modified version of Neeman's forcing in presence of huge cardinals. We will then discuss the similarity between Neeman's forcing and Kunen-style forcing construction in the realm of hugeness.
• #### 09.10. - Dominik Adolf. Determinacy from threadability at and below $$\Theta$$

Abstract: In this talk we consider the hypothesis that all cardinals less or equal to $$\Theta$$ are threadable. (As usual $$\Theta$$ refers to the supremum of prewellorders on the set of reals.) The hypothesis follows easily from $$\mathrm{AD}_\mathbb{R}$$. We will present a way to perform the "next Woodin" step of the core model induction from the hypothesis under some light extra assumptions. We feel confident that the proof can be extended into an equiconsistency.
• #### 26.10. - Gabriel Fernandes. Local core models with more Woodin cardinals without the measurable

Abstract: Assuming $$\kappa$$ is a singular cardinal such that $$\kappa = \beth_{\kappa}$$ and that there is no premouse $$\mathcal{M}$$ such that $$\mathcal{M} \cap \mathrm{OR} = \kappa$$ and $$\mathcal{M} \models \forall \alpha \exists \delta > \alpha \colon \delta \text{ is a Woodin cardinal}$$, we isolate the core model up to $$\kappa$$. If time permits we will mention a few applications. These reults are part of my Ph.D. thesis.
• #### 30.10. - Stefan Hoffelner. Coding, Forcing Axioms and the Definability of the non stationary Ideal

Abstract: We will introduce and discuss a coding technique which is a combination of a result of Shelah and David’s trick. This method can be used in the context of inner models which have a certain amount of generic absoluteness and works for arbitrarily long iterations. We plan to sketch some applications.

Fortsetzung

• #### 20.11. - Farmer Schlutzenberg. Grounds of mice via $$\sigma$$-closed forcings

Abstract: Abstract: If $$W \models\mathrm{ZFC}$$ and $$W$$ is a non-trivial ground of a mouse $$M$$ via a forcing $$\mathbb{P} \in W$$ which is $$\sigma$$-closed in $$W$$ then $$M|\omega_1^M\notin W$$.

Forsetzung

Fortsetzung
• #### 11.12. - Ralf Schindler. Jensen forcing and the Sacks property

Abstract: Jensen introduced a variant of Sacks forcing which adds exactly one generic real. We introduce a variant of Jensen's forcing which also has the Sacks property.
• #### 11.01. - Joel David Hamkins, University of Oxford. An infinitary-logic-free proof of the Barwise end-extension theorem, with new applications

Update: We will meet at 14:15 in lecture hall M5.

Abstract: Abstract. I shall present a new proof, with new applications, of the amazing extension theorem of Barwise (1971), which shows that every countable model of $$\mathrm{ZF}$$ has an end-extension to a model of $$\mathrm{ZFC}+V=L$$.  This theorem is both (i) a technical culmination of Barwise's pioneering methods in admissible set theory and the admissible cover, but also (ii) one of those rare mathematical results saturated with significance for the philosophy of set theory.  My new proof uses only classical methods of descriptive set theory, and makes no mention of infinitary logic. The results are directly connected with recent advances on the universal $$\Sigma_1$$-definable finite set, a set-theoretic version of Woodin's universal algorithm.
• #### 22.01. - Andreas Lietz. The $$<\kappa$$-mantle.

Abstract: Abstract: The $$<\kappa$$ mantle is a restricted version of the usual geologic mantle. Instead of intersecting all grounds, only those which extend to $$V$$ via $$\kappa$$-small forcing are considered. We investigate at which cardinals $$\kappa$$ the $$<\kappa$$ mantle is a model of $$\mathrm{ZFC}$$. This is joint work with Ralf Schindler.

# Vorträge im Sommersemester 2018

• #### 10.04.  - Andreas Lietz. Set-Theoretic Geology

Abstract: Set-Theoretic Geology is the study of grounds, the ground models of forcing extensions, and the generic multiverse and was initially founded by Hamkins and Reitz in an effort to find regular structure under the generic "dust" added by forcing. Although their hope was not quite fulfilled, this investigation left many interesting questions about the nature of forcing open until the recent results of Usuba about the strong downwards directed grounds hypothesis. For example the mantle, the intersection of all grounds, turned out to be a model of ZFC and the largest forcing invariant definable class. I want to give an exposition of this subject and discuss the interplay between the generic multiverse, its mandtle and large cardinals, including Usubas hyper-huge cardinals.
• #### 17.04.  - Stefan Mesken. Varsovian Models with more Woodin cardinals part I

Abstract: Let $$M$$ be the least pure extender model with a strong cardinal that itself is a limit of Woodin cardinals. We calculate the mantle of $$M$$ and show that it is in fact a fine structural bedrock.

• #### 5.6. - Jinglun Cai: On the consistency bound of a square partition instance

Abstract: We show that certain square partition instance follows from a second order Löwenheim-Skolem statement, which is in turn implied by the existence of a huge cardinal. Hence hugeness is found to be a better consistency bound of the square partition instance.
• #### 12.6. - Stefan Hoffelner: A model for $$\mathrm{NS}_{\omega_{1}}$$ saturated, $$\Delta_{1}$$-definable and Martin's Axiom

Abstract: I will present a sketch of a new coding method which can be used over an arbitrary ground model and will result in a model where the nonstationary ideal on $$\omega_{1}$$ will become $$\Sigma_{1} (C)$$-definable,  where $$C$$ is a ladder system on $$\omega_{1}$$. When applied in the context of universes of "$$\mathrm{NS}_{\omega_{1}}$$ is saturated" it is possible, under the assumption of stationarily many Woodin cardinals, to build a model where additionally Martin's Axiom will hold.
• #### 19.6. - Farmer Schlutzenberg: Extending Iteration Strategies

Abstract: Let $$\kappa$$ be regular uncountable and $$\Sigma$$ be an $$(n,\kappa+1)$$-strategy (for normal trees) for an $$n$$-sound premouse
$$M$$. If $$\Sigma$$ has a certain natural condensation property then $$\Sigma$$ can be extended to an $$(n,\kappa,\kappa+1)$$-strategy (for
stacks of normal trees), and can also be extended to an $$(n,\kappa+1)$$-strategy $$\Sigma'$$ of $$V[G]$$, whenever $$G$$ is $$V$$-generic
for a $$\kappa$$-cc forcing. The condensation property follows from the (weak) Dodd-Jensen property. We will discuss such condensation and how
the extensions of $$\Sigma$$ are constructed.
• #### 26.06. - William Chan (Univ. Northern Texas): The Destruction of the Axiom of Determinacy by Forcing on $$\mathbb{R}$$ when $$\Theta$$ is Regular.

Abstract: Ikegami and Trang have shown that many known forcings, such as Cohen forcing, can not preserve $$\mathrm{AD}$$. They showed that in natural models of determinacy satisfying $$\mathrm{ZF} + \mathrm{AD}^+ + V = L(\mathcal{P}(\mathbb{R}))$$, forcings that preserve $$\mathrm{AD}$$ must preserve $$\Theta$$. However, they also showed it is consistent (with some additional assumptions) that a forcing can preserve $$\mathrm{AD}$$ and increase $$\Theta$$, which implies a new set of reals has been added.
We will show, assuming $$\mathrm{ZF}$$ and $$\mathrm{AD}$$, that every nontrivial wellorderable forcing of cardinality less than $$\Theta$$ forces the failure of $$\mathrm{AD}$$. Assuming $$\mathrm{ZF}$$, $$\mathrm{AD}$$, and $$\Theta$$ is regular, we will show that every nontrivial forcing which is a surjective image of the reals forces the failure of $$\mathrm{AD}$$. This is joint work with Stephen Jackson.

• #### 03.07. - William Chan (Univ. Northern Texas): Suslin lines under $$\mathrm{AD}$$

Abstract: A Suslin line is a nonseparable complete dense linear ordering without endpoints which has the countable chain condition. We will show that $$\mathrm{ZF} + \mathrm{AD}^+ + V = L(\mathcal{P}(\mathbb{R}))$$ proves there are no Suslin lines. In particular, if $$L(\mathbb{R})$$ is a model of $$\mathrm{AD}$$, then $$L(\mathbb{R})$$ has no Suslin lines, which answers a question of Foreman. This is joint work with Stephen Jackson.

# Vorträge im Wintersemester 2017

• #### 16.10.  - Farmer Schlutzenberg. $$\mathrm{HOD}^{M[g]}$$.

Abstract: Let $$M = M_{SWSW}$$ be the least pure extender model with $$\delta_0^M < \kappa_0^M < \delta_1^M < \kappa_1^M$$, where $$\delta_i^M$$ is a Woodin cardinal and $$\kappa_i^M$$ is a strong cardinal in $$M$$. We aim to analyze $$\mathrm{HOD}^{M[g]}$$, where $$g$$ is a generic for $$\mathrm{Coll}(\omega, < \kappa_0^M)$$, and show that in fact $$\mathrm{HOD}^{M[g]} = \mathcal{V_0^M} = \mathcal{M}_{\infty}[*]$$ - the $$0$$-order Varsovian model of $$M$$.

• #### 07.11. - Yizheng Zhu. The internal structure of $$\mathrm{HOD}^{L[x]}$$ up to its Woodin [retracted]

Abstract: Assume $$\boldsymbol{\Delta}^1_3$$-determinacy. It is shown that for any $$x \geq_T M_1^{\#}$$, $$\mathrm{HOD}^{L[x]}$$ is a model of GCH, and in fact, it is a Jensen-Steel core model up to $$\omega_2^{L[x]}$$.

• #### 19.12. - Ralf Schindler. The possible number of Woodin cardinals in core models.

Abstract: There are only three such numbers: 0, 1, and strongly many. This is joint work with Grigor Sargsyan.
• #### 09.01. - Aleksandra Kwiatkowska. Ramsey theory, topological dynamics and Ważewski dendrites

Abstract: We study homeomorphism groups of compact connected spaces called Ważewski dendrites, focusing on their universal minimal flows. For each $$P \subset \{3,4,...,\omega\}$$ there is a unique up to homeomorphism Ważewski dendrite $$W_P$$. If $$P$$ is finite, we prove that the universal minimal flow of $$\mathrm{Hom}(W_P)$$ is metrizable and we compute it explicitly. This answers a question of B. Duchesne. If $$P$$ is infinite, we show that the universal minimal flow of $$\mathrm{Hom}(W_P)$$ is not metrizable. This provides examples of topological groups which are Roelcke precompact and have a non-metrizable universal minimal flow.

• #### 16.01. - Grzegorz Plebanek. Mardešić problem and free productsof Boolean algebras

Abstract: Years ago Mardešić posed a certain problem on continuous images of linearly ordered compact spaces.  In a particular case, it amounts to asking what kind of free products of Boolean algebras can be embedded into free products of interval algebras.
In a recent paper written with Gonzalo Martínez Cervantes we introduced a new kind of dimension of compacta, combinatorial in nature. This concept enabled us to answer Mardešić's question and present  generalizations of some results on Boolean algebras due to Heindorf.
• #### 23.01. - Stefan Hoffelner.  A model where NS is saturated and $$\Delta_1$$-definable

Abstract: Questions which investigate the interplay of the saturation of the nonstationary ideal on $$\omega_1$$, NS, and definability properties of the surrounding universe can yield surprising and deep results. Woodins theorem that in a model with a measurable cardinal where NS is saturated, CH must definably fail is the paradigmatic example. It is another remarkable theorem of H. Woodin that given $$\omega$$-many Woodin cardinals there is a model in which NS is saturated and $$\omega$$-dense, which in particular implies that NS is (boldface) $$\Delta_1$$-definable. The latter statement is of considerable interest in the emerging field of generalized descriptive set theory, as the club filter is known to violate the Baire property. With that being said the following question, asked first by S.D. Friedman and L. Wu seems relevant: Is it possible to construct a model in which NS is both $$\Delta_1$$-definable and saturated from less than $$\omega$$-many Woodins? In this talk I will outline a proof that this is indeed the case: Given the existence of $$M_1^{\#}$$, there is a model of ZFC in which the nonstationary ideal on $$\omega_1$$ is saturated and $$\Delta_1$$-definable with parameter $$\omega_1$$. In the course of the proof I will present a new coding technique which seems to be quite suitable to obtain definability results in the presence of iterated forcing constructions over inner models for large cardinals. See arXiv:1701.07230