Stefan Mesken
Set Theorist
Currently a PhD student at the WWU MÃ¼nster under supervision
of Ralf
Schindler. My research interests lie in Inner Model Theory,
especially Inner Model Theoretic Geology,
see G. Fuchs,
R. Schindler. Inner Model Theoretic Geology
and G. Sargsyan,
R. Schindler. Varsovian Models I.
You can reach me at my email: s (dot) mesken (at) wwu (dot) de
You can reach me at my email: s (dot) mesken (at) wwu (dot) de
Research
Current Projects
 Inner Model Theoretic Geology
 Woodin Cardinals in Core Models
Publications
Talks
 Varsovian Models with more Woodin Cardinals, Research Seminar WWU Münster, Germany, April 17th, April 24th & May 5th, 2018
 Inner Model Theoretic Geology, 4th Münster, Conference on Inner Model Theory, Germany, July 27, 2017
 The Mantle of $L[x]$, Young Researcherâ€™s Seminar Week, Barcelona, Spain, November 7 & 11, 2016
 Set Theoretical Aspects of Category Theory, Research Seminar of the Workgroup Prof. Schneider, Münster, Germany, June 21, 2016
 A Brief Introduction to the Stationary Tower, Young Set Theory Workshop, Copenhagen, Denmark, June 16, 2016
Teaching
2019
Descriptive Set Theory Seminar
Continuation of last year's seminar with Andreas Lietz.
2018
Descriptive Set Theory Seminar
Motivated by our recently sparked interest in Core Model
Induction, Gabriel Fernandes, Andreas Lietz and I decided to
organize an informal weekly seminar on Descriptive Set
Theory. Our goal is to introduce our students to the basic
concepts of classical descriptive set theory and then,
building on the previous fine structure seminar, show up some
links to Inner Model Theory.
Normalizing Iteration Trees
Together with Gabriel Fernandes and Dan Saattrup Nielsen, I'm
organizing a weekly, informal seminar
on John
Steel. Normalizing iteration trees and comparing iteration
strategies.
Fine Structure Seminar
Continuation of last year's weekly seminar on fine structure with Gabriel Fernandes. This year's topics include:
 premice and the MitchellSteel indexing,
 iteration trees,
 some large cardinals and their representation via extender sequences,
 genericity iterations via the extender algebra,
 copying constructions,
 the DoddJensen lemma,
 Set Theoretic Geology and
 The Mantle of $L[x]$.
2017
Fine Structure Seminar
Together with Gabriel Fernandes I'm giving an informal, weekly seminar on the basics of fine structure leading to the study of small mice. Some of the topics we've discussed:
 acceptable $\mathcal{J}$structures,
 iterated projecta and mastercodes,
 $\Sigma^*$formulae, their interpretation and the correspondence to $\Sigma$formulae over sound structures,
 extenders and their iterated, fine structural ultrapowers,
 "baby" mice and their comparison.
Problems in Set Theory
Girona Meeting 2018
Problem  By  Solved? 

Is there an extender model $L[E]$ such that the mantle of $L[E]$ is not a fully iterable (from the point of view of $L[E]$) hod mouse?  Ralf Schindler  
A reducible ultrafilter is a countably complete ulterfilter
whose ultrapower embedding doesn't factor as a finite linear
ultrapower embedding by internal ultrafilters.
SchlutzenbergSteel: In $L[E]$ the irreducible ultrafilters are exactly the measures on the sequence. Is it consistent with a $\kappa^{+}$ supercompact cardinal that the Mitchell order restricted to irreducible ultrafilters is a wellordering? What about fine structural models? In $L[E]$, if there is a normal measure concentrating on $\alpha$ which are $\alpha^{+}$supercompact then there is a normal measure which is not on the sequence. Reference: Fine structure at the finite levels of supercompactness. 
Gabriel Goldberg  
Let $(M, \Sigma)$ be a mouse pair, $S$ an iteration tree of
successor length on $M$ (not necessarily by $\Sigma$), with a
nice $m$strategy $\Lambda$. Is $\Lambda = \Psi^{*}_{S}$ for
some $\Psi$ such that $(M, \Psi)$ is a mouse pair?
Analogous question for type $2$ $m$trees? 
Benny Siskind  
What is the consistencty strength of the determincacy of the
least $\sigma$algebra containing all the projective sets?
Lower bound: $\mathrm{PD}$. Upper bound: Let $N^{\#}_{\omega + 1}(x)$ be the least active $x$mouse $P$ with a Woodin cardinal $\delta$ such that $\forall n < \omega \colon M^{\#}_{n}(P  \delta) \triangleleft P$. An upper bound is then $\forall x \in \mathbb{R} \colon N^{\#}_{\omega + 1}(x) \text{ exists}$. 
Juan Aguilera  
What is the consistencty strength of $\mathrm{ZF} +
\text{every Suslin set is } \boldsymbol{\Sigma}^{1}_{2}$?
(Add $\mathrm{DC}$ if you want.) Upper bound: A generic Vopěnka cardinal. Lower bound: $\mathrm{ZFC}$ 
Trevor Wilson  
Let $\kappa \ge \aleph_{1}$ be a regular cardinal. Suppose $M$
has a $\kappa + 1$ iteration strategy with strong hull condensation.
Fact: If $\mathbb{P}$ has the $\kappa$c.c., then this is still true in $V^{\mathbb{P}}$. For which other forcings is this also true (or false)? Example (Schindler): If $\kappa = \aleph_{1}$, can have a proper forcing $\mathbb{P}$ s.t. $V^{\mathbb{P}} \models M^{\#}_{1} \text{ is not } \omega_{1} + 1 \text{iterable }$ but $V \models M^{\#}_{1} \text{ is } \omega_{1} + 1 \text{iterable}$. What about $\omega$closed forcings? See "Iterability for Stacks" on Farmer's homepage. 
Farmer Schlutzenberg  
Let $\Gamma$ be a definable class of forcings. The
$\Gamma$mantle is the intersection of all grounds $W$ with
$W[g] = V$ for some $g$ which is $\mathbb{P}$ generic oveer $W$
and $\mathbb{P} \in \Gamma^{W}$

Stefan Mesken  
Suppose $M^{\#}_{1} \le_{T} x$ and work in $L[x]$. A proper
class inner model $N$ is $M_{1}$like if $N \models \text{ I'm }
M_{1} + \delta^{N} < \omega_{1}$.
Let $H$ be the result of simultaneously pseudo comparing all such $N$. Fact: $H = L[\mathcal{M}(\mathcal{T})]$ for any iteration tree $\mathcal{T}$ from that comparison. $\delta^{H} = \omega_{2}$. Is $H  \omega_{2} = \mathrm{HOD}^{L[x]} \omega_{2}$? 
John Steel  
Let $M_{\mathrm{refl}}$ be the least mouse with $\lambda$, a
limit of Woodins an $< \lambda$strong and some $\kappa
< \lambda$ reflecting the set of $<
\lambda$strongs, i.e. $\kappa$ is $A$ strong up to $\lambda$
for $A := \{\mu < \lambda \mid \mu \text{ is } <
\lambda \text{strong }\}$. Assume $M_{\mathrm{refl}}$ has a
good $\omega_{1} + 1$ iteration strategy.
Does $M_{\mathrm{refl}}$ have a uniquely assigned derived model? E.g. does $\mathbb{D}(M, <) \models \mathrm{LSA}$? Steel: Let $M_0, M_1$ be $\mathbb{R}$genericity iterates of $M_{\mathrm{refl}}$, i.e. $M_{\mathrm{refl}} \overset{i_j}{\longrightarrow} M_j$. Let $g_j$ be $\mathrm{Coll}(\omega, < i(\lambda))$generic over $M_j$ such that $\mathbb{R}^{*}_{g_{j}} = \mathbb{R}^{V}$. Do we have $\mathrm{Hom}^{*}_{g_{0}} = \mathrm{Hom}^{*}_{g_{1}}$? 
Dominik Adolf  
Does $\mathrm{AD}^{+} + \text{ no long extender}$ imply hodpair capturing?  John Steel  
Is there an $L[E]$ model which has an inner model $N$ such that $N[g] = L[E]$, where $g$ is $\mathbb{P}$ generic for some $\mathbb{P} \in N$ such that $N \models \mathbb{P} \text{ is } \sigma \text{closed (/ distributive)}$?  Gabriel Goldberg  
Assume $\mathrm{AD}^{L(\mathbb{R})}$. Does $L(\mathbb{R})$ satisfy the "ground axiom"?
We put "ground axiom" in quotation marks here because we aren't sure whether this is necessarily first order expressible over a model of $\mathrm{ZF} + \neg \mathrm{C}$. 
Gabriel Goldberg 
Münster Meeting 2017
Problem  By  Solved? 

What is $\mathrm{HOD}^{L(\mathbb{R})}$ in the lbr
hierarchy?
Same for $L[x,g]$. Same under $\mathrm{AD}^{+} + \neg \mathrm{AD}_{\mathbb{R}} + \mathrm{HPC}$ (hod pair capturing) 
John Steel  
Let $M$ be the least extender model with a $\lambda$ which is a limit of strong cardinals. What is the mantle?
Does it have a bedrock? Is it the $\mathrm{HOD}$ of the $\mathrm{AD}$model construction from $M$ by LarsonSargsyanWilson (the least model containing $\mathbb{R}^{*}$, $\mathrm{Hom}^{*}$ in which every set of reals is universally Baire)? 
Ralf Schindler  
Assume $\kappa$ is a singular, strong limit, $\mathrm{card}(V_{\kappa}) = \kappa$ and $\neg \square_{\kappa}$. Is there a proper class model containing all reals satisfying $\mathrm{AD}^{+} + \mathrm{LSA}$?
Known:

Dominik Adolf  
Let $\mathcal{M}_{1, \infty}$ be the direct limit of countable interates of $M_1$. Let $A \in \mathcal{M}_{1, \infty}$. Must there be some $n < \omega$ such that either $\{ u_m \mid n \le m < \omega\} \subseteq A$ or $\{ u_m \mid n \le m < \omega\} \cap A = \emptyset$?
Background: We have a characterization of $\mathcal{M}_{1, \infty}$ in terms of descriptive set theory. $\mathcal{M}_{1, \infty} = L_{u_{\omega}}[0_{\Sigma^{1}_{3}}]$, where $0_{\Sigma^{1}_{3}} = \{ (\phi, \alpha) \in \text{fml} \times u_{\omega} \mid \exists w \phi(w) \wedge w \text{ codes } \alpha \text{ in the sharp coding} \}$. Best knowledge thus far: For each $n < \omega$: $u_n \in \mathrm{ran}(\pi_{M_1, \mathcal{M}_{1, \infty}})$. References:

Yizheng Zhu  
Assume $\mathrm{AD}$. Let $\kappa$ be an inaccessible Suslin cardinal. Let $f \colon \kappa \to \kappa$ and let $\mu$ be a normal measure on $\kappa$ (e.g. $\mu$ can be any $\delta$cofinality measure on $\kappa$). Let $\alpha = [f]_{\mu}$. Is there a characterization of the property $\alpha = w(\Delta)$, where $\Delta$ is a pointclass closed under $\exists^{\mathbb{R}}, \forall^{\mathbb{R}}$?
In particular, for $\mu =$ the $\omega$cofinality measure on $\kappa$, can one find an $f \colon \kappa \to \kappa$ $f > \mathrm{id}$ such that $f(\beta) = w(\Delta)$ for all $\beta$ and $[f] = w(\Delta)$? Motivation: If the answer is positive, then Steel's pointclass conjecture fails. 
Steve Jackson 
Contact
Stefan Mesken
Einsteinstraße 62
48149 Münster
Germany
email: s (dot) mesken (at) wwu (dot) de
phone: +492518333767
orcid.org/0000000211797049
Einsteinstraße 62
48149 Münster
Germany
email: s (dot) mesken (at) wwu (dot) de
phone: +492518333767
orcid.org/0000000211797049