This conference will be a sequel to the 1st Conference on the core model induction and hod mice that was held in Münster (FRG), July 19 -- August 06, 2010, to the 2nd Conference on the core model induction and hod mice that was held in Münster (FRG), August 08 -- 19, 2011, to the AIM Workshop on Descriptive Inner Model Theory, held in Palo Alto (CA), June 02 -- 06, 2014, to the Conference on Descriptive Inner Model Theory, held in Berkeley (CA) June 09 -- 13, 2014 to the 3rd Münster Conference on inner model theory, the core model induction, and hod mice that was held in Münster (FRG), July 20 -- 31, 2015, to the 1st Irvine conference on descriptive inner model theory and hod mice that was held in Irvine (CA), July 18 -- 29, 2016, to the 4th Münster Conference on inner model theory, the core model induction, and hod mice that was held in Münster (FRG), July 17 -- August 01, 2017, to the 1st Girona conference on inner model theory that was held in Girona (Catalonia), July 16 -- 27, 2018, as well as to the Berkeley conference on inner model theory that was held in Berkeley (California), July 08 -- 19, 2019.

Once more, this conference will now draw together researchers and advanced students with an interest in inner model theory, in order to communicate and further explore recent work. There will be courses and single talks.

We will meet Monday--Friday, with 2 1/2 hours of lectures in the morning and 2 1/2 hours of lectures in the afternoon. This will leave ample time for problem sessions, informal seminars, and other interactions.

We gratefully acknowledge generous support through the Münster Cluster of Excellence (Mathematics Münster: Dynamics - Geometry - Structure), through the National Science Foundation (NSF), and through the Marianne and Dr. Horst Kiesow-Stiftung, Frankfurt a.M.

Location: Department of Mathematics and Computer Science, University of Münster, Orléansring 12, MM (= Münster Mathematics) conference center. During the first week, the talks will be in room SRZ 203, the coffee breaks will be in the lobby of the building; during the second week the talks will be in room SRZ 216-217, the coffee breaks will be in the common area of the MM conference center. Coffee/tea and cookies will be available before the morning/afternoon sessions as well as during the breaks. There are 35 seats reserved for us in the Mensa am Ring, Einsteinstr. 70, to have lunch (they are located in the upper level, on the back right side when facing the building from outside).

This time, there will be a conference dinner (paid by MM); it's going to take place in "Himmelreich," Annette-Allee 9, Münster, on June 23 at 7pm. There will also be a guided tour of the Jewish Cemetry which is next door to the math building (many thanks to Marie-Theres Wacker and Ludger Hiepel!); it's going to take place on June 22 at 4pm. Related to that, you're also invited to explore the Stolpersteine project, here is a list of the ones in Münster. You might also be interested in the ongoing debate about our name patron. As a special event, there will be a colloqium talk by Menachem Magidor on June 30 at 4:30pm (location: Einsteinstr. 62, lecture hall M2).

Mon, June 20 | Tue, June 21 | Wed, June 22 | Thu, June 23 | Fri, June 24 | |

9:30--10:45 | Schlutzenberg | Schlutzenberg | Fuchs | Woodin | Ben-Neria |

11:15--12:30 | Goldberg | Woodin | Woodin | Woodin | Woodin |

14:30--15:45 | Lietz | Levinson | Minden | Adolf | Adolf |

16:15--17:30 | Hoffelner | Chan | 16:00-18:00: guided tour of the Jewish Cemetry. | Siskind | Siskind |

17:30--∞ | Problems and Discussions | Problems and Discussions | --- | 18:30: conference dinner, Himmelreich | free |

Mon, June 27 | Tue, June 28 | Wed, June 29 | Thu, June 30 | Fri, July 01 | |

9:30--10:45 | Cox | Sargsyan | Sargsyan | Kruschewski | Fatalini |

11:15--12:30 | Sargsyan | Wilson | Müller | Trang | Trang |

14:30--15:45 | Sargsyan | Zeman | Trang | Schlicht | N.N. |

16:15--17:30 | Zeman | Welch | Gappo | 16:30: Math Colloquium talk by M. Magidor, lecture hall M2 | N.N. |

17:30--∞ | Problems and Discussions | Problems and Discussions | Problems and Diskussions | --- | free |

- Dominik Adolf:
**Scales from Inner Models, a covering lemma**. We will present a method of constructing (pre-)scales (from PCF theory!) not necessarily in, but around or you might say from inner models. These scales have many nice properties, but this will not be the focus of the talk. As might be expected, the scales derived from the core model are of special interest. We shall present a covering type lemma establishing a relationship between these scales and overlaps of extenders. All of this we will do using λ-indexing. This will provide some interesting challenges similar to the ones from the λ-indexing version of the "Zipper-Lemma". - Omer Ben-Neria:
**Diamond, Compactness, and Scales**. We report on recent progress in the ongoing study on the interaction between guessing and compactness principles. We consider variations of the diamond principle given in terms of (pcf-type) scales and their interaction with Jensen's diamond and various compactness principles such as weak compactness. This is joint work with Jing Zhang. - William Chan:
**Cardinalities Below Omega Sequences of Ordinals**. Under the axiom of determinacy, the perfect set property gives a complete understanding of the cardinalities below ω^{ω}. Woodin established the complete structure of the cardinalities below ω_{1}^{ω}assuming dependent choice and the axiom of real determinacy. We will sketch an argument for this structure below ω_{1}^{ω}under half-real determinacy (equivalently, AD and Uniformization). We will discuss how these methods can be extended to establish the structure of the cardinalities below the set of omega-sequences through some higher cardinals under the same assumption. We may also discuss some results concerning these structures in L(R) where uniformization fails. - Sean Cox:
**Set theory and deconstructibility in homological algebra**. The notion of a*deconstructible class of modules*arose from the work of Eklof and Trlifaj around the year 2000, as part of their contribution to the solution of the Flat Cover Conjecture. Deconstructible classes are very useful in doing "relative" homological algebra, where they generalize certain properties of projective modules. I noticed a convenient characterization of deconstructibility in terms of set-theoretic elementary submodels of the universe, and used similar arguments to answer or partially answer a couple of questions that had appeared in the homological algebra literature: Salce's Problem about cotorsion pairs is independent of ZFC (assuming consistency of Vopenka's Principle); and ("Maximum Deconstructibility") Vopenka's Principle implies that a certain easy consequence of deconstructibility ("eventual almost everywhere closure under quotients") actually characterizes deconstructibility. This has many consequences for the well-studied class of Gorenstein projective modules. - Azul Fatalini:
**Partitions of R**. It is known that R^{3}in unit circles and the Axiom of Choice^{3}can be partitioned in circles, there is even an explicit construction of such a partition. Moreover, it has been proved that R^{3}can be partitioned in circles with radius 1. However, this last proof relies on the Axiom of Choice, particularly, on having a well-order of the reals. In the talk, we will show a model of ZF without a well-order of the reals but in which there is a partition of R^{3}in unit circles. This is joint work with Ralf Schindler. - Gunter Fuchs:
**On the structure of the blurry HODs**. The basic idea of this talk is that one can blur the concept of the definability of a set x, which can be formulated by saying that x is the unique object satisfying some property, to just ask that it is one of fewer than κ objects satisfying some property, for a given cardinal κ. If one allows ordinal parameters in the formulation of the relevant property, one arrives at the concept of <κ-blurry ordinal definability. The collection of all sets which are hereditarily blurrily definable in this way is what I call <κ-HOD. The cases where κ is ω or ω_{1}have been studied by Hamkins-Leahy and Tzouvaras, respectively, with some intricate results on the latter case by Kanovei and Lyubetsky. I will focus in this talk on some maybe somewhat unexpected structural results about the resulting hierarchy of inner models, indexed by the cardinals. - Takehiko Gappo:
**Determinacy in the Chang model**. We will show that if there exist an lbr hod pair (M, Σ) and an ordinal δ such that Σ is universally Baire and δ is a Woodin limit of Woodin cardinals in M, then the Chang model satisfies AD in any generic extension. This is a joint work with Sargsyan. - Gabriel Goldberg:
**Constructibility in stationary logic**. Stationary logic is obtained by augmenting the usual first-order logical connectives and quantifiers with the "aa quantifier," which is defined roughly as follows: a structure M satisfies "aa x Φ" if for a club of countable subsets x of M, the structure (M,x) satisfies the formula Φ. The inner model C(aa) is the analogue of Gödel's constructible universe formed by iterating definability in stationary logic instead of first-order definability. This talk concerns a number of questions about C(aa) raised in Kennedy-Magidor-Väänänen's "Inner models from extended logics, II." For example, I will present a proof that if there is a proper class of Woodin cardinals, C(aa) satisfies the generalized continuum hypothesis. This is joint work with Kennedy, Larson, Magidor, Schindler, Steel, Väänänen, and Wilson. - Stefan Hoffelner:
**The uniformization property, the reduction property and forcing**. Projective determinacy yields a complete and global description of the behaviour of the uniformization property, and consequentially of the reduction and the separation property for projective pointclasses. Yet the question of universes which display an alternative behaviour of these regularity properties has remained in large parts a complete mystery, mostly due to the absence of forcing techniques to produce such models. Consequentially, a lot of very natural problems have remained wide open ever since. In my talk, I want to outline some recently obtained tools, which turn the question of forcing a universe with the Π^{1}_{n}-uniformization property into a fixed point problem for certain sets of forcing notions. This fixed point problem can be solved, yielding a specific set of forcing notions which in turn can be used to force the Uniformization property (for n>2) over fine structural inner models with large cardinals (for n=3, the inner model is just L). For even n, these universes witness for the first time the consistency (relative to the existence of n-3 many Woodin cardinals) of the Π^{1}_{n}-uniformization property, and, for odd n, give new lower bounds in terms of consistency strength. A similar fixed point problem corresponds to the reduction property, whose solution yields a universe with the Π^{1}_{n}-reduction property, for n>2, in which the uniformization property fails, thus separating these two notions for the first time. - Jan Kruschewski:
**On a Conjecture about the Mouse Order on Weasels**. We investigate Steel's conjecture in 'The Core Model Iterability Problem' (p.28), that if R and W are Ω+1-iterable weasels, then R ≤^{*}W iff for club many α < Ω, if α is regular, then α^{+R}≤ α^{+W}.'' We will show that the conjecture fails, assuming that there is a premouse of the form J(Q), where J denotes the J-hierarchy, such that J(Q) ⊨ δ is Woodin, Q ⊨ KP, and δ is the largest cardinal of Q. On the other hand, we show that assuming there is no transitive model of KP with a Woodin cardinal, the conjecture holds. This is joint work with Farmer Schlutzenberg. - Derek Levinson:
**Unreachability of Pointclasses in L(R)**. I will prove in L(R) there is no sequence of distinct Σ^{2}_{1}sets of length (δ^{2}_{1})^{+}. I will also discuss extensions of this theorem, with the goal of proving an analogous result for pointclasses S(κ) where κ is a regular Suslin cardinal. This is joint work with Neeman and Sargsyan. - Andreas Lietz:
**More instances of MM**. We show that the "MM^{++}implies (*)^{++}implies (*) picture" remains intact when conditioned on a variety of Σ_{2}-sentences over (H_{ω2};∈, NS_{ω1}). Among them are "There is a strongly homogeneous Suslin", b=ℵ_{1}and d=ℵ_{1}. In each case, there is a consistent (from a supercompact) MM^{++}-style forcing axiom that implies the (*)-axiom corresponding to a P_{max}-variant conditioned on the Σ_{2}-sentence. We can also find a consistent MM^{++}-style axiom from which the (*)-axiom associated to Woodin's S_{max}follows. The same is true for Woodin's Q_{max}, but the full consistency of the forcing axiom is open in this case. - Kaethe Minden:
**Showing generalized diagonal Prikry forcing is subcomplete**. I plan to discuss the methodology of showing a forcing notion is subcomplete. Specifically I plan to give an idea as to how to show that a simplified version of what I refer to as generalized diagonal Prikry forcing (which adds a point below each cardinal in an infinite discrete set of measurable cardinals) is subcomplete, after giving an overview of Jensen's methods for Namba and Prikry forcing. I may also outline Fuchs' proof that Magidor forcing (up to ω_{1}) is subcomplete. - Sandra Müller:
**A stationary-tower-free proof of Woodinâ€™s Sealing Theorem**. I will present a proof of Sealing from a supercompact and a class of Woodin cardinals using genericity iterations. The proof is joint work with Sargsyan and Wcisło, and builds on the work of Sargsyan-Trang. - Grigor Sargsyan:
**Combinatorial structures on ω**. I will present a forcing construction that produces a model of ¬ □_{3}in Pmax extensions of the Chang model_{ω3}+ ¬ □(ω_{3}) + 2^{ω2}=ω_{3}by forcing over a model of determinacy. As a consequence we get that the above mentioned theory has a consistency strength below a Woodin cardinal that is a limit of Woodin cardinals. It then follows from a work of Jensen-Schindler-Schimmerling-Steel that the K^{c}constructions in the universe mentioned above do not converge. The main forcing construction is a joint work with Paul Larson. - Philipp Schlicht:
**Determinacy and iterated forcing**. We study the robustness of determinacy principles under well behaved forcings. Regarding the history of this problem, it was known that besides properness, one should assume that the forcings are simply definable, since it is consistent that a Σ^{1}_{3}c.c.c. forcing destroys Π^{1}_{1}-determinacy by a result of David. Woodin proved that Cohen and random forcing preserve Π^{1}_{n}-determinacy for any fixed n≥ 1. This was extended to Σ^{1}_{2}absolutely c.c.c. forcings in my dissertation and to several classical tree forcings with Castiblanco. We show that Π^{1}_{n}-determinacy is preserved by virtually all countable support iterations of simply definable proper forcings which add reals. This is joint work with Jonathan Schilhan and Johannes Schürz. - Farmer Schlutzenberg:
**The extent of determinacy in ω-small mice**. Rudominer and Steel conjectured, assuming large cardinals, that for every omega-small mouse M which models ZF^{-}+ "ω_{1}exists", the reals of M are in a sense "optimally correct": that letting R^{M}be the set R ∩ M of reals in M, there an ordinal α such that L_{α}(R^{M}) is a model of determinacy, but L_{α +1}(R^{M}) has a wellorder of R^{M}, and in fact, that there is an integer n such that L_{α}(R^{M}) can be Σ_{n}-elementarily embedded into some L_{β}(R), and there is a wellorder of R^{M}which is Δ_{n+1}-definable over L_{α}(R^{M}) from a parameter in R^{M}. They verified the conjecture or a weaker variant under extra assumptions, but the full conjecture has remained open. We will discuss progress toward a positive resolution of the conjecture. This is joint work with John Steel. - Benjamin Siskind:
**The uniqueness of the core model**. Jensen and Steel identified the core model K as an inner model whose levels are ms-indexed (pure-extender) premice. Of course, other varieties of premice have been studied, for example Jensen-indexed premice or Steel's recent pfs premice and least branch strategy mice. These other varieties could give rise to ostensibly different versions of K. However it is expected that all these versions are actually the same. One reason for this expectation is that it should be possible to translate premice of one variety into premice of another variety in such a way that preserves the associated version of K. But translation methods seem like they must be carefully tailored to the varieties one is translating between, so such methods don't seem like they can yield the kind of general result one would really like to show: any successful notion of premouse must give rise to the same core model. We take a different approach, establishing sufficiently general results along these lines. We show that, in some contexts, abstract properties of the core model uniquely determine it, i.e. there is at most one inner model with these properties. We haven't managed to do this in full generality, however, and will talk about some basic problems around the theory of K without a measurable cardinal which still need to be resolved. - Nam Trang:
**Condensation for mouse pairs**. We will sketch a proof of our condensation theorem for pfs, type 1 mouse pairs. We will make precise the content of the theorem and give the main ideas of the proof. This is a condensation theorem not just for models but also for strategies. Applications include characterizing □ in HOD of AD^{+}models. This is joint work with John Steel. - Martin Zeman:
**Fine structure and Distributivity of iterations of club shooting posets**. This is a continuation of the joint work of M.Foreman, M.Magidor and M.Zeman on games with filters. The main result concerns the distributivity of iterations of club shooting posets, which is also of independent interest, and very likely has broader applications. In our situation, this kind of result can be used to gain more control over winning strategies constructed for Player II in the Welch's variant of Holy-Schlicht games with filters. The active stages in the iterations in question are typically, but not necessarily inaccessible cardinals, at each active stage α a closed unbounded set is added through the complement of a carefully chosen non-reflecting stationary subset of α^{+}, and the supports are sufficiently large. For instance, Easton supports would be suitable here (but the result seems to hold for larger supports as well). The conclusion is that if the first active stage is δ then the entire iteration is (δ^{+},∞)-distributive. The main point in the argument is passing through inverse limits. Whereas passing through inverse limits of small cofinalities can be done in ZFC using methods known for a long time (and most likely the result has been known for a long time), passing through inverse limits of large cofinalities seems to be less clear, and the only way we know how to do it at this point is using fine structure of extender models. In this talk a simple instance of such an argument is presented which nevertheless features all essential combinatorial aspects of the construction. The model used will be the constructible universe L. I will then explain how to generalize the argument for arbitrary extender models. - Trevor Wildon:
**The least strong cardinal is a Löwenheim-Skolem number**. We characterize the least strong cardinal as the Löwenheim-Skolem number of a fragment of infinitary second-order logic. - Philip Welch:
**Free subsets of internally approachable structures**. If A is a first order structure, a subset X of dom(A) is*free*if no element of X can be defined in A from other elements of X. In general, finding infinite free subsets of infinite structures, requires large cardinals. We survey the field here, and examine extensions of this property due to Pereira, distilled from his work on the pcf conjecture, and recently another due to Adolf and Ben Neria. Work of the latter now establishes, with some older work of the speaker, an equiconsistency between inner models with sequences of measures and their extension of Pereira's "Approachable Free Subset Property". - W. Hugh Woodin:
**Failues of UBH and CBH**.

Dominik Adolf (UNT) | (--) |

Omer Ben-Neria (Jerusalem) 🚲 | June 23-July 01, Jellentrup (fu) |

William Chan (CMU) 🚲 | (fu) |

Sean Cody (UCLA) | (p.d.) |

Sean Cox (VCU) | June 25-July 01 (fu) |

Azul Fatalini (Münster) | |

Gunter Fuchs (CUNY) 🚲 | 1st week (--) |

Takehiko Gappo (Gdansk) | (fu) |

Gabriel Goldberg (UC Berkeley) 🚲 | (--) |

Stefan Hoffelner (Münster) | |

Ronald Jensen (HU Berlin) | had to cancel |

Obrad Kasum (Paris) | (p.d.) |

Jan Kostrzon (Warsaw) | (p.d.) |

Jan Kruschewski (Münster) | |

Derek Levinson (UCLA) | (-u) |

Andreas Lietz (Münster) | |

Menachem Magidor (Jerusalem) | June, EH (mmf) |

Kaethe Minden (Simon's Rock, MA) | (??) |

Sandra Müller (Wien) | (--) |

Otto Rajala (Helsinki) | (p.d.) |

Hossein Lamei Ramandi (Münster) | |

Grigor Sargsyan (IMPAN, Sopot) | (fu) |

Ralf Schindler (Münster) | |

Philipp Schlicht (Bristol) | (--) |

Farmer Schlutzenberg (Münster) | |

Xianghui Shi (Beijing) | had to cancel due to Covid-19 |

Benjamin Siskind (CMU) | (??) |

Shervin Sorouri (Münster) | |

John Steel (UC Berkeley) | had to cancel |

Xiuyuan Sun (Münster) | |

Sebastiano Thei (Udine) 🚲 | (--) |

Nam Trang (UNT) | June 26-July 02 (--) |

Philip Welch (Bristol) | (--) |

Trevor Wilson (Miami University) | (--) |

Bartosz Wcisło (Warsaw) | (--) |

W. Hugh Woodin (Harvard University) | EH, June 19-25 (-u) |

Taichi Yasuda (Tokyo University) | had to cancel |

Liang Yu (Nanjing University) | had to cancel due to Covid-19 |

Martin Zeman (UC Irvine) | June, EH (fu) |