Un castell. Copyright: Joana Schindler López, 2017

1st Girona conference on

inner model theory

July 16--27, 2018

Organizers: Ralf Schindler (Münster), John Steel (Berkeley), and Joan Vergés (Girona).

The Eiffel bridge, Girona.

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, as well as 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.
The university of Girona (above three pictures)

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.

Location: On Thu + Fri, July 19 + 20, 18, the conference will take place in the seminar room A1 of the Facultat de Lletres of the University of Girona, Plaça Josep Ferrater i Móra, 1, see here for a map. How to get to A1: From Plaça Ferrater Móra, enter thru the main door of the Facultat de Lletres; go straight and then up the staircase, then make a left. During the 2nd week, the conference will take place in the seminar room R3. How to get to R3: From Plaça Ferrater Móra, enter thru the main door of the Facultat de Lletres and make a left thru a smaller door (before the stairs) to get into the cloister; R3 will then be on your left side.

We are most grateful to Joan Vergés, to Ramon Fàbrega, Secretari de la Cŕtedra Ferrater Mora, and to Marga López Arpí for hosting us and for their help in organizing this meeting!
There is no conference fee. However, we will ask for a donation of 20 € for drinks and cookies which we will provide and for a security service fee. There is a café within the building with good coffee; it is open 8am - 3pm. There's also the "Campus Bar," a coffee place, Plaça de Sant Domènec, 1, which is right next door.

Warning: Wed, July 25, 2018, is a holiday in Girona (Sant Jaume)! Most shops will be closed on that day. As having been scheduled, our meeting will take place on that day also, though.

Above: view of the old part of Girona.


Each day, the building needs to get locked at 7:30pm. We are thus asked to leave at 7pm, so please keep that in mind. There are plenty of cafés, bars, and restaurant within very short walking distance, where discussions may continue.

Mon, July 16 Tue, July 17 Wed, July 18 Thu, July 19 Fri, July 20
10:00--11:15 John Steel Farmer SchlutzenbergJohn SteelTrevor WilsonJohn Steel
11:45--13:00 Benjamin Siskind Farmer SchlutzenbergBenjamin SiskindTrevor WilsonBenjamin Siskind
15:30--16:45 Gabriel Goldberg Sean CoxSteve JacksonFarmer SchlutzenbergDan Saattrup Nielsen
17:15--18:30 Gabriel Goldberg Sandra MüllerWilliam ChanGunter FuchsProblem Session
18:30--∞ Problems and Discussions Problems and DiscussionsProblems and DiscussionsProblems and Discussionsfree

Mon, July 23 Tue, July 24 Wed, July 25 Thu, July 26 Fri, July 27
10:00--11:15 John Steel W. Hugh WoodinW. Hugh WoodinMenachem MagidorPhilipp Schlicht
11:45--13:00 Benjamin Siskind W. Hugh WoodinW. Hugh WoodinMenachem MagidorRalf Schindler
15:30--16:45 Gabriel Goldberg Ralf SchindlerRalf SchindlerJohn Steelfree
17:15--18:30 Trevor Wilson Farmer SchlutzenbergDominik AdolfRalf Schindlerfree
18:30--∞ Problems and Discussions Problems and DiscussionsProblems and DiskussionsProblems and Discussionsfree

There will be a city tour, but you won't know even the night before that it's actually gonna take place the very next day and who will be the guide.

Talks, abstracts, and problem list:

  • List of open problems, see here.

  • Dominik Adolf: "Improved results on a mutual stationarity property." A sequence (Sn: m < n < ω) (Sn ⊂ ℵn) is called mutually stationary iff there exist stationarily many X ⊂ ℵω such that sup(X ∩ ℵn) ∈ Sn for all m < n < ω. We are interested in sequences where each Sn concentrates on points of a fixed cofinality, but these fixed cofinalities do not stabilize as n approaches ω. In earlier work (jointly with S.Cox and P. Welch) it was shown that the existence of a mutually stationary sequences with blocks of size m (m >1) implies the existence of infinitely many cardinals κn which have Mitchell order at least (κ+(m+1))K in the Core Model (assuming 0P does not exist) under the additional assumption, (*), that all Sn concentrate on points of uncountable cofinality. We will offer two ways to improve the above result: (a): We show that the hypothesis actually the imply the existence of a sort of "pseudo" Prikry sequence; (b): We achieve the same result, but eliminate assumption (*). Time will likely only permit an in-depth discussion of one of these results; which one will be decided by an audience vote.

  • William Chan (Denton TX): "Destruction of the Axiom of Determinacy by Forcing on R when Θ is Regular. Ikegami and Trang showed that many known forcings, such as Cohen forcing, cannot preserve AD. They also showed that ZF + AD+ + V = L(P(R)) can prove that forcings which preserve AD must also preserve Θ. We will show, assuming ZF and AD, that every nontrivial wellorderable forcing of cardinality less than Θ forces the failure of AD. Assume ZF, AD, and Θ is regular, we will show that every nontrivial forcing which is a surjective image of the reals forces the failure of AD. This is joint work with Stephen Jackson. notes by W. Chan

  • Sean Cox (Richmond VA): "Reflection Principles and Strong Chang's Conjecture." We will discuss some relationships between stationary reflection principles, Strong Chang's Conjecture, and fragments of the (♰) principle ("every ω1-stationary set preserving forcing is semiproper"). For example, Doebler-Schindler isolated a version of Strong Chang's Conjecture that is equivalent to the Semi-Stationary Reflection Principle (SSR). Fuchino-Usuba isolated a similar (but stronger) version of Strong Chang's Conjecture that sits between two stationary reflection principles. I will separate some of these principles, while also providing a simplification and strengthening of a theorem of Krueger. notes taken by rds

  • Gunter Fuchs (CUNY): "Subversions of iterable forcing classes." In my recent research, I have worked on Jensen's class of subcomplete forcing notions. These forcing notions do not add reals, but may change uncountable cofinalities to become countable, and they are iterable with revised countable support. Jensen has shown that the forcing axiom for subcomplete forcing (SCFA) is consistent, assuming the consistency of a supercompact cardinal. The natural model for SCFA is obtained by performing a Baumgartner style iteration of subcomplete forcing notions, in a way reminiscent of how one produces models of the proper forcing axiom or Martin's Maximum. Since subcomplete forcing does not add reals, CH will hold in the resulting model, and this is a feature that sets it apart from other, more well-known forcing axioms. I studied the effects of SCFA on the failure of square principles, and showed that SCFA has almost exactly the same effects as MM or PFA have, and it is a natural question to ask whether this difference can be attributed to the fact that SCFA is consistent with CH. There are many variations of this question, all of the form: given a statement that is consistent with SCFA but inconsistent with MM, is it equivalent (modulo ZFC + SCFA) to CH? In order to approach these questions, one is led to investigate models of SCFA in which CH fails, and this leads to the study of iterable classes of forcing notions that comprise the subcomplete ones but also include ones that may add reals, or may destroy Souslin trees, etc. I will present some such classes and answer some questions of the flavor mentioned above. This is joint work with Corey Switzer. Slides.

  • Gabriel Goldberg (Harvard): "Supercompact cardinals and the Mitchell order, II." The Ultrapower Axiom (UA) is large cardinal theoretic principle that is both an abstract formulation of the comparison lemma and a generalization of the statement that the Mitchell order is linear on normal ultrafilters. We outline a proof of the equivalence of strong compactness and supercompactness under UA, subject to the exceptions from Menas's thesis. notes taken by rds

  • Steve Jackson (Denton TX): "AD and the Suslin Hypothesis. We show assuming AD+ + V=L(P(R)) that SH holds, that is, there are no Suslin lines. In particular, L(R) satisfies SH assuming AD. This answers a question of M. Foreman. We discuss this and some related results on Aronszajn trees in determinacy models, and mention also a result of Woodin on SH in Solovay models. This is joint work with William Chan. notes taken by rds

  • Menachem Magidor (Jerusalem): "Inner models from strong logics." notes taken by rds

  • Sandra Müller (Wien): "Long games and Woodin cardinals." Itay Neeman proved the determinacy of analytic games of length ω ∙ θ for countable θ > ω from a sharp for θ Woodin cardinals. We aim for a converse at successor ordinals and show how to obtain ω+1 Woodin cardinals from the determinacy of analytic games of length ω ∙ (ω + 1). This is joint work with Juan P. Aguilera. notes by S. Müller

  • Dan Saattrup Nielsen (Bristol): "Game-theoretic Ramsey-like cardinals." Holy and Schlicht recently introduced a new hierarchy of Ramsey-like cardinals, the (strategic) α-Ramsey cardinals, which can be seen as 'mini-versions' of measurable cardinals and which also admit a game-theoretic definition. We show that strategic ω1-Ramseys are measurable in K and that they admit precipitous ideals (in V), as well as showing that the α-length game associated to the (strategic) α-Ramseys is determined when α ≤ ω and not determined when α ≥ ω1. This is joint work with Philip Welch. The argument presented in the talk had a bug which got fixed in these notes.

  • Ralf Schindler (Münster): Tutorial on "Inner model theoretic geology." If P ⊆ W ⊆ V, P and W being inner models of V, then P is called a ground of W iff W is a forcing extension of P. The mantle of W is the intersection of all gounds of W. The analyses of the mantle of given (pure) extender models W=L[E] has led to interesting and surprising results. E.g., the mantle of the least inner model with a strong cardinal above a Woodin cardinal is the least strategic model with a Woodin cardinal; the mantle of the least inner model with a strong cardinal above a Woodin cardinal above a strong cardinal above a Woodin cardinal is a strategic model with two Woodin cardinals; etc. In this tutorial, we will focus on presenting key techniques of this area rather than listing known results. This is joint work with Sargsyan and Schlutzenberg. Suggested reading: Schindler, Steel, "The self-iterability of L[E]," JSL 74 (2009), pp. 751-779; Fuchs-Schindler, Inner model theoretic geology," JSL 81 (2016), pp. 972-996; Sargsyan-Schindler, "Varsovian models I," JSL, to appear. Plus the notes on "Varsovian models, II": 1st set, 2nd set, 3rd set, 4th set, 5th set, 6th set, 1st appendix, 2nd appendix. Notes taken by Stefan M.: part 1, part 2, part 3, part 4.

  • Philipp Schlicht (Bristol): "Definable subsets of generalized Baire spaces and long games." In the last few years, people have tried to generalize regularity properties and determinacy results to generalized Baire spaces κκ, where κ is regular. I plan to talk about the consistency of the definable version of the open coloring axiom for uncountable regular κ from an inaccessible cardinal. notes by P. Schlicht

  • Farmer Schlutzenberg (Münster): "Scales from mice." In approximately 1999, Neeman announced a construction of scales directly from mice, avoiding determinacy arguments. His work remains unpublished. We give a new such construction of scales. The method is a variant of that used for semiscales presented in Münster 2017, which admits a proof of lower semicontinuity. notes taken by rds [incomplete]

  • Benjamin Siskind (Berkeley): "On the combinatorics of normalizing, etc." notes taken by rds [incomplete]

  • John Steel (Berkeley): "Hod pair capturing." notes taken by rds

  • Trevor Wilson (Oxford, OH): "Generic Vopenka cardinals and models with few Suslin sets." A generic Vopenka cardinal is an inaccessible cardinal κ such that for every κ-sequence of structures in Vκ for the same first-order language, an elementary embedding between two of the structures exists in some generic extension of V. Because the elementary embedding is not required to exist in V, this is a rather weak large cardinal property: if 0# exists, then every Silver indiscernible is a generic Vopenka cardinal in L. We show that generic Vopenka cardinals are closely related to models of ZF with few Suslin sets of reals. In particular, we show that ZFC + "there is a generic Vopenka cardinal" is equiconsistent with ZF + "there is no injection from P(ω1) to the pointclass of Suslin sets." notes taken by rds

  • W. Hugh Woodin (Harvard): "Ultimate-L." notes taken by rds

  • Martin Zeman (Irvine): "On consistency strength of mutual stationarity with fixed uncountable cofinalities." This is a joint work with Omer Ben Neria which is still in progress. We investigate the consistency strength of mutual stationarity at small singular cardinals of countable cofinality. The precise formulation for instance at ℵω reads that, given some uncountable γ=ℵk and a sequence Sn such that each Sn is a stationary subset of ℵn concentrating on ordinals of cofinality γ there is a stationary set of substructures X of Hθ (θ large given in advance) such that X ∩ ℵn ∈ Sn on a tail-end of n's. Lower bounds of this variant of mutual stationarity was investigated by several authors, predominantly Koepke and Welch. It turns out to have high consistency strength, and we are working on the core model induction in L(R) to obtain AD (we believe the actual consistency strength is much higher). In this talk I will present an argument which takes care of successor step of the core model induction. More precisely, an argument which shows the existence of an inner model with a Woodin cardinal. Here we use a covering type of argument, which is an approach different from the one used by Koepke and Welch, and our lower bound is an improvement of their result. [cancelled]

  • Yizheng Zhu (Beijing): "Some regularity properties on projective equivalence relations." Assme PD. Then every thin Σ12n+1 equivalence relation admits a Δ12n+2 wellordering of the equivalence classes. [cancelled]

  • Above and below: the streets of Girona.


    If you want to learn a bit of Catalan before travelling to Girona, you may go here. Once you're more fluent in Catalan, you may go ahead and go here or there and read a Catalan newspaper.

    Tourist information:

  • City guide in Chinese.
  • For general info, see here.

    Coffee places:

  • Café Boira on Plaça Independència.
  • Café Royal, same square.
  • Café Context, Carrer del Pou Rodó, 21.
  • König, Carrer dels Calderers, 16.
  • Café L'arc, by the cathedral.
  • Xurros amb xocolate (a must if you like chocolate): L'Antiga, Plaça del Vi 8, 17004 Girona.
  • Café Artusi.

    Lunch and dinner:

  • Rule no. 1: Avoid La Rambla de la Llibertat, it's only for tourists!
  • Rule no. 2: Forget about finding a reasonable place for lunch before 1pm and a place for dinner before 7:30pm.
  • Le Bistrot.
  • L'Alqueria.
  • Sushi Bar Girona.
  • Zanpanzar, Carrer de la Cort Reial 12, 17004 Girona: Basque cuisine.
  • Arròs i Peix: fish.
  • Mimolet: top end Catalan cuisine.
  • Creperie Bretonne.
  • Tasca Galega: Galician cuisine.
  • Casa Moner: best bakery in town, plus coffee house.
  • El Celler de Can Roca, three Michelin stars, named the best restaurant in the world in 2013 and 2015 by the magazine Restaurant. Here's a recent article on them.

  • Housing information:

    Girona is very popular among tourists. You should book your accomodation as soon as possible! The meeting will take place here. Housing suggestions:
    Below: Monestir de Sant Daniel.

  • apartments in Girona.
  • Hotel Museu Llegendes de Girona.
  • Monastery of Sant Daniel.
  • Hotel Històric.
  • Casa Cúndaro.
  • Palau dels Alemanys.
  • Pensió Bellmirall.
  • Hotel Ciutat de Girona.
  • Airbnb.
  • Youth hostel.

  • Travel information:

  • Girona has an airport, IATA code: GRO. Ryanair and some others fly there. A pretty short bus or cab ride takes you from the Girona airport to the center.
  • Another excellent option to reach Girona is to fly to Barcelona, IATA code: BCN, take a bus from there to the center of Barcelona, and then a train from Bercelona Sants (fast trains leave from here) or from Barcelona Passeig de Gracia to Girona; there is also a direct bus from BCN to the center of Girona; total travel time in both cases 1 1/2 hours or more. See also the info provided here.
  • Renfe has fast trains coming from France which make a stop at Girona. E.g. a direct train from Lyon to Girona is 4 1/2 hrs. You can also take a TGV directly from Paris Gare de Lyon to Girona, travel time 5 hrs 46 min.

  • Below: Les fires.


    Dominik AdolfJuly 14-27✓ K
    Juan P. Aguilera (TU Vienna)
    Alessandro Andretta (Torino)July 16-25
    Joan Bagaria (Barcelona)July 23-27
    Douglas Blue (Harvard Univ., Cambridge, MA)
    Fabiana Castiblanco (Münster) July 19-?✓ K
    Justin Cavitt (Harvard Univ., Cambridge, MA)
    William Chan (Denton, TX)July 15-28
    Sean Cox (Richmond, VA) July 14-21
    Gabriel Fernandes (Münster)
    Gunter Fuchs (CUNY)July 14-26
    Elliot Glazer (Harvard Univ., Cambridge, MA)
    Gabriel Goldberg (Harvard Univ., Cambridge, MA)
    Stefan Hoffelner (Münster)
    Steve Jackson (Denton, TX)
    Andreas Lietz (Bonn)✓ K
    Menachem Magidor (Jerusalem)July 23-30
    Stefan Mesken (=Miedzianowski) (Münster)
    Sandra Müller (=Uhlenbrock) (Vienna)
    Dan Saattrup Nielsen (Bristol)July 15-30
    Ralf Schindler (Münster) July 12-Aug 23
    Philipp Schlicht (Bristol)July 21-27
    Farmer Schlutzenberg (Münster)
    Noah Schoem (UIC)
    Xianghui Shi (Beijing)
    Benjamin Siskind (UC Berkeley)
    John Steel (UC Berkeley, CA)July 14-30
    Trevor Wilson (Miami Univ., Oxford OH)
    W. Hugh Woodin (Harvard Univ., Cambridge, MA)July 22-28
    Yizheng Zhu (Beijing) [cancelled]
    Martin Zeman (UC Irvine, CA) [cancelled]July 14-?

    Above: The town hall of Girona.
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