# Category Archives: Inner model theory series

## First Girona inner model theory conference, Girona, July 16-27 2018

The first Girona inner model theory conference will take place on July 16-27 at the Philosophy Department of the University of Girona, Catalonia.

The conference is a sequel to previous conferences on inner model theory in Münster, Palo Alto, Berkeley and Irvine. Once more, the meeting will 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 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.

The conference is organized by Ralf Schindler (Münster), John Steel (Berkeley) and Joan Vergés (Girona). Please contact Ralf Schindler (rds@wwu.de) if you intend to participate.

## 1st Irvine Conference on Descriptive Inner Model Theory and HOD Mice, July 18-29 2016

1st Irvine Conference on Descriptive Inner Model Theory and HOD Mice

July 18 — 29, 2016
Department of Mathematics, UC Irvine

Supported by: NSF Grants DMS-x, DMS- 1044150, DMS-y, and UCI CORCL

Organizers: Grigor Sargsyan (Rutgers), Nam Trang (Irvine), Martin Zeman
(Irvine)

This workshop is a sequel to a series of conferences and workshops on
descriptive inner model theory including 1st Conference on the core model
induction and hod mice that was held in Münster (FRG), July 19 — August
06, 2010, the 2nd Conference on the core model induction and hod mice that
was held in Münster (FRG), August 08 — 19, 2011, the AIM Workshop on
Descriptive Inner Model Theory held in Palo Alto (CA), June 02 — 06,
2014, and to the Conference on Descriptive Inner Model Theory, held in
Berkeley (CA) June 09 — 13, 2014, and the 3rd Conference on the core
model induction and hod mice, held in Münster (FRG), July 20 — 31, 2015.

The main purpose of the workshop is to disseminate and communicate results
and recent development in descriptive inner model theory and related
subjects. The workshop consists of single talks by experts in the field on
their recent work as well as lectures aimed at advanced graduate students
interested in inner model theory and related fields.

Following past workshops, the first week of the workshop meets M–F; each
day consists of 4 lectures (each is 75 minutes long), 2 in the morning and
2 in the afternoon. Between the lectures, we will leave plenty of time for
discussions, lunch, and informal seminars. The second week will be more
informal; as in the past, the topics and speakers for the second week will
be decided during the first week of the meeting.

All lectures will take place in Natural Scienes II building, room 1201.
map

The organizers gratefully acknowledge the financial support from the
National Science Foundation (NSF).

# inner model theory, the core model induction, and hod mice

## July 20 — 31, 2015

Organizers: Ralf Schindler (Münster), John Steel (Berkeley)
This conference will be a sequel to the 1st Conference on the core model induction and hod mice that was held in Münster (FRB), 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, as well as to the AIM Workshop on Descriptive Inner Model Theory, held in Palo Alto (CA), June 02 — 06, 2014, and to the Conference on Descriptive Inner Model Theory, held in Berkeley (CA) June 09 — 13, 2014.

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

We will meet Monday–Friday, with 2 hours of lecture in the morning and 2 hours of lecture in the early afternoon. This will leave ample time for problem sessions, informal seminars, and other interactions in the late afternoons and evenings. The week July 20 — 24, 2014, will be the week when the formal part of the conference will take place; the organizers will decide beforehand about speakers and talks for the 1st week. There will be a second week, July 27 — 31, 2014, which will be much less formal; speakers and talks for the 2nd week will be decided about during the first week.

The conference will take place at the Department of Mathematics and Computer Science, Univ. of Muenster, Orléans-Ring 10, 48149 Muenster, FRG, lecture halls N2 and N3. We also have lecture hall N2 available for the week June 13 — 17 for pre–conference activities.

The conference organizers gratefully acknowledge financial support from the Marianne and Dr. Horst Kiesow-Stiftung, Frankfurt a.M., from the DVMLG (Deutsche Vereinigung für mathematische Logik und Grundlagenforschung), as well as from the NSF (National Science Foundation).

## Schedule:

 Mo, July 20 Tue, July 21 We, July 22 Thu, July 23 Fr, July 24 9:30–10:45 Yizheng Zhu Trevor Wilson Nam Trang Ralf Schindler Martin Zeman 11:15–12:30 Gunter Fuchs Farmer Schlutzenberg Daniel Rodrigues Steve Jackson Chris Le Sueur 14:30–15:45 John Steel 1 Hugh Woodin 1 Hugh Woodin 3 John Steel 3 Philipp Schlicht 16:45–17:30 John Steel 2 Hugh Woodin 2 Hugh Woodin 4 John Steel 4 N.N. 17:30–∞ Problems and Discussions Problems and Discussions Problems and Diskussions Problems and Discussions free

 Mo, July 27 Tue, July 28 We, July 29 Thu, July 30 Fr, July 31 9:30–11:00 N.N. N.N. N.N. N.N. N.N. 11:30–12:30 N.N. N.N. N.N. N.N. N.N. 14:30–16:00 N.N. N.N. N.N. N.N. free 16:30–17:30 N.N. N.N. N.N. N.N. free 17:30–∞ Problems and Discussions Problems and Discussions Problems and Diskussions Problems and Discussions free

## Talks and abstracts:

1. Gunter Fuchs: Geology, inner models and the solid core. I will present some results due to Ralf Schindler and myself, isolating situations in which certain models of set theory that arise in the context of set theoretic geology turn out to be fine structural. For example, if the universe is constructible from a set and there is an inner model with a Woodin cardinal, then the mantle, i.e., the intersection of all inner models of which the universe is a forcing extension by set forcing, is fine structural. A similar result concerns the concept of the solid core, which is the collection of all sets that are constructible from a set of ordinals that is solid, meaning that it cannot be added to an inner model by set forcing. Again, the solid core is fine structural if there is an inner model with a Woodin cardinal.
2. Steve Jackson: Some closure properties of pointclasses under AD.
3. Daniel Rodriguez: L(R,μ) is unique. We show that under various appropriate hypotheses there is only one determinacy model of the form L(R,μ) in which mu is a supercompact measure on Pω1(R).
4. Grigor Sargsyan: The Largest Suslin Axiom. We will state the Largest Suslin Axiom. Show its consistency relative to large cardinals. Show how to derive it from combinatorial principles (in the presence of mild large cardinals), and if time permits, make some comments about how to go beyond. (Cancelled.)
5. Ralf Schindler: AD from ℵω is a strong limit and 2ω > ℵω1. Using a cheat, we get an inner model with ω1 Woodin cardinals from said hypothesis and also from the assumption that κ is a limit of cofinality ω1 s.t. the set of μ < κ with 2μ = μ+ is stationary and costationary. This is joint with Gabriel Fernandes.
6. Philipp Schlicht: M1 and infinite time machines. A subset of an ordinal is recognizable if it is the unique subset for which an infinite time machine has a certain output. We consider reals as inputs to infinite time machines with finitely many ordinal parameters, and define the recognizable hull by iterating relative recognizability for reals. In joint work with Merlin Carl, we connect this with M1 by showing that a real is in the recognizable hull if and only if it is in M1.
7. Farmer Schlutzenberg: Fine structure from normal iterability. For short extender premice, the solidity of the standard parameter and related fine structural properties, follow from normal iterability. That is, let k< ω and let M be a k-sound, (k, ω1+1)-iterable premouse. Then M is k+1-solid and k+1-universal, degree k+1-condensation holds for M, and if k>0 then M is Dodd-sound. Further, any (0, ω1+1)\$-iterable pseudo-premouse is a premouse. I will explain at least the main ideas in the proofs of these results. (The results hold for Mitchell-Steel indexing, permitting extenders of superstrong type to appear on the extender sequence.)
8. John Steel: Normalizing iteration trees and comparing iteration strategies.
9. Chris Le Sueur: Determinacy a bit beyond co-analytic. It is quite well-known result of Martin that the existence of a measurable cardinal is enough to prove the determinacy of all boldface Π11 sets. The argument nicely modifies to get the determinacy of all lightface Π11 sets from the existence of O. This argument has since been pushed to go beyond Π11, and I will discuss how this is done. In doing so, I’ll introduce a generalised effective descriptive set theory suitable for studying uncountable spaces, and describe a ramified forcing (akin to Cohen’s original conception) that allows class forcing over weak models.
10. Nam Trang: Square in hod mice and applications. We discuss ideas for proving square in hod mice up to lsa hod mice. We describe how the straightforward adaptation of the Schimmerling-Zeman construction fails for hod mice and how to overcome it. Applications of this type of theorems include improving lower-bound consistency strength of various principles. For instance, I prove that PFA implies the sharp for the minimal model of “ADR + Θ is strong” exists. We hope to build on this to construct a model of “AD+ + LSA” from principles like PFA.
11. Trevor Wilson: Determinacy from strong compactness of ω1. Determinacy axioms are well-known to imply large cardinal properties of ω1 such as measurability and certain degrees of strong compactness and supercompactness. Conversely, from large cardinal properties of ω1 one can obtain inner models of determinacy axioms using the core model induction. This approach leads to some equiconsistencies between determinacy axioms and their consequences in terms of degrees of strong compactness of ω1. For a set X, we say that ω1 is X-strongly compact if there is a fine, countably complete measure on Pω1(X). We obtain the following equiconsistencies modulo ZF plus DC: (1) the P(ω1)-strong compactness of ω1 is equiconsistent with AD, and (2) the P(R)-strong compactness of ω1 is equiconsistent with ADR (plus DC). This talk is based on joint work with Nam Trang.
12. Hugh Woodin: The Ultimate-L Conjecture. I discuss progress on proving the Ultimate-L Conjecture focusing on two obstructions. The first obstruction concerns comparison by least disagreement and the second concerns the necessary failure of amenable or even weakly amenable levels. The latter is just the necessary failure of weak square principles at singular cardinals translated into fine-structural terms. Both these obstructions provide key clues as to how to pass the “ω-barrier”.
13. Yizheng Zhu: The higher sharp. We establish the descriptive set theoric representation of the mouse Mn#, which is called 0(n+1)#. At even levels, 02n#is the higher level analog of Kleene’s O; at odd levels, 02n+1# is the unique iterable remarkable level-2n+1 blueprint. notes taken by rds
14. Martin Zeman: The true core model with a Woodin cardinal. We show that in certain universes one can construct the true core model using Steel’s CMIP construction. The model will be the true core model in the sense that it will satisfy the rigidity theorem, generic absoluteness, and weak covering. This is a joint work with Grigor Sargsyan.

## Descriptive Inner Model Theory

2-6 Jun 2014
Palo Alto CA, U.S.A.

The AIM at Palo Alto will host a workshop on descriptive inner model
theory, June 02-06, 2014, organized by R. Schindler (Münster) and J. Steel
http://aimath.org/workshops/upcoming/innermodel/ If you would like to
participate, then you may fill out the online form provided by AIM. The
deadline to apply is Feb 02, 2014.

## Mitchellfest: 70th birthday of William Mitchell

7-8 Jun 2014
Berkeley CA, U.S.A.

June 07 and 08, 2014, UC Berkeley will host a conference honoring William
Mitchell on the occasion of his 70th birthday. See
http://wwwmath.uni-muenster.de/logik/Personen/Uhlenbrock/MitchellConference/
and J. Steel ( coremodel@berkeley.edu ).

## Workshop on descriptive inner model theory

9-13 Jun 2014
Berkeley CA, U.S.A.

UC Berkeley will host a workshop on descriptive inner model theory June
09-13, 2014, organized by G. Sargsyan (Rutgers), R. Schindler (Münster),
and J. Steel (Berkeley). See
http://wwwmath.uni-muenster.de/logik/Personen/Uhlenbrock/DIMTConference/
) and J. Steel ( coremodel@berkeley.edu ).