# July 18 -- 29, 2016

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

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

Local Organizers: Nam Trang and Martin Zeman

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.

Losely following past workshops, the first week of the workshop meets M--F; each day consists of 3 lectures (each is 75 minutes long), 2 in the morning and 1 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).

Conference Pictures:   Pictures MP01-MP23 were taken by William Mitchell.
T-shirt:   T-shirt
Group pictures:   GP1   GP2
More pictures:   MP01     MP02     MP03     MP04     MP05     MP06     MP07     MP08     MP09     MP10     MP11     MP12     MP13     MP14     MP15     MP16     MP17     MP18     MP19     MP20     MP21     MP22     MP23     MP24

# Participants

1. Juan Aguilera (TU Wien, Austria)
2. Sean Cox (VCU)
3. Scott Cramer (Rutgers)
4. Cody Dance (U. of North Texas, Denton)
5. Derrick Dubose (U. of Nevada, Las Vegas)
6. Elliot Glazer (Rutgers)
7. Gabe Goldberg (Harvard)
8. Stephen Jackson (
U. of North Texas, Denton)
9. Jean Larson (U. of Florida, Gainesville)
10. William Mitchell (U. of Florida, Gainesville)
11. Dan Nielsen (Copenhagen)
12. Grigor Sargsyan (Rutgers) (tentative)
13. Ralf Schindler (Münster)
14. Farmer Schlutzenberg (Münster)
15. Xianghui Shi (Beijing Normal)
16. Benjamin Siskind (Berkeley)
17. John Steel (Berkeley)
18. Ryan Sullivant (Irvine)
19. Nam Trang (Irvine)
20. Sandra Uhlenbrock (Münster)
21. Sean Walsh (UCI)
22. Trevor Wilson (Miami, OH)
23. Martin Zeman (Irvine)

# Scheduled talks

First week M, July 18 T, July 19 W, July 20 Th, July 21 F, July 22
9:30--10:45 Steel Cramer Schlutzenberg Uhlenbrock Steel
11:15--12:30 Steel Cramer Schlutzenberg Schindler Steel
14:30--15:45 Wilson Uhlenbrock Dance Schindler Dance
16:15--∞ Problems and discussions Problems and discussions Schlutzenberg + Prob and Dis Problems and discussions Steel + Prob and Dis

Second week M, July 25 T, July 26 W, July 27 Th, July 28 F, July 29
9:30--10:45 Steel Steel Jackson Informal discussions Informal discussions
11:15--12:30 Steel Steel Jackson Informal discussions Informal discussions
14:30--15:45 Chan Trang Trang Informal discussions Informal discussions
16:00--∞ Problems and discussions Schindler + Prob and Dis Schlutzenberg + PrbDis Problems and discussions Problems and disucssions

# Talk titles and abstracts

Willam Chan: When an Equivalence Relation with All Borel Classes Will Be Borel Somewhere?
Abstract: Kanovei, Sabok, and Zapletal asked whether it is possible that every analytic equivalence relation with all Borel classes could be a Borel equivalence relation on some I-positive Borel set, where I is a sigma-ideal whose associated forcing is proper. Using homogeneous tree representations of sets, this talk will show that if there is a measurable cardinal with infinitely many Woodin cardinals below it, then every equivalence relation in L(R) with all Borel classes is a Borel equivalence relation when restricted to some I-positive Borel set, where I is an ideal whose forcing is proper. Also under AD_R, every equivalence relation with all Borel classes is a Borel equivalence relation when restricted to some comeager set. This is joint work with Magidor.
Notes:   Talk

Scott Cramer: Woodin's AD-conjecture for I_0 and generic absoluteness from j-Suslin representations
Abstract: I will discuss some consequences and aspects of the proof of Woodin's AD-conjecture for I_0. In particular I will introduce j-Suslin representations and contrast them with U(j)-representations, which were introduced by Woodin in the context of the very large cardinal I_0. I will also sketch a proof of a generic absoluteness result which can be obtained from the existence of j-Suslin representations.
Notes:   Talk 1   Talk 2

Cody Dance: The External Ultrapower of HOD via the club measure on \omega_1
Abstract: We work in AD+V=L(R). Given an inner model M and a measure \mu in L(R), one can define the external ultrapower of M via \mu by taking all function in L(R) from crit(\mu) into M. In this talk, we analyze the external ultrapower of HOD via the club measure on \omega_1 from the point of view of inner model theory. We use our techniques to answer a question of Jackson-Ketchersid about codes for ordinals less than \omega_\omega.
Notes:   Talk 1   Talk 2

Steve Jackson: The spectrum of a pointclass and Steel's conjecture.
Abstract: We introduce the notion of the spectrum of a pointclass, which is a set or ordinals associated to the pointclass. Using this notion we prove two results (one joint with Woodin). One of these results relates to a conjecture of Steel on the closure properties of pointclasses.
Slides:   Talk 1   Talk 2

Ralf Schindler: Varsovian models
Abstract: In joint work with Gunter Fuchs we show that if L[E] is tame, has no strong cardinal, and does not know how to fully iterate itself, then L[E] has class many grounds and their intersection is a lower part model, the minimal core'' of L[E]. In sharp contrast, in joint work with Grigor Sargsyan, building upon earlier work of himself and Martin Zeman, we show that if L[E] is least with a strong cardinal above a Woodin cardinal, then L[E] has only set many grounds. Its smallest ground is of the form L[E?,?], where L[E?] is the fully iterable (in L[E]) core model of L[E] and ? is a partial iteration strategy for L[E?]. In particular, L[E?,?] has no proper ground. This research also led to a long extender version of Woodin's extender algebra which corresponds to an old forcing of Bukovský's that gets exploited in our arguments.
Notes:   Talk 1   Talk 2   Talk 3

Farmer Schlutzenberg: HOD^L[E] above omega_3^L[E] below a Woodin limit of Woodins.
Abstract: In this talk I will present some theorems regarding HOD_X^M, where M=L[E] is an iterable mouse satisfying ZFC and X=M|alpha for some alpha <= omega_1^M. I will discuss (1) some criteria on M,X which guarantee that M=HOD^M_X, and (2) for M tame, or M below a Woodin limit of Woodins, some structural facts regarding HOD^M above omega_2^M, or omega_3^M, respectively.
Notes:   Talk 1   Talk 2   Talk 3   Talk 4

John Steel: Least branch hod pairs
Abstract: We outline the proof of a general comparison theorem for iteration strategies. This leads to a fine structural analysis of HOD^M under AD^+. HOD is the the direct limit of all "least branch hod pairs". The analysis requires an assumption we call "hod pair capturing" (HPC), which basically says that the iteration strategies of least branch hod pairs are Wadge cofinal. How to prove HPC, assuming AD^+ and no iteration strategies for mice with superstrongs, is the main open problem.
Notes:   Talk 1   Talk 2   Talk 3   Talk 4   Talk 5   Talk 6   Talk 7   Talk 8   Talk 9

Nam Trang: The HOD computation in the non-AD_R case
Abstract: We outline the construction a direct limit system giving rise to HOD in models of AD^+ + not-AD_R, under HPC and Mouse Capturing for short tree mice. This is joint work with John Steel.
Notes:   Talk 1   Talk 2

Sandra Uhlenbrock: Producing M_n^#(x) from optimal determinacy hypotheses.
Abstract: In this talk we will outline a proof of Woodin's result that boldface Sigma^1_{n+1} determinacy yields the existence and omega_1-iterability of the premouse M_n^#(x) for all reals x. This involves first generalizing a result of Kechris and Solovay concerning OD determinacy in L[x] for a cone of reals x to the context of mice with finitely many Woodin cardinals. We will focus on using this result to prove the existence and omega_1-iterability of M_n^# from a suitable hypothesis. Note that this argument is different for the even and odd levels of the projective hierarchy. This is joint work with Ralf Schindler and W. Hugh Woodin.
Notes:   Talk 1   Talk 2

Trevor Wilson: Generic absoluteness and universally Baire sets of reals
Abstract: We discuss some results and open questions about one-step and two-step generic absoluteness for the pointclass exists-reals (\Pi^2_1)^uB, assuming a proper class of Woodin cardinals. The determination of the consistency strength of two-step generic absoluteness for this pointclass is joint work with Grigor Sargsyan.
Slides:   Slides

# Problem sessions

Problem session 1:   Board 1   Board 2   Board 3   Board 4   Board 5   Board 6   Board 7   Board 8   Board 9
Problem session 2:   Board 1   Board 2   Board 3   Board 4   Board 5   Board 6   Board 7
Misc problems:   Board

# Accommodation

Important: If you are intersted in Option 1 or 2 below please let Martin or Nam know.

1. Hotel. Here is a link to UCI webpage with recommended hotels. Note that you can request UCI rate which should be a bit lower than the usual rate. We recommend Ayres Hotel & Suites, Costa Mesa, map (approximately \$100 per night).
2. On campus housing.
3. Airbnb.

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