__1st IRVINE ____CONFERENCE____ on DESCRIPTIVE INNER
MODEL THEORY and HOD MICE__

# 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 Bukovskı'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.**

**HOME**

Last Modified: August 15, 2016