Workshop in Inner Model Theory & Descriptive Set Theory | Department of Mathematics

# Workshop in Inner Model Theory & Descriptive Set Theory

We will be running a workshop in Inner Model Theory and Descriptive Set Theory in the summer.
The workshop will run June 1 - June 3, 2012, starting around 9am June 1, and finishing by around 5pm June 3.

Participants
Dominik Adolf, Scott Cramer, Logan Crone, Monroe Eskew, Daisuke Ikegami, Steve Jackson, Richard Ketchersid, John Krueger,
William Mitchell, Grigor Sargsyan, Farmer Schlutzenberg, John Steel, Nam Trang, Trevor Wilson, Martin Zeman

Talks - Program
All talks will be held in the GAB (General Academic Building), room 461.
The first talk is scheduled for Friday 9:30am.

Scott Cramer (Berkeley): Inverse Limit Reflection and the Structure of L(V_{\lambda+1})
The structure L(V_{\lambda+1}) was first studied by Woodin to prove the consistency of AD^{L(R)} from large cardinals. He later showed that many of the same structural properties of L(R) under determinacy hold for L(V_{\lambda+1}) under large cardinals. Laver first introduced the tool of inverse limits in this context to tackle the problem of reflecting large cardinals at this level. In this talk we will extend Laver's results on inverse limit reflection, and then use this technique to analyze further the structure of L(V_{\lambda+1}) and its relationship to models of determinacy.

Daisuke Ikegami (Berkeley): Boolean valued second order logic.
Two semantics of second order logic have been exclusively investigated. One is full semantics where one lets second order quantifiers range over the full power sets of the first order universes while the other is Henkin semantics where one considers second order structures satisfying certain comprehension axioms. While the former expresses many properties such as well foundedness of partial orders but does not enjoy most of the properties first order logic has, the latter is very weak and is close to first order logic in terms of logical properties.
In this talk, we introduce Boolean valued semantics for second order logic which sits between the two semantics and compare it with Woodin's \Omega logic and with full semantics of second order logic.
This is joint work with Jouko Väänänen.

Steve Jackson (UNT): Jonsson cardinals in L(R)

Grigor Sargsyan (Rutgers): More on the strength of PFA

Farmer Schlutzenberg (UNT): +0 iteration trees on countable substructures of V
Two iterability theorems proven in Martin and Steel's paper Iteration Trees'' depend on the tree in question being +1 or +2. We will discuss how the proofs can be adapted to yield the same conclusion, without the +1/+2 hypotheses.

John Steel (Berkeley): More playful universes

Nam Trang (Berkeley): Structure theory of L(R,\mu)
For each \alpha < \omega_1, let X_\alpha = \{f : \omega^\alpha \rightarrow \powerset_{\omega_1}(\mathbb{R})| f is increasing and continuous} and \mu_\alpha be a normal fine measure on X_\alpha. We identify X_0 with \powerset_{\omega_1}(R). Martin and Woodin independently showed that these measures exist assuming (ZF + DC_\mathbb{R}) + AD + Every set is Suslin (\mu_0's existence was originally shown by Solovay from AD_\mathbb{R}). We sketch the proof of the derived model construction giving the existence of these measures (+ AD^+) from large cardinals and the Prikry forcing construction which gives back the exact large cardinal strength from AD^+ and the measure. We also survey some theorems on the structure theory of the model L(\mathbb{R},\mu_\alpha) assuming the model satisfies \Theta > \omega_2 and \mu_\alpha is a normal fine measure on X_\alpha. Here the main theorem is that our assumption implies L(\mathbb{R},\mu_\alpha) satisfies AD^+. If time allows, we'll discuss open questions concerning extending this hierarchy of measures.

Trevor Wilson (Berkeley): The next Suslin cardinal in ZF + DC
The "next Suslin cardinal" analysis is generalized to work without full AD: the envelope of an inductive-like pointclass Gamma is defined in such a way that its basic properties can be established assuming only Delta determinacy. An application of this is the nonexistence of "divergent envelopes," which gives a simple proof of Woodin's theorem that divergent models of AD^+ satisfy AD_R in their intersection. As another application, the Martin-Woodin construction of weakly homogeneous trees and Jackson's proof of stability of such trees are adapted in a new approach to the "gap in scales" case of the core model induction.

Martin Zeman (Irvine): Characterization of \Box(\kappa^+) in extender models

Getting to UNT / Honors Hall
The public transportation from the airport to UNT is somewhat limited. We suggest either taking a taxi / shuttle bus or renting a car at the airport. If you plan to take a shuttle bus it's best to book one in advance (e.g. at supershuttle.com) to avoid wait time. UNT is about 25 miles from DFW airport. Make sure to retain all receipts for reimbursement.
Housing is at UNT's Honors Hall, which is located on the north-west corner of the intersection of Eagle Drive and Avenue C. See below for parking information.

Local Information
See here for local information including a campus map and parking.
The above link does not include information on places to eat near Honors Hall.
(See the campus map through the link above to clarify the following.)
On Friday a store on the 2nd floor of the union building, on campus, will be open; this is roughly on the way between Honors Hall and the GAB.
There are also several eating places along Ave C, south of the intersection of Eagle and Avenue C.
There is a supermarket there also, on the west side of Ave C.
There is a laundromat on Eagle, just east of its intersection with Ave C.

Wireless
There will be wireless access available in Honors Hall; you will receive instructions for connecting to this when you check-in.
There will also be wireless access available in the GAB (General Academic Building), but this will be through a different connection, which we'll give you instructions for on Friday.

Accommodation
We have made a preliminary reservation for several suites in UNT's Honors Hall, with check-in Thursday May 31 (evening) and check-out Monday June 4 (morning).
These are student apartments. They are minimal 1-bedroom/1-bath apartments, with a small lounge area. We will cover the cost of these accommodation arrangements for all participants.
If you are interested in other accommodation arrangements, we will provide partial support. If you're interested in staying for longer, we can look into extending reservations.

Travel funding
We also have some travel grants available, which would partially cover travel expenses. Women and underrepresented groups are especially encouraged to apply. Please contact the organizers to apply for funding.

Organizers
The organizers are Steve Jackson (UNT), Grigor Sargsyan (Rutgers) and Farmer Schlutzenberg (UNT). Please send an email to "workshop-imt-dst AT unt DOT edu" for further information.

Conference in Logic, Dynamics and their Interactions
The workshop immediately precedes the conference Logic, Dynamics and their Interactions, also to be held at UNT, June 4 - June 8.

The workshop is being supported by the National Science Foundation and the University of North Texas.