Department of Mathematics, UC Irvine

July 10 -- 21, 2023

Supported by NSF Grant DMS-1945592

Organizers: Nam Trang (UNT Denton), Martin Zeman (UC Irvine)

Special thanks to Claudia Cheffs for her invaluable help with logistics


One level at a time

This conference follows up on the successful previous conferences on Inner Model theory held in Muenster (Germany) (2010, 2011, 2015..., 2022), Berkeley (2014, 2019), Irvine (2016), Girona (Spain) (2018). Its purpose is to bring together researchers working in the field as well as graduate students, and present and discuss the recent developments.

See the bottom of this website for some logistics information

The following people expressed their interest to participate (alphabetically)
Adolf, Dominik (Technische Universitaet Wien, Vienna, Austria)
Chan, William (Denton, Texas, USA)
Cody, Sean (Berkeley, Califonia, USA)
Cox, Sean (Virginia Commonwealth University, Virginia, USA)
Dubose, Derrick (University of Nevada, Las Vegas, USA)
Eshkol, Julian (University of California Irvine, USA)
Gappo, Takehiko (Technische Universitaet Wien, Vienna, Austria)
Goldberg, Gabriel (Berkeley, California, USA)
Habic, Miha (Simons Rock, USA)
Koshat, Lukas (Technische Universitaet Wien, Vienna, Austria)
Kruschewski, Jan (Muenster University, Muenster, Germany)
Levinson, Derek (University of California, Los Angeles, USA)
Minden, Kaethe (Simmons Rock, USA)
Schlutzenberg, Farmer (Muenster University, Muenster, Germany)
Shi, Xianghui (Beijing Normal University, Beijing, China)
Siskind, Benjamin (Carnegie Mellon, Pittsburgh, Pennsylvania, USA)
Steel, John (Berkeley, California, USA)
Trang, Nam (University of North Texas Denton, Texas, USA)
Wilson, Trevor (Miami University Oxford, Ohio, USA)
Zeman, Martin (University of California Irvine, USA)
Zhang, Jiaming (Carnegie Mellon, Pittsburgh, Pennsylvania, USA)

Picture Week 1    Picture Week 2    Problem List 2023

Scheduled talks

All talks are in Natural Sciences II, Room 1201

First week Monday, July 10 Tuesday, July 11 Wednesday, July 12 Thursday, July 13 Friday, July 14
9:30--10:45 Goldberg Chan Schlutzenberg Schlutzenberg Cox
11:15--12:30 Gappo Siskind Steel Steel Steel
14:30--15:45 Chan Kruschewski Gappo Koschat Trang
16:15--17:30 Problems + Discussions Problems + Discussions Problems + Discussions Problems + Discussions Problems + Discussions

Second week Monday, July 17 Tuesday, July 18 Wednesday, July 19 Thursday, July 20 Friday, July 21
9:30--10:45 Schlutzenberg Wilson Adolf Schlutzenberg Problem Session
11:15--12:30 Steel Steel Wilson Zeman Zeman
14:30--15:45 Levinson Adolf Trang Adolf Informal Discussions
16:00--17:30 Problems + Discussions Problems + Discussions Problems + Discussions Problems + Discussions Informal Discussions

Titles and abstracts of the talks

Adolf: Title: Universally Baire Presentations from Generic Ultrapowers
Abstract: Let $j:V \rightarrow M$ elementary with $\crit(j) = \kappa$ where $M \subset V\left[G\right]$ with $G$ generic over the universe. We will show how to construct an universally Baire representation for a reasonable mouse/strategy operator with base in $H_\kappa$ over the generic ultrapower. This has applications in Core Model Inductions.

Chan: Title: Ultrapowers by the Partition Measures on $\omega_1$
Abstract: We will show under the axiom of determinacy that the ultrapower of the first uncountable strong partition cardinal $\omega_1$ by its strong partition measure is strictly less than the second strong partition cardinal $\omega_{\omega + 1}$, which answers a question of Goldberg. In the process, we will formulate club uniformization principles and prove continuity for functions on the strong partition space. Variations of Martin's good coding system will be developed to prove club uniformization and serve as a mechanism to code the ultrapower as a relation on the reals.

Cox: Title: On the universality of the nonstationary ideal
Abstract: Douglas Burke, building on a previous result of Matt Foreman, proved that the generalized nonstationary (NS) ideal is universal in the following sense: every normal ideal, and every tower of normal ideals of inaccessible height, is a canonical Rudin-Keisler projection of the restriction of NS to some stationary set. There is a natural way to make sense of NS canonically projecting to an "ideal extender" in the sense of Claverie, and it is natural to wonder if Burke's theorem extends to this setting. But it does not; for example, if $\kappa$ is strong but not supercompact, then (most) of its extenders cannot be canonical projections of NS. In fact, ``$\kappa$ is supercompact" is equivalent to "for every $\lambda$ there is a short $(\kappa,\lambda)$ extender that is a canonical projection of NS". Curiously, the characterization makes no mention whatsoever of the strength of the extenders. Burke's theorem also does not extend to towers of normal ideals, if the height of the tower is accessible (such as towers that yield I3 embeddings). We can also characterize "$\aleph_\omega$ is Jonsson" in terms of I3 towers that happen to be canonical projections of NS.

Gappo: Title: Chang models over derived models with supercompact measures
Abstract: We provide a new construction of Chang-type models of determinacy with supercompact measures in a hod mouse. If the given hod mouse has a cardinal \delta that is a limit of Woodin cardinals and <\delta-strong cardinals, then our model satisfies AD^+ + AD_R + \Theta is regular + \omega_1 is <\delta_\infty -supercompact for some regular cardinal \delta_\infty > \Theta.
This complements Woodin's generalized Chang model, which satisfies AD^+ + AD_R + \omega_1 is supercompact, assuming a proper class of Woodin cardinals that are limits of Woodin cardinals.
This is joint work with Sandra Müller and Grigor Sargsyan.

Goldberg: Title: Generalizations of the Ultrapower Axiom, II. Notes
Abstract: The Ultrapower Axiom (UA) is a structure principle for countably complete ultrafilters that provides a convenient setting for large cardinals theory, particularly for large cardinals that can be formulated in terms of ultrafilters (e.g., strongly compact and supercompact cardinals). Unsurprisingly, UA has not proved quite as useful in the domain of large cardinals formulated in terms of extenders (e.g., strong cardinals and Woodin cardinals), which motivates the attempt to find generalizations of UA that yield more information in this context. In this talk, I'll extract such a generalization of UA from the proof of Woodin's recent theorem that UA holds in the HOD of any model of AD+ + V = L(P(R)) and use this principle to attack some of the problems that seem intractable assuming UA alone.

Koschat: Title: \omega-sequences in the uB-power set
Abstract: (.pdf)

Kruschewski: Title: On a Conjecture Regarding the Mouse Order for Weasels
Abstract: We investigate Steel's conjecture in 'The Core Model Iterability Problem', that if W and R are \Omega+1-iterable, 1-small weasels, then W\le^*R iff there is a club C\subseteq\Omega such that for all \alpha\in C, if \alpha is regular, then the cardinal successor of \alpha in W is less or equal than the cardinal successor of \alpha in R . We will show that the conjecture fails, assuming that there is an iterable premouse which models KP and which has a \Sigma_1-Woodin cardinal. On the other hand, we show that assuming there is no transitive model of KP with a Woodin cardinal the conjecture holds. In the course of this we will also show that if M is an iterable admissible premouse with a largest, regular, uncountable cardinal \delta, and P is a forcing poset with the \delta-c.c. in M, and g is M-generic, but not necessarily Sigma_1-generic, M[g] is a model of KP. Moreover, if M is such a mouse and T is maximal normal iteration tree on M such that T is non-dropping on its main branch, then the last model of T is again an iterable admissible premouse with a largest regular and uncountable cardinal.

Levinson: Title: Unreachability of Inductive-Like Pointclasses in L(R)
Abstract: We show in L(R) that if \Gamma is an inductive-like pointclass, then there is no sequence of distinct \Gamma sets of length \delta_\Gamma^+. This is joint work with Itay Neeman and Grigor Sargsyan.

Siskind: Title: Mouse operators and order-preserving Martin's Conjecture
Abstract: We'll explain an approach for proving Martin's Conjecture for order-preserving functions which relies on establishing analogues of computability-theoretic theorems for (essentially) arbitrary mouse operators. These analogues are known to hold for many familiar mouse operators but it is open whether they even hold for all operators which map x to a level of L[x]. Time permitting, we'll explain how far we can go at present. This is joint work with Patrick Lutz

Schlutzenberg: Title: Full normalization and the initial segment condition for mice with long extenders
Abstract: Mice M containing ordinals $\kappa$ such that M models "$\kappa$ is $\kappa^+$-supercompact” were introduced and analysed by Woodin in the early 2010s. Neeman and Steel also introduced an alternate but equivalent hierarchy. While the theory is largely parallel to that for short extender mice, there are some notable differences. For example, the initial segment condition for extenders in the extender sequence fails. We will describe an alternate hierarchy of mice at this level for which the initial segment condition holds in a manner very analogous to that in the short extender realm. Full normalization of stacks of normal trees holds (under natural strategy condensation hypotheses), and we will describe the key new feature that distinguishes full normalization at this level from that for short extender mice. Reference

Steel: Title:

Trang: Title: Forcing over models of determinacy
Abstract:Built on earlier work of Chan and Jackson, we have completed the proof of the following theorem. \begin{theorem}[Chan-Jackson, Trang-Ikegami] Assume $\sf{AD}$ and $\mathbb{P}\subseteq \mathbb{R}$ is a non-trivial forcing. Then whenever $g\subset \mathbb{P}$ is $V$-generic, $V[g]\models \neg \sf{AD}$. \end{theorem} This result in some sense is optimal. Assuming ``$\sf{AD}^+ + V = L(\powerset(\mathbb{R})) + \Theta$ is regular", we construct a forcing of size $\Theta$ that preserves determinacy. This is joint work with D. Ikegami.

Wilson: Title: Virtual Woodin cardinals
Abstract: Large cardinal notions defined in terms of elementary embeddings can be virtualized by allowing the elementary embeddings to exist in generic extensions of the universe. We define virtually strong and virtually Woodin cardinals as well as "weak" (but equiconsistent) variants of these. There may be multiple equivalent definitions of large cardinals having inequivalent virtualizations, so we must take care to select the right definitions. In particular, in the definitions of virtually strong and virtually Woodin cardinals we do not require the codomains of the generic elementary embeddings to be well-founded beyond their rank initial segments that are required to agree with V. We show that the existence of a weakly virtually Woodin cardinal is equivalent to certain Vopenka-like statements about trees. More specifically, we show that a beth-fixed point is weakly virtually Woodin if and only if the Vopenka Principle holds below it for wellfounded 2-labeled trees and their homomorphisms. (The labels can be omitted if the homomorphisms are required to be embeddings.) Moreover, we mention the related result that the least weakly virtually Woodin cardinal is in fact virtually Woodin and equals the least cardinal such that any well-founded tree of that cardinality is isomorphic to a proper subtree of itself. We also characterize weakly virtually Woodin cardinals by a partition property that weakens the definition of omega-Erdos (or more precisely a variant of it due to Silver) by weakening the existence of a shift-invariant omega-sequence to the existence of many shift-invariant finite sequences -- enough to form a tree of large rank.

Zeman: Title: A bounding principle for products of cardinals
Abstract: We introduce a bounding principle for products of cardinals which can be used to prove distributivity for iterated club shooting with Easton support at increasing sequence of cardinals. We give a proof of this principle in fine structural extender models. Some instances of the principle can be proved in ZFC, but it is not known whether this is the case for the principle at its full generality.

Some food places on Campus
- Cafe Espresso (In front of Reines Hall; Coffee, Bakery, Sandwiches, Fruits, Drinks)
- Starbucks (Between Natural Sciences I and Biological Sciences III)
- University Town Center (Restaurants, Peets, Trader Joe's, Target. Accessible on foot via the bridge over Campus Drive next to Irvine Barclay Theatre
- Student Center (Mostly fast food style; many places may be closed in summer)
- Garbanzo (mediterranean-like) in Interdisciplinary Sciences and Engineering Building
- Starship

Campus maps
Interactive Campus Dining Map

Some housing options If you prefer to stay in a hotel instead of AirBnb or , you may consider the following.
More affordable hotels
- Extended Stay America (.pdf)
- Sonesta Simply Suites (.pdf)
- Sonesta Irvine (.pdf)
- MainStay Suites (.pdf)

More expensive hotels
- Ayres hotel:  (https)
- Atrium hotel:  (https)


Last Modified: July 24, 2023