at the

American Institute of Mathematics, San Jose, California

organized by

Ralf Schindler and John Steel

- the core model induction technique,
- the analysis of the hereditarily ordinal definable sets in models of $AD$, and
- inner models with long extenders.

The first broad front concerns the construction of iteration strategies. Such constructions can be organized by the core model induction method. Here one combines core model theory and descriptive set theory so as to produce logically complicated iteration strategies and scales, via an induction on their logical complexity. The core model induction is our most powerful method for obtaining consistency strength lower bounds, and in many important cases (for example, in the lower bound problem for the Proper Forcing Axiom), it seems indispensable.

The core model induction method relies heavily on results which connect the hierarchy of scaled pointclasses in determinacy models (more precisely, models of $ZF + AD^+$) with iteration strategies and mice, via an analysis of the $HODs$ of models of $AD^+$. Grigor Sargsyan made important progress on this problem in his 2009 Ph.D. thesis. Sargsyan's work has been further developed, and applied in core model inductions, by several people since then. Nevertheless, the basic theory needed here is very much unfinished.

This workshop will explore its frontier. One key test question is whether the $HOD$ of an $AD^+$ model must always satisfy GCH; another is whether it can it satisfy moderately strong large cardinal hypotheses, such as "there is a Woodin limit of Woodin cardinals".

The second broad front on which there has been recent progress concerns the form that canonical inner models take when the large cardinals hypotheses are so strong that the embeddings associated to them must be represented by long extenders. W. Hugh Woodin has published some dramatic results on inner models with long extenders, in a series papers that won the Hausdorff medal of the European Set Theory Society in July 2013. We hope to communicate some of the recent work by Woodin and others in this area.

The workshop schedule.

A report on the workshop activities.

A list of open problems.

Papers arising from the workshop:

Equiconsistencies at subcompact cardinals

by Itay Neeman an John Steel, *Arch. Math. Logic 55 (2016), no. 1-2, 207–238 * MR3453585

The higher sharp I

by Yizheng Zhu

Universally Baire sets and generic absolutness

by Trevor M. Wilson, *J. Symb. Log. 82 (2017), no. 4, 1229–1251* MR3743609

Inner model theoretic geology

by Gunter Fuchs and Ralf Schindler, *J. Symb. Log. 81 (2016), no. 3 * MR3569115

A weak (?) consequence of determinacy

by G. Fuchs, E. Schimmerling, R. Schindler, F. Schultzenberg, S. Uhlenbrock, and T. Wilson

The analysis of HOD below the theory AD^+ + "The largest Suslin cardinal is a member of the Solovay sequence"

by Grigor Sargsyan

$L(R,\mu)$ is unique

by Daniel Rodriguez and Nam Trang, *Adv. Math. 324 (2018), 355–393 * MR3733890

Scales in hybrid mice over R

by Farmer Schlutzenberg and Nam Trang