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## From $\aleph_2$ to infinity

May 15 to May 19, 2023

at the

American Institute of Mathematics, San Jose, California

organized by

James Cummings, Itay Neeman, and Dima Sinapova

This workshop, sponsored by AIM and the NSF, is devoted to combinatorial problems about infinite cardinals. There are two types of infinite cardinals to investigate: successors of regular cardinals, most notably $\aleph_2$, and successors of singular cardinals, for example ${\aleph_{\omega+1}}_{\omega+1}$. The workshop will focus on combinatorial principles such as the tree property, stationary reflection and the effect of consequences on forcing axioms on cardinal arithmetic, in particular what implications they have on the continuum, and the singular cardinal hypothesis (SCH).

The main topics of the workshop are

• The tree property, its strengthening ITP, stationary reflection how these combinatorial principles interact with SCH;
• Which consequences of PFA and MM require the continuum to be $\aleph_2$, and more generally, their effect on cardinal arithmetic;
• Forcing techniques such as proper iterated forcing and Prikry type forcing.

This event will be run as an AIM-style workshop. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.

Space and funding is available for a few more participants. If you would like to participate, please apply by filling out the on-line form no later than January 15, 2023. Applications are open to all, and we especially encourage women, underrepresented minorities, junior mathematicians, and researchers from primarily undergraduate institutions to apply.

Before submitting an application, please read the description of the AIM style of workshop.