### Saturday, June 2, 2018

#### 8:30 a.m. – 4:30 p.m. with coffee and lunch breaks

### Carnegie Mellon University

#### Refreshments in Wean Hall 6220

#### Breakfast and coffee starting at 7:30

# Dilip Raghavan : “Boolean ultrapowers and iterated forcing”

## Description

In joint work with Saharon Shelah, we develop a new method for proving consistency results on cardinal invariants, particularly results involving the invariant . This method can be used with a wide range of forcing notions, including arbitrary ccc posets. However the method always requires a supercompact cardinal κ in the ground model and produces forcing extensions in which the desired invariants sit above κ. Another feature of our method is that it generalizes to cardinal invariants above ω, and can be used to give uniform consistency proofs that work at any regular cardinal. It can also be used to treat situations where three cardinal invariants must be separated. In particular, our technique solves various long standing open problems about cardinal invariants at uncountable regular cardinals. All the results use Boolean ultrapowers, studied by Keisler and other model theorists in the 1960s. I will aim to give a fairly self contained introduction to this method and to some to its applications to the theory of cardinal invariants.

## Suggested reading

- Beginning: Kunen “Set theory”, Chang and Keisler “Model theory” (particularly Chapter 6).
- Intermediate: Andreas Blass, “Combinatorial characteristics of the continuum”, Handbook of set theory.
- Advanced:
- Saharon Shelah, “Two cardinal invariants of the continuum ( < ) and FS linearly ordered iterated forcing”, Acta Mathematica, 2004.
- Marayanthe Malliaris and Saharon Shelah, “Existence of optimal ultrafilters and the fundamental complexity of simple theories”, Advances in Mathematics, 2016.

## Local information

The nearest airport is Pittsburgh International Airport. The Supershuttle shared van service is cheaper but slower than taking a taxi from the airport.

## Lodging

VERY IMPORTANT NOTE ABOUT LODGING: A block of rooms earmarked for attendees has been set aside at a local hotel (the Shadyside Inn). If we are covering your lodging expenses then we will need to make a reservation for you. Please don’t make your own reservation if we have promised you support, this will cause confusion and may make it impossible for us to reimburse you.

## Participant travel support

Funds provided by the National Science Foundation will be used to reimburse some participant transportation and lodging expenses. Priority will be given to students and faculty who do not hold federal research grants. Please request such funds as far in advance of the meeting as possible by sending the following information to the email address appalachiansettheory@gmail.com

- Your name, university affiliation, mailing address, phone number and email address
- Your nationality and visa status
- Your professional status and some additional information:
- Undergraduate students: please describe your background in set theory
- Graduate students: please tell us your year and the name of your thesis advisor if you have one
- Faculty: please tell us whether you hold a federal research grant

- A brief statement about your interest in the workshop