**List of participants in this workshop**

**Lecture notes from this workshop by Menachem Magidor and Chris Lambie-Hanson**

Jensen also defined
□_{κ}^{*},
which is a weaker principle than □_{κ}.
Between
□_{κ}
and □_{κ}^{*}
there is Schimmerling's hierarchy of principles
□_{κ,λ}
for 1 ≤ λ ≤ κ.
Even weaker than □_{κ}^{*}
are Shelah's approximation properties.

There are several ways in which weak squares are useful. It is of great interest to identify the weakest form of square that suffices for a given application. As the strongest possible forms of square hold in the core models constructed for large cardinals, the strongest weak square that holds in a given model of ZFC is an intuitive measure of its distance from being a canonical inner model. The problem of determining the consistency strength of the failure of weak square (especially at singular cardinals) is a good litmus test for the success of the inner model program. Another fascinating subject is the impact of forcing axioms like PFA, MRP and MM on the existence or absence of weak squares.

In these lectures, we shall study some of the problems related to weak squares. A broad outline of the talks is:

- The definitions of weak squares and a few examples of applications in infinite combinatorics, topology and algebra.
- The conneciton between weak squares and the existence of good and very good scales.
- Weak squares at singular cardinals are difficult to avoid.
The Džamonja-Shelah result that changing the cofinality of an
inaccessible cardinal κ without collapsing cardinals
automatically creates a model of
□
_{κ,cf(κ)}. (This generalizes a result of Cummings and Schimmerling about Prikry forcing and weak squares.) - Weak squares and large cardinals.
- Weak squares and forcing axioms.

A good place to study weak squares and some of their applications is
Todd Eisworth's article in __The Handbook of Set Theory__.
See chapter 15 in volume 2. The relevant section is on pages 1287-1317.
Some of this material will be covered in the talks.

The large cardinal notions that will be used are all introduced
in Kanamori's __The Higher Infinite__.
The most relevant chapter is 5 on __Strong Hypotheses__.

A good reference for forcing axioms is Jech's __Set Theory__.
The relevant chapters are 31 on __Proper Forcing__ and 37 on
__Martin's Maximum__.

- Mirna Džamonja and Saharon Shelah,
__On squares, outside guessing of clubs and__I_{f}[λ], Fundamenta Mathematicae 148 (1995), no. 2, 165-198 [PDF] - Matthew Foreman and Menachem Magidor,
__A very weak square principle__, The Journal of Symbolic Logic 62 (1997), no. 1, 175-196 [JSTOR] - James Cummings, Matthew Foreman and Menachem Magidor,
__Squares, scales and stationary reflection__, Journal of Mathematical Logic 1 (2001), no. 1, 35-98 [WorldScientific] - Ernest Schimmerling and Martin Zeman,
__Characterization of__□_{κ}__in core models__, Journal of Mathematical Logic 4 (2004), no. 1, 1-72 [WorldScientific] - James Cummings and Menachem Magidor,
__Martin's Maximum and weak square__, Proceedings of the American Mathematical Society, in press

- Your name, university affiliation, mailing address, phone number and email address
- Your professional status and
- 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
- An itemized estimate of your expected transportation expenses

The Shadyide Inn is less than 3/4 mile from CMU; you could walk, take a bus, or call the Shadyside Inn to pick you up.

(If you arrive early, you may want to meet others in the Mathematical Sciences Department lounge, 6220 Wean Hall.)

Taxis from the airport cost about $50.

Another door-to-door option is SuperShuttle at $27 per person in a shared van.