**LOGIC
IN SOUTHERN CALIFORNIA
**

Saturday, May 21st, 2016

**UCLA Math Sciences 6627**

**Funded
by NSF Grant DMS-1044604**

**Schedule:**

2:00 - 3:00 Ronnie Chen (Caltech), Structurable equivalence relations.

3:15 - 4:15 Garrett Ervin (UC-Irvine), Every linear order isomorphic to its cube is isomorphic to its square.

4:45 - 5:45 Bill Chen (UCLA), Can mutual and tight stationarity agree anywhere?

**Abstracts**
**Driving
directions and parking** **Organizers
Previous
meetings**

Ronnie Chen: **Structurable equivalence relations**.

**Abstract:**
For a class K of countable structures, a countable Borel equivalence relation is K-structurable if there is a Borel way to put a structure in K on each equivalence class. We will present some results in the abstract theory of structurable equivalence relations, including: their relationship with various kinds of Borel homomorphisms and reductions; the order-theoretic structure of the poset of classes of K-structurable equivalence relations for various K; and the relevance of certain model-theoretic properties of the class K. This is joint work with Alexander Kechris.

Garrett Ervin: **Every linear order isomorphic to its cube is isomorphic to its square**.

**Abstract:**
In the early 1950s, Sierpinski asked whether there exists a linear order
that is isomorphic to its lexicographically ordered cube but not
isomorphic to its square. The analogous question, whether for a given
class of structures \(C\), there exists \(X\in C\)
isomorphic to \(X^3\) but not to
\(X^2\), has been answered positively for many different classes of structures,
including groups, Boolean algebras, topological spaces, graphs, partial
orders, and Banach spaces. However, the answer to Sierpinski's question
turns out to be negative: any linear order that is isomorphic to its cube
is isomorphic to its square, and thus to all of its finite powers. I will
present an outline of the proof and give some related results.

Bill Chen: **Can mutual and tight stationarity agree anywhere?**

**Abstract:**
Mutual and tight stationarity are two notions of stationarity defined
on certain products associated to a singular cardinal, introduced by
Foreman and Magidor. Tight stationarity is closely related to the
structure of scales at the singular cardinal, whereas mutual stationarity
has a more mysterious, model-theoretic character. In this talk, I will
investigate the question of Cummings, Foreman, and Magidor of whether
every mutually stationary sequence can be tightly stationary.
The main result is a model where mutual and tight stationarity are
distinct everywhere, and where this property is quite indestructible
under further forcing (based on joint work with Itay Neeman).

Driving directions and parking:

From the 405 North

- Take 405 (San Diego Freeway) to Wilshire Blvd. East
- Travel east three blocks to Westwood Blvd.
- Turn Left on Westwood Blvd.
- Travel five blocks to Information & Parking Booth

From the 405 South

- Take 405 (San Diego Freeway) to Sunset East
- Take Sunset east to Westwood Plaza
- Turn right on Westwood Plaza
- Proceed straight to Information & Parking Booth

From the east via the 10 (Santa Monica Freeway)

- Take 10 (Santa Monica Freeway) to 405 (San Diego Freeway) North
- Take 405 (San Diego Freeway) to Wilshire Blvd. East
- Travel east three blocks to Westwood Blvd.
- Turn left on Westwood Blvd.
- Travel five blocks to Information & Parking Booth

To park on campus you will need to purchase a daily parking permit at the Information & Parking Booth. The nearest parking lots to the Department of Mathematics are 9, 8, and 2.

Organizers: Alexander Kechris, Itay Neeman, Martin Zeman

**UCLA, May
12, 2012**