**LOGIC
IN SOUTHERN CALIFORNIA
**

Saturday, June 1, 2013

**UCLA Math Sciences room 6627**

**Funded
by NSF Grant DMS-1044604**

**Schedule:**

2:00 - 3:00 Monroe Eskew (UCI), Measurability properties on small cardinals

3:30 - 4:30 Athipat Thamrongthanyalak (UCLA), Michael's selection theorem and Whitney's extension problem in o-minimal structures

5:00 - 6:00 Jay Williams (Caltech), Cone measures and biembeddability of Kazhdan groups

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

Monroe Eskew: **Measurability properties on small cardinals.**

**Abstract:**
Following the discovery that the continuum cannot be a measurable
cardinal, Ulam asked, can it be in some sense close to measurable? Is
there a "small" collection of countably complete, two-valued measures that
collectively measure all subsets of R? Results of Alaoglu-Erdos, Taylor,
and Gitik-Shelah shed light by giving bounds on the answer in ZFC, and
consistency results of Woodin show some possible values from large
cardinals. A result of Taylor characterizes a positive solution under CH:
a solution via aleph_1 many normal measures is equivalent to the existence
of a dense ideal on aleph_1, which is in turn equivalent to the existence
of an ideal without the disjoint refinement property. We ask, what about
when CH fails—does this equivalence generalize to higher cardinals? By
combining a generalization of Woodin's consistency result with Foreman's
recently published Duality Theorem, we give a negative answer. This
separates some saturation properties of ideals on aleph_2 which are
equivalent on aleph_1. Lastly, we explore questions about dense ideals on
consecutive cardinals.

Athipat Thamrongthanyalak: **Michael's selection theorem and Whitney's extension problem in o-minimal structures.**

**Abstract:**
In 1956, E. Michael proved Michael's Selection Theorem which provides a sufficient condition that guarantees the existence of a continuous selector.
In this talk, I'll discuss this problem in o-minimal context and use it to give an answer to a special case of Whitney's Extension Problems:
how can one determine whether a definable continuous function on a closed subset of R^n is the restriction of a C^1-function. This is joint work with Matthias Aschenbrenner.

Jay Williams: **Cone measures and biembeddability of Kazhdan groups.**

**Abstract:**
It is a result of Martin that for every Borel Turing-invariant
set X in 2^N, either X or its complement contains a Turing cone, i.e. a set
consisting of every Turing degree above a specific degree. We show there
is no analog of Martin's theorem in the context of embeddability of
finitely generated groups. Along the way we prove some results on groups
which are bi-embeddable with Kazhdan groups. Joint work with Simon Thomas.

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, 6, 8, and 2.

Organizers: Alexander Kechris, Itay Neeman, Martin Zeman

**Local organizers:**
Donald A. Martin and Yiannis Moschovakis

**UCLA, May
12, 2012**