# Borel Uniformization

## Speaker:

## Institution:

## Time:

## Location:

I will present an application of Borel Uniformization theorem for "large sections", and if time permits give a proof of the theorem.

Sean Xue

UCI

Monday, November 15, 2010 - 4:00pm

RH 440R

I will present an application of Borel Uniformization theorem for "large sections", and if time permits give a proof of the theorem.

Dr Dima Sinapova

UCI

Monday, November 22, 2010 - 4:00pm

RH 440R

We show that given $\omega$ many supercompact cardinals, there is a

generic extension in which there are no Aronszajn trees at

$\aleph_{\omega+1}$. This is an improvement of the large cardinal

assumptions. The previous hypothesis was a huge cardinal and $\omega$ many

supercompact cardinals above it, in Magidor-Shelah.

Dr Dima Sinapova

UCI

Monday, November 8, 2010 - 4:00pm

RH 440R

We show that given $\omega$ many supercompact cardinals, there is a

generic extension in which there are no Aronszajn trees at

$\aleph_{\omega+1}$. This is an improvement of the large cardinal

assumptions. The previous hypothesis was a huge cardinal and $\omega$ many

supercompact cardinals above it, in Magidor-Shelah.

Dr Dima Sinapova

UCI

Monday, November 1, 2010 - 4:00pm

RH 440R

We show that given $\omega$ many supercompact cardinals, there is a

generic extension in which there are no Aronszajn trees at

$\aleph_{\omega+1}$. This is an improvement of the large cardinal

assumptions. The previous hypothesis was a huge cardinal and $\omega$ many

supercompact cardinals above it, in Magidor-Shelah.

Dr. Martin Zeman

UCI

Monday, October 25, 2010 - 4:00pm

RH 440R

Monroe Eskew

UCI

Monday, October 4, 2010 - 4:00pm

RH 440R

Dr Sean Cox

Munster University, Germany

Monday, October 18, 2010 - 4:00pm

RH 440R

I will present the proofs of some recent results of Viale

and Weiss. Weiss introduced the notion of a slender function in his

dissertation: roughly, a function $M \mapsto F(M) \subset M$ (where

$M$ models a fragment of set theory) is slender iff for every

countable $Z \in M$, $Z \cap F(M) \in M$; i.e. $M$ can see countable

fragments of $F(M)$. Viale and Weiss proved that under the Proper

Forcing Axiom, for every regular $\theta \ge \omega_2$, there are

stationarily many $M \in P_{\omega_2}(H_{(2^\theta)^+})$ which

``catch'' $F(M \cap H_\theta)$ whenever $F$ is slender (i.e. whenever

$F$ is slender then there is some $X_F \in M$ such that $F(M \cap

H_\theta) = M \cap X_F$). The stationarity of this collection implies

many of the known consequences of PFA; e.g. failure of weak square at

every regular $\theta \ge \omega_2$; and separating internally

approachable sets from sets of uniform uncountable cofinality.

Dr Sean Cox

Munster University, Germany

Monday, October 11, 2010 - 4:00pm

RH 440R

I will present the proofs of some recent results of Viale

and Weiss. Weiss introduced the notion of a slender function in his

dissertation: roughly, a function $M \mapsto F(M) \subset M$ (where

$M$ models a fragment of set theory) is slender iff for every

countable $Z \in M$, $Z \cap F(M) \in M$; i.e. $M$ can see countable

fragments of $F(M)$. Viale and Weiss proved that under the Proper

Forcing Axiom, for every regular $\theta \ge \omega_2$, there are

stationarily many $M \in P_{\omega_2}(H_{(2^\theta)^+})$ which

``catch'' $F(M \cap H_\theta)$ whenever $F$ is slender (i.e. whenever

$F$ is slender then there is some $X_F \in M$ such that $F(M \cap

H_\theta) = M \cap X_F$). The stationarity of this collection implies

many of the known consequences of PFA; e.g. failure of weak square at

every regular $\theta \ge \omega_2$; and separating internally

approachable sets from sets of uniform uncountable cofinality.

Andres Caicedo

Boise State University

Wednesday, June 2, 2010 - 3:30pm

RH 340P

The graph G_0 was introduced by Kechris-Solecki-Todorcevic in the late 90s,

and has since turned into an essential object in descriptive set theory. In

joint work with Richard Ketchersid, we prove a version of the G_0-dichotomy

in models of AD^+. This is then used to establish that the quotient by the

equivalence relation E_0 is a successor of R, a result previously known

under AD_R, but (perhaps surprisingly) not in L(R).

Sean Xue

UCI

Monday, May 24, 2010 - 3:20pm

RH 440R