# Category Archives: Logic In Southern California

## 2016 Graduate summer school in set theory, Irvine, July 25 – August 5 2016

A graduate summer school in set theory will be held July 25 – August 5 2016 at the University of California Irvine.

The subject of the summer school will be singular cardinal combinatorics, with a focus on PCF theory. The school will begin with a thorough treatment of PCF and related topics in combinatorial set theory, including club guessing and approachability. After this the school will treat interactions between PCF and other areas of set theory (including for example large cardinals, forcing axioms, reflection principles and squares).

The instructors will include James Cummings (CMU) and Christopher Lambie-Hanson (Hebrew University of Jerusalem). Lectures will be held Monday-Friday in each of the weeks July 25-29 and August 1-5. Each day will include two 90-minute lectures and two 90-minute discussion periods.

Participants will be housed in student housing on the Irvine campus. The school will fully support US citizens and permanent residents: this includes housing, meals, and travel costs (from US cities using US carriers).

If you are a graduate student and are interested in attending the summer school, please write to 2016gsst@gmail.com with a short statement about your background in set theory and your interest in attending.

The web page for the summer school is

http://www.math.cmu.edu/users/jcumming/summer_school/

This summer school is supported by the National Science Foundation grant DMS-1044150 as part of the program EMSW21-RTG: Logic in Southern California.

## Logic in Southern California Caltech, May 10, 2014

LOGIC IN SOUTHERN CALIFORNIA

Saturday, May 10, 2014

Rowland Hall 306

Funded by NSF Grant DMS-1044150

Schedule:

2:00 – 3:00  Sherwood Hachtman (UCLA)

3:30 – 4:30  Bill Chen (UCLA)

5:00 – 6:00  Monroe Eskew (UCI)

Abstracts of the talks:
Sherwood Hachtman:   The Strength of Borel Determinacy

It is a landmark result of Martin that Borel determinacy is a theorem of ZF. Interestingly, Martin’s inductive proof uses transfinitely many iterations of the Powerset axiom, and an analysis due to Friedman shows that these are necessary.  In this talk, we will present a refinement of these results, giving level-by-level equiconsistencies between determinacy and a novel family of weak reflection principles.  We will also discuss how these results adapt to the Borel hierarchy on coanalytic sets, where the inner model theory for measurable cardinals of high Mitchell order comes into play.

Bill Chen:   Tight stationarity and careful sets

Mutual stationarity is a notion of stationarity for certain sequences of subsets of a singular cardinal $\lambda$ (possibly even of countable cofinality) which was defined by Foreman and Magidor. They isolated tight stationarity as a version of mutual stationarity that is easier to analyze. We use a pcf-theoretic scale to relate sequences of subsets of $\lambda$ to subsets of $\lambda^+$, translating from tightly stationary sequences to stationary subsets of $\lambda^+$. Then we will define careful sets, which are the subsets of $\lambda^+$ that are involved in this translation. The main result is the construction of a model where $\lambda$ is a strong limit and every subset of $\lambda^+$ is careful. This construction uses the combinatorics of tree-like scales and a diagonal supercompact Prikry forcing.

Monroe Eskew:   Applications of the anonymous collapse

Abstract: We will discuss a “universal” forcing for collapsing a large cardinal to be the successor of smaller regular cardinal.  It absorbs most of the effects of a wide class of “standard” collapsing posets, yet is incomparable with all of them, hence the name, anonymous collapse. Using this forcing, we show how to construct many different models with the same reals and same cardinals but very different cardinal characteristics of the continuum.  We show how it can be used with almost-huge cardinals to achieve minimal solutions to Ulam’s measure problem, and to obtain successor cardinals which are “generically supercompact” in a strong sense.  This will enable a solution to a question of Foreman related to an old conjecture in model theory about the cardinality of ultrapowers.  In contrast to the traditional variety, we show that these generically supercompact cardinals are compatible with squares.

## Logic in Southern California Caltech, March 8, 2014

### Schedule:

2:00-3:00 Erik Walsberg (UCLA)
3:15-4:15 Henry Macdonald (Caltech)
4:15-5:00 Coffee Break
5:00-6:00 Andres Forero (UCI)
Abstracts:

Talk Title: Metric geometry in the O-minimal setting.
Abstract: I will discuss the geometry of those metric spaces which are definable in o-minimal expansions of fields.

Henry Macdonald (Caltech).

Talk Title: Descriptive combinatorics
Abstract: In combinatorics, the axiom of choice is often used to justify the existence of a certain object – a graph coloring, say, or a matching. In descriptive combinatorics, we ask: what happens if we place definability restrictions from descriptive set theory on these objects? I will discuss various situations where these considerations give rise to some interesting questions. In particular, I will discuss “Borel chromatic numbers”, and a result about extending a combinatorial argument from cardinal arithmetic to the field of Borel equivalence relations.

Andres Forero (UCI).

Talk Title: Consistency strength of Stationary Catching
Abstract: In this talk we will give a brief overview of Generic large cardinal axioms and their motivation. In concrete we consider certain collections of structures that behave nicely with respect to a fixed ideal on omega_2, and introduce axioms asserting that these collections are large. We will specifically consider the consistency strength of the Stationary Catching Axiom (which is a weakening of the saturation of an ideal), in terms of Woodin cardinals. For this purpose, we will describe two important techniques used: the core model induction, and covering arguments.

## Logic in Southern California Caltech, February 18, 2012

### Schedule:

Title: Improving the Consistency Strength of Reflection at אω+1
Title: The group of all homeomorphisms of the Cantor set has ample generics
Abstract: There are various notions of largeness of conjugacy classes in topological groups: the Rokhlin property, the strong Rokhlin property, and ample genericity. We give examples and state properties of groups with large conjugacy classes . Then, we focus on the group of all homeomorphisms of the Cantor set, H(K). Answering a question of Kechris and Rosendal, we show that H(K) has ample generics, that is, we show that for every m the diagonal conjugacy action of H(K) on H(K)^m has a comeager orbit. I plan to sketch the proof of this result in as many details as time permits.
4:15-5:00 Coffee Break
5:00-6:00 Justin Palumbo (UCLA)
Title: Unbounded and dominating reals in Hechler extensions
Abstract: We present results exploring the relationship between dominating and unbounded reals in the Hechler extension and its variations, as well as the relationships between the extensions themselves. We describe the impact of these results on some recent work of Brendle and Loewe. The main structural result is a representation theorem for the dominating reals in the Hechler extension: every dominating real eventually dominates the sandwich composition of the Hechler real with two ground model reals that monotonically converge to infinity. From this we derive the answer to a question due to Laflamme