## Clinton Conley: Amenability and μ-hyperfiniteness

Mathematical logic seminar – January 12, 2016
Time:     12:30 – 13:30

Room:     Wean Hall 8220

Speaker:         Clinton Conley
Department of Mathematical Sciences
CMU

Title:     Amenability and μ-hyperfiniteness

Abstract:     Ornstein-Weiss showed in the early 80s that any measure-preserving Borel action of a countable amenable group on a standard probability space generates, after deleting a null set, a hyperfinite orbit equivalence relation. This was soon generalized by Connes-Feldman-Weiss to handle non-measure-preserving actions. After some background on amenability, we discuss a modern graph-theoretic approach to proving these theorems, building on work of Elek, Kaimanovich, and Kechris-Miller. This talk includes joint work with Gaboriau, Marks, and Tucker-Drob.

## Andy Zucker: Universal metrizable minimal flows

Mathematical logic seminar – December 8, 2015
Time:     12:30 – 13:30

Room:     Wean Hall 7201

Speaker:         Andy Zucker
Department of Mathematical Sciences
CMU

Title:     Universal metrizable minimal flows

Abstract:     Let G be a Polish group. We consider the following question: is it possible for the universal minimal flow M(G) to be non-metrizable, but for there to exist a metrizable minimal G-flow which maps onto every other metrizable minimal G-flow? We show that this cannot be the case. As a corollary, we also show that if G admits only countably many metrizable minimal flows, then M(G) is metrizable.

## Andy Zucker: Universal metrizable minimal flows

Mathematical logic seminar – November 24, 2015
Time:     12:30 – 13:30

Room:     Wean Hall 7201

Speaker:         Andy Zucker
Department of Mathematical Sciences
CMU

Title:     Universal metrizable minimal flows

Abstract:     Let G be a Polish group. We consider the following question: is it possible for the universal minimal flow M(G) to be non-metrizable, but for there to exist a metrizable minimal G-flow which maps onto every other metrizable minimal G-flow? We show that this cannot be the case. As a corollary, we also show that if G admits only countably many metrizable minimal flows, then M(G) is metrizable.

## Jacob Davis: Some results in set-theoretic geology

Mathematical logic seminar – November 10, 2015
Time:     12:30 – 13:30

Room:     Wean Hall 7201

Speaker:         Jacob Davis
Department of Mathematical Sciences
CMU

Title:     Some results in set-theoretic geology

Abstract:     Given V a universe of set theory, we might be interested in whether it was formed by forcing from some smaller inner universe; such an inner universe is called a ground of V, and the intersection of all such universes is called the mantle ofV. Alternatively we can access a larger class of universes by moving to some forcing extension V[G] and then taking a ground of V[G]. A universe that can be reached in this way is called a generic ground of V, and the intersection of all such universes is the generic mantle of V. It is clear that all grounds are generic grounds, and so the generic mantle of V must be a sub-class of its mantle. It is much less clear, and indeed an open question, whether the mantle and generic mantle are necessarily equal. We shall explore some results in this direction.

## Bill Chen: Can every mutually stationary sequence be tightly stationary?

Mathematical logic seminar – October 27, 2015
Time:     12:30 – 13:30

Room:     Wean Hall 7201

Speaker:         Bill Chen
Department of Mathematics
UCLA

Title:     Can every mutually stationary sequence be tightly stationary?

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 (joint with Itay Neeman).

## Clinton Conley: Borel marker sets and hyperfiniteness

[this talk is a warmup for the Appalachian set theory workshop at CMU on October 24,
where Su Gao (UNT) will speak on “Countable abelian group actions”]

Mathematical logic seminar – September 29, 2015
Time:     12:30 – 13:30

Room:     Wean Hall 7201

Speaker:         Clinton Conley
Department of Mathematical Sciences
CMU

Title:     Borel marker sets and hyperfiniteness

Abstract:     A classical tool in ergodic theory is the Rokhlin lemma, which more or less states that any ergodic measure-preserving automorphism of a standard probability space is the uniform limit of periodic automorphisms. At its combinatorial core, the lemma’s proof relies on the ability to find measurable sets which intersect every orbit in a reasonably spaced out fashion. We discuss analogs of this in the purely Borel context, and use such marker sets to prove the Slaman-Steel / Sullivan-Weiss-Wright result that every Borel action of the integers on a standard Borel space generates a hyperfinite orbit equivalence relation. Time permitting, we discuss the (still open) problem of extending this to actions of arbitrary countable amenable groups, in preparation for Su Gao’s Appalachian Set Theory workshop this October.

## James Cummings: Forcing Pi_2 statements II

Mathematical logic seminar – September 22, 2015
Time:     12:30 – 13:30

Room:     Wean Hall 7201

Speaker:         James Cummings
Department of Mathematical Sciences
CMU

Title:     Forcing Pi_2 statements II

Abstract:

We often find ourselves needing to force assertions of the general form “for every x there exists y….” where typically x and y are subsets of some regular cardinal κ

We will consider three general cases (of increasing difficulty)

κ is ω1
κ is the successor of an uncountable regular cardinal
κ is the successor of a singular cardinal

## James Cummings: Forcing ∀ ∃ statements

The logic seminar resumes next week. This term we are
meeting in Wean Hall 7201 on Tuesdays at 12:30.

Mathematical logic seminar – September 15, 2015
Time:     12:30 – 13:30

Room:     Wean Hall 7201

Speaker:         James Cummings
Department of Mathematical Sciences
CMU

Title:     Forcing ∀ ∃ statements

Abstract:

We often find ourselves needing to force assertions of the general form “for every x there exists y….” where typically x and y are subsets of some regular cardinal κ

We will consider three general cases (of increasing difficulty)

κ is ω1
κ is the successor of an uncountable regular cardinal
κ is the successor of a singular cardinal

## Sebastien Vasey: Independence in abstract elementary classes

Mathematical logic seminar – April 28, 2015
Time:     12:30 – 13:30

Room:     Wean Hall 8201

Speaker:         Sebastien Vasey
Department of Mathematical Sciences
CMU

Title:     Independence in abstract elementary classes

Abstract:

Forking is one of the central notion of modern model theory. Roughly speaking, it is a notion of independence generalizing both linear independence in vector spaces and algebraic independence in fields. In the first-order framework, it was introduced by Shelah and is one of the main devices of his book. One can ask whether there is such a notion for classes that are not first-order axiomatizable, such as classes of models of a sentence in infinitary logic. We will focus on abstract elementary classes (AECs), a very general axiomatic framework introduced by Shelah in 1985. In Shelah’s book on AECs, the central concept is again a local generalization of forking: good frames.

We will attempt to explain why good frames are so useful by surveying recent applications to problems like existence of saturated models or stability transfer. Time permitting, we will sketch a proof of the following approximation to a conjecture of Shelah:

Theorem (modulo a claim of Shelah whose proof has yet to appear): Assume there are unboundedly many strongly compact cardinals and the weak generalized continuum hypothesis holds. Then an AEC which for a high-enough cardinal λ has a single model of size λ will have a single model of size μ for every high-enough μ.

## Nam Trang: Covering and more covering

Mathematical logic seminar – April 14, 2015
Time:     12:30 – 13:30

Room:     Wean Hall 8201

Speaker:         Nam Trang
Department of Mathematical Sciences
CMU

Title:     Covering and more covering

Abstract:

We present a couple of basic arguments of getting sharps for operators with nice properties from certain failures of covering. These arguments are featured in various constructions of canonical models of large cardinals from forcing axioms like PFA, the existence of strongly compact measures in ZF+DC etc.