Compactness of \omega_1

Speaker: 

Nam Trang

Institution: 

UCI

Time: 

Monday, April 3, 2017 - 4:00pm to 5:30pm

Host: 

Location: 

RH440R

We investigate various aspects of compactness of \omega_1 under ZF+ DC (the Axiom of Dependent Choice). We say that \omega_1 is X-supercompact if there is a normal, fine, countably complete nonprincipal measure on \powerset_{\omega_1}(X) (in the sense of Solovay). We say \omega_1 is X-strongly compact if there is a fine, countably complete nonprincipal measure on \powerset_{\omega_1}(X). A long-standing open question in set theory asks whether (under ZFC) "supercompactness" can be equiconsistent with "strong compactness. We ask the same question under ZF+DC. More specifically, we discuss whether the theories "\omega_1 is X-supercompact" and "\omega_1 is X-strongly compact" can be equiconsistent for various X. The global question is still open but we show that the local version of the question is false for various X. We also discuss various results in constructing and analyzing canonical models of AD^+ + \omega_1 is X-supercompact.

 

Definability aspects of the Denjoy integral

Speaker: 

Sean Walsh

Institution: 

UCI

Time: 

Monday, February 27, 2017 - 4:00pm

Location: 

RH 440R

The Denjoy integral is an integral that extends the Lebesgue integral and can integrate any derivative. In this talk, it is shown that the graph of the indefinite Denjoy integral $f\mapsto \int_a^x f$ is a coanalytic non-Borel relation on the product space $M[a,b]\times C[a,b]$, where $M[a,b]$ is the Polish space of real-valued measurable functions on $[a,b]$ and where $C[a,b]$ is the Polish space of real-valued continuous functions on $[a,b]$. Using the same methods, it is also shown that the class of indefinite Denjoy integrals, called $ACG_{\ast}[a,b]$, is a coanalytic but not Borel subclass of the space $C[a,b]$, thus answering a question posed by Dougherty and Kechris. Some basic model theory of the associated spaces of integrable functions is also studied. Here the main result is that, when viewed as an $\mathbb{R}[X]$-module with the indeterminate $X$ being interpreted as the indefinite integral, the space of continuous functions on the interval $[a,b]$ is elementarily equivalent to the Lebesgue-integrable and Denjoy-integrable functions on this interval, and each is stable but not superstable, and that they all have a common decidable theory when viewed as $\mathbb{Q}[X]$-modules.

Definability aspects of the Denjoy integral

Speaker: 

Sean Walsh

Institution: 

UCI

Time: 

Monday, February 13, 2017 - 4:00pm

Location: 

RH 440R

The Denjoy integral is an integral that extends the Lebesgue integral and can integrate any derivative. In this talk, it is shown that the graph of the indefinite Denjoy integral $f\mapsto \int_a^x f$ is a coanalytic non-Borel relation on the product space $M[a,b]\times C[a,b]$, where $M[a,b]$ is the Polish space of real-valued measurable functions on $[a,b]$ and where $C[a,b]$ is the Polish space of real-valued continuous functions on $[a,b]$. Using the same methods, it is also shown that the class of indefinite Denjoy integrals, called $ACG_{\ast}[a,b]$, is a coanalytic but not Borel subclass of the space $C[a,b]$, thus answering a question posed by Dougherty and Kechris. Some basic model theory of the associated spaces of integrable functions is also studied. Here the main result is that, when viewed as an $\mathbb{R}[X]$-module with the indeterminate $X$ being interpreted as the indefinite integral, the space of continuous functions on the interval $[a,b]$ is elementarily equivalent to the Lebesgue-integrable and Denjoy-integrable functions on this interval, and each is stable but not superstable, and that they all have a common decidable theory when viewed as $\mathbb{Q}[X]$-modules.

Definability aspects of the Denjoy integral

Speaker: 

Sean Walsh

Institution: 

UCI

Time: 

Monday, February 6, 2017 - 4:00pm

Location: 

RH 440R

The Denjoy integral is an integral that extends the Lebesgue integral and can integrate any derivative. In this talk, it is shown that the graph of the indefinite Denjoy integral $f\mapsto \int_a^x f$ is a coanalytic non-Borel relation on the product space $M[a,b]\times C[a,b]$, where $M[a,b]$ is the Polish space of real-valued measurable functions on $[a,b]$ and where $C[a,b]$ is the Polish space of real-valued continuous functions on $[a,b]$. Using the same methods, it is also shown that the class of indefinite Denjoy integrals, called $ACG_{\ast}[a,b]$, is a coanalytic but not Borel subclass of the space $C[a,b]$, thus answering a question posed by Dougherty and Kechris. Some basic model theory of the associated spaces of integrable functions is also studied. Here the main result is that, when viewed as an $\mathbb{R}[X]$-module with the indeterminate $X$ being interpreted as the indefinite integral, the space of continuous functions on the interval $[a,b]$ is elementarily equivalent to the Lebesgue-integrable and Denjoy-integrable functions on this interval, and each is stable but not superstable, and that they all have a common decidable theory when viewed as $\mathbb{Q}[X]$-modules.

Stability and sparsity in sets of natural numbers

Speaker: 

Gabriel Conant

Institution: 

Notre Dame

Time: 

Monday, March 13, 2017 - 4:00pm

Location: 

RH 440R

The additive group of integers is a well-studied example of a stable group, whose definable sets can be easily and explicitly described. However, until recently, very little has been known about stable expansions of this group. In this talk, we examine the relationship between model-theoretic stability of expansions of the form (Z,+,0,A), where A is a subset of the natural numbers, and the number theoretic behavior of A with respect to sumsets, asymptotic density, and arithmetic progressions.

The absolute Vaught conjecture and randomizations

Speaker: 

Isaac Goldbring

Institution: 

UCI

Time: 

Monday, March 6, 2017 - 4:00pm

Location: 

RH 440R

Morley introduced the notion of a scattered sentence of $L_{\omega_1,\omega}$.  Roughly speaking, $\varphi$ is scattered if it does not have a perfect set of countable models.  He then showed that scattered sentences have at most $\aleph_1$ many countable models (up to isomorphism) whilst non-scattered sentences have continuum many nonisomorphic countable models.  The absolute Vaught conjecture states that scattered sentences have only countably many countable models up to isomorphism.  Unlike the original Vaught conjecture (which holds trivially under CH), the absolute Vaught conjecture does not depend on the model of set theory in question and is in fact equivalent to the original Vaught conjecture under the negation of CH.

Keisler connected the notion of scattered sentences with randomizations of structures.  Randomizations are models of a certain continuous theory, called the pure randomization theory, and in such a randomization, one can define the probability that $\varphi$ holds.  A randomization of $\varphi$ is a randomization in which $\varphi$ holds with probability one.  An example of a randomization of $\varphi$ is a basic randomization of $\varphi$, which consists of a collection of "measurable" random variables taking values in a countable family of models of $\varphi$.  $\varphi$ is is said to have few separable randomizations if every randomization of $\varphi$ is a basic randomization of $\varphi$.   

Keisler showed that if $\varphi$ has few separable randomizations, then $\varphi$ is scattered.  Moreover, he showed that, assuming Martin's axiom for $\aleph_1$, the converse holds.  Andrews, Goldbring, Hachtman, Keisler, and Marker were able to remove the use of Martin's axiom by an absoluteness argument.  Thus, it is a theorem of ZFC that $\varphi$ is scattered if and only if $\varphi$ has few separable randomizations.

In this talk, I will try to define most of the above results in more detail and sketch the ideas behind the proofs of the theorems alluded to above.

Linear dynamics and recurrence properties defined via essential idempotents of $\beta\N$

Speaker: 

Yunied Puig de Dios

Institution: 

Ben Gurion

Time: 

Monday, January 9, 2017 - 4:00pm

Location: 

RH 440R

Consider $\mathscr{F}$ a non-empty set of subsets of $\N$.  An operator $T$ on $X$ satisfies property $\p_{\mathscr{F}}$ if for any $U$ non-empty open set in $X$, there exists $x\in X$ such that $\{n\geq 0: T^nx\in U\}\in \mathscr{F}$. Let $\overline{\mathcal{BD}}$ the collection of sets in $\N$ with positive upper Banach density. Our main result is a characterization of sequence of operators satisfying property $\p_{\overline{\mathcal{BD}}}$, for which we have used a result of Bergelson and McCutcheon in the vein of Szemer\'{e}di's theorem. It turns out that operators having  property $\p_{\overline{\mathcal{BD}}}$ satisfy a kind of recurrence described in terms of essential idempotents of $\beta \N$. We will also discuss the case of weighted backward shifts. Finally, we obtain a characterization of reiteratively hypercyclic operators.

Applications of Descriptive Set Theory in Ergodic Theory III

Speaker: 

Matthew Foreman

Institution: 

UCI

Time: 

Monday, November 21, 2016 - 4:00pm to 5:30pm

Host: 

Location: 

RH 440R

Recent years have seen an increasing number of applications of descriptive set theory in ergodic theory and dynamical systems. We present some set theoretic background and survey some of the applications.

Slides for this series of talks can be found here:

https://www.dropbox.com/sh/om8efuv6ez10ysb/AADOA4SPbdjXKoDajEftFb2pa?dl=0

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