Applications of Descriptive Set Theory in Ergodic Theory I

Speaker: 

Matthew Foreman

Institution: 

UCI

Time: 

Monday, October 31, 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

Embedding problems in C*-algebras I

Speaker: 

Isaac Goldbring

Institution: 

UCI

Time: 

Monday, October 10, 2016 - 4:00pm to 5:30pm

Host: 

Location: 

RH 440R

Some prominent conjectures in the theory of C*-algebras ask whether or not every C*-algebra of a particular form embeds into an ultrapower of a particular C*-algebra.  For example, the Kirchberg Embedding Problem asks whether every C*-algebra embeds into an ultrapower of the Cuntz algebra O_2.  In this series of lectures, we show how techniques from model theory, most notably model-theoretic forcing, can be used to give nontrivial reformulations of these conjectures.  We will start from scratch, assuming no knowledge of C*-algebras nor model theory.

 

Strong reductions between combinatorial problems

Speaker: 

Damir Dzhafarov

Institution: 

University of Connecticut

Time: 

Monday, October 3, 2016 - 4:00pm to 5:30pm

Host: 

Location: 

RH 440R

I will discuss recent investigations of various reducibility notions between Pi^1_2 principles of second-order arithmetic, the most familiar of which is implication over the subsystem RCA_0. In many cases, such an implication is actually due to a considerably stronger reduction holding, such as a uniform (a.k.a. Weihrauch) reduction. (Here, we say a principle P is uniformly reducible to a principle Q if there are fixed reduction procedures Phi and Gamma such that for every instance A of P, Phi(A) is an instance of Q, and for every solution S to Phi(A), Gamma(A + S) is a solution to A.) As an example, nearly all the implications between principles lying below Ramsey's theorem for pairs are uniform reductions. In general, the study of when such stronger implications hold and when they do not gives a finer way of calibrating the relative strength of mathematical propositions, and has led to the development of a number of new forcing techniques for constructing models of second-order arithmetic with prescribed combinatorial properties. In addition, this analysis sheds light on several open questions from reverse mathematics, including that of whether the stable form of Ramsey's theorem for pairs (SRT^2_2) implies the cohesive principle (COH) in \omega (standard) models of RCA_0.

 

Weak Squares and Very Good Scales

Speaker: 

Maxwell Levine

Institution: 

University of Illonois at Chicago

Time: 

Monday, September 26, 2016 - 4:00pm to 5:30pm

Host: 

Location: 

RH 440R

Abstract: The combinatorial properties of large cardinals tend to clash with those satisfied by G\"odel's constructible universe, especially the square property (denoted $\square_\kappa$) isolated by Jensen in the seventies. Strong cardinal axioms refute the existence of square, but it is possible with some fine-tuning to produce models that exhibit some large cardinal properties together with weakenings of square. In this talk we will exhibit some results along these lines and will outline the techniques used to produce them.

The distance between HOD and V

Speaker: 

Omer Ben Neria

Institution: 

UCLA

Time: 

Monday, May 23, 2016 - 4:00pm to 5:30pm

Host: 

Location: 

RH 440R

The pursuit of better understanding the universe of set theory V motivated an extensive study of definable inner models M whose goal is to serve as good approximations to V. A common property of these inner models is that they are contained in HOD, the universe of hereditarily ordinal definable sets. Motivated by the question of how ``close" HOD is to V, we consider various related forcing methods and survey known and new results. This is a joint work with Spencer Unger.

Every linear order isomorphic to its cube is isomorphic to its square VIII

Speaker: 

Garrett Ervin

Institution: 

UCI

Time: 

Monday, May 9, 2016 - 4:00pm to 5:30pm

Host: 

Location: 

RH 440R

We complete the proof of the main theorem by showing that if X^3 is isomorphic to X, then X^{\omega} has a parity-reversing automorphism. By our previous results this implies X^2 is isomorphic to X as well. The proof generalizes to show that for any n > 1, if X^n is isomorphic to X, then X^2 is isomorphic to X. Time permitting we will discuss related results, including the existence of an A and X such that A^2X is isomorphic to X, while AX is not.

Every linear order isomorphic to its cube is isomorphic to its square VII

Speaker: 

Garrett Ervin

Institution: 

UCI

Time: 

Monday, April 25, 2016 - 4:00pm to 5:30pm

Host: 

Location: 

RH 440R

We complete the proof of the main theorem by showing that if X^3 is isomorphic to X, then X^{\omega} has a parity-reversing automorphism. By our previous results this implies X^2 is isomorphic to X as well. The proof generalizes to show that for any n > 1, if X^n is isomorphic to X, then X^2 is isomorphic to X. Time permitting we will discuss related results, including the existence of an A and X such that A^2X is isomorphic to X, while AX is not.

 

Ordinal definable subsets of singular cardinals

Speaker: 

Dima Sinapova

Institution: 

University of Illinois Chicago

Time: 

Monday, May 2, 2016 - 4:00pm to 5:30pm

Host: 

Location: 

RH 440R

A remarkable theorem of Shelah states that if $\kappa$ is a singular strong limit cardinal of uncountable cofinality, then there is a subset $x$ of $\kappa$, such that $HOD_x$ contains the powerset of $\kappa$. We show that in general this is not  the case for countable cofinality. Using a version of diagonal supercompact extender Prikry forcing, we construct a generic extension in which there is a singular cardinal $\kappa$ with countable cofinality, such that $\kappa^+$ is supercompact in $HOD_x$ for all $x\subset\kappa$. This result was obtained during a SQuaRE meeting at AIM and is joint with Cummings, Friedman, Magidor, and Rinot.

Every linear order isomorphic to its cube is isomorphic to its square VI

Speaker: 

Garrett Ervin

Institution: 

UCI

Time: 

Monday, April 18, 2016 - 4:00pm to 5:30pm

Host: 

Location: 

RH 440R

Our eventual goal is to see that if X is any linear order that isomorphic to its cube, then X^{\omega} has a parity-reversing automorphism. Then by the results of last week, X will also be isomorphic to its square. This week, I will describe a method for building partial parity-reversing automorphisms on any A^{\omega}, and give structural conditions on A under which these partial automorphisms can be stitched together to get a full p.r.a. We will see in particular that if A is countable, then A^{\omega} has a p.r.a.

 

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