Archives of: UC Irvine Set Theory Seminar

Zach Norwood: Coding along trees and remarkable cardinals

Time: Mon, 02/12/2018 – 4:00pm – 5:30pm
Location: RH 440R

Speaker: Zach Norwood (UCLA)

Title: Coding along trees and remarkable cardinals

Abstract. A major project in set theory aims to explore the connection between large cardinals and so-called generic absoluteness principles, which assert that forcing notions from a certain class cannot change the truth value of (projective, for instance) statements about the real numbers. For example, in the 80s Kunen showed that absoluteness to ccc forcing extensions is equiconsistent with a weakly compact cardinal. More recently, Schindler showed that absoluteness to proper forcing extensions is equiconsistent with a remarkable cardinal. (Remarkable cardinals will be defined in the talk.) Schindler’s proof does not resemble Kunen’s, however, using almost-disjoint coding instead of Kunen’s innovative method of coding along branchless trees. We show how to reconcile these two proofs, giving a new proof of Schindler’s theorem that generalizes Kunen’s methods and suggests further investigation of non-thin trees.

Scott Cramer: Algebraic properties of elementary embeddings

Time: Mon, 12/04/2017 – 4:00pm – 5:30pm
Location: RH 440R

Speaker: Scott Cramer (California State University San Bernardino)

Title: Algebraic properties of elementary embeddings

Abstract. We will investigate algebraic structures created by rank-into-rank elementary embeddings. Our starting point will be R. Laver’s theorem that any rank-into-rank embedding generates a free left-distributive algebra on one generator. We will consider extensions of this and related results. Our results will lead to some surprisingly coherent conjectures on the algebraic structure of rank-into-rank embeddings in general.

Toby Meadows: A Step Back from Forcing

Time: Mon, 11/27/2017 – 4:00pm – 5:30pm
Location: RH 440R

Speaker: Toby Meadows (University of Queensland)

Title: A Step Back from Forcing

Abstract. In this talk, I’ll sketch a way of unifying a wide variety of set theoretic approaches for generating new models from old models. The underlying methodology will draw from techniques in Sheaf Theory and the theory of Boolean Ultrapowers.

Matthew Foreman: Applications of Descriptive Set Theory in Ergodic Theory

Attached are slides of a series of talks given by Matt Foreman at the UCI seminar.

1st Irvine Conference on Descriptive Inner Model Theory and HOD Mice, July 18-29 2016

1st Irvine Conference on Descriptive Inner Model Theory and HOD Mice

July 18 — 29, 2016
Department of Mathematics, UC Irvine

Supported by: NSF Grants DMS-x, DMS- 1044150, DMS-y, and UCI CORCL

Organizers: Grigor Sargsyan (Rutgers), Nam Trang (Irvine), Martin Zeman
(Irvine)

This workshop is a sequel to a series of conferences and workshops on
descriptive inner model theory including 1st Conference on the core model
induction and hod mice that was held in Münster (FRG), July 19 — August
06, 2010, the 2nd Conference on the core model induction and hod mice that
was held in Münster (FRG), August 08 — 19, 2011, the AIM Workshop on
Descriptive Inner Model Theory held in Palo Alto (CA), June 02 — 06,
2014, and to the Conference on Descriptive Inner Model Theory, held in
Berkeley (CA) June 09 — 13, 2014, and the 3rd Conference on the core
model induction and hod mice, held in Münster (FRG), July 20 — 31, 2015.

The main purpose of the workshop is to disseminate and communicate results
and recent development in descriptive inner model theory and related
subjects. The workshop consists of single talks by experts in the field on
their recent work as well as lectures aimed at advanced graduate students
interested in inner model theory and related fields.

Following past workshops, the first week of the workshop meets M–F; each
day consists of 4 lectures (each is 75 minutes long), 2 in the morning and
2 in the afternoon. Between the lectures, we will leave plenty of time for
discussions, lunch, and informal seminars. The second week will be more
informal; as in the past, the topics and speakers for the second week will
be decided during the first week of the meeting.

All lectures will take place in Natural Scienes II building, room 1201.
map

The organizers gratefully acknowledge the financial support from the
National Science Foundation (NSF).

Sean Cox: Guessing models

Time: Mon, 01/04/2016 – 4:00pm5:30pm
Location: RH 440R

Speaker: Sean Cox
Institution: Virginia Commonwealth University

Abstract: Many consequences of the Proper Forcing Axiom (PFA) factor through the stationarity of the class of guessing models. Such consequences include the Tree Property at $\omega_2$, absence of (weak) Kurepa Trees on $\omega_1$, and failure of square principles.  On the other hand, stationarity of guessing models does not decide the value of the continuum, even when one requires that the guessing models are also indestructible in some sense.  I will give an introduction to the topic and discuss some recent results due to John Krueger and me.

Monroe Eskew: Global Chang conjectures and generic supercompactness

Time: Mon, 03/09/2015 – 4:00pm – 5:30pm
Location:  RH 440R

Speaker: Monroe Eskew (Tsukuba University, Japan)

Title: Global Chang conjectures and generic supercompactness

Abstract: Starting from a 2-huge cardinal, we construct a model where for all pairs of regular cardinals kappa<lambda, (lambda^+,lambda) –> (kappa^+,kappa) and there is a lambda^+ saturated ideal on P_{kappa^+}(lambda). Then using a modified Radin forcing we get similar global principles involving singular cardinals but with only finite jumps.

Dana Bartosova: Approximate Ramsey properties and topological dynamics

Time: Mon, 08/Dec/2014 – 4:00pm – 5:30pm
Speaker: Dana Bartosova (University of Sao Paulo)
Location: RH 440R

Title: Approximate Ramsey properties and topological dynamics

Abstract: The interplay between structural Ramsey theory and topological dynamics of automorphism groups has been extensively studied since their connection was established in a paper by Kechris-Pestov-Todorcevic, while earlier works of Pestov, and Glasned and Weiss exhibited the phenomena in special cases. This line of research was extended to metric structures and approximate Ramsey property by Melleray and Tsankov. We establish the approximate Ramsey property for the class of finite-dimensional normed vector spaces and deduce that the group of linear isometries of the universal approximately homogeneous Banach space, the Gurarij space, is extremely amenable, that is, every continuous action on a compact Hausdorff space has a fixed point. Dualizing our ideas, we show that the class of finite-dimensional simplexes with a distinguished extreme point and affine surjections satisfies the approximate Ramsey property. As a consequence, we find that the universal minimal flow of the group of affine homeomorphisms of the Poulsen simplex is its natural action on the Poulsen simplex. This is a joint work (in progress) with Aleksandra Kwiatkowska (UCLA), Jordi Lopez Abad ICMAT Madrid and USP) and Brice Mbombo (USP).

Paul Larson: Automorphisms of $P(\omega_1)/Fin$

Speaker: Paul Larson
Institution: Miami University, Oxford, Ohio
Time: Mon, 10/13/2014 – 4:00pm – 5:30pm
Location: RH 440R

It appears to be an open question whether for every regular uncountable regular $\lambda$, every automorphism of $P(\lambda)/fin$ is trivial on a co-countable set. We will show that a small fragment of Martin’s Axiom implies that if $\lambda$ is at most the continuum then every automorphism of $P(\lambda)/fin$ which is trivial on sets of cardinality less than $\lambda$ is trivial.

Andres Forero: Self-genericity axioms

Time: Mon, 02/10/2014 – 4:00pm – 5:30pm
Location: RH 440R
Speaker: Andres Forero
Title: Self-genericity axioms