## 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

## Monroe Eskew: Clubs, diamonds, and saturated ideals

Speaker: Monroe Eskew
Time: Mon, 01/27/2014 – 4:00pm – 5:30pm
Location: RH 440R

Title: Clubs, diamonds, and saturated ideals

Abstract: We will present two different ways to generically add a club subset of a successor cardinal, under some GCH. The first one is designed to destroy a given stationary set, and we show that it also forces diamond. The second adds a club with “small” conditions and destroys saturated ideals. We will discuss the open problem of whether this can be done without any cardinal arithmetic assumptions.

## Peter Vojtas: A logician journey from set theory to preference learning benchmarks

Speaker: Peter Vojtas
Institution: Charles University, Prague
Time: Mon, 12/16/2013 – 11:00am – 12:00pm

Title: A logician journey from set theory to preference learning benchmarks

Abstract: The talk will consist of two loosely connected parts: set-theoretic and computer science. We give an overview (no technical details) of our results on Galois-Tukey connections as a general framework for problem reduction. Boolean structure of absolutely divergent series gives rise to several Boolean-like asymptotic structures. Second part deals with applications of many valued logic to preference modeling, querying top-k answers and learning each individual user preferences from behaviour data (especially we mention lack of real world benchmarks)

## Sean Walsh: The Constructible Universe, the Naive Conception, and Intensional Logic

Time: Mon, 11/25/2013 – 4:00pm – 5:30pm
Location: RH 440R

Speaker: Sean Walsh
Institution: Logic and Philosophy of Science, UC Irvine
Abstract: This talk looks at the relationship between three foundational systems: Goedel’s Constructible Universe of Sets, the naive conception of set found in consistent fragments of Frege’s Grundgesetze, and the intensional logic of Church’s Logic of Sense and Denotation. One basic result shows how to use the constructible sets to build models of fragments of Frege’s Grundgesetze from which one can recover these very constructible sets using Frege’s definition of membership. This result also allows us to solve the related consistency problem and joint consistency problems for abstraction principles with limited amounts of comprehension. Another basic aim of this paper is to show how to “factor” this result via a consistent fragment of Church’s Logic of Sense and Denotation: so one may use the constructible sets to build models of Church’s Logic of Sense and Denotation, from which one may then define models of the consistent fragments of Frege’s Grundgesetze.