## Appalachian Set Theory workshop: Benjamin Miller, January 21, 2017

Appalachian set theory

# Benjamin Miller : “Applications of the open graph dichotomy”

## Description

The open graph dichotomy is a generalization of the perfect set theorem, ensuring that every open graph on an analytic set has either a countable coloring or a perfect clique. As the proof of this result is essentially the same as that of the perfect set theorem, it can be viewed as one of the very simplest descriptive set-theoretic dichotomy theorems. Nevertheless, there is an infinite-dimensional analog of the open graph dichotomy (whose proof is essentially the same) that has recently proven particularly useful in studying Borel functions, graphs, and sets of low complexity.

We will begin by stating and proving the infinite-dimensional analog of the open graph dichotomy. We will then describe how it can be used to give particularly simple proofs of several well-known facts, such as the Hurewicz dichotomies, the Jayne-Rogers theorem, and Lecomte’s characterization of the existence of countable Borel colorings of low complexity. Finally, we will turn our attention to the new result that there is a twenty-four element basis, under closed continuous embeddability, for the class of Borel functions that are not Baire class one.

## Marcos Mazari Armida: Categoricity of an AEC in three successive cardinals, Part 1

Date: Monday, October 31, 2016

Time: 5:00 – 6:30 PM
Location: Wean 8201

Model Theory Seminar, CMU

Marcos Mazari Armida

Title:   Categoricity of an AEC in three successive cardinals, Part 1
Abstract:  In this series of talks we will be working on Abstract Elementary Classes (AECs), a semantic generalization of first order Model Theory. One of the first things one looses when stepping out of the first order setting is the compactness theorem, which in particular assures us that if there is an infinite model then there is a model in each cardinality. What we will do in this series of talks is prove a theorem in this direction for AECs. More specically (under some cardinal arithmetic hypothesis) we will show that if an AEC K is categorical in $\lambda$, $\lambda^+$ and $\lambda^{++}$ then there is a model in K of size $\lambda^{+++}$In order to do that we will have to develop many key concepts in the study of AECs like the concept of Galois Type, Reduced Type and Minimial Type. This talks will follow Saharon Shelah paper  “Categoricity of an Abstract Elementary Class in two successive cardinals” [Sh576].

## James Cummings: Cardinal invariants of the continuum

Mathematical logic seminar – October 18 2016
Time:     3:30pm – 4:30 pm

Room:     Wean Hall 8220

Speaker:         James Cummings
Department of Mathematical Sciences
CMU

Title:     Cardinal invariants of the continuum

Abstract:

The cardinal invariants of the continuum are cardinals which measure properties of the continuum more subtle than its cardinality. We will define some of the important ones and discuss their properties.

Note: This seminar will provide some background for Mayanthe Malliaris’ forthcoming Appalachian Set Theory workshop on November 5, see workshop web page at http://www.math.cmu.edu/users/jcumming/Appalachian/malliaris_cmu_2016.html for details.

## Andy Zucker: Topological dynamics and Devlin’s theorem

Mathematical logic seminar – October 4 2016
Time:     3:30pm – 4:30 pm

Room:     Wean Hall 8220

Speaker:         Andy Zucker
Department of Mathematical Sciences
CMU

Title:     Topological dynamics and Devlin’s theorem

Abstract:

If G is a topological group, a G-flow is a compact Hausdorff space X with a continuous right action a: X x G to X. A G-ambit is just a flow X along with a point x_0 in X whose orbit is dense. After a brief introduction to the world of G-flows and G-ambits, we will define a new dynamical object, a completion ambit, and discuss some of their basic properties. Of course, given any category of dynamical systems, it’s natural to ask whether the category has a universal object, and if so, if it is unique. Time permitting, we will connect these questions to a striking theorem of D. Devlin from his 1979 thesis about the partition properties of the countable dense linear order.

## Joe Zielinski: Compact metrizable structures and classification problems II

Mathematical logic seminar – September 27 2016
Time:     3:30pm – 4:30 pm

Room:     Wean Hall 8220

Speaker:         Joe Zielinski
Department of Mathematical Sciences
CMU

Title:     Compact metrizable structures and classification problems II

Abstract:

We consider compact metrizable spaces with equipped with closed relations. Two such structures are considered equivalent when there is a homeomorphism between their domains that respects the relational structure. By representing other classes of objects as compact structures, we establish bounds for classification problems in Borel reducibility. Portions of this talk are based on joint work with C. Rosendal.

## Joe Zielinski: Compact metrizable structures and classification problems

Mathematical logic seminar – September 20 2016
Time:     3:30pm – 4:30 pm

Room:     Wean Hall 8220

Speaker:         Joe Zielinski
Department of Mathematical Sciences
CMU

Title:     Compact metrizable structures and classification problems

Abstract:

We consider compact metrizable spaces with equipped with closed relations. Two such structures are considered equivalent when there is a homeomorphism between their domains that respects the relational structure. By representing other classes of objects as compact structures, we establish bounds for classification problems in Borel reducibility. Portions of this talk are based on joint work with C. Rosendal.

## James Cummings: Dowker filters

Seminar will generally be meeting at 3:30pm in Wean Hall 8220 this semester.

Mathematical logic seminar – September 6 2016
Time:     3:30pm – 4:30 pm  (PLEASE NOTE CHANGED TIME FROM LAST YEAR)

Room:     Wean Hall 8220

Speaker:         James Cummings
Department of Mathematical Sciences
CMU

Title:     Dowker filters

Abstract:

Dowker filters are a class of filters introduced by Dowker in connection with a problem in topology. Their properties are quite mysterious. We will discuss some recent progress (in joint work with Charles Morgan) and mention a number of outstanding open questions.

## Appalachian Set Theory workshop: Maryanthe Malliaris, November 5, 2016

The next meeting of the Appalachian Set Theory workshop series will take place
at CMU and may be of interest to the Pittsburgh logic community.

Maryanthe Malliaris will lead a workshop on “Cofinality spectrum
problems: p, t, and model theory”.

For details please see the workshop web page at http://www.math.cmu.edu/users/jcumming/Appalachian/malliaris_cmu_2016.html

The Appalachian Set Theory workshop series is supported by the National
Science Foundation.

## Daisuke Ikegami: On a class of maximality principles

Mathematical logic seminar – May 13 2016
Time:     12:30 – 13:30Room:     Wean Hall 8220

Speaker: Daisuke Ikegami
Tokyo Denki University
Japan

Title:     On a class of maximality principles

Abstract:

The Maximality Principle (MP) states that for any first-order sentence φ in the language of set theory, if it is forced by a set forcing that φ is true in any further set generic extension, then φ must be true. The basic idea of MP was proposed by Chalons and its basic theory was developed by Hamkins. In this talk, we will discuss several variants of maximality principles and their relations with forcing axioms, bounded forcing axioms, and large cardinals. This is joint work with Nam Trang.

## Andy Zucker: Algebra in the Samuel compactification II

Mathematical logic seminar – April 19 2016
Time:     12:30 – 13:30

Room:     Wean Hall 8220

Speaker:         Andy Zucker
Department of Mathematical Sciences
CMU

Title:     Algebra in the Samuel compactification II

Abstract:

To every topological group G we can associate its Samuel compactification (S(G), 1). This is the largest point-transitive G-flow according to a suitable universal property. Using the universal property, we can endow S(G) with the structure of a compact left-topological semigroup. While the algebraic properties of S(G) are an active area of research for G a countable discrete group, less attention has been paid to other topological groups. In this talk, we will discuss a method of characterizing S(G) when G is an automorphism group of a countable structure. We will then take a closer look at the case G = S∞ and answer several questions about the algebraic structure of S(G). This is joint work with Dana Bartošová.