## Clinton Conley: Measure-theoretic unfriendly colorings II

Mathematical logic seminar – Nov 14 2017
Time:     3:30pm – 4:30 pmRoom:     Wean Hall 8220

Speaker:         Clinton Conley
Department of Mathematical Sciences
CMU

Title:     Measure-theoretic unfriendly colorings II

Abstract:

Given a graph with vertices painted red and blue, we say the coloring is unfriendly if every red vertex has at least as many blue neighbors as red, and vice versa. Every finite graph admits an unfriendly coloring, but (ridiculously) it remains open whether every countable graph does. Rather than tackle that problem, we consider measure-theoretic analogs associated with probability-measure-preserving actions of finitely generated groups. We don’t really answer any questions here, either, but we do obtain such colorings up to weak equivalence of actions. Time permitting, we also discuss recent constructions of unfriendly colorings of acyclic hyperfinite graphs. The talk may include joint work with Kechris, Marks, Tucker-Drob, and Unger.

## Clinton Conley: Measure-theoretic unfriendly colorings

Mathematical logic seminar – Oct 31 2017
Time:     3:30pm – 4:30 pm

Room:     Wean Hall 8220

Speaker:         Clinton Conley
Department of Mathematical Sciences
CMU

Title:     Measure-theoretic unfriendly colorings

Abstract:

Given a graph with vertices painted red and blue, we say the coloring is unfriendly if every red vertex has at least as many blue neighbors as red, and vice versa. Every finite graph admits an unfriendly coloring, but (ridiculously) it remains open whether every countable graph does. Rather than tackle that problem, we consider measure-theoretic analogs associated with probability-measure-preserving actions of finitely generated groups. We don’t really answer any questions here, either, but we do obtain such colorings up to weak equivalence of actions. Time permitting, we also discuss recent constructions of unfriendly colorings of acyclic hyperfinite graphs. The talk may include joint work with Kechris, Marks, Tucker-Drob, and Unger.

## Vahagn Aslanyan: Ax-Schanuel and related problems

Mathematical logic seminar – Oct 24 2017
Time:     3:30pm – 4:30 pm

Room:     Wean Hall 8220

Speaker:         Vahagn Aslanyan
Department of Mathematical Sciences
CMU

Title:     Ax-Schanuel and related problems

Abstract:

Ax proved a functional analogue of Schanuel’s conjecture in 1971. I will show how one can use it to axiomatise the first-order theory of the exponential differential equation in analogy with Zilber’s pseudo-exponentiation. Then I will discuss the possibility of Ax-Schanuel type results for other functions (differential equations), and some related problems. If time permits, I will show how Ax-Schanuel can be applied to prove a weak version of the Conjecture on Intersections with Tori.

## Vahagn Aslanyan: Schanuel’s conjecture, pseudo-exponentiation, and Ax’s theorem

Mathematical logic seminar – Oct 17 2017

Time:     3:30pm – 4:30 pm

Room:     Wean Hall 8220

Speaker:         Vahagn Aslanyan
Department of Mathematical Sciences
CMU

Title:     Schanuel’s conjecture, pseudo-exponentiation, and Ax’s theorem

Abstract:

Schanuel’s conjecture captures the transcendence properties of the complex exponential function, and is considered out of reach. An interesting, novel approach to it was given by Zilber which led to the construction of pseudo-exponentiation. This gave rise to more conjectures related to Schanuel’s conjecture and the complex exponential field C_exp. One of those, known as Zilber-Pink, is purely number theoretic and generalises many known conjectures (and results) in diophantine geometry such as Mordell-Lang and Andree-Oort. I will describe Zilber’s construction and the Zilber-Pink conjecture. If time permits, I will also discuss a functional analogue of Schanuel’s conjecture proven by Ax in 1971.

## Chris Lambie-Hanson: A forcing axiom deciding the generalized Souslin Hypothesis

Mathematical logic seminar – Oct 3 2017
Time:     3:30pm – 4:30 pm

Room:     Wean Hall 8220

Speaker:         Chris Lambie-Hanson
Department of Mathematics
Bar-Ilan University

Title:     A forcing axiom deciding the generalized Souslin Hypothesis

Abstract:

Given a regular, uncountable cardinal $\kappa$, it is often desirable to be able to construct objects of size $\kappa^+$ using approximations of size less than $\kappa$. Historically, such constructions have often been carried out with the help of a $(\kappa,1)$-morass and/or a $\diamondsuit(\kappa)$-sequence.
We present a framework for carrying out such constructions using $\diamondsuit(\kappa)$ and a weakening of Jensen’s $\square_\kappa$. Our framework takes the form of a forcing axiom, $\textrm{SDFA}(\mathcal P_\kappa)$. We show that $\textrm{SDFA}(\mathcal P_κ)$ follows from the conjunction of $\diamondsuit(\kappa)$ and our weakening of $\square_\kappa$ and, if $\kappa$ is the successor of an uncountable cardinal, that $\textrm{SDFA}(\mathcal P_\kappa)$ is in fact equivalent to this conjunction. We also show that, for an infinite cardinal $\lambda$, $\textrm{SDFA}(\mathcal P_{\lambda^+})$ implies the existence of a $\lambda^+$-complete $\lambda^{++}$-Souslin tree. This implies that, if $\lambda$ is an uncountable cardinal, $2^\lambda =\lambda^+$, and Souslin’s Hypothesis holds at $\lambda^{++}$, then $\lambda^{++}$ is a Mahlo cardinal in $L$, improving upon an old result of Shelah and Stanley. This is joint work with Assaf Rinot.

## Garrett Ervin: The Cube Problem for linear orders II

Mathematical logic seminar – Sep 26 2017
Time:     3:30pm – 4:30 pm

Room:     Wean Hall 8220

Speaker:         Garrett Ervin
Department of Mathematical Sciences
CMU

Title:     The Cube Problem for linear orders II

Abstract:

In the 1950s, Sierpiński asked whether there exists a linear order that is isomorphic to its lexicographically ordered Cartesian cube but not to its square. The analogous question has been answered positively for many different classes of structures, including groups, Boolean algebras, topological spaces, graphs, partial orders, and Banach spaces. However, the answer to Sierpinski’s question turns out to be negative: any linear order that is isomorphic to its cube is already isomorphic to its square, and thus to all of its finite powers. I will present an outline of the proof and give some related results.

## Garrett Ervin: The Cube Problem for linear orders

Mathematical logic seminar – Sep 19 2017
Time:     3:30pm – 4:30 pm

Room:     Wean Hall 8220

Speaker:         Garrett Ervin
Department of Mathematical Sciences
CMU

Title:     The Cube Problem for linear orders

Abstract:

In the 1950s, Sierpiński asked whether there exists a linear order that is isomorphic to its lexicographically ordered Cartesian cube but not to its square. The analogous question has been answered positively for many different classes of structures, including groups, Boolean algebras, topological spaces, graphs, partial orders, and Banach spaces. However, the answer to Sierpinski’s question turns out to be negative: any linear order that is isomorphic to its cube is already isomorphic to its square, and thus to all of its finite powers. I will present an outline of the proof and give some related results.

## Marcos Mazari Armida: Introduction to good frames in Abstract Elementary Classes

Hello,

The seminar will continue to meet on Mondays in WeH 8201 at 5PM, the talks usually last 90 minutes.
Marcos Mazari Armida will give at least three talks, introducing Shelah’s good frames which the generalization to Abstract Elementary Classes of forking, he will focus on obtaining exists theorem of models when model theoretic assumptions will be replacing rather article non-ZFC axioms used by Shelah.
Information on this seminar is posted on the departmental web page http://www.math.cmu.edu/math/modeltheoryseminars/modeltheoryseminar.php?SeminarSelect=1548  or see below.
Best,
Rami Grossberg.
——————————————————-

Marcos Mazari Armida

Carnegie Melllon University
Title: Introduction to good frames in Abstract Elementary Classes, Part 1

Abstract:
The central notion of Shelah’s book on Abstract Elementary Classes [Sh:h] is the notion of a good $\lambda$-frame. It is a forking like notion for types over models of size $\lambda$ and the existence of it implies that the class is well-behaved in $\lambda$. In this series of talks we will focus on the question of the existence of extensions to models of size greater than $\lambda$. We will prove that under some reasonable hypothesis it is always possible to extend a frame. One interesting corollary of this is the existence of arbitrary large models, this is done within ZFC. The first couple of lectures will be based on [Sh:h] Chapter II, while our main theorem is the main theorem of [Bon14a].

References:
[Bon14a] Will Boney, Tameness and extending frames, Journal of Mathematical Logic 14, no. 2
[Sh:h] Saharon Shelah, Classication Theory for Abstract Elementary Classes, vol. 1 & 2, Mathematical Logic and Foundations, no. 18 & 20, College Publications, 2009.

Date: Monday , September 18, 2017.
Time: 5:00 pm
Location: Wean Hall 8201

Marcos Mazari Armida

Carnegie Melllon University
Title: Introduction to good frames in Abstract Elementary Classes, Part 2

Abstract:
The central notion of Shelah’s book on Abstract Elementary Classes [Sh:h] is the notion of a good $\lambda$-frame. It is a forking like notion for types over models of size $\lambda$ and the existence of it implies that the class is well-behaved in $\lambda$. In this series of talks we will focus on the question of the existence of extensions to models of size greater than $\lambda$. We will prove that under some reasonable hypothesis it is always possible to extend a frame. One interesting corollary of this is the existence of arbitrary large models, this is done within ZFC. The first couple of lectures will be based on [Sh:h] Chapter II, while our main theorem is the main theorem of [Bon14a].

References:
[Bon14a] Will Boney, Tameness and extending frames, Journal of Mathematical Logic 14, no. 2
[Sh:h] Saharon Shelah, Classication Theory for Abstract Elementary Classes, vol. 1 & 2, Mathematical Logic and Foundations, no. 18 & 20, College Publications, 2009.

Date: Monday , September 25, 2017.
Time: 5:00 pm
Location: Wean Hall 8201

Marcos Mazari Armida

Carnegie Melllon University
Title: Introduction to good frames in Abstract Elementary Classes, Part 3

Abstract:
The central notion of Shelah’s book on Abstract Elementary Classes [Sh:h] is the notion of a good $\lambda$-frame. It is a forking like notion for types over models of size $\lambda$ and the existence of it implies that the class is well-behaved in $\lambda$. In this series of talks we will focus on the question of the existence of extensions to models of size greater than $\lambda$. We will prove that under some reasonable hypothesis it is always possible to extend a frame. One interesting corollary of this is the existence of arbitrary large models, this is done within ZFC. The first couple of lectures will be based on [Sh:h] Chapter II, while our main theorem is the main theorem of [Bon14a].

References:
[Bon14a] Will Boney, Tameness and extending frames, Journal of Mathematical Logic 14, no. 2
[Sh:h] Saharon Shelah, Classication Theory for Abstract Elementary Classes, vol. 1 & 2, Mathematical Logic and Foundations, no. 18 & 20, College Publications, 2009.

Date: Monday , October 2, 2017.
Time: 5:00 pm
Location: Wean Hall 8201

## Andy Zucker: A direct solution to the Generic Point Problem II

Mathematical logic seminar – Sep 12 2017
Time:     3:30pm – 4:30 pm

Room:     Wean Hall 8220

Speaker:         Andy Zucker
Department of Mathematical Sciences
CMU

Title:     A direct solution to the Generic Point Problem II

Abstract:

We provide a new proof of a recent theorem of Ben-Yaacov, Melleray, and Tsankov. If G is a Polish group and X is a minimal, metrizable G-flow with all orbits meager, then the universal minimal flow M(G) is non-metrizable. In particular, we show that given X as above, the universal highly proximal extension of X is non-metrizable.

## Andy Zucker: A direct solution to the Generic Point Problem

Mathematical logic seminar – Sep 5 2017
Time:     3:30pm – 4:30 pm

Room:     Wean Hall 8220

Speaker:         Andy Zucker
Department of Mathematical Sciences
CMU

Title:     A direct solution to the Generic Point Problem

Abstract:

We provide a new proof of a recent theorem of Ben-Yaacov, Melleray, and Tsankov. If G is a Polish group and X is a minimal, metrizable G-flow with all orbits meager, then the universal minimal flow M(G) is non-metrizable. In particular, we show that given X as above, the universal highly proximal extension of X is non-metrizable.