# Category Archives: Seminars

## Simon Cho: A Category Theoretic Perspective on Continuous Logic

Thursday, September 21, 2017, from 4 to 5:30pm
East Hall, room 3096

Speaker: Simon Cho (University of Michigan)

Title: A Category Theoretic Perspective on Continuous Logic

Abstract:

Although classical model theory is largely formulated in terms of the framework of sets, there is a rich theory that casts model theoretic structures in a category theoretic setting, a project which began with Lawvere’s thesis on “functorial semantics of algebraic theories” and has since grown into an important subfield of category theory. This interface between classical model theory and category theory continues to be an active area of research today.

In parallel, Lawvere also showed that structures – such as metric spaces – seemingly unrelated to categories arose naturally as examples of categories with appropriate enrichments V (for example V=R in the case of metric spaces). Now continuous logic/metric model theory is a generalization of classical model theory that, roughly, replaces sets with metric spaces and equality with the metric; a natural question to ask is whether the above perspective on metric spaces combines with the way of interpreting classical logic into category theory to produce a way to interpret continuous logic into enriched category theory. This talk will answer this in the affirmative, under reasonable conditions. The talk will make every effort to be self-contained, and as such will assume little to no prior knowledge of category theory.

## 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

## Yuan Yuan Zheng: Moderately-abstract parametrized Ellentuck theorem

Place: Fields Institute (Room 210)

Date: September 15, 2017 (13:30-15:00)

Speaker: Yuan Yuan Zheng

Title: Moderately-abstract parametrized Ellentuck theorem

Abstract: Mimicking the parametrized Ellentuck theorem in the Ellentuck
space and the parametrized Milliken theorem in the Milliken space, we
present a ‘moderately abstract’ parametrized theorem for ‘moderate’
topological Ramsey spaces. It is a parametrization of the abstract
Ellentuck theorem with infinitely many perfect sets of real numbers,
implying that essentially all infinitely-dimensional Ramsey properties
proven using topological Ramsey space theory can be parametrized by
products of infinitely many perfect sets.

## Miha Habič: The grounded Martin’s axiom

Dear all,

The seminar meets on Wednesday September 13th at 11:00 in the Institute
of Mathematics CAS, Zitna 25, seminar room, 3rd floor, front building.

Program:
Miha Habič — The grounded Martin’s axiom

We will examine the notion of a grounded forcing axiom, which asserts
that the universe is a forcing extension by a forcing notion from a
particular class and that the usual forcing axiom holds for forcings
from that class coming from the ground model of the extension. We shall
focus in particular on the grounded Martin’s axiom, where the universe
is a ccc extension. The principle has some of the combinatorial strength
of MA, but allows for more flexibility (for example, a singular
continuum). Furthermore, it is more robust under mild forcing extensions
than full MA, since it is often preserved after adding a Cohen or a
random real. We will also briefly glance at grounded versions of other
forcing axioms, such as grounded PFA, and outline some open questions in
the area.

Best,
David

## Micheal Pawliuk: The Perfect Expansion Property

Place: Fields Institute (Room 210)

Date: September 8, 2017 (13:30-15:00)

Speaker: Micheal Pawliuk

Title: The Perfect Expansion Property

Abstract: The expansion property for classes of finite structures is a well studied Ramsey property for homogeneous structures. Recently, a quantitative version of this property was introduced to answer questions related to amenability and unique ergodicity of automorphism groups of homogeneous structures. A typical way to check this property involves fine estimates and the probabilistic method.

We introduce an even stronger expansion property that is purely combinatorial, while not being so strong as to be impossible. We will then classify which completely n-partite directed graphs have this property. Remarkably, the property is able to isolate the geometry of completely n-partite directed graphs.

This provides a step in the right direction towards the goal of showing that the semigeneric digraph has a uniquely ergodic automorphism group (which is still open).

## 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.

## Stefan Hoffelner: NS saturated and Δ_1-definable

The seminar meets on Wednesday September 6th at 11:00 in the Institute
of Mathematics CAS, Zitna 25, seminar room, 3rd floor, front building.

Program:
Stefan Hoffelner — NS saturated and Δ_1-definable

Abstract:
Questions which investigate the interplay of the saturation of the
nonstationary ideal on ω_1, NS, and definability properties of the
surrounding universe can yield surprising and deep results. Woodins
theorem that in a model with a measurable cardinal where NS is
saturated, CH must definably fail is the paradigmatic example. It is
another remarkable theorem of H. Woodin that given ω-many Woodin
cardinals there is a model in which NS is saturated and ω-dense, which
in particular implies that NS is (boldface) Δ_1-definable. The latter
statement is of considerable interest in the emerging field of
generalized descriptive set theory, as the club filter is known to
violate the Baire property.
With that being said the following question, asked first by S.D.
Friedman and L. Wu seems relevant: is it possible to construct a model
in which NS is both Δ_1-definable and saturated from less than ω-many
Woodins? In this talk I will outline a proof that this is indeed the
case: given the existence of M_1^#, there is a model of ZFC in which the
nonstationary ideal on ω_1 is saturated and Δ_1-definable with parameter
ω_1. In the course of the proof I will present a new coding technique
which seems to be quite suitable to obtain definability results in the
presence of iterated forcing constructions over inner models for large
cardinals.

## 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.

## Juris Steprans: Groups with sub exponential growth and actions on the integers with unique invariant means in the Cohen model

Place: BA6183, Bahen Centre

Date: August 11, 2017 (13:30-15:00)

Speaker: Juris Steprans

Title: Groups with sub exponential growth and actions on the integers with unique invariant means in the Cohen model

Abstract: It will be shown that the objects mentioned in the title do not exist.