# Category Archives: Seminars

## 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.

## Alejandro Poveda: Woodin’s HOD-Dichotomy

Date: Wednesday 27 September 2017

Time: 16:00

Place: Room S-3
Department of Mathematics & Computer Science
University of Barcelona
Gran Via de les Corts Catalanes 585
08007 Barcelona

Speaker: Alejandro Poveda (Universitat de Barcelona)

Title: Woodin’s HOD-Dichotomy

Abstract: We shall give a complete proof of W. H.
Woodin’s remarkable result that if there exists an
extendible cardinal, then either the set-theoretic universe
V is very “close” to HOD (the class of Hereditarily Ordinal
Definable sets), or it is very “far” from it.

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

Thursday, September 28, 2017, from 4 to 5:30pm
East Hall, room 3096

Speaker: Simon Cho (University of Michigan)

Title: A Category Theoretic Perspective on Continuous Logic, II

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.

## Peter Vojtáš: Galois Tukey connections and reductions of (finite) combinatorial search problems

The seminar meets on Wednesday September 27th at 11:00 in the Institute
of Mathematics CAS, Zitna 25, seminar room, 3rd floor, front building.

Program:
Peter Vojtáš — Galois Tukey connections and reductions of (finite)
combinatorial search problems

We start from paper of A. Blass Query-Answer category … and present
some results and problems connected to K. Weihrauch reduction from
constru

## Yinhe Peng: PFA implies a class of hereditarily Lindelof spaces are D spaces

Place: Fields Institute (Room 210)

Date: September 22, 2017 (13:30-15:00)

Speaker: Yinhe Peng

Title: PFA implies a class of hereditarily Lindelof spaces are D spaces

Abstract: For a space X, OSM_X asserts that for any open neighbourhood assignment (or open set mapping) N, there is a partition of X into countably many pieces such that for each x, y in the same piece, either x is in N(y) or y is in N(x).

We introduce a property that will force OSM under PFA. We then use OSM to imply D, assuming additional properties (e.g., sub-Sorgenfrey).

## 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