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Logic Seminar

Contact Person:   Professor Sue Gao

Logic Seminar Archives (incomplete)

John Krueger

October 23, 2009

WRP and the Velickovic Game

We will prove some consequences of the Weak Reflection Principle using the Velickovic Game, applying the determinacy of open games. We also discuss the relationship between the Weak Reflection Principle and the Reflection Principle.

Vincent Kieftenbeld   (UNT)

September 25, 2009

Maps which preserve complete metrizability (Part 2)

Separable, completely metrizable spaces are ubiquitous in mathematics. These spaces are the natural setting for descriptive set theory. It is therefore of interest to find conditions which imply that a separable metrizable space is completely metrizable.

 After a quick review of Hurewicz's criterion and Sierpinski theorem, I will prove Vainstein's result.  Then I will define the class of resolvable maps, and prove that these preserve complete metrizability. This answers a recent question of Ostrovsky and is in some sense the best possible theorem along these lines.

Vincent Kieftenbeld (UNT)

September 18, 2009

Title: Maps which preserve complete metrizability

Separable, completely metrizable spaces are ubiquitous in mathematics. These spaces are the natural setting for descriptive set theory. It is therefore of interest to find conditions which imply that a separable metrizable space is completely metrizable.

 In this logic seminar I will look at maps which preserve complete metrizability. First I will discuss Hurewicz's criterion and use it to prove classical theorems by Sierpinski and Vainstein. Then I will define the class of resolvable maps, and prove that these preserve complete metrizability. This answers a recent question of Ostrovsky and is in some sense the best possible theorem along these lines.

September 10, 2008

Chuang Shao  (UNT)

Universality in Ultrametric spaces

 We consider the existence of universal spaces for certain classes of ultrametric spaces.

We show that, while there is no universal space for all separable ultrametric sapces, the universal spaces do exist for some subcollections.

March 28, 2008

J.B. Nation (University of Hawaii)

MATHEMATICS ON OTHER PLANETS!

How do Aliens do Mathematics? This talk gives a gentle introduction to universal algebra through a tour of mathematics throughout the solar system. All the major planets and some Kuiperbelt objects are included for the same low fare. We leave it to the audience to distinguish real facts from the facts the speaker makes up!

Jan 25,  2008

Mingzhi Xuan

On Steinhaus set problem of 4 points

We shall review the Steinhaus set problem of finite points and prove that there is no Steinhaus set for 4 points on R^2.

2006 - 2007

October 19, 2007

Brandon Seward (UNT)

A coloring property for countable groups (continued) 

October 5, 2007

Brandon Seward (UNT)

A coloring property for countable groups

May 4, 2007  (continued)

Su Gao (UNT)

Coloring properties of countable groups (continued)

This is the concluding talk of the series. We prove that all countably infinite solvable groups have the (2^{\aleph_0},2)-coloring property.

April 27, 2007 (continuation)

Su Gao (UNT)

We say that a countable group G has the coloring property if there is a {0,1}-coloring c on G such that for all s in G there is a finite subset T of G so that for all g in G there is t in T with c(gt) different from c(gst). This seminar is a workshop on the following question: does every countable group have the coloring property? I will talk about the motivation for this concept, what we know about the question so far, and what needs to be done to answer the question completely.

April 20, 2007 (continuation)

Su Gao (UNT)

April 13, 2007

Su Gao (UNT)

Coloring properties of countable groups

We say that a countable group G has the coloring property if there is a {0,1}-coloring c on G such that for all s in G there is a finite subset T of G so that for all g in G there is t in T with c(gt) different from c(gst). This seminar is a workshop on the following question: does every countable group have the coloring property? I will talk about the motivation for this concept, what we know about the question so far, and what needs to be done to answer the question completely.

March, 2, 2007

Bunyamin Sari

The poset of spreading models as a Borel order (continued)

This talk is concerned with the order structure of the set of spreading models of a Banach space (generated by weakly null sequences), where the order is the usual domination of bases. In particular, I will present a recent solution of a problem (posed in our joint work with Dilworth and Odell) by P. Dodos. It turned out that one needs to look at this structure as a Borel order and use Descriptive Set Theory tools. This is one excellent example of showing the importance of communication between different areas of mathematics.

February 20, 2006

Bunyamin Sari (UNT)

The poset of spreading models as a Borel order

This talk is concerned with the order structure of the set of spreading models of a Banach space (generated by weakly null sequences), where the order is the usual domination of bases. In particular, I will present a recent solution of a problem (posed in our joint work with Dilworth and Odell) by P. Dodos. It turned out that one needs to look at this structure as a Borel order and use Descriptive Set Theory tools. This is one excellent example of showing the importance of communication between different areas of mathematics.

January 26, 2007

Su Gao (UNT)

A compactness theorem for complete sections (continued)

January 26, 2007

Su Gao (UNT)

A compactness theorem for complete sections

January 19, 2007

A complete section for an equivalence relation is a set which meets every equivalence class. Descreasing sequences of Borel complete sections with an empty intersection play an important role in hyperfiniteness proofs. In this talk (or series of talks) I will consider the shift action of Z on 2^Z and prove a "compactness" theorem on its free part. The theorem states that a decreasing sequence of closed complete sections must have a non-empty intersection.

2005 - 2006

November 10, 2006

Steve Jackson (UNT)

The Borel Boundedness Property and Hyperfiniteness Part 4

This is the conclusion of a series of talks.

October 27, 2006

Steve Jackson (UNT)

The Borel Boundedness Property and Hyperfiniteness Part III.

Abstract: This is a contribution to the theory of countable Borel equivalence relations. We define the Borel boundedness property and show its connections to the hyperfiniteness of the equivalence relations involved. We will present some recent results of Miller and investigate the possibility of generalizations.

October 20, 2006

Steve Jackson (UNT)

The Borel Boundedness Property and Hyperfiniteness Part II

October 13, 2006

Vincent Kieftenbeld (UNT)

Borel structures on ordinals, Part 3

This is a continuation of the previous talks.

September 29, 2006

Vincent Kieftenbeld (UNT)

Borel structures on ordinals, Part 2

Abstract: We consider the order topology on ordinals and the induced Borel structure.  We give a complete classification of all ordinals up to Borel isomorphism. This is joint work with Gao and Jackson. 

Friday, September 22, 2006

Vincent Kieftenbeld (UNT)

Borel structures on ordinals

Abstract: We consider the order topology on ordinals and the induced Borel structure.  We give a complete classification of all ordinals up to Borel isomorphism. This is joint work with Gao and Jackson. 

Friday, April 28

Steve Jackson

Title: A Borel coloring of 2^{Z^2} and related problems

Friday, April 21, 2006

Steve Jackson (UNT)

Some problems related to the continuous embedding into E_0

Abstract:   This week's seminar will be a workshop and we will work on some problems related to the continuous embedding of 2^Z into E_0 presented in the previous talks. One of the problems is whether there is a recursive such embedding.

Friday,  April 7, 2006

Steve Jackson (UNT)

Continuous embeddings into E_0, Part 4  We will present some new techniques which allow us to show that 2^Z continuously embeds into E_0, and more generally E_0 is universal for the continuous actions of Z on the 0-dimensional Polish spaces. In particular, any free self-homeomorphism of a 0-dimensional Polish space is topologically isomprphic to the standard action of E_0 on a subspace of 2^omega.

Friday,  March 24, 2006

Measurable chromatic numbers

Ben Miller (UCLA)

We will discuss measurable chromatic numbers of graphs of Borel functions, with a particular focus on a strong anti-basis theorem for the collection of such graphs with Borel chromatic number 3.

Friday, March 10, 2006

Continuous embeddings into E0, Part 3

Professor Steve Jackson (UNT)

Abstract: We will present some new techniques which allow us to show that 2^Z continuously embeds into E_0, and more generally E_0 is universal for the continuous actions of Z on the 0-dimensional Polish spaces. In particular, any free self-homeomorphism of a 0-dimensional Polish space is topologically isomprphic to the standard action of E_0 on a subspace of 2^omega.

Friday,  March 3, 2006

Continuous embeddings into E_0, Part 2

Professor Steve Jackson (UNT)

Abstract: We will present some new techniques which allow us to show that 2^Z continuously embeds into E_0, and more generally E_0 is universal for the continuous actions of Z on the 0-dimensional Polish spaces. In particular, any free self-homeomorphism of a 0-dimensional Polish space is topologically isomprphic to the standard action of E_0 on a subspace of 2^omega.

February 17, 2006

Continuous embeddings into E_0

Professor Steve Jackson (UNT)

Abstract: We will present some new techniques which allow us to show that 2^Z continuously embeds into E_0, and more generally E_0 is universal for the continuous actions of Z on the 0-dimensional Polish spaces. In particular, any free self-homeomorphism of a 0-dimensional Polish space is topologically isomprphic to the standard action of E_0 on a subspace of 2^omega.

February 10, 2006

New metrics on the free group

Professor Su Gao

Abstract:  I will give the definition of some new metrics on the free group with continuum many generators. These metrics are not two-sided invariant and the resulting topological groups (as a collection) have the surjective universality among all Polish groups.

February 10, 2006

Computing Graev  metrics

Professor Su Gao

Abstract: This week I will present some technical results on the computation of Graev metrics. The purpose of doing this is twofold. On the one hand, we will be able to deduce the theorems mentioned in the previous talk in full details. On the other hand, this will be a preparation for some new definitions of metrics on the free group.

February 3, 2006

A representation for the Graev metric group

Professor Su Gao

Abstract:  I will define the Graev metrics and the Graev metric group as well as give a representation of it as a Polishable subgroup of the permutation group. Such representations are desirable since the complex combinatorics makes the study of this important group very complicated. As an application we prove that the Graev metric group also satisfies Hjorth's conjecture. This is joint work with Longyun Ding

January 27, 2006

Hjorth's conjecture and the Polishable subgroups problem

Professor Su Gao

Abstract:  I will talk about some recent joint work with Longyun Ding. We prove a conjecture of Hjorth of Polish groups with no discrete abelian subgroups. We also prove a theorem on a related problem on Polishable subgroups of Polish groups

2004 - 2005

October 21, 2005                                                                                                             

Title: Ehrenfeucht-Frasse Games on Ordinals      

 Ross Bryant

Abstract:  Ehrenfeucht-Frasse (EF) games are used to analyze the strength of logical equivalence between any two structures. More specifically, two structures satisfy the same sentences of a given quantifier-rank if and only if Player II has awinning strategy in some EF game. We will compute exactly the quantifier-rank equivalence between certain pairs of ordinal structures. The ultimate goal of this investigation is a master formula which exactly computes the quantifier-rank equivalence between any pair ordinals.

October 7, 2005 

Title:  The classification of ordinal topologies

Vincent Kieftenbeld

Nov. 19, 2004

Title: On Generalizations of Lavrentieff's Theorem for Polish Group Actions

Dr. Su Gao
Abstract: I will review the classical theorem of Lavrentieff in topology and the Becker-Kechris generalization of the theorem to the context of actions by locally compact Polish groups. For Polish groups which are not locally compact we will investigate the possibility of further generalizations of the theorem and give a full answer. This is recent joint work with Longyun Ding.