Forthcoming events in this series

Wed, 13 Nov 2019
16:00
C2

Eliza Jabłońska
(Krakow)
Wed, 15 May 2019
16:00
C2

### Category versus measure: Kendall's characterization of sequential regular variation

(London School of Economics)
Wed, 08 May 2019
16:00
C2

Joel Mitchell
(Birmingham)
Wed, 06 Mar 2019
16:00
C2

### "Large continua via ultracoproducts"

Paul Bankston
(Milwaukee and Aberystwyth)
Abstract

It is known that every continuum X is a weakly confluent image of a continuum Y which is hereditarily indecomposable and of covering dimension one.  We use the ultracoproduct construction to gain information about the number of composants of Y.  For example, in ZFC, we can ensure that this number is arbitrarily large.  And if we assume the GCH, we can arrange for Y to have as many composants at it has points.

Thu, 21 Feb 2019

14:00 - 16:00
L3

### Analytic moduli spaces

Kobi Kremnizer
Wed, 20 Feb 2019
16:00
C2

Chris Good
(Birmingham)
Abstract

Abstract:   Let $f$ be a continuous surjection from the compact metric space $X$ to itself.

We say that the dynamical system $(X,f)$ has shadowing if for every $\epsilon>0$ there is a $\delta>0$ such that every $\delta$-pseudo orbit is $\epsilon$-shadowed.  Here a sequence $(x_n)$ is a $\delta$-pseudo orbit provided the distance from $f(x_n)$ to $x_{n+1}$ is less than $\delta$ and $(x_n)$ is $\epsilon$-shadowed if there is a point $z$ such that the distance from $x_n$ to $f^n(z)$ is less than $\epsilon$.

If $f$ is a homeomorphism, $(X,f)$ is said to be expansive if there is some $c>0$, such that if the distance from $f^n(x)$ and $f^n(y)$ is less than $c$ for all $n\in \mathbb Z$, then $x=y$.

In his proof that a homeomorphism that is expansive and has shadowing is stable, Walters shows that in an expansive system with shadowing, a pseudo orbit is shadowed by exactly one point.  It turns out that the converse is also true: if the system has unique shadowing (in the above sense), then it is expansive.

In this talk, which is joint work with Joel Mitchell and Joe Thomas, we explore this notion of unique shadowing.

Wed, 06 Feb 2019
16:00
C2

(La Reunion)
Wed, 16 Jan 2019
16:00
C2

### Ramsey Theory and Infinite Graphs

Natasha Dobrinen
(Denver)
Abstract

Abstract:  It is a central question in the theory of infinite relational structures as to which structures carry analogues of Ramsey’s Theorem.  This question, of interest for several decades, has gained recent momentum as it was brought into focus by Kechris, Pestov, and Todorcevic, when they proved a deep correspondence between Ramsey theory and topological dynamics.

In this talk, we provide background on the Ramsey theory of the Rado graph, solved by Sauer.  A longstanding open question was whether Henson graphs, the k-clique-free analogues of the Rado graph, have similar features.  We present the speaker’s recent work solving the Ramsey theory of the Henson graphs.  The techniques developed open new lines of investigation for other relational structures with forbidden configurations.  As a byproduct of these methods, we may obtain Ramsey properties for Borel colorings on copies of the Rado graph, with respect to a certain topology.

Wed, 16 Jan 2019
16:00
C2

### Ramsey Theory and Infinite Graphs

Natasha Dobrinen
(Denver)
Abstract

Abstract:  It is a central question in the theory of infinite relational structures as to which structures carry analogues of Ramsey’s Theorem.  This question, of interest for several decades, has gained recent momentum as it was brought into focus by Kechris, Pestov, and Todorcevic, when they proved a deep correspondence between Ramsey theory and topological dynamics.

In this talk, we provide background on the Ramsey theory of the Rado graph, solved by Sauer.  A longstanding open question was whether Henson graphs, the k-clique-free analogues of the Rado graph, have similar features.  We present the speaker’s recent work solving the Ramsey theory of the Henson graphs.  The techniques developed open new lines of investigation for other relational structures with forbidden configurations.  As a byproduct of these methods, we may obtain Ramsey properties for Borel colorings on copies of the Rado graph, with respect to a certain topology.

Wed, 13 Jun 2018

16:00 - 17:00
C4

### Metric aspects in topology

Ittay Weiss
(Portsmouth)
Abstract

Every topological space is metrisable once the symmetry axiom is abandoned and the codomain of the metric is allowed to take values in a suitable structure tailored to fit the topology (and every completely regular space is similarly metrisable while retaining symmetry). This result was popularised in 1988 by Kopperman, who used value semigroups as the codomain for the metric, and restated in 1997 by Flagg, using value quantales. In categorical terms, each of these constructions extends to an equivalence of categories between the category Top and a category of all L-valued metric spaces (where L ranges over either value semigroups or value quantales) and the classical \epsilon-\delta notion of continuous mappings. Thus, there are (at least) two metric formalisms for topology, raising the questions: 1) is any of the two actually useful for doing topology? and 2) are the two formalisms equally powerful for the purposes of topology? After reviewing Flagg's machinery I will attempt to answer the former affirmatively and the latter negatively. In more detail, the two approaches are equipotent when it comes to point-to-point topological consideration, but only Flagg's formalism captures 'higher order' topological aspects correctly, however at a price; there is no notion of product of value quantales. En route to establishing Flagg's formalism as convenient, it will be shown that both fine and coarse variants of homology and homotopy arise as left and right Kan extensions of genuinely metrically constructed functors, and a topologically relevant notion of tensor product of value quantales, a surrogate for the non-existent products, will be described.

Wed, 06 Jun 2018

16:00 - 17:00
C4

### Locally Finite Trees and Topological Minor Relation

Jorge Bruno
(Winchester)
Abstract

Nash-Williams showed that the collection of locally finite trees under the topological minor relation results in a BQO. Naturally, two interesting questions arise:

1.      What is the number \lambda of topological types of locally finite trees?

2.       What are the possible sizes of an equivalence class of locally finite trees?

For (1), clearly, \omega_0 \leq \lambda \leq c and Matthiesen refined it to \omega_1 \leq \lambda \leq c. Thus, this question becomes non-trivial in the absence of the Continuum Hypothesis. In this paper we address both questions by showing - entirely within ZFC - that for a large collection of locally finite trees that includes those with countably many rays:

- \lambda = \omega_1, and

- the size of an equivalence class can only be either 1 or c.

Wed, 17 Jan 2018

16:00 - 17:30
C4

### 'Continuity of Betweenness'

Paul Bankston (Milwaukee and Aberystwyth)
Wed, 01 Nov 2017
16:00
C4

### Prof. Chris Good - title tbc

Prof. Chris Good
((Oxford University))
Thu, 13 Jul 2017
13:30
C1

### The universal triangle-free graph has finite big Ramsey degrees

Natasha Dobrinen
(Denver)
Abstract

A main part of the proof uses forcing to establish a Ramsey theorem on a new type of tree, though the result holds in ZFC.  The space of such trees almost forms a topological Ramsey space.

Mon, 10 Jul 2017
14:30

### Restrictions on the size of some kinds of locally compact spaces

Peter Nyikos
(South Carolina)
Abstract

The talk will focus on five items:

Theorem 1. It is ZFC-independent whether every locally compact, $\omega_1$-compact space of cardinality $\aleph_1$  is the union of countably many countably compact spaces.

Problem 1. Is it consistent that every locally compact, $\omega_1$-compact space of cardinality $\aleph_2$  is the union of countably many countably compact spaces?

[`$\omega_1$-compact' means that every closed discrete subspace is countable. This is obviously implied by being the union of countably many countably compact spaces, but the converse is not true.]

Problem 2. Is ZFC enough to imply that there is  a normal, locally countable, countably compact space of cardinality greater than $\aleph_1$?

Problem 3. Is it consistent that there exists a normal, locally countable, countably compact space of cardinality greater than $\aleph_2$?

The spaces involved in Problem 2 and Problem 3 are automatically locally compact, because by "space" I mean "Hausdorff space" and so regularity is already enough to give every point a countable countably compact (hence compact) neighborhood.

Theorem 2. The axiom $\square_{\aleph_1}$ implies that there is a normal, locally countable, countably compact space of cardinality $\aleph_2$.

This may be the first application of $\square_{\aleph_1}$ to construct a topological space whose existence in ZFC is unknown.

Wed, 17 May 2017
16:00
C4

Dr Mate Puljiz
(Birmingham)
Wed, 03 May 2017
16:00
C2

### Ultra-arcs

Paul Bankston
(Milwaukee and Aberystywyth)
Wed, 01 Mar 2017
16:00
C2

### Towards the solution of the Alexandroff-Archangelskij problem

Dimitrina Stavrova
(Leicester)
Wed, 22 Feb 2017
16:00
C2

### Connectedness of remainders of non-locally-compact spaces

Rolf Suabedissen
(Oxford University)
Wed, 15 Feb 2017
16:00
C2

### Topological properties of some subsets of ßN

Dona Strauss
(Leeds)
Abstract

Abstract:  Anyone who has worked in $\beta$N will not be surprised to learn that some of the algebraically defined subsets of $\beta N$ are not topologically simple, even though their algebraic definition may be very simple.  I shall show that the following subsets of $\beta N$ are not Borel: $N^*+N^*$; the smallest ideal of $\beta N$; the set of idempotents in $\beta N$; any semiprincipal right ideal in $\beta N$; the set of idempotents in any left ideal in $\beta N$.

Wed, 15 Feb 2017
16:00
C2

### Topological properties of some subsets of ßN

Dona Strauss
(Leeds)
Abstract

Anyone who has worked in $\beta$N will not be surprised to learn that some of the algebraically defined subsets of $\beta N$ are not topologically simple, even though their algebraic definition may be very simple.  I shall show that the following subsets of $\beta N$ are not Borel: $N^*+N^*$; the smallest ideal of $\beta N$; the set of idempotents in $\beta N$; any semiprincipal right ideal in $\beta N$; the set of idempotents in any left ideal in $\beta N$.

Wed, 16 Nov 2016
16:00
C2

(LSE)
Wed, 26 Oct 2016
16:00
C2

### On shadowing in compact dynamical systems

Jonathan Meddaugh
(Birmingham)
Thu, 21 Jul 2016
16:00
C3

Steve Watson
(Toronto)
Thu, 21 Jul 2016
14:30
C3

Sergio Macias
(UNAM)