Birmingham

It is quite easy to see that the sobrification of a
topological space is a dcpo with respect to its specialisation order
and that the topology is contained in the Scott topology wrt this
order. It is also known that many classes of dcpo's are sober when
considered as topological spaces via their Scott topology. In 1982,
Peter Johnstone showed that, however, not every dcpo has this
property in a delightful short note entitled "Scott is not always
sober".

Weng Kin Ho and Dongsheng Zhao observed in the early 2000s that the
Scott topology of the sobrification of a dcpo is typically different
from the Scott topology of the original dcpo, and they wondered
whether there is a way to recover the original dcpo from its
sobrification. They showed that for large classes of dcpos this is
possible but were not able to establish it for all of them. The
question became known as the Ho-Zhao Problem. In a recent
collaboration, Ho, Xiaoyong Xi, and I were able to construct a
counterexample.

In this talk I want to present the positive results that we have about
the Ho-Zhao problem as well as our counterexample. 

Point-free topology can often seem like an algebraic almost-topology, 
> not quite the same but still interesting to those with an interest in 
> it. There is also a tradition of it in computer science, traceable back 
> to Scott's topological model of the untyped lambda-calculus, and 
> developing through Abramsky's 1987 thesis. There the point-free approach 
> can be seen as giving new insights (from a logic of observations), 
> albeit in a context where it is equivalent to point-set topology. It was 
> in that tradition that I wrote my own book "Topology via Logic".
> 
> Absent from my book, however, was a rather deeper connection with logic 
> that was already known from topos theory: if one respects certain 
> logical constraints (of geometric logic), then the maps one constructs 
> are automatically continuous. Given a generic point x of X, if one 
> geometrically constructs a point of Y, then one has constructed a 
> continuous map from X to Y. This is in fact a point-free result, even 
> though it unashamedly uses points.
> 
> This "continuity via logic", continuity as geometricity, takes one 
> rather further than simple continuity of maps. Sheaves and bundles can 
> be understood as continuous set-valued or space-valued maps, and topos 
> theory makes this meaningful - with the proviso that, to make it run 
> cleanly, all spaces have to be point-free. In the resulting fibrewise 
> topology via logic, every geometric construction of spaces (example: 
> point-free hyperspaces, or powerlocales) leads automatically to a 
> fibrewise construction on bundles.
> 
> I shall present an overview of this framework, as well as touching on 
> recent work using Joyal's Arithmetic Universes. This bears on the logic 
> itself, and aims to replace the geometric logic, with its infinitary 
> disjunctions, by a finitary "arithmetic type theory" that still has the 
> intrinsic continuity, and is strong enough to encompass significant 
> amounts of real analysis.

CANCELLED - CANCELLED - CANCELLED

Abstract: Joint work with Syahida Che Dzul-Kifli

Let $f:X\to X$ be a continuous function on a compact metric space forming a discrete dynamical system. There are many definitions that try to capture what it means for the function $f$ to be chaotic. Devaney’s definition, perhaps the most frequently cited, asks for the function $f$ to be topologically transitive, have a dense set of periodic points and is sensitive to initial conditions.  Bank’s et al show that sensitive dependence follows from the other two conditions and Velleman and Berglund show that a transitive interval map has a dense set of periodic points.  Li and Yorke (who coined the term chaos) show that for interval maps, period three implies chaos, i.e. that the existence of a period three point (indeed of any point with period having an odd factor) is chaotic in the sense that it has an uncountable scrambled set.

The existence of a period three point is In this talk we examine the relationship between transitivity and dense periodic points and look for simple conditions that imply chaos in interval maps. Our results are entirely elementary, calling on little more than the intermediate value theorem.

 Voevodsky asked what the topology of the universe is in a 
continuous interpretation of type theory, such as Johnstone's 
topological topos. We can actually give a model-independent answer: it 
is indiscrete. I will briefly introduce "intensional Martin-Loef type 
theory" (MLTT) and formulate and prove this in type theory (as opposed 
to as a meta-theorem about type theory). As an application or corollary, 
I will also deduce an analogue of Rice's Theorem for the universe: the 
universe (the large type of all small types) has no non-trivial 
extensional, decidable properties. Topologically this is the fact that 
it doesn't have any clopens other than the trivial ones.

In this talk, I will explain part of the programme of Gorenstein, Lyons

and Solomon (GLS) to provide a new proof of the CFSG. I will focus on

the difference between the initial notion of groups of characteristic

$2$-type (groups like Lie type groups of characteristic $2$) and the GLS

notion of groups of even type. I will then discuss work in progress

with Capdeboscq to study groups of even type and small $2$-local odd

rank. As a byproduct of the discussion, a picture of the structure of a

finite simple group of even type will emerge.

We prove that if $\frac{\log^{117} n}{n} \leq p \leq 1 -

n^{-1/8}$, then asymptotically almost surely the edges of $G(n,p)$ can

be covered by $\lceil \Delta(G(n,p))/2 \rceil$ Hamilton cycles. This

is clearly best possible and improves an approximate result of Glebov,

Krivelevich and Szab\'o, which holds for $p \geq n^{-1 + \varepsilon}$.

Based on joint work with Daniela Kuhn, John Lapinskas and Deryk Osthus.