Abstracts

Symbolic dynamics of the tent map (Chris Good)

We consider a tent map to be an inverted modulus map, \$t(x)= -|x-c|+1\$. The dynamical behaviour of points in \$[0,1]\$ under \$t\$ is sensitive to the value of \$c\$. One way to study the dynamics is symbolically: there is a \$1-1\$ correspondence between the points of the unit interval and their itineraries, where the itinerary of \$x\$ is the sequence whose \$n^{th}\$ element is \$0\$ is \$t^n(x)<c\$ and \$1\$ if \$t^n(x)>c\$.
We will discuss the symbolic dynamics of the tent map and show how we have used the analysis of the itinerary of \$t(c)\$ to prove some new results about tent maps.

Digital Jordan curves (Josef Slapal)

We introduce and study a new topology on the digital plane Z^2. As a criterion of convenience for applications in digital topology, we prove an analogue of the Jordan curve theorem for the topology introduced. The new topology is discussed in relation to the two classical topologies used in digital topology, namely the Khalimsky and Marcus topologies, and its advantages over them are shown. We also investigate another convenient topology on Z^2 which is obtained as a quotient topology of the topology studied.

Construction of two-point sets (Ben Chad)

A subset of the plane is said to be a two-point set iff it meets every line in exactly two points. We will discuss how to construct two-point sets with various analytic and topological properties, in particular, showing that they can be invariant under rotations.

The use of convergence spaces in topology (Rolf Suabedissen)

Since the very beginnings of general topology, the search was on for the "right" concepts. Usually this only examined slightly different definitions of properties (e.g. there are a large number of compactness type properties) but in the 1970s a thorough categorical study of topologies and topology like structures was undertaken. As a result various alternative topological categories were invented to address the perceived shortcomings of topological spaces. As so often, the new concepts never really caught on and sank into oblivion.
In this talk I will try to make a case of why some of these categories can be very useful for a topologist and deserve more attention. I will focus on convergence spaces and illustrate their use by examples from my own research.
Note that this talk is not a presentation of research but rather an expository talk which strays from the narrow mathematical perspective into more philosophical areas.

Kolmogorov spaces and Hilbert's 13th problem (Andrew Marsh)

In the 13th problem of his celebrated list of 23 problems Hilbert asks whether it is possible to construct a solution to the general seventh degree equation x^7+ax^3+bx^2+cx+1=0 using a finite number of two variable functions. This problem was solved in the late 1950's by Arnold and Kolmogorov who showed that any continuous function from I^n to I can be decomposed as a composition of functions of two variables.

In this talk we will discuss these results of Arnold and Kolmogorov and define the Kolmogorov property for an arbitrary topological space. Some very recent work of the speaker and Paul Gartside on the Kolmogorov property will be presented.

Close to normality: examples towards a theory? (Brian McMaster)

The Sorgenfrey plane and spaces resembling it provide an informative source of objects that fail to be normal but by fairly narrow margins. We [Chris Calder and BMcM] have been examining how narrow some of these failures are, and trying to identify what key features such spaces possess that contribute to the margins' widths. This talk will be a partial overview of work in progress