Pure Mathematics Colloquium

In the course of understanding invariants of spaces with an action of the circle group, I was led to formulate some bits of algebra in a robust (homotopy invariant) fashion. The talk will concentrate on the algebra: the theme is that certain problems with finite dimensional non-commutative algebra can be translated into problems with nice commutative rings and solved there. (Actually, I learnt one of the nicest examples from Bill Dwyer during the Leeds BMC, and much of the talk will be about joint work with him). Algebraists might find that the title 'some amusing derived Morita equivalences' gives some idea of the content.

I started to record systematically and analyse logical difficulties experienced by so-called “mathematically able” children in their learning of elementary mathematics. In my talk, I will describe some of the hidden (and highly non-trivial) structures of elementary mathematics which may intrigue and – like shadows in the night – sometimes scare an inquisitive child. For more detail, see an online draft of my forthcoming book "Shadows of the Truth", http://www.maths.manchester.ac.uk/~avb/ST.pdf (or a much smaller file without graphics: http://www.maths.manchester.ac.uk/~avb/ST_Without_Images.pdf).

In this talk I shall first clarify what is meant by constructive mathematics (a la Bishop). Then I shall introduce the notion of apartness: first between points and sets, and secondly between sets and sets. The axiomatic theory of apartness spaces, begun by Luminiţa Simona Vîţă and myself in 2000 and subsequently taken up by a number of other mathematicians, provides a robust constructive framework for topology and the theory of uniform spaces. I will present the fundamentals of the theory, leading to the more interesting set-set case and revealing the interplay between apartness and uniform structures.