Category Archives: exposition

a post aimed to teach something

Finite simple groups

Posted on 15/01/2013 by

The blog Gödel’s Last Letter and P=NP is always worth reading. The two bloggers, RJ Lipton and KW Regan, are computer scientists, but they have interesting things to say about developments in mathematics, and other areas too. For example, last year they … Continue reading →

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A permutation group challenge

Posted on 19/12/2012 by

Long ago, in the distant past before the Classification of Finite Simple Groups, Peter Neumann, Jan Saxl and I investigated the class of permutation groups acting on sets of even cardinality n = 2k, with the following interchange property: Any subset of … Continue reading →

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A talk

Posted on 14/12/2012 by

I was asked, at rather short notice, to give a talk at the Colégio Planalto in Lisbon. We arrived there to a very friendly welcome and an excellent lunch and conversation before my talk. This school has some very able … Continue reading →

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Mathematical Structures, 10

Posted on 07/12/2012 by

I will be in Portugal next week, and one of my colleagues has offered to take the lectures in the final week of term. But I managed to get to where I wanted to get; the last few lectures will … Continue reading →

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Homomorphisms modulo a prime

Posted on 03/12/2012 by

Two of my colleagues have been doing interesting things with counting homomorphisms modulo a prime; Thomas Müller with group homomorphisms, and Mark Jerrum with graph homomorphisms. I may get round to discussing Thomas’s work later. Here I want to discuss … Continue reading →

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Mathematical Structures, 9

Posted on 01/12/2012 by

The last two weeks of the course are about proofs: how to construct them, how to read them, how to spot false proofs, and so on. In keeping with the spirit of the course, we have seen many proofs along … Continue reading →

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Fibonacci numbers, 7

Posted on 27/11/2012 by

I set a question about Fibonacci numbers in the coursework for Mathematical Structures. I was taken to task by John Bray for starting the sequence in the wrong place. He claims that the “official” Fibonacci numbers begin F0 = 0, F1 = 1, and … Continue reading →

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Mathematical Structures, 8

Posted on 23/11/2012 by

Complex numbers this week bring us to the end of our journey through the number systems. I found this much easier going, and I hope the students did too. Complex numbers provided conceptual difficulties for mathematicians, who were reluctant to … Continue reading →

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Mathematical Structures, 7

Posted on 15/11/2012 by

This week, the test is over, and it’s back to work, on the real numbers. In keeping with the general theme of the course, real numbers are not defined as Dedekind cuts or Cauchy sequences, but as something much more … Continue reading →

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Everything and nothing

Posted on 03/11/2012 by

This is a response to Volkan’s comment. Nothing The mathematician says: Theorem There is only one empty set. Proof Two sets are equal if they have exactly the same elements. (This is a version of Leibniz’s principle of identity of … Continue reading →

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