Caroline Series, Indra's Pearls: Geometry and Symmetry
This talk is based on the book of the same title by David Mumford, Caroline Series and David Wright, published by Cambridge University Press. Here is an extract from the book jacket:
Felix Klein, one of the great nineteenth-century geometers, discovered in mathematics an idea prefigured in Buddhist mythology: the heaven of Indra contained a net of pearls, each of which was reflected in its neighbours, so that the whole Universe was mirrored in each pearl. Klein studied infinitely repeating reflections, each simple in itself, but whose interactions produce fractals on the edge of chaos. In the 1980's, the authors embarked on the first computer exploration of Klein's vision, and in doing so, found further extraordinary images of their own.
Illustrated with many pictures, this talk will be an introduction to the mathematics and algorithms behind the fractal pictures explained in the book.
Ian Stewart FRS, All the world's a network
Networks are all the rage in today's science. Food webs in ecology, gene transcription networks in biology, the Internet, Google, the London Underground, even the solar system---they all can be viewed as networks.
Recent advances include the idea of a 'small world' network, which explains the famous 'six degrees of separation' between any two people on the planet, and can be seen in the Kevin Bacon game and the mathematicians' Erdos Number.
At Warwick we have studied the dynamics of networks, with special emphasis on the effects of symmetry. Applications include the human visual system, animal movement, and the formation of new species---all included in the talk.
This is an illustrated talk with no formulas and no technical maths. It does include a pig.
Oh, and it does tell you what a network is, too.
Samir Siksek, Beyond Fermat's Last Theorem
Wiles' proof of Fermat's Last Theorem is one of the happiest memories of the 20th century. Adaptations of Wiles' ideas enable us to solve not only Fermat's Last Theorem, but many other problems that have fascinated and baffled generations of mathematicians (both professionals and amateurs alike). We take a look at three of these:
(i) Perfect powers in Pascal's triangle.
(ii) Perfect powers in the Fibonacci sequence.
(iii) An equation due to the Indian genius Srinivasa Ramanujan.
Biologists tend to study their systems by taking them apart and them down into components, and by knocking out bits and seeing what changes. In this way they have been remarkably successful in revealing a huge amount about how living things work. On the other hand, the actual biological processes that they study nearly always involve lots of interacting components that work together as a system and to understand this one needs more than this reductive approach. In particular, one needs ways of describing and analysing such systems that are likely to be highly mathematical. Moreover, the classical way of breaking down a system and studying it component by component is very slow. Given that typical organisms contain tens of thousands of genes and have huge complexity it will be a very long time before we can really say we understand things unless there is a new approach. To circumvent this there has been a huge interest in developing high-throughput technologies to collect data not just on one gene or protein but on large numbers of them, even all of them. However, these approaches also can only be successful if they are backed by mathematical techniques to analyse the huge amounts of data produced and to relate it to the underlying biological questions. The cross-disciplinary activity around these areas has become known as Systems Biology.
David Rand, Systems biology and its mathematical challenges
In my lecture I will introduce Systems Biology and discuss some of the problems it is being used to tackle. Then I will discuss some of the mathematical challenges that it poses.