MSI Weekly Bulletin - Week starting Monday 19 June, 2006

How should one analyse court rulings to extract key voting patterns? In 2003 an analysis by Sirovich of the Second Rehnquist U.S. Supreme Court was published in the prestigious Proceedings of the U.S.A. National Academy of Sciences and received a lot of attention in the American media. In this presentation Sirovich's results will be reviewed and it will be shown how to get an even clearer picture of the voting pattern. Other approaches that have been attempted also will be described.

I shall give an account in modern terms of Cartan’s celebrated paper on projective connections (E. Cartan ‘Sur les varietes a connexion projective’ Bull Soc Math France 52 (1924) 205-241). The account will include an explanation of what Cartan meant by the term ‘connection’; how to attach at each point of a manifold a projective space of the same dimension (to use Cartan’s description of his construction); how the Cartan projective connection is related to the projective geometry of geodesics in the affine case; and how to extend the construction to encompass the geometry of a system of arbitrary second-order ordinary differential equations (in his paper Cartan covers only the case of a single equation).