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Details of Grant 

EPSRC Reference: EP/C549074/1
Title: Computer assisted calculations of the borel regulator
Principal Investigator: Snaith, Professor V
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Department: Pure Mathematics
Organisation: University of Sheffield
Scheme: Standard Research (Pre-FEC)
Starts: 01 October 2005 Ends: 31 December 2008 Value (£): 141,898
EPSRC Research Topic Classifications:
Algebra & Geometry
EPSRC Industrial Sector Classifications:
No relevance to Underpinning Sectors
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Summary on Grant Application Form
(Number theory is an old branch of mathematics which is studied by many diverse methods. Nineteenth century mathematicians learnt deep facts about numbers by encoding their properties into zeta functions - the most famous being Riemann's zeta function Z(s). In 1849 Kummer proposed a 2-stage programme to solve Fermat's Last Theorem. In 1934Vandiver made a conjecture based on the fact that he had proved the first stage of Kummer's programme. The KummerVandiver conjecture became famous because it implied Fermat's Last Theorem.In fact, Vandiver died in 1973 and around 1980 the late Walter Feit found a gap in the 1934 paper. In 2000 Snaith reduced the problem to showing that a certain set of real numbers are all one -- they are the ratio of the derivative of Z(s) at s= -2r to the Borel regulator for $r=1, 2, .... This result had been proved in 1998 by Kolster, Quang Do and Fleckinger, but Snaith's proof is simpler and comes with a numerical formula suitable for estimation by computer.These numerical estimates will only establish a small number of new cases of the Kummer-Vandiver conjecture - but this will represent an enormous amount of new evidence in its favour.
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Organisation Website: http://www.shef.ac.uk