DescriptionEuler showed that the sum of the inverse squares over all natural numbers equals π2/6 and more generally, for any even integer n>1, that the corresponding sum of the inverses of n-th powers is a rational multiple of πn. This rational multiple is essentially a so-called Bernoulli number---discovered and analysed independently by J Faulhaber and by S Takakazu in the early 17th century---arise naturally when studying sums of k-th powers of successive integers. But they also figure prominently in many other areas and theories, like in the Euler-MacLaurin formula (approximating a given integral by a related finite sum), in Kummer's criteria for Fermat's Last Theorem, in Lie theory (e.g. the Baker-Campbell-Hausdorff formula), algebraic geometry (e.g. the formula for the so-called `Todd class'), algebraic K-theory (e.g. stable homotopy groups of spheres), differential topology (in connection with exotic spheres), in special values of the Riemann zeta function (at positive even integers), and countless others.In this project, you are challenged to try and find as well as understand, and ultimately relate, as many of those instances as possible--albeit many of them will doubtless be beyond reach--and to come up with reasons, proofs or intuition as to why there might be common threads behind those appearances. As a starting point, one might perhaps want to have a look at the Bernoulli page or the discussion on mathoverflow (see links below). PrerequisitesFamiliarity with analysis of sums and series, as well as partial fraction expansions, is recommended; many connections will require a decent background in Algebra, and knowledge of Elementary Number Theory will likely be quite helpful.ResourcesMathoverflow discussion on why Bernoulli numbers are to be found `everywhere'. A book by Arakawa, Ibukiyama and Kaneko (with an appendix by Zagier) Bernoulli numbers and Zeta functions. |
email: Herbert Gangl