Mathematics Colloquia and Seminars

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In 1988 Hitchin introduced the concept of Higgs pairs over a compact Riemann surface, and discover a map (nowadays called Hitchin map) from the moduli space of such objects to an affine space, which turned out to be a completely integrable Hamiltonian system. He address the following question to the scientific community: can we find in some concrete way the differential equations?I will try to give an answer to this question using the Krichever map and the Sato Grassmannian (from an algebro-geometric point of view), computing the equations in terms of residue identities. For the case when the so called spectral cover is totally ramified at a fixed point, the equations will be given in terms of the coefficients of the characteristic polynomial of Higgs field. The first part of the talk will be concerned with an introduction to the Sato Grassmannian and the Krichever construction for such a moduli space, reviewing the known case for vector bundles over smooth curves, almost no background will be assume at this stage. The second part (computing equations) will be a bit more hard, but I will do my best trying to avoid as many technical details as possible, giving the equations by using "linear algebra".