Sumários

The Hitchin Integrable Systems

29 Maio 2019, 10:30 • Davide Masoero

The definition of Hitchin Integrable system, when D=0 and D \neq 0. The case of D-twisted rank 2 vector bundles of the sphere, when D has degree 4 and it is unreduced. Separation of variables and Hilbert scheme.

Moduli space of Vector Bundles

22 Maio 2019, 10:30 • Davide Masoero

Modulis space of stable vector bundles, D-twisted Higgs bundle, spectral curve, BNR correspondence, D=4\infty twisted rank 2 Higgs bundles on the sphere

More on vector bundles on Riemann surfaces

15 Maio 2019, 10:30 • Davide Masoero

Jacobian variety, Push Forward of Vector Bundles, Stability and Semi-Stability

Hitchin Integrable Systems I

8 Maio 2019, 10:30 • Davide Masoero

This is the first of a sereis of 4 lectures on the Hitchin Integrable Systems. The first lecture consisted in a review of the theory of Vector Bundles on Riemann Surfaces, sheaf cohomology, Riemann-Roch, Serre Duality

Line bundles on a Riemann surface

24 Abril 2019, 10:30 • Davide Masoero

A short inotrduction to sheaf cohomology

Classification of isomorphism classes of line bundles on a Riemann surfaces: H^1(M,O*)=Pic(M), Pic_0(M)=Jac(M)

Classification of spin bundles by means of Riemann's vector of constants and theta characteristics.

Exercise: genus 1 algebro-geometric solution of KdV (travelling waves) and their degeneration (solitons). Necessary and sufficient conditions for reality and smoothness.