Manchester Geometry Seminar 2013/2014

Thursday 20 February 2014. The Frank Adams Room (Room 1.212), the Alan Turing Building. 4.15pm

Linearly Degenerate PDEs and Quadratic Line Complexes

Eugene Ferapontov (Loughborough University)

A quadratic line complex is a three-parameter family of lines in the projective space $\mathbf{P}^3$ specified by a single quadratic relation in the Plücker coordinates. Fixing a point $p$ in $\mathbf{P}^3$ and taking all lines of the complex passing through $p$ we obtain a quadratic cone with vertex at $p$. This family of cones supplies $\mathbf{P}^3$ with a conformal structure, with which we canonically associate a three-dimensional second order quasi-linear PDE. We show that any PDE arising in this way is linearly degenerate; furthermore, any linearly degenerate PDE can be obtained by this construction. This provides a classification of linearly degenerate wave-type equations into eleven types, labeled by Segre symbols of the associated quadratic complexes. We classify Segre types for which the corresponding PDE is integrable. (Based on joint work with Jonathan Moss.)