# Manchester Geometry Seminar 2013/2014

**Thursday 20 February 2014. ** *The Frank Adams Room (Room 1.212), the Alan Turing Building. 4.15pm*
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Linearly Degenerate PDEs and Quadratic Line Complexes

Eugene Ferapontov (Loughborough University)

`E.V.Ferapontov@lboro.ac.uk`

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.)