Geometry and Integrability
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Fri, 25/04/2008 12:00 |
Dr Maciej Dunajski (Cambridge) |
Geometry and Integrability  |
L3 |
| Cover a plane with curves, one curve through each point in each direction. How can you tell whether these curves are the geodesics of some metric? This problem gives rise to a certain closed system of partial differential equations and hence to obstructions to finding such a metric. It has been an open problem for at least 80 years. Surprisingly it is harder in two dimensions than in higher dimensions. I shall present a solution obtained jointly with Robert Bryant and Mike Eastwood. | |||
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Fri, 02/05/2008 12:00 |
Dr Galina Filipuk |
Geometry and Integrability |
L1 |
| Any nonlinear equation of the form y”=\sum_{n=0}^N a_n(z)y^n has a (generally branched) solution with leading order behaviour proportional to (z-z_0)^{-2/(N-1)} about a point z_0, where the coefficients a_n are analytic at z_0 and a_N(z_0)\ne 0. Jointly with R.G. Halburd we consider the subclass of equations for which each possible leading order term of this form corresponds to a one-parameter family of solutions represented near z_0 by a Laurent series in fractional powers of z-z_0. For this class of equations we show that the only movable singularities that can be reached by analytic continuation along finite-length curves are of the algebraic type just described. This work generalizes previous results of S. Shimomura. The only other possible kind of movable singularity that might occur is an accumulation point of algebraic singularities that can be reached by analytic continuation along infinitely long paths ending at a finite point in the complex plane. This behaviour cannot occur for constant coefficient equations in the class considered. However, an example of R. A. Smith shows that such singularities do occur in solutions of a simple autonomous second-order differential equation outside the class we consider here. | |||
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Thu, 12/06/2008 14:15 |
Prof D Korotkin (Montreal) |
Geometry and Integrability |
L1 |
