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Lie Groups and Symmetry in Differential Equations and Geometry


Professor A. Kara, University of the Witwatersrand

Dr J. Alt, University of the Witwatersrand


Groups and symmetry play a central role in geometry, from Euclidean geometry to its modern descendants. Felix Klein’s famous Erlangen Program (1872) proposed Lie groups and their homogeneous spaces as a framework for unifying various non-Euclidean geometries that had been developed in the 19th century and before. While Klein’s program failed to subsume the other important generalisation of Euclidean geometry at the time, Riemannian geometry, in the early 1900s Elie Cartan discovered the framework for unifying both the geometries of Klein and those of Riemann. In the process, Cartan developed many of the most important tools of modern differential geometry. These include principal fiber bundles and connections – the setting for gauge theory – where Lie groups and symmetries again play a central role.

Furthermore, Sophus Lie had demonstrated the application of invariance methods to difference equations which has been enthusiastically been developed following the works of Ovsiannikov in the 1950’s. Applications abound in areas such as mathematical physics, engineering, relativity and economics. At the invitation of Klein, Noether, in 1918, had established the relationship between symmetries and differential equations that arise in variational problems (Euler-Lagrange equations) and, recently, the results have been generalized and developed for broader classes of equations. Newer areas such as difference and fractional differential equations have recently been studied using invariance methods.

The special session will provide an opportunity for presentation of current research in these and other areas of geometry and differential equations related to Lie groups and symmetry.