Mathematics Colloquium
3:30 p.m., Friday
Math 100
Professor Elemer E. Rosinger
Department of Mathematics and Applied Mathematics
University of Pretoria
Global parametric Lie group actions, nonlinear PDEs
and Hilbert's fifth problem
Since the work of Chevalley, four or five decades ago, the
importance of the global action of Lie groups has been well
established. In the context of PDEs however, such a globality
has only been achieved in the special and rather particular case
of the so called `fibre preserving' or `projectable' Lie groups
of symmetries. On the other hand, even some of the most simple and
basic PDEs, and even more so in the nonlinear case, have Lie
symmetry groups which are not projectable, a well known example
being that of the shock wave equation. In the speaker's recently
published Kluwer book, global actions are constructed in the case
of arbitrary Lie group symmetries which act on classical or
generalized solutions of a very large class of smooth nonlinear
PDEs. This construction is obtained by using parametric
representations for Lie group actions, as well as for the respective
solutions. One of the effects of the above is that Hilbert's fifth
problem obtains a solution which is more in line with the general
initial formulation of that problem, than the various earlier
claimed solutions given in the 1940s and 1950s. These results
extend earlier ones published by the speaker since 1992 in a
joint NorthHolland research monograph and in three joint papers,
results which only dealt with global projectable Lie group actions
on generalized solutions, and which did not use the parametric method.
