12 June 2009; Mat-Nat Faculty (UiTø), room A228. Friday 12:15-14:00
[notice the date/time!]
Mark Fels (Utah State University, USA)
"Group Actions and Differential Equations".
A
symmetry group of a differential equation is a group which acts on the set of solutions to the differential equation. Therefore I will begin by reviewing some aspects from the theory of group actions on sets (orbits, stabilizers, etc.) using some basic examples. The notions developed for group actions on sets will then be considered in the context of symmetry groups of differential equations.I will then address how group actions can be used to possibly simplify finding solutions to the equation. For example the notion of a fixed point of a group action leads to the so-called invariant or equivariant solutions. A number of examples will again be given.
Utilizing symmetry to study the solution space to a differential equation also leads to a number of interesting geometric problems such as the principle of symmetric criticality, and the inverse problem of quotients. I will explain what some of these problems are, and demonstrate them with some examples.
Finally I will introduce a quotient theory of differential equations which provides a proper mathematical setting in which to study symmetry groups of differential equations. Applications and examples will be given.
Remark
: The symmetry group approach to studying differential equations originated with the work of the great Norwegian mathematician Sophus Lie. This talk can be considered a consequence of his ideas.