M. Golubitsky and D. Schaeffer
A Discussion of Symmetry and Symmetry Breaking
Singularity Theory (P. Orlik, ed.) Proc. Symp. Pure Math.
40 (1983) 499-516
There is an intimate relationship between singularity theory and steady
state bifurcation theory. For the past several years we have been trying
to make this relationship precise (see [9,10]) and have written several
surveys on this material [11, 12, 13, 21]. For the most part these
reviews have been written for an applied audience as has the review by
Ian Stewart [28] which includes several of our applications. In this
review we want to emphasize those theoretical problems whose resolution
would lead to interesting applied mathematics. These are problems about
which we have limited knowledge and have made limited calculations. In
particular, the problems revolve about the interaction of linear
reprsentations of compact Lie groups with the study of singularities of
mappings and the notions of "symmetry breaking" that they engender. This
review is divided into four parts: one states variable problems (the
basic theory), bifurcation problems with symmetry, spontaneous symmetry
breaking, and symmetry breaking in the equations. We shall give few
proofs. The references include a complete listing of the applications
which have followed from this point of view. It seems to us that the
study of singularities of mappings which commute with a given
representation of a compact Lie group is a rich field in need of further
investigation.