M. Golubitsky and D. Schaeffer
A discussion of symmetry and symmetry breaking
In: 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.