| Abstract: | Courant proved that the zeros of the nth eigenfunction of the Laplace operator on a compact manifold M divide this manifold into at most n parts. He conjectured that a similar statement is also valid for any linear combination of the first n eigenfunctions. However, later it was found out that some corollaries to this generalized statement contradict the results
of quantum field theory. Later, explicit counterexamples were constructed by O. Viro. Nevertheless, the one-dimensional version
of Courant’s theorem is apparently valid; to prove it, I.M. Gel’fand proposed a method based on the ideas of quantum mechanics
and the analysis of the actions of permutation groups. This leads to interesting questions of describing the statistical properties
of group representations that arise from their action on eigenfunctions of the Laplace operator. The analysis of these questions
entails, among other things, problems of singularity theory. |
