More specifically, one hopes to understand the rational cohomology of the mapping class group using suitably defined critical points of the systole function. The cohomology in question has been calculated by Madsen and Weiss, settling a conjecture of Mumford's. The idea is that the generators of rational k-cohomology should correspond to critical points of index k, where the index is defined in a suitable topological framework.
If correct, this would mean that for sufficiently high g, the number of critical points of fixed index k (on the moduli space of Riemann surfaces of genus g) is independent of g.
See also the