Given k1 smooth functions fi over a common manifold M, the Lagrangian multiplier method identifies constraint maxima at locations where the gradients are linearly dependent. We call the points of M with this property simultaneous critical points of the fi. This project studies the topological properties of the set of simultaneous critical points, it develops algorithms for PL functions fi, and it considers applications of these ideas in the sciences and in engineering.

For k smooth functions, this set is a (k-1)-manifold that is smoothly embedded in M. To define and construct for piecewise linear functions, we think of the fi as limits of series of smooth functions, and we construct the limit of the resulting series of ’s. As it turns out, the algorithm reduces to a large number of Betti number computations, stressing the importance of fast software for Betti numbers.

A number of applications of these ideas are based on the possibility to model a time-varying function by two static ones. The second function maps space-time to time so that the 1-manifold becomes the path the critical points of the original function take. We plan to use this idea to study the behavior of critical points of the electrostatic and other potentials driving molecular motion. Another application of is to express the correlation between functions as an integral over the 1-manifold. We expect this concept to be useful in understanding the interaction of potentials and to understand how similarities and differences manifest themselves as physical or biochemical phenomena.