Gradient Problems in Biology

Improvement and optimization are consistent themes throughout science. Entropy increases, an object falls to earth to minimize its potential energy, light rays refract in a way that minimizes the time required to travel from a point in one medium t o a point in another. The notion of optimization is useful in biology as well. We shall describe how to capture some of these ideas about optimization using the concept of the gradient. The slight generalization that we will use in our discussion of population genetics requires us t o introduce a simple idea from differential geometry.