The squared-mean-curvature integral was introduced two centuries
ago by Sophie Germain to model the bending energy and vibration patterns of
thin elastic plates. By the 1920s the Hamburg geometry school realized
this energy is invariant under the Möbius group of conformal
transformations, and thus that minimal surfaces in R^3 or S^3 are
equilibria. In 1965 Willmore observed that the round sphere minimizes the
bending energy among all closed surfaces, and he conjectured that a certain
torus of revolution - the stereographic projection of the Clifford minimal
torus in S^3 - minimizer for surfaces of genus 1. This conjecture was
proven this spring by Fernando Coda and Andre Neves; they use the
Almgren-Pitts minimax construction, the Hersch-Li-Yau notion of conformal
area, and Urbano's characterization of the Clifford torus by its Morse
index. We will discuss what is known or conjectured for other topological
types of bending-energy minimizing or equilibrium surfaces, and how
bending-energy gradient flow might be applied to problems in
low-dimensional topology.