mean-curvature flow
Picture a wobbly closed loop of string, or a lumpy soap bubble. Now let every point of the surface move inward at a speed proportional to how sharply it is curved there: tightly-curved bumps rush in fast, gently-curved parts barely move. The surface smooths and shrinks, like surface tension pulling it taut. This is mean-curvature flow — the most natural way for a shape to evolve so as to reduce its area as efficiently as possible, the gradient flow of area.
Concretely, for a moving surface let H be its mean curvature (the average of how it bends in the two principal directions) and let n be the inward normal. Mean-curvature flow is the rule that each point moves with velocity = H times n: the surface moves in its normal direction at speed equal to its mean curvature. This is precisely the steepest-descent flow for the area functional — it decreases area as fast as geometrically possible. In the plane (curve-shortening flow) the beautiful Gage-Hamilton-Grayson theorem says any embedded closed curve, however wiggly, first becomes convex and then shrinks to a round point — order emerges from chaos. A round sphere of radius R stays round and collapses, with R(t)^2 decreasing linearly, vanishing in finite time. The connection to PDE: writing the surface as a graph u, the flow becomes the parabolic PDE u_t = sqrt(1 + |grad u|^2) div( grad u / sqrt(1 + |grad u|^2) ), and the fixed points of the flow (where nothing moves) are exactly the minimal surfaces.
Why does it matter? Mean-curvature flow is the workhorse of geometric analysis and the model for surface tension, grain growth in metals, and image smoothing; it is also the exact sharp-interface limit of the Allen-Cahn equation. The honest difficulty, and what makes it deep, is SINGULARITIES: in three dimensions the flow can pinch off — a dumbbell-shaped surface develops a neck that shrinks to zero before the whole thing disappears, so the flow stops being smooth in finite time. Understanding and continuing the flow through such singularities (via weak 'level-set' or 'Brakke' formulations) is a major and still-active research area, and it is the same circle of ideas Hamilton's Ricci flow used to prove the Poincare conjecture.
A circle of radius R in the plane shrinks under curve-shortening flow with dR/dt = -1/R, so R(t)^2 = R(0)^2 - 2 t and it disappears at t = R(0)^2 / 2 — collapsing to a point. A wiggly closed curve does something subtler but just as clean: it first rounds out into a near-circle and then collapses the same way (Grayson's theorem).
Move every point inward at its curvature: area shrinks as fast as possible.
In the plane curve-shortening is wonderfully clean, but in three dimensions mean-curvature flow can form SINGULARITIES in finite time (a neck pinching off) before the surface disappears — so a smooth flow need not exist for all time, and weak formulations are needed to continue past the pinch.