The heat equation shrinks embedded plane curves to round points. Matthew A. Grayson shows that smooth embedded curves in the plane, under the curvature flow, shrink smoothly to points with round limiting shapes. The main theorem states that for any smooth embedded curve in the plane, there exists a smooth evolution satisfying the curvature flow, which converges to a point as time approaches T, with the limiting shape being a round circle. The proof involves showing that embedded curves remain smooth and embedded, and that curvature remains bounded, preventing singularities. Key results include the nonexistence of corners, the δ-whisker lemma, and the proof that curves shrink to convex shapes. The paper also discusses the behavior of curves under the curvature flow, showing that they remain smooth and embedded, and that curvature remains bounded, preventing singularities. The main theorem is proven by showing that curves either become convex before shrinking to a point or shrink to a point with a round limiting shape. The paper also addresses the behavior of curves under the curvature flow, showing that they remain smooth and embedded, and that curvature remains bounded, preventing singularities.The heat equation shrinks embedded plane curves to round points. Matthew A. Grayson shows that smooth embedded curves in the plane, under the curvature flow, shrink smoothly to points with round limiting shapes. The main theorem states that for any smooth embedded curve in the plane, there exists a smooth evolution satisfying the curvature flow, which converges to a point as time approaches T, with the limiting shape being a round circle. The proof involves showing that embedded curves remain smooth and embedded, and that curvature remains bounded, preventing singularities. Key results include the nonexistence of corners, the δ-whisker lemma, and the proof that curves shrink to convex shapes. The paper also discusses the behavior of curves under the curvature flow, showing that they remain smooth and embedded, and that curvature remains bounded, preventing singularities. The main theorem is proven by showing that curves either become convex before shrinking to a point or shrink to a point with a round limiting shape. The paper also addresses the behavior of curves under the curvature flow, showing that they remain smooth and embedded, and that curvature remains bounded, preventing singularities.