Differential geometry studies smooth surfaces in all dimensions, from curves to the universe. Riemannian geometry shows that curvature is the key to understanding shape, from the curvature of a curve in calculus to the curvature of space in general relativity. Sharp corners and black holes are singularities that require extensions of the theory. We'll look at some open questions. Format: lecture/discussion. Evaluation will be based on homework, classwork, problem sets, projects, and exams. Prerequisites: Mathematics 301 or 305. No enrollment limit (expected: 12).