Open questions in Differential Geometry

External Websites

• Hilbert's Problems. Of the 23 problems stated in 1900, 8, 13, 16 are completely unresolved, although some of the other ones are not really completely resolved (sometimes, because the original question was vague). Number 8 is the Riemann hypothesis, which is a number theory question, but it is strongly related to geometric ideas and may be solved using them. Numbers 13 and 16 are really algebraic geometry questions.
• Of the Millenium Prize problems, one has been solved (the Poincaré conjecture). The proof of this topological question by G. Perelman (see arXiv:math/0211159 [math.DG], arXiv:math/0303109 [math.DG], arXiv:math/0307245 [math.DG]) utilized differential geometry (the Ricci flow), which actually proved the Geometrization Conjecture, which implies the Poincaré conjecture. Another one of the problems, the Yang-Mills existence and mass gap problem, which involves mathematical physics, involves differential geometry heavily. Also, the Hodge conjecture is really in the intersection of differential and algebraic geometry.
• S.T.Yau's list of open problems. He keeps updating it. Here is the latest one.
• See the responses to the question What are some major open problems in Riemannian Geometry? on mathoverflow. Some of the other questions in this list are in there.
• See this list of mostly plane geometry questions, although it has some big ones like the Jacobian conjecture and the Hodge conjecture.
• Frank Morgan and Pierre Pansu's List of Open Problems in Differential Geometry