Unsolved problems in mathematics

Since the Renaissance, every century has seen the solution of more mathematical problems than the century before, and yet many mathematical problems, both major and minor, still remain unsolved. Prizes are often awarded for the solution to a long-standing problem, and lists of unsolved problems (such as the list of Millennium Prize Problems) receive considerable attention. Unsolved problems remain in multiple domains, including physics, computer science, algebra, additive and algebraic number theories, analysis, combinatorics, algebraic, discrete and Euclidean geometries, graph, group, model, number, set and Ramsey theories, dynamical systems, partial differential equations, and miscellaneous unsolved problems.

Over the course of time, several lists of unsolved mathematical problems have appeared.

Of the original seven Millennium Prize Problems set by the Clay Mathematics Institute in 2000, six have yet to be solved, as of 2017:

The seventh problem, the Poincaré conjecture, has been solved. The smooth four-dimensional Poincaré conjecture—that is, whether a four-dimensional topological sphere can have two or more inequivalent smooth structures—is still unsolved.