Smale's Problems

Smale's problems are a list of 18 challenging problems for the twenty-first century proposed by Stephen Smale, a recipient of the Fields Medal. These problems were inspired in part by Hilbert's famous list of problems presented in 1900 (Hilbert's problems), and in part in response to a suggestion by V. I. Arnold on behalf of the International Mathematical Union that mathematicians describe a number of outstanding problems for the 21st century.

1. The Riemann hypothesis: does every nontrivial zero of the Riemann zeta function have real part ? Status. Open.

2. The Poincaré conjecture: is every simply connected closed manifold of dimension 3 homeomorphic to the 3-sphere? Status. Proved affirmatively by Perelman (2002, 2003).

3. The P versus NP problem: does , i.e., are P-problems equivalent to NP-problems? Status. Open.

4. Integer zeros of a polynomial: is the number of distinct integer zeros of a univariate polynomial bounded by a polynomial in its straight-line complexity ? Status. Open.

5. Height bounds for Diophantine curves: for , can the existence of an integer solution to be decided in time exponential in the input size? For algebraic curves of positive genus, a polynomial height bound for a smallest solution would imply an affirmative answer. Status. Open; even decidability is unknown.

6. Relative equilibria in celestial mechanics: for every choice of positive masses, are there only finitely many planar relative equilibria of the -body problem, up to symmetry? Status. Open in general. Finiteness is known for four bodies (Hampton and Moeckel 2006), and for five bodies outside a codimension-two exceptional set of masses (Albouy and Kaloshin 2012).

7. Distribution of points on the 2-sphere: can a real-number algorithm, in polynomial time, produce distinct points whose logarithmic energy is at most above the minimum for a universal constant ? Status. Open.

8. Dynamics in economic theory: extend static general equilibrium theory to include price adjustment in several interacting markets, ideally deriving the dynamics from the actions of individual agents. Status. An open research program, with solutions only in restricted models.

9. The linear programming problem: is there a real-number algorithm running in polynomial time for deciding the feasibility of ? The closely related rational formulation asks for a strongly polynomial algorithm for general linear programming. Status. Open. A strongly polynomial algorithm is known when each row or each column has at most two nonzero entries (Dadush et al. 2024).

10. The closing lemma: if is a nonwandering point of a diffeomorphism of a compact manifold, can the diffeomorphism be approximated in the topology by one for which is a periodic point? Status. Proved for (Pugh 1967); open in general for . A closing lemma is known for Hamiltonian diffeomorphisms of closed surfaces (Asaoka and Irie 2016).

11. Hyperbolicity in one-dimensional dynamics: can every complex polynomial be approximated by a polynomial of the same polynomial degree for which every critical point tends to a periodic sink? Status. The complex problem is open. The analogous density of hyperbolicity for smooth real interval maps was proved by Kozlovski et al. (2007).

12. Centralizers of diffeomorphisms: can every diffeomorphism of a compact manifold be approximated by one whose only commuting diffeomorphisms are its iterates? Status. Proved in the topology in the generic, residual sense by Bonatti et al. (2009); open in general for higher .

13. Problem 16 in Hilbert's problems: for a planar polynomial vector field of polynomial degree at most , is the number of limit cycles bounded by for some universal constant ? Status. Open.

14. The Lorenz attractor: is the dynamics of the Lorenz equations that of the geometric Lorenz attractor of Williams, Guckenheimer, and Yorke? Status. Proved affirmatively by Tucker (2002).

15. The Navier-Stokes equations: do the equations on a three-dimensional domain have a unique smooth solution for all time for all admissible smooth initial conditions? Status. Open.

16. The Jacobian conjecture: must a polynomial map with nonzero constant Jacobian determinant be an invertible polynomial map? Status. Refuted in dimension 3, and hence in every dimension , by a 2026 counterexample credited to the AI system Fable (Alpöge 2026, Zhang 2026). The plane case remains open.

17. Solving polynomial equations: can a zero of complex polynomial equations in unknowns be found approximately, on average, in polynomial time by a uniform algorithm? Status. Solved affirmatively. Beltrán and Pardo (2008) gave a randomized algorithm, and Lairez (2017) gave a deterministic algorithm.

18. Limits of intelligence: develop mathematical models that characterize and compare the capabilities and limitations of artificial and human intelligence. Smale specifically emphasized problem solving, Turing machines versus computation over the real numbers, approximation and roundoff error, randomness, complexity theory, learning, and interaction with an environment. Advances in machine learning, automated theorem proving, and AI-assisted mathematical discovery have made these questions experimentally accessible; notably, an AI-assisted construction supplied the 2026 counterexample to problem 16 above (Alpöge 2026, Zhang 2026). Status. This remains an open-ended research program: there is no generally accepted mathematical definition of intelligence or single criterion by which the problem could be considered solved.