# Smale's problems

Smale's problems are a list of eighteen unsolved problems in mathematics that was proposed by Steve Smale in 1998, republished in 1999. Smale composed this list in reply to a request from Vladimir Arnold, then president of the International Mathematical Union, who asked several mathematicians to propose a list of problems for the 21st century. Arnold's inspiration came from the list of Hilbert's problems that had been published at the beginning of the 20th century.

## List of problems

# Formulation Status
1 Riemann hypothesis (see also Hilbert's eighth problem)
2 Poincaré conjecture Proved by Grigori Perelman in 2003 using Ricci flow.
3 Does P = NP?
4 Shub–Smale τ-conjecture on the integer zeros of a polynomial of one variable
5 Height bounds for Diophantine curves
6 Finiteness of the number of relative equilibria in celestial mechanics Proved for five bodies by A. Albouy and V. Kaloshin in 2012.
7 Distribution of points on the 2-sphere A noteworthy form of this problem is the Thomson Problem of equal point charges on a unit sphere governed by the electrostatic Coulomb's law. Very few exact N-point solutions are known while most solutions are numerical. Numerical solutions to this problem have been shown to correspond well with features of electron shell-filling in Atomic structure found throughout the periodic table. A well-defined, intermediate step to this problem involving a point charge at the origin has been reported.
8 Introduction of dynamics into economic theory
9 The linear programming problem: find a strongly-polynomial time algorithm which for given matrix A ∈ Rm×n and b ∈ Rm decides whether there exists x ∈ Rn with Ax ≥ b.
10 Pugh's closing lemma (higher order of smoothness)
11 Is one-dimensional dynamics generally hyperbolic?
12 Centralizers of diffeomorphisms Solved in the C1 topology by C. Bonatti, S. Crovisier and Amie Wilkinson in 2009.
13 Hilbert's 16th problem
14 Lorenz attractor Solved by Warwick Tucker in 2002 using interval arithmetic.
15 Do the Navier–Stokes equations in R3 always have a unique smooth solution that extends for all time?
16 Jacobian conjecture (equivalently, Dixmier conjecture)
17 Solving polynomial equations in polynomial time in the average case C. Beltrán and L. M. Pardo found a uniform probabilistic algorithm (average Las Vegas algorithm) for Smale's 17th problem. A deterministic algorithm for Smale's 17th problem has not been found yet, but a partial answer has been given by F. Cucker and P. Bürgisser who proceeded to the smoothed analysis of a probabilistic algorithm à la Beltrán-Pardo, and then exhibited a deterministic algorithm running in time $N^{O(\log \log N)}$ .
18 Limits of intelligence