Smale's problems

Template:Short description Smale's problems is a list of eighteen unsolved problems in mathematics proposed by Steve Smale in 1998 and republished in 1999. Smale composed this list in reply to a request from Vladimir Arnold, then vice-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.

Table of problems

Problem	Brief explanation	Status	Year Solved
1st	Riemann hypothesis: The real part of every non-trivial zero of the Riemann zeta function is 1/2. (see also Hilbert's eighth problem)	Template:No	–
2nd	Poincaré conjecture: Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.	Template:Yes	2003
3rd	P versus NP problem: For all problems for which an algorithm can verify a given solution quickly (that is, in polynomial time), can an algorithm also find that solution quickly?	Template:No	–
4th	Shub–Smale tau-conjecture on the integer zeros of a polynomial of one variable	Template:No	–
5th	Can one decide if a Diophantine equation ƒ(x, y) = 0 (input ƒ ∈ $\mathbb{Z}$ [u, v ]) has an integer solution, (x, y), in time (2s)c for some universal constant c? That is, can the problem be decided in exponential time?	Template:No	–
6th	Is the number of relative equilibria (central configurations) finite in the n-body problem of celestial mechanics, for any choice of positive real numbers m1, ..., mn as the masses?	Template:Partial	2012
7th	Algorithm for finding set of $(x_1,...,x_N)$ such that the function: $V_N(x)=\sum _{1\le i<j\le N}\log \frac{1}{\Vert x_i-x_j\Vert }$ is minimized for a distribution of N points on a 2-sphere. This is related to the Thomson problem.	Template:No	–
8th	Extend the mathematical model of general equilibrium theory to include price adjustments	Gjerstad (2013) extends the deterministic model of price adjustment by Hahn and Negishi (1962) to a stochastic model and shows that when the stochastic model is linearized around the equilibrium the result is the autoregressive price adjustment model used in applied econometrics. He then tests the model with price adjustment data from a general equilibrium experiment. The model performs well in a general equilibrium experiment with two commodities. Lindgren (2022) provides a dynamic programming model for general equilibrium with price adjustments, where price dynamics are given by a Hamilton-Jacobi-Bellman partial differerential equation. Good Lyapunov stability conditions are provided as well.
9th	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.	Template:No	–
10th	Pugh's closing lemma (higher order of smoothness)	Template:Partial	2016
11th	Is one-dimensional dynamics generally hyperbolic? (a) Can a complex polynomial Template:Math be approximated by one of the same degree with the property that every critical point tends to a periodic sink under iteration? (b) Can a smooth map Template:Math be Template:Math approximated by one which is hyperbolic, for all Template:Math?	Template:Partial	2007
12th	For a closed manifold $M$ and any $r\ge 1$ let ${\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}}^r(M)$ be the topological group of $C^r$ diffeomorphisms of $M$ onto itself. Given arbitrary $A\in {\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}}^r(M)$, is it possible to approximate it arbitrary well by such $T\in {\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}}^r(M)$ that it commutes only with its iterates? In other words, is the subset of all diffeomorphisms whose centralizers are trivial dense in ${\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}}^r(M)$?	Template:Partial	2009
13th	Hilbert's 16th problem: Describe relative positions of ovals originating from a real algebraic curve and as limit cycles of a polynomial vector field on the plane.	Template:No	–
14th	Do the properties of the Lorenz attractor exhibit that of a strange attractor?	Template:Yes	2002
15th	Do the Navier–Stokes equations in R3 always have a unique smooth solution that extends for all time?	Template:No	–
16th	Jacobian conjecture: If the Jacobian determinant of F is a non-zero constant and k has characteristic 0, then F has an inverse function G : kN → kN, and G is regular (in the sense that its components are polynomials).	Template:No	–
17th	Solving polynomial equations in polynomial time in the average case	Template:Yes	2008-2016
18th	Limits of intelligence (it talks about the fundamental problems of intelligence and learning, both from the human and machine side)	Some recent authors have claimed results, including the unlimited nature of human intelligence and limitations on neural-network-based artificial intelligence	–

In later versions, Smale also listed three additional problems, "that don't seem important enough to merit a place on our main list, but it would still be nice to solve them:"

1. Mean value problem
2. Is the three-sphere a minimal set (Gottschalk's conjecture)?
3. Is an Anosov diffeomorphism of a compact manifold topologically the same as the Lie group model of John Franks?

See also

• Millennium Prize Problems
• Simon problems
• Taniyama's problems
• Hilbert's problems
• Thurston's 24 questions