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.
In his original article Smale written that he chosen the problems using three criteria: simple statement that is preferably precise, personal acquaintance with the problem and belief that solution (or just partial results) for the problem would have great importance for development of mathematics in 21st century.
Four problems from Smale's list were later included in the list of seven Millennium Prize problems by the Clay Institute: the Riemann hypothesis, the Poincaré conjecture, the P versus NP problem, and the Navier–Stokes existence and smoothness problem.
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) – 2nd Poincaré conjecture: Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere. 2003 3rdP versus NP problem: For all problems for which an algorithm can verify a given solution in polynomial time, can an algorithm also find that solution in polynomial time? – 4th Shub–Smale tau-conjecture: let and by denote the minimal number of steps required to generate using addition, subtraction or multiplication from and .
Is there a constant independent on such that number of integer zeros of is bounded by ? – 5th Can one decide if a Diophantine equation f(x, y) = 0 (input f ∈ [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? – 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? – 7th Algorithm for finding set of such that the function: is minimized for a distribution of N points on a 2-sphere. This is related to the Thomson problem. – 8th Develop dynamical mathematical model of market equilibrium to include price adjustments. The excess demand function in such model is required to satisfy economic-justified conditions: be independent on equal inflation of all prices satisfy Walras's law return positive demand on free goods The model may incorporate other economic variables along with the prices but it should be compatible with existing general equilibrium theory. 2013, 2022? 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. – 10th Pugh's closing lemma (higher order of smoothness) –11th Is one-dimensional dynamics generally hyperbolic?(a) Can a complex polynomial 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 be approximated by one which is hyperbolic, for all ? – 2007 12th For a closed manifold and any let be the topological group of diffeomorphisms of onto itself. Given arbitrary , is it possible to approximate it arbitrary well by such that it commutes only with its iterates? In other words, is the subset of all diffeomorphisms whose centralizers are trivial dense in ? – 13th Having a polynomial vector field on a real plane defined by polynomials:
is the number of limit cycles in that field limited by a bound :
where constant is independent on the choice of polynomials? (See also: Hilbert's sixteenth problem) – 14th Do the properties of the Lorenz attractor exhibit that of a strange attractor? 2002 15th Do the Navier–Stokes equations for incompressible flow and constant but non-zero viscosity and no external force applied to the flow in always have a unique smooth solution that extends for all time? – 16th Jacobian conjecture: If F : kN → kN is regular mapping and the Jacobian determinant of F is a non-zero constant and k has characteristic 0, then F has an inverse function G : kN → kN that is also regular. This counterexample induces counterexamples for .
It is unsure if this problem should be thought as resolved completely, since the case remains open, with specialist literature distinguishing its status from the case . 2026? 17th Solving polynomial equations in polynomial time in the average case 2008–2016 18th Limits of intelligence (it talks about the fundamental problems of intelligence and learning, both from the human and machine side) –
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": Mean value problem Is the three-sphere a minimal set (Gottschalk's conjecture)? 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
References