Smale's problems
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Smale's problems are a list of eighteen unsolved problems in mathematics that was proposed by Steve Smale in 1998,^{[1]} republished in 1999.^{[2]} 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]}^{[4]}^{[5]} | |
3 | Does P = NP? | ||
4 | Shub–Smale τ-conjecture on the integer zeros of a polynomial of one variable^{[6]}^{[7]} | ||
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.^{[8]} | |
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.^{[9]} A well-defined, intermediate step to this problem involving a point charge at the origin has been reported.^{[10]} | |
8 | Introduction of dynamics into economic theory | ||
9 | The linear programming problem: find a strongly-polynomial time algorithm which for given matrix A ∈ R^{m×n} and b ∈ R^{m} decides whether there exists x ∈ R^{n} 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 C^{1} topology by C. Bonatti, S. Crovisier and Amie Wilkinson^{[11]} in 2009. | |
13 | Hilbert's 16th problem | ||
14 | Lorenz attractor | Solved by Warwick Tucker in 2002 using interval arithmetic.^{[12]} | |
15 | Do the Navier–Stokes equations in R^{3} 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.^{[13]}^{[14]} 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 .^{[15]} | |
18 | Limits of intelligence |
See also
References
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