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 vicepresident 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.
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Transcription
Table of problems
Problem  Brief explanation  Status  Year Solved 

1st  Riemann hypothesis: The real part of every nontrivial zero of the Riemann zeta function is 1/2. (see also Hilbert's eighth problem)  Unresolved.  – 
2nd  Poincaré conjecture: Every simply connected, closed 3manifold is homeomorphic to the 3sphere.  Resolved. Result: Yes, Proved by Grigori Perelman using Ricci flow.^{[3]}^{[4]}^{[5]}  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?  Unresolved.  – 
4th  Shub–Smale tauconjecture on the integer zeros of a polynomial of one variable^{[6]}^{[7]}  Unresolved.  – 
5th  Can one decide if a Diophantine equation ƒ(x,y) = 0 (input ƒ ∈ [u,v]) has an integer solution, (x,y), in time (2^{s})^{c} for some universal constant c? That is, can the problem be decided in exponential time?  Unresolved.  – 
6th  Is the number of relative equilibria finite, in the nbody problem of celestial mechanics, for any choice of positive real numbers m_{1}, ..., m_{n} as the masses?  Partially resolved. Proved for five bodies by A. Albouy and V. Kaloshin in 2012.^{[8]}  2012 
7th  Distribution^{[clarification needed]} of points on the 2sphere  Partially resolved. 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 Npoint solutions are known while most solutions are numerical. Numerical solutions to this problem have been shown to correspond well with features of electron shellfilling in Atomic structure found throughout the periodic table.^{[9]} A welldefined, intermediate step to this problem involving a point charge at the origin has been reported.^{[10]}  – 
8th  Extend the mathematical model of general equilibrium theory to include price adjustments  Gjerstad (2013)^{[11]} extends the deterministic model of price adjustment 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.  2013 
9th  The linear programming problem: Find a stronglypolynomial time algorithm which for given matrix A ∈ R^{m×n} and b ∈ R^{m} decides whether there exists x ∈ R^{n} with Ax ≥ b.  Unresolved.  – 
10th  Pugh's closing lemma (higher order of smoothness)  Partially Resolved. Proved for Hamiltonian diffeomorphisms of closed surfaces by M. Asaoka and K. Irie in 2016.^{[12]}  2016 
11th  Is onedimensional dynamics generally hyperbolic? (a) Can a complex polynomial T 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 T : [0,1] → [0,1] be C^{r} approximated by one which is hyperbolic, for all r > 1? 
(a) Unresolved, even in the simplest parameter space of polynomials, the Mandelbrot set. (b) Resolved. Proved by Kozlovski, Shen and van Strien.^{[13]} 
2007 
12th  Can a diffeomorphism of a compact manifold M onto itself be C approximated, all r ≥ 1, by one T : M → M which commutes with only its iterates? In other words, what are the centralizers of a diffeomorphism?  Partially Resolved. Solved in the C^{1} topology by Christian Bonatti, Sylvain Crovisier and Amie Wilkinson^{[14]} in 2009. Still open in the C^{r} topology for r > 1.  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.  Unresolved, even for algebraic curves of degree 8.  – 
14th  Do the properties of the Lorenz attractor exhibit that of a strange attractor?  Resolved. Result: Yes, solved by Warwick Tucker using interval arithmetic.^{[15]}  2002 
15th  Do the Navier–Stokes equations in R^{3} always have a unique smooth solution that extends for all time?  Unresolved.  – 
16th  Jacobian conjecture: If The Jacobian determinant of F is a nonzero constant and k has characteristic 0, then F has an inverse function G : k^{N} → k^{N}, and G is regular (in the sense that its components are polynomials).  Unresolved.  – 
17th  Solving polynomial equations in polynomial time in the average case  Resolved. C. Beltrán and L. M. Pardo found a uniform probabilistic algorithm (average Las Vegas algorithm) for Smale's 17th problem^{[16]}^{[17]} F. Cucker and P. Bürgisser made the smoothed analysis of a probabilistic algorithm à la BeltránPardo and then exhibited a deterministic algorithm running in time .^{[18]} Finally, P. Lairez found an alternative method to derandomize the algorithm and thus found a deterministic algorithm which runs in average polynomial time.^{[19]} All these works follow Shub and Smale's foundational work (the "Bezout series") started in^{[20]} 
20082016 
18th  Limits of intelligence (it talks about the fundamental problems of intelligence and learning, both from the human and machine side)^{[21]}  Unresolved.  – 
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:"^{[22]}^{[23]}
 Mean value problem
 Is the threesphere 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
References
 ^ Smale, Steve (1998). "Mathematical Problems for the Next Century". Mathematical Intelligencer. 20 (2): 7–15. CiteSeerX 10.1.1.35.4101. doi:10.1007/bf03025291.
 ^ Smale, Steve (1999). "Mathematical problems for the next century". In Arnold, V. I.; Atiyah, M.; Lax, P.; Mazur, B. (eds.). Mathematics: frontiers and perspectives. American Mathematical Society. pp. 271–294. ISBN 9780821820704.
 ^ Perelman, Grigori (2002). "The entropy formula for the Ricci flow and its geometric applications". arXiv:math.DG/0211159.
 ^ Perelman, Grigori (2003). "Ricci flow with surgery on threemanifolds". arXiv:math.DG/0303109.
 ^ Perelman, Grigori (2003). "Finite extinction time for the solutions to the Ricci flow on certain threemanifolds". arXiv:math.DG/0307245.
 ^ Shub, Michael; Smale, Steve (1995). "On the intractability of Hilbert's Nullstellensatz and an algebraic version of "NP≠P?"". Duke Math. J. 81: 47–54. doi:10.1215/S0012709495081058. Zbl 0882.03040.
 ^ Bürgisser, Peter (2000). Completeness and reduction in algebraic complexity theory. Algorithms and Computation in Mathematics. 7. Berlin: SpringerVerlag. p. 141. ISBN 9783540667520. Zbl 0948.68082.
 ^ Albouy, A.; Kaloshin, V. (2012). "Finiteness of central configurations of five bodies in the plane". Annals of Mathematics. 176: 535–588. doi:10.4007/annals.2012.176.1.10.
 ^ LaFave, T., Jr (2013). "Correspondences between the classical electrostatic Thomson Problem and atomic electronic structure" (PDF). Journal of Electrostatics. 71 (6): 1029–1035. arXiv:1403.2591. doi:10.1016/j.elstat.2013.10.001. Archived from the original (PDF) on 22 February 2014. Retrieved 11 Feb 2014.
 ^ LaFave, T., Jr (2014). "Discrete transformations in the Thomson Problem" (PDF). Journal of Electrostatics. 72 (1): 39–43. arXiv:1403.2592. doi:10.1016/j.elstat.2013.11.007. Retrieved 11 Feb 2014.
 ^ Gjerstad, Steven (2013). "Price Dynamics in an Exchange Economy". Economic Theory. 52 (2): 461–500. CiteSeerX 10.1.1.415.3888. doi:10.1007/s0019901106515.
 ^ Asaoka, M.; Irie, K. (2016). "A C^{∞} closing lemma for Hamiltonian diffeomorphisms of closed surfaces". Geometric and Functional Analysis. 26 (5): 1245–1254. doi:10.1007/s0003901603863.
 ^ Kozlovski, O.; Shen, W.; van Strien, S. (2007). "Density of hyperbolicity in dimension one". Annals of Mathematics. 166: 145–182. doi:10.4007/annals.2007.166.145.
 ^ Bonatti, C.; Crovisier, S.; Wilkinson, A. (2009). "The C^{1}generic diffeomorphism has trivial centralizer". Publications Mathématiques de l'IHÉS. 109: 185–244. arXiv:0804.1416. doi:10.1007/s102400090021z.
 ^ Tucker, Warwick (2002). "A Rigorous ODE Solver and Smale's 14th Problem" (PDF). Foundations of Computational Mathematics. 2 (1): 53–117. CiteSeerX 10.1.1.545.3996. doi:10.1007/s002080010018.
 ^ Beltrán, Carlos; Pardo, Luis Miguel (2008). "On Smale's 17th Problem: A Probabilistic Positive answer" (PDF). Foundations of Computational Mathematics. 8 (1): 1–43. CiteSeerX 10.1.1.211.3321. doi:10.1007/s1020800502110.
 ^ Beltrán, Carlos; Pardo, Luis Miguel (2009). "Smale's 17th Problem: Average Polynomial Time to compute affine and projective solutions" (PDF). Journal of the American Mathematical Society. 22 (2): 363–385. Bibcode:2009JAMS...22..363B. doi:10.1090/s0894034708006309.
 ^ Cucker, Felipe; Bürgisser, Peter (2011). "On a problem posed by Steve Smale". Annals of Mathematics. 174 (3): 1785–1836. arXiv:0909.2114. doi:10.4007/annals.2011.174.3.8.
 ^ Lairez, Pierre (2016). "A deterministic algorithm to compute approximate roots of polynomial systems in polynomial average time". Foundations of Computational Mathematics. to appear.
 ^ Shub, Michael; Smale, Stephen (1993). "Complexity of Bézout's theorem. I. Geometric aspects". J. Amer. Math. Soc. 6 (2): 459–501. doi:10.2307/2152805. JSTOR 2152805..
 ^ "Tucson  Day 3  Interview with Steve Smale". Recursivity. February 3, 2006.
 ^ Smale, Steve. "Mathematical Problems for the Next Century" (PDF).
 ^ Smale, Steve. "Mathematical problems for the next century, Mathematics: Frontiers and perspectives". American Mathematical Society, Providence, RI: 271–294.