# Smale's problems

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 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.

## List of problems

# Formulation Status
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[clarification needed] 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 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.
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) Proved for Hamiltonian diffeomorphisms of closed surfaces by M. Asaoka and K. Irie in 2015.[12]
11 Is one-dimensional dynamics generally hyperbolic? Smale states two variants of this problem: the complex-variable one ("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?") and the real-variable version ("Can a smooth map T: [0,1] → [0,1] be Cr approximated by one which is hyperbolic, for all r > 1?"). The former remains open even in the simplest parameter space of polynomials, the Mandelbrot set. The latter was proved by Kozlovski, Shen and van Strien[13] in 2007.
12 Centralizers of diffeomorphisms Solved in the C1 topology by Christian Bonatti, Sylvain Crovisier and Amie Wilkinson[14] in 2009. Still open in the Cr topology for r > 1.
13 Hilbert's 16th problem
14 Lorenz attractor Solved by Warwick Tucker in 2002 using interval arithmetic.[15]
15 Do the Navier–Stokes equations in R3 always have a unique smooth solution that extends for all time?
16 Jacobian 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[16][17] F. Cucker and P. Bürgisser made the smoothed analysis of a probabilistic algorithm à la Beltrán-Pardo, and then exhibited a deterministic algorithm running in time ${\displaystyle N^{O(\log \log N)}}$.[18] Finally, P. Lairez found an alternative method to de-randomize the algorithm and thus found a deterministic algorithm which runs in average polynomial time.[19] The problem is now considered as fully solved. All these works follow Shub and Smale's foundational work (the "Bezout series") started in [20]
18 Limits of intelligence (it talks about the fundamental problems of intelligence and learning, both from the human and machine side)[21]

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]

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

## References

1. ^ Smale, Steve (1998). "Mathematical Problems for the Next Century". Mathematical Intelligencer. 20 (2): 7–15. CiteSeerX . doi:10.1007/bf03025291.
2. ^ Smale, Steve (1999). "Mathematical problems for the next century". In Arnold, V. I.; Atiyah, M.; Lax, P.; Mazur, B. Mathematics: frontiers and perspectives. American Mathematical Society. pp. 271–294. ISBN 0821820702.
3. ^ Perelman, Grigori (2002). "The entropy formula for the Ricci flow and its geometric applications". arXiv: [math.DG].
4. ^ Perelman, Grigori (2003). "Ricci flow with surgery on three-manifolds". arXiv: [math.DG].
5. ^ Perelman, Grigori (2003). "Finite extinction time for the solutions to the Ricci flow on certain three-manifolds". arXiv: [math.DG].
6. ^ 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/S0012-7094-95-08105-8. Zbl 0882.03040.
7. ^ Bürgisser, Peter (2000). Completeness and reduction in algebraic complexity theory. Algorithms and Computation in Mathematics. 7. Berlin: Springer-Verlag. p. 141. ISBN 3-540-66752-0. Zbl 0948.68082.
8. ^ 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.
9. ^ LaFave, T., Jr (2013). "Correspondences between the classical electrostatic Thomson Problem and atomic electronic structure" (PDF). Journal of Electrostatics. 71 (6): 1029–1035. arXiv:. doi:10.1016/j.elstat.2013.10.001. Archived from the original (PDF) on 22 February 2014. Retrieved 11 Feb 2014.
10. ^ LaFave, T., Jr (2014). "Discrete transformations in the Thomson Problem" (PDF). Journal of Electrostatics. 72 (1): 39–43. doi:10.1016/j.elstat.2013.11.007. Retrieved 11 Feb 2014.
11. ^ Gjerstad, Steven (2013). "Price Dynamics in an Exchange Economy". Economic Theory. 52 (2): 461–500. doi:10.1007/s00199-011-0651-5.
12. ^ Asaoka, M.; Irie, K. (2016). "A C closing lemma for Hamiltonian diffeomorphisms of closed surfaces". Geometric and Functional Analysis. 26: 1245–1254. doi:10.1007/s00039-016-0386-3.
13. ^ 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.
14. ^ Bonatti, C.; Crovisier, S.; Wilkinson, A. (2009). "The C1-generic diffeomorphism has trivial centralizer". Publications Mathématiques de l'IHÉS. 109: 185–244. doi:10.1007/s10240-009-0021-z.
15. ^ Tucker, Warwick (2002). "A Rigorous ODE Solver and Smale's 14th Problem" (PDF). Foundations of Computational Mathematics. 2 (1): 53–117. doi:10.1007/s002080010018.
16. ^ Beltrán, Carlos; Pardo, Luis Miguel (2008). "On Smale's 17th Problem: A Probabilistic Positive answer" (PDF). Foundations of Computational Mathematics. 8 (1): 1–43. doi:10.1007/s10208-005-0211-0.
17. ^ 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: 363–385. Bibcode:2009JAMS...22..363B. doi:10.1090/s0894-0347-08-00630-9.
18. ^ Cucker, Felipe; Bürgisser, Peter (2011). "On a problem posed by Steve Smale". Annals of Mathematics. 174 (3): 1785–1836. doi:10.4007/annals.2011.174.3.8.
19. ^ Lairez, Pierre (2016). "A deterministic algorithm to compute approximate roots of polynomial systems in polynomial average time". Foundations of Computational Mathematics. to appear.
20. ^ 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..
21. ^ http://recursed.blogspot.jp/2006/02/tucson-day-3-interview-with-steve.html Friday, February 03, 2006 Tucson - Day 3 - Interview with Steve Smale
22. ^ Smale, Steve. "Mathematical Problems for the Next Century" (PDF).
23. ^ Smale, Steve. "Mathematical problems for the next century, Mathematics: Frontiers and perspectives". American Mathematical Society, Providence, RI: 271–294.