# 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 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
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)
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[11] in 2007.
12 Centralizers of diffeomorphisms Solved in the C1 topology by C. Bonatti, S. Crovisier and Amie Wilkinson[12] in 2009.
13 Hilbert's 16th problem
14 Lorenz attractor Solved by Warwick Tucker in 2002 using interval arithmetic.[13]
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.[14][15] 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 $N^{O(\log\log N)}$.[16]
18 Limits of intelligence

Smale also listed 3 additional problems:[17]

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. doi:10.1007/bf03025291. CiteSeerX: 10.1.1.35.4101.
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/0211159 [math.DG].
4. ^ Perelman, Grigori (2003). "Ricci flow with surgery on three-manifolds". arXiv:math.DG/0303109 [math.DG].
5. ^ Perelman, Grigori (2003). "Finite extinction time for the solutions to the Ricci flow on certain three-manifolds". arXiv:math.DG/0307245 [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. doi:10.1016/j.elstat.2013.10.001. 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. ^ 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.
12. ^ 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.
13. ^ 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.
14. ^ 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.
15. ^ 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. doi:10.1090/s0894-0347-08-00630-9.
16. ^ 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.
17. ^ Smale, Steve (1998). "Mathematical Problems for the Next Century". Mathematical Intelligencer 20 (2): 7–15. doi:10.1007/bf03025291. CiteSeerX: 10.1.1.35.4101.