Smale's problems are a list of eighteen unsolved problems in mathematics that was proposed by Steve Smale in 1998, republished in 1999. 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
|1||Riemann hypothesis (see also Hilbert's eighth problem)|
|2||Poincaré conjecture||Proved by Grigori Perelman in 2003 using Ricci flow.|
|3||Does P = NP?|
|4||Shub–Smale τ-conjecture on the integer zeros of a polynomial of one variable|
|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.|
|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. A well-defined, intermediate step to this problem involving a point charge at the origin has been reported.|
|8||Introduction of dynamics into economic theory|
|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?|
|12||Centralizers of diffeomorphisms||Solved in the C1 topology by C. Bonatti, S. Crovisier and Amie Wilkinson in 2009.|
|13||Hilbert's 16th problem|
|14||Lorenz attractor||Solved by Warwick Tucker in 2002 using interval arithmetic.|
|15||Do the Navier–Stokes equations in R3 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. 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 .|
|18||Limits of intelligence|
- Steve Smale (1998). "Mathematical Problems for the Next Century". Mathematical Intelligencer 20 (2): 7–15. doi:10.1007/bf03025291.
- Steve Smale (1999). "Mathematical problems for the next century". In V. I. Arnold, M. Atiyah, P. Lax, B. Mazur. Mathematics: frontiers and perspectives. American Mathematical Society. pp. 271–294. ISBN 0821820702.
- Perelman, Grigori (2002). "The entropy formula for the Ricci flow and its geometric applications". arXiv:math.DG/0211159 [math.DG].
- Perelman, Grigori (2003). "Ricci flow with surgery on three-manifolds". arXiv:math.DG/0303109 [math.DG].
- Perelman, Grigori (2003). "Finite extinction time for the solutions to the Ricci flow on certain three-manifolds". arXiv:math.DG/0307245 [math.DG].
- 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.
- 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.
- A. Albouy, V. Kaloshin (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.
- T. LaFave Jr (2013). "Correspondences between the classical electrostatic Thomson Problem and atomic electronic structure". Journal of Electrostatics 71 (6): 1029–1035. doi:10.1016/j.elstat.2013.10.001. Retrieved 11 Feb 2014.
- T. LaFave Jr (2014). "Discrete transformations in the Thomson Problem". Journal of Electrostatics 72 (1): 39–43. doi:10.1016/j.elstat.2013.11.007. Retrieved 11 Feb 2014.
- C. Bonatti, S. Crovisier, A. Wilkinson (2009). "The C1-generic diffeomorphism has trivial centralizer". Publications Mathématiques de l'IHÉS 109: 185–244. doi:10.1007/s10240-009-0021-z.
- Warwick Tucker (2002). "A Rigorous ODE Solver and Smale's 14th Problem". Foundations of Computational Mathematics 2 (1): 53–117. doi:10.1007/s002080010018.
- Carlos Beltrán, Luis Miguel Pardo (2008). "On Smale's 17th Problem: A Probabilistic Positive answer". Foundations of Computational Mathematics 8 (1): 1–43. doi:10.1007/s10208-005-0211-0.
- Carlos Beltrán, Luis Miguel Pardo (2009). "Smale's 17th Problem: Average Polynomial Time to compute affine and projective solutions". Journal of the American Mathematical Society 22: 363–385. doi:10.1090/s0894-0347-08-00630-9.
- Felipe Cucker, Peter Bürgisser (2011). "On a problem posed by Steve Smale". Annals of Mathematics 174 (3): 1785–1836. doi:10.4007/annals.2011.174.3.8.