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 |
---|---|---|
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 | 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 ![]() |
18 | Limits of intelligence |
Smale also listed 3 additional problems:[17]
- Mean value problem
- Is the three-sphere a minimal set?
- Is an Anosov diffeomorphism of a compact manifold topologically the same as the Lie group model of John Franks?
See also
References
- ↑ 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" (PDF). 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" (PDF). Journal of Electrostatics 72 (1): 39–43. doi:10.1016/j.elstat.2013.11.007. Retrieved 11 Feb 2014.
- ↑ O. Kozlovski, W. Shen, S. van Strien (2007). "Density of hyperbolicity in dimension one". Annals of Mathematics 166: 145–182. doi:10.4007/annals.2007.166.145.
- ↑ 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" (PDF). 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" (PDF). 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" (PDF). 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.
- ↑ Steve Smale (1998). "Mathematical Problems for the Next Century". Mathematical Intelligencer 20 (2): 7–15. doi:10.1007/bf03025291.