Smale's problems
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Smale's problems refers to a list of eighteen unsolved problems in mathematics that was proposed by Steve Smale in 2000.[1] 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.
Contents |
[edit] List of problems
| # | Formulation | Status |
|---|---|---|
| 1 | Riemann hypothesis (see also Hilbert's eighth problem) | |
| 2 | Poincaré conjecture | Proved by Grigori Perelman. |
| 3 | Does P = NP? | |
| 4 | 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 | |
| 7 | Distribution of points on the 2-sphere | |
| 8 | Introduction of dynamics into economic theory | |
| 9 | The linear programming problem | |
| 10 | Pugh's closing lemma | |
| 11 | Is one-dimensional dynamics generally hyperbolic? | |
| 12 | Centralizers of diffeomorphisms | Solved in the C1 topology by C. Bonatti, S. Crovisier and A. Wilkinson.[2] |
| 13 | Hilbert's 16th problem | |
| 14 | Lorenz attractor | Solved by Warwick Tucker using interval arithmetic.[3] |
| 15 | Navier-Stokes equations | |
| 16 | Jacobian conjecture (equivalently, Dixmier conjecture) | |
| 17 | Solving polynomial equations in polynomial time in the average case | Carlos Beltrán Alvarez and Luis Miguel Pardo found a uniform (Average Las Vegas algorithm) algorithm for Smale's 17th problem, see [4] [5]. A deterministic algorithm for Smale's 17th problem has not been found yet, but a partial answer has been given by Felipe Cucker and Peter Bürgisser who proceeded to the smoothed analysis of a probabilistic algorithm à la Beltrán-Pardo, and then exhibited a deterministic algorithm running in time NO(log log N).[6] |
| 18 | Limits of intelligence |
[edit] See also
[edit] References
- ^ Steve Smale (2000). "Mathematical problems for the next century". Mathematics: frontiers and perspectives (Providence, RI: American Mathematics Society): 271–294. http://www6.cityu.edu.hk/ma/people/smale/pap104.pdf.
- ^ C. Bonatti, S. Crovisier, A. Wilkinson (2009). "The C1-generic diffeomorphism has trivial centralizer". Publications mathématiques de l'IHÉS 109: 185–244.
- ^ Warwick Tucker (2002). "A Rigorous ODE Solver and Smale's 14th Problem". Foundations of Computational Mathematics 2 (1): 53–117. doi:10.1007/s002080010018. http://www.math.cornell.edu/~warwick/main/rodes/JFoCM.pdf.
- ^ 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. http://personales.unican.es/beltranc/archivos/AffSmale17JAMS.pdf.
- ^ Felipe Cucker, Peter Bürgisser (2010). "Solving Polynomial Equations in Smoothed Polynomial Time and a Near Solution to Smale's 17th Problem". Proc. 42nd ACM Symposium on Theory of Computing. arXiv:0909.2114.
[edit] See Also
- Weisstein, Eric W., "Smale's Problems" from MathWorld.
