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Smale's problems

Smale's problems are a list of eighteen unsolved problems in mathematics proposed by Steve Smale in 1998[1] and 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.

Table of problems

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:"[25][26]

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

See also

References

  1. ^ Smale, Steve (1998). "Mathematical Problems for the Next Century". Mathematical Intelligencer. 20 (2): 7–15. CiteSeerX 10.1.1.35.4101. doi:10.1007/bf03025291. S2CID 1331144.
  2. ^ Smale, Steve (1999). "Mathematical problems for the next century". In Arnold, V. I.; Atiyah, M.; Lax, P.; Mazur, B. (eds.). Mathematics: frontiers and perspectives. American Mathematical Society. pp. 271–294. ISBN 978-0-8218-2070-4.
  3. ^ Perelman, Grigori (2002). "The entropy formula for the Ricci flow and its geometric applications". arXiv:math.DG/0211159.
  4. ^ Perelman, Grigori (2003). "Ricci flow with surgery on three-manifolds". arXiv:math.DG/0303109.
  5. ^ Perelman, Grigori (2003). "Finite extinction time for the solutions to the Ricci flow on certain three-manifolds". arXiv:math.DG/0307245.
  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. Vol. 7. Berlin: Springer-Verlag. p. 141. ISBN 978-3-540-66752-0. 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. ^ Gjerstad, Steven (2013). "Price Dynamics in an Exchange Economy". Economic Theory. 52 (2): 461–500. CiteSeerX 10.1.1.415.3888. doi:10.1007/s00199-011-0651-5. S2CID 15322190.
  10. ^ Hahn, Frank (1962). "A theorem on non-tatonnement stability". Econometrica. 30: 463–469.
  11. ^ Lindgren, Jussi (2022). "General Equilibrium with Price Adjustments—A Dynamic Programming Approach". Analytics. 1 (1): 27–34. doi:10.3390/analytics1010003.
  12. ^ Asaoka, M.; Irie, K. (2016). "A C∞ closing lemma for Hamiltonian diffeomorphisms of closed surfaces". Geometric and Functional Analysis. 26 (5): 1245–1254. doi:10.1007/s00039-016-0386-3. S2CID 119732514.
  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. arXiv:0804.1416. doi:10.1007/s10240-009-0021-z. S2CID 16212782.
  15. ^ Tucker, Warwick (2002). "A Rigorous ODE Solver and Smale's 14th Problem" (PDF). Foundations of Computational Mathematics. 2 (1): 53–117. CiteSeerX 10.1.1.545.3996. doi:10.1007/s002080010018. S2CID 353254.
  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. CiteSeerX 10.1.1.211.3321. doi:10.1007/s10208-005-0211-0. S2CID 11528635.
  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 (2): 363–385. Bibcode:2009JAMS...22..363B. doi:10.1090/s0894-0347-08-00630-9.
  18. ^ Beltrán, Carlos; Pardo, Luis Miguel (2011). "Fast Linear Homotopy to Find Approximate Zeros of Polynomial Systems". Foundations of Computational Mathematics. 11 (1): 95–129. doi:10.1007/s10208-010-9078-9.
  19. ^ Cucker, Felipe; Bürgisser, Peter (2011). "On a problem posed by Steve Smale". Annals of Mathematics. 174 (3): 1785–1836. arXiv:0909.2114. doi:10.4007/annals.2011.174.3.8. S2CID 706015.
  20. ^ Lairez, Pierre (2016). "A deterministic algorithm to compute approximate roots of polynomial systems in polynomial average time". Foundations of Computational Mathematics. to appear (5): 1265–1292. arXiv:1507.05485. doi:10.1007/s10208-016-9319-7. S2CID 8333924.
  21. ^ 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. JSTOR 2152805..
  22. ^ "Tucson - Day 3 - Interview with Steve Smale". Recursivity. February 3, 2006.
  23. ^ Acharjee, S.; Gogoi, U. (2024). "The limit of human intelligence". Heliyon. 10: e32465. arXiv:2310.10792. doi:10.1016/j.heliyon.2024.e32465.
  24. ^ Colbroke, M. J.; Vegard, A.; Hansen, A. C. (2022). "The difficulty of computing stable and accurate neural networks: On the barriers of deep learning and Smale's 18th problem". Proceedings of the National Academy of Sciences. 12: e2107151119. arXiv:2101.08286. doi:10.1073/pnas.2107151119.
  25. ^ Smale, Steve. "Mathematical Problems for the Next Century" (PDF).
  26. ^ Smale, Steve. "Mathematical problems for the next century, Mathematics: Frontiers and perspectives". American Mathematical Society, Providence, RI: 271–294.