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斯梅尔问题英语:Smale's problems)是美国数学家斯蒂芬·斯梅尔于1998年提出的18个当时未解决的数学问题[1][2]时任国际数学联盟副主席、俄国数学家弗拉基米尔·阿诺尔德当时参照20世纪初希尔伯特的23个问题而向世界上的主要数学家征集面向21世纪的数学问题,斯梅尔便是在此背景下提出了斯梅尔问题,作为对阿诺尔德的答复。

斯梅尔的18个问题编辑

# 問題 進展
1 黎曼猜想 未解決
2 庞加莱猜想 2003年,格里戈里·佩雷爾曼證明完成[3][4][5]
3 P=NP是否成立? 未解決
4 单变量多项式的整数零点[6][7] 未解決
5 丢番图曲线高度的界 未解決
6 天体力学中相对平衡状态数量的有限性 2012年,A. Albouy與V. Kaloshin證明五個天體相对平衡狀態[8]
7 二维球面上点的分布 未解決
8 将动力学引入经济学理论 未解決
9 找出一个强多项式时间的线性规划解法 未解決
10 封闭引理英语Pugh's closing lemma M. Asaoka、K. Irie部分解決[9]
11 一维动力系统一般是否为双曲型? Kozlovski、Shen、van Strien部分解決[10]
12 微分同胚中心化子 Christian Bonatti、Sylvain Crovisier、Amie Wilkinson部分解決[11]
13 希尔伯特第十六问题 未解決
14 洛伦兹吸引子 2001年,沃里克·塔克爾英语Warwick Tucker(Warwick Tucker)證明完成[12]
15 纳维-斯托克斯方程 未解決
16 雅可比猜想 未解決
17 多项式方程组 已解決,C. Beltrán、L. M. Pardo、F. Cucker、P. Bürgisser、P. Lairez各自完成部分證明[13][14][15][16][17]
18 智能的极限[18] 未解決

参见编辑

参考文献编辑

  1. ^ Smale, Steve. Mathematical Problems for the Next Century. Mathematical Intelligencer. 1998, 20 (2): 7–15doi=10.1007/bf03025291. 
  2. ^ Smale, Steve. Mathematical problems for the next century. (编) Arnold, V. I.; Atiyah, M.; Lax, P.; Mazur, B. Mathematics: frontiers and perspectives. American Mathematical Society. 1999: 271–294. ISBN 0821820702. 
  3. ^ Perelman, Grigori. The entropy formula for the Ricci flow and its geometric applications. 2002. arXiv:math.DG/0211159 [math.DG]. 
  4. ^ Perelman, Grigori. Ricci flow with surgery on three-manifolds. 2003. arXiv:math.DG/0303109 [math.DG]. 
  5. ^ Perelman, Grigori. Finite extinction time for the solutions to the Ricci flow on certain three-manifolds. 2003. arXiv:math.DG/0307245 [math.DG]. 
  6. ^ Shub, Michael; Smale, Steve. On the intractability of Hilbert’s Nullstellensatz and an algebraic version of "NP≠P?". Duke Math. J. 1995, 81: 47–54. Zbl 0882.03040. doi:10.1215/S0012-7094-95-08105-8. 
  7. ^ Bürgisser, Peter. Completeness and reduction in algebraic complexity theory. Algorithms and Computation in Mathematics 7. Berlin: Springer-Verlag. 2000: 141. ISBN 3-540-66752-0. Zbl 0948.68082. 
  8. ^ Albouy, A.; Kaloshin, V. Finiteness of central configurations of five bodies in the plane. Annals of Mathematics. 2012, 176: 535–588. doi:10.4007/annals.2012.176.1.10. 
  9. ^ Asaoka, M.; Irie, K. A C closing lemma for Hamiltonian diffeomorphisms of closed surfaces. Geometric and Functional Analysis. 2016, 26: 1245–1254. doi:10.1007/s00039-016-0386-3. 
  10. ^ Kozlovski, O.; Shen, W.; van Strien, S. Density of hyperbolicity in dimension one. Annals of Mathematics. 2007, 166: 145–182. doi:10.4007/annals.2007.166.145. 
  11. ^ Bonatti, C.; Crovisier, S.; Wilkinson, A. The C1-generic diffeomorphism has trivial centralizer. Publications Mathématiques de l'IHÉS. 2009, 109: 185–244. arXiv:0804.1416. doi:10.1007/s10240-009-0021-z. 
  12. ^ Tucker, Warwick. A Rigorous ODE Solver and Smale's 14th Problem (PDF). Foundations of Computational Mathematics. 2002, 2 (1): 53–117. doi:10.1007/s002080010018. 
  13. ^ Beltrán, Carlos; Pardo, Luis Miguel. On Smale's 17th Problem: A Probabilistic Positive answer (PDF). Foundations of Computational Mathematics. 2008, 8 (1): 1–43. doi:10.1007/s10208-005-0211-0. 
  14. ^ Beltrán, Carlos; Pardo, Luis Miguel. Smale's 17th Problem: Average Polynomial Time to compute affine and projective solutions (PDF). Journal of the American Mathematical Society. 2009, 22: 363–385. Bibcode:2009JAMS...22..363B. doi:10.1090/s0894-0347-08-00630-9. 
  15. ^ Cucker, Felipe; Bürgisser, Peter. On a problem posed by Steve Smale. Annals of Mathematics. 2011, 174 (3): 1785–1836. doi:10.4007/annals.2011.174.3.8. 
  16. ^ Lairez, Pierre. A deterministic algorithm to compute approximate roots of polynomial systems in polynomial average time. Foundations of Computational Mathematics. 2016,. to appear. 
  17. ^ Shub, Michael; Smale, Stephen. Complexity of Bézout's theorem. I. Geometric aspects. J. Amer. Math. Soc. 1993, 6 (2): 459–501. doi:10.2307/2152805. .
  18. ^ http://recursed.blogspot.jp/2006/02/tucson-day-3-interview-with-steve.html Friday, February 03, 2006 Tucson - Day 3 - Interview with Steve Smale