斯梅爾問題

斯梅爾問題(英語:Smale's problems)是美國數學家史蒂芬·斯梅爾於1998年提出的18個當時未解決的數學問題[1][2]時任國際數學聯盟副主席、俄國數學家弗拉基米爾·阿諾爾德當時參照20世紀初希爾伯特的23個問題而向世界上的主要數學家徵集面向21世紀的數學問題,斯梅爾便是在此背景下提出了斯梅爾問題,作為對阿諾爾德的答覆。

斯梅爾的18個問題

# 問題 進展
1 黎曼猜想 未解決,參見希爾伯特第八問題
2 龐加萊猜想 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 單變量多項式的整數零點[6][7] 未解決
5 丟番圖曲線高度的界 未解決
6 天體力學中相對平衡狀態數量的有限性 2012年,A. Albouy與V. Kaloshin證明五個天體相對平衡狀態[8]
7 二維球面上點的分布 未解決
8 將動力學引入經濟學理論 未解決
9 找出一個強多項式時間的線性規劃解法 未解決
10 封閉引理英語Pugh's closing lemma M. Asaoka、K. Irie部分解決[9]
11 一維動力系統一般是否為雙曲型? 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 希爾伯特第十六問題 未解決
14 洛倫茲吸引子 2001年,沃里克·塔克爾英語Warwick Tucker(Warwick Tucker)證明完成[12]
15 納維-斯托克斯方程 未解決
16 雅可比猜想 未解決
17 多項式方程組 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 智能的極限[18] 未解決

參見

參考文獻

  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. Arnold, V. I.; Atiyah, M.; Lax, P.; Mazur, B. (編). 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  |class=被忽略 (幫助). 
  4. ^ Perelman, Grigori. Ricci flow with surgery on three-manifolds. 2003. arXiv:math.DG/0303109  |class=被忽略 (幫助). 
  5. ^ Perelman, Grigori. Finite extinction time for the solutions to the Ricci flow on certain three-manifolds. 2003. arXiv:math.DG/0307245  |class=被忽略 (幫助). 
  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 [2018-09-30]. doi:10.1007/s10208-005-0211-0. (原始內容存檔 (PDF)於2020-06-07). 
  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 [2018-09-30]. Bibcode:2009JAMS...22..363B. doi:10.1090/s0894-0347-08-00630-9. (原始內容存檔 (PDF)於2020-06-07). 
  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頁面存檔備份,存於網際網路檔案館) Friday, February 03, 2006 Tucson - Day 3 - Interview with Steve Smale