斯梅爾問題
斯梅爾問題（英語：Smale's problems）是美國數學家斯蒂芬·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 | 二維球面上點的分佈 | 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 | 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 | 微分同胚的中心化子 | 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（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 | 解多項式方程組 | 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
- ^ Smale, Steve. Mathematical Problems for the Next Century. Mathematical Intelligencer. 1998, 20 (2): 7–15doi=10.1007/bf03025291.
- ^ 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.
- ^ Perelman, Grigori. The entropy formula for the Ricci flow and its geometric applications. 2002. arXiv:math.DG/0211159
|class=被忽略 (幫助). - ^ Perelman, Grigori. Ricci flow with surgery on three-manifolds. 2003. arXiv:math.DG/0303109
|class=被忽略 (幫助). - ^ 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). - ^ 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.
- ^ 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.
- ^ 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.
- ^ 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.
- ^ 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.
- ^ 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.
- ^ 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.
- ^ 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）.
- ^ 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）.
- ^ 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.
- ^ Lairez, Pierre. A deterministic algorithm to compute approximate roots of polynomial systems in polynomial average time. Foundations of Computational Mathematics. 2016,. to appear.
- ^ 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..
- ^ http://recursed.blogspot.jp/2006/02/tucson-day-3-interview-with-steve.html （頁面存檔備份，存於互聯網檔案館） Friday, February 03, 2006 Tucson - Day 3 - Interview with Steve Smale
