Smale's problems

Smale's problems is 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

Problem	Brief explanation	Status	Year Solved
1st	Riemann hypothesis: The real part of every non-trivial zero of the Riemann zeta function is 1/2. (see also Hilbert's eighth problem)	Unresolved.	–
2nd	Poincaré conjecture: Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.	Resolved. Result: Yes, Proved by Grigori Perelman using Ricci flow. [3] [4] [5]	2003
3rd	P versus NP problem: For all problems for which an algorithm can verify a given solution quickly (that is, in polynomial time), can an algorithm also find that solution quickly?	Unresolved.	–
4th	Shub–Smale tau-conjecture on the integer zeros of a polynomial of one variable [6] [7]	Unresolved.	–
5th	Can one decide if a Diophantine equation ƒ(x, y) = 0 (input ƒ ∈ ${\textstyle \mathbb {Z} }$ [u, v ]) has an integer solution, (x, y), in time (2s)c for some universal constant c? That is, can the problem be decided in exponential time?	Unresolved.	–
6th	Is the number of relative equilibria (central configurations) finite in the n-body problem of celestial mechanics, for any choice of positive real numbers m1, ..., mn as the masses?	Partially resolved. Proved for almost all systems of five bodies by A. Albouy and V. Kaloshin in 2012. [8]	2012
7th	Algorithm for finding set of ${\displaystyle (x_{1},...,x_{N})}$ such that the function: ${\displaystyle V_{N}(x)=\sum _{1\leq i<j\leq N}\log {\frac {1}{\|x_{i}-x_{j}\|}}}$ is minimized for a distribution of N points on a 2-sphere. This is related to the Thomson problem.	Unresolved.	–
8th	Extend the mathematical model of general equilibrium theory to include price adjustments	Gjerstad (2013) [9] extends the deterministic model of price adjustment by Hahn and Negishi (1962) [10] to a stochastic model and shows that when the stochastic model is linearized around the equilibrium the result is the autoregressive price adjustment model used in applied econometrics. He then tests the model with price adjustment data from a general equilibrium experiment. The model performs well in a general equilibrium experiment with two commodities. Lindgren (2022) [11] provides a dynamic programming model for general equilibrium with price adjustments, where price dynamics are given by a Hamilton-Jacobi-Bellman partial differerential equation. Good Lyapunov stability conditions are provided as well.
9th	The linear programming problem: Find a strongly-polynomial time algorithm which for given matrix A ∈ Rm×n and b ∈ Rm decides whether there exists x ∈ Rn with Ax ≥ b.	Unresolved.	–
10th	Pugh's closing lemma (higher order of smoothness)	Partially resolved. Proved for Hamiltonian diffeomorphisms of closed surfaces by M. Asaoka and K. Irie in 2016. [12]	2016
11th	Is one-dimensional dynamics generally hyperbolic? (a) Can a complex polynomial T be approximated by one of the same degree with the property that every critical point tends to a periodic sink under iteration? (b) Can a smooth map T : [0,1] → [0,1] be C r approximated by one which is hyperbolic, for all r > 1?	(a) Unresolved, even in the simplest parameter space of polynomials, the Mandelbrot set. (b) Resolved. Proved by Kozlovski, Shen and van Strien. [13]	2007
12th	For a closed manifold ${\displaystyle M}$ and any ${\displaystyle r\geq 1}$ let ${\displaystyle \mathrm {Diff} ^{r}(M)}$ be the topological group of ${\displaystyle C^{r}}$ diffeomorphisms of ${\displaystyle M}$ onto itself. Given arbitrary ${\displaystyle A\in \mathrm {Diff} ^{r}(M)}$, is it possible to approximate it arbitrary well by such ${\displaystyle T\in \mathrm {Diff} ^{r}(M)}$ that it commutes only with its iterates? In other words, is the subset of all diffeomorphisms whose centralizers are trivial dense in ${\displaystyle \mathrm {Diff} ^{r}(M)}$?	Partially resolved. Solved in the C1 topology by Christian Bonatti, Sylvain Crovisier and Amie Wilkinson [14] in 2009. Still open in the C r topology for r > 1.	2009
13th	Hilbert's 16th problem: Describe relative positions of ovals originating from a real algebraic curve and as limit cycles of a polynomial vector field on the plane.	Unresolved, even for algebraic curves of degree 8.	–
14th	Do the properties of the Lorenz attractor exhibit that of a strange attractor?	Resolved. Result: Yes, solved by Warwick Tucker using a computer-assisted proof combined with normal form techniques. [15]	2002
15th	Do the Navier–Stokes equations in R3 always have a unique smooth solution that extends for all time?	Unresolved.	–
16th	Jacobian conjecture: If the Jacobian determinant of F is a non-zero constant and k has characteristic 0, then F has an inverse function G : kN → kN, and G is regular (in the sense that its components are polynomials).	Unresolved.	–
17th	Solving polynomial equations in polynomial time in the average case	Resolved. C. Beltrán and L. M. Pardo found two uniform probabilistic algorithms (average Las Vegas algorithm) for Smale's 17th problem [16] [17] [18] F. Cucker and P. Bürgisser made the smoothed analysis of a probabilistic algorithm à la Beltrán-Pardo and then exhibited a deterministic algorithm running in time ${\displaystyle N^{O(\log \log N)}}$. [19] Finally, P. Lairez found an alternative method to de-randomize the algorithm à la Beltrán-Pardo and thus found a deterministic algorithm which runs in average polynomial time. [20] All these works follow Shub and Smale's foundational work (the "Bezout series") started in [21]	2008-2016
18th	Limits of intelligence (it talks about the fundamental problems of intelligence and learning, both from the human and machine side) [22]	Some recent authors have claimed results, including the unlimited nature of human intelligence [23] and limitations on neural-network-based artificial intelligence [24]	–

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

• Millennium Prize Problems
• Simon problems

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. doi:10.2307/1909889. JSTOR   1909889.
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 (12): e32465. arXiv: 2310.10792 . Bibcode:2024Heliy..1032465A. doi: 10.1016/j.heliyon.2024.e32465 . PMID   38975068.
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 . 119 (12): e2107151119. arXiv: 2101.08286 . Bibcode:2022PNAS..11907151C. 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.