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Smale's problems

From Wikipedia, the free encyclopedia
18 unsolved problems in mathematics (published 1999)

Smale's problems is a list of eighteenunsolved problems in mathematics proposed bySteve Smale in 1998[1] and republished in 1999.[2] Smale composed this list in reply to a request fromVladimir Arnold, then vice-president of theInternational Mathematical Union, who asked several mathematicians to propose a list of problems for the 21st century. Arnold's inspiration came from the list ofHilbert's problems that had been published at the beginning of the 20th century.

Table of problems

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ProblemBrief explanationStatusYear Solved
1stRiemann hypothesis: Thereal part of every non-trivialzero of theRiemann zeta function is 1/2. (see alsoHilbert's eighth problem)Unresolved.–
2ndPoincaré conjecture: Everysimply connected, closed 3-manifold ishomeomorphic to the3-sphere.Resolved. Result: Yes, Proved byGrigori Perelman usingRicci flow.[3][4][5]2003
3rdP versus NP problem: For all problems for which analgorithm canverify a given solution quickly (that is, inpolynomial time), can an algorithm alsofind that solution quickly?Unresolved.–
4thShub–Smale tau-conjecture on theinteger zeros of apolynomial of one variable[6][7]Unresolved.–
5thCan one decide if aDiophantine equationƒ(x, y) = 0 (inputƒ ∈ Z{\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 inexponential time?Unresolved.–
6thIs the number of relative equilibria (central configurations) finite in then-body problem ofcelestial mechanics, for any choice of positivereal numbersm1, ..., mn as the masses?Partially resolved. Proved for almost all systems of five bodies by A. Albouy and V. Kaloshin in 2012.[8]2012
7thAlgorithm for finding set of(x1,...,xN){\displaystyle (x_{1},...,x_{N})} such that thefunction:VN(x)=∑1≤i<j≤Nlog⁡1‖xi−xj‖{\displaystyle V_{N}(x)=\sum _{1\leq i<j\leq N}\log {\frac {1}{\|x_{i}-x_{j}\|}}} is minimized for a distribution ofN points on a2-sphere. This is related to theThomson problem.Unresolved.–
8thExtend themathematical model ofgeneral equilibrium theory to include price adjustmentsGjerstad (2013)[9] extends the deterministic model of price adjustment by Hahn and Negishi (1962)[10] to astochastic 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.
9thThelinear programming problem: Find astrongly-polynomial time algorithm which for givenmatrixA ∈ Rm×n andb ∈ Rm decides whether there existsx ∈ Rn withAx ≥ b.Unresolved.–
10thPugh's closing lemma (higher order of smoothness)Partially resolved. Proved for Hamiltoniandiffeomorphisms of closed surfaces by M. Asaoka and K. Irie in 2016.[12]2016
11thIs one-dimensional dynamics generally hyperbolic?

(a) Can acomplex polynomialT be approximated by one of the samedegree with the property that every critical point tends to a periodic sink under iteration?

(b) Can a smooth mapT : [0,1] → [0,1] beC r approximated by one which is hyperbolic, for allr > 1?		(a) Unresolved, even in the simplest parameter space of polynomials, theMandelbrot set.

(b) Resolved. Proved by Kozlovski, Shen and van Strien.[13]		2007
12thFor aclosed manifoldM{\displaystyle M} and anyr≥1{\displaystyle r\geq 1} letDiffr(M){\displaystyle \mathrm {Diff} ^{r}(M)} be thetopological group ofCr{\displaystyle C^{r}}diffeomorphisms ofM{\displaystyle M} onto itself. Given arbitraryA∈Diffr(M){\displaystyle A\in \mathrm {Diff} ^{r}(M)}, is it possible to approximate it arbitrary well by suchT∈Diffr(M){\displaystyle T\in \mathrm {Diff} ^{r}(M)} that it commutes only with its iterates?

In other words, is the subset of all diffeomorphisms whosecentralizers are trivial dense inDiffr(M){\displaystyle \mathrm {Diff} ^{r}(M)}?

Partially resolved. Solved in theC1topology by Christian Bonatti, Sylvain Crovisier andAmie Wilkinson[14] in 2009. Still open in theC r topology forr > 1.2009
13thHilbert's 16th problem: Describe relative positions of ovals originating from arealalgebraic curve and aslimit cycles of a polynomialvector field on the plane.Unresolved, even for algebraic curves of degree 8.–
14thDo the properties of theLorenz attractor exhibit that of a strange attractor?Resolved. Result: Yes, solved byWarwick Tucker using acomputer-assisted proof combined with normal form techniques.[15]2002
15thDo theNavier–Stokes equations inR3 always have aunique smooth solution that extends for all time?Unresolved.–
16thJacobian conjecture: If theJacobian determinant ofF is a non-zero constant andk hascharacteristic 0, thenF has aninverse functionG : kN → kN, andG isregular (in the sense that its components are polynomials).Unresolved.–
17thSolvingpolynomial equations inpolynomial time in the average caseResolved. C. Beltrán and L. M. Pardo found two uniform probabilistic algorithms (averageLas Vegas algorithm) for Smale's 17th problem[16][17][18]

F. Cucker andP. Bürgisser made thesmoothed analysis of a probabilistic algorithmà la Beltrán-Pardo and then exhibited a deterministic algorithm running in timeNO(log⁡log⁡N){\displaystyle N^{O(\log \log N)}}.[19]

Finally,P. Lairez found an alternative method to de-randomize the algorithmà 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
18thLimits ofintelligence (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 thethree-sphere aminimal set (Gottschalk's conjecture)?
  3. Is anAnosov diffeomorphism of acompact manifold topologically the same as theLie group model of John Franks?

See also

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References

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  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 (3):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). "AC∞ 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.PMC 11226777.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.PMID 35294283.
  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.
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