Smale's problems

<h2 id="table-of-problems">Table of problems</h2>

<table><tbody><tr><th>Problem</th><th>Brief explanation</th><th>Status</th><th>Year Solved</th></tr><tr><td>1st</td><td><a href="/facts/Riemann_hypothesis/Td8Jq5r0">Riemann hypothesis</a>: The <a href="/facts/Complex_number/2w7mqI7e">real part</a> of every non-trivial <a href="/facts/Zero_of_a_function/mlK3dhoT">zero</a> of the <a href="/facts/Riemann_zeta_function/MnfRu7lY">Riemann zeta function</a> is 1/2. (see also <a href="/facts/Hilbert%2527s_eighth_problem/qbldYArn">Hilbert's eighth problem</a>)</td><td>Unresolved.</td><td>–</td></tr><tr><td>2nd</td><td><a href="/facts/Poincar%25C3%25A9_conjecture/mxk2Pju3">Poincaré conjecture</a>: Every <a href="/facts/Simply_connected/xFlta63J">simply connected</a>, closed 3-<a href="/facts/Manifold/rXPERJtu">manifold</a> is <a href="/facts/Homeomorphic/jMfWg3Mn">homeomorphic</a> to the <a href="/facts/3-sphere/u67nImpY">3-sphere</a>.</td><td>Resolved. Result: Yes, Proved by <a href="/facts/Grigori_Perelman/SmISWW7x">Grigori Perelman</a> using <a href="/facts/Ricci_flow/J6rQd3dm">Ricci flow</a>.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a><a class="footnote-ref" id="fnref:4" href="#fn:4">4</a><a class="footnote-ref" id="fnref:5" href="#fn:5">5</a></td><td>2003</td></tr><tr><td>3rd</td><td><a href="/facts/P_versus_NP_problem/xoC7Ruda">P versus NP problem</a>: For all problems for which an <a href="/facts/Algorithm/fnl5NmRt">algorithm</a> can verify a given solution quickly (that is, in <a href="/facts/Polynomial_time/77T62gmf">polynomial time</a>), can an algorithm also find that solution quickly?</td><td>Unresolved.</td><td>–</td></tr><tr><td>4th</td><td>Shub–Smale tau-conjecture on the <a href="/facts/Integer/PUwwrawx">integer</a> zeros of a <a href="/facts/Polynomial/Lzak8VVx">polynomial</a> of one variable<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a><a class="footnote-ref" id="fnref:7" href="#fn:7">7</a></td><td>Unresolved.</td><td>–</td></tr><tr><td>5th</td><td>Can one decide if a <a href="/facts/Diophantine_equation/mYaHp7oX">Diophantine equation</a> ƒ(x, y) = 0 (input ƒ ∈  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 in <a href="/facts/Exponential_time/77T62gmf">exponential time</a>?</td><td>Unresolved.</td><td>–</td></tr><tr><td>6th</td><td>Is the number of relative equilibria (<a href="/facts/Central_configuration/DfxMTNsa">central configurations</a>) finite in the <a href="/facts/N-body_problem/5goGsW5x">n-body problem</a> of <a href="/facts/Celestial_mechanics/FA3uR78J">celestial mechanics</a>, for any choice of positive <a href="/facts/Real_number/R02gw5Pb">real numbers</a> m1, ..., mn as the masses?</td><td>Partially resolved. Proved for almost all systems of five bodies by A. Albouy and V. Kaloshin in 2012.<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a></td><td>2012</td></tr><tr><td>7th</td><td>Algorithm for finding set of ( x 1 , . . . , x N ) {\displaystyle (x_{1},...,x_{N})} such that the <a href="/facts/Function_(mathematics)/CdReEKCh">function</a>: V N ( x ) = ∑ 1 ≤ i < j ≤ N log ⁡ 1 ‖ x i − x j ‖ {\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 <a href="/facts/2-sphere/jywYLNM2">2-sphere</a>. This is related to the <a href="/facts/Thomson_problem/eyDlFwy4">Thomson problem</a>.</td><td>Unresolved.</td><td>–</td></tr><tr><td>8th</td><td>Extend the <a href="/facts/Mathematical_model/CGhyuxgz">mathematical model</a> of <a href="/facts/General_equilibrium_theory/WY0zX5UD">general equilibrium theory</a> to include price adjustments</td><td>Gjerstad (2013)<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a> extends the deterministic model of price adjustment by Hahn and Negishi (1962)<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a> 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)<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a> 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.</td></tr><tr><td>9th</td><td>The <a href="/facts/Linear_programming/GduXFQxT">linear programming</a> problem: Find a <a href="/facts/Strongly-polynomial_time/tLJqbAxJ">strongly-polynomial time</a> algorithm which for given <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrix</a> A ∈ Rm×n and b ∈ Rm decides whether there exists x ∈ Rn with Ax ≥ b.</td><td>Unresolved.</td><td>–</td></tr><tr><td>10th</td><td><a href="/facts/Pugh%2527s_closing_lemma/TgW5L7I9">Pugh's closing lemma</a> (higher order of smoothness)</td><td>Partially resolved. Proved for Hamiltonian <a href="/facts/Diffeomorphism/855N5YYj">diffeomorphisms</a> of closed surfaces by M. Asaoka and K. Irie in 2016.<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a></td><td>2016</td></tr><tr><td>11th</td><td>Is one-dimensional dynamics generally hyperbolic?(a) Can a <a href="/facts/Complex_number/2w7mqI7e">complex</a> polynomial T be approximated by one of the same <a href="/facts/Degree_of_a_polynomial/Bf8vEIhf">degree</a> 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?</td><td>(a) Unresolved, even in the simplest parameter space of polynomials, the <a href="/facts/Mandelbrot_set/2NVUgrVG">Mandelbrot set</a>.(b) Resolved. Proved by Kozlovski, Shen and van Strien.<a class="footnote-ref" id="fnref:13" href="#fn:13">13</a></td><td>2007</td></tr><tr><td>12th</td><td>For a <a href="/facts/Closed_manifold/jMG6G6Bl">closed manifold</a> M {\displaystyle M} and any r ≥ 1 {\displaystyle r\geq 1} let D i f f r ( M ) {\displaystyle \mathrm {Diff} ^{r}(M)} be the <a href="/facts/Topological_group/cAgNMMRU">topological group</a> of C r {\displaystyle C^{r}} <a href="/facts/Diffeomorphism/855N5YYj">diffeomorphisms</a> of M {\displaystyle M} onto itself. Given arbitrary A ∈ D i f f r ( M ) {\displaystyle A\in \mathrm {Diff} ^{r}(M)} , is it possible to approximate it arbitrary well by such T ∈ D i f f r ( M ) {\displaystyle T\in \mathrm {Diff} ^{r}(M)} that it commutes only with its iterates?In other words, is the subset of all diffeomorphisms whose <a href="/facts/Centralizer_and_normalizer/AKhrEUGR">centralizers</a> are trivial dense in D i f f r ( M ) {\displaystyle \mathrm {Diff} ^{r}(M)} ?</td><td>Partially resolved. Solved in the C1 <a href="/facts/Topological_space/v2ut7CbK">topology</a> by Christian Bonatti, Sylvain Crovisier and <a href="/facts/Amie_Wilkinson/WC4VKMgC">Amie Wilkinson</a><a class="footnote-ref" id="fnref:14" href="#fn:14">14</a> in 2009. Still open in the C r topology for r > 1.</td><td>2009</td></tr><tr><td>13th</td><td><a href="/facts/Hilbert%2527s_sixteenth_problem/SUR6JBKB">Hilbert's 16th problem</a>: Describe relative positions of ovals originating from a <a href="/facts/Real_number/R02gw5Pb">real</a> <a href="/facts/Algebraic_curve/LnWsPmDP">algebraic curve</a> and as <a href="/facts/Limit_cycle/LmM9EVoA">limit cycles</a> of a polynomial <a href="/facts/Vector_field/ccFNuPPp">vector field</a> on the plane.</td><td>Unresolved, even for algebraic curves of degree 8.</td><td>–</td></tr><tr><td>14th</td><td>Do the properties of the <a href="/facts/Lorenz_attractor/GsSfHShq">Lorenz attractor</a> exhibit that of a strange attractor?</td><td>Resolved. Result: Yes, solved by <a href="/facts/Warwick_Tucker/w47LIMUT">Warwick Tucker</a> using a <a href="/facts/Computer-assisted_proof/Dnk4uESy">computer-assisted proof</a> combined with normal form techniques.<a class="footnote-ref" id="fnref:15" href="#fn:15">15</a></td><td>2002</td></tr><tr><td>15th</td><td>Do the <a href="/facts/Navier%25E2%2580%2593Stokes_equations/48pQugEa">Navier–Stokes equations</a> in R3 always have a <a href="/facts/Navier%25E2%2580%2593Stokes_existence_and_smoothness/5aDcIPCw">unique smooth solution</a> that extends for all time?</td><td>Unresolved.</td><td>–</td></tr><tr><td>16th</td><td><a href="/facts/Jacobian_conjecture/J2NbRNk4">Jacobian conjecture</a>: If the <a href="/facts/Jacobian_matrix_and_determinant/0tNv6o8j">Jacobian determinant</a> of F is a non-zero constant and k has <a href="/facts/Characteristic_(algebra)/D2VzcQaG">characteristic</a> 0, then F has an <a href="/facts/Inverse_function/Tg5nnXlu">inverse function</a> G : kN → kN, and G is <a href="/facts/Regular_map_(algebraic_geometry)/5jBzIgps">regular</a> (in the sense that its components are polynomials).</td><td>Unresolved.</td><td>–</td></tr><tr><td>17th</td><td>Solving <a href="/facts/Polynomial_equation/1QauIGxs">polynomial equations</a> in <a href="/facts/Polynomial_time/77T62gmf">polynomial time</a> in the average case</td><td>Resolved. C. Beltrán and L. M. Pardo found two uniform probabilistic algorithms (average <a href="/facts/Las_Vegas_algorithm/K43MRwQz">Las Vegas algorithm</a>) for Smale's 17th problem<a class="footnote-ref" id="fnref:16" href="#fn:16">16</a><a class="footnote-ref" id="fnref:17" href="#fn:17">17</a><a class="footnote-ref" id="fnref:18" href="#fn:18">18</a> <a href="/facts/Felipe_Cucker/vVy53r0X">F. Cucker</a> and <a href="/facts/Peter_B%25C3%25BCrgisser/cXvrJvrc">P. Bürgisser</a> made the <a href="/facts/Smoothed_analysis/aRpTTGDF">smoothed analysis</a> of a probabilistic algorithm à la Beltrán-Pardo and then exhibited a deterministic algorithm running in time N O ( log ⁡ log ⁡ N ) {\displaystyle N^{O(\log \log N)}} .<a class="footnote-ref" id="fnref:19" href="#fn:19">19</a>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.<a class="footnote-ref" id="fnref:20" href="#fn:20">20</a>All these works follow Shub and Smale's foundational work (the "Bezout series") started in <a class="footnote-ref" id="fnref:21" href="#fn:21">21</a></td><td>2008-2016</td></tr><tr><td>18th</td><td>Limits of <a href="/facts/Intelligence/pRmAeRsw">intelligence</a> (it talks about the fundamental problems of intelligence and learning, both from the human and machine side)<a class="footnote-ref" id="fnref:22" href="#fn:22">22</a></td><td>Some recent authors have claimed results, including the unlimited nature of human intelligence<a class="footnote-ref" id="fnref:23" href="#fn:23">23</a> and limitations on neural-network-based artificial intelligence<a class="footnote-ref" id="fnref:24" href="#fn:24">24</a></td><td>–</td></tr></tbody></table>
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:"<a class="footnote-ref" id="fnref:25" href="#fn:25">25</a><a class="footnote-ref" id="fnref:26" href="#fn:26">26</a>

<ol><li><a href="/facts/Mean_value_problem/z5uLyTgN">Mean value problem</a></li>
<li>Is the <a href="/facts/Three-sphere/u67nImpY">three-sphere</a> a minimal set (Gottschalk's conjecture)?</li>
<li>Is an <a href="/facts/Anosov_diffeomorphism/8SudeRhj">Anosov diffeomorphism</a> of a <a href="/facts/Compact_manifold/jMG6G6Bl">compact manifold</a> topologically the same as the <a href="/facts/Lie_group/a9jCQyjF">Lie group</a> model of John Franks?</li></ol>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Millennium_Prize_Problems/WWSB5ddp">Millennium Prize Problems</a></li>
<li><a href="/facts/Simon_problems/ylC91CLP">Simon problems</a></li>
<li><a href="/facts/Taniyama%2527s_problems/6oqKru8S">Taniyama's problems</a></li>
<li><a href="/facts/Hilbert%2527s_problems/HaQWIxvV">Hilbert's problems</a></li>
<li><a href="/facts/Thurston%2527s_24_questions/Nl0upss5">Thurston's 24 questions</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn: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. <a href="/wiki/CiteSeerX_(identifier)" target="_blank">/wiki/CiteSeerX_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="978-0-8218-2070-4" target="_blank">978-0-8218-2070-4</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Perelman, Grigori (2002). "The entropy formula for the Ricci flow and its geometric applications". arXiv:math.DG/0211159. <a href="/wiki/Grigori_Perelman" target="_blank">/wiki/Grigori_Perelman</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Perelman, Grigori (2003). "Ricci flow with surgery on three-manifolds". arXiv:math.DG/0303109. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Perelman, Grigori (2003). "Finite extinction time for the solutions to the Ricci flow on certain three-manifolds". arXiv:math.DG/0307245. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="978-3-540-66752-0" target="_blank">978-3-540-66752-0</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="https://doi.org/10.4007%2Fannals.2012.176.1.10" target="_blank">https://doi.org/10.4007%2Fannals.2012.176.1.10</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="/wiki/CiteSeerX_(identifier)" target="_blank">/wiki/CiteSeerX_(identifier)</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Hahn, Frank (1962). "A theorem on non-tatonnement stability". Econometrica. 30 (3): 463–469. doi:10.2307/1909889. JSTOR 1909889. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Lindgren, Jussi (2022). "General Equilibrium with Price Adjustments—A Dynamic Programming Approach". Analytics. 1 (1): 27–34. doi:10.3390/analytics1010003. <a href="https://doi.org/10.3390%2Fanalytics1010003" target="_blank">https://doi.org/10.3390%2Fanalytics1010003</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="/wiki/Geometric_and_Functional_Analysis" target="_blank">/wiki/Geometric_and_Functional_Analysis</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="https://doi.org/10.4007%2Fannals.2007.166.145" target="_blank">https://doi.org/10.4007%2Fannals.2007.166.145</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="/wiki/Publications_Math%C3%A9matiques_de_l%27IH%C3%89S" target="_blank">/wiki/Publications_Math%C3%A9matiques_de_l%27IH%C3%89S</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="http://www.math.cornell.edu/~warwick/main/rodes/JFoCM.pdf" target="_blank">http://www.math.cornell.edu/~warwick/main/rodes/JFoCM.pdf</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="http://personales.unican.es/beltranc/archivos/Smale17FoCM.pdf" target="_blank">http://personales.unican.es/beltranc/archivos/Smale17FoCM.pdf</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="http://personales.unican.es/beltranc/archivos/AffSmale17JAMS.pdf" target="_blank">http://personales.unican.es/beltranc/archivos/AffSmale17JAMS.pdf</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="https://link.springer.com/article/10.1007/s10208-010-9078-9" target="_blank">https://link.springer.com/article/10.1007/s10208-010-9078-9</a> <a href="#fnref:18" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="/wiki/Annals_of_Mathematics" target="_blank">/wiki/Annals_of_Mathematics</a> <a href="#fnref:19" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:20" class="footnote-back-ref">↩</a></li>
<li id="fn: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.. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:21" class="footnote-back-ref">↩</a></li>
<li id="fn:22">"Tucson - Day 3 - Interview with Steve Smale". Recursivity. February 3, 2006. <a href="https://recursed.blogspot.com/2006/02/tucson-day-3-interview-with-steve.html" target="_blank">https://recursed.blogspot.com/2006/02/tucson-day-3-interview-with-steve.html</a> <a href="#fnref:22" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11226777" target="_blank">https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11226777</a> <a href="#fnref:23" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="https://doi.org/10.1073%2Fpnas.2107151119" target="_blank">https://doi.org/10.1073%2Fpnas.2107151119</a> <a href="#fnref:24" class="footnote-back-ref">↩</a></li>
<li id="fn:25">Smale, Steve. "Mathematical Problems for the Next Century" (PDF). <a href="http://www6.cityu.edu.hk/ma/doc/people/smales/pap104.pdf" target="_blank">http://www6.cityu.edu.hk/ma/doc/people/smales/pap104.pdf</a> <a href="#fnref:25" class="footnote-back-ref">↩</a></li>
<li id="fn:26">Smale, Steve. "Mathematical problems for the next century, Mathematics: Frontiers and perspectives". American Mathematical Society, Providence, RI: 271–294. <a href="#fnref:26" class="footnote-back-ref">↩</a></li>
</ol>

Smale's problems open-in-new

Smale's problems