Smale's problems refers to a list of eighteen unsolved problems in mathematics, proposed by Steve Smale in 2000.^[1] Smale composed this list in reply to a request from Vladimir Arnold, then 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. Stephen Smale Stephen Smale (born July 15, 1930) is an American mathematician from Flint, Michigan, and winner of the Fields Medal in 1966. ... Vladimir I. Arnold (Moscow, December 2001). ... The International Mathematical Union is an international non-governmental organization devoted to international cooperation in the field of mathematics. ... Hilberts problems are a list of twenty-three problems in mathematics put forth by German mathematician David Hilbert at the Paris conference of the International Congress of Mathematicians in 1900. ...

List of problems

1. The Riemann hypothesis
2. The Poincaré conjecture
3. Does P = NP?
4. Integer zeros of a polynomial of one variable
5. Height bounds for Diophantine curves
6. Finiteness of the number of relative equilibria in celestial mechanics
7. Distribution of points on the 2-sphere
8. Introduction of dynamics into economic theory
9. The linear programming problem
10. The closing lemma
11. Is one-dimensional dynamics generally hyperbolic?
12. Centralizers of diffeomorphisms
13. Hilbert's 16th problem
14. Lorenz attractor
15. Navier-Stokes equations
16. The Jacobian conjecture
17. Solving polynomial equations
18. Limits of intelligence

Unsolved problems in mathematics: Is the real part of a non-trivial zero of the Riemann zeta function always ½? In mathematics, the Riemann hypothesis (also called the Riemann zeta-hypothesis), first formulated by Bernhard Riemann in 1859, is one of the most famous unsolved problems. ... In mathematics, the Poincaré conjecture (IPA: [])[1] is a conjecture about the characterization of the three-dimensional sphere amongst three-dimensional manifolds. ... Diagram of complexity classes provided that P ≠ NP. The existence of problems outside both P and NP-complete in this case was established by Ladner. ... In mathematics, linear programming (LP) problems are optimization problems in which the objective function and the constraints are all linear. ... Hilberts sixteenth problem was posed by David Hilbert at the Paris conference of the International Congress of Mathematicians in 1900, together with the other 22 problems. ... A plot of the trajectory Lorenz system for values ρ=28, σ = 10, β = 8/3 A trajectory of Lorenzs equations, rendered as a metal wire to show direction and three-dimensional structure The Lorenz attractor is a chaotic map, noted for its butterfly shape. ... The Navier-Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, are a set of equations that describe the motion of fluid substances such as liquids and gases. ... In mathematics, the Jacobian conjecture is a celebrated problem on polynomials in several variables. ... Intelligence is a property of mind that encompasses many related mental abilities, such as the capacities to reason, plan, solve problems, think abstractly, comprehend ideas and language, and learn. ...

Status

Since Smale proposed the list, several problems have been solved. The first one is problem 14, which was cracked by Warwick Tucker. The Poincaré conjecture (problem 2) has been proved by Grigori Perelman. In mathematics, the Poincaré conjecture (IPA: [])[1] is a conjecture about the characterization of the three-dimensional sphere amongst three-dimensional manifolds. ... Grigori Yakovlevich Perelman (Russian: ), born 13 June 1966 in Leningrad, USSR (now St. ...

References

1. ^ Steve Smale, "Mathematical problems for the next century". Mathematics: frontiers and perspectives, pages 271–294, American Mathematics Society, Providence, RI, 2000.