Smale's problems refers to a list of eighteen unsolved problems in mathematics that was 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.
Contents |
# | Formulation | Status |
---|---|---|
1 | Riemann hypothesis (see also Hilbert's eighth problem) | |
2 | Poincaré conjecture | Proved by Grigori Perelman. |
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 | Pugh's closing lemma | |
11 | Is one-dimensional dynamics generally hyperbolic? | |
12 | Centralizers of diffeomorphisms | Solved in the C1 topology by C. Bonatti, S. Crovisier and A. Wilkinson.[2] |
13 | Hilbert's 16th problem | |
14 | Lorenz attractor | Solved by Warwick Tucker using interval arithmetic.[3] |
15 | Navier-Stokes equations | |
16 | Jacobian conjecture (equivalently, Dixmier conjecture) | |
17 | Solving polynomial equations in polynomial time in the average case | Carlos Beltrán Alvarez and Luis Miguel Pardo found a uniform (Average Las Vegas algorithm) algorithm for Smale's 17th problem, see [4] [5]. A deterministic algorithm for Smale's 17th problem has not been found yet, but a partial answer has been given by Felipe Cucker and Peter Bürgisser who proceeded to the smoothed analysis of a probabilistic algorithm à la Beltrán-Pardo, and then exhibited a deterministic algorithm running in time .[6] |
18 | Limits of intelligence |