Smale's problems

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.

List of problems

#	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	Partially solved by Carlos Beltrán Alvarez and Luis Miguel Pardo who proposed a uniform probabilistic algorithm with polynomial complexity.^[4] Another 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 $N^{O(\log \log N)}$ .^[5]
18	Limits of intelligence

^ Steve Smale (2000). "Mathematical problems for the next century" (PDF). Mathematics: frontiers and perspectives. Providence, RI: American Mathematics Society: 271–294.
^ C. Bonatti, S. Crovisier, A. Wilkinson (2009). "The C¹-generic diffeomorphism has trivial centralizer". Publications mathématiques de l'IHÉS. 109: 185–244.{{cite journal}}: CS1 maint: multiple names: authors list (link)
^ Warwick Tucker (2002). "A Rigorous ODE Solver and Smale's 14th Problem" (PDF). Foundations of Computational Mathematics. 2 (1): 53–117. doi:10.1007/s002080010018.
^ Carlos Beltrán, Luis Miguel Pardo (2008). "On Smale's 17th Problem: A Probabilistic Positive answer" (PDF). Foundations of Computational Mathematics. 8 (1): 1–43. doi:10.1007/s10208-005-0211-0.
^ Felipe Cucker, Peter Bürgisser (2010). "Solving Polynomial Equations in Smoothed Polynomial Time and a Near Solution to Smale's 17th Problem". Proc. 42nd ACM Symposium on Theory of Computing.