Smale's problems

Smale's problems

Smale's problems refers to a list of eighteen unsolved problems in mathematics, proposed by Steve Smale in 2000. [Steve Smale, " [ Mathematical problems for the next century] ". "Mathematics: frontiers and perspectives", pp. 271–294, American Mathematics Society, Providence, RI (2000).] 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

# The Riemann hypothesis
# The Poincaré conjecture
# Does P = NP?
# Integer zeros of a polynomial of one variable
# Height bounds for Diophantine curves
# Finiteness of the number of relative equilibria in celestial mechanics
# Distribution of points on the 2-sphere
# Introduction of dynamics into economic theory
# The linear programming problem
# The closing lemma
# Is one-dimensional dynamics generally hyperbolic?
# Centralizers of diffeomorphisms
# Hilbert's 16th problem
# Lorenz attractor
# Navier-Stokes equations
# The Jacobian conjecture
# Solving polynomial equations
# Limits of intelligence


Since Smale proposed the list, several problems have been solved. The first one is problem 14, which was cracked by Warwick Tucker using interval arithmetic. [Warwick Tucker, " [ A Rigorous ODE Solver and Smale's 14th Problem] ", "Foundations of Computational Mathematics" 2 (2002), pp. 53–117.] The Poincaré conjecture (problem 2) has been proved by Grigori Perelman. Beltran and Pardo partially solved problem 17: the problem asks for an algorithm that numerically solves systems of polynomial equations in polynomial time in the average case (in the framework of real computation) and Beltran and Pardo constructed a uniform probabilistic algorithm with polynomial complexity. [Carlos Beltran and Luis Miguel Pardo, " [ On Smale`s 17th Problem: A Probabilistic Positive answer] ", "Foundations of Computational Mathematics", to appear, doi|10.1007/s10208-005-0211-0.]


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