- Smale's problems
Smale's problems refers to a list of eighteen unsolved problems in mathematics, proposed by Steve Smale in 2000. [Steve Smale, " [http://www6.cityu.edu.hk/ma/people/smale/pap104.pdf 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 theInternational Mathematical Union , who asked several mathematicians to propose a list of problems for the 21st century. Arnold's inspiration came from the list ofHilbert's problems .List of problems
# The
Riemann hypothesis
# ThePoincaré 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
# Thelinear programming problem
# The closing lemma
# Is one-dimensional dynamics generally hyperbolic?
# Centralizers of diffeomorphisms
# Hilbert's 16th problem
#Lorenz attractor
#Navier-Stokes equations
# TheJacobian conjecture
# Solvingpolynomial equations
# Limits ofintelligence Status
Since Smale proposed the list, several problems have been solved. The first one is problem 14, which was cracked by
Warwick Tucker usinginterval arithmetic . [Warwick Tucker, " [http://www.springerlink.com/content/myglw1pwkhu9r5g2/?p=be29e46519bb4ee9aa627a942501b772&pi=1 A Rigorous ODE Solver and Smale's 14th Problem] ", "Foundations of Computational Mathematics" 2 (2002), pp. 53–117.] ThePoincaré conjecture (problem 2) has been proved byGrigori 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 ofreal computation ) and Beltran and Pardo constructed a uniform probabilistic algorithm with polynomial complexity. [Carlos Beltran and Luis Miguel Pardo, " [http://beltranc.googlepages.com/Smale17finalcorregida.pdf On Smale`s 17th Problem: A Probabilistic Positive answer] ", "Foundations of Computational Mathematics", to appear, doi|10.1007/s10208-005-0211-0.]References
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