- Smale's problems
**Smale's problems**refers to a list of eighteen unsolved problems in mathematics, proposed by Steve Smale in 2000. [*Steve Smale, " [*] Smale composed this list in reply to a request from*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).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, " [*] The*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.Poincaré 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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