Smale's problems

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

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 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 N^{O(\log\log N)}.[6]
18 Limits of intelligence

See also

References

  1. ^ Steve Smale (2000). "Mathematical problems for the next century". Mathematics: frontiers and perspectives (Providence, RI: American Mathematics Society): 271–294. http://www6.cityu.edu.hk/ma/people/smale/pap104.pdf. 
  2. ^ C. Bonatti, S. Crovisier, A. Wilkinson (2009). "The C1-generic diffeomorphism has trivial centralizer". Publications mathématiques de l'IHÉS 109: 185–244. 
  3. ^ Warwick Tucker (2002). "A Rigorous ODE Solver and Smale's 14th Problem". Foundations of Computational Mathematics 2 (1): 53–117. doi:10.1007/s002080010018. http://www.math.cornell.edu/~warwick/main/rodes/JFoCM.pdf. 
  4. ^ Carlos Beltrán, Luis Miguel Pardo (2008). "On Smale's 17th Problem: A Probabilistic Positive answer". Foundations of Computational Mathematics 8 (1): 1–43. doi:10.1007/s10208-005-0211-0. 
  5. ^ Carlos Beltrán, Luis Miguel Pardo (2009). "Smale's 17th Problem: Average Polynomial Time to compute affine and projective solutions". Journal of the American Mathematical Society 22: 363–385. http://personales.unican.es/beltranc/archivos/AffSmale17JAMS.pdf. 
  6. ^ 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. arXiv:0909.2114. 

See Also