Smale's problems - AbsoluteAstronomy.com

Smale's problems refers to a list of eighteen unsolved problems in mathematics

Unsolved problems in mathematics

This article lists some unsolved problems in mathematics. See individual articles for details and sources.- Millennium Prize Problems :Of the seven Millennium Prize Problems set by the Clay Mathematics Institute, six have yet to be solved:* P versus NP...

that was proposed by Steve Smale

Stephen Smale

Steven Smale a.k.a. Steve Smale, Stephen Smale is an American mathematician from Flint, Michigan. He was awarded the Fields Medal in 1966, and spent more than three decades on the mathematics faculty of the University of California, Berkeley .-Education and career:He entered the University of...

in 2000. Smale composed this list in reply to a request from Vladimir Arnold

Vladimir Arnold

Vladimir Igorevich Arnold was a Soviet and Russian mathematician. While he is best known for the Kolmogorov–Arnold–Moser theorem regarding the stability of integrable Hamiltonian systems, he made important contributions in several areas including dynamical systems theory, catastrophe theory,...

, then president of the International Mathematical Union

International Mathematical Union

The International Mathematical Union is an international non-governmental organisation devoted to international cooperation in the field of mathematics across the world. It is a member of the International Council for Science and supports the International Congress of Mathematicians...

, 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

Hilbert's problems

Hilbert's problems form a list of twenty-three problems in mathematics published by German mathematician David Hilbert in 1900. The problems were all unsolved at the time, and several of them were very influential for 20th century mathematics...

List of problems

#	Formulation	Status
1	Riemann hypothesis Riemann hypothesis In mathematics, the Riemann hypothesis, proposed by , is a conjecture about the location of the zeros of the Riemann zeta function which states that all non-trivial zeros have real part 1/2... (see also Hilbert's eighth problem Hilbert's eighth problem Hilbert's eighth problem is one of David Hilbert's list of open mathematical problems posed in 1900. It concerns number theory, and in particular the Riemann hypothesis, although it is also concerned with the Goldbach Conjecture... )
2	Poincaré conjecture Poincaré conjecture In mathematics, the Poincaré conjecture is a theorem about the characterization of the three-dimensional sphere , which is the hypersphere that bounds the unit ball in four-dimensional space...	Proved by Grigori Perelman Grigori Perelman Grigori Yakovlevich Perelman is a Russian mathematician who has made landmark contributions to Riemannian geometry and geometric topology.In 1992, Perelman proved the soul conjecture. In 2002, he proved Thurston's geometrization conjecture... .
3	Does P = NP?
4	Integer zeros of a polynomial of one variable
5	Height bounds for Diophantine curve Diophantine equation In mathematics, a Diophantine equation is an indeterminate polynomial equation that allows the variables to be integers only. Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for all equations... s
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 Linear programming Linear programming is a mathematical method for determining a way to achieve the best outcome in a given mathematical model for some list of requirements represented as linear relationships... problem
10	Pugh's closing lemma Pugh's closing lemma In mathematics, Pugh's closing lemma is a result that links periodic orbit solutions of differential equations to chaotic behaviour. It can be formally stated as follows:-Interpretation:...
11	Is one-dimensional dynamics generally hyperbolic?
12	Centralizers of diffeomorphism Diffeomorphism In mathematics, a diffeomorphism is an isomorphism in the category of smooth manifolds. It is an invertible function that maps one differentiable manifold to another, such that both the function and its inverse are smooth.- Definition :... s	Solved in the C¹ topology by C. Bonatti, S. Crovisier and A. Wilkinson.
13	Hilbert's 16th problem Hilbert's sixteenth problem Hilbert's 16th problem was posed by David Hilbert at the Paris conference of the International Congress of Mathematicians in 1900, together with the other 22 problems....
14	Lorenz attractor Lorenz attractor The Lorenz attractor, named for Edward N. Lorenz, is an example of a non-linear dynamic system corresponding to the long-term behavior of the Lorenz oscillator. The Lorenz oscillator is a 3-dimensional dynamical system that exhibits chaotic flow, noted for its lemniscate shape...	Solved by Warwick Tucker using interval arithmetic Interval arithmetic Interval arithmetic, interval mathematics, interval analysis, or interval computation, is a method developed by mathematicians since the 1950s and 1960s as an approach to putting bounds on rounding errors and measurement errors in mathematical computation and thus developing numerical methods that... .
15	Navier-Stokes equations Navier-Stokes equations In physics, the Navier–Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of fluid substances. These equations arise from applying Newton's second law to fluid motion, together with the assumption that the fluid stress is the sum of a diffusing viscous...
16	Jacobian conjecture Jacobian conjecture In mathematics, the Jacobian conjecture is a celebrated problem on polynomials in several variables. It was first posed in 1939 by Ott-Heinrich Keller... (equivalently, Dixmier conjecture Dixmier conjecture In algebra the Dixmier conjecture, asked by , is the conjecture that any endomorphism of a Weyl algebra is an automorphism. showed that the Dixmier conjecture is equivalent to the Jacobian conjecture.... )
17	Solving polynomial equations in polynomial time P (complexity) In computational complexity theory, P, also known as PTIME or DTIME, is one of the most fundamental complexity classes. It contains all decision problems which can be solved by a deterministic Turing machine using a polynomial amount of computation time, or polynomial time.Cobham's thesis holds... in the average case	Carlos Beltrán Alvarez and Luis Miguel Pardo found a uniform (Average Las Vegas algorithm Las Vegas algorithm In computing, a Las Vegas algorithm is a randomized algorithm that always gives correct results; that is, it always produces the correct result or it informs about the failure. In other words, a Las Vegas algorithm does not gamble with the verity of the result; it gambles only with the resources... ) algorithm for Smale's 17th problem, see . 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 Smoothed analysis Smoothed analysis is a way of measuring the complexity of an algorithm. It gives a more realistic analysis of the practical performance of the algorithm, such as its running time, than using worst-case or average-case scenarios.... of a probabilistic algorithm à la Beltrán-Pardo, and then exhibited a deterministic algorithm running in time .
18	Limits of intelligence Intelligence Intelligence has been defined in different ways, including the abilities for abstract thought, understanding, communication, reasoning, learning, planning, emotional intelligence and problem solving....

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