Smale's problems are a list of eighteen
unsolved problems in mathematics proposed by
Steve Smale in 1998 and republished in 1999. Smale composed this list in reply to a request from
Vladimir Arnold
Vladimir Igorevich Arnold (alternative spelling Arnol'd, russian: link=no, Влади́мир И́горевич Арно́льд, 12 June 1937 – 3 June 2010) was a Soviet and Russian mathematician. While he is best known for the Kolmogorov– ...
, then vice-president of the
International Mathematical Union
The International Mathematical Union (IMU) is an international non-governmental organization devoted to international cooperation in the field of mathematics across the world. It is a member of the International Science Council (ISC) and supports ...
, 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 are 23 problems in mathematics published by German mathematician David Hilbert in 1900. They were all unsolved at the time, and several proved to be very influential for 20th-century mathematics. Hilbert presented ten of the pr ...
that had been published at the beginning of the 20th century.
Table of problems
In later versions, Smale also listed three additional problems, "that don't seem important enough to merit a place on our main list, but it would still be nice to solve them:"
#
Mean value problem
In mathematics, the mean value problem was posed by Stephen Smale in 1981. This problem is still open in full generality. The problem asks:
: For a given complex polynomial f of degree d \ge 2 and a complex number z, is there a critical point c o ...
#Is the
three-sphere a
minimal set
Minimal may refer to:
* Minimal (music genre), art music that employs limited or minimal musical materials
* "Minimal" (song), 2006 song by Pet Shop Boys
* Minimal (supermarket) or miniMAL, a former supermarket chain in Germany and Poland
* Minim ...
(
Gottschalk's conjecture)?
#Is an
Anosov diffeomorphism
In mathematics, more particularly in the fields of dynamical systems and geometric topology, an Anosov map on a manifold ''M'' is a certain type of mapping, from ''M'' to itself, with rather clearly marked local directions of "expansion" and "cont ...
of a
compact manifold topologically the same as the
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addit ...
model of John Franks?
See also
*
Millennium Prize Problems
The Millennium Prize Problems are seven well-known complex mathematical problems selected by the Clay Mathematics Institute in 2000. The Clay Institute has pledged a US$1 million prize for the first correct solution to each problem. According ...
*
Simon problems
References
{{reflist, 2
Unsolved problems in mathematics