Millennium Prize Problems, P vs NP Problem
Suppose that you are organizing housing accommodations for a
group of four hundred university students. Space is limited and
only one hundred of the students will receive places in the
dormitory. To complicate matters, the Dean has provided you with a
list of pairs of incompatible students, and requested that no pair
from this list Read More
AWARD: $1,000,000
Millennium Prize Problems, Birch and Swinnerton-Dyer Conjecture
Mathematicians have always been fascinated by the problem of
describing all solutions in whole numbers x,y,z to algebraic
equations like
x2 + y2 = z2
Euclid gave the complete solution for that equation, but for
more complicated equations this becomes extremely difficult.
Indeed, in 1970 Yu. V. Matiyasevich showed that Hilbert's tenth
problem is unsolvable, i.e., there Read More
AWARD: $1,000,000
Millennium Prize Problems, Hodge Conjecture
In the twentieth century mathematicians discovered powerful ways
to investigate the shapes of complicated objects. The basic idea is
to ask to what extent we can approximate the shape of a given
object by gluing together simple geometric building blocks of
increasing dimension. This technique turned out to be so useful
that it got generalized Read More
AWARD: $1,000,000