Grant Ayvazyan's and Thomas Gerhart's Unsolved Problems
-=The
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 glueing
together simple geometric building blocks of increasing dimension.
This technique turned out to be so useful that it got generalized
in many different ways, eventually leading to powerful tools that
enabled mathematicians to make great progress in cataloging the
variety of objects they encountered in their investigations. Unfortunately,
the geometric origins of the procedure became obscured in this generalization.
In some sense it was necessary to add pieces that did not have any
geometric interpretation. The Hodge conjecture asserts that for
particularly nice types of spaces called projective algebraic varieties,
the pieces called Hodge cycles are actually (rational linear) combinations
of geometric pieces called algebraic cycles.
