  In 1997 Richard Pink has clarified the concept of Hodge structures over
function fields in positive characteristic, which today are called Hodge-Pink
structures. They form a neutral Tannakian category over the underlying function
field. He has defined Hodge realization functors from the uniformizable abelian
$t$-modules and $t$-motives of Greg Anderson to Hodge-Pink structures. This
allows one to associate with each uniformizable $t$-motive a Hodge-Pink group,
analogous to the Mumford-Tate group of a smooth projective variety over the
complex numbers. It further enabled Pink to prove the analog of the
Mumford-Tate Conjecture for Drinfeld modules. Moreover, based on unpublished
work of Pink and the first author, the second author proved in her Diploma
thesis that the Hodge-Pink group equals the motivic Galois group of the
$t$-motive as defined by Papanikolas and Taelman. This yields a precise analog
of the famous Hodge Conjecture, which is an outstanding open problem for
varieties over the complex numbers.
  In this report we explain Pink's results on Hodge structures and the proof of
the function field analog of the Hodge conjecture. The theory of $t$-motives
has a variant in the theory of dual $t$-motives. We clarify the relation
between $t$-motives, dual $t$-motives and $t$-modules. We also construct
cohomology realizations of abelian $t$-modules and (dual) $t$-motives and
comparison isomorphisms between them generalizing Gekeler's de Rham isomorphism
for Drinfeld modules.
