Algebraic geometry seeks to describe the spaces defined
by polynomial equations. A basic issue is the relation
between continuous functions (topology) and polynomial functions over the complex numbers (algebraic geometry). The Hodgeconjecture is a major problem about this relation: roughly,when can an even-dimensional real submanifold of a complex algebraic manifold be moved continuously to a complex algebraic submanifold? We discuss the history, including some counterexamples to overly optimistic versions
of the question.
A variant of the Hodge conjecture is to ask how much
of the cohomology of a finite group can be represented
by algebraic cycles. The question was made precise
by defining the “Chow ring of algebraic cycles
for a finite group $G$ (1999). Examples suggest
that the Chow ring has closer links to representation
theory than the cohomology ring does. We will survey
what is known about the structure of the Chow ring
and the cohomology of a finite group.