Hodge Realizations of Motivic Cohomology

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***Hodge Realizations of Motivic Cohomology*** Let us imagine for the moment that you are on a star trek expedition to discover civilizations on different planets in our solar system. You first encounter a civilization on one planet that has ``earthly'' overtones, and as your subsequent encounters of civilizations on other planets indicate, there is a sort of profound ``deja vu'' feeling about your discoveries. One might at this point, imagine that all such planetary civilizations are mere ``incarnations'' of an ``advanced universal civilization of the solar system'', which you can neither see nor touch, but you instinctively know exists. In mathematics, this is the current world of algebraic cycles and motivic cohomology. The analog of the advanced universal civilization of the solar system is precisely motivic cohomology (a universal cohomology theory), which is a reflection of what all cohomology theories (analogous to planetary civilizations) have in common. All known candidates of motivic cohomology use the datum of algebraic cycles as building blocks. A regulator (or realization) is analogous to one particular planetary civilization. In more precise terms, a realization is a map from motivic cohomology to a more ``earthly'' cohomology theory. Hodge realizations are simply a particular class of realizations. Our only way to understand the complexity of motivic cohomology is via its realizations.