Eli5, What is the Hodge Conjecture problem? Whats a real world application?
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At a *very* high level, think of the difference between describing a meal from the standpoint of a critic in terms of presentation, taste, etc. vs. describing a meal from the standpoint of a chef in terms of the process, ingredients, etc.
What the Hodge Conjecture argues is that, given a variety of constraints, there is some process by which we can match those two different descriptions so one can be transformed into another.
The value in doing this is that certain problems can be more easily solved with one description rather than another – consider the value of a critic’s description for knowing where to dine vs. a chef’s description for knowing what supplies to order. So we can take problems described in one way, transform them into an easier description to solve and then transform them back.
The real world application of solving the Hodge Conjecture is, like virtually all pure math problems, none. We know the Hodge Conjecture is *mostly* true and that’s good enough for any real world application we might have. If we built bridges uses the Hodge Conjecture, we’d just go ahead and use it, check the results to see they’re satisfactory and not be too chuffed if there existed edge cases somewhere that our process didn’t work.
However, the process of solving the Hodge Conjecture might, for mathematicians, open up different avenues of inquiry that may lead to further work that, in a few centuries or so, someone might apply to bridge-building (or some other real world application) usefully.
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