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.
You know how there are often 2 ways of describing a curve? Parametric equation: describe the curve as the result of tracing it with a pen. And level curve equation: describe the curve as the result of soaking a piece of fabric into water and mark where the fabric touch the surface of the water. This is the same for surface/space/hyperspace as well, albeit harder to visualize.
The first method can be considered “analytic”, and the second method is “algebraic”. The first method is very flexible, the second is not as flexible: trying to fold a fabric according to a drawn curve is harder than marking the water line with a pen. Abstractly, there is a method of converting 2nd type of description to 1st kind of description.
However, the downsides of being flexible is that when you deal with problems where you have to consider an arbitrary unknown curve/surface/space/hyperspace, the problem becomes very complicated. So sometimes you want to do the opposite. When you deal with a problem, sometimes you want to convert the 1st type of description into the 2nd type, because there are much more limitation on what the 2nd type can do.
Hodge conjecture basically said that, for certain type of problems, even though it seems you need to deal with arbitrary shape described by the 1st method, you actually only need to deal with shape that comes from the 2nd method. There might be extra shapes not covered by the 2nd method, but they do not cause additional difficulty to that kind of problem.
Real world applications? Not anything directly.
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