Eli5, What is the Hodge Conjecture problem? Whats a real world application?
In: 3
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.
Latest Answers