The Algebraic Analog of P vs NP

Mathematicians often isolate themselves within their specialties (the algebraic geometers vs the analysts vs the probabilists vs the theorists and so on). But some of the most brilliant results arise from applying tools from one field to problems in another.

One of the largest open problems in mathematics and computer science is the P vs NP problem. It asks whether a problem whose solution can be checked in polynomial time is actually solvable in polynomial time.  You can learn the basics about P vs NP here. Since the problem was formalized by Stephen Cook in 1971, many people in complexity theory have spent their time trying to make progress on it. So why is P vs NP unsolved? Fundamentally, some people think that the problem cannot be proved or disproved from our standard set of mathematical assumptions (i.e P vs NP might be independent of ZFC and other axiomatic systems). Additionally, several proof techniques have been proven insufficient (specifically relativizing proofs and natural proofs).

(Debatably) The most viable track for progress on P vs NP that’s still on the table (for now, anyway) is actually rooted in algebra! The idea, introduced by Leslie Valiant in 1979, is to study two classes of polynomials, VP and VNP, and see whether they’re equal. (For those whose ears are ringing at that name, this is the same Valiant who would later come up with the “probably approximately correct” PAC learning model.) Many believe that a proof for whether or not VP equals VNP would be a huge step in solving P vs NP.

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P vs NP

If it is easy to verify whether a solution to a problem is correct, is the problem easy to solve? This question is posed as the P vs NP problem and is considered one of the most important unsolved problems in Mathematics. This post will be brief and forgo many technical details surrounding the question for the sake of accessibility.

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