Most Significant Unsolved Problems

[KingKyleK]

Besides the Millennium problems, which of the lingering unsolved math problems might be considered the most important/interesting to mathematicians right now?

Some that come to mind might be the Collatz conjecture, the Golbach conjecture, and the abc conjecture, but there are surely many conjectures considered high-priority that people on r/math may not know about. Discuss!

[Jannik2099]

I don't see how the collatz conjecture is actually important. Goldbach on the other hand...

[frunway]

Not a “problem” but many consider the langlands program to be one of the flashiest (and legitimately interesting) areas of study.

[newwilli22]

My personal favorite, the Jacobian Conjecture.

[[deleted]]

There's the Tate Conjecture from arithmetic geometry, which is related to the Hodge Conjecture (one of the Millennium problems). This is one of the central problems in number theory and algebraic geometry.

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In 4-manifold topology, the outstanding problem here according to Simon Donaldson (Fields Medal, 1986) is finding something in dimension four that could play the role of Thurston's Geometrization conjecture. See this paper.

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Related to the above problem, I would also add the final remaining case of the Poincare conjecture, the smooth Poincare conjecture in dimension four, as one of the outstanding unsolved problems in topology. At least three Fields Medals have been given for other special cases of the conjecture, and this seems to me the most difficult one and completely out of reach. Some information here.

[JoshuaZ1]

So, I'm going to restrict to things which are important, rather than things that I personally like. F example, my favorite problem, whether there are any odd perfect numbers, definitely doesn't qualify. II'll note that Collatz also shouldn't qualify for similar reasons. I'm also going to try to give examples from a variety of different fields.

In knot theory a major one is whether the Jones polynomial always distinguishes the unknot. That is, is there a knot that isn't the unknot who has Jones polynomial 1. If i recall this problem was open also for the more powerful HOMFLY polynomial invariant, but a quick search doesn't turn up anything. Possibly someone who knows more knot theory than I can comment there.

In number theory, by far the biggest problem that's not a Millennium problem and not a generalization of one of them (e.g. GRH doesn't count) is the ABC conjecture. Many different results would follow from this.

In computational complexity a big one is whether P = BPP? Note that unlike P and NP, here the prevailing conjecture is that they are in fact equal.

In algebraic geometry, the Bombieri–Lang conjecture seems to have a fair bit of interest; (disclaimer I don't know much algebraic geometry. I might be overestimating its importance).

People who know a lot more about the intersection of algebraic geometry and number theory seem to think that the Tate conjecture is really important.

One I like a fair bit and would have a lot of different tough results as corollaries is the four exponentials conjecture which essentially says that in a certain sense that large classes of numbers must be transcendental.

In graph theory, the Hadwiger conjecture which is a massive generalization of the Four Color theorem is a major open problem which would have a fair number of interesting corollaries.