what are the biggest unsolved problems in mathematics all about, and where will it get us if we solve them?
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The list would be a bit large. There are a few famous lists, such as the 7 (6 remaining) Millennium Prize Problems and Hilbert’s 23 problems (most of which have been answered or found to be truly unanswerable).
But those are far from all important central conjectures in mathematics. Each area of research has quite a few, and the importance varies from “it solves most of our issues” to “it sounds intriguing and any method to prove this marvel should give us deep insights”.
However, let me list a few, trying to catch many different fields, very roughly arranged by them. I will often simplify their meaning extremely as there is no way to fit even one of them into a single post, especially as ELI5, even less so many of them.
– P versus NP: are certain problems, such as finding the shortest path through all cities, inherently more complex than others, such as sorting numbers by size? Relevant on a theoretical level in computer science; practical relevance strongly depends on the details.
– Collatz: if odd n become 3n+1 and even n become n/2, does this process always reach the number 1, regardless of the starting value? Abstract.
– Riemann: are all zeros of certain _zeta_ functions on a so-called critical line? Despite its nature, this question turns out to be all about prime numbers. Mostly abstract, but it has some consequences on encryption.
– Birch & Swinnerton-Dyer: can the aforementioned types of zeta functions predict how elliptic curves behave? Abstract with very minor effects on encryption.
– Hodge: do all “cycles” come from ones described by algebraic equations? Abstract.
– Kontsevich: if two integrals give the same value, can we prove that only using the basic rules such as substitution, partial integration or Gauß-Stokes? Abstract.
– Standard conjectures (on cycles): are a lot of analytic things ultimately even algebraic (“described by equations”)? Abstract.
– Navier-Stokes: how to properly deal with turbulence and flow within fluids? Relevant for many things aerodynamic.
– Yang-Mills: Are there always good quantum field theories with certain properties? Relevant for theoretical physics.
– ABC: if a+b=c, is can c really be significantly larger than the product of the different prime factors of those numbers? Purely abstract, but there was recently a lot of drama around it.
– Fermat primes: how many primes of the form 2^^n + 1 are there? Abstract. Heuristic arguments say only finitely many, and the largest found is 65537, despite having searched extremely large values already.
– Mersenne primes: are there infinitely many primes of the form 2^^n – 1? Abstract, but search for new ones is used to measure advances in computing and algorithms.
– Rudin: how many square numbers can there really be in an arithmetic sequence of given length? Abstract.
– Schinzel: if there is no obvious reason against it, do certain expressions (“polynomials”) evaluate to prime numbers for infinitely many inputs? Quite abstract with _very_ slight effects on encryption. It implies several famous other conjectures such as Goldbach’s (is every even integer larger than 2 a sum of two prime numbers?), the Twin Prime Conjecture (are there infinitely many primes such that the next odd number is prime, too?) or Landau’s (are there infinitely many primes which when reduced by 1 becomes perfect squares?).
– Whitehead: if certain algebraic things have no holes, is the same true for their sub-things? Abstract.
– Borel: if two things are smooth, have no holes of any kind, and can be deformed into each other, are they already essentially the same? Purely abstract and related to the recently solved Poincare conjecture (if it looks like a sphere and quacks like a sphere, is it a sphere? Now proven answer is: yes).
– Hopf: if something only curves inwards, can you build it using less even- than odd-dimensional parts? Abstract.
– Carathéodory: does every smooth convex body have at least two points that curve just like a sphere does?
– Erdős–Gyárfás: if everyone has at least three friends, can you find some people and get them to hold hands with two friends each, such that they form a single closed loop with size a power of two? Abstract.
– Quantum-PCP: the proven PCP theorem states that if things can be checked within at reasonable but still high time (“polynomial time”), then we can do so extremely fast (“logarithmic time”) within any given positive error range. Does a similar result hold, but quantum? Has some potential applications for yet theoretical quantum encryption/communication issues.
– Hadamard: when are there matrices with only entries +1 and -1 and such that the rows are orthogonal, and similar for the columns? It has some very minor applications in encoding (making data less error-prone) theory, but mostly of purely abstract interest.
– Serre: If two _quotients_ of a ring (numbers where we can add, subtract and multiply with all the usual rules) add in size (“length”) to that of the ring, do they “intersect”? Abstract. We barely know that their number of intersections is not negative (yes this makes sense in context).
Okay, I hope that’s enough…
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