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I'm currently writing an article in which I connect two open problems from different fields of mathematics (group theory and combinatorics). While writing the introduction, I was trying to think of other examples of two problems from different fields of mathematics that are connected.

I could only think of the modularity theorem (formerly known as the Taniyama–Shimura–Weil conjecture) and Fermat's Last Theorem. Both are now solved, but historically, it was first shown that they are related; more specifically, that a negative answer to Fermat's Last Theorem would yield a negative answer to the Taniyama-Shimura-Weil conjecture.

My question is: What are other examples of such connections?

Edit: based on coLaidernotte's suggestion, the type of connections I'm looking for can better be described as follows: examples of two (or more) open problems that are not considered related at first, but which are eventually proved in a uniform way or proved to be closely related.

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    $\begingroup$ Riemann hypothesis is equivalent to like 30 things at least. $\endgroup$ Commented Jul 20, 2024 at 15:15
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    $\begingroup$ In what sense do you think the TSW conjecture and Fermat’s Last Theorem are from different fields of mathematics? For me they are both from the algebraic number theory. $\endgroup$ Commented Jul 20, 2024 at 16:51
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    $\begingroup$ Often people do interesting work translating a famous open problem to very different language from a different field (see e.g. the discrepancy theory reinterpretation of Kadison-Singer, the quantum games interpretation of Connes’ embedding problem, or work of Kronheimer-Mrowka on the four color theorem). But it would be really amazing if two famous open problems from different fields were “secretly the same” (although I guess coLaideronnette provided an example!) $\endgroup$ Commented Jul 20, 2024 at 19:35
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    $\begingroup$ This is way too broad. $\endgroup$ Commented Jul 20, 2024 at 19:47
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    $\begingroup$ At the risk of shameless self-promotion (oh well, here goes...) I recently published a paper with John Greenlees, "Modules with finitely generated cohomology, and singularities of C*BG" showing that a conjecture in algebraic topology is equivalent to a conjecture in modular representation theory, in quite a non-trivial way. But I agree with Andy, the question is way too broad. $\endgroup$ Commented Jul 20, 2024 at 20:02

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