Are there any big controversies in contemporary mathematical research?
Other domains contain big controversial research topics (for example string theory in physics).
The specific nature of mathematics however, makes me suspect that there isn't much room for any serious disagreement, or at least less room in comparison..?
Is this intuition right, or completely ignorant and absurd?
— This is not a question about mathematics per se, but more a question about the people doing it. I hope the question is still appropriate in this stackexchange. —-
Best Answer
Yes. There is a controversy in symplectic geometry/symplectic topology about whether some proofs are complete as they have been presented. Also there is some controversy about whether Mochizuki has actually solved ABC conjecture or not because IUTT is too tough to verify right now.
Edit: After a long wait I have decided to edit this answer. It appears that IUT is false, although this isn't conclusively confirmed (and might not ever be). The fields medalist Scholze has raised some big questions that Mochizuki inadequately (in my opinion) addressed.
Edit 2: There was also controversy about who was the first to prove the prime number theorem, although Selberg was given the credit in the end. See here.