Are there any big controversies in contemporary mathematical research?

Question

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. ----

Answer 1

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.

Answer 2

Maybe "controversy" isn't the word, but like the Gallic village, the constructivism continues alive and defying the mainstream mathematics.

The entry Constructive Mathematics of the Stanford Encyclopedia of Philosophy is very interesting. In the bibliography you can find examples of recent constructivist research.

Answer 3

I want to address the question/comments with this answer, which contains some links/quotes and at the end I put in my two cents.

Mathematician Doron Zeilberger:

“For most people, the computer is only a tool, like a vacuum cleaner. For me, it’s like a colleague. Traditional mathematics is based on the notion of rigorous formal proof, and I think this is going to be obsolete, I think since computers are so powerful, they open up new vistas, and the old agenda of proving everything so rigorously is not as exciting as it was before.”

See In Mathematics, Mistakes Aren’t What They Used To Be

Vladimir Voevodsky who won the Fields Medal in mathematics is also interested in computers:

“The world of mathematics is becoming very large, the complexity of mathematics is becoming very high, and there is a danger of an accumulation of mistakes,” Voevodsky said. Proofs rely on other proofs; if one contains a flaw, all others that rely on it will share the error.

See Will Computers Redefine the Roots of Math?
When a legendary mathematician found a mistake in his own work, he embarked on a computer-aided quest to eliminate human error. To succeed, he has to rewrite the century-old rules underlying all of mathematics.

This video is also interesting,

What if Current Foundations of Mathematics are Inconsistent?
Vladimir Voevodsky

My two cents:

Five years ago few researchers thought that robotics/AI would be able to play the game of Go or Poker at world class levels. The progress that has been made in both games has been awesome, and AI may very soon dominate these two very different types of games. It is reasonable to expect increasing interplay between theoretical computer science and pure mathematics.

So, what is the big controversy? Well, if I am allowed to hold an opinion, perhaps that the dividing line between computers and pure mathematics has not really blurred at all over the past forty or so years.