The only breakthrough in mathematics that could "threaten humanity" would be proving P vs NP for P=NP. That would end the digital economy overnight, as it would render all cryptography broken. But that is hardly enough to end humanity.
That could be adapted to in short order, a few weeks at most. It only effects RSA derived encryptions. You could still just switch over to elliptic curve or lattice public key schemes to replace it.
All you have to do is build a quantum computer for that. Which people are already doing, so it seems like that happens first. You never know when mathematical advancements come along, but people have been working on this for a long time.
The only breakthrough in mathematics that could "threaten humanity" would be proving P vs NP for P=NP. That would end the digital economy overnight, as it would render all cryptography broken. But that is hardly enough to end humanity.
That could be adapted to in short order, a few weeks at most. It only effects RSA derived encryptions. You could still just switch over to elliptic curve or lattice public key schemes to replace it.
All you have to do is build a quantum computer for that. Which people are already doing, so it seems like that happens first. You never know when mathematical advancements come along, but people have been working on this for a long time.