The way I've understood this whole thing so far, the main reason why the mathematicians are so energized right now is because the AI companies' problem solving spree frames mathematics as a discipline that exists to solve open problems. Mathematicians don't fully share this view even though they value problem solving, and they're afraid that this framing will produce the following chain of thought in the public: Mathematicians get paid to solve problems -> AI solves problems better -> we don't need to pay mathematicians. So they're trying to reframe their work to shift problem solving to the background and keep getting paid. I leave it to the mathematicians to explain what their value is, as they would do a better job of it, and it remains to be seen if this reframing will work.

Regardless of that, what happens long-term if AI gets good enough to solve all the hard open problems?

Millenium problems may give AI companies some temporary hype, but advancing research in some narrow directions to find new problems is probably not worth it to them. In addition, current problems have clear, human-defined acceptance criteria. Problems that are both found and solved by AI will be opaque enough not to generate interest in anyone unless they are found to have immediate applications. Mathematicians will somewhat distrust 1M+ LOC Lean formalizations as well as any original AI output in the form of creating a bunch of new concepts. There will surely be heated discussions around that.

Afterwards, mathematicians take some time to catch up with the solutions to existing problems, use AI to understand them and map out further directions. Problem solving by AI companies slows down since there are no cool problems left. They move on, mathematicians get back to whatever they were doing before, now with the help of AI.

With every mathematician having a strong mathematical AI model, a new problem arises. At any point of the current mathematical frontier, a mathematician can use AI to explore a new direction at a high speed. If all mathematicians do that, each will have their very advanced understanding of their direction until they fail to catch up the AI, and nobody knows whose direction is the one that others should check up with. Many possible connections with other branches and explored directions will be lost, unless there's a unified map of all these explorations. It will be hard to create this map, because while exploring one direction may have a moderate cost for a mathematician, having an AI do an overview of all other explorations to find some commonalities and synthesize the findings into a unified theory will take serious amounts of resources only available to AI companies, which they might be reluctant to spend.

Individual mathematicians' explorations will slow down to human level, too, with their unique findings being hard to publish because nobody will care and the mathematical community wants to do things that other mathematicians can appreciate.

In the end, the mathematicians' work does not change that drastically, unless AI gets good enough to one-shot some new crazy branch of mathematics that reveals something with high real-world impact. But that's a discussion for another time.