I agree that AI can do marvels in math, but math quickly departs from reality (like string theory) and being able to prove things about some beautiful intricate mathematical structure most of the times doesn't help you much with the world of matter.
I recommend Culture novels written by Iain M. Banks. There are vastly powerful AIs but they spend as much time as they can afford in "Infinite Fun Space" which is basically math taken to insane depths.
I know that there's a saying that all math is eventually applied but I think this saying is popular because it is sort of paradoxical that some of the most bizarre math eventually found their application. What I think is for math just a bit deeper than what human can grasp, "eventually" quickly becomes larger than the age of universe.
About stealing...
Hackers are successful not because they are ungodly smart. They are successful because they have the will to look for and exploit vulnerabilities. Intelligence, again, doesn't help all that much. I don't think people who wrote stuxnet were able to do this because they were of superior intelligence. They were just usually intelligent people (which means barely more intelligent than average human) that were sufficiently motivated by huge budget and interesting problems.
Super-intelligence won't help you with stock market because it's purely random game. You can see it in results of active management funds. HFT gives impression of algorithms beating people at trading but the thing they use to extract the value is not intelligence, it's speed. If you can trade faster you can beat slower guys because you are playing slightly different random game than they do. If you compare one HFT with other HFT then you are back to random results. So AI won't have upper hand on stock market unless we allow it to trade faster then our non-conscious software.
In theory I could image AI persuading people to give it what it wants. Because people are stupid and have flaws that are recurrent and easily exploitable. But again, I don't think intelligence would make such a huge difference. 1000 times more intelligent conmen could be just twice as effective because of chaotic nature of how human flaws interact.
> If it's so intelligent, it could design working nanotech. The only reason we don't have it is because it's so hard to engineer. So much complexity and moving parts,
It's hard to engineer because it's so damn small, not because it's complex. To build nanotech you'll need to build tools to build tools to build tools to build nanotech. You could do all that but even if you are 1000 times smarter, reality has a speed limit. You can't take a shovel of sand from the beach and build CPU, no matter how intelligent you are. You have to build fab first.
> Human brains weren't evolved to be engineers after all, it's a lucky coincidence we are capable of it at all. An AI brain optimized for this task should far exceed humans. The same way chess computers far exceed human chessmasters.
Yes. But it can exceed humans at consciousness, charity and compassion even faster because those are purely intellectual things, like chess while engineering is limited by moving atoms and energy around.