07/11/2022, 08:56 PM
I'd say that's a very good minimalist description of what we do, lmao. It's also very important to add, that more times than not the different super functions do not agree. And if they do agree, it's probably a large global solution.
This isn't really a mathematically quantifiable statement. But a lot of the time the super function either works for a tiny domain, or it works almost everywhere. Very rarely is there a middle ground. The trouble tends to be the global solutions are ugly and useless, the local solutions are awesome but so restricted they can become useless. Very rarely do we have some breakthrough case like Kneser which has the best of both worlds.
I highly suggest a complex analysis book too, if you are going into milnor. It definitely helps to read complex analysis before hand, and at least have a moderate grasp of how it works. My vote, for a somewhat beginner text, is John B. Conway Functions Of One Complex Variable. Super clean and well written book (And as a fun piece of trivia it's a different John Conway that made game of life, this is John B Conway, lol)
It's not my favourite book, but it'd be an easier read then Reinhold Remmert's two part series on complex analysis (which is a fucking beautiful tome). But it's very very German style math. Conway's book is lighter and more concise.
This isn't really a mathematically quantifiable statement. But a lot of the time the super function either works for a tiny domain, or it works almost everywhere. Very rarely is there a middle ground. The trouble tends to be the global solutions are ugly and useless, the local solutions are awesome but so restricted they can become useless. Very rarely do we have some breakthrough case like Kneser which has the best of both worlds.
I highly suggest a complex analysis book too, if you are going into milnor. It definitely helps to read complex analysis before hand, and at least have a moderate grasp of how it works. My vote, for a somewhat beginner text, is John B. Conway Functions Of One Complex Variable. Super clean and well written book (And as a fun piece of trivia it's a different John Conway that made game of life, this is John B Conway, lol)
It's not my favourite book, but it'd be an easier read then Reinhold Remmert's two part series on complex analysis (which is a fucking beautiful tome). But it's very very German style math. Conway's book is lighter and more concise.