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What kind of prerequisites would be required for [Inter-Universal] Teichmuller theory or at least the closest generally known area near Mochizuki's work? (Starting from undergraduate math).

I'd assume something like this - though of course I don't know enough to say myself or I would not be asking.I'm writing these as more modules than subjects, I know that if I were listing subjects some would be contained in others.

$\textbf{Undergraduate}$:

Basic Abstract Algebra

Basic Linear Algebra

Galois Theory

Basic Algebraic Number Theory

Basic Algebraic Geometry

Basic Homological Algebra

Basic Theory of Elliptic Curves

$\textbf{Graduate}$:

Category Theory

Homological Algebra

Sheaf Theory

Algebraic Number Theory

Algebraic Geometry

Elliptic Curves

Am I right in my rough assessment? What else would be required? Any analysis? No clue about the tags or even if this is the right place for the question, if not could I have another site recommended?

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    $\begingroup$ I'm sorry, but this is absurd. You shouldn't try to figure out all the things you need to know so that you can understand Mochizuki's new ideas, which are in a highly specialized bit of a highly specialized field. Just learn some math! As you go, you will find what interests you in a more natural way than from knowing that Mochizuki's work is widely talked about nowadays. (These sorts of reading lists, where one tries to plan out what they learn years in advance, are almost always doomed to fail; your interests change, especially early in your education.) $\endgroup$ – user98602 May 4 '15 at 17:26
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    $\begingroup$ To fill out Mike's comment: there are at this point only four people who claim to understand Mochizuki's work, all of whom spent between one and ten years focusing solely on it after they were at least ten years beyond the introductory graduate material you've listed here. This is more of a question to ask after you finish your Ph.D. in arithmetic geometry. $\endgroup$ – Kevin Carlson May 4 '15 at 17:57
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Classical Teichmuller theory is a topic in complex analysis. So you would need complex analysis and probably real analysis before that. Depending on the approach you take to Teichmuller theory, some knowledge of manifolds and differential geometry would also be helpful. That's really all you need to pick up an introduction to the subject.

However, I fear you are really asking about Interuniversal Teichmuller Theory, which is an entirely different subject.

Edit: Regarding your edit, you will need everything you listed (and much more). (I'm not a number theorist, but everything you listed is the essential to the study of modern arithmetic geometry.)

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  • $\begingroup$ I was not aware! I apologise, I'll make an edit $\endgroup$ – Nethesis May 4 '15 at 7:57

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