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Assuming that a person has taken standard undergraduate math courses (algebra, analysis, point-set topology), what other things he must know before he can understand the Langlands program and its geometric analogue?

What are the good books for learning these topics?

Is there any book which can explain the Langlands program to an undergraduate with very few prerequisites?

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Re: the last question, it depends on what exactly you mean by "explain." –  Qiaochu Yuan Jul 2 '11 at 5:19
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By "explain" I mean understanding the meaning of the Langland conjectures, why they are important and why is it being called the grand unified theory of mathematics. –  Norman Jul 2 '11 at 5:31
    
Do the standard undergraduate courses include Galois and representation theory? –  jspecter Jul 2 '11 at 5:51
    
Well, consider they don't. –  Norman Jul 2 '11 at 5:58
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I can't give an answer, but I want to mention the book Introduction to the Langlands program by Bernstein, Gelbart et al. –  Soarer Jul 2 '11 at 6:19

4 Answers 4

I am going to contradict the answers given and say: do not read any introductions to the Langlands program at this stage. Instead, learn the following things first (and take your time over them!) and do lots of exercises:

  • Complex representation theory of finite groups, character theory (e.g. the book by Isaacs, or my lecture notes)
  • Algebraic number theory, starting with the basic theory of number fields, Dedekind domains, class numbers, and leading up to class field theory (that's a project for at least a year, you can start with any introductory book on Galois theory, then go on to an introductory book on algebraic number theory)
  • Some basics on algebraic groups and Lie groups. I suspect that you will need to learn some very basic things about manifolds and about varieties first.
  • An introductory course on modular forms.

When you have that covered (two or three years down the line), then you will benefit from reading about the Langlands program. In the meantime, once you have learned representation theory and Galois theory (can be done in one or two months if you are very bright), you should approach a faculty member at your university. He will be able to give you a very rough overview of the general Langlands philosophy, so that you very roughly know where you are heading.

All this is not supposed to discourage you, but rather to excite you about all the fascinating things that lie ahead of you, and to warn you not to skip any of the essentials if you really want to appreciate the beauty of the whole edifice.

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I'd partly second this answer, since a large part of what the Langlands program aims to do is organize a large body of seemingly disparate, wildly different phenomena/things. Until those phenomena are already familiar, formulating big conjectures about them can all too easily turn into a ghastly, oppressive exercise in definitions... answering questions that haven't been asked, in terms of things previously unmentioned. In addition to the topics above, the infinite_dimensional repn theory of Lie groups is necessary. and this necessitates some functional analysis. The phenomena come first. –  paul garrett Jul 3 '11 at 16:39
    
Thanks for your notes. –  Andrew Feb 18 '13 at 15:42
    
@Andrew, you are very welcome! –  Alex B. Feb 19 '13 at 21:46
    
This is an interesting response, and perhaps an even better question @Norman. It really raises a more fundamental question of the training necessary to get comfortable and competent with pursuing your own questions and interests. There is a certain sense in which the "standard undergraduate curriculum" becomes tediously formulaic and more about processing students than really conveying mathematics and developing, and implicitly finding those who are so interested, the right foundations for the student to learn for themselves. –  Erik G. Jan 27 at 12:40

To get started on this check out this paper:

http://www.ams.org/journals/bull/1984-10-02/S0273-0979-1984-15237-6/home.html

That's a good survey article on the Langland's program. It's takes you through the historical perspective to Artin reciprocity and it's implications. If you look that over you'll get a good overview of what you need to know; class-field theory, L-functions, p-adic numbers, adeles, automorphic forms, and group representations. To get a good handle on this stuff check out Frohlich and Taylor:

http://www.amazon.com/Algebraic-Cambridge-Studies-Advanced-Mathematics/dp/052136664X

I have a copy of this book and in my opinion it is accessible to someone with a good course in Number Theory and perhaps two courses in Algebra at the grad level. I don't think an undergrad is going to get there on his own but with guidance from an advisor working in number theory you can do it.

Now I'm no expert in Langlands but (I believe) the Langland's program is a series of conjectures that are basically consequences and generalizations of Artin reciprocity.

Give it a shot. The Langlands program is some deep stuff that took on a role of the same magnitude as Klein's Erlanger program before it. You really can't go wrong getting to the bottom of this one.

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I don't think the Langlands program is accessible to the average undergraduate student. In particular, you won't understand a thing if you have no prior knowledge of algebraic number theory (and class field theory). But, once you passed through the requirements spelled out in Alex B's answer (and in this paper), then here's a list of interesting references:

  1. I second both Gelbart's Introduction to the Langlands program and the book Introduction to the Langlands program.
  2. More advanced is the paper of Knapp called Introduction the Langlands program (yes, another one!), in the Edinburgh Proceedings.
  3. In fact, there is a whole website full of different resources related to automorphic forms, automorphic representations, and the Langlands program. You can find it here.

But again, remember this: One does not simply learn what the Langlands program is about.

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Peter Scholze, a young man recently made a Clay Mathematics Institute research fellow, wrote some readable papers on the Langlands correspondence.

http://www.math.uni-bonn.de/people/scholze/

If you read them, please share your experience (if it was the right level / style for you).

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