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Hello,

I am the 3rd year undegraduate student of mathematics. After I obtain a bachelor degree I want to study maths at graduate level, especially algebraic geometry and complex analysis. This fields of mathematics are well-represented at my univeristy, so at the first glance this plan looks fine.

Unfortunately almost all of my collegues are not interested in this fields (they think that AG and CA are too technically involved; most of them are interested mainly in functional analysis and topology). It will have such unpleasant effect on my studies, that most probably the most (or all) of courses in AG and CA in the upcoming year won't start at my university.

So I'll end up learning alone, from books. It's not that it's a big problem to learn from the book, it's not a problem at all. But I think such learning have no comparison with the regular course, where I could discuss problems and see another approches of other (more gifted) students.

To avoid such situation I may try to convince my collegues to study CA and AG, but I don't have many arguments since I'm still an ignorant in this fields. And here arise my questions:

  1. How would You encourage graduate students to learn algebraic geometry?
  2. How would You encourage graduate students to learn complex analysis?
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Surely if these fields are well represented you can do an independent study with a prof, and take graduate courses as an undergrad? Are there any related background courses like commutative algebra and galois theory that you can take? And finally, go talk to a professor who does work in AG or CA. He'll probably know much better than us the available options at your university. If these other students are already graduate students, I don't think the best strategy is to convince the ones that haven't been convinced already to take such courses. –  Jason Polak Apr 1 '10 at 15:14
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If you're interested in the subjects there's nothing stopping you from learning them. Moreover, evangelising for a subject can do as much harm as good. Maybe your fellow students are too busy with their work to get interested in what you're interested in. –  Ryan Budney Apr 1 '10 at 15:20
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Are you saying that you plan to continue on for graduate studies at the same university you are now enrolled as an undergraduate? I would recommend against that on general principle, and doubly so in your case. As you point out, it would be better for you to be in a graduate program where some of the other students share your interests. –  Pete L. Clark Apr 1 '10 at 15:24
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My personal experience: I made several attempts to read Hartshorne in my 3rd and 4th years as an undergraduate. I failed miserably each time and got absolutely nowhere and understood nothing. Advice: Don't try to read Hartshorne on your own; take a course. –  Kevin H. Lin Apr 1 '10 at 16:58
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@Pete: the OP is Polish and studying at Jagielloni. Your suggestion basically requires him to leave the country (which is not to say is a bad thing). –  Willie Wong Apr 1 '10 at 19:38

1 Answer 1

up vote 5 down vote accepted

This feels really like an "Ask Professor Nescio" question.

Let me ask you a question: if you feel like you cannot learn what you want to learn, why are you staying at the same university? I see from your profile that you are studying a Jagielloni, and you are Polish, so I understand somewhat that, if you want to remain in Poland, you feel that you should stay. But mathematics being the international field as it is, I would recommend going to a different university in a different country. (For example, in the US there seems to be no lack of graduate students who share your interest. I'm sure if you ask around a bit you can find out about other places in Europe.)

Now, with that said, if you decide that you want to stay:

  1. Don't over sell it. Being too pushy will have a negative effect on the other students.
  2. Don't be evangelical. You should not tell them why they should be interested or why they ought to study the subject with you. That'll have the opposite effect.
  3. From your descriptions you need to first dispel the myth that complex analysis and algebraic geometry is too technical. I guess the best thing to do is to introduce them to partial differential equations. (That's a joke.) But you need to be able to show them some examples of how sometimes, things becomes much more clear when viewed in the right framework. Show them a nice theorem or two with relatively simple proofs. A nice forum for this could be an informal seminar organized by the students for themselves: try to run a seminar where each student presents a result (not due to himself) that he finds interesting (don't just hijack it for your ulterior motives). When it is your turn talk about something really pretty from algebraic geometry. It may win you some converts.
  4. Find out what your fellow students like to do. You said functional analysis and topology. Anything else? You need to sell to your audience. For the topologists, at least some introductory complex analysis and algebraic geometry should be that hard to sell: tell them about (Hirzebruch-)Riemann-Roch! Tell them about the works of Kodaira! Complex analysis in one-variable is basically just topology anyway. (Can't help you with the functional analysts there.)
  5. An extension of the above: convince your fellow students that those subjects are useful for them. So a good idea is to find some theorems in their field that was first proven, or has nice interpretations, using the tools of complex analysis or algebraic geometry.

If all else fail, and you cannot get another person to study with you, you can always ask questions here or on sci.math.

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5  
Also, a general tip is that it's better to ask people things than to tell people things. Say things like: "You're really good at functional analysis, right? Could you help me understand a part of this proof of the Riemann existence theorem..." -- or "Hey, you know some topology. Do you have any intuition for Chern classes -- they are a total mystery to me". Or whatever. –  Dan Petersen Apr 1 '10 at 21:01
    
Thanks for Your advice. I'm certain that I should leave the country sooner or later (maybe sooner) and I will, but I'm still not ready for such decision for many reasons (the main are money and language; also I'm afraid that it's a little too late for such decision as leaving in next semester). –  ifk Apr 1 '10 at 21:25
    
As to the another interests of fellow students - one of them is interested in number theory and I think it should help. Another one said me that he want to have good general knowledge about mathematics, so there is a hope. Another one study phisics and mathematics simultanously. Two others study theoretical computer science and mathematics. Nevertheless I need six students to run the course –  ifk Apr 1 '10 at 21:39
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Well, if the physics student is into modern stuff (string-/M-theory), you should have no trouble convincing him that algebraic geometry is worthwhile. There is not much we can do to help at this point, besides wishing you very good luck. –  Willie Wong Apr 1 '10 at 23:44
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I agree with the seminar suggestion, a great way to learn a subject is talk to your friends in a seminar setting about it. You will have to work hard to understand it on your own, but you can always ask your professors for help when you get stuck. If you get lucky you cab convince each of them to give a talk, then you will have a pretty good seminar going. Even if they only give a few talks each and you really shake them for what they know, you can learn a lot. Plus remember that the best way to learn mathematics (IMO)is conversationally with a board or piece of paper. Good luck. –  B. Bischof Apr 2 '10 at 0:50

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