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I am sure that I will be finding out first hand as I am entering a PhD program, but I will ask my question anyway. Say, for instance, a student has worked through the majority of a textbook like Dummit and Foote. How far removed is the aforementioned student from current research in Algebra? In the same vein, how far removed from finding a research topic in Analysis is the student who has worked through the majority of a book like, say, Royden?

I know my question is difficult to answer for a number of reason. However, I am sure that you get the "gist" of what I am asking. Thank you for you time and input!

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2 Answers 2

Generally speaking, you're naming texts that are senior- or first-year graduate level. To start to be prepared for research, you'll generally want to take more advanced and focused second- and third-year courses and start reading some research papers, typically under the guidance of a faculty member in that area.

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I don't think the point of reading classic references is to pick open problems, rather - to pick up tools you need to solve problems :).

To get an open problem, it is more helpful to talk with people active in the field. On top of that, Analysis and Algebra are a bit too broad, you may want to pick a subtopic and try to work with someone who is doing active research in that subtopic...

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Thank you for your response! I suppose what I am trying to ask is how much more technical knowledge is needed to solve an open problem. For example, I browsed this website and found a few posts that suggest Algebraic Topology might be a branch where there is a long road to current open problems. In other words, finishing the majority of Hatcher's text may not be bring you very close. But maybe there is a subtopic topic in Algebra that is hot and the basic grad sequence in algebra already takes you pretty close in terms of technical knowledge. That's what I am curious about! :) –  dagaco Jul 10 '13 at 18:26
    
@dagaco This is very field dependent - there are some fields where this is definitely true, basic grad school is enough to prepare for active research. I am not an algebraist, let's either wait for the professional algebraists to answer, or try to speak to someone in your department doing current research. –  gt6989b Jul 10 '13 at 19:15

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