What can a first year mathematics undergraduate, who wants to pursue research in pure mathematics, learn in 67 days that will help him in the future?
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Depends where you want to wind up within mathematics. If you don't already know (as you probably don't), then explore: read a little analysis and a little algebra. The time will be gone before you know it. – Ian May 24 '16 at 14:35
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4While not math, learning LaTeX can help your writing to a large degree, which is indispensable for research, and can definitely be done in 67 days. – Hayden May 24 '16 at 14:37
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@Ian I have completed the basic course in real analysis - sequences and series, from Bartle and sherbet (along with almost all of the proofs). And I have done linear algebra. If you could suggest other topics that I can explore. Or should I study the above topics in greater depth, perhaps from a different book(eg. Walter Rudin). Please advice. – rhombicosicodecahedron May 24 '16 at 14:54
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@Hayden I already have an elective course on LaTex. Thank you – rhombicosicodecahedron May 24 '16 at 14:57
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@RohanRajagopal Have you learned anything about function spaces, e.g. uniform convergence of functions? – Ian May 24 '16 at 14:57
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@RohanRajagopal It would help for you to list your background and what potential topics you're interested in researching. – Hayden May 24 '16 at 14:59
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@Ian No I haven't. – rhombicosicodecahedron May 24 '16 at 15:11
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@RohanRajagopal Pursuing that (which is in most undergrad real analysis books) would be a good way to assess whether you would enjoy research in analysis. – Ian May 24 '16 at 15:15
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@Hayden I haven't studied enough to confidently say the topics I would like researching in. But I do like real analysis, algebra and algebraic topology(having attended seminars I am probably interested in it, but I know this is absolutely not at all enough, can't emphasise this anymore). Again this brings out the fact that I haven't studied enough. So what can I do to understand what pure mathematics research is like. Thank you for your time – rhombicosicodecahedron May 24 '16 at 15:27
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I think the suggestion by @Ian is excellent. One way of going about this is to check out several (more than 10 if you can) advanced undergraduate level real analysis texts and go through the material on uniform convergence in each of them. The idea I have in mind is to become an expert on this single topic, at the level presented in the books. Many of the exercises and examples will be similar, so with enough books at your disposal, you'll see the same things from many points of view. Then, using this material, write up your own detailed set of notes on the topic. – Dave L. Renfro May 24 '16 at 15:27
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@DaveL.Renfro Alright thank you. Can you suggest some of these books. – rhombicosicodecahedron May 24 '16 at 15:30
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Continuing my previous comment, this laser focus on a single topic will be useful in one aspect of research (becoming an expert on a certain topic, unless you're John von Neumann in which case you don't need this aspect) and the topic itself (uniform convergence) will pay huge dividends in pretty much any later study in analysis (real, complex, functional, PDE's, harmonic, or whatever). – Dave L. Renfro May 24 '16 at 15:32
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@Ian : in my opinion, learning a programming language (instead of mathematics) could be incredibly useful too – reuns May 24 '16 at 15:38
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If you have access to a university library, just look on the shelves where real analysis texts appear (in the U.S.A. it will be mostly in the QA300 to QA330 vicinity). Some authors that occur to me now are Osgood, Bressoud, Bruckner/Thomson, Abbott, Rudin, Apostol, Pugh, Bartle, . . . I could easily (if I had the time and motivation) list over 100 books, but what is going to matter the most is which books are available to YOU, and I don't know this without knowing anything about library holdings near you. – Dave L. Renfro May 24 '16 at 15:42
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@DaveL.Renfro I have, I think, access to books by all the authors you have mentioned. Thank you – rhombicosicodecahedron May 24 '16 at 15:50
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Please do not use the [tag:undergraduate-research] tag. It is being removed. – Caleb Stanford Jul 24 '16 at 17:57
