Undergraduate research in math

Graduate School
1

<p>I've finished my first year of college. I've done the whole calculus sequence, differential equations, and linear algebra with perfect grades. I'm taking analysis and abstract algebra next semester, although I know a good deal about both subjects from self-studying.</p>

<p>Mathematics is my passion, and I want to be a research mathematician, so I have high hopes of getting into a good graduate school in three years. I want to have done as much research as possible upon graduating. I plan on spending the next two summers at REU programs, but in the mean-time I feel like I'm not quite prepared to do research, but I also feel like I'm wasting valuable time by not trying.</p>

<p>I realize my pace is rather slow for someone aspiring to top graduate schools. However, I can say that I'm going considerably faster than everyone else here (analysis and abstract algebra are typically taken by juniors here, which seems incredibly slow-paced to me). Also, I came from a very poor high school whose math offerings were very poor. So I can say I am pushing myself very hard to achieve this pace, even if it doesn't match that of schools like Harvard.</p>

<p>I know this is perhaps not the most appropriate question for this forum, but I was unsure as to where I should put it. I also know there are some people here who are knowledgeable about this topic.</p>

<p>So I'm basically asking for advice. Ideally I'd like to get at least one paper published or under review by the end of the coming semester. For someone in my position, what would you recommend I do to try to accomplish this? My professors know me because I got perfect grades and such and dropped by their office hours a few times, but that's it, so I obviously also need to significantly improve my relationships with my professors. I've spoken with the director for undergraduate studies here a few times, so he knows me. Would it be appropriate it to simply go see him and ask him about research opportunities in math?</p>

<p>My school actually has a strong undergraduate research program, but it offers nothing in math, and I've never heard of any other undergrads here doing math research. However, there is a bright side to this: if I can get some research done soon, I'll stand out here a lot.</p>

<p>Thanks much!</p>

2

<p>You should find a professor who does stuff that interests you and either stop by or send an e-mail stating your interest and desire to do research. I suspect you’ll need a lot more math before you can do anything worthwhile (I highly doubt a paper is very probable–especially that quickly), but definitely talk with professors to know what you should be doing.</p>

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<p>what if you do research in something math related, like engineering and physics. That might help you get your foot in the door in terms of research. And maybe even write about applied math</p>

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<p>Thanks for the replies!</p>

<p>Well, here’s an idea. There are a few journals that exhibit expository articles rather than research papers. The most prestigious of these is probably the Harvard College Mathematics Review. So perhaps as a way to get my “foot in the door,” as Art is Melodic said, I could aim first to get an expository article published.</p>

<p>I do have a good deal of knowledge extending beyond the classes I’ve taken because I’ve spent a lot of time reading myself, so this idea seems fairly realistic to me. Would it be appropriate for me to request a professor here to guide me in writing such an article? I’m really very new to the idea of undergraduate research simply because nobody at my university does it in math, so I’ve got no fellow students to consult about this.</p>

<p>Finally, I have another question about graduate school but not necessarily about research. How many graduate courses do applicants to top graduate schools typically take during their undergraduate years? By pushing myself this coming year, I’ll be ready for graduate courses by the beginning of my junior year–thus I’d have two years to take only graduate courses in math. Two a semester would therefore bring the count up to eight upon graduating. Is this a reasonable pace? What is the typical pace of applicants who apply to top schools?</p>

<p>Thanks again!</p>

5

<p>I have a question Nilk, </p>

<p>I recently decided to be a math major but i did not take calculus my first semester of college so I am 1 semester behind. I was wondering how hard is basic linear algebra. I was thinking of taking this class at UCI over the summer: </p>

<p>2J Infinite Series and Basic Linear Algebra (4) F, W, S, Summer. Lecture, three hours;
discussion, two hours. </p>

<p>Systems of linear equations: matrix operations; determinants; eigenvalues, and eigenvectors. Infinite sequences and series. Complex numbers. Prerequisites: Mathematics 2A-B. (V)</p>

<p>I got A’s in the calculus series but i don’t want to jump into a class thats too theoretical input on this class would be great.</p>

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<p>Regarding 2J: I took a quick look at the second midterm from spring 2006 and the final from spring 2003. It looked pretty easy (especially the linear algebra). It shouldn’t be any more difficult than the calculus series (albeit quite different). There didn’t seem to be any proofs, only computational stuff.</p>

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<p>thank you, I’m a transfer student who is starting at UCI, and its a bit intimidating. </p>

<p>In high school i was not much of a student, and now i feel as if I am am facing goliath.</p>

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<p>Well I can’t speak for any other math programs, but I know that my friends who plan on being math majors or even math/physics majors here at Harvard will be taking graduate level math courses starting next year (rising sophomores). Many of them came in already having done linear analysis and other topics in advanced math, either through colleges or self-study. That said, the fact that they will be taking classes with graduate students (I think the courses will be post-complex analysis, but I couldn’t be wrong) means that graduate students come in from other school without having already mastered that material. So clearly you don’t need to have worked your way through abstract algebra and non-euclidean geometry by freshman spring to stand a chance at top schools :)</p>

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<p>Thanks for the insight, White_Rabbit!</p>

<p>Let’s see, with the current schedule I’ve set up for myself I’ll be doing complex analysis the semester after next despite the fact that I read a textbook in complex analysis as a junior in high school.</p>

<p>I honestly feel like there’s no appropriate place for me at my current school. I’d done so much reading in my spare time while in high school that the undergraduate classes so far have been incredibly easy and largely wastes of my time. But at the same time I’m not sure I’m ready for graduate courses just yet since there are gaps here and there in my knowledge of the upper-level subjects, and graduate courses would assume those gaps were filled.</p>

<p>Undergraduate courses of course fill in those gaps, but they’re incredibly inefficient at it. Imagine spending 95% of the semester retreading familiar territory and 5% of the time filling in gaps. There is no accelerated curriculum here for people like me. </p>

<p>I haven’t discussed any of this with my math advisor because I don’t know what he could possibly do. My school simply doesn’t offer anything that would truly be appropriate for me at this stage, and I’m afraid it’s going to hurt me in the long run by, for example, hindering my chances at getting into a top graduate school. Would it be worthwhile discussing this with him?</p>

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<p>I recommend that you talk first to your math advisor; do not assume that he cannot do anything. (You may be surprised!) If it turns out that he has no solutions for you, then your next step would be to talk to the chairman of the Math Dept. </p>

<p>Both faculty members would be familiar with the specific course offerings at your school and also would be aware of how your background stacks up with other students interested in applying to graduate programs. If current course offerings are not appropriate for you, they might be able to come up with some independent study topics and work directly with you to fill in any gaps in your knowledge of higher-level math.</p>

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<p>While you may talk to your advisor and/or the chairman of the dept., do not in any way suggest that their courses are too easy or that the other students aren’t as accelerated. Do not even hint that you think the offerings are not appropriate for your level. </p>

<p>Explain that you want to go to graduate school some day and that you’re interested in challenging yourself beyond the coursework.</p>

<p>If that doesn’t work, start figuring out where you want to transfer to.</p>

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<p>Isn’t there some kind of independent study option or something you could do? And if you know 95% of the material of most of these classes you’re taking, it seems like you’d be fine taking more advanced classes. Enroll in a graduate class or two–you can drop out if it’s too much.</p>

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<p>for a semester or year. There’s a well-known math program in Budapest for talented Canadian and American college students.</p>

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<p>This also looks like a great program that I looked at once upon a time.</p>

<p>[Math</a> in Moscow](<a href=“Math in Moscow – Study Abroad Program in Mathematics”>Math in Moscow – Study Abroad Program in Mathematics)</p>

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<p>Yeah, I’ve actually been pretty interested in doing either the Budapest or Moscow programs. I’ve read quite a lot about them. I especially want to do summer REUs for the next two summers. I hear competition for those is fierce, though.</p>

<p>I’m not sure if there’s an independent study option here or not. Thanks for the idea, though, sarbruis!</p>

<p>Momwaitingfornew, it’s hard to tell, but based on your tone I think you may have taken what I said wrongly. However, in hindsight I seem to have had a misleading tone in my previous post, so it’s my fault. I certainly do not think I’m any smarter than anyone else here. In fact, I often feel pretty dumb. I work part-time at a software company where I’m surrounded by people with degrees in computer science; the owner is a professor of sociology here; and one of my coworkers/friends does research with the owner in sociology. He’s just a year ahead of me, but he has several publications, has presented at conferences, and was just named a Goldwater scholar.</p>

<p>So trust me when I say I don’t think I’m smarter than anyone here. If anything, it’s because I’m surrounded by such people that I feel compelled to push myself harder.</p>

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<p>Depending on where you’re going, your graduate level analysis and algebra courses might be on par with honors level undergraduate courses at a more advanced institution. That is to say, you might be in a decent position to complete these courses. What texts do the grad courses at your school use?</p>

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<p>I’m not sure what books they use. That information does not appear to be online anywhere. If I find out I’ll post here.</p>

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<p>OP, at my school, top Ivy, math majors have the option of taking 500 level courses instead of 300 level courses to complete a Math Masters in 4 years on top of a BA. Instead of taking 2 semesters of 300 level analysis and 300 level algebra, students take 2 semesters each of 500 level analysis and 500 level algebra. They then take 4 more 500 level courses and get a Masters. I think thats the path generally taken by students wishing to go on to graduate school. I would strongly suggest you try to tack on graduate level courses as an undergrad and even try to get a masters.</p>

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<p>In fact, next semester my school is offering the “basic” graduate courses in algebra and analysis. Right now I’m signed up for undergraduate algebra and analysis next semester. I’d have to get special permission to take the graduate courses (especially since they’ve filled up by now).</p>

<p>Assuming I could get permission to take them, I’m not totally sure I’m ready to take them. It’s very difficult to tell because there are no resources for these courses released online. I can only see a rough outline of the topics to be covered.</p>

<p>The descriptions for “basic algebra” and “basic analysis” follow:</p>

<p>“The major topics covered in this course will be groups, rings, fields, vector spaces, linear mappings, rational and Jordan canonical forms. The topics will be presented in an accelerated fashion. This course should help prepare students for the algebra qualifying exam, however, it will also be of interest and value to any mathematically inclined undergraduate or graduate student.”</p>

<p>“The major topics covered in this course include sequences and series of functions, elementary Fourier series, integration, and differentiation in one and several real variables, the Stone-Weierstrass theorem, the inverse function and the implicit function theorem, elementary properties of holomorphic functions, Cauchy’s theorem, power series representation, calculus of residues, the maximum modulus principle and conformal mappings. This course should help prepare students for the analysis qualifying exam, however, it will also be of interest and value to any graduate student.”</p>

<p>I should also mention that these courses list as prerequisites several undergraduate courses in analysis and algebra (in fact, the ones I’m signed up for next semester).</p>

<p>These are designed to be taken simultaneously. I’d be more comfortable if I could just take one of them and then another undergraduate math course for starters. However, neither is offered spring/winter semester of 2010, so if I took only one, I couldn’t take the other until the fall semester of 2010.</p>

<p>Of the topics listed, my strengths are linear algebra and complex analysis, about which I have read the most. I’m familiar with everything else mentioned but perhaps not to the point that I’d feel completely comfortable going at it at an accelerated pace and at the graduate level.</p>

<p>Again, it’s a hard decision to make since that’s the only information I have to go on. Both descriptions say the courses are intended to prepare students for the qualifying exams in algebra and analysis. Looking at past exams, if their difficulty is an indication of the difficulty of the classes then I’d be comfortable taking the analysis course but probably not the algebra course.</p>

<p>At your school, do you know how much preparation the students have who take graduate versions of analysis and algebra as opposed to the undergraduate versions?</p>

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<p>When you say complex analysis, are you talking about a proofs based understanding or a more applied knowledge? When you think of math, do you think of proofs or numbers(complex numbers, matrices, vectors, whatever)? Analysis and Algebra are going to be extensively proof based.</p>

<p>Looking back, I see you’ve finished your first year. If you’re not sure about the graduate courses–and it does sound like they’re a little advanced–I’ll recommend taking the undergrad offerings this year, and the graduate offerings the next. Even for smart people, analysis and algebra are often killer courses. Taking your undergrad courses sophomore year is a decent enough boost, and graduate work in your junior year will likewise put you ahead.</p>

<p>If you really want to try the graduate courses, and already have some good proof work under your belt, you might just have to talk to the professors to see if you’re up to it.</p>