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I find myself particularly drawn to the subjects of analysis and linear algebra. Although I am currently a rising Sophomore and unsure about the specific field I'd like to pursue for my graduate studies, I am eager to enhance my profile by gaining research experience through REUs (Research Experiences for Undergraduates). Additionally, I see this as an opportunity to determine whether research is genuinely a passion of mine or not.

But the common understanding seems to be that research in analysis-related fields is inaccessible to people like me without background knowledge of at least measure theory and Lp Spaces which I guess would take me at least 1.5 to 2 years to get by self-study which would be my Junior year summer time. Till then I am not sure how should I get involved enough in this field to make myself capable and indulged in the research experience. Research in discrete math seems a viable option till then (but extremely tough to get) but that's not what I would be aiming for in grad school. I am in NYC and discrete math seems to be popular here with probability.

Also an important point, I am an international student so REUs are very limited and I have part-time work to do, so can't fast-track and spend my all free time accelerating the learning process.

So my question now is:

Are there projects in analysis-related fields where a basic first course in real analysis and Linear algebra would be good enough to get started for my Sophomore summer?

My Real analysis study from Zorich Mathematical Analysis 1 should get me to Chapter 7/8 by March 2024. My linear Algebra is from Titu Andreescu's book (till Chapter 7/8 by March 2024) and lectures online.

OR

Is accessible research topics like those from Combinatorics/ Graph theory the only research experience I can think of? (keeping in mind that even these are extremely hard for me to get)

If so would it be wise to dedicate some of my time here to be prepared enough to apply for the projects? I have liked the idea of Algebraic Graph theory but yeah I only have an idea about it not even a preview of the subject.

My mind is toying with a rather silly notion: to prioritize a minor increase in my research experience chance over dedicating time to real analysis self-study. Keeping in mind that real analysis is crucial to learn, not only due to my personal interest but also because it is a fundamental requirement for almost every graduate admission, demanding proficiency in this subject.

Also, an important sub-question, do people usually do REU in a field say A, and go on to do their graduate school in say field B? How does graduate admission look at this fact?

TLDR: Confused and just overthinking due to spare time.

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  • In what country (or countries) do you hope to study in graduate school?
    – Buffy
    Jul 29, 2023 at 11:12
  • Have you taken a look at nsf.gov/crssprgm/reu/list_result.jsp?unitid=5044 or similar lists of REUs? Although there is a general slant toward discrete topics, you'll see there are quite a few whose topics are closer to analysis and related fields, e.g. the ones at Cornell, Fairmont, Indiana, Oregon State, U of Maryland. Jul 29, 2023 at 18:20
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    As a quick note, "REU" is a protected term that indicates NSF funding. You may looking for something similar, but using this term technically always indicates NSF funding, which as you indicate you are not eligible for. I would search instead for "outreach" programs and see if that yields results. Jul 30, 2023 at 13:37
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    Have you thought about simply talking to a professor at your current institution, to see if they have a summer project they could recommend and/or could supervise you on? Summer research doesn't have to be part of a large-scale organized program, and I think this would give you better direction than unstructured problem-solving on your own. Jul 30, 2023 at 14:50
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    @NateEldredge, I submit many proposals with REU supplements and outreach activities. Yes, "outreach" will be broad, but the reason I give this term is because this is the term used by most sponsors to support activities that are focused on education/training. Here's a list of opportunities that may be of interest (undergraduate research is another good term to use): math.harvard.edu/undergraduate/undergraduate-research Jul 30, 2023 at 16:46

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Currently, many of the summer REUs do indeed attempt to provide a stimulating environment for U.S. undergrads who don't actually have much mathematical background (since, in the U.S., many math majors won't see serious abstract algebra or analysis until their last year, and the summer REUs are for the summer prior).

This inevitably leads to at least two sorts of "easier" summer programs: seat-of-the-pants graph theory, and various naive computational projects.

To my perception, the benefit of such programs is not really the literal mathematical content, but, rather, the possibility for math faculty to become acquainted with you, and gauge your future potential (regardless of your summer project). And, specifically, to write letters of recommendation for you to express that appraisal.

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  • Do you think REUs are predictive of success in a PhD program? If not, then what is?
    – cgb5436
    Jul 29, 2023 at 22:33
  • That's what I am currently leaning toward. Doing projects in the two sorts of summer programs you mentioned. It's just that I would have to churn out a little bit of time from analysis self-study to accommodate for that which I for no reason why find it stressing instead of exciting (to learn new maths).
    – G.Yasuo
    Jul 30, 2023 at 5:52
  • @cgb5436, it's not so much that participation in an REU is indicative of success in a grad program, but that the faculty involved in the REU may be able to assess your future potential... and put that into the letters of recommendation that in the U.S. are often the decisive factor in grad admissions. There are (and have always been) other ways to become sufficiently acquainted with faculty... And, pseudo-ironically, "trying to make a good impression" is not the way to make a good impression... There's no guaranteed algorithm... Jul 30, 2023 at 18:15
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    Apparently a long time ago Nick Katz said, "Since when did we expect undergraduates to have done math research?" But now it's apparently hard to get into a "top" PhD program without an REU.
    – cgb5436
    Jul 30, 2023 at 18:21
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    @cgb5436, I am not at all persuaded that it's true that it's hard to get into a top PhD program "without an REU", especially considering that "an REU" is a hugely variable thing. Higher-end approaching-genuine-research projects are not the typical REU fodder. "Senior thesis/project" stuff (with concommitant faculty endorsement/appraisal) is more likely to be construed as evidence for future potential. And/or taking zillions of graduate courses and doing very well... Jul 30, 2023 at 18:26
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Are there projects in analysis-related fields where a basic first course in real analysis and Linear algebra would be good enough to get started for my Sophomore summer ?

No. Your ambitions are way too high by your level of know-how at present. As an undergrad of engineering science our freshman course went far deeper than Titu Andreescu's book (we used Noble & Daniel) and there was no way I'd see myself (even then!) of use to anyone in a math job, least of all in research.

Frankly, you must - for the present - postpone any thought of math research till you are much more proficient not only with those parts, e.g. linear algebra or analysis, where you are interested but also the other core areas of a good math BS course. Employers will fairly expect this from candidates.

I would think that, unless a student is by unanimous consensus of classmates and professors very talented, these REU programmes could do more harm than good - both to their career development and personal confidence. They are competitive, both officially as only the best candidates are selected and unofficially as a result of "natural" competition between candidates within the program. Not every mathematician thrives on this environment.

As an alternative - and this is purely for post Y3 vacation work - you might start looking into local internships in places like government bureaus, large companies, meteorological centers, etc that might have a use for math interns.

Is accessible research topics like those from Combinatorics/Graph theory the only research experience I can think of ? (keeping in mind that even these are extremely hard for me to get)

Look, you are just finished Y1 now. Stop looking so far ahead, stop reading articles and interviews by "famous" math people about new research areas.

Enjoy what's left of your summer. Make as much dough as you can before going back to college. Hang loose, man. That's the way to deal with the manifold options of our career and our life's future.

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    "and unofficially as a result of "natural" competition between candidates within the program": I think your view here is rather narrow. A lot of math REUs these days are set up as team projects and are carefully structured to encourage a collaborative rather than competitive atmosphere. Many are also designed to be accessible and encouraging to students who do not necessarily fit a conventional measure of "very talented". Jul 29, 2023 at 18:14
  • Yeah, I guess I am overlooking how vast knowledge a research mathematician has. It's just the fact that in the US more often than not many students have some sort of research experience under their belt when applying for grad admission, so I was just feeling behind.
    – G.Yasuo
    Jul 30, 2023 at 5:55
  • Please enquire in your department about any undergraduate summer schools in math that they may be holding. This would give you a chance to get more insights, meet like-minded students and meet some positive faculty from other institutions and overseas. Math is becoming a more extrovert profession and is attracting the interest of the sort of people who up until now might have studied languages or philosophy. So be positive and pursue math in a way that fits in with the normal life of a person of your age.
    – Trunk
    Jul 30, 2023 at 11:56
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    I definitely plan to utilize the conferences around the NYC area even though I will not understand almost all of the talks. Right now I would just work on my problem-solving ability through journal problems like MAA CMJ and Putnam books.
    – G.Yasuo
    Jul 30, 2023 at 13:00
  • I don't understand the down votes. Real analysis and Linear algebra are considered only part of math courses. The OP has a lot to learn if they want to study Math. It's too early for them to do research. Concentrate on study algebra, ODE/PDE, Probability/Stats, ..., etc. may be to the best interest for the OP. I would stay home for the summer, grab D&F's Abstract Algebra to study and go outside to play footballs/basketballs with others when bored with studying If I were the OP.
    – Nobody
    Aug 1, 2023 at 6:37

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