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I am a student of Pure Mathematics interested in Linear Algebra and Abstract Algebra. I was browsing the Internet for good research topics containing both a flavor of linear algebra and also having some real life applications when I came to learn about Spectral Graph Theory.

I have searched for professors who are actively involved in this field, but I have only found a few names whose topic of interest is Spectral Graph Theory. There are also a very few number of books available in this field.

Is this a good topic to work on? Is it very difficult to find an advisor in this field? Though I am able to learn the topic on my own, I am finding it difficult to approach someone in case I am having a problem.

Also, is it wise to first decide a topic and then find an institute which has professors in this field or it should be the other way round i.e., I should select a good institute according to QS rankings first and then decide a topic accordingly?

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  • I think the second question is fine, but the first seems a bit of a "shopping question" (as per help center).
    – Kimball
    Commented Jun 30, 2016 at 4:59

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In brief: no, in mathematics in the U.S., you do not "have to choose" a research topic prior to beginning your PhD. At the same time it is _of_course_ good to have something to be enthusiastic about, curious about, fixated upon. But that can change, and probably should change as you learn more. It is apparently the case that in the U.S., and to some degree elsewhere, in mathematics, undergrads and even M.S. students really don't get enough background to understand contemporary mathematics at all, except very superficially. Fine, that's not a moral failing... maybe just evidence that there's a lot happening in mathematics... (!?!) But, then, in that light, the enthusiasms that novices have are necessarily (by-far-typically) ill-informed... and will change if/when they learn more.

This is not to say that one should not have an opinion, or should not admit interest in things because one knows one's interest will change... but only that one should anticipate that change, rather than believe that one "has arrived" at age 22 or so, and there's nothing left to learn, and now it's just "trying to solve problems/do research". I understand the appeal of that, but, if we think about it, it's better in the long run if mathematics is not that shallow or superficial.

More specifically, spectral graph theory is a quite viable research area. "Expander graphs" are popular and google-able. Lots of people work on this, both from the "spectral side", and from the "graph theory" side.

And, perhaps counter-intuitively, the possible fact that there are not so many "books" treating a topic can in fact be a very positive sign... it's unclear. That is, if there are zillions of books on something, then chances are good that it's an older topic and has been worked over pretty hard... leaving less room for beginners to contribute.

So, sure, it's entirely reasonable to have specific and fine-grained interest... as opposed to "general" interest. Like having a specific lunch rather than a lunch-in-general. :) But be utterly open to changes in your viewpoint... as you learn more, which will be for many years. :)

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  • Good answer. Also, I can concur spectral graph theory is a popular area. I've done a little work in myself, despite being a number theorist (yet not quite from the expander/Ramanujan point of view that you might think is most closely related to my other work). Incidentally, my impression was there were quite a lot of books that discuss spectral graph theory, though perhaps books with "spectral graph theory" in the title.
    – Kimball
    Commented Jun 30, 2016 at 5:05
  • Thank you very much;So what should be my next step?Should I now look for an institute of good repute and then see whether there is an advisor in this subject?If not then I should work in some other topic?
    – Learnmore
    Commented Jun 30, 2016 at 10:04
  • @learnmore, in the U.S., in mathematics, the general reputation of a math dept (which is usually roughly correlated with the reputation of the university) is probably the best indicator of the quality of the graduate education and preparation for research you'd receive there. That is, thinking in terms of specific research directions is probably not the wisest criterion, given that one's interests will most often evolve enormously while in grad school. Apparently unlike other subjects (???), the further education one receives in grad math is very important... if done well.
    – paul garrett
    Commented Jun 30, 2016 at 15:57
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True research is driven by passion. It would be better to select a topic that interests you and then search for an advisor who could work on that topic. All that matters is how fine-grained your initial topic ought to be. If it is too narrow, it might be difficult to find a suitable advisor.

Coming back to your problem, Spectral Graph Theory is actually a subtopic of Graph Theory. There are many professors researching on that topic. Just because it is not stated in their research profile doesn't necessarily mean they would not work on it.

The key for a good PhD candidate is to be flexible and open to new ideas. Your initial idea may be readily accepted for further research by your prospective supervisor, or you may be assigned with a new one. But the field of the topic is your choice to make.

Best wishes on your PhD journey!

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    I added a 'mathematics' tag to this; it's a good question but I think the answer depends on the field of the project.
    – arboviral
    Commented Jun 29, 2016 at 9:06
  • @arboviral: I've just approved your edit. Nevertheless, the OP ought to re-look with respect to the topic graph theory itself rather than being specific to its sub-topic
    – Ébe Isaac
    Commented Jun 29, 2016 at 9:16
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    @Ebe I just want piece of advice. I am thinking pf phd on cloud security specifically homomorphic encryption. Homomorphic encryption requires Abstract Algebra
    – user3464093
    Commented Jun 29, 2016 at 16:27
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    Would I need someone from mathematics department to guide me in additional to my security supervisor.
    – user3464093
    Commented Jun 29, 2016 at 16:28
  • @vivek: That won't be always necessary. Computer security comes from the principles of Theoretical Computer Science. Many professors in this field come from a major mathematics background.
    – Ébe Isaac
    Commented Jun 30, 2016 at 13:07
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It depends if you seriously want to stay in academia after or not.

  • If you just want a PhD diploma, you will manage with almost any senior advisor, but it might take a while, you might run into funding limits and so on with an advisor who does not play the "publish or perish" game.

  • If you want to stay in academia after, it is more difficult: you do need to find your own subject, and create your own "niche" (many senior researchers now advocate for such a specialization very early on). But you also need to be sure you will publish papers, in good journals, and there is the importance of the advisor: someone who has good (and long) publishing records is more likely to be an efficient advisor who will not let you extend your PhD time forever without publishing.
    Finally, academia is competitive, and a big name on your CV can sometimes make the difference between the "interview" pile and the trash for a postdoc. In fact I discussed that with 2 seniors in my field, and they admitted having a soft spot for people coming from "big names" because "you anyway get brainwashed by your advisor". They have a point there I think, but only if the advisor really works in collaboration with his PhDs. However, big name on CV with no publications will lead you nowhere.

Personality wise, it is a gamble, and you cannot really avoid it.

I would personally go for a good advisor (forget the QS rankings: look at the specific advisor's records by going to Web of Science and find the h-index of the professors), for the simple reason that after my master I thought I knew what I wanted to do, but in fact working with an excellent competitive advisor during my PhD convinced me to slightly change my overall interests (in terms of modeling approaches).

For the specificity of your topic, I don't know, I am in environmental sciences.

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    I'm not sure about the suggestion to use h-index to judge quality of an advisor. Firstly, it's widely acknowledged that citation metrics are highly flawed representations of the quality of someone's research. Secondly, being a good researcher is not the same thing as being a good advisor.
    – user2390246
    Commented Jun 29, 2016 at 7:46
  • I absolutely agree on the last sentence, that is why I said "They have a point there I think, but only if the advisor really works in collaboration with his PhDs". Quality of someone's research not being linked with metrics: yes, I agree, and everybody agrees. Yet you get hired based on that. One of the many contradictions in academia.To me the most striking example of academic contradiction is Nature/Science papers: not necessarily good, many are in fact rather bad, but if you have one as 1st author in your CV at the PhD level, you are almost guaranteed a postdoc.
    – Lucile
    Commented Jun 29, 2016 at 7:50
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    I'm afraid I cannot see this as good advice at all: it's not true that "any senior advisor will do", nor that h-index or any other software-generated ranking will help you choose a good advisor. That is, this answer sounds as though it comes from someone who's pretty unfamiliar with graduate education, and certainly with mathematics (which was an aspect of the question).
    – paul garrett
    Commented Jun 30, 2016 at 2:28
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    For one thing, in mathematics, for hiring at both postdoc and tenure-track, I've never looked at any "impact factors", nor have I ever heard a single mention of them. Sure, things may evolve otherwise in the future... Then there's the question of what "publication" means these days: must it be traditional? (to affect "impact factors", which are commercially motivated products of traditional for-profit publishers, etc?) Or could it be arXiv? Or web-page? And what foolish advisor would make the mistake of not "pushing a student to publish"? [cont'd]
    – paul garrett
    Commented Jun 30, 2016 at 13:35
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    [cont'd] In mathematics, apparently unlike some other fields, advisors rarely "work in collaboration with their students", but the alternative is not to ignore them or fail to help them. Such a (false) bifurcation is dangerously simplistic. And the issues of "publication" for senior tenured faculty are wildly different than those for a beginner, so senior people who may conform less to ultra-traditional "rules" may nevertheless well-appreciate what young people should do to have reasonable careers. Finally, tenure at a well-ranked university is at least a sign that one understands The Game.
    – paul garrett
    Commented Jun 30, 2016 at 13:41

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