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I'm a first year PhD student and my research is related to Quiver representation and recently I have been looking at semi- invariant rings in the context of Quiver representation.

Sometimes I feel like nobody cares about this particular topic that I'm working on. I recently attended a conference where many eminent mathematicians from this field of representation theory were present and I felt like nobody really cares about semi-invariant rings for quiver representations. My question is:

  1. Will this make it difficult for me to find a post- doc position?
  2. Will it make it difficult for me to do research in the future, because people might not value what I'm doing

One thing that I would like to add is that I find this field and semi-invariant rings for quiver representation interesting and I like studying about it.

It'sMe
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There are at least one or two dozen professors at research universities who are interested in semi-invariant rings of quiver representations.

That's as many as you will find for almost any specific research topic in mathematics.

I wonder if you went to the right conference. Representation theory is a big area, and of course most representation theorists won't be interested in semi-invariants of quiver representations - but that will be the case for almost any topic in representation theory.

In addition, there are another several dozen professors (including me! - but we don't have postdoc positions) who might be interested in your research if it has implications in their area (using known connections between quiver representations and their area).

Now if I were your advisor, I would indeed be worried that I was leading you into some dead end that no one was interested in. Hopefully your advisor is a better mathematician (in this regard) than me and has some good reasons to suggest the problems you are working on, including some good reasons to believe connections to related areas might be uncovered.

How geographically mobile you are for a postdoc also matters; some countries are more interested in this stuff than others.

Alexander Woo
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  • I'm not 100% sure that there is no one interested in semi-invariant rings for quiver representation or that it is a "dead end", it's just that I feel that way sometimes. As I said, I'm a first year PhD student, so my knowledge is very limited, not only in mathematics, but also in other generic things in the world of mathematics. For e.g. I'm really surprised to know that there are at least one or two dozen of mathematicians interested in this particular topic. – It'sMe Jun 27 '21 at 23:11
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    It will matter what specifically you work on. For example, trying to figure out the semi-invariant ring of some moderately complicated random quiver of wild type is probably a dead end (but maybe there is something special about that specific quiver of wild type that makes it interesting or important?) Other problems might be more promising. – Alexander Woo Jun 27 '21 at 23:27
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The more tightly you focus the search for a post-doc tied to any specific research thread, the harder it will be to land a post-doc, just because there are fewer opportunities in narrow areas. You need to be flexible in the short term. You can use mathematical skills and insight to do many things.

To get autonomy you need to first have a secure position. Tenure track is probably secure enough in most fields, but in some, you really need tenure before you are free. (Fields that touch on political questions, for example.)

To keep your interest in your current specialty alive, spend some time on it periodically while you are doing things to please some PI. Take a lot of notes. Especially notes about questions that might later be explored. Once you have the needed autonomy, the world is yours to explore.

I finished my degree in the era before ubiquitous post-docs, but my field was so narrow that there were only two or three people outside my working group that were qualified to understand my research or interested in it at all. And, I wasn't interested in moving halfway around the world anyway. Had I been too narrowly focused at that moment, I'd have failed to move forward. But I had deep insight that I could apply to other things.

Buffy
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  • Thanks for your answer. So, are you suggesting me that I should research on other topics as well, as a part of my PhD thesis, other than semi-invariant rings? Maybe find a connection of these with other topics. – It'sMe Jun 27 '21 at 22:39
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    I disagree with this answer. Some postdoc opportunities are indeed very specific, and will only hire someone who is tied to that specific thread. So specialization can win as well as lose. – Alexander Woo Jun 27 '21 at 22:45
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    No, not at all. Doctoral research is necessarily very narrow. But don't make post doc applications contingent on being related to your dissertation. Best to focus closely on finishing. But a broad search for a post doc later. – Buffy Jun 27 '21 at 22:46
  • @AlexanderWoo, yes, and if you hit one that matches it is pure serendipity. But the closer you focus, the less you see. I don't understand why you disagree with that. Being inflexible lessens your options. – Buffy Jun 27 '21 at 22:47
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    @Buffy - the question isn't about being flexible or not - it's about being ultra-specialized or not. You have to be ultra-specialized to produce a thesis. Whether this leaves you with enough time/energy to learn other bits of mathematics depends on your abilities. – Alexander Woo Jun 27 '21 at 23:03