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I'm finishing my undergraduate studies and I've been interested in several branches of math to research in the future, to say: functional analysis, differential equations, dynamical systems, differential and riemannian geometry, (general, differential and algebraic) topology, algebraic geometry, theory of groups and rings, etc.

My question is if is it possible to learn about all those things and to do research using all? or are there imncompatible areas between them? Because I have seen that almost all of the researchers of my faculty are centered to research in a very specific topic and that makes me believe that is not possible to learn all of the math that i want.

Thanks in advance for your answers!

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    It’s certainly possible for some people. Many years ago I heard that it was said that the day the great mathematician John Von Neumann died, twenty mathematicians became the top people in their fields. – Dan Romik Apr 10 '20 at 17:26
  • It's interesting to note that most modern mathematics is interdisciplinary in the sense that you will need to know/use tools from several fields of math. While this doesn't mean that one is working in several different areas of math, it does require a level of knowledge in those fields. My current problem is on geometric function theory in complex analysis but requires me to know some surgery theory, some harmonic analysis, some conformal geometry, etc. – tangentbundle Apr 11 '20 at 18:31
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It is definitely possible to do research in several branches (if you are good enough that is). But the people do so generally took decades to get to where they are. If your plan is to start in all of them at once, you are doomed to fail.

For the sake of the argument, let me arbitrarily distinguish different levels of knowledge in a topic.

  1. Textbook knowledge (what you learn in classes; definitions, big theorems, general ideas)
  2. Working knowledge (given a new problem (not a textbook problem, those are designed to be solved), knowing what could be done to solve it and the ability to do so)
  3. State of the art knowledge (what are the current problems in a topic, what are other people doing, what are new ideas, etc.)

Since you just finished undergrad, 1. will likely be the highest level you are at in any topic. Getting there is by far the easy part. It does not seem so in the beginning, but learning a new topic to that level gets easier the more you know. Level 2 is the minimal level needed to do actual research if you have someone to guide you. Getting you there in a single topic will take you the next few years. Level 3 is what you need to be an independent researcher. Even assuming that you are a dedicated, above average student, getting you there in a single topic will take you your whole Phd, if not a bit longer. The last level is also one that requires upkeep. Expect to spend several hours each week just to keep up with what other people are doing in a single topic.

Now the simple truth about academic careers is that you can't be a PhD student or post-doc forever. You'll need some sort of permanent position to survive, which requires you to be an independent researcher, which requires state of the art knowledge. This is why you see so many people specializing on a single topic, to get to that level and to stay there.

I don't think that is the only way, but note that you'll need some focus. If you try to work in all topics on your list at once, you will never get anywhere and likely not even finish your PhD before you have to drop out. But if you first focus to get one to the highest level, you might still at some point find that you finally have enough free time to look at some of the others. From there things can develop naturally.

Personally I like to read textbooks on different topics in my free time. I even have some collaborations with other people on topics where I have working knowledge but wouldn't consider myself to be at the state of the art. But the reason that they work with me is that I can contribute ideas from the topic I spent most of my time in and where I am at the state of the art.

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Learning each branch is one thing, learning them to the level needed for research is another - that is why people specialise.

I went to the professor who taught us supersonic flow and had some detailed questions that came up from our lab.

He looked at me and the questions and said “I can answer these for you, but go and see Mr X because he worked on that in industry and he can give you a better answer.”

My respect for that professor increased at that point because some professors can’t or won’t admit others might be better.

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  • how does this answer the question – Praphulla Koushik Apr 10 '20 at 7:52
  • @PraphullaKoushik what do you think the answer should be? – Solar Mike Apr 10 '20 at 7:54
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    I added what I think is an answer :) – Praphulla Koushik Apr 10 '20 at 8:06
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While I agree with the other answers, I'd add:

Some of the most exciting mathematical research comes from combining techniques from different fields of mathematics. So while "doing research in multiple fields", in the sense of "today, I'll write a paper on this neat topic in topology; after that's done, I'll think about this completely unrelated question in number theory for a while" is not very realistic, there's a lot of opportunity out there to pursue a research topic that uses techniques from multiple areas of math. Or even a topic that lies in the intersection of several fields.

My own Ph.D. topic, now several decades ago, dealt with group [=algebra] actions on manifolds [topology/geometry]. There is lots of dynamical systems lurking underneath the surface, and a (messy) part of my dissertation had to do some combinatorial counting of tuples of integers satisfying certain properties to classify low dimensional examples. That did not mean I was an expert dynamical systems theorist or topologist, but I got to play with (and stress out over...) all sorts of disparate mathematical ideas to Prove Something Neat.

I would add that even the best undergraduate mathematical courses only scratch the surface of the cutting edge in their topic areas. In many graduate programs, you will in your first year revisit topics in comprehensive courses that will both broaden your perspective as well as link them together (as well as perhaps introduce new topics you've never heard of...). This can feel it's getting in the way if you're burning to drill down on One Specific Area, but actually is very helpful to setting you up to navigate cross-field linkages.

If I were you, I'd focus less on deciding "I want to be functional analyst" vs "topologist", versus picking a few interesting subtopics in each for now. Then let things mature as you learn more at a good comprehensive math graduate program. Then you will need to choose a topic (specialize!) whose main home will probably be in one area; but your research may well lead you to continue to learn -- in a targeted manner -- different areas, and that may well allow you to pivot later. So: go broad now, narrow down for the Ph.D. itself (but follow specific questions to other fields where needed), and broaden again later.

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I know people who work on different topics. So, it is technically possible to work on different topics.

Now, coming to the question, "is it possible to learn about all those things and to do research using all"... I don't think this is a reasonable idea.

You fix a topic, work on that. If you work really hard to learn some topic, it would take negligible amount of time to learn a neighbour topic, when you compare learning that neighbour topic from scratch. This is what I observed from people who work on different topics.

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    does the down voter care to explain what can be improved? – Praphulla Koushik Apr 10 '20 at 15:21
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    Few downvoters will be so kind, actually. – Buffy Apr 10 '20 at 15:29
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    @Buffy :D I do not know how to read your sentence as :P – Praphulla Koushik Apr 10 '20 at 15:30
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    Sorry, I'm so old that I'm not up on my internet abbreviations. Say what? The best I can do is :-) and :-( though I try for the former, mostly. – Buffy Apr 10 '20 at 15:43
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    @Buffy :D Thanks for your comment.. :) – Praphulla Koushik Apr 10 '20 at 16:06
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The answer is yes, of course, but it is difficult. And if you spread yourself too thin while learning you might not be successful in any field.

The real issue is that to do research in some field of mathematics you need insight, not just knowledge in that field. And insight is expensive and hard to achieve. Some people have deep insight into several areas of math of course. Paul Erdős comes easily to mind, of course.

But that insight doesn't easily transfer from one subfield to another. I once had deep insight into classical real analysis and fairly good insight into general topology. But my insight was almost totally lacking in most of algebra. Especially ring theory.

But, from where you currently sit, I'd suggest that you pick a single (at most two) fields and really seek insight. Mere knowledge isn't enough. It may be enough to tell you why theorem proofs work, but you need insight to be able to propose things that (a) might be non obviously true and (b) worth the effort and time of exploring whether they are indeed true.

Fermat's Last Theorem is a case in point. Whether or not he did have an "elegant" proof, the statement itself was a leap requiring deep insight.

Almost any doctoral program will require you to take a deep dive into a narrow area. In the US, that will be preceded by a broader view of several areas to get you ready for comprehensive exams, but also to help you find that narrow area of research. You will probably need to do that to be successful and even begin a career.

But, once you have a credential and a secure income stream, there is nothing to prevent you from seeking insight into other areas. Your initial specialization will require insight, but it doesn't bind your future. Just don't try to do it all at once, though there are extremely rare exceptions.


In some sense, knowledge tells you what did work. But insight tells you what might work.

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When it comes to doing research, you don't (generally) choose a field, you choose a problem, so don't worry about which fields you can or will do your research.

Look for open the problems that interest you.

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    I disagree with this. You start out focusing on a (set of) subfield(s) that interest you. Then, once you have the grounding allowing you to do research, you start focusing on specific problems. Starting with a problem does not seem reasonable to me. – Tobias Kildetoft Apr 11 '20 at 11:59
  • @Tobias Kildetoft: Picking problems first worked for Paul Cohen --- The Littlewood Conjecture, a uniqueness question for the Cauchy problem in linear PDEs, the Continuum hypothesis, the Riemann hypothesis (not solved), etc. This will probably work for 4 or 5 other mathematicians alive. This will probably not work for the OP. – Dave L Renfro Apr 11 '20 at 12:23

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