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This question seems quite broad to ask...

The situation here is that I'm a second-year undergrad student majoring in math and statistics. I'm really interested in fields like complex system and dynamics on/of networks and would like to plan for doing research in those fields in graduate school.

I've noticed that, speaking of undergraduate-level studies, subjects like statistics/probability, analysis, O/PDE, computational mathematics and linear algebra are essential for entering fields like dynamical systems and complex networks, but there is not much of an emphasis on abstract algebra.

From my personal learning experience, although algebra and analysis are seemingly asking different kinds of questions, I think algebra underpins all areas of mathematics including analysis (as least for what I've learned). This makes me wonder if higher algebra could serve as (potentially important) tools for research-level dynamical systems/complex networks or any other fields which may provide insights to them.

Since I do not have enough credit points to choose all the courses in both pure and applied mathematics, my question goes as:

How much abstract algebra (and higher-level algebra like commutative algebra, representation theory etc.), especially for senior undergraduate studies, is preferable for doing researches in dynamical/complex system and networks in further studies?

Thanks in advance!

J-A-S
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  • You should probably say what country you are from and (if different) what country (or countries) you think you might do your graduate studies and research in, because I've known several people whose Ph.D. work was in algebra and who had nothing more than what's in Fraleigh's A First Course in Abstract Algebra as an undergraduate (indeed, probably not even everything in Fraleigh, such as the Sylow theorems and Galois theory). – Dave L. Renfro Nov 19 '20 at 12:47
  • @DaveL.Renfro Hi Dave, thanks for pointing that out :) I'm currently doing undergraduate in Australia and aiming to read PhD in USA. Nonetheless, there are also some programs in Australia and in the UK that are very favorable – J-A-S Nov 19 '20 at 13:28
  • The reason I asked, and which you likely know (and probably many others, but perhaps not everyone here), is that graduate students in the U.S. typically spend the first two years involved in coursework that students in some other countries would have had as an undergraduate, and true research doesn't begin until one has passed the department's Ph.D. qualifying exams and been formally admitted to candidacy. Regarding your question, I suspect others will say something more specific (I know a little, but not a lot, about certain aspects of your interest), (continued) – Dave L. Renfro Nov 19 '20 at 13:42
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    but my feeling is that you should not worry much about algebra, except maybe enough to understand some technically/formally basic (though conceptually advanced) unifying concepts such as universal mapping properties and other such categorical notions that might be used to describe or frame certain abstract generalizations you may encounter in research. You didn't mention functional analysis, but I would think that the algebraic background needed for (and used in) functional analysis would be more appropriate than, say, heavy doses of commutative algebra and representation theory. – Dave L. Renfro Nov 19 '20 at 13:47
  • @DaveL.Renfro Thank you very much for your detailed information! I do noticed the importance of functional analysis (should have mentioned explicitly that by analysis I mean Real/Complex analysis, measure theory, Fourier, stochastic, functional analysis and some of topology haha), but honestly I know little about functional analysis as far as now (I just heard it is kinda infinite extension of linear algebra..), so regarding to the algebraic background you said for functional analysis, what kinds of algebra do you mean specifically? – J-A-S Nov 19 '20 at 14:17
  • @DaveL.Renfro And also may I ask if you have any recommendations on the categorical notions? – J-A-S Nov 19 '20 at 14:18
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    Not really, as this isn't something I know much about and it probably depends a lot on what you'll later work on anyway. I was thinking that you'd probably want enough algebra background to have been exposed to something like Chapter X in Hungerford's Algebra, which although is at the back of his book, is more like an appendix and is mostly independent of the rest of the book. – Dave L. Renfro Nov 19 '20 at 14:35
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    I originally missed your comment about types of analysis. Regarding algebraic background for functional analysis, I was thinking of what's needed to know study Banach spaces, $C^{}$ algebras, and the like -- not actually knowing a lot about these topics (especially $C^{}$ algebras, which is something you probably don't need to know anything about), but rather knowing enough algebra to begin studying these areas if you wanted to, as an example of how much algebra. Probably nothing more than the generally required 2-semester graduate algebra sequence (continued) – Dave L. Renfro Nov 19 '20 at 15:17
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    in the U.S. As for my comment about "categorical notions that might be used to describe or frame certain abstract generalizations", see this blog entry by Terence Tao (who grew up in Australia, by the way). Incidentally, I'm surprised no one has said anything here yet. I would have thought there were several people in Math SE fairly knowledgeable about the topics you're interested in. – Dave L. Renfro Nov 19 '20 at 15:17
  • @DaveL.Renfro This may be off the topic but may I also ask how much geometry (especially differential geometry) is needed for studying fractal (and/or fractal-related dynamical systems)? – J-A-S Nov 20 '20 at 08:04
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    I think the amount of geometry will vary hugely depending on what you work on: very little to none for much of T. H. Steele's work, a nontrivial amount for much of B. R. Hunt's work (and others, such as Newhouse, Yorke, Palis, etc.), and a huge amount for much of Jerrold E. Marsden's work. (FYI, I personally know Steele, Hunt, Newhouse a little but I only know the names Yorke, Palis, Marsden.) – Dave L. Renfro Nov 20 '20 at 09:23
  • @DaveL.Renfro wow that's pretty cool :) thank you for your informations! I'll probably know how much geometry I need after I learn more – J-A-S Nov 20 '20 at 11:06
  • This questionw as also posted to MO, and I answered it there: https://mathoverflow.net/questions/376859/how-much-algebra-is-preferable-for-studying-doing-research-in-dynamical-complex/467968#467968 – David White Mar 29 '24 at 12:52

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