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I'd be applying for a Ph.D. at various grad schools in the U.S. in the coming months and while I know I'd like to pursue research in the field of Algebraic Topology, I am not knowledgeable enough yet to figure out the exact subfield that would suit me best. I would like to know the best and quickest way to get a brief overview of the major active research areas in Topology (especially Algebraic Topology) so that I can start reading up in the areas that interest me and get in touch with the relevant professors in that area, while also meeting the application submission deadlines.

I've taken an introductory course in Homotopy Theory and Fundamental Groups and another in Simplicial Homology Theory in my Masters, besides a basic and advanced courses in General Topology. I thoroughly enjoyed my General Topology courses, especially the problems on compactness, connectedness, the separation axioms etc. I also liked the concept of Homotopy more than simplicial Homology, mostly because the construction of the simplicial complex seemed too geometric in nature. The book we used was mostly this: https://www.amazon.com/Introduction-Topology-Tej-Bahadur-Singh-ebook/dp/B07ZS1D3H8/ and Topology by Munkres.

However, while browsing through the profiles of professors, I see their areas of interests mentioned as symplectic topology, stable homotopy theory, Floer homology etc. most of which I am unfamiliar with and would like to know which amongst these would most align with my interests. Also, almost nobody seems to mention General Topology as their broad area of research which makes me wonder if it is not an active area of research and that Algebraic Topology is the natural progression that everyone moves on to.

While consulting my past professors would be ideal for this and I am trying that as well, they seem simply too busy for elaborate discussions.

I've gone through the suggestions here (Research in algebraic topology) but the plan laid out there is too long for my situation.

ZSMJ
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    This might suit MO which is full of professionals, undoubtedly including algebraic topologists – FShrike Sep 17 '22 at 14:31
  • @FShrike Thanks for the suggestion. I'll check for duplicates and post it there. Do you think this question is also suited here? – ZSMJ Sep 17 '22 at 14:44
  • Yes I think this question is fine here, too. By the way there is also a research tag which you might include to the post – FShrike Sep 17 '22 at 15:31
  • @FShrike Updated. – ZSMJ Sep 17 '22 at 15:43
  • Cross-posted: https://mathoverflow.net/questions/430646/most-efficient-way-of-getting-a-brief-overview-of-the-current-active-research-ar – RobPratt Sep 17 '22 at 17:04
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    My general response to this type of question is: the advisor makes a much greater difference than the field. First, you will be working closely with your advisor for several years while you work on a thesis, and if you don't get along with them, that's going to be a serious issue. After you finish your degree, you can change your area of research, but your advisor will be tied to you for the rest of your career: on your c.v., writing letters for you, etc. – John Palmieri Sep 17 '22 at 18:16
  • Did you write a thesis for your masters? – Daniel Teixeira Sep 22 '22 at 14:44
  • @DanielTeixeira Unfortunately not. My M.Sc. was purely coursework based without a thesis component. To make matters worse, the second year was conducted online due to covid and I missed out on the various seminars and other research opportunities. I only wrote an exposition on an MAA article on Completely Regular spaces. – ZSMJ Sep 23 '22 at 06:57
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    So in addition to what the others and John said, you should go popping messages to potential advisors whose research sparks anything to you. When mailing them you should explain your background - most advisors don't expect or want someone too especialized, rather just whom they can do research with. Look especially for researchers that write that they're looking for PhD students in their websites. And don't be shy to send dozens os emails (to different people). The safest way you'll get into a program is by finding someone who likes you and will vouch for you during admissions – Daniel Teixeira Sep 26 '22 at 13:52

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