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: [Introduction to Topology by Tej Bahadur Singh](https://rads.stackoverflow.com/amzn/click/com/B07ZS1D3H8) and [Topology by Munkres](https://rads.stackoverflow.com/amzn/click/com/0131816292).


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](https://math.stackexchange.com/q/158171/116321) but the plan laid out there is too long for my situation.



Most efficient way of getting a brief overview of the current active research areas in Algebraic Topology
Algebraic topology is a research area where often even the statements (and certainly the proofs) of active research areas require a lot of background to understand. For that reason, the answer to the linked stack exchange question from 2012 is really spot on. The best way to decide which area you might want to research is to read graduate textbooks in that area (or talk to experts, or go to seminars, but it sounds like this is more difficult for the OP) and see if you like the types of objects being studied, enjoy the nature of the proofs, etc.


For example, you said you prefer homotopy over more geometric arguments. If you start reading a book in symplectic topology or knot theory, you might find you don't like it because it's too geometric. Others might start reading graduate textbooks (or survey papers or research papers, as Tyler Lawson recommended in the linked thread) in homotopy theory and not like when it starts to draw tools from category theory. This kind of trial and error is the best way to settle on a research discipline where you actually enjoy working.


That said, it's also wise early in your career to think a bit about which areas are "hot." As someone who loved learning General Topology from the Munkres book, I was sad to learn that point-set topology is not anywhere near as active now as it was 50 years ago. There are very few departments doing point-set topology research, and it's difficult to get a job if your expertise is point-set topology. Careerwise, it would be better to be in a "hotter" area. I think there are several such areas that might be of interest based on what you've written about your background so far, including homotopy theory, knot theory, symplectic geometry, topological data analysis, differential topology, differential geometry/analysis, etc.


In US PhD programs, you are not generally expected to know what you will research before you show up. Your first year is usually spent taking graduate courses to get a firm foundation across pure mathematics (think: algebra, analysis, and topology) and then pass your qualifying exams. After that, you take more advanced courses and reading courses with potential advisors, where you get closer to the edge of human knowledge. In those courses, you learn the necessary background to understand modern proofs and the statements of interesting open problems. There's a lot of trial and error. In the first two years of my PhD program, I considered going into research in algebraic geometry, knot theory, ergodic theory, and homotopy theory, eventually settling on the latter because I enjoyed proofs and conferences in that area the most. Breadth is a good thing, because it makes you a better mathematician, expands what you can teach, and you never know when you'll find a surprising connection (see, e.g., recent work of Blumberg and Abouzaid on "homotopy Floer homology").


So, even though you are looking for the "most efficient" way, I think the best way is to peruse the books recommended in the link you already had. Even just reading the table of contents, preface, and first chapter of each of those would give you a sense of which you liked and which you didn't. It would help you a bit to have a rough idea of which type of topology you find most interesting so far, so that you can apply to graduate programs with that specialty, but don't expect to learn in the next few months what thing you'll be researching for your whole career. Expect that you might change your mind once you're already in grad school or even later: lots of people have switched into topological data analysis from other things, for example, even years after their PhD.


Most of all, try to pick a PhD program where you think you have the best chance of being successful and happy. When you look at a program, in addition to asking yourself if they have research in the subspecialty of topology that you think you want to study, also ask: do they nurture their students? Do lots of people drop out? Are students spending huge amounts of time teaching and getting no attention from faculty? Are their students getting the types of jobs afterwards that you currently think you might want? Is it in a place you can see yourself living? Etc. Think about the culture. To do good work, it helps to be in a good working environment, and leading a happy life as much as possible.


