1

  1. When a university says they research in "combinatorial topology" what does that mean?

    • I've seen a university in Country A list "combinatorial topology" in its math department's research areas, but I cannot find a professor who researches "combinatorial topology" or any course offered by the university under the name "combinatorial topology".

    • "Combinatorial topology" does not refer to "algebraic topology" because the university's math department has a course "algebraic topology", and "algebraic topology" is listed in the research interests.

    • I could email the university, but I'm asking in general so I won't have to email every university

  2. What "combinatorial topology" does Munkres Topology cover?

I think Section 64 "Imbedding Graphs in the Plane" and all of Chapter 14 are "combinatorial" because they involve graphs.

My idea of "combinatorial topology" based on my 1 course on discrete mathematics 8 years ago and 1 my course on operations research 2 years ago and based on the Wikipedia "See also" for Combinatorial topology is "topological combinatorics" or "topological graph theory". Therefore, when I see "graphs" (as in "vertices" and "edges") in this textbook, I think it is combinatorial topology.

Also, based on the Wikipedia page for Topological graph theory, I think this is exactly what is covered in Chapter 14 wherein graphs are seen as topological spaces which are unions of arcs, spaces homeomorphic to $[0,1]$.

  1. What are some "combinatorial topology" textbooks?

I think numbers 1 and 2 are answered if 3 is answered.

  • 2
    I would just look at the research papers written by the faculty members, as this will give you MUCH more specific information about what research is done there. (Keep in mind that most useful are papers written by faculty when they were at that university.) Indeed, the various terms they use to characterize the areas of research (such as "combinatorial topology") were determined on the basis of what the faculty are actually doing. After all, it's certainly not the case that the description of the research came first and then the faculty tried to write papers to fit that area of research! – Dave L. Renfro Nov 16 '18 at 15:14
  • @DaveL.Renfro And the ones who might be doing "combinatorial topology" likely are the ones who list "topology", "graph", "discrete" or "combinatorics" ? I mean, I would look at the research papers but of which faculty? –  Nov 16 '18 at 15:34
  • 1
    In the U.S., at least, I don't think it's all that important specifically what the faculty do research in as long as it matches your interests in a general way. This is because at the undergraduate level you'll have almost no real conception of what is involved in various areas (just like in high school you probably had almost no real conception of what was involved with group theory or metric spaces), and your interests will likely change while taking graduate courses and preparing for your Ph.D. qualifying exams (probably the main thing you're going to be worrying about the first 2 years). – Dave L. Renfro Nov 16 '18 at 15:43
  • @DaveL.Renfro I have a master's degree, but it's in applied mathematics. Ok the context is I want to claim in my cv or statement of purpose that I have self-studied combinatorial topology citing Munkres Chapter 14. Can I claim that? –  Nov 16 '18 at 15:53
  • 1
    You could, but make sure the purpose of bringing it up is to indicate a possible interest and support for why you think you'd be interested in it, and not make it look like you think this will impress them (it won't, being only one chapter in a beginning graduate level text). If your goal is just getting accepted, then I think your main focus should be on presenting yourself as someone who has a relatively high probability of success in getting through the Ph.D., since you already have a Masters degree. (For what it's worth, I was in this situation TWICE when applying to graduate programs.) – Dave L. Renfro Nov 16 '18 at 16:08
  • 1
    By the way, despite having a Masters degree, you'll still have to take and pass the Ph.D. exams at the university you're going to. This even applies to those with a Masters degree and who have passed the Ph.D. exams at another university, even an ABD person. At least, this is the case for virtually all graduate math programs in the U.S. – Dave L. Renfro Nov 16 '18 at 16:13
  • @DaveL.Renfro Not trying to impress. Just trying to say I've looked at the subject material and that I've had a taste of it and that I like the flavour. Not that I'm applying to a master's where virtually no taste or even knowledge is required. I spoke to a professor last June who helped me understand why my application for 2018 was unsuccessful. studied GRE for 4 months –  Nov 16 '18 at 16:47
  • @DaveL.Renfro then I tried to get a taste of various areas for the past 3 weeks. This is so I'll have basis to say "I'm interested", not so I'll have basis to say "I know" (because I obviously don't with just a few chapters). I tried functional analysis Kreyszig Chapters 2-3 and Munkres Chapter 4 and realized I'm more interested in algebra and topology than analysis. I studied Munkres Part Two and finished ("finished" means I studied definitions theorems and did a few proofs. I tried to understand most of the exercises but did almost none) yesterday. now I'm about to do geometry. thank you! –  Nov 16 '18 at 16:50
  • 1
    Adding to what I previously said, the admissions committee will probably mainly want evidence that you can do will in their basic introductory graduate courses in measure theory, algebra, topology, maybe complex analysis, etc. and pass their Ph.D. Qualifying exams, rather than how much of a headstart you have on topics of your interest. But you definitely want to use these topics of interest to explain WHY you want to attend graduate school, and more specifically, THEIR graduate program (where "THEIR" is whichever places you apply to). – Dave L. Renfro Nov 16 '18 at 16:59
  • @DaveL.Renfro I will have had only 5 weeks to do this because the deadlines are December 1-6, and it's already been 3 weeks. If I'm not ready now, I know I'll be ready next year because if I can do this much in 5 weeks, I can do a lot in 4 more months (There is only one pure math master here, and deadline is March) and of course I can do a hell of a lot in 47 weeks. –  Nov 16 '18 at 17:14
  • @DaveL.Renfro Thank you for the unsolicited advice on top of helping me if this is "combinatorial topology" not for the sake of claiming to know but for the sake of claiming interest. –  Nov 16 '18 at 17:15
  • @DaveL.Renfro Based on my application, my talks and my own understanding, I think the admissions committee believes I "can do will in their basic introductory graduate courses", but the other side to it is interest. Another professor I spoke to (in an informal graduate talk, not in an interview) said the admissions committee might think that I might go back to (the nightmare that is called) applied math, and I think this was because I have not tasted pure math courses. Now I have so I have non-naive basis to say I am interested in pure math at PhD or at least MPhil level instead of just master –  Nov 16 '18 at 17:18
  • @DaveL.Renfro How do you suggest I present my self-study in a way that provides basis for a claim in interest rather than a naive claim in knowledge or even arrogant claim in understanding? –  Nov 16 '18 at 17:20
  • (I was in a Skype meeting until a few moments ago.) I don't know about December applications, but for later application dates perhaps you could present a specific topic (find some mathematical twist or historical issue to base it on) in a MAA Section meeting, plus doing this will likely get you introduced to others interested in the subject who would be willing to advise you when you do apply. For ideas about what to research for a talk, a google search for "comtinatorial topology" + "honors thesis" might be useful. – Dave L. Renfro Nov 16 '18 at 18:09
  • @DaveL.Renfro Can I claim interest if I know only one chapter? To emphasize, I am not claiming knowledge from knowing one chapter. I am claiming interest from knowing one chapter. –  Nov 20 '18 at 13:40
  • 1
    Based ONLY on this (one chapter read on your own), I would probably just briefly mention having a possible interest in combinatorial topology. The faculty will know that this much study is not all that good of an indication that your interest will continue as you delve more deeply into the subject (they'll know this from personal experience and from observation of students), so what you are now able to say about your interest in this specific topic will not carry much weight, other than being an indication of your interest in mathematics. – Dave L. Renfro Nov 20 '18 at 17:48
  • @DaveL.Renfro Thank you so much! –  Nov 22 '18 at 01:51

0 Answers0