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Timeline for Need for support and guidance for my near future as a PhD student (or: has stable homotopy theory become an overly algebraic theory?)

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Feb 27 at 18:47 history edited David White CC BY-SA 4.0
I tidied up this question since it was on the front page anyway.
Feb 27 at 18:44 answer added David White timeline score: 4
Feb 27 at 18:02 answer added David White timeline score: 3
Oct 20, 2020 at 19:16 comment added Dev Sinha You might enjoy and/or find useful in these contemplations some of my lectures on algebraic topology from a geometric viewpoint: pages.uoregon.edu/dps/GeometricAlgebraicTopology And while these are at an "intermediate" level - between a first course and research - there are plenty of questions which interface with geometry at the research level. If you're interested it would probably be most fruitful to contact me directly.
Oct 20, 2020 at 6:30 comment added Simon Pepin Lehalleur Here is a very recent paper which answers concrete, long-standing questions in differential topology using a lot of modern homotopy theory: arxiv.org/abs/1910.14116
Oct 20, 2020 at 3:08 comment added user51223 I also have to correct that the 2008 Warwick talk is a talk by Milnor; I rather remembered one of its references!
Oct 19, 2020 at 18:15 comment added Fedya One area that hasn't yet been mentioned here is the relationship between topology and metric geometry. There is a lot that remains to be understood about how the topology of spaces and maps constrains their geometry, and some nice theory already built. See for example this paper of Larry Guth: doi.org/10.1007/s00039-013-0246-3 and I suggest browsing through some of his other work as well.
Oct 19, 2020 at 17:53 comment added user51223 There are many schools around the world. For instance, I presume, the Japanese school still would have many people whom continued line of thoughts of Toda, hence doing some geometric work. I probably can name a few people doing unstable homotopy, but I prefer not to as this may leave some others out. Perhaps, looking at papers, recent and old, and using math reviews would uncover more people doing sorts of maths that you like and tastes nice to you. Since the algebraic topology that we know these days is not very old, I hope a look into this could be done in a couple of months.
Oct 19, 2020 at 17:49 comment added user51223 centers of topology around the world. I am not sure if this has been updated or not, since as you can imagine some centers might have disappeared due to retirement of their members. Now, this is important since different places, often, collect people belonging to similar school of practice and philosophy in doing maths. So, for a throughout search, I suggest you look at some school around the world.
Oct 19, 2020 at 17:47 comment added user51223 @buck I apologise if my comment has hurt you. The reason for my suggestion is that I have seen a good number of people, who have quit doing maths right after their PhD or after doing some postdocs in respected places. I presume, these people, all loved what they had done already and chose their field or research in accordance with that. I think there is always a matter of taste. But, you can also see that some people start doing very geometric topology and after a couple of years tend to do a very algebraic one. Neil Strickland's homepage use to have a section with a map pointing out at ....
Oct 19, 2020 at 14:18 comment added buck Come on, how can you tell someting like "I would suggest you quit" without knowing me ? Maybe I haven't expressed enough the fact that I do love what I'm doing in general. I'm just trying to find the right subject of algebraic topology/homotopy theory to do a PhD, and not a PhD in biology or medecine. But I know it was said with a good intention. Otherwise, I've been looking at people doing unstable homotopy theory and maybe it could be more adapted to my interests and the style of maths I want to discover. Would you have some suggestions of people in woking this field ?
Oct 19, 2020 at 7:26 comment added user51223 Maybe doing mathematical biology would be a more appropriate thing to do. In this fields you will see how some patterns lead people to formulate problems in terms of maths and I personally find it very satisfying! One last thing is that a 2008 talk of Shahshahni, at Warwick I guess, addresses using cobordism to model evolution. Isn’t that exciting? Anyway, good luck in your future academic life!
Oct 18, 2020 at 22:11 comment added Dmitri Pavlov @DenisNardin: I did not create any neologisms. The term has been in use for a very long time and has been included in titles of various papers by Peter May. Searching for it does produce useful results, e.g., the article cited above, which is very much on topic. (It does include some stable homotopy theory, but it also has plenty of genuine algebraic topology.)
Oct 18, 2020 at 20:43 comment added Denis Nardin @DmitriPavlov I don't have alternative proposals, but I think that it ought to be mentioned when one creates a neologism, to make clear that googling the name won't produce useful results. That said, I think our conversation is no longer of much use to the OP, so I won't continue it.
Oct 18, 2020 at 20:09 comment added buck I mainly see people doing modern stable homotopy theory, and I don't know many name of people, or subjects of research going towards the topological and geometrical direction. The topics mentionned by C. Malin and D. Pavlov seem great indeed as well as the people, I'd love to know other topics if possible. I'm realizing that I'm pretty ignorant about what people do in "geometric algebraic topology/homotopy theory".
Oct 18, 2020 at 20:08 comment added buck It seems that I really needed to clarify these distinctions, indeed. So, maybe I am more into stable algebraic topology then, if it is a thing. But reading May, this "stable algebraic topology" seems to actually more of a topological ancester of stable homotopy theory right?
Oct 18, 2020 at 19:45 comment added Dmitri Pavlov @DenisNardin: May's survey was published in 1999, so predates the terminological split, and, in any case, a historical survey like this one necessarily has to cover stable homotopy theory, since at that time almost all of its applications were in topology, and the two subjects were researched together. But if you agree on the distinction, then I don't think that adding the adjective “stable” in from of “algebraic topology” creates any confusion or ambiguity as to what its subject matter could be, and I don't know any better name for it (do you?).
Oct 18, 2020 at 19:23 comment added Denis Nardin @DmitriPavlov I was referring specifically to the expression "stable algebraic topology", that I never heard before. For example most of the topics in the note by May you refer to I would classify as homotopy theory rather than algebraic topology (I'm not saying it doesn't exist -- that's why I'm asking for references). I, of course, broadly agree with Haynes and Clark on the distinction between topology and homotopy theory :).
Oct 18, 2020 at 19:13 comment added Dmitri Pavlov @DenisNardin: Here is what Haynes Miller has to say about this: “This volume may be regarded as a successor to the “Handbook of Algebraic Topology,” edited by Ioan James and published a quarter of a century ago. In calling it the “Handbook of Homotopy Theory,” I am recognizing that the discipline has expanded and deepened, and traditional questions of topology, as classically understood, are now only one of many distinct mathematical disciplines in which it has had a profound impact and which serve as sources of motivation for research directions within homotopy theory proper.”
Oct 18, 2020 at 19:03 comment added Dmitri Pavlov @DenisNardin: What terminology are you referring to? See Peter May's Stable Algebraic Topology, 1945-1966, for example. If you are referring to my making distinction between homotopy theory and algebraic topology, this is clearly articulated in the writings of Haynes Miller and Clark Barwick cited above. In fact, the main question by the OP clearly illustrates the desirability of making such a distinction.
Oct 18, 2020 at 18:36 comment added Dmitri Pavlov @buck: Part of the confusion stems from the fact that (homotopy) spaces can be modeled by topological spaces up to a weak homotopy equivalence. With respect to this, it is useful to keep in mind that results in (stable) homotopy theory remain unaffected if you replace topological spaces by simplicial sets up to a simplicial weak equivalence, or even more algebraic models of ∞-groupoids, so the fact that topological spaces can be used to model ∞-groupoids is a red herring. Some additional comments can be found here: ncatlab.org/nlab/show/homotopy%20theory%20FAQ
Oct 18, 2020 at 18:36 comment added Denis Nardin @DmitriPavlov I'll have to confess, I never heard such terminology. Can you show where it is used?
Oct 18, 2020 at 18:31 comment added Dmitri Pavlov @buck: Stable algebraic topology studies manifolds (and closely related geometric and topological objects such as polyhedra, for example) by applying tools and methods from stable homotopy theory. Stable homotopy theory studies spectra and, more generally, stable (∞,1)-categories. These have purely algebraic and categorical nature, and can be applied in many areas: algebraic geometry, algebraic topology, differential geometry, quantum field theory, etc. Of course, algebraic topology motivated many developments in (stable) homotopy theory, but this doesn't make the latter part of the former.
Oct 18, 2020 at 14:51 comment added buck It is indeed what I had in mind, D. Nardin, when I asked if stable homotopy theory hadn't become an overly algebraic theory. It seems to me that this theory gets close to a kind of homological algebra of spaces, whereas I was at start seduced by its applications really rather than by the "theory" itself. I may still have a mistaken point of view though. Also, maybe it is worth saying that I have real interest in the different geometric motivations for stable homotopy theory (the several historical ones, Kapil, should I develop?) rather than by the more algebraic later (>=1970) ones.
Oct 18, 2020 at 14:42 comment added buck Thanks to all of you. Indeed I did not know that their could be so many interactions between stable homotopy theory and geometry, thank you for all the names and also for the references on what is homotopy theory. I am realizing that maybe, I shoudn't try to persue pure stable homotopy theory and should start looking at some subjects where stable homotopy theory interacts with more geometric theories. I did not understand the difference between stable homotopy theory and stable algebraic topology though, could you (D. Pavlov) develop on that quickly ?
Oct 18, 2020 at 10:25 comment added Denis Nardin I think this question arises from a misconception that stable homotopy theory has a "geometric" perspective. Stable homotopy theory has much more in common with homological algebra (albeit being much more difficult) than with geometric topology. That said, spectra can and are used to study manifolds, in the same way as chain complexes sure do show up in many places across math. The aforementioned applications to surgery theory are one prominent example.
Oct 18, 2020 at 3:06 comment added usr0192 I will say I had your same experience - started studying algebraic topology (without a mentor) because of the stunning/surprising proofs of understandable geometric/topological questions that have made it into textbooks are 50+ years old. I gave up so I am not qualified to answer your question. Others gave great suggestions for current geometric work in homotopy theory. I wanted to mention something outside of what you are thinking: Teichmuller theory - a active area with pretty pictures, lots of connections with other areas of math. e.g. Alex Wright and others (I don't work in this area)
Oct 17, 2020 at 22:23 comment added Dmitri Pavlov There are many geometric branches of stable algebraic topology, e.g., check out the recent work of Lurie, Schommer-Pries, Galatius, Randal-Williams and others on bordism spectra, or the work on homological stability, or the Stolz–Teichner program, which provides geometric models for various spectra. Knowledge of algebraic geometry is not required for any of these.
Oct 17, 2020 at 22:17 comment added Dmitri Pavlov Homotopy theory is not a branch of algebraic topology, and stable homotopy theory is not a branch of stable algebraic topology. See, for example, Haynes Miller's preface to the Handbook of Homotopy Theory, or Clark Barwick's manifesto. You seem to indicate a clear preference for stable algebraic topology, and it is perfectly fine to do research in that area learning the necessary parts of (stable and unstable) homotopy theory on your way, as you need them.
Oct 17, 2020 at 19:58 comment added Connor Malin I think you should check out the works of Kupers, Randall-Williams, and Galatius (there are many more one could name). These guys are all able to do the hard homotopy theory that you mention to obtain geometric results, but from what I have seen they most often try to proceed through very geometric arguments. It is a very beautiful interplay between the geometric and algebraic parts of topology.
Oct 17, 2020 at 18:26 comment added Kapil What are the "initial geometric goals" of stable homotopy theory from your perspective? More generally, first ask yourself what properties of what spaces you want to study, then worry about what areas of mathematics will help you to study them. Personally, I feel that there is a lot of geometric intuition behind the papers of Kan on simplicial sets even though the presentation is abstract and algebraic.
Oct 17, 2020 at 18:17 review First posts
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Oct 17, 2020 at 18:15 comment added Thomas Rot Noone should tell you who to marry, but people can suggest a good partner. I think i’m paraphrasing arnold, but it could be someone else. I’m not a homotopy theorist, but I think there should be many problems on the geometric side of things.
Oct 17, 2020 at 18:10 history asked buck CC BY-SA 4.0
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