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In my limited perspective on the history of mathematics, I can name at least two big revolutions in Topology and Geometry (broadly construed): the introduction of Schemes in Algebraic Geometry, and the introduction of $\infty$-categories in Homotopy Theory. By "revolution" I mean a new formal framework to approach problems in a given branch of mathematics such that:

  1. Previous results can be reformulated in the new framework, at least partially.
  2. Its introduction brings substantial insight and advancement to that area.
  3. It usually requires much effort to be studied, but once obtained it provides an efficient and systematic toolkit to approach problems in the same branch.

Some of these revolutions divide the experts, whether temporarily or permanently; it could be the case that an alternative perspective makes some aspects of the theory clearer so that two frameworks can (usually with struggle) cohabit. I still count that as a 'revolution'. Let me also specify that (sub-)sub-branches are allowed and encouraged. For example, a similar shift can be seen in Knot Theory when Vassiliev began to study the cohomology of the (infinite-dimensional) space of embeddings $S^1 \to S^3$ instead of single, hand-crafted invariants on such space. It is not as radical as schemes though, since some aspects are not (yet) satisfactorily reformulated in this context.

Since revolutions are to some extent unexpected contributions, it is hard to predict them. However, I think they are not (only) the invention of a bright mathematician, but existing conditions in the mathematical 'ground' help them to stem.

To limit the subjectivity of the answers, which will be inevitable to a certain degree, I have chosen two indicators that the ground is fertile for such a revolution to happen:

  1. A sub-branch that has several interesting examples, tricks, and results, which lacks at the moment a clear pattern of strategies and a coherent formalism to explain them;
  2. A sub-branch in which formal aspects became a subject of research on its own, but no satisfactory results in terms of cohesion and efficacy have been found.

Regarding (1), I guess algebraic geometry felt a bit like that before Grothendieck. On the other hand, I see (2) happening often, e.g., in category theory, and sometimes there is a struggle to find a nice reformulation. However, some axiomatizations become accepted in time, and several models are discovered to fit the same axiomatization. I find e.g. the development of cohomology theories very rewarding from this point of view, with the study of representing spectra becoming a new focus and boosting (or initiating?) the study of the stable homotopy category.

Remark. I did my best to make this question objective, and I would like to avoid advertisements of some areas of mathematics over some others. I understand if the question will be voted to close since big-list are quite controversial, but I think I would learn a lot from answers, so I'd give anyway a try. I also have a practical motivation: new theories usually require a lot of hours of studying to be mastered, and I'd be happier to avoid that... despite being curious about their content if they are satisfactory!

Regarding branches, I am mostly interested in algebraic topology and anything in this neighborhood, but I would be happy to listen from other geometers (e.g. differential). I think opening to other areas would make the question too broad.

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    $\begingroup$ I'd add, to your list of past revolutions in Geometry: Euclid's Elements, coordinates (analytic geometry), projective geometry, unprovability of the parallel postulate, geometry over fields other than the reals. $\endgroup$
    – Andreas Blass
    Commented Jan 26 at 16:58
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    $\begingroup$ @DavidWhite No, that's not true. You can't mark a question CW. Only moderators can. You can mark an answer that you are posting as CW. But that's different. $\endgroup$
    – Asaf Karagila ♦
    Commented Jan 26 at 20:19
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    $\begingroup$ Isn't it a little odd to have a question about geometry that doesn't have a single tag with the word "geometry"? $\endgroup$
    – Deane Yang
    Commented Jan 27 at 17:12
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    $\begingroup$ From my limited perspective, revolutions come out of nowhere and blindside everyone. So they're rather hard to predict. I'm also older and I don't know that much about algebraic geometry. But it seems to me that the revolution in algebraic geometry and topology really originated in the introduction of category theory itself. It originated as an abstract seemingly empty framework for organizing proofs in algebraic topology. But it turned into much more than I think anyone imagined it would. There certainly have been revolutions within this, but I think you have to start there. $\endgroup$
    – Deane Yang
    Commented Jan 27 at 17:21
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    $\begingroup$ The title doesn’t seem to have much to do with the non-geometric examples the OP has in mind. Also, “revolutions” are almost by definition not things you can predict. All you are going to get as answers are breathless invocations of whatever is being hyped at this moment. I entered grad school 22 years ago, and there is almost no relationship between what was being hyped then and what actually lead to advances in the intervening years. I have therefore voted to close. $\endgroup$
    – Andy Putman
    Commented Jan 27 at 19:57

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I agree with the OP that it's difficult to predict revolutions. But I think condensed mathematics is an area worth paying careful attention to, given the powerful results it's already been used to prove, and the potential for even deeper results connecting geometry, topology, and number theory.

This MO thread gives some good motivation for condensed mathematics. Quoting from Tim Chow:

The starting point is the observation that the traditional way of endowing something with both a topology and an algebraic structure has some shortcomings. The simplest example is that topological abelian groups do not form an abelian category.

The nLab page explains several uses of this new field, within the past couple of years, including to "the geometrization of the local Langlands correspondence" and to analytic geometry. A bunch of references can be found on this MO thread. Also, these lecture notes are a good place to learn. I should also mention that the theory of pyknotic sets by Barwick and Haine is related.

Condensed mathematics has recently been invented by Peter Scholze and co-authors. Scholze won a Fields Medal in 2018 for "transforming arithmetic algebraic geometry over p-adic fields through his introduction of perfectoid spaces, with application to Galois representations, and for the development of new cohomology theories." He also wrote a great answer on MO about perfectoid spaces, which is another potential revolution.

Lastly, unrelated to the above, recent work of Abouzaid and Blumberg provide a new way of thinking about Floer homology. Here's an article in Quanta about it that seems to suggest it could be revolutionary.

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    $\begingroup$ Regarding the last paragraph: Floer theory as a whole seems to me to be an under-appreciated revolution in (a certain part of) geometry. For that matter, Morse theory many decades earlier was its own revolution. But I suppose a list like this risks including all techniques... $\endgroup$
    – Sam Hopkins
    Commented Jan 27 at 15:05
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    $\begingroup$ Oh yeah, I probably should have mentioned Morse theory. Also, all the stuff about topological and geometric data analysis! But, my MO time for the day is over and now I have to go do much less fun work :-( $\endgroup$
    – David White
    Commented Jan 27 at 15:07
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    $\begingroup$ Your "[recent work] of Abouzaid and Blumberg" link goes to https://Arnold%20Conjecture%20and%20Morava%20K-theory, which seems probably unintentional. Was it meant to be arxiv.org/abs/2103.01507, Abouzaid and Blumberg - Arnold conjecture and Morava K-theory? $\endgroup$
    – LSpice
    Commented Jan 27 at 15:49
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    $\begingroup$ @LSpice Yes, thanks. I fixed this. $\endgroup$
    – David White
    Commented Jan 27 at 16:29
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    $\begingroup$ I +1'd this answer, but in terms of constructive criticism: I prefer when MO answers, even to "asking for a big list of examples" questions, only contain a single example. So the answer might be improved simply by removing the last paragraph and focusing solely on condensed math (which I believe is a good example of "a new perspective that might prove revolutionary in the future, although we don't know yet"). But I also think this question as a whole is off-topic, so maybe the whole discussion is moot... $\endgroup$
    – Sam Hopkins
    Commented Jan 29 at 14:35

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