Source
r/math•
AngelTC

What makes algebraic geometry prone to revolutions?

While Im sure there's a lot of people who might consider the last homotopical shift in algebraic geometry to be part of the original categorical shift of Grothendieck in the 60s, I think that the fact that there is at the very least a cultural divide in mathematicians who work at the level of schemes and those who have seen a need to expand into the higher categorical realms is a good argument to say there's been two. There are of course also people who work with honest algebraic varieties, perhaps just with the compromise of using modern language but mostly caring about the more classical problems and techniques.

So, what is it with algebraic geometry? Are there other branches of math that have gone through something like that somewhat recently ? ( I mean, post 1900's formalization )

Are there branches of math that might go there?

Posted May 25, 2021, 10:07 PM UTC · edited

3457
asaltzGeometric Topology•

easy subjects like algebraic geometry are easier to revolutionize

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AngelTCOPAlgebraic Geometry•

shots fired

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PaboBormotAnalysis•

Math is so easy.

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XkF21WNJ•

I'm not convinced something similar isn't happening with any other active area of research.

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AngelTCOPAlgebraic Geometry•

Could you elabroate? Im very aware all branches of math move pretty fast and there are paradigm shifts and some problems gain focus over other problems all the time, but the cultural divide in AG where the language itself makes it hard to communicate with others is to a degree Im not aware something so sudden happens in other areas.

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XkF21WNJ•

Well it's possible the situation in AG truly is unique, to truly prove otherwise would require me to be up to date with the cutting edge of several branches of mathematics, which I'm not. So let's just call me an outside observer.

From an outside perspective the thing you're describing doesn't seem any different from how math has worked in any field since forever. You already mentioned the 1900's, where you've suddenly got formalists going in a complete new direction and quickly spinning of various forms of set theory, formal logic, model theory which seems to still be going on with category theoretical models of (intuionistic) set theory, various shenanigans with increasingly large ordinals. etc.

And I can't really think of an active area of mathematical research (or indeed any research) where something similar isn't happening. It seems to be very rare for scientists to all collectively agree that something is the right way. That said I'm far from up to date the cutting edge of all (or really any) branches of mathematics, so it's possible they all decided that some research should be done in the same way from now on, but I doubt it.

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Carl_LaFong•

I agree that algebraic geometry has gone through much bigger revolutions than other fields.

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hobo_stewDynamical Systems•

in partial differential equations the transition to considering weak solutions and so on seems comparable.

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grothendieck1•

I don't really agree with that. Literally every sub-field of math has its own language, terms, and toolkit that makes it hard for someone outside that field to understand at first.

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AngelTCOPAlgebraic Geometry•

For sure, the point here is the degree and how sudden it happened. I have the impression that people working on PDEs still work on PDEs, and that a random researcher in combinatorics could give a soft pitch of the structures they're interested about to another random combinatorics person in relatively short time.

This is not to say that there are very specific paradigms and techniques that cant be explained at all in a short time, it all takes time to process.

But in algebraic geometry there is not only a huge technical barrier that divides groups of people, its a whole cultural thing. People have a shared preocupation for some sort of problems but use a wide range of abstractions to study them, while still keeping some of the spirit of what they're doing in the background.

Is it true, in other branches of mathematics, that at some point there was a movement to rewrite everything in new terms, and that event produced a fracture in the field? Has that happened multiple times to 'literally every sub-field' of math?

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TazerenixComplex Geometry•

Symplectic geometry has gone through several revolutions of a similar flavour to what AG went through. First Gromov's developments showing that symplectic geometry had a distinctly topological flavour, and then the more recent move towards a purely categorical approach to symplectic topology in terms of the Fukaya category, A-inf categories and so on. The field now consists of people who study fundamental problems about symplectic geometry/topology from quite varying perspectives.

I'd also say that homotopy theory is also a good example of a similar example. Originally it was very classical (studying the homotopy theory of topological spaces) and then over the 20th century a paradigm shift occured towards a more homological algebraic approach, and eventually to the realisation that topological spaces weren't actually the natural playground of homotopy theory at all. Fundamentally the field is still concerned with the most basic important question: what are the higher homotopy groups of spheres, but it is now a vast field with many different techniques.

In my opinion algebraic geometers just have the best marketing. It helps that the invention of the scheme was a genuinely brilliant abstraction, but I'm not convinced that it was a singular event and other areas don't see similar transformations.

In a way I think algebraic geometry may have just been the last big field which went through a transformative process where a new highly abstract but very powerful definition was made. For example the definition of a topological space is in many ways as abstract and as transformative as the definition of a scheme, but its impact was felt in the 20s and 30s. Similarly for the definition of a smooth manifold, or earlier the definition of a group or ring, and so on. Perhaps it is simply the case that AG was the last big field to go through this transformation, and the subjects that are left are too esoteric to be marketed to the general pure maths audience as such.

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_felani_•

Just curious, could you expand a bit on "the realisation that topological spaces weren't actually the natural playground of homotopy theory at all"?

For reference, I'm a grad student working in symplectic geometry. I have friends who do "homotopy theory" but to me it just looks like wacky category theory voodoo that I don't quite understand. In my mind, "homotopy theory" means Hatcher chapter 4 (which I haven't even studied that thoroughly) and I know that's not representative or indicative of modern "homotopy theory"

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TazerenixComplex Geometry•

So people realised you could neatly package the information of all paths and homotopies between them with the notion of a "2-category" which is a category with objects, morphisms, and morphisms between morphisms (i.e. Map(x,y) is itself a category for every x,y in Ob(C)). Here you take the objects to be points of X, 1-morphisms to be paths, and 2-morphisms to be homotopies between paths.

If you replace every homotopy class of 1-morphisms by a single representative, you obtain a type of category called a groupoid, where every morphism is an isomorphism, so this thing is called a 2-groupoid.

But you can continue this process by also including "morphisms between morphisms between morphisms" or homotopies between homotopies. These have complicated composition rules with homotopies and with paths, but what you get is a category with 1,2, and 3-morphisms, which encodes more of the homotopy theory of the underlying space X.

And so on, until you arrive at whats called an infinity groupoid, where you record all possible homotopies between homotopies between ... and so on. Grothendieck conjectured that this groupoid should determine the underlying topological space X up to weak homotopy equivalence (homotopy hypothesis) and along the way homotopy theorists realised that really all the information of homotopy theory is encoded in the structure of these sorts of categories with higher morphisms. Now homotopy theory is concerned with the abstract study of these sorts of categories and the various ways of interpreting them (words like Kan complex, simplicial sets, etc. that I don't understand).

This also lead (along with parallel developments in homological algebra) to the general theory of infinity-categories, stacks, and so on. The general principle behind this whole area is "remember all possible morphisms between objects" rather than "throw all morphisms away" which is the more "classical" approach to many problems in mathematics. The latter is recording just the points of the quotient whereas the former remembers the entire quotient structure. If you formalise this in geometry you end up with coarse moduli spaces (quotient) and stacks (remember how you got the quotient), but it also appears in other contexts. For example if you formalise it in classical homotopy theory you get the fundamental group (quotient) v.s. the fundamental infinity-groupoid (remember the homotopies). This is now a huge area of research and there are lots of philosophical questions people have posed in various contexts (homotopy theory, homotopy type theory; a proposed new foundation of maths, homological algebra, derived categories, symplectic topology/Fukaya categories, topological QFTs, even algebraic QFT which hopes to formalise the axioms of quantum field theory in curved spacetimes).

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AngelTCOPAlgebraic Geometry•

Thanks for the answer, I do agree that probably it has a lot to do with marketing and that AG is just the more recent one that has gone through that kind of movement, Im pretty ignorant about sympletcic geometry but I do have the impression it fits in what I'm looking for as theres people working on old and new approaches alike. With topology and algebra however I think people have moved on from the earlier conceptions of how to approach them, we all agree on what a ring, and the ways to study them are more or less different in spirit.

This is not the case in AG, the main object of study comes in different flavors, and I do not believe once we will all be settling for one.

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yatima2975•

PDE's are over the reals/complexes, combinatorics is over the naturals/integers. The operative word being 'the' here, as trite as that may sound.

AG, and number theory, is being done 'over' a whole load of domains (in the loosest sense) as the same time: PIDs/UFDs/p-adics/finite fields/Noetherian or not/and I'm missing a couple here :-)

The machinery for dealing with all that generality at once is more abstract and abstruse by necessity.

There's also a historical part: sure you can do analysis over S^1 but that's just called Fourier Analysis.

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AngelTCOPAlgebraic Geometry•

Coefficients are certainly a part of why you need a lot of machinery, but its certainly not the whole reason. You could agree to work on C and require your spaces all to have nice properties, you would still experience the differences if you talk with somebody who works on pure algebraic varieties, people who will prefer to speak in terms of schemes, or some who will say things about derived stacks totally unfazed.

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[deleted]•

[deleted]

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AngelTCOPAlgebraic Geometry•

Yeah maybe, I shouldve picked a better example from an area I actually know about.

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SilchasRuinLogic•

I'd argue that the pre-Grothendieck Italian school algebraic geometry still has a successor in model theory, which has generalized in a different way.

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jheavner724Arithmetic Geometry•

Could you elaborate on what you mean?

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SilchasRuinLogic•

Varieties are the solution sets to systems of equations. If you relax the notion of equation enough to involve "elements that satisfy a first order formula in a structure" you start doing model theory.

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jheavner724Arithmetic Geometry•

Sorry, I should have been more clear. My confusion wasn’t the connection between algebraic geometry and model theory. That much is clear. But how is the model-theoretic approach a successor to the Italian school in particular? Or do you just mean variety vs scheme focused?

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SilchasRuinLogic•

I'm not an expert on the Italian school in particular, but you're right in that it's varieties vs schemes that I'm speaking of here. Elementary equivalence as the notion of equivalence works when speaking of varieties / fields / rings, but it does not function well in a scheme theoretical situation.

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Snuggly_Person•

I think algebraic geometry has an unusual amount of distance between its core subject and its main tools: you care about solutions to polynomial equations, but understand them primarily through an auxiliary geometry. It makes sense then that you could completely swap out that geometry for a 'better' one while still preserving what the field is about. On the other hand something like differential geometry seems to be more deeply about manifolds fundamentally, so swapping to some better behaved category like diffeological spaces is viewed less favourably.

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aginglifter•

It's an interesting question. It stands in stark contrast with Different Geometry.

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PaboBormotAnalysis•

Different geometry is my new favourite name for any weird as heck differential geometry.

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FinitelyGeneratedCombinatorics•

You know, we have so many things in math named "normal"; we could stand to name a few more things "different."

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hobo_stewDynamical Systems•

Different ideals exist in algebraic number theory. Literally the only place i have seen that kind of usage

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cereal_chickUndergraduate•

The different distribution just wants to be accepted for who they are.

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cereal_chickUndergraduate•

How so? I'm very interested in differential geometry, and some insight into the state of the field would be very nice.

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drzewka_mpComplex Geometry•

My view (as a grad student in the area) is that differential geometers are still asking questions fundamentally about smooth manifolds with some extra structure, be it symplectic, Riemannian, complex, Kahler, etc. Many open questions end up being PDE problems, and various methods of analysis are applied to try and solve these. For example, the famous Calabi conjecture asking about the existence of Kahler forms with special properties was settled by Yau by a very extensive analysis of a complex Monge-Ampere equation.

Which isn't to say that there are no "revolutions" in the field. The use of geometric flows (parabolic PDE applied to geometric objects) has been very successful in obtaining solutions, most notably in the Poincare conjecture proof by Perelman, but of course in many contexts since then.

That being said, it's impossible for me to comment on the entirety of the field.

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cereal_chickUndergraduate•

Ah. Thank you very much for your insight.

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HeilKaibaDifferential Geometry•

At the end of the day, to a differential geometer there is still a manifold somewhere that we care about. There are useful abstractions and plenty of abstract tools to use but ultimately we're investigating the properties or structures of that manifold (or doing some geometry on it anyway).

Algebraic geometry, as a rule, has no such hangups. You start by looking at zero sets of polynomials but this can be abstracted to discussions about categories and functors. Then you can happily study algebraic geometry with no underlying space that you care about.

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cereal_chickUndergraduate•

I know I'm on the right track, because the first sounds appealing and the second does not. Thank you!

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HeilKaibaDifferential Geometry•

Haha, I can definitely agree there. One of the main reasons I ended up in differential geometry is that I didn't understand a word of what I read about algebraic geometry.

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willbellMathematical Biology•

Algebra? The isomorphism theorems are a 20th century discovery.

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Tlapoualmatini•

Perhaps it has to do with the depth and breadth of the concept of a "space"? The richer the spaces one is interested in, the more sophisticated the tools that seem to be needed to study them. Other than algebraic geometry, many subjects that deal with variants of the concept of "space" have undergone similar "foundational revolutions" (some flavours of geometry and topology have already been mentioned in this thread).

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drzewka_mpComplex Geometry•

Perhaps the question is then whether geometry and topology are more prone to "revolutions", precisely because the concept of "space" feels vaguely-defined. Algebraic geometry may be singled out more simply because it seems to deal with anything that's not a manifold (or something close to it).

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Tlapoualmatini•

Mmm... I guess my point was not so much that the concept of "space" is vaguely defined, but rather that its scope was broadened significantly during the last century, not least due to the introduction of schemes and topoi. In some sense, one could see the current "homotopical revolution" of algebraic geometry as a natural continuation of these ideas. Indeed, for various reasons, one may want to consider sheaves not of sets or rings, but of derived such things (homotopy types, DG algebras, ring spectra, etc) which is what some people in derived algebraic geometry and stable homotopy theory seem to be interested in nowadays.

But I agree with you that, in a way, the concept of a space is not well defined (and it might not matter much what it is).

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drzewka_mpComplex Geometry•

Being outside AG, these abstractions seem to only make it harder for me to understand what this group of geometers is working on, but I understand that such problems may be inevitable.

As for the last comment, I suspect you're right. There's an advantage to having flexible notions of space, if only to be able to study lots of interesting things. In this sense, I think something like arithmetic geometry can study some strange geometric objects.

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Tlapoualmatini•

There's an advantage to having flexible notions of space, if only to be able to study lots of interesting things.

Definitely. It is remarkable the kind of mathematical objects which are amenable to "geometric" methods.

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derp_trooper•

I would guess most mathematicians who aren't algebraic geometers or work in related fields won't have a deep understanding of the revolution you are talking about. I know Grothendieck is said to have caused a paradigm shift in the field, but I know that more as a piece of trivia than truly understand the significance of it. Ofcourse I shouldn't project my ignorance onto everyone else! So only someone who has a good understanding of modern algebraic geometry and is an expert in some unrelated field, could claim with some authority whether such a revolution ever took place in that field.

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AngelTCOPAlgebraic Geometry•

That's fair, probably you have to be a bit on the loop to know the extent of what I'm talking about.

To give you what I think is a good example, what Grothendieck did was alike the formalism that analysis went through with Cauchy and the whole formalism of set theory all rolled into one, then 40 years later inspired again by that first movement were now going through another big change of similar fashion.

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agguy12•

I am not really convinced that this higher-categorical stuff is going to be a "revolution" for all of AG so much as an interesting new direction that some people work on, and the rest of us ignore.

I work on kind of old-fashioned stuff, some kind of birational geometry. Many of the questions are totally classical, but of course everyone uses Grothendieckian technology. You can't get anywhere without schemes and cohomology. It's totally unavoidable, even if all you care about is something like curves in P^3.

The higher categorical stuff is not like this. I have yet to see a paper I care about that uses any of this newfangled higher category stuff; it just hasn't had much impact on the really classical areas despite having been around for awhile. (This is in strong contrast to the first revolution, which was swift and complete.) I have some vague understanding of it, and don't deny that it's interesting, but I don't really think it's going to rewrite the foundations for those of us working on old-fashioned topics in the way that the last revolution did. (And I do not think I'm in the minority here; if you look through the most recent talks at general algebraic geometry conferences like AGNES, few if any of them are going to use the high-tech stuff; it's remained fairly niche. Of course this is partially reflective of who organizes these events, but I think the so-called "revolution" hasn't touched the vast majority of algebraic geometers.)

Derived categories are a bit of an in-between place for me. I have a couple papers where they appear, and they have surely had some nice applications to classical questions, but for my purposes it feels more an extra technique that's available, rather than some grand rewrite of how I think about things. Basically just an additional invariant with more than the usual technical overhead. Maybe there are a few infiltrations by fancier things (Bridgeland stability conditions for example) but overall, I am not feeling revolutionized. I guess maybe I'm just an old fogey by now.

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AngelTCOPAlgebraic Geometry•

Its perfectly fair to say there's a large number of algebraic geometers who wont need higher categorical machinery in the future, that we can agree on. But Im skeptical in that there is not a barrier in abstraction before that, I think there's still plenty of work produced on AG that is truly more classical in spirit and where you wouldnt even see the word scheme or stack featured that much if at all. The formalism is probably there in the background doing its thing but its safe to think and work classically without caring about the inner workings of the theory.

Im not here to try to convert people into the homotopical ways, you dont need to feel revolutionized.

On the other hand

Derived categories are a bit of an in-between place for me. I have a couple papers where they appear, and they have some applications to classical questions, but to me it feels more an extra technique that's available, rather than some grand rewrite of how I think about things

The heavy use of derived categories is a givaway that the homotopical ideas are there, you dont need to care about the formalism and the role that they play at large, but* maybe the ideas are much closer than one can think :)

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sunlitlakeRepresentation Theory•

The emergence of condensed mathematics, if successful, will probably teach us that it’s not algebraic geometry that’s prone to revolutions, but the main modern tools. In representation theory, for example, importing algebro-geometric techniques in the 1980s has lead to a similar cultural divide.

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RevolutionaryMoney•

Can you be more precise about what you are trying to describe?

For instance, minimal surfaces can be thought of as parameterized surfaces and this point of view quickly leads to relationships with complex analysis while they can also be thought of as subsets or in terms of their Hausdorff measure and this leads to relationships with geometric measure theory. The two points of view are quite distinct (even though they are studying the same concrete object) with very different emphasis on techniques. Is this an example of the phenomena you are describing?

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AngelTCOPAlgebraic Geometry•

Sort of but it all happens internally withing algebraic geometry itself. As I said, maybe I care about algebraic varieties and then I chose to work with classical ones, zeros of polynomials and what not and then I use somewhat classical techniques, I could care about the same object and study them as integral schemes of finite type, same object but different model, then on spirit I could be using some similar techniques as before but I'd be saying the word cohomology a lot, and then I could still care about algebraic varieties but then I'd be talking about a derived stack version, not the same object but very close, considerably more complicated model and the techniques are also there in spirit but I don't even know what a line is anymore, I'm all about using Yoneda and now everything is an infinity category, but yeah still doing geometry somehow.

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RevolutionaryMoney•

I think this happens in many areas of mathematics, there is just usually less nee or opportunity for abstraction. For instance, a harmonic function on a two dimensional domain, can be viewed from the point of view of complex analysis, probability theory, harmonic analysis, various flavors of PDE theory (e.g. maximum principle, energy methods or mean value properties) and can be generalized in a number of different directions. However, a lot of this stuff tends not to be formalized, but left more at the level of heuristics that one can try to apply to a wide range of problems -- that might be one distinction.

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[deleted]•

[deleted]

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linusrauling•

It seems to me that Topology goes through a revolution every other month...

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AngelTCOPAlgebraic Geometry•

How so?

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linusrauling•

I don't really keep up in the subject, but every time I turn around it seems the paradigm has shifted. I'm dating myself here, but I got off at the Spectra stop (which seemed some vast generalization of representable functor...)

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SigurdtheVolsung•

I think it is because of Grothendieck. He gave algebraic geometry a new language. If you look at other fields there are certainly revolutions happening but it is hard to find someone who explicitly thinks the foundations of this field need to be rewritten in much more abstract language. Grothendieck and really his school along with him made this possible. And even then Grothendieck was not just trying to purposefully make things abstract but to find the right definitions. I think the other thing is that the concept of space that algebraic geometry deals with is deeply close to many other fields and the kind of structures that exist are amenable to category theory. I think the only way for this to happen to other fields is when the people working in these fields want to explicitly change the foundations. In Esquisse d'un Programme Grothendieck proposes a new kind of topology the tame or moderated topology. He says that he sent it to other topologists but was dismissed. This is much closer to what most mathematicians think. And to their defense redoing the foundations of a field is extremely difficult.

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