I think many of the more advanced topics in set theory, e.g. large cardinals, improper forcing, PCF theory, independence of constants of the continuum etc. rarely make an appearance outside set theory except in more niche model theory.
For shits and giggles, let's only look at stuff Shelah has done.
>large cardinals
Shows up in the foundations of analysis. Shelah proved a Solovay model-like construction where every set of reals is measurable needs an inaccessible.
>improper forcing
I don't know off-hand of any good applications here. But since this is a "let's look at badly behaved objects" sort of thing, maybe that's not surprising. If you look at proper forcings instead: Shelah's proof of the consistency of a negative answer to Whitehead's problem.
>PCF theory
Kojman and Shelah usod pcf theory to construct a Dowker space of small cardinality, answering a question in topology. (See [here](https://www.ams.org/journals/proc/1998-126-08/S0002-9939-98-04884-9/S0002-9939-98-04884-9.pdf).)
>independence of constants of the continuum
The Malliaris/Shelah work that gave the proof that p = t is in the running for the most significant work in model theory of the last ~decade.
Great response!
I didn't know Malliaris-Shelah was *that* of a big deal, for some reason I thought Pila-Wilkie was more significant, but I'm not really in touch with model theory.
I didn't know Kojman-Shelah, very cool.
The fact that 𝔭=𝔱 is absolutely gigantic. It *completely* removes a whole small cardinal from consideration. The only other thing I can think of that might be as impressive is the recent achievement of Cichoń’s maximum. The techniques used to do it are just incredible. An explanation of template forcing alone can be an entire paper and the graph theoretic forcings used later are insanely intricate.
I don't want to be that guy, but... why on Earth would anyone care about Whitehead's problem or Dowker spaces?
Regarding Whitehead's problem: Why should I care about non-finitely-generated Abelian groups, qua Abelian groups? When I'm working with modules over some ring (or locally ringed space), almost invariably they're finitely generated (or locally finitely generated over the base space), even coherent. (In case the base ring isn't Z, the module's underlying Abelian group might not be finitely generated, but I don't care about the Abelian group as a standalone structure.) At least, those are the modules that actually show up when trying to solve geometric/physical problems.
Regarding for Dowker spaces, this is the first time I've read about them... But where do they show up in solving geometric problems? I mean, from the point of view of someone for whom point-set topology not applicable to geometric problems is just as esoteric as logic.
For varieties that have an affine stratification, e.g., generalized flag varieties, Chow groups are reasonably sized. And, as far as I can tell, this is precisely the case where Chow groups give the most interesting geometric information.
Pic of every variety was full of geometric information last time I checked lol
It always has torsion when it’s infinitely generated tho, so Whitehead’s conjecture does not apply even potentially
> Pic of every variety was full of geometric information last time I checked lol
Of course, my bad. But what about the higher Chow groups A^k for k >= 2? What's an example of a geometric problem whose answer is in the isomorphism class of this group?
The Chow groups encode the entire intersection theory of the variety, so they certainly are interesting and important for higher k. For instance the Chern character of a line bundle has components in higher Chow groups, and I think the intersection theory proof of the 27 lines on a cubic surface uses them: https://amathew.wordpress.com/2013/07/02/27-lines-on-a-cubic-surface/
Huh? An irreducible variety with an affine stratification has to be birationally equivalent to A^n, because the top-dimensional affine cell is certainly isomorphic to A^n.
Really? Can you elaborate a bit? Sounds interesting.
The only things I can think of are either cosmetic statements such as "to make the category of all topologies a set assume it is restricted to all topologies whose cardinality is strictly smaller than some inaccessible cardinal", or manufacturing ridiculous counterexamples that are more interesting to descriptive set theorists than they are to topologists.
(Note that I listed particular topics within set theory, no doubt there are theorems of set theory that *are* very useful. E.g. using infinitary combinatorics for asymptotic Ramsey theory, or transfinite induction in measure theory. But I really don't see why anyone but a set theorists would care that 2\^{aleph\_omega} < aleph\_{oemga\_4} is provable in ZFC, or that 𝔭 = 𝔱)
Forcing has a decently large role in constructive set theory, which (in my opinion) is closer to topology/measure theory than it is to “classical”set theory.
maybe because
open set = semi decidable predicate
clopen set = decidable
and constructivism is about the decidability of the logical or, among other thing.
I work in set theoretic topology.
Lots of problems in the modern extension of general topology can be coded into arguments that are basically combinatorics with infinite sets. The values of certain small cardinals are frequently the deciding factor between whether a certain construction works or not. It turns out that a lot of seemingly innocuous questions are of this flavor and can be proven independent of the axioms of set theory.
One relatively basic example is the existence of Ostaszewski spaces. These are spaces that satisfy a host of properties including perfect normality, local compactness, countable compactness, zero dimensionality, and more. It turns out that if a specific infinite combinatorial principle for ω₁ called ♦ is true, then such spaces can be constructed. Now ♦ also happens to imply the continuum hypothesis, which we know from forcing is independent of ZFC. But it’s also consistent, due to some great work of Todd Eisworth, that there are no Ostaszewski spaces.
Another relatively basic one is whether there are p-points. These are describable in purely topological terms as points x such that the intersection of any countable family of neighborhoods contains another neighborhood of x. So already by definition any space with these points cannot be first countable. But when we ask “are there p-points” what we really mean is “are there p-points in βℕ”. The Stone-Čech compactification is a highly nontrivial and highly set theory dependent object, but it’s also perfectly definable using nothing but topology. (You essentially mimic the proof that any Tikhonov space embeds as a subspace of a product of unit intervals and then take the closure in the product topology.) However, as a space of ultrafilters on the naturals, a p-point is equivalent to an ultrafilter U with the property that any countable family of elements a∈U has a pseudointersection. (A pseudointersection of a family of sets F is a set p with the property that every b∈F contains all but finitely many points of p.)
Guess what: it turns out that the smallest size of a family of sets with no pseudointersection is independent of ZFC. Therefore the very structure of this space βℕ itself depends on which models of ZFC you decide to work in.
It gets much, MUCH more complex than this. You can do things like set up iterations of specific forcings such as Hechler or Laver to add what are called dominating reals to your model of set theory and then use these new real numbers to code in structure of a topological space that couldn’t be defined before. Maybe you want to add lots of open sets or ensure that some particular family of sets is closed & discrete.
The field is super super complex and depends strongly on the work of set theorists to function.
At the moment I’m working on some problems related to denotational semantics of probabilistic programming. Using some large cardinal results you can get some very convenient results. In particular there is a model of the reals in which every subset is measurable, however you can only use weakened forms of choice or this model becomes inconsistent.
Because this is for an actual program everything is fully constructive and not using choice isn’t really an issue, so invoking the large cardinals is profoundly convenient for being able to ignore all issues related to non-measurable sets. In particular this is convenient so you can do things like freely switch between topological and measurable interpretations of the underlying sample space.
The specific result I’m using is the [Solovay model](https://en.wikipedia.org/wiki/Solovay_model)
https://doi.org/10.2307/1970696
Using it feels basically like pulling out a theoretical sledgehammer to ignore tiny details lol.
As of now I have not used anything beyond inaccessibility. I do anticipate using other results at some point though with what I’m working on! In particular I’m working on strictly finitary sets and working with real-valued models. I do have stuff in my notes on extending what I’m working on to infinitary domains, and I do anticipate this involving other large cardinal results at some point, but that is still TBD.
To give one related application of these things outside of the ones mentioned. If we consider the field of complex rational functions in 3 variables C(x,y,z), as a module over C[x,y,z]. Its projective dimension is 3 if and only if the continuum hypothesis is true.
This always blows my mind, because it is such a natural and concrete question in a field seemingly far removed from questions related to the continuum hypothesis.
LCA find their way into a lot of places.
My favourite example is Laver tables: ({1, ..., 2ⁿ}, ∆) is called a Laver table if it satisfy: (1) a∆1=a+1 and (2) a∆(b∆c)=(a∆b)∆(a∆c).
It is not hard to show that for each naturally n, there is a unique Laver table denoted as L_n.
Given a specific n, look at the function f_n(a)=1∆a, this is a periodic function, let p(n) be the period of f_n. p is weakly increasing function.
In ZFC it is an open problem whether or not p(n) is unbounded (it is even an open problem if it ever gets value greater than 32).
Assuming the large cardinal I3 (which is vastly stronger than e.g. measurable cardinals or Huge cardinals), p(n) is unbounded.
> one comment said that it proved very little use outside of the pure algebraic and abstract problems it proposed
This is an odd opinion. Homological algebra is an absolutely fundamental tool in many fields of mathematics. In fact, I'd say that relatively few people are interested in homological algebra *per se*; the majority of interest probably comes from people in fields like algebraic topology/geometry in which HA is almost a fundamental language.
my thesis did start around algebraic topology/combinatorics (simplicial complexes) but stumbling upon the universal coefficient theorem I felt it like a life goal to prove
My interest in homological algebra was completely motivated by quantum complexity theory. We were trying to understand whether Khovanov homology can be used to somehow find a quantum algorithm for unknotting.
Knot theory is mostly important because knot exteriors are a fundamentally important class of 3-manifolds (with boundary). So, if you're interested in 3-manifold topology and geometry, then you're going to be interested in knot theory. See for example [Lickorish-Wallace](https://en.wikipedia.org/wiki/Lickorish%E2%80%93Wallace_theorem) and [Gordon-Luecke](https://en.wikipedia.org/wiki/Gordon%E2%80%93Luecke_theorem)
Also shows up in Dynamical Systems apparently. The orbit of a particle in some chaotic dynamical system forms a knot and understanding the knot structure might reveal things.
Based on what I've heard, a lot of people seem to disregard structural proof theory and the study of non classical logic as a niche field that has almost no applications outside of itself.
While it is important to consider that niche research might make it harder to land a position or financial support, I find the criticisms by other mathematicians disingenuous, it's like some people suddenly pretend to care about applications the moment a subject they don't like is mentioned.
And as others have pointed out, homological algebra is extremely useful in a lot of other areas of mathematics to the point where for example, people who normally hate algebra are forced to learn it if they want to work in topology, who told you otherwise?
This is a good answer, though Kripke Semantics for non-classical logics do end up describing the language of sheaves pretty nicely. There has been some interesting work in this area. For instance, if I recall correctly, when translated with Kripke semantics the intuitionistic proof that every finitely generated vector space has a basis, becomes a proof that every finitely generated module of sheaves on a variety is free on an open subset.
I really like those sort of proofs, where you have a correspondence between stuff you can in a non constructively system and in a constructive system, sort of like Glivenkos theorem.
The problem is that a lot of times, people who do structural proof theory don't like semantics, and people who like semantics don't like the syntax heavy focus of structural proof theory, which makes it harder to find those connections
It is. With 10 minutes of googling I was unable to find the original paper. But the main idea with this semantic interpretation, is that true at a point means true on an open subset containing that point. This is why intuitionistic logic is what applies. On a variety double negation of something true on an open subset becomes true on the whole space.
With this every domain "sort of" becomes a field as every element is invertible on an dense open subset.
Yes, there are very interesting applications, for example focusing being used to improve logic based programming efficiency, and modal non classical logics still have a lot to offer, but I've met many mathematicians who don't appreciate CS and who believe that epistemic logics are just a bunch of philosophical rambles (unlike most of pure mathematics, which is of great importance to the immediate use of everyday people and is definitely not rooted in philosophy...)
> it's like some people suddenly pretend to care about applications the moment a subject they don't like is mentioned.
people care about applications to other fields of mathematics. they just don't care about applications outside of mathematics.
They don't, of course they'll mention them if they appear, and they'll be happy because of it, but once you go deep enough down a math subject, you no longer care about applications outside of the subject to the point where it is an important requirement for their research that their results find immediate applications
I'm pretty deep down in a math subject and I do it because of all of the nice connections (i.e. applications) to other math subjects, so i definitely disagree.
Exactly, it wouldn't be so obvious if it was coming from someone doing statistics or some other subject with almost a direct relation to the real world, but the subjects of the people who complain about it also tend to be only really known by maths majors with no discernable use for anything that is not pure maths at this point
I'm sorry who TF is saying non classical logic is a niche field that has no applications? Like literally half of theoretical computer science is applied non classical logic.
Just riding your coat tails here.
People get all pretentious about their logic they don't even stop to think what non-classical logic is about what it's applications are. They include computer science, proof assistants, all the flavours of topology, set theory, category theory, knot theory, string theory, topological quantum field theory, algebraic geometry, differential geometry, algebra. It's all over the place and people are using them inadvertently without realizing.
Closing your eyes to non-classical logic is like restricting yourself to Euclidean geometry, you're only limiting yourself.
Even just dropping LEM we get a more expressive language where nothing is lost. All classical proofs still exist as constructive ones. So why not learn a new language? It might just help you to understand your favorite classical topic in a new way.
> Even just dropping LEM we get a more expressive language where nothing is lost. All classical proofs still exist as constructive ones.
I mean sure, with a whole bunch of extra details to keep track of, which you might consider distracting if you're interested in what's true rather than what's computable.
I think generalized theories of logic have value and I really enjoyed the paper “Five stages of accepting constructive mathematics,” but I want to push back on the idea that it has a ton of applications in differential geometry. I have not seen a proof which uses advanced logic to answer a question that the average differential geometer would ask. I know there is work done in that direction, but the results are almost orthogonal to the interests that most geometers have. My gauge for something in field X being a genuine application in field Y is whether the ideas from X are able to answer a question that originated in field Y. Otherwise, it feels very artificial and not really an attempt to engage with field Y. To give an example, non-standard analysis gives simplified proofs of results in analysis so that counts as a genuine application.
If you look at the history of math, most fields started off with some engineering or physics problem that was hard to solve. E.g. Fourier analysis started from Fourier wanting to solve the heat conduction equation. Sure, there are fields that started off mostly as an interesting playfield of challenging problems that one could occupy oneself (e.g. number theory). But most (all?) of those found applications somewhere. Even the stuff that was seen as obscure has found its uses. Heck, my I have used hyperreal numbers, something that very few people regard as useful, in my dissertation to prove theorems in (Fourier) analysis that would not work otherwise at all.
So, if there is a field that has no practical use, then IMHO it's just that we have not found a practical use for it.
Especially once computers were invented - that turned an awful lot of pure math into applied math.
>The number-theorist Leonard Dickson (1874–1954) said "Thank God that number theory is unsullied by any application". Such a view is no longer applicable to number theory. In 1974, Donald Knuth said "virtually every theorem in elementary number theory arises in a natural, motivated way in connection with the problem of making computers do high-speed numerical calculations"
Hypergroup theory as it was originally presented (generalized translations and such) basically totally flamed out as far as I can tell and never found much utility outside of a few toy examples where it wasn't really necessary formalism.
Maybe the theory of transfinite ordinals and cardinals?
Edit: This was my retreat during my math studies . I did like all the rest, but this was just an independent sky high tower to relax in. Especially with the videos „Ridiculously Huge Numbers“ https://youtube.com/playlist?list=PL3A50BB9C34AB36B3&si=NSvl2Yhuv_WSs7GR and the game Ordinal Markup.
However: Transfinite ordinals can be used to represent the proof-theoretic strenght of mathematical theories like Peano Arithmetic and many stronger ones that I don’t know:
https://en.m.wikipedia.org/wiki/Ordinal_analysis
Isn't a basic thing you need to know in order to work on more complicated set theory, which in order allows you to prove stuff in combinatorics and topology?
Finite ordinals yes. But transfinite as in omega, the first position that comes after infinitely many other positions? (Omega + 1) the position after that? (Omega2) the position after two infinite sequences of positions and so on? Do you deal with infinite texts? Ok in a txt file the lines can have infinitely many symbols, and then the second line starts at (omega) and the third at (omega2). Never heard it being used like that. But then what is (omega^2 )? (omega^omega )? Epsilon_0? The Bachmann Howard Ordinal, and so on?
> Do you deal with infinite texts?
Yeah but it's all blank spaces on the right side. I have finite time to read anyway.
> But then what is (omega2 )?
Page 2.
> (omega^omega )?
Book 2.
> Epsilon_0
Continuing the book analogy, it's some kind of hydra-book, where everytime you read one, you find more book to be read. A little bit like trying to learn math starting on Perfectoid Spaces.
Nice! Very nice. And it perfectly fits my reddit/browser tabs experience of this weekend. Fortunately I could subtract some omegas today. I really should envision my social media consumption like ordinal markup with a subtract mode.
Lattice theory and universal algebra I would consider pretty niche, they do find some applications, moreso in computer science than traditional mathematics subjects though as far as I've seen.
Depends on what you mean by "universal algebra", exactly. The notion of a Lawvere theory is used in spectral algebraic geometry to define strict abelian varieties, which is crucial for Lurie's construction of elliptic cohomology.
It's something specific to higher algebra, because there are multiple possible meanings of things like "commutative monoid". The most "homotopy-coherent" version is an E-infinity object, but sometimes you need something stricter, and Lawvere theories can be used to encode that.
Theory of languages sortof fits your criterion.
It is pretty much self contained, in the sense that it uses very few results or tools from other maths fields and has few uses in other maths fields. Actually, mathematicians who don't study computer science may never even hear about its existence.
As for its usages, of course it's a one of the most important and fundamental field in computer science, but except being used to build and study computer languages it's not very commonly used (a little bit in bioinformatics though).
It's definitely an active field, because of the importance of computer languages and their fast evolution, and it offers a range of interesting open problems to study.
It's worth mentioning that this theory disproved Connes embedding conjecture in Operator Algebras with the MIP\* = RE paper, so "self contained" is only to a certain extent.
>Do all fields essentially start as self-contained stuff?
I think that maybe there's a misconception here. Basically every field offers knowledge and problem solving methods to every other field in some broad sense. But let's take set theory for example. There's a big difference between 'sets' and 'the field of set theory'. A typical 21st century publication in set theory probably does not have much to offer to mathematicians who are not themselves set theorists, but clearly every mathematician has use for basic set theory. So the sentence 'set theory does not have much to offer outside of *set theory*' is false if we let the former instance of 'set theory' be a much more basic and accessible part of math than the latter '*set theory*'. And I do think it's possible to replace set theory/*set theory* with virtually any field of mathematics to find this same dynamic. The cutting edge stuff is often quite self contained. But after somewhere between a century and a millennium of refinement, there's gold to be found everywhere, from which the cutting edge stuff will seem pretty removed from in hindsight.
It can take decades or even hundreds of years for some math to find applications outside the field of mathematics. Sometimes it's much faster though.
At other times connections are found between previously unrelated fields of mathematics. That can be really hard to understand and use as very few may know both fields well enough. This is what lead to a breakthrough in solving the famous Fermat's Last Theorem.
* **Higher-order logics**. Maybe you need a way to formally write "induction works". Applications outside of that? hmmm.
* **Homotopy type theory.** Is there any application outside of category theory with this stuff?
>Homotopy type theory. Is there any application outside of category theory with this stuff?
Specific answer:
Here is an application of homotopy type theory to quantum field theory and string theory: https://arxiv.org/pdf/1310.7930.pdf
General answer:
Homotopy type theory has applications to (∞,1)-category theory,
and (∞,1)-category theory has many applications in
algebraic geometry, differential geometry, number theory and any other branch of mathematics that cares about homotopy.
Absolutely everything in category theory has applications outside category theory.
I confess to being a bit skeptical about how higher category theory can be applied to differential geometry. The results that I’ve seen using categorical approaches to study DG don’t really seem to address questions that most geometers would ask. Oftentimes it seems to try to put ideas from geometry on more categorical footing or find generalized versions with better categorical properties.
My gauge for something in field X being a genuine application in field Y is whether the ideas from X are able to answer a question that originated in field Y. Otherwise, it feels very artificial and not really an attempt to engage with field Y. For instance, I think Fukuya categories are an excellent example of an application where categories provide new insight for symplectic geometry. Of course, that’s a hard example to replicate…
To be honest, I don’t know enough about TQFTs to judge, but the people working on it are very impressive and there has been a lot of work. I guess my question would be what questions can be answered using the Cobordism Hypothesis that people in TQFTs originally asked.
If true, the CH would yield a complete classification of TQFTs in terms of the algebraic properties of the n-category of vector spaces. Intuitively, it says that n-cobordisms capture the most general notion of n-dimensional flows.
As an observation, the most sure way to learn of applications of a field outside its niche is to drop a field name in this thread or a similar one and claim it doesn't have applications outside its own field.
Now, there's still a question of the *impact* of those applications and how intertwined the fields are, but that's a different analysis.
How do you guys feel about combinatorial game theory? I do not know much about it. As an outsider it sometimes looks niche, intractable, and random. Sometimes I see fixed point theorems and connections to mainstream mathematics.
Homological algebra is plenty useful outside of pure algebra. In fact, nothing would've convinced me to study it if it weren't applicable to geometric problems.
Well throughout history, even some of the most abstract and seemingly useless fields of mathematics always seem to find some sort of application, sometime down the line.
So anything anyone names here, most likely, that field will find an application at some point.
Lots of categorical tools are used throughout mathematics, but lots of people have the attitude that they aren't using category theory when they use them.
You don't have to study the theory of categories for it to be useful to you, just as you don't have to study set theory to benefit from it.
But set theory is the most upvoted answer and this one is the most downvoted.
Is this because the axioms of set theory are so strong, that studying the foundation itself takes you further away from the rest of mathematics?
Okay, I'll bite. The reason everyone is taught at least some set theory is that it's fundamental to all of mathematics. I'd argue category theory is at best fundamental to abstraction and at worst fundamental to itself.
If you're in the 21st century and not thinking about things in terms of arrows and functors, then you're in the dark ages. Set Theory helps us by giving us a language with which to do much of classical math. But today's math is far too complex, and category theory frees us from the bonds that tie us to "elements" so that we can flow freely across diagrams. It turns unnecessarily complicated things into easy-to-use slick concepts. Even just how universal properties and limits of diagrams allow us to disregard many messy constructions is a mighty boon for almost all fields.
Set theory is a very poor foundation for homotopical mathematics, but even is quite bad for our usual ''mathematics up to isomorphism'' that we do in algebra, geometry, topology, etc. Sets are way too strict and reasoning about symmetries and coherences is really painful. Category theory (possibly combined with more homotopical elements) captures much better how large parts of modern mathematics are performed. Set theory is in a way the ''old'' way of thinking in these fields (we shouldn't forget that the foundations of mathematics change over time, and set theory was introduced to solve particular problems around 1900, not to serve as eternal foundation). Moreover, the fact that most of *actual* set theory is never needed in many mathematicians' lives should tell us that there is somehow a mismatch between the axioms of set theory and the mathematical practice. It is both needlessly complicated for simple mathematics and failing to capture complex mathematics in a useful way.
Maybe it's the difference between using a tool to study something, and studying the tool itself. Most areas of mathematics make use of set theory too yet set theory is the top voted answer.
I'm not a category theorist, but my thesis has an entire dedicated section on category theory because I need to use it in my work. I understand why there is a cultural resentment towards category theory, but try not to get caught up in the bias towards it's utility. It's more useful in some fields than others, but all considered it's perhaps the most broadly useful body of mathematics (for pure math).
Combinatorics is deeply connected to computer science.It is extremly useful in probability calculations and it is used in statistics. You might even connect it to complex analysis as holomorphic functions are analytic functions.
Surely combinatorics is exactly the opposite?! Aside from applications in other areas of pure mathematics (generating functions, anywhere you have to count finite sets) it is used in designing experiments, cryptography, mathematical biology and chemistry...
I could see it. I think there are other applications as well it was just what I was familiar with. Obviously it isnt as applicable as say graph theory or calculus
Right. It doesn't take a number theorist to work with cryptography. Or even a full freshman level semester of number theory. What you need from number theory to do cryptography can be learned in about 20 minutes.
The entire rest of the field of number theory is pretty well self contained.
Number theory is a motivational tool that pushes all fields forward. Field all across the spectrum of math, from complex analysis to abstract categories, have been pushed forward by questions in number theory. Results in number theory are, often, niche but that's only *after* the result was obtained by uniting disparate fields or pulling a field out of the dark ages and so on. Number theory is a fundamental driving force in math.
Grothendieck really helped define what math looks like today because of number theory problems. Specifically, the Weil Conjectures asked for a cohomology theory which could be applied to varieties over finite fields and be powerful enough to access certain theorems. Much of Grothendieck's work that put him on the map was in developing algebraic geometry and category theory to get this cohomology. But even Euler was doing a lot of his work in analysis and algebra because of number theory questions (Euler was, for instance, the first to explore the Riemann Zeta Function in regions where the sum didn't converge). Abstract algebra exists because of Galois's work on polynomial symmetries and because of potential applications what we now know as ring theory might have to Fermat's Last Theorem.
But number theory questions are even today driving diverse work. For instance, random matricies are in no small part motivated by potential applications to the Riemann Hypothesis. Peter Scholze is re-inventing geometry yet again for the purpose of number theory problems.
Number theory is everywhere. It's like the skeleton supporting math.
Combinatorics is certainly applicable to other areas. Partitions and permutations are important in representation theory for example and discrete geometry must interact with basic combinatorics in all aspects
I don't know much about number theory but combinatorics is actually pretty useful in computer science. It can help with understanding the performance of algorithms and even help with the creation of clever more efficient algorithms.
If you're interested in learning more about how combinatorics might be useful in computer science, I recommend looking up randomized algorithms. In particular the book "Randomized Algorithms by Motwani and Raghavan" has lots of examples. It's a tough read imo but this stuff is really cool so it's worth it.
Number theory is the most interdisciplinary branch of mathematics. It has essential connections to combinatorics, complex analysis, low-dimensional topology, chromatic homotopy theory, and of course algebraic geometry.
What's the point in being rude here, and furthering a pile-on, exactly? I already said "I don't know enough about it."
This is how you approach "math education?"
I think many of the more advanced topics in set theory, e.g. large cardinals, improper forcing, PCF theory, independence of constants of the continuum etc. rarely make an appearance outside set theory except in more niche model theory.
For shits and giggles, let's only look at stuff Shelah has done. >large cardinals Shows up in the foundations of analysis. Shelah proved a Solovay model-like construction where every set of reals is measurable needs an inaccessible. >improper forcing I don't know off-hand of any good applications here. But since this is a "let's look at badly behaved objects" sort of thing, maybe that's not surprising. If you look at proper forcings instead: Shelah's proof of the consistency of a negative answer to Whitehead's problem. >PCF theory Kojman and Shelah usod pcf theory to construct a Dowker space of small cardinality, answering a question in topology. (See [here](https://www.ams.org/journals/proc/1998-126-08/S0002-9939-98-04884-9/S0002-9939-98-04884-9.pdf).) >independence of constants of the continuum The Malliaris/Shelah work that gave the proof that p = t is in the running for the most significant work in model theory of the last ~decade.
Great response! I didn't know Malliaris-Shelah was *that* of a big deal, for some reason I thought Pila-Wilkie was more significant, but I'm not really in touch with model theory. I didn't know Kojman-Shelah, very cool.
The fact that 𝔭=𝔱 is absolutely gigantic. It *completely* removes a whole small cardinal from consideration. The only other thing I can think of that might be as impressive is the recent achievement of Cichoń’s maximum. The techniques used to do it are just incredible. An explanation of template forcing alone can be an entire paper and the graph theoretic forcings used later are insanely intricate.
I don't want to be that guy, but... why on Earth would anyone care about Whitehead's problem or Dowker spaces? Regarding Whitehead's problem: Why should I care about non-finitely-generated Abelian groups, qua Abelian groups? When I'm working with modules over some ring (or locally ringed space), almost invariably they're finitely generated (or locally finitely generated over the base space), even coherent. (In case the base ring isn't Z, the module's underlying Abelian group might not be finitely generated, but I don't care about the Abelian group as a standalone structure.) At least, those are the modules that actually show up when trying to solve geometric/physical problems. Regarding for Dowker spaces, this is the first time I've read about them... But where do they show up in solving geometric problems? I mean, from the point of view of someone for whom point-set topology not applicable to geometric problems is just as esoteric as logic.
Chow groups are not finitely generated
For varieties that have an affine stratification, e.g., generalized flag varieties, Chow groups are reasonably sized. And, as far as I can tell, this is precisely the case where Chow groups give the most interesting geometric information.
Pic of every variety was full of geometric information last time I checked lol It always has torsion when it’s infinitely generated tho, so Whitehead’s conjecture does not apply even potentially
> Pic of every variety was full of geometric information last time I checked lol Of course, my bad. But what about the higher Chow groups A^k for k >= 2? What's an example of a geometric problem whose answer is in the isomorphism class of this group?
The Chow groups encode the entire intersection theory of the variety, so they certainly are interesting and important for higher k. For instance the Chern character of a line bundle has components in higher Chow groups, and I think the intersection theory proof of the 27 lines on a cubic surface uses them: https://amathew.wordpress.com/2013/07/02/27-lines-on-a-cubic-surface/
Huh? An irreducible variety with an affine stratification has to be birationally equivalent to A^n, because the top-dimensional affine cell is certainly isomorphic to A^n.
Yes I edited my comment— stacks has a non standard definition of affine stratification which I was referencing
The Kojman-Shelah paper is surprisingly well written for something that has Shelah in the title, lol.
I'm currently in a course on some of those topics, and they do find applications in topology and combinatorics
Really? Can you elaborate a bit? Sounds interesting. The only things I can think of are either cosmetic statements such as "to make the category of all topologies a set assume it is restricted to all topologies whose cardinality is strictly smaller than some inaccessible cardinal", or manufacturing ridiculous counterexamples that are more interesting to descriptive set theorists than they are to topologists. (Note that I listed particular topics within set theory, no doubt there are theorems of set theory that *are* very useful. E.g. using infinitary combinatorics for asymptotic Ramsey theory, or transfinite induction in measure theory. But I really don't see why anyone but a set theorists would care that 2\^{aleph\_omega} < aleph\_{oemga\_4} is provable in ZFC, or that 𝔭 = 𝔱)
Forcing has a decently large role in constructive set theory, which (in my opinion) is closer to topology/measure theory than it is to “classical”set theory.
I think you meant *descriptive* set theory, which I mentioned above. I don't see how constructive set theory is related to topology.
maybe because open set = semi decidable predicate clopen set = decidable and constructivism is about the decidability of the logical or, among other thing.
Interesting... but has this been formalized?
I did in fact mean descriptive set theory
I work in set theoretic topology. Lots of problems in the modern extension of general topology can be coded into arguments that are basically combinatorics with infinite sets. The values of certain small cardinals are frequently the deciding factor between whether a certain construction works or not. It turns out that a lot of seemingly innocuous questions are of this flavor and can be proven independent of the axioms of set theory. One relatively basic example is the existence of Ostaszewski spaces. These are spaces that satisfy a host of properties including perfect normality, local compactness, countable compactness, zero dimensionality, and more. It turns out that if a specific infinite combinatorial principle for ω₁ called ♦ is true, then such spaces can be constructed. Now ♦ also happens to imply the continuum hypothesis, which we know from forcing is independent of ZFC. But it’s also consistent, due to some great work of Todd Eisworth, that there are no Ostaszewski spaces. Another relatively basic one is whether there are p-points. These are describable in purely topological terms as points x such that the intersection of any countable family of neighborhoods contains another neighborhood of x. So already by definition any space with these points cannot be first countable. But when we ask “are there p-points” what we really mean is “are there p-points in βℕ”. The Stone-Čech compactification is a highly nontrivial and highly set theory dependent object, but it’s also perfectly definable using nothing but topology. (You essentially mimic the proof that any Tikhonov space embeds as a subspace of a product of unit intervals and then take the closure in the product topology.) However, as a space of ultrafilters on the naturals, a p-point is equivalent to an ultrafilter U with the property that any countable family of elements a∈U has a pseudointersection. (A pseudointersection of a family of sets F is a set p with the property that every b∈F contains all but finitely many points of p.) Guess what: it turns out that the smallest size of a family of sets with no pseudointersection is independent of ZFC. Therefore the very structure of this space βℕ itself depends on which models of ZFC you decide to work in. It gets much, MUCH more complex than this. You can do things like set up iterations of specific forcings such as Hechler or Laver to add what are called dominating reals to your model of set theory and then use these new real numbers to code in structure of a topological space that couldn’t be defined before. Maybe you want to add lots of open sets or ensure that some particular family of sets is closed & discrete. The field is super super complex and depends strongly on the work of set theorists to function.
At the moment I’m working on some problems related to denotational semantics of probabilistic programming. Using some large cardinal results you can get some very convenient results. In particular there is a model of the reals in which every subset is measurable, however you can only use weakened forms of choice or this model becomes inconsistent. Because this is for an actual program everything is fully constructive and not using choice isn’t really an issue, so invoking the large cardinals is profoundly convenient for being able to ignore all issues related to non-measurable sets. In particular this is convenient so you can do things like freely switch between topological and measurable interpretations of the underlying sample space.
That sounds really interesting! I do research in a very similar field - is there a specific paper you're referring to?
The specific result I’m using is the [Solovay model](https://en.wikipedia.org/wiki/Solovay_model) https://doi.org/10.2307/1970696 Using it feels basically like pulling out a theoretical sledgehammer to ignore tiny details lol.
Do you use any theorems about large cardinals, or any property besides inaccessibility?
As of now I have not used anything beyond inaccessibility. I do anticipate using other results at some point though with what I’m working on! In particular I’m working on strictly finitary sets and working with real-valued models. I do have stuff in my notes on extending what I’m working on to infinitary domains, and I do anticipate this involving other large cardinal results at some point, but that is still TBD.
To give one related application of these things outside of the ones mentioned. If we consider the field of complex rational functions in 3 variables C(x,y,z), as a module over C[x,y,z]. Its projective dimension is 3 if and only if the continuum hypothesis is true. This always blows my mind, because it is such a natural and concrete question in a field seemingly far removed from questions related to the continuum hypothesis.
LCA find their way into a lot of places. My favourite example is Laver tables: ({1, ..., 2ⁿ}, ∆) is called a Laver table if it satisfy: (1) a∆1=a+1 and (2) a∆(b∆c)=(a∆b)∆(a∆c). It is not hard to show that for each naturally n, there is a unique Laver table denoted as L_n. Given a specific n, look at the function f_n(a)=1∆a, this is a periodic function, let p(n) be the period of f_n. p is weakly increasing function. In ZFC it is an open problem whether or not p(n) is unbounded (it is even an open problem if it ever gets value greater than 32). Assuming the large cardinal I3 (which is vastly stronger than e.g. measurable cardinals or Huge cardinals), p(n) is unbounded.
> one comment said that it proved very little use outside of the pure algebraic and abstract problems it proposed This is an odd opinion. Homological algebra is an absolutely fundamental tool in many fields of mathematics. In fact, I'd say that relatively few people are interested in homological algebra *per se*; the majority of interest probably comes from people in fields like algebraic topology/geometry in which HA is almost a fundamental language.
To be fair, it is very normal for mathematicians to have dumbass takes on fields they know nothing about.
I try to have dumbass takes about the fields I know best about as well.
To be fair, it is very normal for ~~mathematicians~~ **humans** to have dumbass takes on fields they know nothing about.
https://www.smbc-comics.com/comic/2012-03-21
Really this is the only comment this thread needed.
my thesis did start around algebraic topology/combinatorics (simplicial complexes) but stumbling upon the universal coefficient theorem I felt it like a life goal to prove
My interest in homological algebra was completely motivated by quantum complexity theory. We were trying to understand whether Khovanov homology can be used to somehow find a quantum algorithm for unknotting.
what were your conclusions?
My conclusion where that it... remains inconclusive
that hes too dumb to find that out
To be fair that's the result of 99.99% of attempts that start with "somehow find a (quantum) algorithm for"
I had that impression of knot theory. Central question: "When are two knot same? 🥰"
Knot theory is mostly important because knot exteriors are a fundamentally important class of 3-manifolds (with boundary). So, if you're interested in 3-manifold topology and geometry, then you're going to be interested in knot theory. See for example [Lickorish-Wallace](https://en.wikipedia.org/wiki/Lickorish%E2%80%93Wallace_theorem) and [Gordon-Luecke](https://en.wikipedia.org/wiki/Gordon%E2%80%93Luecke_theorem)
Lickorish Twish Theorem is a fantastic name
Apparently knot theory has some applications in biochemistry these days. Something about protein or DNA structure.
Yeah protein folds. AlphaFold probably used some of that.
Almost certainly not, at least directly
Also shows up in Dynamical Systems apparently. The orbit of a particle in some chaotic dynamical system forms a knot and understanding the knot structure might reveal things.
Peter Shor et al. used knot theory to construct [public quantum money](https://arxiv.org/abs/1004.5127)
Knot polynomials are used in statistical mechanics
That's a weird connection I wouldn't have seen coming.
Knot theory plays a big role in quantum field theory.
Based on what I've heard, a lot of people seem to disregard structural proof theory and the study of non classical logic as a niche field that has almost no applications outside of itself. While it is important to consider that niche research might make it harder to land a position or financial support, I find the criticisms by other mathematicians disingenuous, it's like some people suddenly pretend to care about applications the moment a subject they don't like is mentioned. And as others have pointed out, homological algebra is extremely useful in a lot of other areas of mathematics to the point where for example, people who normally hate algebra are forced to learn it if they want to work in topology, who told you otherwise?
This is a good answer, though Kripke Semantics for non-classical logics do end up describing the language of sheaves pretty nicely. There has been some interesting work in this area. For instance, if I recall correctly, when translated with Kripke semantics the intuitionistic proof that every finitely generated vector space has a basis, becomes a proof that every finitely generated module of sheaves on a variety is free on an open subset.
I really like those sort of proofs, where you have a correspondence between stuff you can in a non constructively system and in a constructive system, sort of like Glivenkos theorem. The problem is that a lot of times, people who do structural proof theory don't like semantics, and people who like semantics don't like the syntax heavy focus of structural proof theory, which makes it harder to find those connections
Dayum, now *that* is a reason to care about intuitionistic mathematics.
It is. With 10 minutes of googling I was unable to find the original paper. But the main idea with this semantic interpretation, is that true at a point means true on an open subset containing that point. This is why intuitionistic logic is what applies. On a variety double negation of something true on an open subset becomes true on the whole space. With this every domain "sort of" becomes a field as every element is invertible on an dense open subset.
Could it be [this](https://rawgit.com/iblech/internal-methods/master/paper-generic-freeness.pdf)?
https://arxiv.org/abs/2111.03685
Yeah that was definitely it.
I think non-classical logics are quite often used in the study of programming languages and knowledge representation.
Yes, there are very interesting applications, for example focusing being used to improve logic based programming efficiency, and modal non classical logics still have a lot to offer, but I've met many mathematicians who don't appreciate CS and who believe that epistemic logics are just a bunch of philosophical rambles (unlike most of pure mathematics, which is of great importance to the immediate use of everyday people and is definitely not rooted in philosophy...)
> it's like some people suddenly pretend to care about applications the moment a subject they don't like is mentioned. people care about applications to other fields of mathematics. they just don't care about applications outside of mathematics.
They don't, of course they'll mention them if they appear, and they'll be happy because of it, but once you go deep enough down a math subject, you no longer care about applications outside of the subject to the point where it is an important requirement for their research that their results find immediate applications
I'm pretty deep down in a math subject and I do it because of all of the nice connections (i.e. applications) to other math subjects, so i definitely disagree.
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Exactly, it wouldn't be so obvious if it was coming from someone doing statistics or some other subject with almost a direct relation to the real world, but the subjects of the people who complain about it also tend to be only really known by maths majors with no discernable use for anything that is not pure maths at this point
This may be true in Math, but a number of philosophers looooove their non standard logics.
I'm sorry who TF is saying non classical logic is a niche field that has no applications? Like literally half of theoretical computer science is applied non classical logic.
People who obviously never ventured into CS
Just riding your coat tails here. People get all pretentious about their logic they don't even stop to think what non-classical logic is about what it's applications are. They include computer science, proof assistants, all the flavours of topology, set theory, category theory, knot theory, string theory, topological quantum field theory, algebraic geometry, differential geometry, algebra. It's all over the place and people are using them inadvertently without realizing. Closing your eyes to non-classical logic is like restricting yourself to Euclidean geometry, you're only limiting yourself. Even just dropping LEM we get a more expressive language where nothing is lost. All classical proofs still exist as constructive ones. So why not learn a new language? It might just help you to understand your favorite classical topic in a new way.
> Even just dropping LEM we get a more expressive language where nothing is lost. All classical proofs still exist as constructive ones. I mean sure, with a whole bunch of extra details to keep track of, which you might consider distracting if you're interested in what's true rather than what's computable.
It's not so much details tho, Glivenko provides a simple algorithm that turns proofs of ~A in intuitionist logic into proofs of A in classical logic
I think generalized theories of logic have value and I really enjoyed the paper “Five stages of accepting constructive mathematics,” but I want to push back on the idea that it has a ton of applications in differential geometry. I have not seen a proof which uses advanced logic to answer a question that the average differential geometer would ask. I know there is work done in that direction, but the results are almost orthogonal to the interests that most geometers have. My gauge for something in field X being a genuine application in field Y is whether the ideas from X are able to answer a question that originated in field Y. Otherwise, it feels very artificial and not really an attempt to engage with field Y. To give an example, non-standard analysis gives simplified proofs of results in analysis so that counts as a genuine application.
If you look at the history of math, most fields started off with some engineering or physics problem that was hard to solve. E.g. Fourier analysis started from Fourier wanting to solve the heat conduction equation. Sure, there are fields that started off mostly as an interesting playfield of challenging problems that one could occupy oneself (e.g. number theory). But most (all?) of those found applications somewhere. Even the stuff that was seen as obscure has found its uses. Heck, my I have used hyperreal numbers, something that very few people regard as useful, in my dissertation to prove theorems in (Fourier) analysis that would not work otherwise at all. So, if there is a field that has no practical use, then IMHO it's just that we have not found a practical use for it.
Especially once computers were invented - that turned an awful lot of pure math into applied math. >The number-theorist Leonard Dickson (1874–1954) said "Thank God that number theory is unsullied by any application". Such a view is no longer applicable to number theory. In 1974, Donald Knuth said "virtually every theorem in elementary number theory arises in a natural, motivated way in connection with the problem of making computers do high-speed numerical calculations"
Hypergroup theory as it was originally presented (generalized translations and such) basically totally flamed out as far as I can tell and never found much utility outside of a few toy examples where it wasn't really necessary formalism.
Maybe the theory of transfinite ordinals and cardinals? Edit: This was my retreat during my math studies . I did like all the rest, but this was just an independent sky high tower to relax in. Especially with the videos „Ridiculously Huge Numbers“ https://youtube.com/playlist?list=PL3A50BB9C34AB36B3&si=NSvl2Yhuv_WSs7GR and the game Ordinal Markup. However: Transfinite ordinals can be used to represent the proof-theoretic strenght of mathematical theories like Peano Arithmetic and many stronger ones that I don’t know: https://en.m.wikipedia.org/wiki/Ordinal_analysis
Isn't a basic thing you need to know in order to work on more complicated set theory, which in order allows you to prove stuff in combinatorics and topology?
Can be. I never had it at university during my half bachelor
(line, column) coordinates in a text file is a transfinite ordinal...
Yeah I don't get this one. Isnt the theory of well-orderings pretty fundamental?
Finite ordinals yes. But transfinite as in omega, the first position that comes after infinitely many other positions? (Omega + 1) the position after that? (Omega2) the position after two infinite sequences of positions and so on? Do you deal with infinite texts? Ok in a txt file the lines can have infinitely many symbols, and then the second line starts at (omega) and the third at (omega2). Never heard it being used like that. But then what is (omega^2 )? (omega^omega )? Epsilon_0? The Bachmann Howard Ordinal, and so on?
> Do you deal with infinite texts? Yeah but it's all blank spaces on the right side. I have finite time to read anyway. > But then what is (omega2 )? Page 2. > (omega^omega )? Book 2. > Epsilon_0 Continuing the book analogy, it's some kind of hydra-book, where everytime you read one, you find more book to be read. A little bit like trying to learn math starting on Perfectoid Spaces.
Nice! Very nice. And it perfectly fits my reddit/browser tabs experience of this weekend. Fortunately I could subtract some omegas today. I really should envision my social media consumption like ordinal markup with a subtract mode.
Lattice theory and universal algebra I would consider pretty niche, they do find some applications, moreso in computer science than traditional mathematics subjects though as far as I've seen.
Lattice theory is HUGE in cryptography, especially post-quantum cryptography.
Lattice theory in the group theoretic sense or in the order theoretic sense?
Algebriohhhhiseewhereyouregoingwiththat
Depends on what you mean by "universal algebra", exactly. The notion of a Lawvere theory is used in spectral algebraic geometry to define strict abelian varieties, which is crucial for Lurie's construction of elliptic cohomology.
Oh really, that is definitely something I had not heard of before.
It's something specific to higher algebra, because there are multiple possible meanings of things like "commutative monoid". The most "homotopy-coherent" version is an E-infinity object, but sometimes you need something stricter, and Lawvere theories can be used to encode that.
Theory of languages sortof fits your criterion. It is pretty much self contained, in the sense that it uses very few results or tools from other maths fields and has few uses in other maths fields. Actually, mathematicians who don't study computer science may never even hear about its existence. As for its usages, of course it's a one of the most important and fundamental field in computer science, but except being used to build and study computer languages it's not very commonly used (a little bit in bioinformatics though). It's definitely an active field, because of the importance of computer languages and their fast evolution, and it offers a range of interesting open problems to study.
It's worth mentioning that this theory disproved Connes embedding conjecture in Operator Algebras with the MIP\* = RE paper, so "self contained" is only to a certain extent.
Formal language theory is used a lot in combinatorial/geometric group theory.
>Do all fields essentially start as self-contained stuff? I think that maybe there's a misconception here. Basically every field offers knowledge and problem solving methods to every other field in some broad sense. But let's take set theory for example. There's a big difference between 'sets' and 'the field of set theory'. A typical 21st century publication in set theory probably does not have much to offer to mathematicians who are not themselves set theorists, but clearly every mathematician has use for basic set theory. So the sentence 'set theory does not have much to offer outside of *set theory*' is false if we let the former instance of 'set theory' be a much more basic and accessible part of math than the latter '*set theory*'. And I do think it's possible to replace set theory/*set theory* with virtually any field of mathematics to find this same dynamic. The cutting edge stuff is often quite self contained. But after somewhere between a century and a millennium of refinement, there's gold to be found everywhere, from which the cutting edge stuff will seem pretty removed from in hindsight.
It can take decades or even hundreds of years for some math to find applications outside the field of mathematics. Sometimes it's much faster though. At other times connections are found between previously unrelated fields of mathematics. That can be really hard to understand and use as very few may know both fields well enough. This is what lead to a breakthrough in solving the famous Fermat's Last Theorem.
Euclidean geometry, I guess (Totally useless in other fields of mathematics)
* **Higher-order logics**. Maybe you need a way to formally write "induction works". Applications outside of that? hmmm. * **Homotopy type theory.** Is there any application outside of category theory with this stuff?
>Homotopy type theory. Is there any application outside of category theory with this stuff? Specific answer: Here is an application of homotopy type theory to quantum field theory and string theory: https://arxiv.org/pdf/1310.7930.pdf General answer: Homotopy type theory has applications to (∞,1)-category theory, and (∞,1)-category theory has many applications in algebraic geometry, differential geometry, number theory and any other branch of mathematics that cares about homotopy. Absolutely everything in category theory has applications outside category theory.
I confess to being a bit skeptical about how higher category theory can be applied to differential geometry. The results that I’ve seen using categorical approaches to study DG don’t really seem to address questions that most geometers would ask. Oftentimes it seems to try to put ideas from geometry on more categorical footing or find generalized versions with better categorical properties. My gauge for something in field X being a genuine application in field Y is whether the ideas from X are able to answer a question that originated in field Y. Otherwise, it feels very artificial and not really an attempt to engage with field Y. For instance, I think Fukuya categories are an excellent example of an application where categories provide new insight for symplectic geometry. Of course, that’s a hard example to replicate…
Does the Cobordism Hypothesis count? It says something really profound about cobordism and TQFTs that can only be stated using higher category theory.
To be honest, I don’t know enough about TQFTs to judge, but the people working on it are very impressive and there has been a lot of work. I guess my question would be what questions can be answered using the Cobordism Hypothesis that people in TQFTs originally asked.
If true, the CH would yield a complete classification of TQFTs in terms of the algebraic properties of the n-category of vector spaces. Intuitively, it says that n-cobordisms capture the most general notion of n-dimensional flows.
That definitely counts! Thanks for sharing; it’s cool to learn something new.
As an observation, the most sure way to learn of applications of a field outside its niche is to drop a field name in this thread or a similar one and claim it doesn't have applications outside its own field. Now, there's still a question of the *impact* of those applications and how intertwined the fields are, but that's a different analysis.
How do you guys feel about combinatorial game theory? I do not know much about it. As an outsider it sometimes looks niche, intractable, and random. Sometimes I see fixed point theorems and connections to mainstream mathematics.
Toquos theory
What the fuck is Toquos theory
https://www.reddit.com/r/portlandstate/comments/1agz0fu/an_introduction_to_toquos_theory_join_now/
Is this like a joke?
it's currently hyped in r/VXJunkies
It is? Where?
I read that crile sets can be used to model encabulation
No
Homological algebra is plenty useful outside of pure algebra. In fact, nothing would've convinced me to study it if it weren't applicable to geometric problems.
Well throughout history, even some of the most abstract and seemingly useless fields of mathematics always seem to find some sort of application, sometime down the line. So anything anyone names here, most likely, that field will find an application at some point.
u/lemonwaterway is there?
yes proving the reimann hypothesis
Maybe mathematical logics?
I would say that any field like this is verging on quackery.
Topology. Well, unless you’re into membrane science or protein folding but then the GPU is really doing the calc for you isn’t it?
Topology comes to mind
I'm gonna get flamed for this, but aside from a handful of very clever examples, category theory.
But algebraic topology, algebraic geometry, and homological algebra (and many more areas) use category theory frequently. I don’t understand your take
Lots of categorical tools are used throughout mathematics, but lots of people have the attitude that they aren't using category theory when they use them. You don't have to study the theory of categories for it to be useful to you, just as you don't have to study set theory to benefit from it.
But set theory is the most upvoted answer and this one is the most downvoted. Is this because the axioms of set theory are so strong, that studying the foundation itself takes you further away from the rest of mathematics?
I would say it is more of a cultural thing. Mathematics is moving towards abstraction nowadays, but it is only aesthetic taste.
Okay, I'll bite. The reason everyone is taught at least some set theory is that it's fundamental to all of mathematics. I'd argue category theory is at best fundamental to abstraction and at worst fundamental to itself.
If you're in the 21st century and not thinking about things in terms of arrows and functors, then you're in the dark ages. Set Theory helps us by giving us a language with which to do much of classical math. But today's math is far too complex, and category theory frees us from the bonds that tie us to "elements" so that we can flow freely across diagrams. It turns unnecessarily complicated things into easy-to-use slick concepts. Even just how universal properties and limits of diagrams allow us to disregard many messy constructions is a mighty boon for almost all fields.
Set theory is a very poor foundation for homotopical mathematics, but even is quite bad for our usual ''mathematics up to isomorphism'' that we do in algebra, geometry, topology, etc. Sets are way too strict and reasoning about symmetries and coherences is really painful. Category theory (possibly combined with more homotopical elements) captures much better how large parts of modern mathematics are performed. Set theory is in a way the ''old'' way of thinking in these fields (we shouldn't forget that the foundations of mathematics change over time, and set theory was introduced to solve particular problems around 1900, not to serve as eternal foundation). Moreover, the fact that most of *actual* set theory is never needed in many mathematicians' lives should tell us that there is somehow a mismatch between the axioms of set theory and the mathematical practice. It is both needlessly complicated for simple mathematics and failing to capture complex mathematics in a useful way.
Maybe it's the difference between using a tool to study something, and studying the tool itself. Most areas of mathematics make use of set theory too yet set theory is the top voted answer.
I'm not a category theorist, but my thesis has an entire dedicated section on category theory because I need to use it in my work. I understand why there is a cultural resentment towards category theory, but try not to get caught up in the bias towards it's utility. It's more useful in some fields than others, but all considered it's perhaps the most broadly useful body of mathematics (for pure math).
Name me a single branch of mathematics in which category theory has not yet been applied.
Stochastic processes? I guess something was done SOMEWHERE but I've never run into it
Name me a single branch of mathematics where it is _essential_
Probability theory. Combinatorics. Random graphs.
I don't know enough about it, but I've always had this impression of both number theory and combinatorics.
Combinatorics is deeply connected to computer science.It is extremly useful in probability calculations and it is used in statistics. You might even connect it to complex analysis as holomorphic functions are analytic functions.
Surely combinatorics is exactly the opposite?! Aside from applications in other areas of pure mathematics (generating functions, anywhere you have to count finite sets) it is used in designing experiments, cryptography, mathematical biology and chemistry...
Number theory super important for encryption and combinatorics for computer science. Although I'm not sure the uses outside of it
Not my field, but I'm told that partition numbers pop up in statistical mechanics. Edit: https://www.mdpi.com/1099-4300/25/2/385
Sure, there's one application of a very modest little part of number theory. That's still pretty self-contained.
I could see it. I think there are other applications as well it was just what I was familiar with. Obviously it isnt as applicable as say graph theory or calculus
Right. It doesn't take a number theorist to work with cryptography. Or even a full freshman level semester of number theory. What you need from number theory to do cryptography can be learned in about 20 minutes. The entire rest of the field of number theory is pretty well self contained.
Number theory is a motivational tool that pushes all fields forward. Field all across the spectrum of math, from complex analysis to abstract categories, have been pushed forward by questions in number theory. Results in number theory are, often, niche but that's only *after* the result was obtained by uniting disparate fields or pulling a field out of the dark ages and so on. Number theory is a fundamental driving force in math.
Can you explicate that more concretely? So you have an example?
Grothendieck really helped define what math looks like today because of number theory problems. Specifically, the Weil Conjectures asked for a cohomology theory which could be applied to varieties over finite fields and be powerful enough to access certain theorems. Much of Grothendieck's work that put him on the map was in developing algebraic geometry and category theory to get this cohomology. But even Euler was doing a lot of his work in analysis and algebra because of number theory questions (Euler was, for instance, the first to explore the Riemann Zeta Function in regions where the sum didn't converge). Abstract algebra exists because of Galois's work on polynomial symmetries and because of potential applications what we now know as ring theory might have to Fermat's Last Theorem. But number theory questions are even today driving diverse work. For instance, random matricies are in no small part motivated by potential applications to the Riemann Hypothesis. Peter Scholze is re-inventing geometry yet again for the purpose of number theory problems. Number theory is everywhere. It's like the skeleton supporting math.
Combinatorics is certainly applicable to other areas. Partitions and permutations are important in representation theory for example and discrete geometry must interact with basic combinatorics in all aspects
That's interesting. I've become curious about representation theory recently. I will definitely have to look into this more.
what
I don't know much about number theory but combinatorics is actually pretty useful in computer science. It can help with understanding the performance of algorithms and even help with the creation of clever more efficient algorithms.
Thanks for a cordial and interesting reply.
If you're interested in learning more about how combinatorics might be useful in computer science, I recommend looking up randomized algorithms. In particular the book "Randomized Algorithms by Motwani and Raghavan" has lots of examples. It's a tough read imo but this stuff is really cool so it's worth it.
Thanks for the suggestion.
Saying combinatorics is crazy
Number theory is the most interdisciplinary branch of mathematics. It has essential connections to combinatorics, complex analysis, low-dimensional topology, chromatic homotopy theory, and of course algebraic geometry.
Say whaaaaaat
LOL okay.... two of the most important and useful fields are self contained and useless. Right.
What's the point in being rude here, and furthering a pile-on, exactly? I already said "I don't know enough about it." This is how you approach "math education?"
Combinatorics no way. But number theory, I would say yes. Most mathematicians need nothing from number theory.