Obscure branches of math?

[mike-l]

So how are you getting these determinants?

You could look at the induced map on the nth exterior power of the underlying space. (This is really and truly coordinate free) (And in a completely roundabout way, a choice of direction on the nth exterior power is the same as an orientation on the original space)

[Yakk]

Isn't the standard analogue of the cross product into arbitrary finite dimensions given by https://en.wikipedia.org/wiki/Exterior_algebra ?

Ie, it admits that the wedge in R^3 goes to a different R^3 (3 choose 2 instead of 3 choose 1), which we (by convention) align with our original R^3.

The result of the wedge product of two R^4 vectors is in an R^6 space.

But I never got into wedge products myself, so I have only a very shallow understanding.

[Adam Preston]

Reply to a post further back into the forum, I am currently doing A level Maths, and the course is being taught in 2 parts, Pure Mathematics and Statistics, seeing as you said we don't get taught pure maths in the UK, I'd say by now we are 100% getting taught pure maths.

[Yakk]

I tried to google for questions that the UK might call "pure mathematics", and got this:
http://www.cimt.plymouth.ac.uk/interact ... efault.htm
Every one of these questions looks like Engineering-type mathematics to me (ie, it isn't what I would call "Pure Mathematics", as opposed to "Applied Mathematics"). Ie, most of the subjects covered, while taught divorced from applications, are the kind of thing that can actually be attached to a myriad of applications nearly directly.

Glancing at the "further, enriched" materials from Cambridge:
http://www.cie.org.uk/qualifications/ac ... def_id=756
I see at best a small amount of material that would qualify as "Pure Mathematics".

[jestingrabbit]

I tried to google for questions that the UK might call "pure mathematics", and got this:
http://www.cimt.plymouth.ac.uk/interact ... efault.htm
Every one of these questions looks like Engineering-type mathematics to me (ie, it isn't what I would call "Pure Mathematics", as opposed to "Applied Mathematics"). Ie, most of the subjects covered, while taught divorced from applications, are the kind of thing that can actually be attached to a myriad of applications nearly directly.

Equally, some of them are adjacent to the theory of field extensions and linear algebra.

The real difference between pure and applied only really appears at uni imo.

[Yakk]

*nod*, but I suspect that the "pure" mathematics at the UK "A" level means "mathematics that isn't being used as a direct adjunct to an application topic", as opposed to the (at university level) "pure" mathematics being "mathematics that, at this point, isn't designed to be directly adjunct to an application topic outside of mathematics".

And, more directly, this was in response to Adam Preston's post:

Reply to a post further back into the forum, I am currently doing A level Maths, and the course is being taught in 2 parts, Pure Mathematics and Statistics, seeing as you said we don't get taught pure maths in the UK, I'd say by now we are 100% getting taught pure maths.

in which I attempted to see if the UK A level maths "pure mathematics" was pure mathematics outside of a UK high school curriculum. As far as I can tell, nope.

I mean, you could probably teach some basic abstract algebra -- rings, fields, etc -- and/or do a high school level proof-driven course. It wouldn't be easy, and it wouldn't generate material that would be all that useful for application-destined mathematicians (Engineers, non-theoretical Physicists, Biologists, Chemists, etc).

[Zamfir]

Yakk, I am not sure if categorizing maths by its use in other fields is helpful. You already run into the problem with theoretical physics, that you need to keep out of "application-destined", lest too much maths becomes applied.

It's more useful to draw the line between maths that is mostly relevant because of its use in other fields, and maths that is seen as relevant regardless of its use elsewhere. Where things can be the latter category despite being also highly useful in some applied field.

Otherwise you run into the mathematical equivalent of hipsters, people who take pride in the pureness of their math because it lacks the taint of the grubby applications. That's perhaps an OK attitude for some maths departments, but not really for looking at high school curricula.

[Yakk]

I suspect, much like how every field of science was at one point just a part of Philosophy, branches of pure mathematics that find applications will wander into being a subject in the domain of their applications as time goes by. We still call higher degrees PhD's, but that doesn't mean that they are actually Philosophy degrees.

---

"No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world."
Then again:
"No one has yet discovered any warlike purpose to be served by the theory of numbers or relativity, and it seems unlikely that anyone will do so for many years."
(both of which is used heavily in warfare, and are likely to be used increasingly so over the next few decades.)

Yes, pure mathematics seems doomed[1] to becoming applied. Probably because pure mathematics finds in a sense tractable problems (problems that elicit beauty, and we can work our heads around, and work out ways to make them more beautiful), while applied mathematics is constrained by reality to work on actual problems (which might be intractable, at least without a complete swapping out of tools).

"Pure mathematics is on the whole distinctly more useful than applied. [...] For what is useful above all is technique, and mathematical technique is taught mainly through pure mathematics."

But the field of categorizing mathematical subject areas isn't an obscure branch of math.

[1] I mean this in a joking sense, not in a serious "applied is worse than pure".

[Timefly]

Replying to people saying that Pure Math is not taught in the UK.

I'm an A level maths student and in our 'Core' modules we generally cover what might be though of as 'Pure' mathematics; being things like Calculus, functions, iterative formulae, further trigonometry etc...

For real pure mathematics though, you have to take the Further Maths A level, which covers (the following is the list of chapter names from my textbook:
- First Order Differential Equations.
- Lines and Planes
- Linear Differential Equations
- Vector Product
- Complex Numbers in Polar Form
- De Moivre's Theorem.
- Further Trigonometry (Weird mixture of trig with complex variables and other assorted jumble)
- Calculus with complex numbers
- Groups
- Subgroups
- isomorphisms

This can hardly be called applied mathematics.
Also, here's an example of a past paper for the module I just listed.
http://www.ocr.org.uk/download/pp_10_ju ... e_4727.pdf

Edit: Okay, so differential equations can be called applied, but they never come up in questions in applied modules such as mechanics modules.

[pizzazz]

It's not "obscure" but formal logic would do a lot of people some good.

I think a lot of functional analysis (which is more of a subset of set theory and linear algebra) is lacking in many areas, as I never was introduced to the concept of a function in its real form until calculus, and it's a hugely powerful mathematical construct that current algebra and even a lot of pre-calculus courses really don't get into the level of detail that they could, and possibly should (especially if we want more scientists and electrical engineers), and it really takes no calculus to understand, but would help many people understand calculus better, however to cover everything in the subject to its fullest extent does require a lot of math theory usually reserved for algebraic structures courses in college, however the direct applications of the concepts in this type of mathematics don't fully come out until higher level differential equations and signal analysis (however anything you've done in physics dealing with harmonics was derived in some way using this).

Functional analysis has nothing to do with the definition of a function, that's generally just presented straight from set theory. A functional is an operator from a normed linear space to its field.

[doogly]

This can hardly be called applied mathematics.

The fact that you were not taught the applications for those topics says very little about the topics themselves, and more about the narrowness of the presentation.

[Timefly]

The fact that you were not taught the applications for those topics says very little about the topics themselves, and more about the narrowness of the presentation.

I suppose that's fair, but the whole course feels more like a primer for a degree in maths than anything else.

[Zamfir]

Perhaps that's how your school chose to present it, but that list is pretty much standard math for most engineering or physical science curricula. They're all mathematical constructs that are closely related to the modelling of the world.

[jestingrabbit]

They're also pretty good for starting an exploration of pure mathematics. Everything "vector product" and below is stuff that leads very directly to pure topics, and even the first three can be approached from a very pure perspective.

Anyway, if you go back to what was initially said,

At school (in the UK at least)? All of pure mathematics is ignored. The kids only get taught mathematical methods, the kind of thing you'd learn in training to be an economist, physicist or engineer.

That's pretty clearly false. I've known physicists in post grad who wouldn't know a group if you beat them over the head with it.

[Yakk]

Yep -- reading the actual test, just under half of the questions would fall under the umbrella of "pure mathematics done at a high school level", based off rough counting and part marks.

Based off the short list, I'd be more skeptical. The only things in that short list that would show up more in a Pure as opposed to Applied mathematics degree would be the groups/subgroups/isomorphisms.

I've known physicists in post grad who wouldn't know a group if you beat them over the head with it.

They are introverts: they have avoided groups in their education. They spend more time in fields, after all.

[Catmando]

How about cellular automata? It seems pretty obscure to me. I've seen mentions of it here and there by a few people on the Internet, but precious little more than that, and certainly nothing in school, lessons or otherwise.

[Adam Preston]

Reading most of these responses here, it's clear my idea of pure mathematics subjects are inaccurate, could some please post what they would consider pure maths or are widely known as pure maths.

[jestingrabbit]

For research, the answer is pretty easy: the difference between pure and applied research is basically one of motivation. Pure maths is motivated by a desire to know, whereas applied maths is motivated by real world problems. That said, there isn't really a hard line between the two, though some areas are more definitely on one side of the division than others. However, even topics that are strongly considered to be pure or applied, there's typically some crossover.

For instance, number theory is usually considered a pure maths subject. But, some aspects of number theory lend themselves to cryptography that finds application thousands of times a day in securing electronic money stuff.

Likewise, solving differential equations is usually thought of as applied maths, because differential equations usually arise in modelling real world systems. But, I'd say that most of the work being done on solving heat kernels using harmonic analysis is really pure maths, despite it finding application in physics.

In terms of what gets taught as pure maths and what gets taught as applied, I'd say the big difference is what its trying to prepare you to do. If its about proving stuff, its pure. If its about solving real world problems its applied. If its about solving made up problems, it could sensibly be described as either. NB: that's what the meat of any school curriculum will be, so the distinction is kinda meaningless at secondary levels imo.

[doogly]

And proofs can also be applied. A lot of times there are differences of perspective as well. If someone is working on formal mathematics of general relativity... is that an application?

I think there's also a regional thing going on. Very mathy physics is more likely to show up in a physics dept in the US, and in a math dept in the UK.

[Tirian]

If someone is working on formal mathematics of general relativity... is that an application?

It's a floor wax AND a dessert topping. It's an application of differential geometry and an abstraction of physically observable phenomena.

[J Thomas]

Anyway, if you go back to what was initially said,

At school (in the UK at least)? All of pure mathematics is ignored. The kids only get taught mathematical methods, the kind of thing you'd learn in training to be an economist, physicist or engineer.

That's pretty clearly false. I've known physicists in post grad who wouldn't know a group if you beat them over the head with it.

I feel like I'm misunderstanding this. He says future physicists only get taught applied math. You say some actual physicists don't know about groups. How does what you say contradict what he says?

[Yakk]

Because UK students get taught about groups.

If (UK students never get taught pure maths) and (Applied math is what physicists know) and (Groups are not something that physicists know) and (Applied math is math that is not pure) and (UK students are taught groups) and (standard logic rules apply) then true proves false.

While there are applications for group theory (large integer multiplication, cryptography, error correcting codes, grand unification of physics, and other pedestrian pursuits), it is a pretty damn abstract branch to be considered applied mathematics. It is, in a sense, more of a sign that esoteric pure mathematics ends up having applications despite the best efforts of the subject area than evidence that it is applied mathematics.

Another way of looking at it is that the corners of pure mathematics that no application has been found for are relatively far out there, and are often directly adjacent when learning to things that have applications, which makes it difficult to generate a curriculum that is pure mathematics.

On the other hand, a course on taking integrals is going to be considered relatively applied on the pure-applied spectrum beyond the high school level, while a course on group, ring and field theory is going to be considered relatively abstract (and probably relatively "pure"). Then again, at the high school level, being taught mathematics that isn't directly attached to a practical subject area in the course in question might be what the UK considers "pure mathematics".

Dunno.

[J Thomas]

I happen to think though that engineers, economists and the like would learn mathematical methods specific to their trade better at university

But the stuff needed by them is very uniform: good algebraic manipulation skills and basic calculus are needed everywhere, not in one specific field. The more general grounding is, perhaps paradoxically, needed by far fewer people.

I guess the distinction here is between people who need to think, versus people who just need to do.

Like, it makes sense that EMTs don't actually need to learn any of the general principles that MDs learn. They just need to do the right thing instantly. If they have to stop and think then their training is lacking. So they don't need any theory. But MDs do need theory.

Engineers who just apply formulas don't need to know how it works. Just teach them the formulas and let it go at that. But people who advance the frontiers of engineering by designing new things, those people might benefit by actually learning math.

I think there's room for a third area inbetween. Currently we have pure math which focuses on making sure it all works, and applied math which focuses on being able to get results with the tools that have been proven useful. Kind of like the distinction between medical researchers and EMTs. We need an applied math which looks at agile ways to apply the math. Less interest in proving it works than in seeing how to apply it in new ways. Kind of like an MD who gets lots of routine cases where he knows what to do, but who must sometimes apply reason from his general understanding and hope it works.

[J Thomas]

Because UK students get taught about groups.

If (UK students never get taught pure maths) and (Applied math is what physicists know) and (Groups are not something that physicists know) and (Applied math is math that is not pure) and (UK students are taught groups) and (standard logic rules apply) then true proves false.

OK, now I see! If a physicist doesn't know it, it isn't applied math but pure math. Since UK students do know it, then UK students are taught pure math.

Where I got lost was the idea that every physicist learns all of applied math.

[doogly]

Eh, that's just a bias towards certain developments of group theory (like a dummit and foote, f'rexample) where applications just sort of happen. But it was certainly developed with a pretty close relationship to symmetry, making the applications in physics quite intimate and natural.

[Desiato]

It is, in a sense, more of a sign that esoteric pure mathematics ends up having applications despite the best efforts of the subject area than evidence that it is applied mathematics.

This.

For example, one of the areas that I'm rather interested in and have at least a vague concept of an idea about what's going on is set theory, model theory and logic. One would think that such a highly abstract and in some instances bordering on philosophics area would be as pure as you can get. But - it's simply not true these days if you consider that some of the results there have more or less direct effects on areas in, let's say, complexity theory in CS. And that, in turn, can have practical consequences for very applied algorithmic questions. Or consider proof theory and its relation to validation of programs and the implications on what's possible there or not.

Another example just came up in a different discussion on this forum that kinda stunned me: graph theory. Now I know that graph theory, by essence, is a very applied topic. But when doing some research on directed graphs, it turns out that most recent and current results are about finite directed graphs, which often are very directly applied questions.

And then there's what's these days is often used in jokes: Whatever "pure" math comes up with is getting turned into something "applied" by theoretical physicists. It's hard to find a field that does not have some "applications" over in theoretical physics.

A reason that I believe is relevant is something that emerged over (and that's a vague estimate, correct me if you know better!) the last half century or so, which is a focus in pretty much all math towards generalisation and dualities between (a priori seeminly unrelated) fields of mathematics. Whether you look at the infamour proof for Fermat's last theorem (connecting seemingly very basic number theory in a fundamental way to something as abstract as modular forms) or Diophantine equations paired, through algebraic geometry, with cs (Matiyasevich) or consider what economists do, which includes functional analysis and various differential equations.

These days, it's challenging to find an area of mathematical research that doesn't have any known relation to something more "applied" than what "pure mathematics" would have meant 50 years ago. I'm not aware of any, and I'd be curious if anyone could name one.

[mollwollfumble]

For what I've been working on, I find that much of the time I have to go back to papers that are a bit over 100 years old, because the branches of math have been largely ignored since.

For some very simple and useful mathematics that definitely counts as "obscure", I can thoroughly recommend Hardy (1910):
http://www.archive.org/details/ordersof ... 00harduoft
or as a pdf:
http://ia700302.us.archive.org/31/items ... oft_bw.pdf

It's a very easy read, and begins with:
"The notions of the ' order of greatness' or ' order of smallness' of a function f(n) when n is 'large', ..."
and ends the final appendix with:
"... the number of molecules in the Earth is roughly 10.8\times 10^{50}, 42!, e^{e^{4.77}}, (2333)^{e^e}, 3.56^{3.56^{3.56}}.

[Thomas18]

Does this account as well?