No Topology Subforum? Discussing on General Maths

[pivoxa15]

Why not? Does topics on it go under general maths?

 

[jostpuur]

Topology related threads have usually not been moved away from the Calculus & Analysis section, at least.

 

[JasonRox]

I think Topology should have a heading somewhere. I never actually know where to post it. General Math? Calculus and Analyis? Or Linear and Abstract Algebra?

I was thinking that we should rename the headlines as...

General Math
Calculus and Differential Equations
Analysis and Topology
Linear and Abstract Algebra
...and so on...

 

[Moonbear]

I think Topology should have a heading somewhere. I never actually know where to post it. General Math? Calculus and Analyis? Or Linear and Abstract Algebra?

I was thinking that we should rename the headlines as...

General Math
Calculus and Differential Equations
Analysis and Topology
Linear and Abstract Algebra
...and so on...

I hope some other math folks will chime in on this. Basically, you're suggesting we just rearrange two math subforums...instead of having Calculus and Analysis as one subforum, and Differential Equations as another, you're suggesting Analysis be split off from Calculus and combined with Topology, and Diff Eqs be combined with Calculus.

So, I have a few questions, because I'm not a math person and can't just skim through those forums myself to get a sense of this...
Are there currently enough Analysis questions in the Calc and Analysis forum to justify moving them off on their own? From previous discussion, we don't get enough topology questions to sustain a forum, so Analysis would have to be a sufficiently popular discussion topic to justify that split.

Also, does it make sense in terms of the subject matter to combine Analysis and Topology? Or is this more of lumping together two topics for convenience of enough posts to justify a separate forum?

I can see the logic in combining Calc and Diff Eqs, but from a practical, forum management perspective, are these both subjects with enough questions that it would be overwhelming to have a single forum with them combined?

Before we start jumping in and rearranging forums, I think these are the questions that need to be considered.

In the meantime, without having asked Halls or Hurkyl (they'd be the ones to specifically ask), my general impression is that using your best judgement of where a topic that's not specifically listed fits will probably be okay. At worst, they'll move it to a different subforum and you'll know where to put it next time. It would probably be safest to put it in general math, but you could also post it, send them a link to the post by PM and let them know you weren't sure where to place it and if they think it fits better elsewhere, to move it for you.

 

[morphism]

Topology & Analysis is a valid combination; so is Topology & Geometry. So a simple renaming of the Geometry subforum would be the easier solution. Although (sadly), there isn't much topology that gets discussed. The Ask A Topologist forum is much more active, although it's plagued with boring homework problems (and the posters do not show any work, so one is less inclined to help them), and it's somewhat of an eyesore.

Another unrelated question, but one I've wondered about for quite some time, is: why are logic, set theory, statistics and probability all clumped up together? The first two are reasonably detached from the other two. :p

 

[Chris Hillman]

Oh, a chance to classify mathematics!

Due to popular interest in physics, there are too many posts related to differential geometry, so I suggest two headings be "Topology & Analysis", "Manifolds and Geometry", and hope that students with a general topology question will head to the first, while students with a question about de Rham cohomology will head to the second

There are existing subject classifications in mathematics, e.g at the arXiv, AMS Subject Classification, etc., but AFAIK no-one is happy with any of them However, I would stress that good classifications should try to conform to the expected population of submissions. In particular, the topics of PF posts are weighted very different from those of arXiv eprints.

Coming at the same point from a different direction: "Topology & Analysis" covers a lot of ground, but based upon what I've seen at PF, questions in this area are under-represented wrt the importance and breadth of these areas, so such a heading might be appropriate at PF.

No matter what headings one comes up with, newbies will be confused since many questions overlap (there are so many interconnections in mathematics!). Still, one can hope by judicious terminology to minimize gross misplacement of threads. I might make some specific recommendations after I've had a chance to think it over. I'd be interested to hear the thoughts of mathwonk and Matt Grime as well!

OTH, why wait? Here are my initial suggestions, based upon very little thought (!):

• rename "Linear & Abstract Algebra" as "Linear and Modern Algebra" (the latter embraces groups, rings, fields, group actions, and many other topics which are rarely the subject of PF posts--- such as tensor algebra; questions about the latter would probably be misplaced in the next but they are rare so this would be tolerable),
• rename "Tensor Analysis & Differential Geometry" as "Manifolds and Geometry" (theory of manifolds is usually taken to embrace calculus on manifolds, aka exterior calculus, and differential geometry is founded in part upon tensor calculus),
• keep "Differential Equations" (well-defined area which gets a lot of traffic), or rename it "Differential Equations and Dynamical Systems" (this might induce newbies with a question about ergodic theory to put their post here instead of in "Topology and Analysis", but that might be acceptable and even more or less valid),
• rename "Set Theory, Logic, Probability, Statistics" as "Probability Theory and Statistics",
• rename "Number Theory" as "Combinatorics, Graphs, and Number Theory" (number theory gets too much traffic; there are two few questions about graphs and combinatorics, but combinatorics and graph theory are often taught as one course and often with some number theory tossed into the mix),
• rename "General Math" as "Foundations" (embraces mathematical logic, set theory and some category theory),
• split "Calculus and Analysis" into "Topology and Analysis" and, wince wince, "Calculus and All Other Math" (many questions in calculus should probably go to "Homework Help" but that notice could be a sticky).

Some kind of "grab bag" seems necessary, and since Calculus probably attracts the greatest number of frantic newbies...

COI notice: I have an obvious interest in promoting dynamical systems, but I resisted the urge to propose a new forum just for that subject, since I acknowlege that despite its importance it attracts few questions at PF. I can't help predicting that if we renamed "Differential Equations" as "Differential Equations and Dynamical Systems", questions about dynamical systems might in fact appear more often, which would probably be a good thing! IOW, I predict that to some extent changing forum names at PF might actually noticeably change the nature of the traffic!

As a random test, suppose someone hears that the Szemeredi Lemma is the most important result in all of mathematics; where would he inquire about that? Depending upon the context in which he has encountered it, his post could go to "Combinatorics, Graphs, and Number Theory" or "Probability Theory and Statistics" or "Topology and Analysis". Which illustrates the near impossibility of creating any entirely satisfactory classification. Still, my initial feeling is that the categories I tenatively proposed above correspond fairly well to PF traffic over the past year.

 

[Integral]

Chris,
Thanks for your input. I like your suggestions and would support the changes you propose, inculding the addition of "dynamical systems" to the DE forum title.

 

[Chris Hillman]

Thanks, Integral!

I think it's possible that some newbies who are confused about where to place their question may wind up not posting at all That's what I was getting at when I suggested that improving the index might improve traffic.

Another thought occurs: I suspect that the lurkers least likely to post are the cautious thoughtful types who know their own limitations and who don't like to say anything which might prove wrong or even foolish. I sometimes get the impression that the n00bs most likely to delurk are too young/inexperienced/manic to read good responses with sufficient care (given that quite a few simple questions turn out to involve subtle mathematical issues.) If so, I doubt there is any cure for this "self-selection phenomenon".

 

[JasonRox]

I like Chris's layout too although I think "General" Math should remain and not changed to "Foundations".

 

[Chris Hillman]

Jason and I had some further pertinent comments in another thread; see [post=1503282]this post[/post] and my reply.

Moonbear, I have been too lazy to PM mathwonk or Matt Grime but I recommend seeking there input too. I think renaming/reorganizing is probably a good idea, but only after input from these knowledgeable and experienced (at PF) mathematicians.

 

[Kummer]

Eventhough Topology and Analysis are similar I consider them separate. I would put Topology under Geometry section.

 

[Chris Hillman]

Eventhough Topology and Analysis are similar I consider them separate. I would put Topology under Geometry section.

Your honors, I object!

• Most students first encounter topology in a general topology course, which typically focus on constructions whcih are useful in analysis. This is the most likely source of questions from newbies on topology.
• Topological issues are very important in analysis.
• Topological issues are also very important in geometry (which is currently usually understood to encompass the theory of manifolds, differential geometry, vector bundles, and related constructions), but students who know enough to have questions about de Rham cohomology will, I hope, see that their post should go in the proposed "Geometry" forum.

But as Kummer's comment shows, mathematics is far too varied and interconnected to succumb gracefully to any attempt to categorize its body parts.

 

[Chris Hillman]

Another PF user independently (?) suggests reorganizing math forums at PF

Ehrenfest has posted some new comments which I think should be moved to this thread [post=1510174]here[/post] (see also subsequent discussion).

 

[JasonRox]

But as Kummer's comment shows, mathematics is far too varied and interconnected to succumb gracefully to any attempt to categorize its body parts.

But Topology needs to be put somewhere. The topic is a very large topic.

It's still surprising that Topology isn't listed anywhere yet.

 

[Chris Hillman]

I asked the moderators to combine the three threads to no avail, but to repeat what I wrote above and in one of the other threads:

I [post=1491724]proposed[/post] the following reorganized roster of Math subforums

• Calculus and Miscellaneous
• Combinatorics, Graphs, and Number Theory
• Differential Equations and Dynamical Systems
• Linear and Modern Algebra
• Logic and Foundations
• Manifolds and Geometry
• Probability, Information Theory, and Statistics
• Topology and Analysis

The OPs query provides a good example of why it would be useful to have a conveniently found list of possible topics. In this case,

• fractal quantities arise in "chaotic dynamics" and in "holomorphic dynamics (Julia sets and all that--- dynamical systems meets analysis),
• quantities such as entropy arise in dynamics and IT as the Hausdorff dimension of a fractal set (e.g. the set of "plausible sequences" for a given Markov measure), and Hausdorff dimension is defined in terms of Hausdorff measure, which belongs to measure theory, which is part of modern analysis; plausible sequences involves a probability measure, which also is part of probability and information theory,
• quantities such as "correlation dimension" which arise in less rigorous parts of chaotic dynamics can be thought of as approximations to Hausdorff dimension of certain sets

Thus, I'd say that questions about fractals could fit into "PIT&S" or "T&A", but a vague question about fractals might seem odd the proposed "DE&DS" forum, which would probably mostly contain questions about differential equations, not analysis. In any case, if my scheme were adopted, the question would most likely wind up in "C&M", which would work.

So I tentatively proposed "Topology and Analysis". See also my comments in the other threads, where I argued against "Topology and Geometry". And sorry for the three threads but it's not my fault--- I tried! Note that, as it happens, my proposal would also address the issue raised by Ehrenfest.

 

[pivoxa15]

Weyl said the angel of topology and the devil of algebra fights for the heart and soul of every branch of mathematics. So topology must be big to say the least.

 

[JasonRox]

Weyl said the angel of topology and the devil of algebra fights for the heart and soul of every branch of mathematics. So topology must be big to say the least.

Where do you get all these little fun facts?

 

[Chris Hillman]

Possibly from http://www-groups.dcs.st-and.ac.uk/~history/Quotations/Weyl.html.

 

[mathwonk]

i don't really mind if forums are organized based on frequency of posting, but logially it is nonsense. i.e. a separate topic should logically remain mathematically separate even if it is unpopular. i.e. this is aNOTHER INSTANCE OF BUREAUCRacy dictating scientific reality.

but i can live with it. we are dependent on the existence of the forum and someone has to do the grunt work.

 

[pivoxa15]

Where do you get all these little fun facts?

The opening quote to Micheal Artin's 'Algebra' book.

 

[matt grime]

Thanks to Chris for pointing me here.

I don't have any useful suggestions at this stage, since I've only just thought about it. But here are some observations on current posting issues that need to be eradicated.

1. Posting of point-set topology questions in the set theory forum, or general math, when if anything it should be analysis.

2. The dubious position of measure theory posts (general, analysis, probability)

3. Algebraic questions not getting posted in algebra.

My only constructive comment right now is that we should not label the subforums according to the American university undergraduate's expectation of mathematics as a subsidiary subject to engineering.

If you were to force me to make a pie in the sky suggestion right away (and I should be something positive, I suppose), then here's a whacky idea, based upon a 'bigger picture' of mathematics:

1. Introductory mathematics (What currently passes for "analysis")
2. Algebraic Topology and Geometry
3. Differential Equations and Dynamical Systems
4. Measure Theory and Probability
5. Foundations of Mathematics (Logic, sets, categories)
6. Number Theory (if we must - this seems to be the least used forum, and the one with the highest crackpot hit rate).

And possibly a thread called "How do I do this integral, differentiate this, or find the limit as (x,y) tends to (0,0)", which take up a disproportianate amount of space.

 

[pivoxa15]

Matt, where would differential topology go?

More importantly, where would algebra go?

 

[matt grime]

Oh for pity's sake, it wasn't supposed to be exhaustive. If we include representation theory with Alg. geom and Topology that will do. Moreover, any mathematical question on 'differential geometry' can be considered part of alg top or alg geom, and anything on general relativity and tensors can be shipped off to physics instead, where it will have a much better group of people to answer questions on it.I would also propose a ban on pointless debates like which subject is better, whilst we're at it.

 

[Evo]

If you were to force me to make a pie in the sky suggestion right away (and I should be something positive, I suppose), then here's a whacky idea, based upon a 'bigger picture' of mathematics:

1. Introductory mathematics (What currently passes for "analysis")
2. Algebraic Topology and Geometry
3. Differential Equations and Dynamical Systems
4. Measure Theory and Probability
5. Foundations of Mathematics (Logic, sets, categories)
6. Number Theory (if we must - this seems to be the least used forum, and the one with the highest crackpot hit rate).

Those seem good. I don't understand why the math forums lack the descriptions that you see, for instance, with forum headings in physics and engineering. If we did this, it would certainly help students decide which forum might be best to start with.

 

[Chris Hillman]

Those seem good. I don't understand why the math forums lack the descriptions that you see, for instance, with forum headings in physics and engineering. If we did this, it would certainly help students decide which forum might be best to start with.

Yes, I've been arguing that something like this would be an essential component of making the proposed reorganization work better than the current organization.

Let me try to reconcile my proposal with Matt's:

• Calculus and Miscellaneous
• Combinatorics, Graphs, and Number Theory
• Differential Equations and Dynamical Systems
• Linear and Modern Algebra
• Foundations (Logic, Sets, Categories)
• Manifolds and Geometry
• Measure Theory, Probability, Information Theory, Statistics
• Topology and Analysis

The biggest different between our proposals appear to be that my title "Manifolds and Geometry" would mostly subsume his "Algebraic Topology and Geometry", plus as discussed previously I advocate broadening "Number Theory" to include some other topics (combinatorics, graph theory) which are often taught in the same UG courses alongside number theory. Also, as discussed above, I propose to broaden "Measure Theory and Probability" to include information theory and statistics.

As per the discussion above, "Linear and Modern Algebra" and some of the other titles will look a bit odd to working mathematicians; they are compromises intended to head off anticipated misunderstandings by mathematical newbies who barely know what is a "matrix" is probably will not know that groups, rings, fields, modules and vector spaces are all of a piece.

Matt, can you live with this? Mathwonk?

 

[matt grime]

Can we drop either the reference to graphs or to combinatorics. I have little idea why people consider graph theory not to be combinatorics. And one can add in group theory. Thus Bullet Point 2 could become

Combinatorics, Groups and Number theory (purely alphabetical ordering)

These are what would be taught in an introductory course in 'finite mathematics'.

I would like to see 'linear and modern algebra' removed, as it is a nothing title - the word modern conveys nothing, and is arguably unrepresentative of truly modern algebra (a representation of a group is a functor from a category with one object and all maps isomorphism to Vect, anyone? Ok, that's perhaps not 'modern' but just 'obtuse'. Not sure there is a difference some times). However, the alternative of "calculus and linear algebra" is possibly even less attractive. I see no good solution here other than Chris's.

OK, before we keep coming up with more permutations, how about this:

can we agree on who we should be aiming the descriptions at? Undergraduates, (post)graduate students, or mathematicians?

 

[jostpuur]

Suppose there's a question related to the functional analysis and operator theory. Should it go to topology and analysis, or into the linear algebra?

Or I mean... I would probably put such question into linear algebra, because functional analysis and operator theory is infinite dimensional linear algebra, but if that subforum didn't exist, then it would be topology and analysis, probably.

The question about necessity of linear algebra was raised, so I just threw this as a point of view...

Or no! Actually I agree with Matt's comment about not having linear algebra like that. The proofs of functional analysis and operator theory are nearly always based on some concepts of topology. Having the linear algebra besides the topology and analysis would make it more difficult to decide where to put the operator question.

More elementary finite dimensional linear algebra questions fit probably quite well in the calculus and miscellaneous.

 

[Chris Hillman]

All Classification Represents a Compromise

Can we drop either the reference to graphs or to combinatorics. I have little idea why people consider graph theory not to be combinatorics.

Ow! Ow! I specifically said that the proposed titles do not represent an attempt to classify mathematics, but an attempt to help newbies figure out where to put their post.

That said, I could live with deleting "Graphs" from "Graphs, Combinatorics, and Number Theory" if its really that important to you. Ugliness and verbosity was also a factor which I balanced against clarity and correctness, but in the end in several cases trying to guide clue-deprived newbies was consistently my most heavily weighted criterion.

And one can add in group theory. Thus Bullet Point 2 could become

"Combinatorics, Groups and Number theory" (purely alphabetical ordering)

These are what would be taught in an introductory course in 'finite mathematics'.

It could be, but that's not the point: all my suggestions were prompted by actual episodes of newbies posting in the wrong place. I happen to think "Graphs, Combinatorics, and Number Theory" is a bit ugly/long and possibly (as you suggest) redundant, but I chose it to help guide newbies.

I would like to see 'linear and modern algebra' removed, as it is a nothing title - the word modern conveys nothing, and is arguably unrepresentative of truly modern algebra (a representation of a group is a functor from a category with one object and all maps isomorphism to Vect, anyone? Ok, that's perhaps not 'modern' but just 'obtuse'.

Ow, ow! I tried to make it clear that my suggestion of "Linear and Modern Algebra", in particular, represents a conscious compromise between The Truth and Avoiding Confusing Newbies (which would obviate the entire point of the proposed reorganization). I agree that to a working mathematician, some of my proposed titles sound odd/redundant, but I am modeling another target audience.

can we agree on who we should be aiming the descriptions at? Undergraduates, (post)graduate students, or mathematicians?

Good question. That's exactly the issue I raised way up above and also tried to stress in the other threads. My contention is that the audience we should be trying to guide with our reorganization are PF users, who run from high schoolers to Ph.D.s, with a bias toward the group I judge to supply newbies with the greatest frequency, UG students (often panicked students seeking study advice or homework help). My proposed categories were drawn up with this target audience in mind.

I see no good solution here other than Chris's.

OK, good, sounds like we are basically on the same page about priorities and purpose of the proposed reorganization.

Mathwonk, given what Matt and I just said, can you take a second look at my proposed titles?

Suppose there's a question related to the functional analysis and operator theory. Should it go to topology and analysis, or into the linear algebra?

Did you see that I discussed this very issue? (Either in this thread or one of the other two--- again, I asked the mentors to combine these three concurrent threads which all concern the same topic but to no avail, so it's not my fault that everyone has to read three threads.) Basically, I pointed out that those of us who know a lot of mathematics will know that ergodic theory posts could go in several forums under my plan, as could posts on topological vector spaces. I said I see now way of entirely avoiding confusion; the goal is to try to ensure that more posts find reasonably appropriate homes, to maximize the chance that newbie questions will get prompt attention from the most knowledgeable PF members.

I myself pointed out somewhere above (or in one of the other two threads) that questions on topological vector spaces could in principle wind up in "Linear and Modern Algebra" whereas they might better go to "Analysis and Topology". I said that some confusions are unavoidable no matter what titles we adopt. I have already urged (see also Evo's post above) that as part of the reorganization we should write a short sticky listing suitable topics for each forum. If we can agree on the basic titles, we can move on to writing that.

About topology posts, I tried to explain my feeling that posts involving topology of manifolds or Lie groups or homogeneous spaces will hopefully wind up in "Manifolds and Geometry", while posts involving function spaces will wind up in "Analysis and Topology". I said that posts involving ergodic theory would probably fit adequately in either "Analysis and Topology" or "Differential Equations and Dynamical Systems". I pointed out that most newbie questions on Dynamical Systems are very likely to involve the Lorenz attractor or Julia sets. The latter could go to either "Analysis and Topology" or "Differential Equations and Dynamical Systems" and I judged that acceptable. I myself pointed out that newbies posts on something like "box counting dimension" in chaotic dynamics might seem to belong to "Analysis and Topology" (since box dimension approximates Hausdorff dimension which is defined in terms of Hausdorff measure), but such questions would most likely arise in the context of time series, which would imply that "Differential Equations and Dynamical Systems" would be a good home after all.

Our reigning math guru, mathwonk, works (IIRC) on geometric invariant theory, which involves algebraic geometry, algebra, analysis, differential equations, topology, a whole lotta stuff. Under my proposed scheme most questions on that would probably wind up in "Manifolds and Geometry", which I think would probably be OK. In fact posts on algebraic geometry generally would probably wind up there too, which I think would be acceptable.

As [post=1511470]this recent post[/post] reminds us, IT is also interdisciplinary, with strong foundations in ergodic theory and probability, strong connections to analysis, but algebraic coding theory intersects linear algebra and modern algebra. So posts on coset decoding could go to either "Linear and Modern Algebra" or "Measure Theory, Probability, Information Theory, Statistics". Working mathematicians would plump for the former but I think it would be OK if the odd post on coset theory wound up in the latter.

By the way, if we're still discussing what titles to pick, I propose that everyone who wants to complain should find and cite a recent post illustrating the problem they forsee, as I have been doing to critique/amend my own proposal!

 

[Kummer]

• Calculus and Miscellaneous
• Combinatorics, Graphs, and Number Theory
• Differential Equations and Dynamical Systems
• Linear and Modern Algebra
• Foundations (Logic, Sets, Categories)
• Manifolds and Geometry
• Measure Theory, Probability, Information Theory, Statistics
• Topology and Analysis

Place Number theory with Algebra. That makes it better.

 

[Moonbear]

Ow! Ow! I specifically said that the proposed titles do not represent an attempt to classify mathematics, but an attempt to help newbies figure out where to put their post.

That said, I could live with deleting "Graphs" from "Graphs, Combinatorics, and Number Theory" if its really that important to you. Ugliness and verbosity was also a factor which I balanced against clarity and correctness, but in the end in several cases trying to guide clue-deprived newbies was consistently my most heavily weighted criterion.

Looking at those titles from the non-mathematician (a.k.a., clueless newbie) perspective, I think you're better off leaving graphs out of the title. Someone who has reached a high enough level of mathematics to be studying those subjects will not be so clueless about knowing how those subjects fit together if they are traditionally taught together, but if you leave "graphs" in the title, you run the risk of getting the junior high and high school kids who have been told to draw a graphs in their algebra or trig homework showing up to ask how to plot graphs. I suspect that's not what you intend that forum to include.

And since it's been asked who the "target audience" is, keep in mind that for most high school students, they will mostly participate in the homework forums and perhaps in a general math and calculus forums (we need to keep a general forum...partly because we can never be all-inclusive with the other forums, so need a place for questions that don't fit within any other category, and there should be a place where our younger students can ask simpler questions without being intimidated by the higher level discussions in the other forums). Most of our membership are students as far as I know, so I'd mostly keep in mind the upper level undergraduate math majors and graduate students, as well as physics majors who need to learn the math to do their science, as the primary users of those forums. Of course we have mathematicians and it would be wonderful to attract more, but across all our forums, generally those of us who are professionals in our fields come here expecting to assist students, with the occassional higher level discussion.

Keep in mind the number of questions we actually get on subjects. Some things don't make sense put together in a curriculum, but in a forum structure, in order to avoid an under-utilized forum which leads to people losing interest in sticking around, it is sometimes better to combine unrelated topics just to maintain a steady level of activity in a forum so that those who are interested in discussing a topic have other things to read and keep them interested while waiting for someone else to appear. When a particular forum is "bulging at the seams" with one topic dominating all the others, then it's a good time for us to split off the popular topic into its own subforum.

So, if everyone will look through their "my ideal forum would have..." lists, which is a great place to start, and now consider the practical matters of forum traffic and how some keywords in the titles will appear to a struggling high school student, I think the topics can be refined further to keep a reasonably sized forum with subforums that all maintain good activity levels without being overly burdened with too many popular topics all lumped together while less popular topics are split up.