Things math majors should know

[dkotschessaa]

Those of you in your upper level classes, graduate school, docs and post docs - what is it you wish you had known when you were an undergrad in mathematics? Things that nobody told you - but you wish they would have?

I have all sorts of silly questions sometimes on things ranging from what classes to take to what kind of pen and paper I should use sometimes. Perhaps you can help me and others out here by using the question above as a guideline.

-Dave K

 

[Robert1986]

Personally, I think that I did myself a disservice by not taking a topology class. I know some basic stuff from analysis, but not enough. So, I think that everyone should take topology (especially those who are going to grad school.)

 

[transphenomen]

If you can, you should audit at least one or two classes per semester. That way you can learn a lot more without it being too much of a time sink. You can do the homework of those classes when you have time or just ignore those classes when you need to prepare for tests in your real classes.

 

[nonequilibrium]

I mind not having had differential geometry in my undergrad, but the university didn't offer it is an undergrad course, and couldn't take a grad course in it, mostly due to there being too many course requirements already. Differential geometry seems really useful as I want to become a theoretical physicist and now I'll probably have to start learning general relativity without knowing differential geometry, which is presumably worse than studying quantum mechanics without knowing hilbert spaces, and I already deemed the latter as unacceptable...

EDIT:

If you can, you should audit at least one or two classes per semester.

Although that is a good idea, some people don't even have time for that.

 

[dkotschessaa]

I've thought about that transphenomen. I'm afraid I'm not too clear on my universities policy on auditing and it's been hard to find info on it. Does that usually cost something? Or can you just ask a professor if it's ok to sit in and do it unofficially?

Robert - I'm glad you said that. I was pretty sure I was going to take topology just out of interest, but it's seeming more indispensable now.

 

[dkotschessaa]

Does anybody wish they had taken a lighter course load and spent more time understanding the material? Or is the fast pace just "how it is" in undergrad years? (And then you can delve more deeply into your interests in grad school.)

I'm seriously thinking of limiting my course load, even if it takes me a year longer. I don't have any course requirements other than math right now.

 

[transphenomen]

I've thought about that transphenomen. I'm afraid I'm not too clear on my universities policy on auditing and it's been hard to find info on it. Does that usually cost something? Or can you just ask a professor if it's ok to sit in and do it unofficially?

At my college auditing is free. However, you obviously don't get a grade and you don't turn in homework or tests since they won't get graded. You just go there to sit for lectures. Finding information on auditing was very hard for me; there was only a vague paragraph about auditing in my 600+ page catalog. You best bet is to just email the teacher of the class you want to audit and ask if it is ok to audit his class. You don't even need the prerequisites to audit the class. For example, I will be auditing a graduate level general relativity class. When I emailed the teacher, he said he would assume the class knows Lagrangians and tensors. I don't know either, but I am studying up on them and find the probable challenge of the class exciting.

Also, even if you have time constraints, you can do your homework from other classes while auditing a class and just listen if something interesting happens.

 

[dkotschessaa]

Yes, I'm finding the same thing WRT to the search for information about auditing. It is a bit late in this semester anyway, but I'll consider next semester auditing linear algebra or something, which I'll take the semester after.

 

[transphenomen]

That's a good idea. There are two versions of real analysis and abstract algebra. One for applied and theoretical where the theoretical is harder. I am auditing the applied so I can ace the theoretical later.

Also, you can audit at anytime you want, even start in the middle of a semester. I did that with one class and I had a hard time catching up, but I learned enough to not need to try too hard when I take it for real later. For linear algebra, the first few weeks are just matrix multiplication, vectors, Gaussian elimination, and other things you should have known in a high school intermediate algebra class. If you start auditing it now, you should be able to keep up.

 

[moouers]

If you can, you should audit at least one or two classes per semester. That way you can learn a lot more without it being too much of a time sink. You can do the homework of those classes when you have time or just ignore those classes when you need to prepare for tests in your real classes.

I really wish I could do that. At my school, auditing a class costs the same as taking the class for full credit. It's silly.

 

[dkotschessaa]

I suspect it's the same at my university, but I'm double checking.

 

[Sankaku]

I'm seriously thinking of limiting my course load, even if it takes me a year longer.

If this is possible for you, I highly recommend doing it. Many people are constrained by financial considerations that make it difficult to take extra time for their degree. However, if you can take fewer classes you will learn the material much better (and get better grades as well).

 

[redrum419_7]

I really wish I could do that. At my school, auditing a class costs the same as taking the class for full credit. It's silly.

Same at my school

 

[dkotschessaa]

If this is possible for you, I highly recommend doing it. Many people are constrained by financial considerations that make it difficult to take extra time for their degree. However, if you can take fewer classes you will learn the material much better (and get better grades as well).

Yeah, whenever I mention it to people at my school they start talking about their scholarships and such, having to maintain a certain amount of credits, etc. I really don't have that issue. It will make things a bit more expensive as there are costs to every semester outside of just credits, but I'm ok with that.

 

[dkotschessaa]

Ok, here come some of the really silly questions I warned you about:

Paper and pencils.

Seriously - I can't seem to find out what my optimum tools are. I go through notebooks like crazy, and I can't seem to keep organized between class notes and homework. Now I'm even starting to wonder if I should just stop using notebooks and use *blank* paper, since this is what professors seem to use on overheads and this is how we are tested. Makes sense to do my homework the same way, right?

I haven't found the right writing utensil either. I keep getting mechanical pencils whose lead just falls out all the time. I end up having to carry a bunch of them around so I always have "backup."

Becoming more visual: Are there "equation people" and "geometric people?"

I tend to think I'm the former, but it seems that there's a lot of mathematics that depends on some kind of visualization, which I'm not that great at. Some teachers de-emphasize this, some teachers put a lot of emphasis on it. (Perhaps they themselves are in these two groups). Should I make a conscious effort to be more visual or just go with what I'm better at? I mean - knowing the shapes of different graphs, being able to work in three dimensional coordinate planes, etc.

-DaveK

 

[diligence]

Honestly, I think those are the type of thingsyou just need to figure out what works best for you. I don't see any general trends in my classes, some use pens, some use pencils, some use notepads, some use notebooks...I use blank paper and binders. The only thing I would recommend is either use a mechanical pencil or a pen so you don't have to sharpen the tip constantly.

As far as equation people vs geometric people, I think this is true to a degree. But again, you just need to do what works best for you.

 

[20Tauri]

Personally, I use recycled paper for my homework and take my notes in a notebook. My school collects computer paper that is printed on one side only to reuse in a recycle printer, so I just take paper from the collection bins and do my homework on the blank side. I like Pilot pens, but then I end up having to rewrite from the beginning if I screw up. My homework buddy turns in his work on gridded yellow paper and writes in mechanical pencil.

A topologist told me that he didn't really like algebra because it felt like symbolic manipulation to him and he couldn't visualize it very well, so I guess some people do prefer equations over geometry or vice versa.

 

[Sankaku]

Paper and pencils.

...snip...

Becoming more visual: Are there "equation people" and "geometric people?"

...snip...

I take notes on lined paper with a Pentel Kerry mechanical pencil using 0.5mm HB lead. Everyone has to find their own groove, though.

I submit any of my work in Latex. However, this is a time investment to learn properly so don't do it unless you are serious about carrying on in mathematics.

Strangely, I have a slight aversion to Point-Set topology because (to me) it seems like a mess of arbitrary unintuitive spaces with more counterexamples than anything resembling structure. Algebraic Topology is a bit of a different beast, though.

On the other hand Algebra is the most beautiful thing I have encountered in mathematics. I am deeply visual so I have found (admittedly strange) ways to visualize all the structures I have met in Algebra. Whatever way you build intuition is up to you. You will need intuition of some kind, though. If you rely on just cranking through equations there is a ceiling waiting for you...

 

[chiro]

I don't know if this is useful, but I would recommend that you know enough about other areas so that you understand why the other area is useful.

In other words know what a specific area is about in a few sentences but not the actual specifics of the proofs, identities, formulas and so on: just enough so that you know that if you have to figure out something out and you remember what a particular area is all about, then you go to the area later on and use what is already out there for your problem. It's also useful in advising other people who you work with directly or indirectly in your own field since the nature of mathematics is that it's connected and often a different perspective can end up solving a problem (which has happened countless times)

So if you a pure mathematician, know a bit about probability and statistics in terms of how to make a good inference. If you are a statistician, know about analysis in terms of convergence, continuity.

The thing is that this is becoming necessary anyway with number theorists using probabilistic primality testing and other examples of this.

 

[dkotschessaa]

I take notes on lined paper with a Pentel Kerry mechanical pencil using 0.5mm HB lead. Everyone has to find their own groove, though.

I submit any of my work in Latex. However, this is a time investment to learn properly so don't do it unless you are serious about carrying on in mathematics.

Hey, that's pretty cool. Now do you actually do your work in Latex or do you do it on paper and then re-type into Latex? Seems like I'd have a hard time working without physical paper and pencil. What editor specifically?

Strangely, I have a slight aversion to Point-Set topology because (to me) it seems like a mess of arbitrary unintuitive spaces with more counterexamples than anything resembling structure. Algebraic Topology is a bit of a different beast, though.

On the other hand Algebra is the most beautiful thing I have encountered in mathematics. I am deeply visual so I have found (admittedly strange) ways to visualize all the structures I have met in Algebra. Whatever way you build intuition is up to you. You will need intuition of some kind, though. If you rely on just cranking through equations there is a ceiling waiting for you...

I feel sometimes that I am just plugging through formulas, however I'm only in Calc III. I find that it's hard to get a very deep understanding of anything given the pace of the classes. I feel that if and when I go and review the material on my own time I can get a more intuitive/visual grasp of it.

-DaveK

 

[ingenvector]

As an undergrad, I can't add to your main question other than reiterating what I heard from some acquaintances that numerical methods are often times lacking and, as someone had already mentioned, topology is important. Though, because of research in my prospective interests, I could also add geometry as a possible concern.

However, the question of pens and paper is of intimate and immediate concern to me. Though ostensibly subjective, I think that plain white paper makes for excellent work, however, one can easily be caught in an aesthetic trap sacrificing efficiency for aesthetics. This is why, from an academic perspective, lined-paper may be more advantageous. You can also try those large artist pads, writing on them well, filling the entire page with equations and graphs thoughtfully laid out, makes mathematics and aesthetics converge into one wondrous art of physical expression(s). I recommend mechanical pencils, I think the 0.7mm is a good and common range, and the Zebra f-301 pen.

As to "geometric vs. equation" people, often times professors understand certain problems in either extreme and want to press their knowledge of the subject from their perspective. I think that this is only human: to understand something particularly from one frame of mind and then to try and impart the perspective onto others to lead to the same conclusions.

 

[ingenvector]

I submit any of my work in Latex. However, this is a time investment to learn properly so don't do it unless you are serious about carrying on in mathematics.

One doesn't necessarily need to know LaTeX to profit from its benefits, what about LyX? It's a GUI for LaTeX so one doesn't have to learn its language. It's main insufficiency is its limited range of formatting though, but I would think if this was a concern then the point has already come where LaTeX should be learned anyways.

 

[dkotschessaa]

One doesn't necessarily need to know LaTeX to profit from its benefits, what about LyX? It's a GUI for LaTeX so one doesn't have to learn its language. It's main insufficiency is its limited range of formatting though, but I would think if this was a concern then the point has already come where LaTeX should be learned anyways.

I'll have to check that out...Thanks. Still wondering what editors people use to do raw latex though. I downloaded TexWorks awhile ago but it's been slow going.

 

[ingenvector]

LyX actually uses MiKTeX which is what I learned LaTeX in, though it isn't source like TeXworks would be. I quite like MiKTeX, though there doesn't seem to be any definitive standards, and it is so far sufficient. It seems that most editors were made by some group who needed basic LaTeX functionality plus some extra esoteric function that wasn't found before in the previous distributions, so really, in essence they are all basically the same. Actually, TeXworks would probably be quite representative of what you're describing, it may be that you will simply have to suffer through the experience.

But now I think I may have accidentally high-jacked the thread and brought it somewhat off topic...

 

[zif.]

I really wish I could do that. At my school, auditing a class costs the same as taking the class for full credit. It's silly.

Same at my uni, but that's because it's set up as an, "Oh ****! I ****ed up and don't want a 'withdraw' or, worse, 'not complete/fail' on my transcript" for students.

School's like this are playing pretty fast and loose with the term 'audit'.

That said, even if your school charges to 'audit' a class like mine does, there's not a single physics or math prof here that won't let you sit in on the classes (as long as you're not a distraction).

 

[bpatrick]

Here's a few of my thoughts on this thread so far:

auditing is great, I would highly recommend it. I've never had trouble auditing courses. One of my friends tried to audit an actuarial class and was made to pay for it because it was "university policy" to still register even if you were auditing, but I audited 2 pure math classes at the same university and never had to register nor pay. I think the profs, in general, have some sort of code where, if you're pursuing higher, pure math, then they welcome auditing, but if you want to audit stuff that you might benefit directly from (like free actuarial exam prep) or auditing finance courses (free CFA prep) is where you run into more trouble ... since so many people would do that just to pass various career exams. I've never run into any trouble auditing graduate level math at any university I've audited from.

As far as the paper/pencil/work stuff: I roll with a clipboard and use recycled paper (one side already printed) to do all my busy work / general problem solving. I also have a big white board at home with dry erase markers that are made from recycled stuff and can be recycled when they run out.

I don't really take notes, so I can't help you much there. I read ahead in the course so I can pay attention and let the material sink in a second time rather than mindlessly copying what the prof is saying/writing and not really absorbing it. I have some scrap paper handy if they do something interesting/new and I usually keep those around in a notebook until I have the material mastered, then just recycle it like all my used paper.

I found the Zebra M-301 0.5mm mechanical pencil is really nice for me. They are well built enough to not worry about them breaking but also inexpensive enough that I bought 3 packs of them (2 to a pack) about 10 years ago and don't get super worried that I'll lose them or whatever.

I always recommend taking as much linear algebra + algebra as possible, especially if you are thinking about grad school. You can never know enough linear algebra. That being said, even though topology is one of the "big three" that you'll probably take qualifying exams in, take as much analysis and algebra as you can while an undergrad.

If you want reading material that will help a little with some "what is grad school like" moments you might be having, look into: "A Mathematician's Survival Guide" by Steven G. Krantz.

 

[homeomorphic]

Becoming more visual: Are there "equation people" and "geometric people?"

I tend to think I'm the former, but it seems that there's a lot of mathematics that depends on some kind of visualization, which I'm not that great at. Some teachers de-emphasize this, some teachers put a lot of emphasis on it. (Perhaps they themselves are in these two groups). Should I make a conscious effort to be more visual or just go with what I'm better at? I mean - knowing the shapes of different graphs, being able to work in three dimensional coordinate planes, etc.

I should comment on this, since I'm Mr. Visual. Just having intuition is the more fundamental thing, rather than "visualization" or geometry. Most of all, I like to understand. I just find that visualization sometimes makes things totally make sense where they would otherwise be incomprehensible (I can follow the steps, but might not remember any of it, were it not for the pictures). It's also a great aid to memory.

The advantages of visualization seem pretty obvious to me because a lot of times, it took me a long period of not understanding it visually before I came up with a good visual explanation. So, I've done both in a lot of cases, and everything always makes so much more sense once I come up with the pictures. So, I suspect a lot of the people who are more on the equations side just don't know what they are missing.

I attribute my (relatively limited, but quite substantial, by layman's standards) success in math largely to the fact that I am good at visualization. It's the inspiration for a great many of my proofs.

Sometimes, I do wonder if maybe it's good not to convert everyone to my viewpoint because maybe we need people who actually LIKE ugly calculations. Then, I can leave all the dirty work to them.

In any case, visualization does improve with practice. When I read Visual Complex Analysis, I had to rehearse the arguments in my mind for a long time in some cases before I was able to see it clearly in my mind's eye. It doesn't necessarily come instantly. I've gotten a lot better at it, but even now, sometimes, it takes some thought.

 

[dkotschessaa]

LyX actually uses MiKTeX which is what I learned LaTeX in, though it isn't source like TeXworks would be. I quite like MiKTeX, though there doesn't seem to be any definitive standards, and it is so far sufficient. It seems that most editors were made by some group who needed basic LaTeX functionality plus some extra esoteric function that wasn't found before in the previous distributions, so really, in essence they are all basically the same. Actually, TeXworks would probably be quite representative of what you're describing, it may be that you will simply have to suffer through the experience.

But now I think I may have accidentally high-jacked the thread and brought it somewhat off topic...

Nope, it all relates back. It seems like LaTex is something "Math Majors should know" at some point or another. I know the physics department at our school uses it.

-DaveK

 

[dkotschessaa]

I should comment on this, since I'm Mr. Visual. Just having intuition is the more fundamental thing, rather than "visualization" or geometry. Most of all, I like to understand. I just find that visualization sometimes makes things totally make sense where they would otherwise be incomprehensible (I can follow the steps, but might not remember any of it, were it not for the pictures). It's also a great aid to memory.

The advantages of visualization seem pretty obvious to me because a lot of times, it took me a long period of not understanding it visually before I came up with a good visual explanation. So, I've done both in a lot of cases, and everything always makes so much more sense once I come up with the pictures. So, I suspect a lot of the people who are more on the equations side just don't know what they are missing.

I attribute my (relatively limited, but quite substantial, by layman's standards) success in math largely to the fact that I am good at visualization. It's the inspiration for a great many of my proofs.

Sometimes, I do wonder if maybe it's good not to convert everyone to my viewpoint because maybe we need people who actually LIKE ugly calculations. Then, I can leave all the dirty work to them.

In any case, visualization does improve with practice. When I read Visual Complex Analysis, I had to rehearse the arguments in my mind for a long time in some cases before I was able to see it clearly in my mind's eye. It doesn't necessarily come instantly. I've gotten a lot better at it, but even now, sometimes, it takes some thought.

I'm way undergrad, so I think it's too early to know what it is I really "like." But I seem to be an "equation person," and somewhat outside of math I really enjoy learning about symbolic logic. But I think this is not exclusive of visual thinking. Whenever I glance at a topology book the first thing I see is some kind of set theory (which I haven't learned yet).

-DaveK

 

[dkotschessaa]

How about a more practical way of talking about this visual thing?

If you're a math major, you should at least know graphs of basic functions: conic sections, trigonometric functions, the square root function, logarithmic functions and be able to know what happens when you manipulate them. (Functions of one variable). What else?

Then you start getting into mutivariable. I can't draw these damn 3-d graphs. I can barely graph a point, (on ruddy 2 dimensional paper!). But I think I need to get some more facility of this, and familiarity with...the 3-d analogs of the above?

-Dave K

 

[espen180]

How about a more practical way of talking about this visual thing?

If you're a math major, you should at least know graphs of basic functions: conic sections, trigonometric functions, the square root function, logarithmic functions and be able to know what happens when you manipulate them. (Functions of one variable). What else?

Then you start getting into mutivariable. I can't draw these damn 3-d graphs. I can barely graph a point, (on ruddy 2 dimensional paper!). But I think I need to get some more facility of this, and familiarity with...the 3-d analogs of the above?

-Dave K

I am currently learning about manifolds, and since doing just about anything with manifolds has you climbing up and down chart functions (you will see expressions like $D(\psi\circ f \circ \phi^{-1})(\phi(p))w_p$ ), and having a geometric image of what is going on is useful not to get lost in the equations. Although this is in an arbitrary number of dimensions, a 2-to-1 dimensional analog usually suffices.

 

[ingenvector]

Nope, it all relates back. It seems like LaTex is something "Math Majors should know" at some point or another. I know the physics department at our school uses it.

-DaveK

Well, that's good at least. My university's physics department doesn't seem to require LaTeX, though it is the norm and they do provide a preamble for submission formatting. Regardless, LaTeX should be something that everyone in a scientific, mathematical, or engineering field should learn, nearly every science or mathematics book and paper is published with it. I sometimes try and see if I can tell whether or not a publication was made using LaTeX. I suppose I'm somewhat of a typeset nerd.

Also, I think that using LyX, one can be nearly as fast typing notes in math as by hand, the only fall would be graphics would still need to be hand-drawn. What I sometimes do is type in some of my notes, the ones which are probably important and can use for future reference, into LaTeX, and it keeps a high degree of organisation, particularly if all the notes are in one comprehensive collection. I'm at around 250 pages now.

 

[ingenvector]

I'm way undergrad, so I think it's too early to know what it is I really "like." But I seem to be an "equation person," and somewhat outside of math I really enjoy learning about symbolic logic. But I think this is not exclusive of visual thinking. Whenever I glance at a topology book the first thing I see is some kind of set theory (which I haven't learned yet).

-DaveK

You do bring up a good point. Though I personally think it's largely useless, symbolic logic does not lend itself to visualisation readily unless the problem is already defined as a relational problem. It is also important to algorithms, though I don't think people usually visualise flow charts or trees beyond a certain size. What you are seeing in Set Theory I think is principally a similar syntax, but it doesn't use first-order or propositional logic and consigns itself to the more restrictive intuitionistic logic. It's what makes Set Theory so... intuitive.

 

[homeomorphic]

I'm way undergrad, so I think it's too early to know what it is I really "like." But I seem to be an "equation person," and somewhat outside of math I really enjoy learning about symbolic logic. But I think this is not exclusive of visual thinking. Whenever I glance at a topology book the first thing I see is some kind of set theory (which I haven't learned yet).

There are areas of math that are less visual than others. Modern math is all founded on set theory. Actually, set theory is pretty visual to me. A set is just a bunch of dots or sometimes a big blob in my mind's eye (or chalkboard or piece of paper). I can't remember things or reason effectively about sets if I don't visualize something. Even though it's abstract, point-set topology is pretty visual for me. I'm not just talking about topology. Analysis, topology, differential geometry, combinatorics, and even algebra (for example, there are lots of pictures you can draw of "root" systems in the theory of Lie algebras) all have their visual side. I think there's a place for less visual people in math, but there are large chunks of math that I don't see how anyone could make heads or tails of if they are geometrically challenged. Even if they can somehow manage, they'd be missing all the beauty of it. So, I don't think it's possible to be a broad or deep mathematician without being at least somewhat comfortable with visual arguments.

 

[Sankaku]

Hey, that's pretty cool. Now do you actually do your work in Latex or do you do it on paper and then re-type into Latex? Seems like I'd have a hard time working without physical paper and pencil. What editor specifically?

I use Kile (KDE-Linux) but Texmaker seems to be a similar cross platform editor.
http://en.wikipedia.org/wiki/Texmaker

There are plenty of options out there. Just try them and see what you like.

I do very rough pencil work on paper before typing my work up. However, sometimes it is just as fast to work on the screen once you get used to the Latex notation.