Exploring Mathematics with Limited Knowledge

[trancefishy]

so, I'm currently working on my bachelor's in mathematics, and seriously considering going on to pursue a PhD after that. Currently, I'm frustrated (not for long, semester is nearly over) with my calc II course and "matrix theory and linear algebra", though not so much the latter as the former.

I would like to do some research, or independant studies. There is an incredible amount of material though, and 99% seems over my head. I've read some online journals, read a bit on rudimentary knot theory, and have heard of modular forms, though can't find anything that is even remotely accessible by myself.

In short, my question is, what are some interesting mathematics I could play with, research, perhaps even write a paper on, without having much knowledge beyond the two classes I'm taking now? I'm obviously willing to read up on pre-requisite subjects if that is required for the main topic I would be looking into. Thanks.

 

[matt grime]

Without wishing to be a killjoy, there is little to no chance of you being able to do any maths research of an original nature. The best you could do is to be a research assistant to one of the professors and do some of the tedious legwork for her.

Maths research is incredibly hard, and it's not a surprise you can't understand the journals. Nor should it be a concern.

 

[trancefishy]

I don't care whether or not it's original. I'm just looking for suggestions on a topic that is accessible and interesting to me personally.

 

[matt grime]

Sorry, I took "write a paper" to mean something different from your intention.

 

[arildno]

Without being able to give you a detailed and specific programme, you should perhaps take into consideration the following two more "personal" sides:
1) What have particularly fascinated YOU in maths?
Is there some specific area which seems interesting to you personally?
2) How well do you get along with the different lecturers/tutors?
Is there someone you really think you could have a good and constructive advisor/student relation with?

 

[trancefishy]

Well, the things I have mentioned about are interesting to me, but, a bit above and beyond. There looks to be a good book on knot theory recently published by the AMS that I may get my hands on soon. Anything with symmetry or patterns I find interesting.

As for professors I could talk to, there are 2 so far. One I had all last year, and I went to visit him just the other day. He is more along the lines of a teacher than a mathematician. The other, I have for calc2 right now. I don't talk to him very often, and he is a bit inaccessible (personality wise). He does do, or has done, research. I am going to go into his office soon and ask him essentially the same question, and talk about different options for the future. I would like to have a little bit of knowledge of something before I go in there though. I really feel like I'm going to be going in there clueless, but that is why I'm going in the first place. It'd be nice to have a topic or two that looks interesting that we could discuss (on very general terms).

 

[arildno]

Just remember:
You are not the first "clueless" person to enter a professor's office!
They are used to that; in fact, I think most would be rather surprised if a student early on his studies had a clear conception of all the details in his future research.

 

[trancefishy]

You're absolutely correct. I suppose I feel like I'm meeting up to suggest a topic for my final dissertation, when in fact, I'm just going to a teacher to say "hey, i want to do stuff that is beyond what we do in class, where do you suggest i start?"

 

[arildno]

Well, he would probably be pleased to guide you on if you ask him:

"I'm kind of interested in so-and-so (fill in your own interests), and I really would like to know what material I ought to use to study this stuff a bit further on my own".

I think he could give you very good advice on that (but probably with the cautionary remark: Don't neglect your current curriculum..)

 

[mathwonk]

jason cantarella at the university of georgia math dept works on knot theory and has a unmber of downloadable papers on his website.

the basic text on the topic is knots and links by dale rolfsen, long out of print from publish or perish. maybe available again now.

a number of universities including georgia have summer programs called REU's (Research experience for undergraduates) intended to introduce undergrads to research and somehow or other they manage.

there are well known programs at Boston University in number theory for undergrads. check out glenn stevens there. the number theory community has perhaps the best and longest running tradition of good teaching for undergrads, and maybe also grads.


heres a used copy of rolfsen for you. they are rather scarce.

Rolfsen, Dale
Knots and Links (Mathematics Lectures, No. 7)
Houston, TX, U.S.A.: Publish or Perish, Incorporated, 1976*VG, clean, no writing, no highlighting.
ISBN: 0914098160
Bookseller Inventory #ABE-311731660
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As to doing research, get in the habit if asking questions. in your courses when they prove things, ask yourself what they are not proving.

Oh yes, you said also you are interested in modular forms. you need to learn complex analysis first, but there is a wonderful book on the topic that starts slow and gets very far by Serre, called Course in Arithmetic.

there is another nice little book by Robert Gunning, but you probably cannot read it yet.

By the way, what Matt Grime said is true, math research is very hard, so learn all you can and then go with what you love. When something is hard it is crucial to be doing what you love.

 

[trancefishy]

Thanks a bunch for the info. I have already looked into the REUs, and ended up discovering the math in moscow program, which I will be participating in for the fall 2006 semester (ways off, but I'm excited). 50 bucks is a bit steep for a book, though, compared to texts i guess it is not. I am going to go and see if the library has the Course in Arithmetic, or something on complex analysis at the least.

Again, thanks a whole lot

 

[trancefishy]

I have been nosing around on jason cantarella's site, and I love it. the pictures are pretty cool, and now I'm moving on to his papers. thanks again

 

[philosophking]

trance--If you're interested in calc II (probably my favorite of the beginning calc courses, because of sequences of series), then you should seriously take an analysis course.

For instance, my school offers Analysis I and II. What these courses do is they take what you learn in calc I and II and explain the theory behind them. I'm in Analysis I right now and love it. It's nothing like calculus though, there's no numbers really! No more plug and chug as they say. A lot of writing proofs of theorems. It is so fun though. It might turn you off though, as it is very hard and requires quite a different mindset when compared to calculus I II III and linear algebra/matricies/differential equations.

 

[trancefishy]

Analysis sounds great. I like doing that sort of math, and only get a taste of it in matrix theory (I haven't seen a number in there in a long time), and we do lots of proofs. calc 2 for me was really boring. very mechanical, and we took forever getting through techniques of integration. boring. we have no analysis course here (northern michigan university), though I'm sure something makes up for it (inadequately). the next 3 semesters i will be taking calc 3, abstract algebra, and diff eq.

Also, i found the William S. Massey book, Algebraic Topology: An Introduction. near those, i also ran into some books that look really good. Geometry from a Differentiable Viewpoint by John McCleary, and The Shape of Space, 2nd ed, by Jeffrey R. Weeks. on top of those, i ran across this book in the new books section (which i was unaware existed until today at the university library), called Realistic Rationalism by Jerrold J. Katz of MIT. I thought at first it was just another philosophy treatise, but, it's a book about the philosophy of mathematics. reading the intro and a few pages, it looks to be interesting.

 

[mathwonk]

try "topology from the differentiable viewpoint", the alltime classic by john milnor.

and for analysis just get a good calculus book, like courant or spivak, or apostol.

then if you want a great analysis book you will keep all your life, try dieudonne, foundations of modern analysis.

 

[trancefishy]

I've got the James Stewart Calculus textbook, that I've used for 2 semesters and I have mastered nearly everything in it. this is single variable though. for calc 3 next semester i am getting a new and different one. I'm not sure who it is authored by, though. I am debating whether or not I want to keep the one I have, or sell it back.

are the books you recommended textbooks, or more like, for lack of a better work, regular math books?

 

[JasonRox]

try "topology from the differentiable viewpoint", the alltime classic by john milnor.

and for analysis just get a good calculus book, like courant or spivak, or apostol.

then if you want a great analysis book you will keep all your life, try dieudonne, foundations of modern analysis.

I did exactly what trancefishy is going to do.

I went to the professor and said the stuff in the class is too easy and I was getting bored. I told I wanted to know if I was pure math material before dropping the idea of attempting. So, I asked if Stewart's textbook was real math, he said no and recommend Micheal Spivak's text.

I strongly recommend getting Spivak's text because as of now, you are behind. Maybe not behind, but you will sure feel like an idiot. I wish I asked a lot earlier.

 

[mathwonk]

as jason rox says, stewart is a decent middle range text, but nowhere near the level of math majors at top schools, which is what spivak is for. the good books are spivak, courant, and apostol.

(if you think 50 bucks is high wait till you price those.)

have you thought of transferring to ann arbor? that is one of the best math depts in the world.

 

[JasonRox]

as jason rox says, stewart is a decent middle range text, but nowhere near the level of math majors at top schools, which is what spivak is for. the good books are spivak, courant, and apostol.

(if you think 50 bucks is high wait till you price those.)

have you thought of transferring to ann arbor? that is one of the best math depts in the world.

I got Spivak for $10CDN at a used book store. I also got Halliday/Resnic (Physics) for $5CDN. Halliday/Resnick is mildly used and Spivak's is brand spanking new.

I know you weren't speaking to me, but I've been thinking of the University of Waterloo since I'm in Canada. I've been looking up the courses they offer and some of the material they are going through as Calculus I,II,III and its much better than what my school offers. I meet my prof on a weekly basis now and we discuss whatever question in Spivak's text, which is priceless. I have been on the text for only 3 weeks now and its a pain in the ass to work on it when you have all these annoying classes and work on the weekend. I hope I can catch up to where we are so I can just follow along with it for Calculus II next term.

 

[JasonRox]

By the way, I don't think it will happen.

I have a hard time agreeing/understanding the concept of the proof by induction. The steps they take make sense and are normally obvious, but I don't know if I accept it as proof. It doesn't feel like its enough.

 

[shmoe]

I am debating whether or not I want to keep the one I have, or sell it back.

Unless you really need the cash and the resale value is decent, I'd probably keep it. Even if you replace it with something more thorough, you know Stewart well and might find yourself wanting to look something up in it. Plus, the memories...

 

[mathwonk]

suppose some one of a sequence of statements is false, say the 10th one is false. but a proof by induction shows that the first one is true, and that any of them is true provided the previous statements are all true. so since the first statement is true, then the next one must also be true, i.e. then the second statement is also true.

but now since the first and second statements are true, then also the next one after that is true, i.e. the third one is also true. but then the 4th one is true and so on.

you keep going like that until you have proved the 10th one is true, contradicting the assumption it was false.

in a similar way it follows that none of the statements can be false, so they must all be true.

for example: all cats are the same color.
proof by induction: consider the following sequence of statements: the first statement is that any set consisting of one cat, contains only cats of the same color, and the second statement is that any set of two cats are all the same color, and the third statement is that any set consisting of three cats are all the same color. etc...

so let's prove all these statements by induction.

now obviously a set consisting of only one cat, contains only cats of one color, since there is only one cat in there.

now suppose we have given the proof for all sets with up to n-1 cats in them.

so then consider a set of n cats. e.g. think of a set of 10 cats. we claim they are all the same color. if we remove one cat, we get a set of n-1 = 9 cats, which by our inductive hypothesis are all the same color. now remove some other one cat, and then again all those n-1 = 9 cats are the same color. but the two sets of n-1 cats have n-2 = 8 cats in common, which are also all the same color, so all the cats are the same color and therefore all cats are the same color.

now isn't that clear?

 

[trancefishy]

Well, I will certainly keep an eye our for those calc texts. perhaps over break i can do that, though the books i have now are going to keep me busy for the next 6 weeks.

As for Ann Arbor, yes, I've looked into transferring in general, and in fact, moved to flagstaff, arizona, 3 years ago, got accepted to NAU, but decided not to attend and instead took the year off. I don't want to now transfer to ann arbor because I'm almost 23, and I would like to get my bachelor's degree as quickly as possible, so that I can move on to more interesting things. When (and if) I go on to pursue my PhD, then I will certainly choose a premium-quality institution. I have University of Wisconsin, Madison in mind, but I haven't looked at many others, besides MIT, which is a bit beyond my capabilities, I think.