Other Should I Become a Mathematician?

[Mariomaruf]

I'm a freshman in high school who has spent tedious years dealing with the school system and what you might call it's ignorance when evaluating the student body for candidates who want a future career or passion which they are dedicated towards, but this year my high school allowed me to change everything that was holding me back for so many years. Right now, I'm in AP Calc, but you could say I pretty much learn nothing new there because I'm so far down the road of math that calculus is just an elementary tool I use for some higher things. I teach 6 kids: an 8th grader, a high school freshman, two high school juniors, and two high school seniors; and doing so is helping me to understand what the students' individual needs are, and how they choose to interpret math. We're all blazing through the curriculum at just a little faster than university speeds, and it always makes me proud when we can do that and they understand it well even with sprinkles of upper level theory, so well in fact that they go to teach others.

I'm also taking AP Chem and AP Bio if it matters, and my counselor is seeing what he can do to get me into undergrad/grad work at Princeton next year, so it's nice to finally have all of my education set straight for me. I won't say how I got them to recognize me, but I will say that I'm about up past analysis and intermediate topology.

I'm really not sure what to do right now, but I'm really worried that the pure math that I really love to do won't be able to get me a good job or anything, so what I really want to know is

A) Where do I go from topology?
B) What do I do after my Ph.D in math?

 

[Boogeyman]

Does anyone take part in the International Mathematics Olympiad?

 

[tcbh]

I'm a 2nd year math major at a quarter school. I've already taken the first upper division course in Linear Algebra (goes up through IP spaces, Normal and Self-adjoint operators and Diagonalizability) and the second quarter isn't a graduation requirement. But I was wondering, do most grad schools expect that applicants will have covered subjects like Dual Spaces or Jordan Canonical Form?

To be honest, I've found analysis much more interesting, and I'd like to take a few classes on logic. The 2nd quarter of Linear Algebra is only offered once a year, so by the time I take it many quarters will have passed and I'm afraid I'll be a bit rusty. I'm not even sure what would be the most important material to review.

 

[mathwonk]

@ Periotic: Is there a difference?

 

[mathwonk]

@Mariomaruf: Have you read some of the early pages of this thread? There is a lot of general advice there.

Would you say you have mastered calculus at the level of the book by Spivak? And is your topology at the level of Munkres? If so, you are indeed sailing along.

There are always jobs for pure mathematicians at a certain level, as professors in university, and the pay is not terrible, especially at places like Princeton.

As to what to do next, if you only know calculus at the high school AP level, then read Spivak. If you are already really past that and know only some beginning analysis, you might try Rudin, or a complex analysis book like the one of Cartan, or Lang, or easier, the one of Frederick Greenleaf.

Since your studies seem specialized in topology and analysis, you might start learning some algebra, such as from Theodore Shifrin, or even Michael Artin. Or maybe you should begin with linear algebra, such as from Friedberg Insel and Spence, or Shilov, or Hoffman and Kunze.

There are a number of free sets of course notes on my website as well, whatever they are worth.

 

[mathwonk]

tcbh: yes jordan form is always tested on the algebra prelim exam at UGA. and dual spaces are fundamental in many areas of math including analysis.

You might get some use out of the free course notes for math 4050 or 845, on my web page:

http://www.math.uga.edu/~roy/

 

[mathwonk]

I'm sorry Boogeyman, I wasn't in the Olympiad. You might email Malcolm Adams, or Valery Alexeev at university of georgia math dept for information, or Valery's son Boris, who is a grad student at Princeton:

http://www.math.princeton.edu/~balexeev/

I think Boris at least took the Putnam exam, and Valery may have been in the Olympiad.

 

[sponsoredwalk]

Hi Mathwonk, just reading some of the notes on your website & I think I've found a small
error, apologies If I'm wrong or if it's been pointed out countless times before

7. Math 4050. Advanced undergraduate linear algebra:

http://www.math.uga.edu/~roy/4050sum08.pdf

On page 2 in the axioms for k x V → V (scalar multiplication axioms)

5) “associativity”: for all a,b, in k, and all x in V, (a+b)x = ax + bx;

7) distributive over addition in k: for all a,b in k, all x in V, (a+b)x = ax + bx;

Shouldn't 5) be:

5) “associativity”: for all a,b, in k, and all x in V, (ab)x = a(bx) ?

I checked the book "Linear Algebra Thoroughly Explained" (page 8)" to make sure,
Either everyone skipped over it as your notes tell them to or I'm just confused, it's late

I'm a 2nd year math major at a quarter school. I've already taken the first upper division course in Linear Algebra (goes up through IP spaces, Normal and Self-adjoint operators and Diagonalizability) and the second quarter isn't a graduation requirement. But I was wondering, do most grad schools expect that applicants will have covered subjects like Dual Spaces or Jordan Canonical Form?

To be honest, I've found analysis much more interesting, and I'd like to take a few classes on logic. The 2nd quarter of Linear Algebra is only offered once a year, so by the time I take it many quarters will have passed and I'm afraid I'll be a bit rusty. I'm not even sure what would be the most important material to review.

https://www.amazon.com/dp/1402054947/?tag=pfamazon01-20

 

[sponsoredwalk]

What you wrote is commutativity. (ab)x = a(bx)

Check page 8 of the linear algebra book I linked to, they say:

The associative property of the multiplication of numbers with respect to scalar multiplication:

(ab)x = a(bx)

They give only 4 porperties for scalar multiplication vector spaces &
commutativity is included seperately, unless there is something
deficient in this book I don't think it's wrong.

edit: lol @ deleted post, thought I just got another response that wasn't
showing up on the thread, common enough to scare me

 

[Shackleford]

Check page 8 of the linear algebra book I linked to, they say:

The associative property of the multiplication of numbers with respect to scalar multiplication:

(ab)x = a(bx)

I immediately deleted my post. It was a knee-jerk post. Matrices do not generally commute.

 

[Ryker]

Check page 8 of the linear algebra book I linked to, they say:

The associative property of the multiplication of numbers with respect to scalar multiplication:

(ab)x = a(bx)

Yeah, you're right about associativity, that is (5), although I'm sure this is just a typo on mathwonk's side.

 

[mathwonk]

thank you sponsored walk! i suspect this means you are the first person to read these notes!

 

[mathwonk]

notice how i cleverly avoided such errors in the defn in my shorter notes "rev lin alg", p.1, line 7,

http://www.math.uga.edu/%7Eroy/rev.lin.alg.pdf

 

[Dougggggg]

First day of DE and I think I have feel in love. It takes everything I learned in Calculus, puts a spin on it, and makes it more useful.

 

[mathwonk]

Douggggg, you might enjoy reading pages 17-20 of those same notes from my web page for math 4050. I.e. "Jordan form", the hardest topic in beginning linear algebra, is nothing but the matrix of a constant coefficient differential operator in a standard basis given there. oops, pardon me I pronounced your name wrong, Dougggggg.

 

[Dougggggg]

Ha, I will check it out, I haven't gotten to Linear Algebra yet, but I will check it out for sure. Can I find it through one of the links above?

 

[mathwonk]

here is the linear algebra book link:https://www.math.uga.edu/sites/default/files/inline-files/4050sum08.pdf

 

[Dougggggg]

Wait, did you write this? It seems like everything is pretty well stated. I do think I may need a bit more studies before I can truly understand all that to a level I would like but it seems really clear and laid out.

 

[raff]

Is the quick way really the quick way...

I have a MBA in Finance and want to eventually get a Master's in Math. I see two learning options:

1- Use standard University Fare: Stewart for Calculus, Boyce for DE, Lay for Linear Algebra, etc.

This way "looks" quickest and easiest. For example, I did all of the problems in a section of Stewart and they were painless. Boyce also seems very clear and well explained.

2- The other way is to choose more demanding texts. For example, Apostol or Spivak for Calculus, Hubbard or Robinson for DE, Halmos and/or Axler for Linear Algebra, etc.

This way will be challenging but much more interesting. I read a section of Apostol, it took me days to fiqure out one of the harder problems.

My purpose in getting the Masters in Math is not to become a Mathematician rather to work in Quantitative Finance.

My question is whether I will really be "saving" time by choosing the 1st path.

Another question, I read Lang's "Basic Mathematics" as a refresher prior to beginning my MBA. It is a great book; he assumes intelligence. His introductory books on Calculus (the initial versions) and Introduction to Linear Algebra seem much shorter than standard books. How does Lang on Calculus compare to Spivak or Apostol?

 

[tcbh]

tcbh: yes jordan form is always tested on the algebra prelim exam at UGA. and dual spaces are fundamental in many areas of math including analysis.

You might get some use out of the free course notes for math 4050 or 845, on my web page:

http://www.math.uga.edu/~roy/

Thanks. I just noticed that it's on the basic exam here too. I guess I have another year to decide

 

[Mariomaruf]

@mathwonk

I already did Shilov's Linear Algebra , tenenbaum and pollard's Diff Eq's, and Rudin's Real/Complex Analysis. Stockton is letting me go there on Tuesdays and Thursdays to learn what they call advanced calculus starting next week, where I spend the first half of the high school day here and the second half at stockton, and I'll also be dropping AP Calc and picking up Physics C. I'll try those books on Abstract Algebra because I've only rarely seen what it is, but I think my understanding of Calculus is fairly full fledged, and this class will sharpen it a little for me.

 

[mathwonk]

those 4050 notes are the lecture notes i wrote for my summer course. If they help I am truly delighted. on my web page are more detailed notes for math 845, in the 843-4-5 sequence that are much more thorough. also the notes for math 8000 are also there, but more sophisticated than the 4050 notes.
mariomaruf, you sound very advanced. as such, you will get lots of help, but if i can help i will be glad to do so.

take a look at artin's algebra, spivak's calculus on manifolds, or loomis and sternberg advanced calculus (free download from sternberg's web page), arnol'd's ODE, and some of my free notes on my web page, like math 843-4-5. check out also milnor's topology from the differentiable viewpoint.

edit years later: you know, unless my notes really call out to you, I have to admit they totally violate my advice to read the masters. However one very good student did tell me they helped him prepare for prelims in algebra.

 

[Dougggggg]

So integration, though I understand the concept completely and know how to do it (at least to my level of learning), seems to keep giving me trouble. It seems mandatory that I make one silly mistake that I shouldn't make per integral.

In Calc I, it was frustrating because while studying for a test, I would miss so many practice problems for integration. I would be very worried going into test day, somehow on test day I didn't make many errors at all. I missed like 1 integration problem on all the tests combined for that class.

Now I'm in Calc II, now my teacher for this class takes up the homework unlike the one I had for Calc I. After we finish a section, the homework is not due the first class after we finish it, but the one after that. The idea is that we are to ask questions during that next class. I worked a bit on the homework, got stuck on one problem, it was an odd problem, so the answer was in the back of the book. I kept getting a different answer. I decide to ask him for help during class.

We have been working on volumes by integrating. I ask my question, he starts working on setting up the integral, which I had already set up just fine but couldn't figure out where I messed up after that. Then soon as he finished that he says "ok well that was the difficult and interesting part of the question, the rest of it is easy and I'm sure you all know how to do it, if not then you probably didn't do well in Calc I."

I was pretty ticked off at this point because I got an A in Calc I, I don't make anything less in math or physics courses. I did finally talk to a friend in my class and saw that I had a minus sign error while simplifying my answer. Which left me with the I don't know who I want to kick in the head more, my teacher or myself.

I realize there wasn't really a question but I will save it here at the end. Number one, am I wrong for thinking that even good mathematicians can make errors while integrating or are true good mathematicians something that I have never actually seen in real life? Secondly, is me being a little ticked about how that played out an understandable response or am I making a mountain out of a mole hill? If I am, I apologize for my whining and crying.

 

[rmalik]

is this the right thread to ask questions about math degrees?

 

[lisab]

is this the right thread to ask questions about math degrees?

If you have a specific question, it's probably best to start a new thread. Find the "New Topic" button in the Academic Guidance Forum.

 

[rmalik]

If you have a specific question, it's probably best to start a new thread. Find the "New Topic" button in the Academic Guidance Forum.

ok thanks, and nice quote =)

Going into my quotes.txt file

 

[mathwonk]

you can ask anything. after asking it we may say we have no clue.

 

[dkotschessaa]

ok thanks, and nice quote =)

Going into my quotes.txt file

Make sure you attribute it properly to Ben Franklin though, and used the less dumbed down version "Dost thou love life? Then do not squander time, for that is the stuff life is made of."

I mean I love Bruce Lee but, c'mon! :)

-DaveKA

 

[sponsoredwalk]

Hi mathwonk, if you don't mind would you have a read of this thread & let me if you already
knew this? I know you understand the Hoffman/Kunze idea that I've mentioned in my latest
post but the ideas in my post speaking about Cayley & his ideas are virtually non-existent
apart from his original papers - I can't find a single textbook that mentions them. Hopefully
you'll learn something new but if not I'd love to know if the idea was ever taught like this
since it's far clearer than the rote memorization technique taking place in schools & really
not an advanced concept.

The first post is just my frustration at being unable to find a satisfactory answer & is
justifiably very confused but luckily I went back to Cayley himself & got an answer,
if I'd read Hoffman/Kunze first I'd probably have missed his wonderful ideas & no doubt
many people are missing it, it's absolutely fascinating though.

 

[mathwonk]

My explanation to my summer 4050 class is on pages 10-13 of the notes linked above in post 2481. I say there, after defining linear maps, that we want a way to compute values of linear maps T. Then I show how this leads to a matrix whose columns are the values of T on a basis, and then evaluation on another vector is given by taking a linear combination of those columns. Then I equate that with a mechanical multiplication procedure involving dot products of rows with columns. The main point is then that this multiplication, applied to two matrices, corresponds to composition of the corresponding maps. In particular that let's you understand why the multiplication is associative.