Should I Become a Mathematician?

[wisvuze]

Thanks all for the clarifications/insights. And this thread is awesome, thanks for keeping it up

 

[mathwonk]

bless you wisvuze. it is the questions that keep it alive. fire away.

 

[sponsoredwalk]

Mathwonk, here is one definition of a differential equation:

"An equation containing the derivatives of one or more
dependent variables, with respect to one of more independent
variables, is said to be a differential equation (DE)",
Zill - A First Course in Differential Equations.

Here is another:

"A differential equation is a relationship between a function
of time & it's derivatives",
Braun - Differential equations and their applications.

Here is another:

"Equations in which the unknown function or the vector function
appears under the sign of the derivative or the differential
are called differential equations",
L. Elsgolts - Differential Equations & the Calculus of Variations.

Here is another:

"Let f(x) define a function of x on an interval I: a < x < b.
By an ordinary differential equation we mean an equation
involving x, the function f(x) and one of more of it's
derivatives",
Tenenbaum/Pollard - Ordinary Differential Equations.

Here is another:

"A differential equation is an equation that relates in a
nontrivial way an unknown function & one or more of the
derivatives or differentials of an unknown function with
respect to one or more independent variables.",
Ross - Differential Equations.

Here is another:

"A differential equation is an equation relating some function
ƒ to one or more of it's derivatives.",
Krantz - Differential equations demystified.

Now, you can see that while there is just some tiny variation between them,
calling ƒ(x) the function instead of ƒ or calling it a function instead of an
equation but generally they all hint at the same thing.

However:

"Let U be an open domian of n-dimensional euclidean space, &
let v be a vector field in U. Then by the differential equation
determined by the vector field v is meant the equation
x' = v(x), x e U.

Differential equations are sometimes said to be equations
containing unknown functions and their derivatives. This is
false. For example, the equations dx/dt = x(x(t)) is not a
differential equations.",
Arnold - Ordinary Differential Equations.

This is quite different & the last comment basically says that all of the
above definitions, in all of the standard textbooks, are in fact incorrect.

Would you care to expand upon this point if it interests you as you know
a lot about Arnold's book & perhaps give some clearer examples than
dx/dt = x(x(t)), I honestly can't even see how to make sense of dx/dt = x(x(t)).

A second question I really would appreciate an answer to would be - is there
any other book that takes the view of differential equations that Arnold does?
I can't find any elementary book that starts by defining differential equations in
the way Arnol'd does & then goes on to work in phase spaces etc...

 

[mathwonk]

there are two kinds of functions on a manifold M, i.e. functions f:R-->M, and functions g:M-->R/ These have as derivatives, either a curve of velocity vectors in M along the curve f(R), or gradient vectors in the domain of g.

Thus a differential equation is a vector field on M, i.e. an assignment of a vector to each point of M. A solution is either a function g:M-->R whose gradient at each point of M is the given vector at that point, or a curve f:R-->M whose velocity vector at each point f(t) is the given vector in M at f(t).The theorem is that the first kind of solution, i.e. gradient solution, usually does not exist, [it exists iff i forget what, something about equality of mixed partials,..], but the second kind, the velocity solution, usually does.

 

[jmblock2]

I'm in the electrical engineering PhD program (in the very early stages) at georgia tech right now. I often feel like I want to go into pure math, but I kind of keep chugging along the engineering track. I have a lot of credit available to take electives and I'm putting them towards mathematics. I was hoping for a bit a feedback on possible routes.

I took undergraduate classes at the university of illinois in linear, complex, and abstract algebra, and differential geometry. I really loved the differential geometry stuff, but it seems like it is important (or fun) to just know everything! So currently I'm stuck in choosing 3 courses out of the following graduate classes:

Linear Algebra
Algebra I (http://www.math.gatech.edu/course/math/6121 )
ODE I
PDE I
Real Analysis I
Algebraic Geometry I (http://www.math.gatech.edu/course/math/6421 )
Algebraic Topology I (http://www.math.gatech.edu/course/math/6441 )
Into to Geometry and Topology I (http://www.math.gatech.edu/course/math/6457 )

Right now I'm leaning towards Algebra I, PDE I, and either Algebraic Geom or Algebraic Top. Maybe it is too specific of a question, but more generally how does a graduate level math course compare to an upper-level undergraduate course? Would it be a grave error to skip on graduate versions of the fundamentals? Thanks!

 

[mathwonk]

the grad course is usually faster paced and repeats the same topics from a more sophisticated, i.e. abstract, point of view. most people find the repetition helpful or essential. e.g. that algebra syllabus is also covered in michael artin's undergraduate algebra book. I like Artin, but Dummit and Foote is also well liked by students, and has an especially extensive problem list. that algebraic geometry syllabus is just a list of basic foundational topics without much interesting geometry of curves or surfaces in it yet, but still constitutes basic language a lot of people find it useful now to know. It sounds more or less like the topics in chapter one of hartshorne. the algebraic topology syllabus sounds a little higher level to me, and sounds quite interesting, roughly the same stuff my second graduate course in topology covered. Hatcher's book is also considered very readable. he writes so well maybe he can make higher level material more accessible.

Why don't you go talk to your professors at Tech. Ernie Croot and Matt Baker are nice guys and experts in algebra and number theory. Jeff Geronimo is an analysis and very nice. These guys should be able to give more pertinent advice. You can say roy smith sent you if you want.

 

[jmblock2]

I'll definitely be doing that. Thanks for the insight!

 

[ECmathstudent]

I'm starting research this summer on a linear algebra problem, I only really have a vague idea of what I'll be doing, outside of setting up algorithms and going over journals. How often should you check with your adviser to make sure you aren't on a stupid tangent, or how do you even figure out where to start? I'm also worried I picked too difficult a topic-optimizing matrix multiplication, like Strassen's algorithm-since far smarter people seemed intimidated by it.

 

[ireallymetal]

If I want to become an actuary or something in finance like a quant what are some good places to go for undergrad(Other than the usual MIT/Caltech/Harvard etc...) and what majors/dualmajors/minors are recommended?

Is it best to major in applied math and minor in CS in this case?

 

[mathwonk]

EC, most advisors see their students once a week.

Any students out there in actuarial science etc with advice for Ireally?

 

[snipez90]

If I want to become an actuary or something in finance like a quant what are some good places to go for undergrad(Other than the usual MIT/Caltech/Harvard etc...) and what majors/dualmajors/minors are recommended?

Is it best to major in applied math and minor in CS in this case?

Creating your own thread in Academic Guidance will get you more responses due to increased visibility.

Being an actuary and being a quant are pretty different. Many universities don't offer an undergraduate actuarial science degree, certainly not the ones you've listed. This is probably because it's essentially a professional degree. If you're good at math (in particular basic probability theory) you can probably pass the first few (out of several) actuarial exams already. The idea is that if you pass these the first few stages of examination and demonstrate potential, actuarial firms will hire you and pay you to take subsequent exams. So if you really want to be an actuary now, apply to a school that has an actuarial science major. But I don't think you need to unnecessarily confine yourself at this point if you're not sure.

Thinking about what the best major for quant related work is again confining yourself. There is no best combination of degrees, but several that would work. Think more carefully about your own interests (which I'm assuming is math related since you're on this forum and asking this question) and what you can do to improve your mathematical abilities and more generally your reasoning abilities. Employers care less about your major and more about what you've learned.

 

[QuantumP7]

I wonder how do math graduate departments, especially those of top schools, feel about students who have self-studied a lot of mathematics. Do they look upon it favorably? Do they see it as a positive indicator of motivation?

 

[hunt_mat]

Not too sure, many of them will go on grades, but if you can impress the person you want to be working for ten all the better.

 

[mathwonk]

we don't know how to measure that. we need somebody who we know is an expert to tell us what they think of your expertise or potential. Failing that, you need to come talk to us, so we can assess it ourselves. So you probably need to schedule an interview.

 

[wisvuze]

lately, I've been being picky between a couple of analysis books: I've been leafing through Apostol's mathematical analysis book, one by Andrew Browder, the advanced calculus book by Loomis and Sternberg and Dieudonne's foundations of mathematical analysis book ( I found this at a school booksale for 10 bucks!)

I have to say that I really like the dieudonne book, and the loomis book, and the apostol book. The browder book seems a bit more mundane to me, but I can tell that it's still very good. Any thoughts on these books? Is there much of a difference between the first and second editions of the apostol book? (I only have a library copy, but if I were to buy my own copy, I can only find the first edition)
thanks!

 

[mathwonk]

forgive me, but I am tempted to say, as one of my advisors said to me: stop "dancing around the fire" and just start reading one of those books. they are all great.

 

[Cod]

I just took my Real Analysis final this morning. We used Lay's "Analysis" as the text and it was pretty straight forward and systematic. The number of examples was a little lax, but easily rectified by searching for examples online. Also, the book was on the "cheap end" at $60 USD.

 

[Dougggggg]

How often do students take graduate courses before they finish their undergraduate program? My academic advisor and I were talking about classes for the next couple semesters and with the year I plan to graduate, there would be a year of getting a bunch of gen eds that got neglected and I would have already taken all the math offered there. She suggested taking a graduate course at Vanderbilt University since my school doesn't have a graduate math program and Vandy is a hop, skip, and a jump away from my campus. So general advice, does this happen often and could I do it with at least 9-15 hours of gen ed courses (easy courses, but time consuming)?

 

[Sethric]

Happens all of the time, to my knowledge. About 1/3 of the students in the entry level graduate courses in my school were undergraduates. Some of them overloaded credits as well. It just means that you will be expected to work harder. And a lot of the classes will specifically require the permission of the instructor for you to join. My experience, at least.

 

[Dougggggg]

Happens all of the time, to my knowledge. About 1/3 of the students in the entry level graduate courses in my school were undergraduates. Some of them overloaded credits as well. It just means that you will be expected to work harder. And a lot of the classes will specifically require the permission of the instructor for you to join. My experience, at least.

Thanks for the response, since I haven't had any of the Vanderbilt professors I am assuming they would have to give the ok after getting letters of recommendation from a professor or two from my university or something to that extent?

 

[Sethric]

It is their class and you would not be a graduate student, so final say rests with them. The letters of recommendation would only serve to help sway their opinions. More than likely, they would be fine with it, but it is still their call.

 

[sphericow]

I'm in first year and taking calc I (well the New Zealand equivalent), I have already covered most of the course at high school; differentiation rules, related rates etc etc etc. The only real difference is an introduction to some proofs and its generally deeper than what i received at high school. So naturally I scored in the high nineties in the first test. I planned on getting an engineering degree, so this semester I'm taking only the one math class. I plan to change my major next year, in the mean time I'm stuck.

Background info aside, should i focus my study time going over material relevant to this class (doing tonnes of practice problems etc). OR should I start to self study analysis (I have spivak and access to the uni library)?

 

[mathwonk]

Here is a survey of attitudes of entering college students in calc I. One point of interest: 94% expect to get at least a B, whereas in fact only 50% do so.

http://maa.org/columns/launchings/launchings_05_11.html

 

[qspeechc]

Mathwonk, you have probably read this article, but I would like to know your thoughts on it. It is http://www.maa.org/devlin/LockhartsLament.pdf" by Paul Lockhart, a mathematician who teaches school mathematics. He writes about what he thinks is wrong with school math.

I agree with him on many things, but I think he goes too far in some instances. In my opinion, it isn't so much the material presented for school math, but the way it is presented that is the problem. Lockhart seems to think that both the material and the methods are seriously at fault.

 

[mathwonk]

I will read it when I have time and comment. For now I recall that I have probably looked at it, and my reaction was that is a negative diatribe of minimal value.
\
\(example: "Sadly, our present system of mathematics education is precisely this kind of nightmare.")

We all know things are in trouble, let's not spend all our time detailing the problems. As one of my respected gurus in yoga said: " when throwing out garbage, it is unnecessary to examine it first".

 

[samuelarnold]

Mathematics is my favorite subject, but i don't want to be a mathematician.

 

[mathwonk]

qspeechc, I tried again but could not wade through that article by Lockhart. Its not that I disagree with him, but I have learned, on this forum e.g., that expressing negative thoughts, even if true ones, tends to be counterproductive.

So I guess I would challenge people who are troubled by the situation Prof Lockhart describes to try to think of solutions. I think this forum makes a contribution to the solution, as does mathoverflow and stackexchange. Best wishes.

 

[qspeechc]

qspeechc, I tried again but could not wade through that article by Lockhart. Its not that I disagree with him, but I have learned, on this forum e.g., that expressing negative thoughts, even if true ones, tends to be counterproductive.

So I guess I would challenge people who are troubled by the situation Prof Lockhart describes to try to think of solutions. I think this forum makes a contribution to the solution, as does mathoverflow and stackexchange. Best wishes.

Indeed, it is always easier to criticize than to come up with solutions. At least there are some people out there willing to try to make a difference, and I am grateful for places like PF for it.

 

[dkotschessaa]

I just completed my first semester of college after having been out of school 13 years, and it has given me an interesting perspective. I see the problem more in the students than the teachers, to be perfectly honest. If you really want to learn something then you'll do it whether the teacher gives it to you or not. This is why Mathwonk's page 1 advice is gold - read great mathematicians, do lots of problems. I believe school can help you with these tasks and make them "official" and give you some structure. But I'm not sure it's the teachers job to pass on the "art" to all of his students, since most of them are not interested. What I saw was a lot of "shortcutting" and finding the quickest way to pass the class to get on to the next one and finish the degree.

As that "art of math" thing goes, I do think there is something to be said for a mentor relationship outside the classroom. If I've surveyed the academic community correctly, this will most likely be the person who helps you with your research.

I agree with the author about some of the attempts to make math "fun" in textbooks. They are mostly humorous. My favorite is a chapter opener in my pre-calc book that shows a picture of the U.S.S. enterprise, gives half a sentence about warp drive before going into a discussion of "mach" speed and then logarithms. Total bait and switch! Other pictures include people on roller coasters (yaay physics!) other fun activities that merely serve to remind you that you are reading a textbook written by some very out of touch people who should probably just stick to the math.

 

[magheera]

I love math. I'm a 18 year old college sophomore and I'm devoting my entire summer to math courses so I can catch up on my degree as a math major. (I was until recently a theater major.) I want to be a mathematician.

 

[Wingeer]

I think I want to become a mathematician. At least I hope so, as mathematics is the only subject I feel comfortable with. That being said, I am not a prodigy, neither am I a hard learner. I am in my second year of an undergraduate course in mathematics and up to now things have been pretty good. In the first year we had calculus, linear algebra, the usual stuff, and though I didn't get the grades I was hoping for it went kind of easy. The main reason is of course lack of effort; I know I can do better than a C. However this year it is getting tougher as we've started to touch upon the upper level math like abstract algebra and such. I know I haven't done my best this year, and I suspect it will show on my final exams. However I don't want to lose my devotion for the subject, but these courses are getting tougher. Is it normal to struggle with these courses? Or am I not predisposed for mathematics?

 

[Ryker]

The main reason is of course lack of effort

However this year it is getting tougher as we've started to touch upon the upper level math like abstract algebra and such. I know I haven't done my best this year, and I suspect it will show on my final exams. However I don't want to lose my devotion for the subject, but these courses are getting tougher. Is it normal to struggle with these courses? Or am I not predisposed for mathematics?

Is this a trick question?

 

[Wingeer]

Not entirely. I know I should put in more effort. However I still struggle with the problem sets. Usually I have to check the solutions, and most of the time I think "Aah, of course!". There are seldom things I have to read more than once to grasp. This is maybe a problem of patience, and something I have to work on myself. Still, I wonder if it is normal to struggle with these subjects, if one compare with the same work effort as earlier courses?

 

[mathwonk]

keep reading books written by the best mathematicians you can enjoy and appreciate, and try to have actual one on one conversations with mathematicians, as these convey more in fewer words than any other mode of learning.

 

[Wingeer]

Great advice. I recently bought "Disquisitiones Arithmeticae" and I think I will attempt to read it this summer.