Should I Become a Mathematician?

[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.

 

[Sankaku]

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?

It can often be a shock that upper level Math courses require a much larger investment of time than the usual Calculus sequence. The types of questions (proofs, etc.) may be different and require a different part of the brain, but doing more practice problems is very important. Essentially, you should know when you have done enough because you will feel (fairly) confident going into exams. If you don't feel confident, you haven't done enough.

If you can't get there on your own, get someone to help (office hours, TA, friends, PF, etc.).

 

[Wingeer]

Thank you for your answer.
Yes. Then the conclusion is clear. I have not done enough. Although the abstract algebra is really starting to come together.

Yes. I will have to do that.

 

[mathwonk]

with gauss, you will benefit from even one page. so do not obsess about about reading it all or even a certain amount, just read some, and think about it.

 

[Shackleford]

My calculus textbook says the three greatest mathematicians are Newton, Gauss, and Archimedes. What are your thoughts?

I may have already mentioned that. Haha.

 

[muppet]

Personally, my thought is probably that {all mathematicians in history} and the integers don't have the same order type...

EDIT: Ever have a moment when you realize you should get out more?

 

[mathwonk]

i don't know whos the greatest, those are certainly great. I appreciate archimedes especially, and i also like riemann a lot.

 

[Shackleford]

i don't know whos the greatest, those are certainly great. I appreciate archimedes especially, and i also like riemann a lot.

My probability professor talked about Cantor once or twice. He said when Cantor developed set theory and so forth, the mathematicians thought he was crazy.

The professor also talked about Riemann and the Lebesgue and how Riemann knew there was something wrong with his integral up until the day he died.

 

[Dougggggg]

If we are on the subject of great mathematicians I would like to put in a note of Euclid. Granted many of the proofs in "Elements" are rather simple to understand, the logic he did them with at the time was years ahead of its time.

So I have been doing a little bit of thinking about different branches of mathematics and have been wondering what branch I could see myself falling into. I have finished a bunch of the lower level foundational courses like Calc I, II, and an intro course to higher math (learning proof methods, set theory, mathematical logic, etc.). There are many branches I know I will get a taste of before I finish my undergraduate degree but I really want to get a feel for what else that I might really really enjoy. So given a list of things I will get a taste of before I'm done, is there anything that someone could suggest me maybe checking out to see how interested in it I am. If you know of a good textbook on it then even better. Below is that list of things that I will be getting a chance to study.

Real Analysis
Graph Theory and Combinatorics
Operations Research
Linear Algebra
Euclidean and Non Euclidean Geometries
Probability and Statistics
Modern Algebra
Differential Equations


I would be so grateful for some suggestions of other branches that have active research.

 

[jbunniii]

Euler certainly merits a mention, both for the quality and the quantity (76 volumes of the Opera Omnia published to date) of his work. All the more astonishing given that he was completely blind for the last 20 years of his life but still averaged one paper per week during much of that period.

 

[Sankaku]

Real Analysis
Graph Theory and Combinatorics
Operations Research
Linear Algebra
Euclidean and Non Euclidean Geometries
Probability and Statistics
Modern Algebra
Differential Equations

Don't forget things like Number Theory, Complex Analysis and Topology.

;-)

 

[elfboy]

it's too hard..too much to learn

 

[Dougggggg]

Don't forget things like Number Theory, Complex Analysis and Topology.

;-)

Thanks, I was talking to one of my professors Friday and he also recommended Number Theory and Topology as well. He also mentioned Differential Geometry but he said I should wait until I have taken an upper level proof course before I try self teaching myself over the summer. So my reading for this summer is probably going to be mostly philosophy.

 

[epsilon>0]

I have a few questions:

How much free time to graduate students and phd's have? Besides mathematics, there are other areas that I would like to be successful in? Is that even possible or is it necessary to prioritize? I've read that you have to want to eat, sleep, and breathe mathematics to be successfull in grad school and beyond, if I did that I know i wouldn't feel fullfilled.

https://www.physicsforums.com/showthread.php?t=148086

I read this thread, but I was wondering if anyone else had any insight.

How important is it to go to a highly ranked school? Do phd's in the top 20 or 30 have an easier time landing an academic position?

 

[kramer733]

I have a few questions:

How much free time to graduate students and phd's have? Besides mathematics, there are other areas that I would like to be successful in? Is that even possible or is it necessary to prioritize? I've read that you have to want to eat, sleep, and breathe mathematics to be successfull in grad school and beyond, if I did that I know i wouldn't feel fullfilled.

https://www.physicsforums.com/showthread.php?t=148086

I read this thread, but I was wondering if anyone else had any insight.

How important is it to go to a highly ranked school? Do phd's in the top 20 or 30 have an easier time landing an academic position?

If you've contributed to any field in science, then that's what matters. It might not be until 200 or 300 years before it gets used in some sort of application but you've still contributed to a pool of knowledge and you're helping humanity understand the world with one more step.

 

[epsilon>0]

If you've contributed to any field in science, then that's what matters. It might not be until 200 or 300 years before it gets used in some sort of application but you've still contributed to a pool of knowledge and you're helping humanity understand the world with one more step.

But you can't really contribute if you are a starving mathematician without a job. Also, for myself the teaching part of academia is important. I noticed that where I went to school, every professor in mathematics had their phd from a top 25 institution.

In professional degrees, it is necessary to go to a top school, if you want to work as an academic or get a prestigious job...is the same true with phds?

 

[tauon]

Don't forget things like Number Theory, Complex Analysis and Topology.

;-)

Speaking of Number Theory. It's one of the courses I have to take but I simply loathe. I already flunked it because I couldn't get motivated enough to study for it. How does one find the necessary excitement for it, what are some interesting results in number theory? And by interesting I mean results with interdisciplinary connections, because I find purely number theoretic results boring as hell.
I kinda feel that, considering all the other maths I'm taking in college, with this number theory course I've reverted back to some level similar to long division in middle-school.
I mean, I don't care that 2^{29} has 9 distinct digits and that you can find which digit is missing (4) without actually calculating the number, I'm completely unmoved by the fact that \phi (n) = \sigma (n) has a finite number of solutions generated by 2, 3 and 5.
In short, I do not like numbers - numbers are for computers. Half the time I was at lectures or seminars (especially seminars) I spent it thinking "you know, I can write code that can solve that much faster than any human can, wtf am I doing here?".
I know standard problem sets are somewhat simplistic and silly for any course, but I feel it's downright ridiculous with number theory. I can't help but think that there are no complex results in number theory, only complicated ones.

Help? :(

 

[mathwonk]

how about mordell's conjecture that if the smooth compact complex surface obtained by smoothing out the zero locus defined by a polynomial with integer coefficients is a doughnut with more than one hole, then there are only a finite number of rational roots?

Or that in the set of all prime numbers, the density of the subsets of those ending in 1,3,7,9 are all equal?

or that a prime > 2 is a sum of two squares iff it has form 4K+1?

or that all primes are sums of at most 4 squares?

and i like euclid a lot too. did you know he described tangents to circles as essentially limits of secants? Prop III.16.

 

[capandbells]

Does anyone know where I can find (or at least find resources to assemble myself) something like a "math roadmap"? That is, some sort of tree that shows the courses one would need to take in order to study other material. Like, if I wanted to study algebraic geometry, I'd need to first study abstract algebra, then commutative algebra first.

 

[dkotschessaa]

I would think that most math departments, like mine, would have something like this they give to their math majors. I have mine hanging on a wall next to my desk that I can stare at so I know what's ahead. I find it oddly inspiring.

Mine is not available electronically but I did find a few from other universities (I know nothing about the programs. I just googled "Prerequisite flow chart for math majors" since that's what mine was called) and they follow more or less the same logic:

http://www2.sfasu.edu/math/programs/advising/0708MathMajPrereq_tree.pdf
http://www.morris.umn.edu/academic/math/advising2.html

The gist (I've discovered) is that math doesn't "start" until at least two semesters of calculus.

-Dave K

 

[mathwonk]

I am reluctant to give lists of prerequisite books for any goal, as they become very long and make the journey seem inaccessible. Rather if you read even part of one excellent book you are well on your way to some interesting stuff.

If you want to learn algebraic geometry, try walker's algebraic curves, maybe the first couple chapters, then fulton's algebraic curves, as much as you enjoy, or miles reid's undergraduate algebraic geometry, and then shafarevich's basic algebraic geometry.

fulton's book is free online, and shafarevich's book is based on an article in Russian math surveys that is available in libraries for free.

an older book of interest is that of semple and roth, and another good modern introduction is joe harris' algebraic geometry.

if you get an old copy of shafarevich one good feature is that it includes the needed commutative algebra.

there are many other excellent well known books, but most of the books above have low entry levels of prerequisites.

 

[Nano-Passion]

In need of urgent help. =/

I never really payed attention to much math, but I started liking it freshmen college year. I took precalculus in the spring semester and put all my effort in it, likewise I got 100s and aced everything so I was pretty content and that gave me confidence that hard work and effort prevails.

But... whenever I go on the forum I hear many many alien mathematical terms that I have never heard before like topology, etc. I don't really know much math at all aside from pre-calculus and I'm only beginning to self teach myself calculus. I'm pretty much only familiar with the basics of algebra, geometry, functions/pre-calc, and some trigonometry.

What are some sources to further spice up my interest in mathematics and challenge me/get me familiar with other concepts?

 

[jbunniii]

This book may be just what you are looking for:

https://www.amazon.com/dp/0195105192/?tag=pfamazon01-20

 

[Nano-Passion]

This book may be just what you are looking for:

https://www.amazon.com/dp/0195105192/?tag=pfamazon01-20

Thanks, I read the reviews and they were amazing. I'm kind of excited but I won't be able to buy it for some time. Any other, but cheaper books?

 

[Ryker]

Thanks, I read the reviews and they were amazing. I'm kind of excited but I won't be able to buy it for some time. Any other, but cheaper books?

But this one's ... 15 bucks

 

[Nano-Passion]

But this one's ... 15 bucks

lmaooo. Why you have to put me on the spot Ryker? =[

Okay well I help support my family so more often than not I end up broke. And I don't get paid until the end of this week but then again I have a lot of stuff to pay for.

But yeah they probably don't get much cheaper then this hehe.

 

[Ryker]

Yeah, sorry, I didn't mean to get at you this way, it's just that, as you mentioned, it's hard to find something cheaper.

 

[zonk]

What geometry book would you recommend for someone who barely remembers high school geometry? I don't remember many of the ratios and facts about circles and triangles. Going through Feynman he uses ratios like this. I also find it hard to do optics problems. And when it talks about how you can or cannot construct certain measurements I have no idea how to do it with a straight edge or compass

 

[jwxie]

Maybe you can find these books at your school library / local library. You'd be surprised

What geometry book would you recommend for someone who barely remembers high school geometry?

I'd say here
http://www.regentsprep.org/regents/math/geometry/math-GEOMETRY.htm

Just pick up a high school geometry book. There is nothing more simpler than a pre book.