Other Should I Become a Mathematician?

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Becoming a mathematician requires a deep passion for the subject and a commitment to problem-solving. Key areas of focus include algebra, topology, analysis, and geometry, with recommended readings from notable mathematicians to enhance understanding. Engaging with challenging problems and understanding proofs are essential for developing mathematical skills. A degree in pure mathematics is advised over a math/economics major for those pursuing applied mathematics, as the rigor of pure math prepares one for real-world applications. The journey involves continuous learning and adapting, with an emphasis on practical problem-solving skills.
  • #1,141
proton said:
how is abstract algebra compared to real analysis in terms of difficulty? - for someone who's mastered upper-div linear algebra

A lot of people find algebra easy and analysis hard or vise verse. I think it has to do with motivations and the students background. It is impossible to tell which will be harder with just the knowledge that you did well in linear algebra. Certainly, you are probably prepared for either one of them. I don't know:

Algebra will be SLIM and what I mean by that is you will not have a lot of tools to use (at least at first) in proving theorems or working problems. This makes the problems sort of easier but more abstract and less intuitive.

Analysis will be FAT. You will have too much knowledge to use on anyone single problem and often it is hard to really figure out what you need in order to solve a problem, but (in a first course) it will be more intuitive and familar.

Finally, I would say linear algebra leads more into abstract algebra for the most part. I can't really say much more than that.
 
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  • #1,142
mathwonk said:
It is so hard, but to me so important, to try to ignore personal issues, fame, competition, etc... and focus on enjoyment, understanding,... when doing math. hang in there!

I was talking to a Professor at my school about what it is like to have a mathematical career in academia yesterday. He mentioned that there is (like in all science fields) a lot of strife between people (some of it underserved). He was mentioning how he at first when he was applying for an NSF grant couldn't get it because of a few people or had problems with his advisor. He used the option to not let these people review his application and got it. But, I think it is interesting that in many ways mathematicians can be decitful, dishonest, and childish. Of course, it is to be expected I guess. You would wish (or at least I would as a young idealist) that mathematics would be this great open community in which everyone collaborates with everyone else and there is mutual respect for every one and so on. To some extent I am sure this exists; but, it hit me that there are a lot of jerks out there (and no matter what career you choose you won't be able to get away from them). His advice was that you only share your ideas with people who you have a commitment in working with and that you keep your ideas until you publish them. That's sad I guess because it closes down discussion to some extent. As I see it, this is directly caused by a high level of compitition.

I am sure Mathwonk as many stories of grudges within departments and between people from diferent universities.
 
  • #1,143
eastside00_99 said:
A lot of people find algebra easy and analysis hard or vise verse. I think it has to do with motivations and the students background. It is impossible to tell which will be harder with just the knowledge that you did well in linear algebra. Certainly, you are probably prepared for either one of them. I don't know:

Algebra will be SLIM and what I mean by that is you will not have a lot of tools to use (at least at first) in proving theorems or working problems. This makes the problems sort of easier but more abstract and less intuitive.

Analysis will be FAT. You will have too much knowledge to use on anyone single problem and often it is hard to really figure out what you need in order to solve a problem, but (in a first course) it will be more intuitive and familar.

Finally, I would say linear algebra leads more into abstract algebra for the most part. I can't really say much more than that.

really? I've heard that at my school at least that analysis is the hardest math course.
 
  • #1,144
I know many people who found that the first analysis course was easily. But, I remember also everyone saying this is such a hard course and everyone was worried about doing well in it. That just contributed to the difficulty of the course more than anything.
 
  • #1,145
That's a bad question. You can't say analysis is harder than algebra or vice versa. Analysis and algebra, along with topology, are the three main fields of math, with other fields on the side of course. If you're considering two classes, one in algebra and one in analysis, then the question of which would be harder depends on the school, the classes, the professors, and the person taking the class.
 
  • #1,146
eastside00_99 said:
I was talking to a Professor at my school about what it is like to have a mathematical career in academia yesterday. He mentioned that there is (like in all science fields) a lot of strife between people (some of it underserved). He was mentioning how he at first when he was applying for an NSF grant couldn't get it because of a few people or had problems with his advisor. He used the option to not let these people review his application and got it. But, I think it is interesting that in many ways mathematicians can be decitful, dishonest, and childish. Of course, it is to be expected I guess. You would wish (or at least I would as a young idealist) that mathematics would be this great open community in which everyone collaborates with everyone else and there is mutual respect for every one and so on. To some extent I am sure this exists; but, it hit me that there are a lot of jerks out there (and no matter what career you choose you won't be able to get away from them). His advice was that you only share your ideas with people who you have a commitment in working with and that you keep your ideas until you publish them. That's sad I guess because it closes down discussion to some extent. As I see it, this is directly caused by a high level of compitition.

I am sure Mathwonk as many stories of grudges within departments and between people from diferent universities.

A perfect example of this at the highest level is Perelman's case.

It comes down to the biology of humans and our evolutionary past. So unfortunately it's only natural for us to be 'bad'. That is why mathswonk said '...it is so hard...'
However I also understand why he also said '...so important...'
 
  • #1,147
In my own experience these cases of jealousy and competition are less common than might be supposed. Since we are human they do exist but they have not at all defined my experience in math. I have met so many generous mathematicians. As a simple example, if you look in Mumford's book on Theta functions, in one footnote he credits me with having described theta functions in a certain way, when actually I myself got that description from a book by Siegel. So I am guilty of not acknowledging Siegel in my talk, but Mumford was so scrupulous as not to want to even give a definition that had been inspired by someone else without crediting it.
In a paper by DeBarre where he proves a certain important Torelli result, he credits me and Robert Varley with having done it first even though our proof was never published. I.e. no one would have known if he had not mentioned us, but he was not willing to do that. This is my general experience, that most people are very generous and kind.
 
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  • #1,148
yes. Terry Tao is also very kind. He answers people's questions on his blog.
 
  • #1,149
as another example, consider my birthday conference last april, http://www.math.uga.edu/~valery/conf07/conf07.html .
as you can see, the speakers who came were much more famous than me, and it was extremely generous of them to come for that occasion. I was really blown away by their kindness, and that of the organizers who planned it and invited them.
 
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  • #1,150
That's pretty sweet.
 
  • #1,151
yes, there are a lot of nice people in the world.
 
  • #1,152
mathwonk said:
It is so hard, but to me so important, to try to ignore personal issues, fame, competition, etc... and focus on enjoyment, understanding,... when doing math. hang in there!

In science especially the applied areas, competition is definitely a positive as mentioned by some scientists. However this is not the case with maths? I admit being driven by competition in my courseworks like striving to achiever higher assignment and exam marks then others. I find it natural but you think it's not good in maths?

I'd imagine competition would a strong driving force in top unis like Harvard? And also to solve top/popular problems?
 
  • #1,153
I've always wanted to become a mathematician, but considering the possibilities:
teaching
living on the streets

and I don't want to teach.
Furthermore, income of mathematician = $0
12 x $0 = $0
Doesn't take a mathematician to figure out. It's just so difficult.
 
  • #1,154
You can have an academic career, ie. a permanent position, just doing research (at least in the UK).

(I know of people who have gone straight into a chair posiion from PhD -- however, this is extremely rare.)
 
  • #1,155
mathwonk said:
In my own experience these cases of jealousy and competition are less common than might be supposed. Since we are human they do exist but they have not at all defined my experience in math. I have met so many generous mathematicians. As a simple example, if you look in Mumford's book on Theta functions, in one footnote he credits me with having described theta functions in a certain way, when actually I myself got that description from a book by Siegel. So I am guilty of not acknowledging Siegel in my talk, but Mumford was so scrupulous as not to want to even give a definition that had been inspired by someone else without crediting it.
In a paper by DeBarre where he proves a certain important Torelli result, he credits me and Robert Varley with having done it first even though our proof was never published. I.e. no one would have known if he had not mentioned us, but he was not willing to do that. This is my general experience, that most people are very generous and kind.

That's nice to hear. You know the person I am talking about had to deal with this stuff right after getting his ph.d. I presume he still has to deal with it although he has won some prizes now. It sounds like UGeorgia is a friendly place. But, I am sure there are some places that have rifts and such: like two research groups fighting over the direction of the university. I guess that is something to avoid when actually working as a mathematician.
 
  • #1,156
Invictious, what you said doesn't make sense. Teaching is a very small burden for professors at research universities. My math professor teaches one class at a time and I believe my physics professor teaches two classes at a time. This amounts to 2-5 hours in the classroom per week. I'm sure you would spend just as much time per week doing necessary chores. And for such a small burden, a tenured professor averages around $70-90k+ a year, and up to mid 100's at top universities (Wikipedia). Plus, they can't get fired. So you see, a tenured professor has a very comfortable position financially and a very small workload.
 
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  • #1,157
ktm said:
Invictious, what you said doesn't make sense. Teaching is a very small burden for professors at research universities. My math professor teaches one class at a time and I believe my physics professor teaches two classes at a time. This amounts to 2-5 hours in the classroom per week. I'm sure you would spend just as much time per week doing necessary chores. And for such a small burden, a tenured professor averages around $100k+ a year, and $160k+ at a top university. Plus, they can't get fired. So you see, a tenured professor has a very comfortable position financially and a very small workload.

I would imagine it is a lot more work to teach a class than to just put in the 2-5 hours that are actually in the classroom.
 
  • #1,158
A little, but not that much more. A professor is usually responsible for the lectures, creating the homework, creating the tests/quizzes, and the final grades.

The homework can just be assigned out of a textbook, though it doesn't take much time to come up with problems at the class level. The tests/quizzes would probably have to change from year to year, but coming up with the problems shouldn't take that long and it's not an everyday thing. The final grades also shouldn't take that long -- the TA's could compute it actually. And then there's office hours, which could amount to 2 hours a week.

So I would estimate a minimum rate of 5 hours per week per class. This is much better than the 40 hour a week norm. And even if I'm off in my estimate, it would still be much better than the 40 hour a week norm. Of course, a lot of professors do more than this because they enjoy teaching.

Also, the TAs usually do the grading of homeworks, tests, and quizzes, which a big proportion of the work involved in teaching a class, especially a big class.
 
  • #1,159
ktm, i take the maximum exception to your remarks. you can only say this if you have never taught yourself. trying to do a good job of teaching a class that meets 3 times a week, is a large order, especially if it has 30-40 students or more. and teaching two of them essentially takes all your time.

there is not only lecturing, but preparation, office hours, exam and test writing, grading (which can take 2 or more full days for one class), note writing, administrative duties, hassles from students who do not attend, then ask for special consideration, or who ask for make up tests, etc, etc,...

if you try to give your students the experience of making presentations, there is also tutoring them in the material in advance, wroting notes for them, scheduling opportunities to hear them practice the presentation,...

I have sometimes spent 2-3 hours with one clueless student, helping them grasp the basic ideas of say integration, only to have the student still decide to quit the course. When a promising but poorly motivated student with bad work habits recently stopped attending class, I emailed him, then called his home, then sent messages via student acquaintances, trying to keep him in the course. he still disappeared without a trace, or a goodbye, or any explanation.

then there are committees which meet regularly and endless paperwork.

all three jobs, teaching, research, and administration, are potentially infinite. you must always make choices and compromises to do them all minimally, much less to do them all well.

then we are not counting trying to have some time for family, not to mention a private life of ones own. one spends literally years with no time even to go out to dinner or read a book.

i once made a pact never to sleep over at the office, no matter how much work i had, in order to at least see my home every night. that semester, i once came home at 5am, slept 45 minutes total, and went back to work at 6:30am.

i also once worked 36 hours straight at the office, trying to go through over 700 job applications.

respectfully, you do not have a clue what you are talking about.

there is a huge difference between teaching a class or two, and having a semester off to do research full time. try it sometime.

of course there are people who do not care about doing a good job, and spend little time or energy on their teaching, but they are very rare in my experience. the hardest thing is to keep your research alive in the face of all these demands on your time.

oh yes, and i do not have a TA in either of my classes this semester, not even for grading, much less for office hours, or lecturing.

in the graduate algebra course i taught last fall, for which i posted lecture notes on my website, (which you are welcome to use for free), the time commitment was 5 hours a week, 3 in lecture and 2 more in a lab session preparing students for prelim exams.

then i had to also write and grade the prelim. that was only one of two courses that semester. teaching is a huge time sink. having a year or even a semester off for research is a tremendous boost to ones productivity.

forgive me for unloading on you, but i am tired, i have taught already 7 classes the first three days of this week, and have scheduled 3 more review classes for friday. listening to know nothings say how easy it is, is just too much to take right now.

i wanted to present a seminar on my recent research this week but had no time to prepare it properly.

of course different departments are different, and biological sciences professors e.g. have much lower teaching responsibilities. But English profs may have more.
 
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  • #1,160
Which institutions if any offer tenure for full time research (no teaching whatsoever) in pure maths (I know there are more for more applied research but all pure maths academics do teaching in my uni)?

How hard is it to get these positions?
 
  • #1,161
mathwonk said:
ktm, i take the maximum exception to your remarks. you can only say this if you have never taught yourself. trying to do a good job of teaching a class that meets 3 times a week, is a large order, especially if it has 30-40 students or more. and teaching two of them essentially takes all your time.

there is not only lecturing, but preparation, office hours, exam and test writing, grading (which can take 2 or more full days for one class), note writing, administrative duties, hassles from students who do not attend, then ask for special consideration, or who ask for make up tests, etc, etc,...

if you try to give your students the experience of making presentations, there is also tutoring them in the material in advance, wroting notes for them, scheduling opportunities to hear them practice the presentation,...

I have sometimes spent 2-3 hours with one clueless student, helping them grasp the basic ideas of say integration, only to have the student still decide to quit the course. When a promising but poorly motivated student with bad work habits recently stopped attending class, I emailed him, then called his home, then sent messages via student acquaintances, trying to keep him in the course. he still disappeared without a trace, or a goodbye, or any explanation.

then there are committees which meet regularly and endless paperwork.

all three jobs, teaching, research, and administration, are potentially infinite. you must always make choices and compromises to do them all minimally, much less to do them all well.

then we are not counting trying to have some time for family, not to mention a private life of ones own. one spends literally years with no time even to go out to dinner or read a book.

i once made a pact never to sleep over at the office, no matter how much work i had, in order to at eklast see my home every night. that semester, i once came home at 5am, slept 45 minutes total, and went back to work at 6:30am.

i also once worked 36 hours straight at the office, trying to go through over 700 job applications.

respectfully, you do not have a clue what you are talking about.

there is a huge difference between teaching a class or two, and having a semester off to do research full time. try it sometime.

of course there are people who do not care about doing a good job, and spend little time or energy on their teaching, but they are very rare in my experience. the hardest thing is to keep your research alive in the face of all these demands on your time.

oh yes, and i do not have a TA in either of my classes this semester, not even for grading, much less for office hours, or lecturing.

in the graduate algebra course i taught last fall, for which i posted lecture notes on my website, (which you are welcome to use for free), the time commitment was 5 hours a week, 3 in lecture and 2 more in a lab session preparing students for prelim exams.

then i had to also write and grade the prelim. that was only one of two courses that semester. teaching is a huge time sink. having a year or even a semester off for research is a tremendous boost to ones productivity.

Marking sucks ass. Some people just don't know how brutal it is to mark, especially papers from clueless first year students!

My question now is... aren't you better off working 9 to 5pm so you just go to work and do what you want after work without hassles? Because it seems like you have to put a great deal of your plans out of the way just to be a professor.
 
  • #1,162
pivoxa15 said:
Which institutions if any offer tenure for full time research (no teaching whatsoever) in pure maths (I know there are more for more applied research but all pure maths academics do teaching in my uni)?

How hard is it to get these positions?

Win a Field's Medal and maybe a school will consider you to do only research every now and then.
 
  • #1,163
JasonRox said:
Win a Field's Medal and maybe a school will consider you to do only research every now and then.

And even then you'll be lucky. Hawking is head of his department at Cambridge, has considerable prestige and research value, has the best possible reason not to teach given his medical condition, and still has to supervise a couple of PhDs. Admittedly not a field medallist... but are you THAT good? :rolleyes:
 
  • #1,164
muppet said:
And even then you'll be lucky. Hawking is head of his department at Cambridge, has considerable prestige and research value, has the best possible reason not to teach given his medical condition, and still has to supervise a couple of PhDs. Admittedly not a field medallist... but are you THAT good? :rolleyes:

Hawking isn't head of DAMTP; a guy called Peter Haynes is. Besides, supervision is not the same as teaching duties; generally researchers like having research students. If anything, it boosts their paper count!
 
  • #1,165
mathwonk said:
ktm, i take the maximum exception to your remarks. you can only say this if you have never taught yourself. trying to do a good job of teaching a class that meets 3 times a week, is a large order, especially if it has 30-40 students or more. and teaching two of them essentially takes all your time.

there is not only lecturing, but preparation, office hours, exam and test writing, grading (which can take 2 or more full days for one class), note writing, administrative duties, hassles from students who do not attend, then ask for special consideration, or who ask for make up tests, etc, etc,...

if you try to give your students the experience of making presentations, there is also tutoring them in the material in advance, wroting notes for them, scheduling opportunities to hear them practice the presentation,...

I have sometimes spent 2-3 hours with one clueless student, helping them grasp the basic ideas of say integration, only to have the student still decide to quit the course. When a promising but poorly motivated student with bad work habits recently stopped attending class, I emailed him, then called his home, then sent messages via student acquaintances, trying to keep him in the course. he still disappeared without a trace, or a goodbye, or any explanation.

then there are committees which meet regularly and endless paperwork.

all three jobs, teaching, research, and administration, are potentially infinite. you must always make choices and compromises to do them all minimally, much less to do them all well.

then we are not counting trying to have some time for family, not to mention a private life of ones own. one spends literally years with no time even to go out to dinner or read a book.

i once made a pact never to sleep over at the office, no matter how much work i had, in order to at eklast see my home every night. that semester, i once came home at 5am, slept 45 minutes total, and went back to work at 6:30am.

i also once worked 36 hours straight at the office, trying to go through over 700 job applications.

respectfully, you do not have a clue what you are talking about.

there is a huge difference between teaching a class or two, and having a semester off to do research full time. try it sometime.

of course there are people who do not care about doing a good job, and spend little time or energy on their teaching, but they are very rare in my experience. the hardest thing is to keep your research alive in the face of all these demands on your time.

oh yes, and i do not have a TA in either of my classes this semester, not even for grading, much less for office hours, or lecturing.

in the graduate algebra course i taught last fall, for which i posted lecture notes on my website, (which you are welcome to use for free), the time commitment was 5 hours a week, 3 in lecture and 2 more in a lab session preparing students for prelim exams.

then i had to also write and grade the prelim. that was only one of two courses that semester. teaching is a huge time sink. having a year or even a semester off for research is a tremendous boost to ones productivity.

forgive me for unloading on you, but i am tired, i have taught already 7 classes the first three days of this week, and have scheduled 3 more review classes for friday. listening to know nothings say how easy it is, is just too much to take right now.

i wanted to present a seminar on my recent research this week but had no time to prepare it properly.

of course different departments are different, and biological sciences professors e.g. have much lower teaching responsibilities. But English profs may have more.

That sounds brutal.
 
  • #1,166
mathwonk,

First of all, I admit I was speculating (while appearing too sure of myself), as I am not a professor myself and so I'm not in a very good position to make such estimates. Your opinion is probably much more valid than my speculation. I'm sorry for speculating, and for most likely saying something very inaccurate.

Second of all, I recognize your analogy of teaching students and teaching yourself, but it's no basis for saying that I've never taught myself anything and that I'm a "know-nothing". I've taught myself a lot and don't consider myself a "know-nothing".

Thirdly, I was only considering teaching, and I didn't mean to include doctoral students, job applications, committees, and department meetings. Regardless, I admit I was still speculating, and that I still probably said something very inaccurate.

Anyways, it's good that this dispelled an incorrect notion of mine.
 
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  • #1,167
JasonRox said:
Marking sucks ass. Some people just don't know how brutal it is to mark, especially papers from clueless first year students!

Tell me about it! I never knew how much work lecturers actually put into teaching until I started as a TA this semester, and has to attend tutorials and mark scripts. As mathwonk says, I can see how at least half a week can easily be taken up with teaching one class.
 
  • #1,168
The decision is yours.
If it wrong then you learn

I used to decide to leave Italy. It WAS my decision. No one has ever taught me to do taht anyway
 
  • #1,169
i'm sorry ktm, as i said i just got home and I'm tired and i unloaded my frustrations on you. please forgive me.

being a professor is certainly easier than most jobs. it beats the meat lugging job i used to do. the heaviest piece of meat i ever lifted there was 305 pounds. now i complain if a calc book weighs 5 pounds.
 
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  • #1,170
I'm currently a freshman in college (at Yale) and am very interested in a major in math (or math/physics) and quite possibly grad school beyond (though it is very premature for me to make such a decision). The math class I am currently taking (Year-long Vector Calc+Linear Algebra with textbook "Multivariable Mathematics" by Mathwonk's colleague Ted Shifrin) is interesting but feels a bit light. I studied the first 7 chapters of Baby Rudin (and a bit of linear algebra too) last summer, which was a much more invigorating experience. In my spare time, what would be the best way to learn more math? Is studying putnam exam type questions a good idea? Should I read textbooks on my own? Should I put more time into my (slightly dull) math class, or read about more advanced related materials? Should I simply study harder for my math class, and if so how? I've heard it said that foundations in linear algebra are essential in more advanced math, so would a linear algebra book like Hoffman and Kunze be a good idea for enrichment?

I feel like I'm really passionate about math right now and not 100% sure what step to take next.
 

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