Other Should I Become a Mathematician?

[mathwonk]

Tronter, if you really want to behave like the teacher, I recall that when i started out teaching calculus from Thomas, I worked every problem at the end of every section before every class, so i could answer any question asked on them. Eventually I realized they were not all different and began to lighten up, but only when I really knew how to do them. I also volunteered to teach extra classes and to give seminars, so I could learn more. Thus in addition to my regular load I also taught a free extra section of Spivak style calculus since we did not offer one, and ran a seminar for faculty on the de Rham theorem using sheaf theory.

 

[tronter]

Thanks mathwonk.

I asked another mathematician if he followed those rules you gave. I was surprised when he said he generally never did any homework except for one course. Now I find that quite hard to believe. Perhaps he understood the concepts extremely well that he didn't need to do any problems?

 

[proton]

what kinds of jobs do pure mathematicians have (besides academia)? what are they like? do they involve a lot of computer programming? is it possible to for their work to involve physics too (like providing the mathematics for engineering)?

 

[mathwonk]

i also have difficulty believing your mathematician friend did not do hw. is he a real mathematician? i.e. does he do research, publish in good journals, and give invited talks at international meetings and bring in grant money? there are always a few exceptions to these rules, but they are very unusual. or is he a liar?

of course i also did not do hw, but i was unsuccessful until i did so.

 

[mathwonk]

if you read this lengthy thread you will see it mentioned that mathematicians do all kinds of jobs because they are able to learn to do anything after learning how to think and how to learn.

 

[tronter]

he is not a mathematician in academia, but he did get a PhD in math. Also what happens if you didn't know the answer to a question. If you don't have a solutions manual then that's good right? Then its a true test of your knowledge.

 

[mathwonk]

it sounds possible to me that your friends refusal to do his homework may have led to his not being able to continue his mathematical career.

 

[pivoxa15]

Thanks for your advice, if only I followed them during my undergrad years. It does explain why my grades are so poor but I'm willing to change myself. What extra advice would you give to students taking grad courses? I assume the lectures are more heavy going and less then undergrad courses so more condensed?

i also have difficulty believing your mathematician friend did not do hw. is he a real mathematician? i.e. does he do research, publish in good journals, and give invited talks at international meetings and bring in grant money? there are always a few exceptions to these rules, but they are very unusual. or is he a liar?

of course i also did not do hw, but i was unsuccessful until i did so.

I read that Ian Stewart said he didn't do much work in college nor did Stephan Hawkings, he did on average one hour a day but then again he didn't do that brilliantly getting a 2nd class honors I think in Oxford before moving to Cambridge for a Phd.

 

[proton]

I've skimmed through this whole thread, but I don't remember all the details. How difficult is it to find jobs after obtaining a phD in math (non-professor jobs)? is it similar to physics, where its extremely hard to unless you specialize in an area that has a lot of applications?

 

[J77]

I read that Ian Stewart said he didn't do much work in college nor did Stephan Hawkings, he did on average one hour a day but then again he didn't do that brilliantly getting a 2nd class honors I think in Oxford before moving to Cambridge for a Phd.

Regarding "homework"...

Back when I did my first degree, I can't remember doing too much homework: new university, new town, new friends -- there wasn't much time for work in the evenings.

However, I did attend nearly all lectures, and worked on assignments during the free periods between lectures.

I worked harder when doing my masters, but then the classes were a lot smaller and we all shared an office -- it felt a bit more grown-up

During my PhD, I think I did more homework, than in my UG, preparing for the teaching jobs we had to do

Now, doing research, it's of course necessary, and part of the job, to go out and seek things on your own -- you certainly don't have to recall everything you've been taught in the past. More, your level of maturity in dedicating yourself has to be there.

I guess that's the bottom-line, as you get older, you become more mature, I think you can see ideas from a higher perspective -- all which means you don't have to strain over a textbook trying to force yourself to understand something for, eg., an exam.

(I did go through the UK system though -- like the names you quote -- and it was a decade ago now. Times have moved on, maybe students are more dedicated to studying these days because they have to pay? I hope not, because uni should also be about enjoying yourself, not all study.)

 

[mathwonk]

Well it is surprizing to me to find people chiming in that did not do hw. In my experience when i did not do homework at harvard, i flunked out. when i went back to utah a decade later i did all homework and went to all lectures, and worked as hard as possible. i was the presidential scholar, and upon graduation, in a tight job market, i applied about 5-10 places and got 4 jobs.

it may be that the people saying they did not study much are just smarter than me, certainly hawking. but if you look back you see several of us seem to agree we did little work in college, but more in grad school. I am just saying it was a mistake not to do more work in college too. My fellow students who are famous mathematicians now, like Cheeger, Bloch, Mather, Hochster, Zimmer...apparently worked hard also in colllege.

If you want to be all you can be, as the army slogan goes, I cannot imagine not working as hard as possible, as i recommend here. there is a big difference between just having a PhD, and solving problems that top people are interested in and trying to solve. Obviously there is also a big advantage to starting to work hard at 18 or 20 as opposed to waiting till you are 30.

 

[pivoxa15]

Do you think it would be bad to have a Phd advisor who hasn't published on the topic he is recommanding as a Phd?

Although it would mean both the advisor and student are learning new stuff so there would be more of a collaboration? It would mean that the supervisor is learning new stuff as well. Can it be successful?

Do you have anything to add for doing successfully in grad school courses?

 

[mathwonk]

apparently there are many different standards, even in grad school. when i was a mature student i went to all classes, did all hw, got almost 100% on all homework assignments.

in topology, the instructor, a world famous topologist, used one of my arguments when his own was faulty for one result. in algebraic geometry, i gave an argument for abels theorem which had apparently eluded some world famous experts.

I worked essentially all the time, except when with my family. In several complex variables seminar I presented Kodaira's proof of his vanishing theorem, which was apparently too daunting for the faculty members in attendance to read.

What can i say? Do your best. Work as hard as it takes to realize your goals. But do not ignore your family or loved ones. read the best experts, original papers, talk to your teachers, and listen to what they advise. Do you hope to have a job at a top school someday? or solve a problem that will impress renowned workers? This will take real effort.

But everyone is different. if you are not this competitive, do not feel bad. Live your own life. But if you want to do as well as possible, to see how good you can be, then you must try as hard as possible.

This attitude of : well i am smart and can get by with only a moderate amount of work, I recommend leaving that behind, or risk feeling unfulfilled.

But not everyone needs to feel that way. There is no requirement to be compulsive about research achievement. Find your own goals, your own happiness, and then try to realize it. My commitment to hard work sometimes makes it hard for me to relate to my students who have other priorities, and hard to teach them.

Successful life is a balancing act, trying to keep all priorities fairly served.

 

[mathwonk]

the answer to question on advisor not having published is yes. any collaboration at all can lead to a fruitful result. the point is to pursue something you care about, and have ideas about.

It is also true that different people will find different paths to success. It is entirely possible for someone to apparently work less and have more success. But I would not take that as a model if I were giving advice to a young person. Indeed as a professor for over 30 years now, I have seen thousands of students, but NONE of them has ever done well without working hard, although hundreds and hundreds have sabotaged the chance to do well by goofing off.

For some reason it seems to upset me to hear people apparently suggesting to young people that there are successful people out there who did not work hard. Frankly I do not believe it. I have been in close contact with many very bright people, including Fields medalists, and believe me, they all work extremely hard, and very consistently. They are also very disciplined in not letting anything get in the way of work. I am even something of an exception in having always given high priority to my family and social time. I.e. as hard as I have worked, it is less so than many successful mathematicians I have known.

When I was in college, some of my friends pretended that the really successful students they knew did not work hard, they were just smart. Looking back these tales seem to have been fables. It seemed more interesting to talk about the people who supposedly did nothing but were still top performers. Sort of like the guys sitting in the poolroom all day doing nothing, talking about the big money they were making or someone else was making doing little work.

Indeed these claims are in the same family as the ones on television ads about getting rich with other peoples money, or a beer that is both less filling and great tasting, or any of the myriad other "something for nothing" stories, i.e. they are simply not true.

I have also known personally some psychologists of science and research, and they confirmed that top research scientists work essentially all the time. They are able to do this in my opinion because they love what they are doing. They have high energy, and lots of enthusiasm for their work. So they are actually happiest when they are working.

If you have lunch with them, they are always talking about math research they are doing. All time spent with colleagues is used for work, but they are having a ball at it. But it may be that mathematicians are unusual in this regard. I have read that some visitors to the Institute for Advanced Study remark that other scholars talk about anything at lunch, but math types seem to only talk math research.

 

[eastside00_99]

If you want to be all you can be, as the army slogan goes, I cannot imagine not working as hard as possible, as i recommend here. there is a big difference between just having a PhD, and solving problems that top people are interested in and trying to solve.

This has been one of the things I have been thinking about lately. I wonder just how consumed you or others who do research at a high level are with their job and mathematics. Sometimes this seems very romantic and in reality at times I am sure is very rewarding. But, often I find much of what I think it would be like very dull and boring. I fear if my goal was to be very consumed with research, then, in the end, I would feel a little bit shafted when or if I finally arrived. It just seems like all I would be doing is preparing myself for a terrific mid-life crisis. I guess this is part of a larger belief that people who strive after a goal but miss the journey are to be pitied not praised. For me, I take classes, do homework, and read books on my own (ahhh, the best part of being a student) not because I have a specific goal in mind. To be honest, I don't have a clue why I do it (aside from the practicalities having a degree). I do know that at times it is very enjoyable; but, the enjoyment usually comes exactly when there is no ulterior motive---just a desire to attend the class, work on a problem, or read which arises through curiosity.

 

[mathwonk]

I think you have a good orientation. Read the supplement I have just added to the previous post, about loving what you do.

 

[mathwonk]

here is the announcement of my impromptu seminar talk yesterday:

Algebraic Geometry Seminar November 28, 2007
Title: Geometric Schottky for g=4.
Abstract: Definition: A principally polarized abelian variety (ppav) of dimension g over C, is a pair (A,D), where A is a complex torus, and D is a divisor on A given up to translation by a point of order 2 (a point of A2), such that D is invariant under the minus map involution of A, and the g - fold self intersection number D^g = g! Then D is necessarily reduced, hence dim(singD) < g-1.

An example is the Picard variety of line bundles of degree g-1, on a curve C of genus g, translated to degree 0 by any square root of the canonical divisor, and with D = (the translate of) the subset of effective divisors of degree g-1. This example is called the Jacobian of C.

The following solution of the Schottky problem of characterizing Jacobians of dimension 4, is an accumulation of work of many people including Riemann, Andreotti-Mayer, Mumford, Beauville, and (most recently) Grushevsky.

Theorem: In dimension g = 4, a ppav (A,D) is
i) a product of Jacobians of lower dimension iff dim(singD) = 2,
ii) a hyperelliptic Jacobian iff dim(singD) = 1,
iii) a non hyperelliptic Jacobian iff dim(singD) = 0, and
either a) A2 contains no singular points of D,
or b) A2 contains a "rank 3" double point of D.

Moreover: If ii) holds, then singD is isomorphic to the unique
hyperelliptic curve C such that (A,D) is isom. to (J(C), D(C)). If iii)a holds
then singD consists of 2 rank 4 double points exchanged by the involution. If iii)b holds, then singD is precisely one rank 3 double point in A2.

The necessity of these conditions for a Jacobian is due to Riemann.
The sufficiency of parts i), ii), and iii)a, are due to Beauville, while that of part iii)b is due to Grushevsky, (conjectured by Hershel Farkas). The proof outlined today for part iii)b is new, joint work with Robert Varley.

 

[tronter]

your advice makes much more sense mathwonk.

 

[pivoxa15]

Mathwonk, you once remarked that you thought physics was more exciting but you were much better at maths. Why didn't you pursue reserach in mathematical phyiscs in the areas that recquire abstract maths?

 

[mathwonk]

well in grad school i fell in love with math, and had no need to go back and learn physics. i.e. by then i was magnetized to think about pure math.

i am still interested in physics, but it takes so long to learn anything (for me), and there is so much to learn!

 

[JasonRox]

the answer to question on advisor not having published is yes. any collaboration at all can lead to a fruitful result. the point is to pursue something you care about, and have ideas about.

It is also true that different people will find different paths to success. It is entirely possible for someone to apparently work less and have more success. But I would not take that as a model if I were giving advice to a young person. Indeed as a professor for over 30 years now, I have seen thousands of students, but NONE of them has ever done well without working hard, although hundreds and hundreds have sabotaged the chance to do well by goofing off.

For some reason it seems to upset me to hear people apparently suggesting to young people that there are successful people out there who did not work hard. Frankly I do not believe it. I have been in close contact with many very bright people, including Fields medalists, and believe me, they all work extremely hard, and very consistently. They are also very disciplined in not letting anything get in the way of work. I am even something of an exception in having always given high priority to my family and social time. I.e. as hard as I have worked, it is less so than many successful mathematicians I have known.

When I was in college, some of my friends pretended that the really successful students they knew did not work hard, they were just smart. Looking back these tales seem to have been fables. It seemed more interesting to talk about the people who supposedly did nothing but were still top performers. Sort of like the guys sitting in the poolroom all day doing nothing, talking about the big money they were making or someone else was making doing little work.

Indeed these claims are in the same family as the ones on television ads about getting rich with other peoples money, or a beer that is both less filling and great tasting, or any of the myriad other "something for nothing" stories, i.e. they are simply not true.

I have also known personally some psychologists of science and research, and they confirmed that top research scientists work essentially all the time. They are able to do this in my opinion because they love what they are doing. They have high energy, and lots of enthusiasm for their work. So they are actually happiest when they are working.

If you have lunch with them, they are always talking about math research they are doing. All time spent with colleagues is used for work, but they are having a ball at it. But it may be that mathematicians are unusual in this regard. I have read that some visitors to the Institute for Advanced Study remark that other scholars talk about anything at lunch, but math types seem to only talk math research.

I totally agree.

I goof off from time to time, but sometimes I count doing mathematics as goofing off. I can sit and read mathematics that has nothing to do with my courses or just out of nowhere start talking about what's "neat" from class with fellow students even though I know they don't care.

I really do just love it.

 

[pivoxa15]

well in grad school i fell in love with math, and had no need to go back and learn physics. i.e. by then i was magnetized to think about pure math.

i am still interested in physics, but it takes so long to learn anything (for me), and there is so much to learn!

I see what you mean. I am starting to feel what you say. My interest in physics whined when pure maths became harder and more involved. When I bring the pure maths mentality into physics, I always ask 'how do you know it's right'. Now I know it is the wrong mentality for physics. It recquires something else.

 

[pivoxa15]

So you think it is a good idea to have a Phd advisor who is also new (although not totally new) to the field he is supervising?

It occurred to me that an academic would always like to expand his/her horizon but it may be difficult to do it on his own as competing against people who are experts in the given field is tough, especially if it a relativly mature field. Hence what better way (apart from collaboration with another expert) is there to expand one's horizon then having a Phd student doing the topic on this field. This way, the academic is able to work at it at a more comfortable pace then if he was to dive into the field himself. Is it the way most academics expand their horizon?

 

[mathwonk]

that may be true!

 

[mathwonk]

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!

 

[pivoxa15]

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!

The hardest thing for me to ignore would be personal issues. As Feyman said 'Physics is not the most important thing, love is.'

It would take an extra human to ignore that one. It's also the thing that nearly destroyed by academic record. How do you get over this one?

 

[mathwonk]

i did not mean to ignore love, just envy.

 

[ROLEX4life]

I hope to be a mathematician and teach as a professor. Any recommendations for textbooks? Also, after thoroughly studying Linear Algebra, would it be wise for me to begin reading a text on Abstract Algebra? Or is there more mathematical preparation required?

Abstract Algebra is very doable with a knowledge of Linear Algebra. In fact, I am taking both classes at the same time. The only prerequisite at my school for Abstract Algebra is MAT 310 which is a course entitled "Introduction to Set Theory and Logic" which basically is just an introductory course in proofs. Some basic of knowledge of Number Theory would also be to your benefit but it is not a necessity. I am signed up for that class next quarter but I wish I had it before Abstract Algebra. However, you will be fine if you are proficient in modular arithmetic.

 

[proton]

how is abstract algebra compared to real analysis in terms of difficulty? - for someone who's mastered upper-div linear algebra

 

[pivoxa15]

i did not mean to ignore love, just envy.

Also ignore infatuation? A combo of that and envy is deadly.