Should I Become a Mathematician?

[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.

 

[eastside00_99]

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.

 

[eastside00_99]

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.

 

[proton]

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.

 

[eastside00_99]

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.

 

[ktm]

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.

 

[pivoxa15]

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...'

 

[mathwonk]

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.

 

[tronter]

yes. Terry Tao is also very kind. He answers people's questions on his blog.

 

[mathwonk]

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.

 

[JasonRox]

That's pretty sweet.

 

[mathwonk]

yes, there are a lot of nice people in the world.

 

[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!

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?

 

[Invictious]

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.

 

[J77]

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.)

 

[eastside00_99]

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.