Other Should I Become a Mathematician?

[mathwonk]

my posts are still not up, due to an error i made. i think i need your email address to send you a copy as pdf file. you could pm it to me if you wish, and i'll try to respond.

 

[mathwonk]

ok the new 100 page algebra book is up on my webpage. 6.a is the course outline or description,
6b is the theory of finitely generated abelian groups and pid's and a little on noetherian rings and modules.
6c is a second course in linear algebra, proving the existence and uniqueness of rational canonical form and its variation the jordan form. i also include a few words on spectral theorems abd duality since some people asked for them.
6d is a treatment of finite galois field extensions, including proofs of extensions theorems for homomorphisms, separability, normality, and existence and uniqueness of algebraic closures up to isomorphism. I did not give full proofs of galois' theorem on the necessary condition for solvability by radicals, but the statements are there, and all the big underlying technical results are actually there so its a good exercise. there is nothing on solutuions formulas for degree 3,4.
6.e is a set of homework problems and tests.

some stuff referred to is on either my webpage or the departmental page under grad student info, like old prelims.

If anybody looks at them i would appreciate any feedback. thanks.

the longer notes for 843-4-5 are better for a first time learner, but these are designed for a more advanced student, or someone willing to spend longer filling in details and making up or looking up illustrative examples. It is often useful to have a shorter version since you can actually get through all the pages of one section in a real life day or three. they were intended for grad students preparing for the algebra prelim, but are not guaranteed to be exhaustive for that purpose, even at UGA.

Interestingly, algebra was always my hardest and weakest subject, well until analysis I guess. I also wrote notes (twice) on complex analysis and riemann surfaces, but that was before personal computers and no magnetic copies exist. maybe someday. i also have several various algebraic geometry notes, plane curves (no magnetic copies), foundations, sheaves and cohomology, surfaces. sighhh... fortunately for you, better ones exist in print, but i still learn a lot by writing mine.

 

[mathwonk]

the book by kaplansky looks to be outstanding as an introduction to metric spaces, sets, and topology.

it was written by a famous algebraist and expositor, based on notes from a course given by the outstanding algebraic topologist edwin spanier.

moreover it is elementary in the sense that it begins the subject with the most natural version of topology, namely metric spaces, where euclidean intuition is most useful.

this is exactly the sort of book i would recommend to any young person. note i have not read it but i know very well the reputations of the author and the original lecturer. in fact i have met kaplansky, but not spanier. i have studied from spanier's great book on algebraic topology however, a very scholarly work indeed.

i had the good luck to audit a class on set theory and metric spaces that sounds much like this one, but mine was from the famous representation theorist george mackey at harvard.

oh yes, the kaplansky book is under $30! amazing in today's world of mediocre books for $150+.

 

[fopc]

the book by kaplansky looks to be outstanding as an introduction to metric spaces, sets, and topology.

it was written by a famous algebraist and expositor, based on notes from a course given by the outstanding algebraic topologist edwin spanier.

moreover it is elementary in the sense that it begins the subject with the most natural version of topology, namely metric spaces, where euclidean intuition is most useful.

this is exactly the sort of book i would recommend to any young person. note i have not read it but i know very well the reputations of the author and the original lecturer. in fact i have met kaplansky, but not spanier. i have studied from spanier's great book on algebraic topology however, a very scholarly work indeed.

i had the good luck to audit a class on set theory and metric spaces that sounds much like this one, but mine was from the famous representation theorist george mackey at harvard.

oh yes, the kaplansky book is under $30! amazing in today's world of mediocre books for $150+.

Thanks. I asked about Kaplansky's book because there was strong recommendation by Mendelson in his book on topology.
Mendelson's book is a beauty (a well written intro). So, I figure his references for concurrent or future reading are
worth a look. Yes, I noticed the $29 price tag. Not too bad.

Incidentely, I came across this Kaplansky quote.

"... spend some time every day learning something new that is disjoint from the problem on which you are currently working (remember that the disjointness may be temporary), and read the masters. "

Read the masters? That rings a bell.

Thanks for posting your new (100 page) algebra text. I'll try to give feedback.
But first I think I'll have to review your 843-845 .pdf's.

 

[mathwonk]

yes i like that quote. the first part too is meaningful to me, having seen so many times how someone using tools from another topic i had ignored, like classification theory, shed light on a problem of interest to me, like singularities of theta divisors.

 

[Werg22]

Question: How successful can someone outside of academia be successful in the filed of pure mathematics? Can one pursue the understanding of pure math with the same comfort than someone who works in academia?

 

[mathwonk]

well fermat was pretty successful while being a jurist. it all boils down to how much time you can spend at it.

 

[mathwonk]

here is a rambling, somewhat cynical, but in my opinion very truthful and representative account, of one mans life as a professor, for those wanting a realistic version of what one can encounter in academics.

he details the frustrations of trying to do a good job in the face of administrative indifference or hostility to good teaching, and frequent student indifference to useful learning. still he kept trying. like many of us he felt sympathetic to lack of ability, but not to lack of interest. that component of student interest is what attracts us to this forum.

http://www.math.hawaii.edu/~lee/education/kline.html there are a lot of free books on his webpage.

 

[Winzer]

wow, a good read mathwonk.
Thanks

 

[threetheoreom]

Yes a truly great read.

Thanks.

 

[J77]

mmm... The guy certainly has some issues.

At the same time, though, I have to realize that I'm simply wasting my life. Yes, a lot of the things I've done during my summers and otherwise have been personally very worthwhile for me. But for the most part, I'm not accomplishing anything. Nobody here at the University of Hawaii has any need of my talents.

I've got to get out of this place.

Other interesting articles of his include: http://www2.hawaii.edu/~lady/faq/why-stop.html

 

[mathwonk]

yes that last sentence surprized me. but i understand it.

if you have spent 30 years or so training yourself to do math and teach it to other like minded people, it is very stressful to be faced with 30 more years of trying to interest students, some of whom hate the topic and just want an A without doing any thinking.

thats why sabbaticals are a good idea. at schools without them, we depend on the rejuvenation of summer activities, conferences, colloquia, physics forum, etc...

another thing that seems to help is to learn not to judge people for their different attitudes towards a subject we love. to care about them and enjoy them as people, and then maybe if they begin to like you and your acceptance of them, they may ask themselves what it is about math that interests you.

this may seem almost somewhat saintly though, and recall he said somewhere "they don't pay me enough to be a saint". actually trying to adopt saintly patience is maybe impossible, but still helpful.

no matter where you are, eventually you may feel used, or underpaid, or unappreciated, or even disrespected. so it is crucial to do what you do for the love and enjoyment of it, not for prestige, nor money.

at Harvard the students are as good as anywhere, and the profesors are also, and they have relatively good pay, good conditions, and time for research, and collegial stimulation that is almost unrivalled.

still at 70, even the most respected professor is forced to retire there, regardless of activity level. if he has been dependent on that title of Harvard professor instead of joy in his work, this is very hard. he realizes he has been considered a commodity by his university, one which has exhausted its value.

but the intangible community of mathematics and mathematicians just continues discussing matters of interest.

maybe this is nonsense. I am just saying i understand Lee Lady's frustrations, have felt them, and have tried for decades to resist giving up the fun of doing math, and also to not give up the sense of community or being a teacher and member of a university myself.

at UGA we have several retired members who continue to come in and do research in the department, something i never saw at Harvard. this is a good sign. in fact Matt Grime is coming next week, currently from Princeton, to chat with some very active members of our group, some retired, some relatively young.

 

[mathwonk]

by the way, since you point out Lee Lady stopped doing research, that suggests to me one reason he may have become discouraged, because research is what "holds our molecules together" in the words of my closest colleague for several decades.

 

[threetheoreom]

The Future Mathematician should be a clever problem-solver but to be a clever problem-solver is not enough. In due time, he should have solved significant mathematical problems; and he should find out for which kind of problem his native gift is particularly suited.

For him, the most important part of his work is to look back at the completed solution. Surveying the course of his work and the final shape of the solution he may find an unending variety of things to observe. He may meditate on the difficulty of the problem and the decisive idea. He might try to see what hampered him and what helped him finally. He may look out for simple intuitive ideas: Can you see it at a glance?. He may compare and develop various methods: Can you derive the result differently? He may try to clarify his present problem by comparing it to problems formerly solved. He may invent new problems which he can solve on the basis of his just completed work: Can you use the result, or the method for some other problems? Digesting the problems he solved as completely as he can, he may acquire well ordered knowledge, ready to use.

The future mathematician learns as does everbody else by imitation and practice. He should look out for the right model to imitate. He should observe a stimulating teacher. He should compete with a capable friend. Then, what maybe more important he should read not only current textbooks but good authors till he finds one whose ways he is naturally inclined to imitate.
He should enjoy and seek what seems to him simple or instructive or beautiful. He should solve problems, choose the problems that are in his line, meditate on his solutions and invent new problems. By these means and by all other means he should endeavor to make his first important discovery: he should discover his likes and his dislikes, his taste, his own line.

taken from How to Solve It - A New Aspect of mathematical Method by G. Poyla.

Just wanted to share a piece of an interseting read from the classic.

I found of that passage this paragraph quite insightful:

The future mathematician learns as does everbody else by imitation and practice. He should look out for the right model to imitate. He should observe a stimulating teacher. He should compete with a capable friend. Then, what maybe more important he should read not only current textbooks but good authors till he finds one whose ways he is naturally inclined to imitate.

 

[mathwonk]

extremely good advice. thanks.

 

[JasonRox]

I plan on working in the finance/accounting field while doing academia as a hobby. It might not be the most productive way to do research, but I'll actually be able to live comfortably, or that's the plan.

The city I plan on moving to is Vancouver which has two universities, UBC and SFU. Which is good. Once I land a job (I need to make money first), I will talk to the professors of interest and see if they'll take me on as a part-time graduate student. I think they would be interested because they wouldn't need any funding. I can pay for my own schooling. They can give the open TA jobs to other TA's which attract more graduate students (or better ones).

I'll see how that works out. I'm really excited to be done my undergraduate. I've been dying to meet people (my age) that are as excited about learning mathematics as I am.

 

[J77]

Careful with the part-time thing.

From what I've seen, the only way to produce an effective PhD is full-time.

For example, you could find it very hard to keep up with developments in the field doing it part-time, plus you don't have the same, crucially important, interaction with your peers.

I'd say, if you really want to do it, live like a pauper and go full-time!

 

[mathwonk]

UBC math dept is great. I know Jim Bryan, James Carrell, Zinovy Reichstein, William Casselman, among others. J77 you may be right, but i think he'll figure out the details as he goes along.

however, once most people start making a living they seldom want to return to poverty!

 

[J77]

however, once most people start making a living they seldom want to return to poverty!

Exactly.

And, making a living really doesn't leave you much time for much else

 

[mathwonk]

i am reminded of the untenured faculty member who lost his job when i was hired, went to industry, and returned to visit about three years later making triple my salary. he was not at all sad, nor desirous of getting his job back.

 

[JasonRox]

Careful with the part-time thing.

From what I've seen, the only way to produce an effective PhD is full-time.

For example, you could find it very hard to keep up with developments in the field doing it part-time, plus you don't have the same, crucially important, interaction with your peers.

I'd say, if you really want to do it, live like a pauper and go full-time!

Well, I'm going for my Master's first. I'll worry about the Ph.D details later.

I think I will have more time for other things if I make a good living. Once I get on my feet, and employers see how good I am, I can start asking for many vacations. If they don't want to offer me that, I'll just start looking for another job that will. My goal is to find a job by the time I'm 30 that will give me 6 weeks of vacation a year. I don't care if 4 of those weeks are not paid.

Also, I'll be able to go see the Olympics coming up rather than be broke and can't afford it. Take trips to California or weekend trips to Whistler or Banff. All kinds of things I can do. Another thing is that I would have the means to travel across the US to all kinds of mathematic conferences and meet all kinds of people. Otherwise, I'd be a graduate student sitting at home eating ramen noodles.

I also would like to retire early. I would like to save and invest and build wealth so I can do the things I enjoy even more. Maybe even hire my own graduate student to do work for me and find articles that would interest me. That would save me lots of time. And since graduate students may be desparate for money, it might not be too expensive either!

I thought about this long and hard, and I don't think I can handle living below poverty or at the poverty. I have health needs that I have to make sure they're met and I doubt being a graduate student will help me do that. What if my hearing aid breaks? Now what? Buying a new one isn't cheap, and so on.

 

[JasonRox]

Exactly.

And, making a living really doesn't leave you much time for much else

That's false. Many people do nothing with their lives... which means they have lots of time.

A 9-5pm job from Monday to Friday is hardly considered life consuming.

 

[mathwonk]

we just don't know how to relate to people making a decent salary. my older son went into industry with an undergraduate math degree, and about two years after i gave him the toyota i had myself been driving at the age of 60, he traded it for a bmw M3 well before he turned 30.

 

[JasonRox]

we just don't know how to relate to people making a decent salary. my older son went into industry with his undergraduate math degree, and about two years after i gave him the toyota i had myself been driving at the age of 60, he traded it for a bmw M3 well before he turned 30.

That's crazy.

 

[Ronnin]

It's good to see I'm not the only finance guy on here. It seems like anyone who majors in anything business related kinda gets different looks from the science/math majors (Everyone in my Uni Phys class just couldn't understand why I would want to take that vs. something easier). I too enjoy math very much and have found that yes there are programs for people like us who like counting and allocating beans AND enjoy higher level math, Financial Math. It seems a few colleges are offering this as a masters program. Hopefully, more will pick this program up.

 

[JasonRox]

It's good to see I'm not the only finance guy on here. It seems like anyone who majors in anything business related kinda gets different looks from the science/math majors (Everyone in my Uni Phys class just couldn't understand why I would want to take that vs. something easier). I too enjoy math very much and have found that yes there are programs for people like us who like counting and allocating beans AND enjoy higher level math, Financial Math. It seems a few colleges are offering this as a masters program. Hopefully, more will pick this program up.

I'm taking a Theory to Financial Mathematics this year. I'll see how that goes.

I plan on doing my Master's in something like Algebraic Topology or something with that area. It's so interesting. Or maybe even Number Theory, I just have to see how I like the course this year. I might do my Honours Thesis in Number Theory. Not sure.

 

[J77]

That's false. Many people do nothing with their lives... which means they have lots of time.

A 9-5pm job from Monday to Friday is hardly considered life consuming.

Have you done such a job? (By which I mean salary based and not what you've done in the holidays.)

Also, and this goes back to what I said before, a PhD should be 9-5. I know a lot of people in the UK who worked much harder than this to get it done within 3-4 years, and they didn't have the extra burden of a lot of courses and teaching which you have over in NA.

I'm not trying to put you off -- I just want to give you my opinion that part-time PhDs are very hard to manage.

we just don't know how to relate to people making a decent salary. my older son went into industry with his undergraduate math degree, and about two years after i gave him the toyota i had myself been driving at the age of 60, he traded it for a bmw M3 well before he turned 30.

He'd off have it before he turned 21 if he'd have gone into banking -- from his golden handshake alone

 

[JasonRox]

Have you done such a job? (By which I mean salary based and not what you've done in the holidays.)

Yes, I've even worked a consistent 35 hours a week for nearly 4 years while going to school full-time. I did it. I still had extra time to dick around.

So, to take one course at a time while working 9-5pm, (MASTER'S not Ph.D) seems entirely doable for sure. I will most likely not take the thesis route, so technically, all I have to do is one course at a time until I'm done. If I can't manage one course while working 9-5pm, that's literally ****ed up because I literally have like 5 hours of study time every night from Monday to Friday and like 12+hours on the weekend. And a friend of mine is taking 3 in the fall, if you can't handle one with all that time, no one can handle 3 no matter what you're doing. I'm honestly not nervous about it at all. He plans on partying too, so honestly no worries. I dicked around in undergrad so much, so if I assume that one grad course is like taking 4 undergraduate courses, I can do it since I already did it while working full-time!

If one is tough, just think how dumb it would sound if you quit a full-time job to manage one Master's course. Seriously, you'll look like a moron. That's just messed up. What do you do when you take 3 since life obviously does not have enough time for that since one requires your full attention, so what does 3 require?

Honestly, I'll ask my mentor/advisor about it. I hardly consider it something to be worried about. Obviously some people might think it's dumb because I'm not doing it like everyone else, but you know what, I want to make a living.

 

[marlon]

I plan on working in the finance/accounting field while doing academia as a hobby.

Great choice on the financial field but why care about doin' research in your spare time. Use it to grow in the financial areas. There is a lot to learn there is well. In your case, don't waste your time doing some obscure research that will get you noweher. Most "fulltime" PhD's out there are not even able to produce something useful so why bother ?

Really, stick to the financial maths and your life will be much nicer !

marlon

 

[J77]

So, to take one course at a time while working 9-5pm, (MASTER'S not Ph.D) seems entirely doable for sure. I will most likely not take the thesis route, so technically, all I have to do is one course at a time until I'm done.

Oh -- you want to take Masters courses not write a thesis :shrugs: