Other Should I Become a Mathematician?

[threetheoreom]

Because Oct 31 is Dec 25.

Old One.. still interseting though

 

[DefaultName]

quick: integral of e^(x^2) dx

 

[ice109]

quick: integral of e^(x^2) dx

not integrable

 

[J77]

uhh... before those last two posts degenerate into a derail -- this isn't a thread for doing sums on!

 

[mathwonk]

just heard a talk on strange duality that has a moral for all young researchers: look for what is natural, as answer to a question.

the setup concerned the study of spaces of vector bundles on a given riemann surface. if we fix the rank r of the vector bundle, and the riemann surface, there is a "moduli" space M(r) parametrizing all these bundles (with some other fixed data).

This moduli space M(r) itself has a distinguished "divisor" the subvariety of codimension one of bundles having non trivial sections, and this in turn defines a line bundle L on M(r).

now we can ask about sections of this line bundle L, and of its powers L^k.

It was noticed early on, again i think by the honoree Beauville, that in some cases there is a duality between sections of L^k on M(r), and section of L^r on M(k), called strange duality. All this has some interest to physicists, since i guess vector bundles on Riemann surfaces probably have some connection to Witten's theory of quantum gravity, or string theory.Anyway, eventually this duality was proved in general for all Riemann surfaces, by linking it to a familiar duality. I.e. in projective n space there is a classical duality between points and hyperplanes, and in general between k planes and n-k planes.

e.g. a point of projective space has coordinates which may be viewed as coefficients of a linear form defining a hyperplane. this is the usual duality defiend by dot products in euclidean space. the more generals ewtup is called grassman duality. (the space of k planes in n space is called a Grassmann variety.)but the point is there is a way of interpreting the strange duality of those exotic sections of bundles as just the duality in a suitable grassmannian.

moral, if something new looks like something familiar, try to see why, i.e. try to relate them somehow. the idea is that there is often a reason an apparently new phenomenon resembles a classical one.

 

[mathwonk]

here is a tiny example:

compare

arctan'(x) = 1/(1+x^2)

tan' = sec^2 = 1 + tan^2,

f' = 1+ f^2 is soved by f = tan(x).

these are all basically the same statement.i.e. if you know that arctan'(x) = 1/(1+x^2), or that sec^2 = 1+tan^2,

then the diff eq f' = 1+f^2 should remind you of those facts, and that enables you to solve the de.

 

[Darkiekurdo]

Sounds like you're having a blast. Have a safe return trip.

Anyhow, I just came across a very light-hearted joke that some of you oughta like:

Why do so many math majors confuse Halloween and Christmas?
Because Oct 31 is Dec 25.

I do not understand your joke.

 

[mathwonk]

i didnt either until i decided the month names were suggestive of modular equivalences.

 

[Darkiekurdo]

I am obviously too stupid for this forum.

 

[Pseudo Statistic]

not integrable

Looks (Riemann) integrable to me; it doesn't have a closed form solution, though.

 

[mathwonk]

no you ain't.

 

[Darkiekurdo]

Haha, I get the joke.

 

[DeadWolfe]

It's really a computer science joke.

 

[mathwonk]

seriously now (belive that?),smart or dumb is our ancestors' doing, achievement is our own.

not to malign my ancestors, who were very bright, but i used to call myself the "rocky" of algebraic geometry, dumb, but persistent.

 

[InbredDummy]

i always hear algebraic geometry being used in terms of string theory and quantum field theory, can you talk a bit about this mathwonk?

 

[mathwonk]

apparently the basic "particle" in string theory is essentially a riemann surface, which is an algebraic curve over the complex numbers, so algebraic geometers since the tim of riemann have been studying them, and the physicists seem to want to take crash courses in this topic for their own ends, which I know little about.

on the receiving end, algebraic geometers have been able to use Witten's formulas to calculate the expected number of maps from the curve of genus zero to various interesting manifolds such as quintic threefolds, but have not been able to prove the formulas calculate actual curves in all cases.

 

[pivoxa15]

Mathswonk, have you read the book 'Letters to a young mathematican" by Ian Stewart? From the opening pages it reads as if it is only the second book to be like the kind of Hardy's "A mathematician's apology". Although it's purpose is a bit different to Hardy's book.

https://www.amazon.com/dp/0465082319/?tag=pfamazon01-20

It seems these days, maths if a lot more social than in the past. Unlike the picture portrayed by Hardy. The ivory Tower image is gone and more and bigger collaborations. Team work seems to be important.

 

[mathwonk]

pivoxa, i have not read that book, and the first few pages indicate that stewart knows a lot more than i do about uses of math in diverse fields.

i was influenced more by hardy's views in the old days, that math was so interesting in itself that applications were irrelevant to my enjoyment of it. I regret somewhat taking hardys view now since it narrowed my experience. we never need a reason to ignore something, since evrything is potentially interesting.

but my own bent was in the direction of hardy's and i have always just tried to understand the internal structure of math itself, or whatever parts i found beautiful. indeed the more i have learned the more parts have become beautiful.

i like learning many proofs of the same result, and hopefully finding my own way to prove and understand things. I like guessing new facts and trying to prove them. I like finding simpler ways to understand things, and ideally getting the point where i can do what we were advised by some wise person, namely be able to explain everything to the man in the street.

I agree there are more interactions than perhaps before, but recall a highlight of hardy's career was a collaboration with ramanujam, or so he makes it sound in his book as i recall. and he has a famous collaborative book i guess with maybe wright?

so like me and some others, he may have fiound his greatest pleasure in at least discussing math with kindred souls. i love talking about math. I think i did some of my best work mostly so i would have something to tell my advisor.

after a while it began to seem like a fascinating exploration and search for exciting patterns and answers to mysteries. just as i never wanted to run my electric train after setting it up, i never careed about applying the theorems after proving them.

except in math, i did always take the approach that specific problems were more interesting than abstract theories, and i only discovered techniques and principles in order to apply them to the solution of specific problems, but they were purely math problems.

after discovering a method or technique to solve a problem, i would not develop it and sometimes others would work it out as a principle later, but if you read closely my work, the principle was already there in the solution of a specific problem.

so i like thinking and talking about and resolving specific interesting problems.

 

[threetheoreom]

but my own bent was in the direction of hardy's and i have always just tried to understand the internal structure of math itself, or whatever parts i found beautiful. indeed the more i have learned the more parts have become beautiful.

So If I understand correctly you were inspired by Hardy, more precisely his veiws of mathematics?

can you talk more about that.

thanks

 

[NathanExplosion]

Sounds like mathwonk was inpsired by mathematics itself, and Hardy's views on it just happened to ring true in his mind. Of course, I don't have any experience in psychoanalysis, so I'll shut up now.

 

[mathwonk]

i agree with nathan. but Hardy encouraged a narrow tendency i had which might have limited my early exposure to other areas of math.

 

[J77]

Mathswonk, have you read the book 'Letters to a young mathematican" by Ian Stewart? From the opening pages it reads as if it is only the second book to be like the kind of Hardy's "A mathematician's apology". Although it's purpose is a bit different to Hardy's book.

https://www.amazon.com/dp/0465082319/?tag=pfamazon01-20

It seems these days, maths if a lot more social than in the past. Unlike the picture portrayed by Hardy. The ivory Tower image is gone and more and bigger collaborations. Team work seems to be important.

Ian's an excellent writer -- most famous for his popular work "Does God play dice?".

Check out his personal page: http://members.aol.com/istewjoat/homepage.html containing his other popular works plus details of his more rigorous journeys into, for example, bifurcation theory and dynamical systems.

 

[pivoxa15]

Mathwonk, Can you shed some light to why Hardy said pure maths is a young man's game. Young being age 40 and under. Do you agree with him?

Also do you ever loss interest in maths or just not want to do maths at times? If so what do you do when that happens?

 

[mathwonk]

well, probably my best work was done when i was 35-40 or 45. It takes a lot of physical energy, long hours, and mental exertion is very strenuous.

So when you get older you have less energy. I guess people who do not do much of it do not realize that mental work takes physical energy, and that diminishes with age.

let me look at my vita and see...

Well i am most well known for that early work, but i am quite proud of some work done as late as 60 years of age, so maybe i am not so sure about that statement. it is just a general fact that we get older and less energetic i guess.

I also have still some hope of pushing on some ideas a bit further, and that would be fun. It does not really matter whether one is as good at 65 as at 25, what matters is to keep trying to achieve ones goals, and enjoy ones work.

I mean some people also say math is a genius's game, so then there would be no reason for us non geniuses to ever do anything. Well, too bad. I like doing math so I am going to keep on.

All that chatter is just part of the psychological detritus one has to ignore to succeed. If you give up everytime someone says you are not going to get aywhere, then you will have a harder time than you would anyway. I can definitely tell you I have seen some only modestly gifted persons, e.g. myself, who succeeded by perseverance on into their later age as researchers.

 

[mathwonk]

when you lose interest try something else for a while, take a break, a vacation, read novels, go to the beach, enjoy life, teach, read great writers and thinkers,... at length inspiration tends to rekindle.

i may be misunderstanding you here, but to me doing math is diferent from learning it, so when i lose interest in doing math i can also take a break by learning it, or by posting here, where i consider i am among friends.

sharing knowedge here is not "doing" math for me, i.e. not new math, but just talking about stuff I already know. Posting here is like an old man telling tales of his youth.

 

[pivoxa15]

well, probably my best work was done when i was 35-40 or 45. It takes a lot of physical energy, long hours, and mental exertion is very strenuous.

So when you get older you have less energy. I guess people who do not do much of it do not realize that mental work takes physical energy, and that diminishes with age.

well let me look at my vita and see...

Well i am most well known for that early work, but i am quite proud of some work done as late as 60 years of age, so maybe i am not so sure about that statement. it is just a general fact that we get older and less energetic i guess.

I also have still some hope of pushing on some ideas a bit further, and that would be fun. It does not really matter whether one is as good at 65 as at 25, what matters is to keep trying to achieve ones goals, and enjoy ones work.

I mean some people also say math is a genius's game, so then there would be no reason for us non geniuses to ever do anything. Well, ***** them. I like doing math so I am going to keep on.

All that chatter is just part of the psychological detritus one has to ignore to succeed. If you give up everytime some a** h*** says you are not going to get aywhere, then you wilol have a harder time than you would anyway. I can definitely tell you I have seen some only modestly gifted persons, perhaps myself, who succeeded by perserverance on into their later age as researchers.

So older people have less energy. Can you describe what it is like to have less energy compared to your youth? Or can one only feel it in order to know what it is like?

What about memory loss? Do older people forget easier and more often? Or is that an exception?

 

[JasonRox]

So older people have less energy. Can you describe what it is like to have less energy compared to your youth? Or can one only feel it in order to know what it is like?

What about memory loss? Do older people forget easier and more often? Or is that an exception?

mathwonk is full of ****. Less energy? If he's still doing mathematics now, he's full of energy. He comes on here to talk about mathematics and share his knowledge. He's amongst the most energetic and euthusiastic mathematicians out there. No energy? What a load of crock.



I know you don't count coming on here as math, and neither would I. But the idea to spread knowlegde and experience is there. That takes work. I'm sure lots of teachers and professors just go home and do their own thing. It takes special people to choose to share it.

 

[mathwonk]

less is relative. i used to work up to 30 or more hours at a stretch, and now i cannot do that. that's less energy than before. but it is true, even now my students say i am among their more energetic profs.

i used to commute to work and back home, 3 hours round trip, and sleep as lttle as one hour or less and go back and work a whole day, but i cannot do that any more, now i need 8 hours sleep like other people.

 

[JasonRox]

less is relative. i used to work up to 30 or more hours at a stretch, and now i cannot do that. that's less energy than before. but it is true, even now my students say i am among their more energetic profs.

i used to commute to work and back home, 3 hours round trip, and sleep as lttle as one hour or less and go back and work a whole day, but i cannot do that any more, now i need 8 hours like normal people.

But with what you passed on, you're now getting all these younger guys working! That's a lot of energy you passed on.

 

[mathwonk]

thank you so much! I hope i can psyche myself up for more theorems too! or maybe my proper role next year is to plan and organize some conference for the younger guys.

 

[quasar987]

My dear Mr. Wonk,

I have 6 credits left to do before completing my undergrad degree. Three of these I have to invest in a first algebra course, and three others I have already arranged to be invested in 'Measure & Integration', which is a graduate level course.

Now, I should be taking more classes because it would be a waste to only take 2. I am strongly inclined to take 'Logic' and am undecided btw 'Differetial Geometry' (another graduate level course!) or the undergrad 'Number Theory' course with Andrew Granville.

The dilemma is that

1) I don't know what graduate level courses are like. Are they much tougher than undergrad ones? Would it be too much to take 2 of them on top of the already hard 'Logic' and 'Algebra'? (N.B. My brain is of moderate proportion )

2) Number theory does not particularly appeal to me, but then again, I know next to nothing of the subject, and it is taught by Andrew Granville, which I remember you recommending me to take him in number theory class. And, I feel my math background would be lacking w/o at least an elementary introduction to number theory.

3) The 'Differential Geometry' class is taught by Octav Cornea, which I got last semester in 'Topology' and he was mighty fine. Plus, differential geometry is one of the two topics that appeal to me the most (with analysis) at the moment and I'm considering doing my masters in this topic (and hopefully with Mr. Cornea).


Got any advice? (And what's the answer to the question in dilema point #1) ?

 

[mathwonk]

talk to granville. he is a friend of mine so say hi. he will give you excellent advice and will teach a great course. my advice is to take his course because he is such a great teacher.

always try to choose teachers not topics.

let me know what transpires.

well now that i read your remarks, talk also to mr cornea since you liked his class and want to work with him. yes grad courses are a lot harder than undergrad onejs, and take more time. sounds like you have a wonderful group of faculty there. you can't go wrong.

ill say more if you want, but those two profs are the best experts to advise you on your exact situation there.

 

[quasar987]

k, I'll do that right now.

What else do you have to say? I'm all ears.

However, I can't imagine Granville's number theory course being less demanding than a grad course. When I had him in 'Applied analysis' (fourier stuff & sturm liouville stuff), he made us work like we had never worked before. lol, I had to lend the first homework set in two massive parts of 60 pages each that I jokingly labelled 'Vol. I' and 'Vol. II' It was great though... that guy understood and applied the principle that math is learned by doing it. Poor TA though.

 

[quasar987]

Hm, Andrew's email has an automatic response:

"I will be trying to avoid my email during the summer of 2007,
so please do not expect a response in the near future."

 

[mathwonk]

ok that means you have to persist. so keep in there. he is greatly in demand because he gets the most out of his students. and mention my name. i carry HUGE clout. harr harr.

or pm me your real name and i'll email him, but it may not matter. your good offices in trying to reach him count with me though. i think you have what it takes, namely good intentions and moxie.

 

[J77]

What about memory loss? Do older people forget easier and more often? Or is that an exception?

Mathematics isn't about remembering things!

This is a common trap -- thinking if you can remember every example in every textbook will make you a genius in your respective field.

Good results come from within, from using ideas from the past, but ultimately coming up with something of your own.

That's research!

 

[JasonRox]

I have a question about graduate school now.

I've been calling the two places that I'm interested in. Seeing how much it is and how it works, and etc...

Awhile ago I decided to do my Master's part-time, but the school I called said they don't offer night courses or anything like that. She said it's possible that they will make exceptions by offering night courses here and there, and we also discussed the possibility of having one class a week as opposed to 3 times a week. Therefore, I can just like book off every Wednesday morning to take a 3 hour class or something. I can do some courses independently while working with a professor here and there, and then write the exam for the credit. So, we talked about all these possibilities. I have to e-mail her back about what we discussed. She will look into it further. It's nice to see that they will consider it if I get in there or what not.

Anyways, my question is how hard would it be to handle one graduate course while working full-time? The industry I'm going in is mainly 9 to 5pm.

Note: I will also be attending part-time undergraduate to complete my business degree (which will be my second degree). I never found business courses hard to handle. It's just a matter of reading the chapter, thinking a little bit, and doing some work. It's not crazy hard, or it hasn't been yet anyways.

 

[mathwonk]

i think one is ok. it should be somewhat interesting and challenging compared to most regular work. but it will require time management. you might start preparing in advance by reading the book and doing exercises.

 

[mathwonk]

there are lots of different ways to make contributions to research. an older more knowledgeable person can often see connections between the work of a young oerson and some other older ideas and questions that are still out there and hence direct the thoughts of the younger person in a useful way. i.e. older persons can still provide valuable guidance even if the younger person provides more energy. In fact it is unfortunate when older persons decline to provide this because of a feeling it is not as exciting as doing the computations, or because they are discouraged by their superiors or funding agencies. Research is a community effort and very collaborative at its best and most productive.

 

[tronter]

does one need an undergrad math degree to go to grad school? Or can he major in something else (and self study the math)?

 

[pivoxa15]

Mathematics isn't about remembering things!

This is a common trap -- thinking if you can remember every example in every textbook will make you a genius in your respective field.

Good results come from within, from using ideas from the past, but ultimately coming up with something of your own.

That's research!

That's true but I have a feeling that having a phenomenal memory will help in some ways. Some of the best had exceptional memory like Euler, Fermi, Riemann, Gauss. At least one can save time such as bypassing time spent searhing through books or relearning old stuff. It is like doing computations. Actually relearning old stuff might be an issue for older people who has spent a lifetime researching. Even for undergrads some relearning is needed when he/she is in her final year.

People say being a good 'calculator' dosen't necessary make a good mathematician but I have a feeling that is because people think it is exceptionally boring and try to avoid it just as most try to avoid memorising. But some of the best were exceptional calculators as well. In fact all of the above. I don't know about Fermi though.

 

[pivoxa15]

does one need an undergrad math degree to go to grad school? Or can he major in something else (and self study the math)?

I use to think self study in maths might be enough since it is a priori and there is no need to get a degree in it. I use to think that for science as well. But decided to do the degree anyway. I think it has been a very good decision as I cannot see myself self study these subjects. So if you are not a top student then try to do a formal degree.

 

[pivoxa15]

mathswonk, is it possible for a student to lift his mark in pure maths by 15 out of 100 when going from third year to fourth year? More specifically going from 60 average to 75 average?

Have you seen it done?

 

[mathwonk]

well yes. but if you want a different result, you must provide different behavior.

I myself went from a 1.2 gpa to a let's see, 3 or 4. i forget. the difference is i stopped skipping class and began attending every one.

And I started reading the materal assigned, and actually trying hard to write the assignments, and rewrite them, and so on.

i guarantee if you double the amount of time and effort you spend working your grades will go way up.

you know that guy you think is a twit, a wonk? try imitating him, going in the libs every day and maybe staying there until it closes. you are every bit as smart as he is, you just have a different list of priorities, you want to be cool, and have free time, and behave as you did in high school.

try putting that off for just a couple years and wonderful changes will occur. i did not learn this until grad school, but i was lucky to get in. then i became a "star" (4.0), just by doing all the work every day.

it is not at all easy, but you can do it. the difficulty is in terms of discipline, not smarts.

 

[J77]

That's true but I have a feeling that having a phenomenal memory will help in some ways. Some of the best had exceptional memory like Euler, Fermi, Riemann, Gauss. At least one can save time such as bypassing time spent searhing through books or relearning old stuff. It is like doing computations. Actually relearning old stuff might be an issue for older people who has spent a lifetime researching. Even for undergrads some relearning is needed when he/she is in her final year.

People say being a good 'calculator' dosen't necessary make a good mathematician but I have a feeling that is because people think it is exceptionally boring and try to avoid it just as most try to avoid memorising. But some of the best were exceptional calculators as well. In fact all of the above. I don't know about Fermi though.

I have no idea to what extent the memories of the people you mentioned are documented.

However, the key issue -- which I think I pointed out in my previous post, I can't remember -- is that when doing research, you don't recall everything which you have learned in the past. The way it works for me is to read recent papers, and if they use a technique I'm not familiar with, which will help towards my own work, then I will go to a library and take out some key works, plus going back over their (and subsequent) references.

Usually, the methods will have only been have remembered by myself -- or completely new to me -- you'll find it doesn't matter so much, ie. there is less pressure to "remember", when you are past any form of testing.

 

[JasonRox]

That's true but I have a feeling that having a phenomenal memory will help in some ways. Some of the best had exceptional memory like Euler, Fermi, Riemann, Gauss. At least one can save time such as bypassing time spent searhing through books or relearning old stuff. It is like doing computations. Actually relearning old stuff might be an issue for older people who has spent a lifetime researching. Even for undergrads some relearning is needed when he/she is in her final year.

People say being a good 'calculator' dosen't necessary make a good mathematician but I have a feeling that is because people think it is exceptionally boring and try to avoid it just as most try to avoid memorising. But some of the best were exceptional calculators as well. In fact all of the above. I don't know about Fermi though.

Really? Most bright people don't have the best memory at all. Most biographies I read said they were terrible at remembering things. Forgetting whether or not they ate, have an appointment somewhere, and so on and so on.

To say Mathematics is a subject of memory is like saying speaking English requires a lot of memory too. Speaking English does require memory and a lot of it too, but when you participate in it on a daily basis, it hardly comes across as something that requires memory. The same thing happens with Mathematics.

 

[threetheoreom]

To say Mathematics is a subject of memory is like saying speaking English requires a lot of memory too. Speaking English does require memory and a lot of it too, but when you participate in it on a daily basis, it hardly comes across as something that requires memory. The same thing happens with Mathematics.

This is the truth.

 

[mathwonk]

to get into grad school in math one must convince the admissions committee (that was me and 5 friends last year) that one has the potential to do strong independent work in math.

now what evidence are you going to offer for this if you do not take a certain number of hard math courses and do well in them?

self study will not do, since there is no test to adequately measure your preparation. the gre is pretty easy, and some admissions people do not even look at them.

i myself do look at them, because i think even though they actually test very little, still in my experience there is a good correlation between high scores on them (785-800) and success in the program.

but people are usually looking for success in a good hard course like abstract algebra, complex and real variables, topology, and a letter from the instructor testifying to the strength of the student.

self study means you have no one to write that letter. even if the letter is there it helps if we know the person writing it, and what they mean when they say "excellent chance to stand out" or "best in 10 years at this school".

there are other ways to make an impression but no substitute for that data. based just on posting experience here, i have said i would recommend hurkyl for our program for instance sight unseen, and there are others who have impressed me by their posts, but the committee would still want some supporting data.

let me mention one thing that can help, and that is exhibited sincere interest by the student. i.e. if we are making offers and stand to lose some of our best candidates to other schools, we want some hope of success. thus we appreciate a student who clearly prefers us to other schools, and that can weight our decision that way, even possibly above a stronger looking candidate on paper.

it is no guarantee, but something to think about. i.e. if you sincerely do prefer a certain program, be sure to let them know that. but do not say so if you do not mean it, as they will remember any deception when you are looking for a job.

 

[Darkiekurdo]

How does one know that one has the intelligence to become a mathematician? I doubt my own mathematical skills, but children of my age (14) do not know the things I know about mathematics, i.e., analyisis and algebra.

 

[JasonRox]

How does one know that one has the intelligence to become a mathematician? I doubt my own mathematical skills, but children of my age (14) do not know the things I know about mathematics, i.e., analyisis and algebra.

I'd say if you're intelligence is atleast above average your fine. If you're average but love mathematics, your fine too.

Note: 700th Thread Post