Other Should I Become a Mathematician?

[mathwonk]

actually it is not that odd to be able to learn math at a young age. our system of "education" in USA is so discouraging to learning that it retards all learning enormously.

my own younger son was able to easily understand and learn geometry from harold jacobs book at the age of 8, but because his peers were not doing the same, I held off teaching him until high school. then at the age of 15 his high school decided to stop using jacobs and begin using the pitiful books of john saxon. hence in high school, he actually learned less than he could have learned in 3rd grade.

my older son, read jacob's algebra when he was 11 and took first place in the state math contest for 8th graders. then he tired of learning so much more difficult material than his peers and tailed off for a while.

thus in my experience the main ingredient is interest. if a kid likes what he is doing it is very easy for a bright kid to learn at a rate 5 or 10 years ahead of the usual USA math courses.

not everyone wants to do it but it isn't that hard for someone who is smart, likes the material, and is exposed to good books or good teachers.

 

[PhysicalAnomaly]

I noticed Spivak doesn't cover vector calculus. Which vector calculus text would you recommend? I know Stewart covers it but I am looking for one similar to Spivak. The only other one I know about is Marsden and Tromba but I've had conflicting opinions on it.

Thanks.

 

[qspeechc]

That's interesting mathwonk- you're raising little mathwonk geniuses
I guess you are correct. The work in school is very much dumbed down. Also, the level of work has seemed to drop over the decades. I remember my maths teacher in high school telling me they did vector geometry/algebra in his days, now they have almost completely erased trigonometry and circle geometry. It's this silly FET education, which has not worked anywhere but they want to try and put it in place here (South Africa). He made the point- what's wrong with the old education system? His generation put people on the moon with it.

Sorry for the digression.

 

[torquerotates]

Although the current educational system dumbs things down a whole lot, it does cater to the majority of students. Just look at how it will be if we soup things up. About half the students in high school wouldn't even graduate let alone go to college. An interesting comparison would be to look a the eduation system in an asian country and compare it to ours. Look a China. The average Chinese student far exceeds the average America student in math and science. But then you have to look at it and say that the average person in China probably was never even a student. Just because they can't make the cut. So it goes to show that the more rigorous an system is, the more exclusive it is,( i.e it weeds more people out). At least in America, most people can graduate and move on in the system, hence giving them more chances.

 

[uman]

That's why they should do it like they do in Europe -- everyone gets educated; people who are able get a high-quality academic education while the rest are prepared for a career.

 

[mathwonk]

i don't see why everyone in high school should take the same courses. what would be so bad about offering higher level material to those who can handle it? they allow this in music, and they have different teams in basketball. political correctness is really kind of stupid in education.

 

[mathwonk]

lets start a list of good free books. i have listed good books, but free books are a special category worth knowing about.

i have found a good linear algebra book today, at least i think so. there are lots of free books out there but most of them are not that great. this is probably for a second course in linear algebra.http://www.math.brown.edu/~treil/papers/LADW/LADW.pdf

 

[morphism]

There's the algebra book by Robert Ash:
http://www.math.uiuc.edu/~r-ash/Algebra.html

 

[torquerotates]

i don't see why everyone in high school should take the same courses. what would be so bad about offering higher level material to those who can handle it? they allow this in music, and they have different teams in basketball. political correctness is really kind of stupid in education.

I agree. But sometimes there really isn't enough students for the schools to make a course for. For example, there was only about 10 people in my high school that was ready for calc BC. As a result, the class wasn't offered. So what ended up happening was that one teachers actually had to teach the class for free at 6:00 in the morning. And this was off the record. But how many teachers are there that are willing to teach for free? Almost none. Most likely the best choice for the students that are really advanced would be to move to a better public or private school. Education is important enough that its worth the cost. Private schools or top publics usually has the more advance classes or special program to cater to gifted students. If worst comes to worse, there's always self study, homeschooling or private tutoring.

 

[mathwonk]

well, i recommend paying teachers who teach better courses, but i have also taught many courses for free. and i have encountered schools who declined to let me teach them for free because they did not like that their students could not all take them successfully. i have not only taught them for free, i have also brought in grants that paid the students to take the course while i have taught it for free. many of us are ready to teach if students are available. the pay on this site is rather low for example.

 

[qspeechc]

It becomes tricky deciding which teachers teach better, but it does provide incentive. Also, at least here in South Africa, teaching attracts very few people- we have a serious shortage. No wonder, the wages are pathetic. Many not-so-good teachers get through.
Also, not everyone has the money to afford better education, or to buy books toteach yourself from, or, for that matter, have access to the internet (again, this is in my country)
I do believe in free edution, up until the end of high school, if the free education can be of quality. Somehow, I don't see this happening in the majority of capitalist nations.
Maybe there should be specialised schools for gifted kids, and those that cannot afford it, the government subsidises their study. But this is my dream world.

A tougher education for all is realisable, in my opinion. Kids today are just lazy, or society has become dumber. I believe (I stand corrected), that in 18th or 19th century Germany high school children were taught real analysis and some pretty hard-core calculus.

The topic of good free e-books seems worthy of a thread of its own, but until then:
Another good linear algebra e-book I found:
http://linear.ups.edu/download.html
I suppose it can be used for both 1st and 2nd courses.

 

[Gib Z]

Well, I've been learning Linear Algebra for about 2 weeks now (after Finishing Calc, I heard that would be beneficial for this) and been using the free textbook, "Linear Algebra" by Jim Hefferon. It is available, complete with exercises (and answers in a different file) in pdf form at http://joshua.smcvt.edu/linearalgebra/. In fact, when looking at that sight again, I found a link to a review at http://www.theassayer.org/cgi-bin/asbook.cgi?book=29. It rates the book very highly, so I guess its a good thing I found it =]

That review sight may have links to other good books, all free. Check it out.

 

[mathwonk]

i didn't say teach courses better, i said teach better courses. that is easier to determine.

 

[RufusDawes]

i don't see why everyone in high school should take the same courses. what would be so bad about offering higher level material to those who can handle it? they allow this in music, and they have different teams in basketball. political correctness is really kind of stupid in education.

Teachers really aren't smart enoguh to make that distinction.

They told me I wasn't cut out for mathematics back in year 9, by the end of high school I scored reasonably well in the highest band of mathematics classes.

In short, teachers are stupid it really needs to be up to the kid to decide what level of course work he or she would like to undertake.

 

[torquerotates]

lol same thing happened to me. They thought I as stupid so they put me in remedial arithmetic classes for years on end. By the time I was 16, my comprehension level was still below 6th grade math. But when I was 17 I was acing pre-calc/trig without even trying. Oh the irony! And now I'm a math major with mostly A's in math and science classes.

 

[mathwonk]

I am a teacher. Are you calling me stupid? That may be true, but it is undiplomatic.

get smart, kids. the game is not against your teachers, but to gain ground against the unsolved problems of the world.

 

[Gib Z]

I think they have generalized the initial statement to the point where it is not true anymore. The more specific statement holds though: Some teachers are not capable of recognising talent in a student and may wrongly label them as failures.

 

[RufusDawes]

I am a teacher. Are you calling me stupid? That may be true, but it is undiplomatic.

get smart, kids. the game is not against your teachers, but to gain ground against the unsolved problems of the world.

teachers... where to start with teachers... they make it hard for good teachers to want to be teachers.

 

[Gib Z]

Well I don't know about that? I want to be just like mathwonk in the future, I am not too sure if I am a good teacher but I can't see any restrictions put in my way? I mean, it's not like they have a tax on good teachers salarys or anything...

 

[RufusDawes]

Well I don't know about that? I want to be just like mathwonk in the future, I am not too sure if I am a good teacher but I can't see any restrictions put in my way? I mean, it's not like they have a tax on good teachers salarys or anything...

wait until you have to meet all the other teachers.

 

[Gib Z]

That will take a while, I don't think I will ever see every single teacher, I'm sure you haven't either. Believe it or not, there ARE good teachers out there.

 

[qspeechc]

Another good linear algebra book:
http://www.mth.uct.ac.za:9080/mathscourses/maths-2/modules/2la/main08.pdf/download

 

[Werg22]

I want to be just like mathwonk in the future

Not even mathwonk would want that.

 

[Werg22]

That will take a while, I don't think I will ever see every single teacher, I'm sure you haven't either. Believe it or not, there ARE good teachers out there.

I, for one, have met them all.

 

[mathwonk]

well i was too flattered to remind gibz to just be himself.

im sure he knows this and was just trying to make me feel good.

(success.)

 

[Darkiekurdo]

Is it possible to be a "good" (let's keep the definition ambiguous) without having a natural talent for mathematics?

 

[mathwonk]

that is the sort of undeterminable philosophical question that can be discussed forever without settling it, but it is clear that no matter what ones talent level is, hard work will make more difference in performance than anything else. I have certainly known people who did not strike me as particularly brilliant who nonetheless achieved significant success in mathematical research. Persistence is crucial, ignoring people who imply one is not brilliant, continuing to pursue ones interests and passions.

The sad thing is not so much lacking gifts, as allowing ones gifts to languish unrealized and undeveloped.

 

[tgt]

Mathwonk, how old were you when you moved out from your parent's home? And how did it affect your mathematics?

 

[mathwonk]

in 1960 i left home at age 18, to go to college and live on campus in boston.

college was my first encounter with real math, as i was taught freshman calc by the famous john torrence tate (check him out on the web).
he chose courant for the first course and we were thus initiated at a high level.

not being at home also allowed me to avoid studying, which started me on a long road of up and down success and lack of it in school, until in 1970 or so I finally began again to study hard and consistently.

 

[mathwonk]

moving away from home is not closely linked in my mind with any effect on my math. moving out has so many other ramifications, like learning to be responsible for your own welfare, learning to accommodate and understand other people who were raised with different assumptions from yours, etc...

learning to stand on your own feet, take up for yourself, clean after yourself, make your own choices for your life, do what is needed to achieve them, become free of the restrictions your parents placed on you, or at least distinguish the ones you value from the ones that go against your own nature and opinions.

i.e. leaving home is often the first step of becoming your own person, it has lots more consequences than just affecting your math. it is part of learning to please yourself rather than just your parents. you begin to set your own goals and go after them, hopefully using the good parts of what your parents gave you.

 

[eastside00_99]

Hey Mathwonk,

what advice can you give to someone who doesn't get into grad school (with support) at the places they have applied? It looks like this may be my situation; although, the verdict is not out. I did not apply to do a masters at my current university, and I didn't took very little applied courses. Do you have any advice here?

 

[mathwonk]

i would try some other places. someone at your school should be able to advise you as to places you will get in.

when i interviewed at columbia, eilenberg thought me unprepared for columbia but recommended maryland to me. as it turned out i got into another one of my schools and did not need to apply further.

later as a new father and more experienced teacher at ellensburg in 1973, i just waltzed over to univ of washington and took their phd prelims. i figured if i beat their own students at their own prelims they had to take me and they did. i prepared for them by teaching the courses beforehand.

again i got a better offer from utah so went there instead. but if you have promise and some data to back up that promise, someone will take you. and there are a lot of places out there where you can learn to do research.

 

[eastside00_99]

Well, as of right now, I have gotten into UIUC, CUNY, and TuftS. I have taken enough graduate course work to have a reasonable attempt at quals in topology, abstract algebra, and real analysis. Of course, I would need to study very hard this summer. I can get partial support from Cuny automatically (an in-state tuition wavier for teaching one course) and am on the wait list for full support. But, I don't think I am going to get it as support for CUNY is rare from what I have heard.

So, I could attempt the Ph.D. quals at any of these schools at the beginning of the fall semester and take out a loan to pay for the first semester. This would take hard work and there are no assurances of success. But, I feel as if I pass the quals in the first semester then not only is a lot of required course work behind me but this should push for support. If I don't get support then I would have about 10,000 in loans to pay back.

This is just me thinking out loud about possible options however dumb they may be.

 

[mathwonk]

well if you got into those schools you are obviously good, and i am not worried about you. you'll be fine. good luck. if you go to UIUC say hi to Sheldon Katz and maybe William Haboush (he may not remember me from the recent summer meeting at Seattle, but we were both old friends of George Kempf), and at Tufts to Mauricio Gutierrez and Loring Tu, and Montserrat Teixidor. I do not know the people at CUNY.

 

[eastside00_99]

I will if I go to UIUC or tufts but right now Cuny is the most affordable option.

 

[mathwonk]

well looking over the faculty, i know of martin bendersky in algebraic topology, josef dodziuk in topology and geometry of manifolds, maybe michel handel in dynamical systems, raymond hoobler in algebraic geometry, leon karp pde, ravi kulkarni diff geom and riemannn surfaces, oh and linda keen in riemann surfaces,. oh and joseph lewittes in riemann surfaces,...oh my, kolyvagin, and an old friend adam koranyi,moreno, roitberg, wow! dennis sulivan, szabo, szpiro,...

hey alphonse vasquez taught me algebraic topology at brandeis in 1965..!

i think i looked on the wrong website before, i know lots of these people at least by reputation and some personally. this is a terrific looking place.

. it should be great. what area interests you?

 

[eastside00_99]

I have been interested in Algebraic Geometry for over a year now. But, it is such large subject that it is hard to be more specific than that. There is one person at CUNY who uses algebraic geometry to study Modal Theory which is fascinating and tickles my more philosophical side. But, CUNY has a good number theory group so I could also study function field theory or algebraic curves over finite fields. I don't know: I guess what will determine the specificity of my research will be my thesis advisor.

 

[mathwonk]

there is a lot of riemann surfaces there, and i see specifically two people in algebraic geomnetry, schoutens and szpiro, schoutens very algebraic, but with some past work on rigid analytic geometry, and szpiro is just very strong and very broad.

did you consider UGA? we have algebraic geometry and number theory as well, indeed they are two of our best groups. and we just received word of an impending vigre grant so we will have lots of funding.

 

[eastside00_99]

My advisor strongly recommended that I apply to UGA, and originally I was going to. I even visited Athens (I have a friend in the philosophy department) but it was late on a friday evening when I got there so I didn't get a chance to talk to anyone in the math department. I don't really know why I eventually decided against it, but there were a few places like that (e.g., UNC and WashingtonU) which because of money/time I had to make some cuts.

 

[eastside00_99]

Looking at the program at UGA, I have to say I like how the quals are done. One has three years to complete the quals and you can take them as often as you like. At the school I go to now for undergrad, you only have two shots and then you are out (you only have two years at that). This is some what like how CUNY does it. If I were going to UGA, I think I would do one qual per year and take a lot Special Topics courses or advanced graduate courses each year: doing topology the first year, real analysis the second, and complex analysis the third. Students there certainly have the opportunity to learn a lot of mathematics before they begin doing specific research. This must partly account for those three people in the math department who have such good doctoral theses (one was published in the Annals of Math!).

Mathwonk, I understand that your work in Algebraic Geometry is more analytical in nature. This may be a dumb question but: Does your work involve working on questions in Several Complex Variables? If so, did you have a chance to take classes in Several Complex Variables as a graduate student because when I look at schools a lot of places do not offer such classes nor have a lot of people who just work in this field.

 

[mathwonk]

we do give a lot of leeway to students who are having difficulty qualifying, but for strong students, i would advise qualifying as quickly as possible. that's because it is notoriously hard to write a thesis, and you should allow as much time for that as possible.

my own work started out more analytical and did take advantage of my brief previous life as a several complex variables specialist. since then i have done some more algebraic work, as in the 1990 paper in compositio, and the 2004 compositio paper, where we work over algebraically closed fields of characteristic not 2.

most of our work is what i would characterize as geometric rather than analytic, but i do have a thorough grounding in several complex variables from hugo rossi and joe taylor, and i learned riemann surfaces from herb clemens.

 

[mathwonk]

actually i started out interested in banach spaces, then algebraic topology then became interested in commutative algebra and categories, then algebraic geometry, then differential topology, then several complex variables, and finally got back into algebraic geometry where i found one could profitably use all those tools.

 

[proton]

what is the bare minimum needed to get into a decent pure math grad program? such as caltech, mit, etc

like would getting a bs in physics, but the only pure math classes are 1 semester each of linear algebra, real analysis and complex analysis be sufficient? I'm aware that you can make up undergrad courses that you didnt take during undergrad later in grad school. but i was just wondering if taking so few classes is ok?

in other words, can i still get into good math programs by having a good physics gpa, letters of rec, gre score to make up for my lack of math courses completed?

 

[eastside00_99]

I know you probably want a qualified position on this, but nevertheless:

what if a math major took one modern physics course, one mechanics course, and one electrodynamics course (didn't even take a year sequence of mechanics and electrodynamics), could that person get into Princeton, MIT, or Harvard for grad school in physics?

Maybe, it is possible, but maybe they would have to have a ~4.0 gpa, a 90% on physics GRE, and published work in mathematics that has applications to physics.

Why do i say that? because my teacher who had an undergraduate math degree got into MIT with 4.0 in math, near perfect math GRE, and published papers in mathematics!

 

[eastside00_99]

Hey Mathwonk, Is the compositio paper on your website? Also, I noticed that UGA has (had) a Vigre grant for work on Tropical Algebraic Geometry. This is a subject that interest me quite a bit as Combinatorics is what drew me into mathematics in the beginning. I actually have book here "Tropical Algebraic Geometry," Oberwolfach Seminars Vol. 35 by Itenberg, Mikhalkin, and Shustin.

 

[MathematicalPhysicist]

Shustin? do you mean evgenii shustin?
he is/was my lecturer in topology and linear algebra 2, If so I hope for you that he is better in book than in lecturing, unless you like handwaving.

 

[eastside00_99]

Yes. It is Evgenii Shustin. Then does this mean you are a undergrad/grad student in math at Tel-Aviv University?

Funny enough, the book contains very few proofs, but this is more because of the nature of the book. It is more like a large research article than a textbook (all proofs that the reader can't do contain references to the articles in which you can find them).

 

[MathematicalPhysicist]

Yes, well I think this lecturer should only lecture at grad courses' level, I don't think that I will take another course with him, although you cannot know who they will place in the next year, and from what I know he also taught in the past an undergraduate course in Groups, which I want to take next year. Let's cross our fingers that he will not teach this course.

 

[mathwonk]

With the proliferation of good young faculty in US from abroad, there are many good programs, and the shortage is that of good students to enroll in them. So although top programs like the famous ones you mention are always competitive, places like UGA are always looking for students who have the ability to do good graduate work, even if not huge numbers of undergrad courses.Yes we have people who can work in tropical geometry. Indeed in my opinion Professor Rumely of our department's number theory group may be arguably said to be one of the founders of tropical curve theory, via his introduction of the concept of directed graphs, as a "non archimedean analog of riemann surfaces" as i understand it (which isn't much).

One Compositio paper on my website is about riemanns singularity theorem for prym varieties. the final conjecture in that paper, suggesting the best possible result, has since been proved by casalaina - martin, now at harvard, to appear in annals of math.

 

[mathwonk]

the cantrell lecture series has begun at UGA featuring the outstanding mathematician Bjorn Poonen of berkeley this week. Yesterday he gave an elementary intro to some of his favorite number theory problems.

can you find integer solutions to this equation:
X^3 + Y^3 + Z^3 = 29?what about this one X^3 + Y^3 + Z^3 = 30?

it was solved by some grad students at UGA some years back, with the smallest solution somewhere around 40,000.

What about X^3 + Y^3 + Z^3 = 31? or 32?

If you know modular arithmetic you can check that a cube is always congruent to 0,1, or -1, mod 9, hence numbers that are 4 away from a multiple of 9 cannot be a sum of three cubes.

so what about X^3 + Y^3 + Z^3 = 33?well no one knows! but poonen conjectures all such equations can be solved, i.e. all ones where the Right hand side is within 3 of a multiple of 9, in fact infinitely often.how about them apples? so simple, so close to known results, and fascinating, but not something i myself had thought about.

we should all remember to take the chance to hear such inspiring speakers when the chance presents itself.