Other Should I Become a Mathematician?

[mathwonk]

i think most students accepted at most schools receive such a stipend. occasionally we accept students without support but it is rare since few students have independent means.

 

[mathwonk]

i have recently written advanced undergraduate linear algebra notes, covering jordan forms and spectral theory, although they technically begin at the beginning of vector spaces. they should be available on my website soon. this is a 66 page version of the 14 page "primer" of linear algebra now on my site. they are my notes for math 4050.

 

[j0k3R_]

mathwonk,

can you recommend to me a book for studying advanced group theory, after studying hungerford ch. 1, 2, etc.

 

[mobe]

Thanks for all the notes mathmonk! They are quite useful.

 

[mathwonk]

thank you!. i will send my new linear algebra notes to anyone who asks, since it may be awhile before they get posted to my website. I feel i have finally understood jordan form, and I hope this comes through in the notes.

 

[quasar987]

I would really love to see those new notes of yours. Any updates other than the Jordan form?

my email in a pm

 

[altcmdesc]

Hey everybody, great thread.

I'm fairly certain I want to major in mathematics. I'm just having trouble deciding where to go in the field. Calculus was really the course that made me truly love mathematics. The idea of a limit and all that proceeds that (sequences, series, integration, differentiation, etc) are things I simply could not get enough of. What current research is being done in this branch of mathematics (Analysis, I assume)? I'm not too familiar with it because the mathematics involved are usually presented in such a way that an undergraduate cannot readily understand the concepts.

Also, the idea of applying mathematics to biological systems seems very interesting to me. What sort of mathematics are frequently used? I can imagine differential equations and stochastic processes are used to death, but what else?

The only problem I see with pursuing a career in Biomathematics is that I would seemingly have to study applied mathematics in lieu of pure mathematics. I guess that question comes down to: how probable is it that a pure mathematician is able to perform applied mathematical research (at a university-level)? Basically, I want to take pure math courses, some biology/physics/chemistry courses and leave behind some of the dryer applied mathematics courses. I like applying math, but for some reason I don't like wasting course time on it.

Thanks

 

[mathwonk]

you sound like someone who is going to be very successful. biomath is a very hot well funded topic today, and as always in applied math, the problems are hard and demand the best you have.

i am not an analyst, and will try to find some more expert advice for you. nice to meet you.

 

[quasar987]

I'm not an expert of course, but from reading the titles of seminars and workshop in analysis, I have the impression that most of the work done in analysis is in connection with differential equations.

For instance, I'm currently doing a master's thesis in analysis and specifically critical point theory. The typical theorem in critical point theory reads "If a function f:V-->R, where V is a (complete) vector space, is such that [hypothesis], then it has a critical point", where a critical point of f is, as you maybe know, a point p such that f '(p)=0.

Now the interest in critical point theory stems from the fact that if we have a differential equation, it is sometimes possible to find a function f:V-->R where V is a space of functions, such that critical points of f correspond to solutions of the differential equation.

It suffices then, to show that f satisfies the hypothesis of a critical point theorem to conclude that the differential admits a solution. This is useful because it means we're not looking in vain for a solution!

 

[altcmdesc]

That actually seems pretty interesting. I guess I'll take the Honors (Real) Analysis course when it comes around as well as Complex and see how I like it.

What about my other question: How probable is it for a pure mathematician to do a bit of work in a more applied field as well? I just hate restricting myself like that.

 

[quasar987]

Have you read the wiki articles on pure and applied math? I just skimmed through them and I believe they might give you an idea of the "odds" involved.

I guess the odds would depend on what you decide to specialize into. If you go into number theory, which is like the flagship of pure math, then the odds of doing something applied in relation to that are quite low.

But if you go into analysis, and consider that everything differential equation-related is applied math, then the odds of doing something applied in relation to your research in pure math are much higher.

In my case above, proving a critial point theorem would be the pure part, and finding a differential equation whose associated functional satisfies the hypothesis of the theorem would be the applied part.

 

[mathwonk]

altcmdesc, here is a response from a colleague of mine in analysis:

"If this student is at UGA, the best advice is to steer him to Andrew
Sornborger and/or Caner Kazanci. You might also mention this term's
remaining VIGRE introductions:

Monday, August 18
3:30 PM – 4:10 PM Room 304
VIGRE Presentation: Neil Lyall, “Geometry, Combinatorics, and Fourier
Analysis”

Tuesday, August 19
2:00 PM – 2:35 PM Room 304
VIGRE Presentation: Caner Kazanci, “Modeling the Health of Ecosystems”
2:40 PM – 3:15 PM Room 304
VIGRE Presentation: Jason Cantarella, “Tabulating Composite Links”

In any event, applied interests need not preclude concentration on
rigorous math courses. In particular, a serious baby Rudin type course
would be valued in any applied program. Also some general info on Math
careers can be found at http://www.math.uga.edu/undergraduate/careers.html"

 

[altcmdesc]

Thanks a lot mathwonk! You have all been a great help.

In the "Intro to Analysis" course at the UMN, the text is, in fact, Rudin's "Principles". I'll definitely make sure to take this.

I've come to the conclusion that I'll stay in what the UMN calls the "Graduate Track" for Mathematics (which is basically a "pure" math track), taking some applied coursework on the way to test the water. Would taking this route harm my chances of doing applied work in graduate school (specifically in Biomathematics, which requires a bit of Biology) should I choose to do so? Would it be difficult to take the necessary Biology in graduate school (I hear most biomathematicians teach themselves)?

 

[mathwonk]

the new notes for my summer course 4050 in advanced linear algebra are up on my webpage. they cover jordan and generalized jordan form, duality, spectral theorems, determinants, finite abelian groups, and constant coefficient linear ode's. they are an expansion to 68 pages of my 14 page linear algebra primer. they are much more explanatory. still they cover in 68 pages more than most books do in several hundred pages. i hope they are readable. there is a table of contents. the introduction got omitted from the notes but appears on the webpage. enjoy!

 

[quasar987]

mathwonk,

I see in your VITA that at the end of the 80s, you and R. Varley received 2 ~90k grants for research.

Did you actually spend all that money on research? If so, how?

-plane tickets to conferences
-subscriptions to journals (?)
-paying grad students for research

What else?

 

[mathwonk]

federal research grants are kind of a mechanism for the government to fund universities. i.e. the university takes about 30-50% off the top for "overhead", claiming reimbursement for the lights in our building, etc...

then we sometimes obtain a grant for a piece of equipment, such as a computer, but often the feds say that should be paid for by the university, although often it isn't.

There is usually money in there to fund graduate students in the summer to work on their research projects, and visitor money to fund airfare for people to come in and talk to us.

the basic grant moneys that fund research are twofold:

1) we ask for salary for two months in the summer so we do not have to teach or go without pay while doing research in the summer. In Canada this is unnecessary since they receive 12 month salaries but in the US we only receive 9 months pay per year, and must either obtain grants for summer work or teach or go without pay. most of the past 10 years i have done my research in the summer without pay, while my wife supported me.

2) travel money so we can visit other universities and learn and collaborate. this buys plane tickets and food and lodging.

so out of a grant of whatever, for one year, each recipient can expect to receive at most 2 months pay per year, plus the right to buy some plane tickets. sometimes we only got one month's pay, or none. one year i wrote a grant that paid a group of students a stipend so they could afford to study with me instead of working. It also paid their teacher while i donated my own time. One of those students, is now a full professor at Brown, and I consider that time well spent.

research is expensive simply because to do it one needs free time. So to buy a month's research one needs to buy a month's free time for a scientist. but it is much more expensive because most of it goes to the university.

the person doing the research and writing the grant receives relatively little of the money, sometimes none at all. for a while i know there were NSF programs, notably topology, that gave grants with no salary in them at all, just travel, visitor and student moneys. the researchers donated all their time. Still there is prestige from the university for bringing in money that benefits the Uni. I.e. you are expected by your university to bring in money for them, not yourself.

the point is to get your name on that money, i.e. to have it on your vita, although you do not get your hands on much of it.

and those sums you read were for multi year (2-3 year) grants.

once as a young person i obtained an NSF grant for about 15K to finance a large conference that has become a famous event in the subject of curves and abelian varieties, the athens conference headed by phillip griffiths, and leading to the book by arbarello cornalba griffiths harris on geometry of algebraic curves. when trying to augment the grant with local university sources i was told that money was tight and i offered to donate my own $700 salary for the conference, which provoked amusement from the research VP at that time, who said that was not needed. Later I learned he had found over $400,000 unspent dollars the day before and given it to other more favored programs immediately. I was asking for $5K, and being stonewalled.

some 30 years ago i read in our university research reporter that in the us, over 50% of all grant dollars go to biological and medical sciences, while less that 2% goes to all physical and mathematical sciences combined. so if you want to be well funded go into genetics, not algebraic geometry. of course nowadays the genome projects are being told to obtain mathematical input to be more competitive but it is not happening to my knowledge.

grant money is awarded by politicians hence for political reasons, not scientific ones. look on our departmental website and see where most of the grant money is coming from: we have a recent renewal for an educational VIGRE grant for millions of dollars, because we are doing a good job of helping train US citizens in math.

at the same time researchers are being denied money for their research, they may be granted money to try to bring US students up to snuff.

 

[serkan]

I've just recently discovered this forum and want to say it is amazing to find such a topic.

I've completed two semesters in financial mathematics program, and am quite confused about the direction of my further study. I find thinking about dynamics and the nature of the markets and formulating them quite interesting, That's why I got into this program, but I think a graduate education is also necessary in order to get into maths as much as I want to. My aim is to make a doctoral study on applied mathematics in an US university and I was wondering if my background would be enough for this, and if not what courses should I take in order to make it so. http://www.bilgi.edu.tr/pages/faculties.asp?fid=3&did=20&curri=true&mfid=2&mdid=92&r=8" is the link to our curriculum. I was eager to make a double major with mathematics but the director of the department said it is beyond human capacity as much as I disagree. Which courses do you think can I overtake to be a good PhD. applicant? Or should I abandon studying financial mathematics and get into mathematic program? I considered this too but although my university's math program is well regarded and one of the most rigorous ones in Turkey, it is a new one and might not be much known by US universities. As there are not much place to study applied mathematics in Turkey, the possibility makes me think. I need some guidance at the moment and any input would be really much appreciated.

By the way, I think at the end of the undergraduate study, my GPA would be close to 4.0 and I would have some good recommendation letters. But what makes me think is that it is hard to get admitted from a US university from Turkey. Especially from a new university.

 

[mathwonk]

with few or no courses in algebra, topology, real and complex analysis, geometry, you have little experience in the areas that are tested in PhD pure math programs. Still you could pick it up if you are very strong.

here are the qualifying requirements at UGA:
The PhD Qualifying Examination System consists of two parts. The first part consists of four Written Qualifying Exams and the second consists of an Oral Qualifying Exam.

Written Qualifying Exams are offered every year in August before the start of Fall semester classes and in January before the start of Spring semester classes. Study guides and copies of previous qualifying exams are available on the Graduate Program website for students to use in preparing for their Written Qualifying Exams.

The Written Qualifying Exams are divided into three groups:

Group 1: Complex Analysis, Real Analysis

Group 2: Algebra; Topology

Group 3: Probability; Numerical Analysis

Each PhD candidate is required to pass four Written Qualifying Exams, including both exams from Group 1 and at least one exam from Group 2. The exams in Group 1 are two hours long, and the other exams are three hours long. Each of the six introductory 8000-level courses (MATH 8000, 8100, 8150, 8200, 8500, and 8600, along with the associated 8xx5 problem session) is designed to help prepare students for the written qualifying exam in the corresponding subject area.

since you have to pass tests in these areas at the graduate level it is advised to have undergraduate courses in the areas you will choose, but very bright students can sometimes make up deficiencies in grad school. it is hard to do though- i myself tried and did not succeed at first.

 

[serkan]

thanks for the input mathwonk. would you say that those are the same criteria of applied mathematics phds?

 

[mathwonk]

i think so, in our dept. it looks as if our analysts voted in a block to force everyone to take both analysis prelims, and then the other pure groups voted to force everyone to take at least one of algebra or topology.

we let the applied people express this preference in their choice of exams from the third part of the syllabus, but apparently do not let them choose the two applied exams and no algebra or topology.

as usual, analysis is still strongly represented over algebra and geometry.

 

[serkan]

thanks a lot mathwork. it is great to ask a question and get the answer in hours. you have been very helpful.

 

[altcmdesc]

This is a question to any PhD holders out there: How was life directly following the PhD? The moving around from university to university, searching for and getting postdoc positions, the salary earned from those positions - what was your experience?

 

[mathwonk]

good question, and one I seem to have stopped short of answering in my general discussion. real world survival is very tough. with all the shortcomings the attractiveness of doing math research for a living is so appealing to many very bright people here and abroad that the job situation is often difficult. perhaps that will change as my generation of baby boomers retires beginning now and continuing for some time. but there are many emigres looking for these jobs too and they are very well trained.i myself had a very fine advisor with some contacts, and I had several offers of temporary jobs, including one at columbia. I myself generated another offer, the very tenure track offer i have tenure in now at UGA, and preferred it to the others because with a family, tenure track seemed very attractive.

however the shortcoming was there was no one else here in my field although at least one person was interested in it. having no one to learn from or work with, my future development was hindered. so i obtained an nsf grant for a regional conference headed by the famous Phillip Griffiths, and this brought a large number of outstanding people here for me to make contact with.

Professor Griffiths also said if i would come to Harvard to visit i could have some fun doing algebraic geometry with his team, so my university gave me leave to do this. i also met David Mumford and Heisuke Hironaka there, learned from all of them, and wound up staying 18 months.
thus i survived by doing things backwards, tenure track first, then postdoc.

others no doubt have different stories.

 

[mathwonk]

for more on my career path and related matters, go back and read posts 166 and 177-186.

 

[altcmdesc]

Hello again,

Now that I've started college, I've had some time to think about my Math career. Right now I'm absolutely loving my current math course (a theoretical treatment of multivariable calculus that uses linear algebra and differential forms) even though we haven't gotten very far. The rigor of the course is very stimulating.

I'm at the U of MN - Twin Cities campus right now. How is the U of MN PhD program in Pure math? Applied? Or does it really matter that much what university I obtain my PhD from?

 

[mathwonk]

Minnesota seems an excellent place, but it is usual to go somewhere else from your undergraduate school for a phd. please go see my friend Joel Roberts for more advice. tell him roy smith sent you.

 

[v.dinesh]

i want to become a good theorotical physicist how far pure maths is useful for that?

 

[mathwonk]

well Ed Witten, who seems to me a fine theoretical physicist is a fields medalist in math. so the two are certainly related. i have also myself been a guest lecturer in math at the International Center for Theoretical Physics in Trieste lecturing on riemann surfaces to physicists and mathematicians. as far as i know physicists are often interested in learning as much math as possible e.g. group representations, operator theory, differentiable manifolds, and riemann surfaces, for application to quantum mechanics, string theory, relativity,...

 

[muppet]

I want to do theoretical physics, so I decided to do half of my degree in maths-the best courses my universty offers on GR or quantum field theory are taught by the maths department. The reason Witten got a fields medal is because the maths he needed to use didn't exist... so he invented it

 

[eastside00_99]

Hey mathwonk. I am taking a year of complex analysis now. Its good stuff!

 

[mathwonk]

Great! are you enjoying a particular book you think others might like too? and would you like to give your prof a plug?

 

[eastside00_99]

Well, its only the first semester so we started out with Complex Analysis Lars V. Ahlfors, secondary sources are Theory of Functions by Knopp and Hyperbolic Geometry from a Local Viewpoint by my teacher Linda Keen.

 

[thrill3rnit3]

I'm just wondering, did I do the right thing in not skipping geometry? I had the choice of testing out of it, but I decided not to and instead I took it freshman year (high school). I should say I didn't really regret it, since my geometry teacher was like the only math teacher in our school who knows his stuff (plus probably the AP "Calc" teacher)

Now I'm a soph, and I'm on Alg. II/Trig Honors class (nevermind the honors label. It's not really "honors", if you know what I mean). I'm thinking about doing Precalculus over the summer (which would cost me - no, my parents - a painstaking 800 bucks), so that I can take 2 years of AP "Calc" (AB and BC) to add to my college application.

Am I doing the right thing??

 

[symbolipoint]

Skipping Geometry might or might not mean much now; in any case, you already studied it instead of skipping it and it probably helped you at least a little bit, certainly did not hurt you. PreCalculus in the summer might be rough going - not always enough time for some people.

If you are truly interested in Mathematics then you shoud definitely study Geometry. You may see how some things are obtained with Calculus while those same things can also be developed in Geometry without resorting to Calculus.

 

[thrill3rnit3]

I'm doing self-studying so I'll probably be almost if not finished with PreCalc by next summer. I'll take the course just to refreshen my memory, ask some questions, and because my school puts PreCalc as a prerequisite for AP Calc.

 

[mathwonk]

i do not know what your geometry course was like, but i seem to recall no one was allowed to enter plato's academy who was ignorant of geometry. the same should hold for college entrance today in my opinion. just today i have been reading archimedes, for more insight on his anticipation of basic facts now considered a part of integral calculus. euclid is also superb training.

 

[thrill3rnit3]

i do not know what your geometry course was like, but i seem to recall no one was allowed to enter plato's academy who was ignorant of geometry. the same should hold for college entrance today in my opinion. just today i have been reading archimedes, for more insight on his anticipation of basic facts now considered a part of integral calculus. euclid is also superb training.

my geometry teacher emphasized proofs and my dad told me that mastering geometry would be really helpful in the long run, I guess what he says is true...

 

[merjalaginven]

Pure mathematics is the way to the underground. I don't get it all but I do know that if you understand how you are doing it and WHY you are doing it- in every way!- then you are able to understand why everything is so- i mean everything. Pure math is what people do not see, it is the foundation. I want to see like them, not just do what they thought of. ahhh that is the beauty of mathematics. :) they see things others do not-

 

[CoCoA]

I want to be a mathematician (sort of), but I don't know if math would still want me. I am more than 15 years removed from undergrad, no major or minor in math/science/engineering. I have taken some math courses for last 3 years, and I am doing research with a Prof this year; I think I have a minor extension of a minor result. But to go into PhD, I would have to quit work (in my good earning years), get through exams (probably not a big deal), get an advisor (may be a big deal) and write a thesis (probably a big deal). Still, I am applying this year.

Unlike the young students here, I don't expect to solve a major problem - that is like picking the best apple from the top of the tree. But in just the little research I have done, I have started to see so many little apples lying on the ground ready to be picked up - like the little problem I am working on. I don't know, meybe this is because my work crosses over with CompSci, and maybe those problems are more accessible.

 

[mathwonk]

although very different, the last two messages seem more insightful than many. best wishes and good luck to you both.

 

[JasonRox]

Pure mathematics is the way to the underground. I don't get it all but I do know that if you understand how you are doing it and WHY you are doing it- in every way!- then you are able to understand why everything is so- i mean everything. Pure math is what people do not see, it is the foundation. I want to see like them, not just do what they thought of. ahhh that is the beauty of mathematics. :) they see things others do not-

I don't see what your point is, or how that is special.

There are many things others do SEE and mathematicians DO not.

There is no advantage to seeing one thing over another. It's all subjective.

 

[merjalaginven]

It was an opinion.
I did not say that mathematicians were the only people who see things differently. I just said some do.
Between applied and pure mathematics, which was the topic, a pure mathematician is most likely going to understand the concepts more in depth than the person who just uses a formula without questioning what you are really doing. I am not knocking applied math- I would much rather do that any day than proof writing!
I agree, everything is subjective based on our perceptions- that was just my opinion.

 

[Helical]

mathwonk, I was just wondering your opinion (should one exist) on Calculus by Hughes-Hallett et al (required text for my university). It seems to have poor ratings, though quite a few do. Should I get another book to learn from and just use this for problem sets? I find it strange that they would use a book that is so bad but the department at my university seems pretty good.

 

[The|M|onster]

Unlike the young students here, I don't expect to solve a major problem - that is like picking the best apple from the top of the tree. But in just the little research I have done, I have started to see so many little apples lying on the ground ready to be picked up.

Wow! I really like your apple analogy. It's very poetic.




And Helical, I used Hughes-Hallet for Calc I, II, and III and I absolutely hated it. Fortunately, I had some really good teachers. I recommend picking up another text to supplement your studies. Try browsing a used bookstore. You'd be surprised what kind of gems you can pick up if you look hard enough.

 

[JasonRox]

It was an opinion.
I did not say that mathematicians were the only people who see things differently. I just said some do.
Between applied and pure mathematics, which was the topic, a pure mathematician is most likely going to understand the concepts more in depth than the person who just uses a formula without questioning what you are really doing. I am not knocking applied math- I would much rather do that any day than proof writing!
I agree, everything is subjective based on our perceptions- that was just my opinion.

That's not true either. Where do you get this from?

A biologists will understand things mathematics will not. I said what I said in a general term. As in, don't try and feel superior or believe something is superior because one is seeing something others do not. There will always be something you don't see and someone else does.

My comment had nothing to do with applied vs. pure either.

 

[merjalaginven]

I did not knock anyone or what they think or how they think it-
I am not disputing an opinion that is mine alone with people that are rude.
I am a grad student and a new mom- Not a philosophy major- you twist my words around- I never said that a 'biologist' wouldn't see things that a mathematician would or whatever trivial example you want to say- all i was implying is that if one has a clearer sense of why you are doing something you are more likely to understand the outcome better.
I came here to read proofs and refresh- not to argue about deductive logic.
If you want to reply to my thoughts please refrain from things such as- well this makes no sense- or what is your point- this is rude. Otherwise feel free to say what you like- just be respectful bc I know I am not hurting anyone here by voicing my opinion-
oh and my comment did have something to do with applied vs pure mathematics- someone brought it up- which one was better to do- so sorry you are so worked over my comment!

 

[mathwonk]

i have not studied hughes hallet's book but she is not even a mathematician as far as i know, so why would anyone use a book by her?

 

[Vid]

It was an opinion.
I did not say that mathematicians were the only people who see things differently. I just said some do.
Between applied and pure mathematics, which was the topic, a pure mathematician is most likely going to understand the concepts more in depth than the person who just uses a formula without questioning what you are really doing. I am not knocking applied math- I would much rather do that any day than proof writing!
I agree, everything is subjective based on our perceptions- that was just my opinion.

What are you saying here? Are you saying that an applied mathematician uses formulas without questioning them because that is definitely not the case.

 

[merjalaginven]

I am not saying that at all- i am not saying ALL of ANYTHING/ANYONE thinks like anything! All i was saying is if you understand WHY you are doing something then you understand the entire concept more thoroughly- omg people get off my case- I was shoutin out to people that take interest in math/science- someone asked - which one should i do- i would do applied over pure but i am just saying i give respect to mathematicians in the past who figured all this stuff out so far- I NOT IMPLYING ANYTHING ELSE- the one you "" was in response to another person- I did not say an applied mathematician- i said a person who uses a formula which could be anyone- i am not knocking anyone.

 

[PowerIso]

I think some of you guys are reading a bit to much into Mer's post.

What she wrote is pretty much common sense. If you understand the root of a subject you will probably understand the subject a lot more. I'm not exactly sure how any of you guys got that she is knocking applied. I study statistics, which is in my opinion, applied math for applied math ;) and I wasn't offended or bother by her post. Relax and take it for what it is.