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for advice on preparing for grad school, from me and others, see my posts 11 and 12 in the thread "4th year undergrad", near this one.
Other Should I Become a Mathematician?
- Thread starter mathwonk
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Becoming a mathematician requires a deep passion for the subject and a commitment to problem-solving. Key areas of focus include algebra, topology, analysis, and geometry, with recommended readings from notable mathematicians to enhance understanding. Engaging with challenging problems and understanding proofs are essential for developing mathematical skills. A degree in pure mathematics is advised over a math/economics major for those pursuing applied mathematics, as the rigor of pure math prepares one for real-world applications. The journey involves continuous learning and adapting, with an emphasis on practical problem-solving skills.
- #152
JasonJo
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how are Summer REUs regarded for graduate admissions?
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They add something, especially if the summer reu guru says you are creative and powerful.
One of my friends (now a full prof at Brown) did one at amherst or williams and actually proved some theorems and got a big boost there. they are also taught by people who may be either refereeing or reviewing letters of grad school application.
One of my friends (now a full prof at Brown) did one at amherst or williams and actually proved some theorems and got a big boost there. they are also taught by people who may be either refereeing or reviewing letters of grad school application.
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tiny comment, possibly superfluous to todays youth: learn to be as computer literate as possible. for example learn to type, and learn to use TEX, and AMS TEX or LATEX.
All papers are written in TEX on computers now, usually by the author him/herself. (I even have students who refuse to read typed class notes that are not written in TEX.)
All NSF grants are submitted online. All courses have or should have webpages to support them, and even grades are submitted online.
And if you have trouble geting an academic job, there are many more openings for tech support people, and they are more essential, than are pure mathematicians.
if you want to be in the wave of the future of education, try to learn to use computers to teach effectively. i have my own doubts abut the vaue of this educationally, but it is inevitable, and can at least enhance regular classroom instruction.
if you have bad handwriting, it can at least render it readable to project your notes on the board. long calculations, like the antiderivatives of
1/[1 + x^20] become trivial work of fractions of a second.
this can help impress on students the folly of merely learning to do such calclations, without understanding the iDEAS.
All papers are written in TEX on computers now, usually by the author him/herself. (I even have students who refuse to read typed class notes that are not written in TEX.)
All NSF grants are submitted online. All courses have or should have webpages to support them, and even grades are submitted online.
And if you have trouble geting an academic job, there are many more openings for tech support people, and they are more essential, than are pure mathematicians.
if you want to be in the wave of the future of education, try to learn to use computers to teach effectively. i have my own doubts abut the vaue of this educationally, but it is inevitable, and can at least enhance regular classroom instruction.
if you have bad handwriting, it can at least render it readable to project your notes on the board. long calculations, like the antiderivatives of
1/[1 + x^20] become trivial work of fractions of a second.
this can help impress on students the folly of merely learning to do such calclations, without understanding the iDEAS.
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thought for the day: students, when learning a theorem, get in the habit of trying to think up a proof by yourself, before reading one. usually if you try hard, you will find on reading it that you have thought of at least the first few lines of the proof. this makes a huge difference in understanding it.
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here is another exercise: if k is any field, and c is any element of k, and p is a prime integer, prove that the equation X^p - c is either irreducible over k, or has a root in k.
hint: if it factors as g(X)h(X), with deg g = r and deg h = s, and the constant terms of g,h respectively are (1-)^r a, (-1)^s b, then show that a is a pth root of c^r and b is a pth root of c^s.
then use the fact that r,s are relatively prime to find a product of powers of them that is a pth root of c. hence X^ p -c has a root in k.
hint: if it factors as g(X)h(X), with deg g = r and deg h = s, and the constant terms of g,h respectively are (1-)^r a, (-1)^s b, then show that a is a pth root of c^r and b is a pth root of c^s.
then use the fact that r,s are relatively prime to find a product of powers of them that is a pth root of c. hence X^ p -c has a root in k.
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hint: if nr+ms = 1, then (c^r)^n . (c^s)^m = c.
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J77
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On the subject of writing papers...
In some of the papers that I've reviewed, even the titles are ungrammatical! That's not a good start...
Being able to write a good description of your work is more important than writing down a mass of equations.
Adding to this, make sure you have a good (not just decent) grasp of English.mathwonk said:
In some of the papers that I've reviewed, even the titles are ungrammatical! That's not a good start...
Being able to write a good description of your work is more important than writing down a mass of equations.
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how to get a phD; get into grad school, then pass prelims, then find a good helpful advisor, then start work as soon as possible on your thesis, [because it will take lots longer than you think it will], believe in your own intuition of what should be true and try to prove it, don't give up, because you WILL finish if you keep at it.
(secret: they really do want everyone to graduate: when they press you they are just trying to get you to extend yourself as much as possible: repeat they are NOT trying to flunk you out).
best of luck! as sylvanus p thompson put it: what one fool can do, another can.
(secret: they really do want everyone to graduate: when they press you they are just trying to get you to extend yourself as much as possible: repeat they are NOT trying to flunk you out).
best of luck! as sylvanus p thompson put it: what one fool can do, another can.
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I do not know how to advise people on how to write a thesis, as I have never had a PhD student complete a thesis under me.
I am not sure why this is, but suspect it is because I was not supportive enough. When I was a student, thesis advisors sort of waited for us to produce a result, then said whether it was enough or not.
I was not too good at this and needed more help, so eventually found an advisor who proposed a specific problem and also an approach to it and then even suggested a conjectural answer and I found the solution proving his guess correct.
Along the way I needed courage and confidence however, as at one point my advisor announced that a famous mathematician had become interested in my problem. He seemed to feel that this was the kiss of death, but I cheekily responded that was fine, when i solved it I would inform the famous man of the answer. This actually occurred fortunately for me.
[I solved three problems before finding a new one. The first had already been done by Hurwitz in the 19th century and the second by deligne in the 1960s. Finally the third made progress on a problem left open by Wirtinger in 1895.]
This solution of mine was actually pretty interesting and led to some significant further work in the area by experts who extended it a lot.
Even this fairly minimal contribution is more than many students produce today, and advisors are expected apparently to essentially outline and design the thesis for them.
I.e. thesis in math is supposed to be new, interesting, non trivial, discovery, and verification of substantive results.
In many cases it consists of reproving more clearly or simply a known result, or clarifying an old solution from ancient times of an interesting tresult, or generalizing a good result to a slightly broader setting.
In mathematics, a thesis is not at all merely the recitation of the results of some experiments, whether they succeeded or not. Failed experiments are a failed thesis in math, they do not count at all, they only give the experience needed to try again more successfully.
In my thesis I partially solved a problem attempted unsuccessfully by some famous mathematicians, and discovered in the process a method that was useful in other settings, and which I used for years afterwards on other questions.
In writing a thesis I can suggest that one must take advantage of everything one has learned or heard, that one must step out on faith and believe in ones intuition, and then work very hard to substantiate the results of ones imagination.
It often takes great stamina, persistence, and help from more knowledgeable people, as well as some luck, to achieve something new and interesting.
But just as in other settings, even if one does not achieve the maximum result hoped for, one can still anticipate graduating. As stated (perhaps by Robin Hartshorne) to a friend of mine, the goal of a thesis is to be the first creative work of ones life, not the last.
you CAN do more than you think, and what you can do is enough.
I am not sure why this is, but suspect it is because I was not supportive enough. When I was a student, thesis advisors sort of waited for us to produce a result, then said whether it was enough or not.
I was not too good at this and needed more help, so eventually found an advisor who proposed a specific problem and also an approach to it and then even suggested a conjectural answer and I found the solution proving his guess correct.
Along the way I needed courage and confidence however, as at one point my advisor announced that a famous mathematician had become interested in my problem. He seemed to feel that this was the kiss of death, but I cheekily responded that was fine, when i solved it I would inform the famous man of the answer. This actually occurred fortunately for me.
[I solved three problems before finding a new one. The first had already been done by Hurwitz in the 19th century and the second by deligne in the 1960s. Finally the third made progress on a problem left open by Wirtinger in 1895.]
This solution of mine was actually pretty interesting and led to some significant further work in the area by experts who extended it a lot.
Even this fairly minimal contribution is more than many students produce today, and advisors are expected apparently to essentially outline and design the thesis for them.
I.e. thesis in math is supposed to be new, interesting, non trivial, discovery, and verification of substantive results.
In many cases it consists of reproving more clearly or simply a known result, or clarifying an old solution from ancient times of an interesting tresult, or generalizing a good result to a slightly broader setting.
In mathematics, a thesis is not at all merely the recitation of the results of some experiments, whether they succeeded or not. Failed experiments are a failed thesis in math, they do not count at all, they only give the experience needed to try again more successfully.
In my thesis I partially solved a problem attempted unsuccessfully by some famous mathematicians, and discovered in the process a method that was useful in other settings, and which I used for years afterwards on other questions.
In writing a thesis I can suggest that one must take advantage of everything one has learned or heard, that one must step out on faith and believe in ones intuition, and then work very hard to substantiate the results of ones imagination.
It often takes great stamina, persistence, and help from more knowledgeable people, as well as some luck, to achieve something new and interesting.
But just as in other settings, even if one does not achieve the maximum result hoped for, one can still anticipate graduating. As stated (perhaps by Robin Hartshorne) to a friend of mine, the goal of a thesis is to be the first creative work of ones life, not the last.
you CAN do more than you think, and what you can do is enough.
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although i have never advised a phd student, i can say what has led to my own best work: namely to read and familiarize yourself with the work of excellent people, and try to understand it as well as possible. speak on it, give a seminar on it, and it will seep into your pores and illuminate you and lead you to something further.
if things are slow, give a seminar on a paper by someone you admire. never stop working, as chern told me, maybe rest for a day or two, then go back to work.
do not be content just learning like a student, as Bill Fulton said to a friend of mine, but try to reprove significant results or extend them. at some point you will find you are going beyond what is known and on your way.
if things are slow, give a seminar on a paper by someone you admire. never stop working, as chern told me, maybe rest for a day or two, then go back to work.
do not be content just learning like a student, as Bill Fulton said to a friend of mine, but try to reprove significant results or extend them. at some point you will find you are going beyond what is known and on your way.
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as students we are often dependent on the simplest explanations to get on the train, but we should always aspire and try to reach the level exemplified by the masters, so do as abel said: read the masters, or prepare until one can do so.
in algebra this means to get to the point where one can read artin, van der waerden, lang, sah, jacobson. do not stop with dummitt and foote, or hungerford, rotman, herstein, or other second level texts, but do use those to get to the point one desires to reach.
(Edit: Actually of course one wants to be able to read original papers.)
in algebra this means to get to the point where one can read artin, van der waerden, lang, sah, jacobson. do not stop with dummitt and foote, or hungerford, rotman, herstein, or other second level texts, but do use those to get to the point one desires to reach.
(Edit: Actually of course one wants to be able to read original papers.)
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courtrigrad
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Do most universities have a time limit for which a student can complete a PhD (like 10 years)? Wouldn't professors want graduate students to stay and work because they are cheap labor?
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well that depends. i heard in the old days it was 3 years at princeton but i think it is longer now. we keep students at UGA much longer.
I was put on notice at Utah to finish or leave after three years, but i entered with a masters.
every place is different so check around. yes grad students are cheap labor for teaching but departments want students to produce research and get on with their lives as scholars.
the cheap labor is of interest not to professors but to administrations. we are not paying the salaries, so we we want talented, they want cheap.
I was put on notice at Utah to finish or leave after three years, but i entered with a masters.
every place is different so check around. yes grad students are cheap labor for teaching but departments want students to produce research and get on with their lives as scholars.
the cheap labor is of interest not to professors but to administrations. we are not paying the salaries, so we we want talented, they want cheap.
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Mickey
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matt grime said:
Aw, we're better than that.
I mean, we got your fancy "Laplace method," raising our pinkies to a "Lagrange" multiplier, and we clean up our denominators with "bordered Hessian matrices" just like the upper class.
But we can drop our constraints and take this outside if you want. We don't need borders on our Hessians. We know what sign our principal minors have!
Well, sometimes. -1 raised to the 3 is... one, two..
You're right. I don't know how to add. :shy:
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when i was a grad student there were very few jobs available so the professors tried to weed out students who seemed less likely to write strong theses, and to push those who could, to get the maximum results feasible. This meant few students graduating. I was lucky and upon graduation applied to about 10 places and got about 4 jobs, 2 firm offers and 2 more possibles that I turned down. My advisors connections opened most of them but I chose the one I generated myself.
He wanted me to go to a place with strong established workers in my field where I could get support and stimulation, but I wanted to go somewhere I thought I would have a stable job. He was right of course, as I found myself isolated as the only person in my area, and had to struggle hard even to survive.
My solution was to take leave at some top places where there were outstanding experts and get the stimulation I needed. This involved significant financial sacrifice on postdoc wages not meant to support a family.
The situation is different today. Instead of applying for 10 jobs it seems many students apply for scores or hundreds of them by internet. I am not sure there is much sense in this. Probably it is better to do as I did and apply places where your advisor has a connection and they will listen to his opinion of you, plus generate a few applications on your own for your own reasons.
One good thing for todays applicants is the coming retirement of baby boomers. In a few years there will be a huge number of retirements from my generation and those somewhat younger. This will leave a large void of jobs needing to be filled. This will not guarantee jobs however for US students, as there may be an influx of foreign Aapplicants for these jobs. In recent history the absence of US PhDs in math has been taken up by applicants from China, Britain, Russia, India, and other places.
But these applicants are having a harder time entering the US in the current political climate. In any case there will be more jobs soon. Also the salaries in some other countries are actually beginning to exceed those here and drawing some of those applicants back to their home countries, lessening competition slightly in the US.
Of course no one has a crystal ball. The current and recent past governments have squandered the money set aside by law for the upcoming retirees and so in fact there is not sufficient money to pay for our retirement. This means many of us will not be able to retire after all, and will try to keep working, or will be forced to do so.
There is also a move to reduce the number of well paid and well supported faculty at many colleges and replace them with temporary positions staffed by people who receive no health care benefits and who teach too many courses to be effective.
It is hard to know the direction these things will take in future. Much depends on who is elected to the offices of president and to congress. Probably a lot also depends eventually on how well teachers and mathematicians do their job of explaining what they do to more people. Politicians who cannot fathom mathematics or its power may be unlikely to vote money to support the study of it.
It is always prudent to be open to interactions with people in other areas. Mathematicians who can talk profitably to physicists, biologists, and educators, and who can use computers effectively, are unlikely to be without work.
He wanted me to go to a place with strong established workers in my field where I could get support and stimulation, but I wanted to go somewhere I thought I would have a stable job. He was right of course, as I found myself isolated as the only person in my area, and had to struggle hard even to survive.
My solution was to take leave at some top places where there were outstanding experts and get the stimulation I needed. This involved significant financial sacrifice on postdoc wages not meant to support a family.
The situation is different today. Instead of applying for 10 jobs it seems many students apply for scores or hundreds of them by internet. I am not sure there is much sense in this. Probably it is better to do as I did and apply places where your advisor has a connection and they will listen to his opinion of you, plus generate a few applications on your own for your own reasons.
One good thing for todays applicants is the coming retirement of baby boomers. In a few years there will be a huge number of retirements from my generation and those somewhat younger. This will leave a large void of jobs needing to be filled. This will not guarantee jobs however for US students, as there may be an influx of foreign Aapplicants for these jobs. In recent history the absence of US PhDs in math has been taken up by applicants from China, Britain, Russia, India, and other places.
But these applicants are having a harder time entering the US in the current political climate. In any case there will be more jobs soon. Also the salaries in some other countries are actually beginning to exceed those here and drawing some of those applicants back to their home countries, lessening competition slightly in the US.
Of course no one has a crystal ball. The current and recent past governments have squandered the money set aside by law for the upcoming retirees and so in fact there is not sufficient money to pay for our retirement. This means many of us will not be able to retire after all, and will try to keep working, or will be forced to do so.
There is also a move to reduce the number of well paid and well supported faculty at many colleges and replace them with temporary positions staffed by people who receive no health care benefits and who teach too many courses to be effective.
It is hard to know the direction these things will take in future. Much depends on who is elected to the offices of president and to congress. Probably a lot also depends eventually on how well teachers and mathematicians do their job of explaining what they do to more people. Politicians who cannot fathom mathematics or its power may be unlikely to vote money to support the study of it.
It is always prudent to be open to interactions with people in other areas. Mathematicians who can talk profitably to physicists, biologists, and educators, and who can use computers effectively, are unlikely to be without work.
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I apologize for writing the last few posts at night over a [make that several] glass[es] of wine.
Is this thread dead? or are there topics we need to cover for you?
would you like more exercises? more job related advice?: more data on the situation in Britain, or Belgium or Austria?
after all it might be a good idea for more people to go to schools there [assuming americans get some language skills!]
anyway, thanks for the participation and even if the thread is moribund to you I may be tempted to enter more posts.
i am about to start teaching grad algebra for prelims, so any interested parties may want to ramp up a prelim topic practice segment.
Is this thread dead? or are there topics we need to cover for you?
would you like more exercises? more job related advice?: more data on the situation in Britain, or Belgium or Austria?
after all it might be a good idea for more people to go to schools there [assuming americans get some language skills!]
anyway, thanks for the participation and even if the thread is moribund to you I may be tempted to enter more posts.
i am about to start teaching grad algebra for prelims, so any interested parties may want to ramp up a prelim topic practice segment.
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courtrigrad
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how is Columbia's math department? Don't hear much about it compared to other colleges.
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gee it is terrific in my opinion, but i have a lot of friends there so i may be biased, but i don't think so.
bob friedman, john morgan, henry pinkham, dave bayer, brian conrad, joan birman, bill fulton, johan de jong, herve jacquet, igor krichever, m. kuranishi, michael thaddeus.oh my word, the riches at these places.
if you have a chance to go there you will never forget it.
bob friedman, john morgan, henry pinkham, dave bayer, brian conrad, joan birman, bill fulton, johan de jong, herve jacquet, igor krichever, m. kuranishi, michael thaddeus.oh my word, the riches at these places.
if you have a chance to go there you will never forget it.
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courtrigrad
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thanks
yeah this is the program I am think of: http://www.apam.columbia.edu/research/am.htm (the applied math department)
yeah this is the program I am think of: http://www.apam.columbia.edu/research/am.htm (the applied math department)
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J77
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I don't know anyone there but I like the way they promote nonlinear dynamics.
Like mathwonk said, it's good to work with physicists, biolodists, chemists, transport people, computer scientests, geologists... the list goes on and on, and there's no better specilisation to have - if you want to collaborate with other fields - than nonlinear dynamics!
Like mathwonk said, it's good to work with physicists, biolodists, chemists, transport people, computer scientests, geologists... the list goes on and on, and there's no better specilisation to have - if you want to collaborate with other fields - than nonlinear dynamics!
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Dragonfall
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I have the opportunity to take some physics courses and possibly minoring in physics (with a major in pure math) or more math courses, possibly some grad courses at the third (final) year. Which do you suggest? My career is being steered towards 'pure' math research but I do have some interest in physics.
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neurocomp2003
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mathwonk: was this thread intended for a student to become a pure mathematician? I take it you don't consider CS apart of mathematics?
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sorry. just my incompetence. my son majored in math and numerical methods and works in the internet world. (Id tell you what he does but i can't understand him when he tells me.) all such info is welcome. please share any insights or suggestions for people going into these areas. and thanks for the reminder.
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to continue that last thought, up to now people have mostly indulged me in my narrow assumption that a mathematician is a university professor of pure mathematics with a PhD who does abstract research in algebra geometry or analysis, maybe even in the US.
Lets hear from others who think of themselves as mathematicians, or of what they do as mathematics, and get a wider view of the mathematical world, its options and participants.
So, new definition of mathematician: if you think you might be one, then you are.
some feeble attempts at humor have been removed.
Lets hear from others who think of themselves as mathematicians, or of what they do as mathematics, and get a wider view of the mathematical world, its options and participants.
So, new definition of mathematician: if you think you might be one, then you are.
some feeble attempts at humor have been removed.
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i see i ignored a question asking what the grad school experience is like, passing prelims, writing a thesis, etc...
so here goes a little on that. of course everyone enters grad school, in US anyway, with a different background. Since we have a shortage of PhD mathematicians in US, we are always scouting for talent and recruiting people to our programs. So many people get in who are less than ideally prepared.
Thus the beginning of the grad experience can be rough for the less prepared. At UGA we have recently begun a testing and placement program, and have courses designed to help people get up to speed on some crucial undergrad material they may not have been fully taught before. This is new, as it used to be more or less sink or swim.
Different schools are different on this matter of sink or swim, and it would be wise to find out whether your school just brings in and watches to see which ones survive, ot whether they try to help everyone make it. I suspect today most try to be helpful, but that may be less true where the school is very popular and the professors are very busy.
The most fortunate people are those who know all the basic stuff and are ready to begin work toward a thesis right away, the primary reason for being there.
At the other extreme, I entered not knowing what an ideal was and was immediately plunged into an algebra course on homological ring theory. I had also never had complex variable (in undergrad they said, "oh you'll learn that in grad school") and began in an analysis course that spent one month on reals and one on complex and moved on to Riemann surfaces!
So you need to come in knowing as much as possible, and also choose a school where the introduction is somewhat sensitive to what people know. (Brandeis was still a great place to be, and has no doubt also changed totally since then.)
So you must talk to people currently at the schools of interest, students as well as professors, to find out the department's expectations, and how those are viewed by students.
the first thing then is to get up to speed as quickly as possible. Writing a thesis will take much longer and be much harder usually than you could have imagined, so you need to get ready to do it and sart doing it as soon as possible.
so since for many students the first big hurdle is the prelims, i will post here the current prelim syllabi from UGA in a few subjects. The requirements may vary, and are constantly changing, but a pure mathematics aspirant should hope to be able to pass prelims in all 3 pure subjects, say topology, algebra, and at least one type of analysis, real or complex.
We have gradually made these syllabi less and less demanding over the years, continually removing material, to where they will probably read like undergraduate syllabi to students from abroad (or elite US schools) now.
Notice for example the algebra syllabus no longer covers noetherian rings and modules, nor tensor products, and the complex syllabus no longer covers elliptic functions or riemann mapping theorem. Still it is quite challenging for an average undergrad from the US to master all this in a short amount of time, i.e. a year or so of grad school.
Be sure to get these syllabi from your target school, as they may be very different at different places.
so here goes a little on that. of course everyone enters grad school, in US anyway, with a different background. Since we have a shortage of PhD mathematicians in US, we are always scouting for talent and recruiting people to our programs. So many people get in who are less than ideally prepared.
Thus the beginning of the grad experience can be rough for the less prepared. At UGA we have recently begun a testing and placement program, and have courses designed to help people get up to speed on some crucial undergrad material they may not have been fully taught before. This is new, as it used to be more or less sink or swim.
Different schools are different on this matter of sink or swim, and it would be wise to find out whether your school just brings in and watches to see which ones survive, ot whether they try to help everyone make it. I suspect today most try to be helpful, but that may be less true where the school is very popular and the professors are very busy.
The most fortunate people are those who know all the basic stuff and are ready to begin work toward a thesis right away, the primary reason for being there.
At the other extreme, I entered not knowing what an ideal was and was immediately plunged into an algebra course on homological ring theory. I had also never had complex variable (in undergrad they said, "oh you'll learn that in grad school") and began in an analysis course that spent one month on reals and one on complex and moved on to Riemann surfaces!
So you need to come in knowing as much as possible, and also choose a school where the introduction is somewhat sensitive to what people know. (Brandeis was still a great place to be, and has no doubt also changed totally since then.)
So you must talk to people currently at the schools of interest, students as well as professors, to find out the department's expectations, and how those are viewed by students.
the first thing then is to get up to speed as quickly as possible. Writing a thesis will take much longer and be much harder usually than you could have imagined, so you need to get ready to do it and sart doing it as soon as possible.
so since for many students the first big hurdle is the prelims, i will post here the current prelim syllabi from UGA in a few subjects. The requirements may vary, and are constantly changing, but a pure mathematics aspirant should hope to be able to pass prelims in all 3 pure subjects, say topology, algebra, and at least one type of analysis, real or complex.
We have gradually made these syllabi less and less demanding over the years, continually removing material, to where they will probably read like undergraduate syllabi to students from abroad (or elite US schools) now.
Notice for example the algebra syllabus no longer covers noetherian rings and modules, nor tensor products, and the complex syllabus no longer covers elliptic functions or riemann mapping theorem. Still it is quite challenging for an average undergrad from the US to master all this in a short amount of time, i.e. a year or so of grad school.
Be sure to get these syllabi from your target school, as they may be very different at different places.
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We have recently upgraded the syllabus by calling half the material "undergraduate" material and trying to offer it prior to the grad course. We still need to make this work in practice, as this is new. Thus we spend only one semester in grad school teaching the alg prep course now, as opposed to the old days when it was a year. This is one justification for deleting some material from the syllabus.
Study Guide for Algebra Qualifying Exam
Proposed: April 2006
UNDERGRADUATE MATERIAL
Group Theory: MATH 6010
subgroups
quotient groups
Lagrange's Theorem
fundamental homomorphism theorems
group actions with applications to the structure of groups such as
the Sylow Theorems
group constructions such as:
direct products
structures of special types of groups such as:
p-groups
dihedral, symmetric and alternating groups, cycle decompositions
the simplicity of An, for n ≥ 5
References: [1,3,5].
Linear Algebra: MATH 6050
determinants
eigenvalues and eigenvectors
Cayley-Hamilton Theorem
canonical forms for matrices
linear groups (GLn , SLn, On, Un)
dual spaces, dual bases, pull back, double duals
finite-dimensional spectral theorem
References: [1,2,5]GRADUATE MATERIAL (MATH 8000)
Foundations:
Zorn's Lemma and its uses in various existence theorems such as that of a basis for a vector space or existence of maximal ideals.
References: [1,4]
Group Theory:
Sylow Theorems
free groups, generators and relations
semi-direct products
solvable groups
References: [1,3,5].
Theory of Rings and Modules:
basic properties of ideals and quotient rings
fundamental homomorphism theorems for rings and modules
characterizations and properties of special domains such as:
Euclidean implies PID implies UFD
classification of finitely generated modules over Euclidean domains
applications to the structure of:
finitely generated abelian groups and
canonical forms of matrices
References: [1,3,4,5].
Field Theory:
algebraic extensions of fields
fundamental theorem of Galois theory
properties of finite fields
separable extensions
computations of Galois groups of polynomials of small degree and cyclotomic polynomials
solvability of polynomials by radicals
References: [1,3,5]As a general rule, students are responsible for knowing both the theory (proofs) and practical applications (e.g. how to find the Jordan or rational canonical form of a given matrix, or the Galois group of a given polynomial) of the topics mentioned.References [Need to be updated; e.g. [3] and [4] are out of print.]
[1] Thomas W. Hungerford, Algebra, Springer, New York, 1974.
[2] Kenneth Hoffman and Ray Kunze, Linear Algebra, Prentice-Hall, 1961.
[3] Nathan Jacobson, Basic Algebra 1, W.H. Freeman, San Francisco, 1974.
[4] Nathan Jacobson, Basic Algebra 2, W. H. Freeman, San Francisco, 1980.
[5] Serge Lang, Algebra, Addison Wesley, Reading Mass., 1970
Study Guide for Algebra Qualifying Exam
Proposed: April 2006
UNDERGRADUATE MATERIAL
Group Theory: MATH 6010
subgroups
quotient groups
Lagrange's Theorem
fundamental homomorphism theorems
group actions with applications to the structure of groups such as
the Sylow Theorems
group constructions such as:
direct products
structures of special types of groups such as:
p-groups
dihedral, symmetric and alternating groups, cycle decompositions
the simplicity of An, for n ≥ 5
References: [1,3,5].
Linear Algebra: MATH 6050
determinants
eigenvalues and eigenvectors
Cayley-Hamilton Theorem
canonical forms for matrices
linear groups (GLn , SLn, On, Un)
dual spaces, dual bases, pull back, double duals
finite-dimensional spectral theorem
References: [1,2,5]GRADUATE MATERIAL (MATH 8000)
Foundations:
Zorn's Lemma and its uses in various existence theorems such as that of a basis for a vector space or existence of maximal ideals.
References: [1,4]
Group Theory:
Sylow Theorems
free groups, generators and relations
semi-direct products
solvable groups
References: [1,3,5].
Theory of Rings and Modules:
basic properties of ideals and quotient rings
fundamental homomorphism theorems for rings and modules
characterizations and properties of special domains such as:
Euclidean implies PID implies UFD
classification of finitely generated modules over Euclidean domains
applications to the structure of:
finitely generated abelian groups and
canonical forms of matrices
References: [1,3,4,5].
Field Theory:
algebraic extensions of fields
fundamental theorem of Galois theory
properties of finite fields
separable extensions
computations of Galois groups of polynomials of small degree and cyclotomic polynomials
solvability of polynomials by radicals
References: [1,3,5]As a general rule, students are responsible for knowing both the theory (proofs) and practical applications (e.g. how to find the Jordan or rational canonical form of a given matrix, or the Galois group of a given polynomial) of the topics mentioned.References [Need to be updated; e.g. [3] and [4] are out of print.]
[1] Thomas W. Hungerford, Algebra, Springer, New York, 1974.
[2] Kenneth Hoffman and Ray Kunze, Linear Algebra, Prentice-Hall, 1961.
[3] Nathan Jacobson, Basic Algebra 1, W.H. Freeman, San Francisco, 1974.
[4] Nathan Jacobson, Basic Algebra 2, W. H. Freeman, San Francisco, 1980.
[5] Serge Lang, Algebra, Addison Wesley, Reading Mass., 1970
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Here is the analysis syllabus, recently divided into reals and complex.Study Guide for Real Analysis Exam
I. Calculus and Undergraduate Analysis, Continuity and differentiation in one and several variables, Compactness and connectedness in analysis , Sequences and series, Uniform convergence and uniform continuity
Taylor's Theorem, Riemann integrals References: [2]
II. Measure and Integration, Measurability:
*** Measures in Rn and on ơ-algebras *** Borel and Lebesgue measures *** Measurable functions Integrability: *** Integrable functions *** Convergence theorems (Fatou’s lemma, monotone* and dominated *** convergence theorems) *** Characterization of Riemann integrable functions, Fubini and Tonelli theorems
Lebesgue differentiation theorem and Lebesgue sets
References: [1] Chapter 1, 2, 3.
[3] Chapter 3, 4, 5, 11, 12.
[4] Chapter 1, 2, 3, 6.
III. Lp and Hilbert Spaces
Lp space: Holder and Minkowski inequalities, completeness, and the dual of Lp
Hilbert space and L2 spaces: orthonormal basis, Bessel’s inequality, Parseval’s identity,
Linear functionals and the Riesz representation theorem.
References: [1] Section 5.5, Chapter 6.
[3] Chapter 6.
[4] Chapter 4.
[1]*G. Folland, Real Analysis, 2nd edition, John Wiley & Sons, Inc.
[2] W. Rudin, Principle of Mathematical Analysis, 3rd edition
[3]* H. Royden, Real Analysis, 3rd edition
[4] E. M. Stein and R. Shakarchi, Real Analysis, Princeton University Press. Study Guide for Complex Analysis Exam
I. Calculus and Undergraduate Analysis Continuity and differentiation in one and several real variables
Inverse and implicit function theorems Compactness and connectedness in analysis Uniform convergence and uniform continuity Riemann integrals
Contour integrals and Green’s theorem References: [3].
II. Preliminary Topics in Complex Analysis
Complex arithmetic
Analyticity, harmonic functions, and the Cauchy-Riemann equations
Contour Integration in C
References: [1] Chapter 1, 2;
[2] Chapter 1, 2, 4;
[4] Chapter 1.
III. Cauchy's Theorem and its consequences
Cauchy's theorem and integral formula, Morera’s theorem
Uniform convergence of analytic functions Taylor and Laurent expansions
Maximum modulus principle and Schwarz’s lemma Liouville's theorem and the Fundamental theorem of algebra
Residue theorem and applications
Singularities and meromorphic functions, including the Casorati-Weierstrass theorem Rouche’s theorem, the argument principle, and the open mapping theorem
References: [1] Chapter 4, 5, 6;
[2] Chapter 5, 7, 8, 9;
[4] Chapter 2, 3, 5.
IV. Conformal Mapping, General properties of conformal mappings , Analytic and mapping properties of linear fractional transformations References:* [1] Chapter 3, 8; [2] Chapter 3, 4; [4] Chapter 8.
References [1]* L. Ahlfors, Complex Analysis, Third Edition, McGraw-Hill
[2]* E. Hille, Analytic Function Theory, Vols. 1, Ginn and Company.
[3]* W. Rudin, Principle of Mathematical Analysis, Third Edition.
[4] E. M. Stein and R. Shakarchi, Complex Analysis, Princeton University Press.PAGE 1
I. Calculus and Undergraduate Analysis, Continuity and differentiation in one and several variables, Compactness and connectedness in analysis , Sequences and series, Uniform convergence and uniform continuity
Taylor's Theorem, Riemann integrals References: [2]
II. Measure and Integration, Measurability:
*** Measures in Rn and on ơ-algebras *** Borel and Lebesgue measures *** Measurable functions Integrability: *** Integrable functions *** Convergence theorems (Fatou’s lemma, monotone* and dominated *** convergence theorems) *** Characterization of Riemann integrable functions, Fubini and Tonelli theorems
Lebesgue differentiation theorem and Lebesgue sets
References: [1] Chapter 1, 2, 3.
[3] Chapter 3, 4, 5, 11, 12.
[4] Chapter 1, 2, 3, 6.
III. Lp and Hilbert Spaces
Lp space: Holder and Minkowski inequalities, completeness, and the dual of Lp
Hilbert space and L2 spaces: orthonormal basis, Bessel’s inequality, Parseval’s identity,
Linear functionals and the Riesz representation theorem.
References: [1] Section 5.5, Chapter 6.
[3] Chapter 6.
[4] Chapter 4.
[1]*G. Folland, Real Analysis, 2nd edition, John Wiley & Sons, Inc.
[2] W. Rudin, Principle of Mathematical Analysis, 3rd edition
[3]* H. Royden, Real Analysis, 3rd edition
[4] E. M. Stein and R. Shakarchi, Real Analysis, Princeton University Press. Study Guide for Complex Analysis Exam
I. Calculus and Undergraduate Analysis Continuity and differentiation in one and several real variables
Inverse and implicit function theorems Compactness and connectedness in analysis Uniform convergence and uniform continuity Riemann integrals
Contour integrals and Green’s theorem References: [3].
II. Preliminary Topics in Complex Analysis
Complex arithmetic
Analyticity, harmonic functions, and the Cauchy-Riemann equations
Contour Integration in C
References: [1] Chapter 1, 2;
[2] Chapter 1, 2, 4;
[4] Chapter 1.
III. Cauchy's Theorem and its consequences
Cauchy's theorem and integral formula, Morera’s theorem
Uniform convergence of analytic functions Taylor and Laurent expansions
Maximum modulus principle and Schwarz’s lemma Liouville's theorem and the Fundamental theorem of algebra
Residue theorem and applications
Singularities and meromorphic functions, including the Casorati-Weierstrass theorem Rouche’s theorem, the argument principle, and the open mapping theorem
References: [1] Chapter 4, 5, 6;
[2] Chapter 5, 7, 8, 9;
[4] Chapter 2, 3, 5.
IV. Conformal Mapping, General properties of conformal mappings , Analytic and mapping properties of linear fractional transformations References:* [1] Chapter 3, 8; [2] Chapter 3, 4; [4] Chapter 8.
References [1]* L. Ahlfors, Complex Analysis, Third Edition, McGraw-Hill
[2]* E. Hille, Analytic Function Theory, Vols. 1, Ginn and Company.
[3]* W. Rudin, Principle of Mathematical Analysis, Third Edition.
[4] E. M. Stein and R. Shakarchi, Complex Analysis, Princeton University Press.PAGE 1
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Study Guide for Topology Exams
General Topology
topological spaces and continuous functions
product and quotient topology
connectedness and compactness
Urysohn lemma
complete metric spaces and function spaces
References: [2]
Algebraic Topology
fundamental group
van Kampen's theorem
classifications of surfaces
classifications of covering spaces
homology:
simplicial, singular and cellular: computations and applications
degree of maps
Euler characteristics
Lefschetz fixed point theorem
References: [1,3]
The weight of topics on the exam should be about 1/3 general topology and 2/3 algebraic topology.
References
[1] W. Massey, Algebraic Topology: An Introduction, Springer Verlag, 1977.
[2] J. Munkres, Topology, A First Course, Prentice-Hall, 1975.
[3] J. Munkres, Elements of Algebraic Topology, Addison-Wesley, 1984.
General Topology
topological spaces and continuous functions
product and quotient topology
connectedness and compactness
Urysohn lemma
complete metric spaces and function spaces
References: [2]
Algebraic Topology
fundamental group
van Kampen's theorem
classifications of surfaces
classifications of covering spaces
homology:
simplicial, singular and cellular: computations and applications
degree of maps
Euler characteristics
Lefschetz fixed point theorem
References: [1,3]
The weight of topics on the exam should be about 1/3 general topology and 2/3 algebraic topology.
References
[1] W. Massey, Algebraic Topology: An Introduction, Springer Verlag, 1977.
[2] J. Munkres, Topology, A First Course, Prentice-Hall, 1975.
[3] J. Munkres, Elements of Algebraic Topology, Addison-Wesley, 1984.
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compare this modest syllabus with that at harvard, where one needs to know all 6 of the following areas, and the exam has one question on each. note their undergraduate algebra syllabus covers more than our graduate and undergradiate algebra syllabi combined.
Harvard:
The syllabus is divided into 6 areas. In each case we suggest (sections of) a book to more carefully define the syllabus. The examiners are asked to limit their questions to major topics covered in (these sections of) these books. We have tried to choose books we think are good. However there are many good books and others might better suit your needs. In each case we divide the syllabus into two sections. Section U is material which are usually covered in our undergraduate, not our graduate, courses. Section G is material usually taught at the graduate level. Where appropriate we list courses which will cover some of this material.
1) Algebra.
U: Dummit+Foote, Abstract Algebra, except chapters 16 and 17. (math 122, 123, 126)
G: Dummit+Foote, Abstract Algebra, chapter 17.
2) Algebraic Geometry
G: Harris, Algebraic geometry, a first course, lectures 1-7, 11, 13, 14, 18.
3) Complex Analysis
(Table of contents)
U: Ahlfors, Complex Analysis (2nd ed), chapters 1-4 and section 5.1. (math 113)
G: Ahlfors, Complex Analysis (2nd ed), section 5.4.
4) Algebraic Topology
U: Hatcher, Algebraic Topology, chapter 1 (but not the additional topics). (math 131)
G: Hatcher, Algebraic Topology, chapter 2 (including additional topics) and chapter 3 (without additional topics). (math 272a)
5) Differential Geometry
(Table of contents)
U: Boothby, An introduction to differentiable manifolds and Riemannian geometry, sections VII.1 , VIII.1 and VIII.2. (math 136)
G: Boothby, An introduction to differentiable manifolds and Riemannian geometry, chapters I - V and VII. (math 134, 135 and 230a)
6) Real Analysis
(Rudin: Table of contents)
(Birkhoff+Rota: Table of contents)
U: Rudin, Principles of mathematical analysis, chapters 1-8.
Birkhoff + Rota, Ordinary differential equations, chapters 1-4 and 6. (math 25, 55, 112)
G: Rudin, Principles of mathematical analysis, chapter 10.
Rudin, Functional analysis, chapters 1, 2, 3.1-3.14, 4, 6, 7.1-7.19 and 12.1-12.15. (math 212a)as usual it seems to be the real analysts who cannot bring themselves to shorten the syllabus.
Harvard:
The syllabus is divided into 6 areas. In each case we suggest (sections of) a book to more carefully define the syllabus. The examiners are asked to limit their questions to major topics covered in (these sections of) these books. We have tried to choose books we think are good. However there are many good books and others might better suit your needs. In each case we divide the syllabus into two sections. Section U is material which are usually covered in our undergraduate, not our graduate, courses. Section G is material usually taught at the graduate level. Where appropriate we list courses which will cover some of this material.
1) Algebra.
U: Dummit+Foote, Abstract Algebra, except chapters 16 and 17. (math 122, 123, 126)
G: Dummit+Foote, Abstract Algebra, chapter 17.
2) Algebraic Geometry
G: Harris, Algebraic geometry, a first course, lectures 1-7, 11, 13, 14, 18.
3) Complex Analysis
(Table of contents)
U: Ahlfors, Complex Analysis (2nd ed), chapters 1-4 and section 5.1. (math 113)
G: Ahlfors, Complex Analysis (2nd ed), section 5.4.
4) Algebraic Topology
U: Hatcher, Algebraic Topology, chapter 1 (but not the additional topics). (math 131)
G: Hatcher, Algebraic Topology, chapter 2 (including additional topics) and chapter 3 (without additional topics). (math 272a)
5) Differential Geometry
(Table of contents)
U: Boothby, An introduction to differentiable manifolds and Riemannian geometry, sections VII.1 , VIII.1 and VIII.2. (math 136)
G: Boothby, An introduction to differentiable manifolds and Riemannian geometry, chapters I - V and VII. (math 134, 135 and 230a)
6) Real Analysis
(Rudin: Table of contents)
(Birkhoff+Rota: Table of contents)
U: Rudin, Principles of mathematical analysis, chapters 1-8.
Birkhoff + Rota, Ordinary differential equations, chapters 1-4 and 6. (math 25, 55, 112)
G: Rudin, Principles of mathematical analysis, chapter 10.
Rudin, Functional analysis, chapters 1, 2, 3.1-3.14, 4, 6, 7.1-7.19 and 12.1-12.15. (math 212a)as usual it seems to be the real analysts who cannot bring themselves to shorten the syllabus.
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