Other Should I Become a Mathematician?

[mathwonk]

i am borrowing this link from the forum on books, but it answers perfectly to the questions posed here on how to become a research mathematician.

I especially recommend the first article in this section by atiyah, on becoming a researcher starting from grad school.

http://press.princeton.edu/chapters/gowers/gowers_VIII_6.pdf

 

[mathwonk]

the problem solving question was asked and partially answered here:

https://www.physicsforums.com/showthread.php?t=277465

 

[Bourbaki1123]

Mathwonk,

I have noticed quite a bit of dismissal of mathematical logic as a field of study. It seems to me though, that model theory and proof theory and recursion theory have elicited some fruitful discoveries in other fields of mathematics.

I was wondering, being as you are a professor and we have the benefit of a disassociated conversation over the internet so I feel that I will get a more pure response from you, how is the field of model theory viewed by most mathematicians? I am asking as someone with an interest in the field.

 

[mathwonk]

i may dismiss logic because i personally do not enjoy logic much. but some of the very smartest people i have known have enjoyed it a lot, my colleague at my first job, a colleague i have now, and a moderator here, hurkyl. these are very very sharp people and they like logic. so maybe i am just not smart enough to be a logician.

so i personally cannot help you much there, but it is certainly a field with limited but dedicated and very accomplished practitioners. another name, is paul cohen, (solver of the continuum hypothesis problem), a man who was described by one of the smartest men i ever knew, maurice auslander, as the smartest man he knew.

try googling model theory and see whether anyone in that area has been a speaker at the ICM, or whether you can find other evidences of high level activity, such as practitioners located at top places.

 

[tgt]

i may dismiss logic because i personally do not enjoy logic much. but some of the very smartest people i have known have enjoyed it a lot, my colleague at my first job, a colleague i have now, and a moderator here, hurkyl. thee are very very sharp people and they like logic. so maybe i am just mot smart enough to be a logician.

But why then has the fields medal only gone to one mathematical logician so far?

 

[mathwonk]

well you are asking the wrong person, but since you asked me, this is consistent with what i have said. It is apparently a narrow field, which appeals mostly to very smart people, but which has only a few very widely appreciated problems. was cohen the last guy to get one?

 

[mathwonk]

If interested in mathematical logic, the university of georgia in athens is a good place to work in it, especially in connection with number theory, due to the presence of Robert Rumely, a number theorist, who is famous for generalizing Hilbert's 10th problem (positively!) to the case of algebraic integers.

I.e. the original problem of whether an algorithm exists to decide existence of solutions to equations in ordinary integers was settled negatively by Putnam and Robinson and ??, but Rumely developed capacity theory on algebraic curves to show there is such an algorithm over the algebraic integers.

To see some of his impact you can search under his name even on Amazon books.we are an attractive place especially for US citizens to apply now because we are looking for about 17 new students next fall, and we have a VIGRE grant that supports US students generously with lower than average teaching. Along with the stipends to students we also support faculty in the teaching of useful seminars introducing research topics to PhD students, especially those getting started.

we have strong programs in algebraic geometry, number theory, geometry/topology, and representation theory, just to mention the ones I am closest to. We also have significant presence in applied subjects, and analysis.

I.e. we are good, and not on everyone's radar, we currently have more money than average, at least for US applicants, and we have more openings than we are likely to fill. So it is a good time to apply to the PhD program.

if interested, check out our website at http://www.math.uga.edu/

If you are more of a larger city person, Emory and Ga Tech in Atlanta are also good. Ga Tech is strong all around, and at Emory I personally know Professor Parimala, for example, who is a world famous algebraist.

 

[tgt]

well you are asking the wrong person, but since you asked me, this is consistent with what i have said. It is apparently a narrow field, which appeals mostly to very smart people, but which has only a few very widely appreciated problems. was cohen the last guy to get one?

Cohen was the one and only person to be awarded in mathematical logic.

I actually asked a guy working in the association associated with the fields medal and he said the medal is simply given to the best mathematician 40 years or under. But since only one mathematical logician has received it, this suggests that the best mathematicians don't work in mathematical logic.

The fact that its narrow probably has something to do with it as the probability of the best mathematicians working in it is small compared to the rest of mathematics.

 

[mathwonk]

it seems we are going to fill about 12 grad slots. we have about 5 of the vigre openings, which pay about $25K per year for 2 of the years one is here. We also have a campus wide competition for some fellowships which pay about $24K per year, maybe for more years.

As a related topic, may i ask people what factors most influence their decision as to where to go to grad school?

1) presence of researchers working in a subject of interest.
2) supportive grad program.
3) availability of adequate/generous student stipends.
4) appealing community/social life.
5) prestigious name/reputation of university.
6) congenial geographic location.
7) large diverse grad program (to maximize choice of specialty)
8) other?

 

[mathwonk]

well, tgt, that guy's answer is kind of meaningless to me. ask him how does he decide who is the best mathematician? i am guessing it has to do with solving problems that are recognized as outstanding. hence the existence of such problems in the field is a necessary condition for deciding someone in the field is outstanding.

of course the existence of such problems also would attract top workers. so a field with no great problems will not have great practitioners. of course there are also people so great that they do great things that are not expected.

so if you work in a field that is a bit boring or stale at the moment, you have to be fantastic to do something that will reveal your ability.

 

[future_phd]

I am thinking of picking up Spivak's Calculus because we used Stewart's Calculus for our calc 1 and 2 and it really doesn't look like it prepares you well for Analysis courses or Pure Math in general.

Am I right in picking Spivak or is there another one I should pick instead? I want to be prepared for when I take my first Real Analysis class (next September). Also I haven't really self studied up to this point so I am wondering if there are any tips on good tips/habbits for self studying and also should I start at page 1 and work through absolutely everything?

 

[mathwonk]

well start wherever you like. its all very helpful. if you start on page 1, and get bogged down, just skip ahead.

 

[mathwonk]

after talking with recent members of the graduate program, it is still hard to give a completely precise description of how to get into our grad program.

Basically we are looking for candidates who will succeed in our program, and we take everything we can find out about them academically, into account. There is a committee making recommendations, so different people look at different things.

This means everything matters to some extent, recommendation letters, grades, gre scores, extra activities, and also a consistent picture should be revealed by all of these taken together.

The most substantive data is perhaps a record of success in substantial courses over time, but letters from professors giving a personal opinion are also important.

Personal qualities can also matter, as there are a few people whose records show gaps or flaws, but who persevere and improve, and eventually come out on top. These cases are harder to recognize but do exist.

A candidate with a strong record of challenging courses and high grades in most or all of them, combined with high gre's and letters that identify the student as outstanding among all those over a number of years, even at a small college, should stand very well in our competition, but not all successful candidates have these qualifications.

Our current stipends range from 24K - 25K for 5 or more top qualifiers, and those are not for every year, but roughly every other year, to the average stipends of 14-15K. And we apparently do manage to support most students also in the summer. A few students are sometimes admitted without support I believe, provisionally, based on demonstrating success, but this is not the norm.

We are one of only a dozen departments in the US whose VIGRE grant has been renewed, which is testimony to our success and commitment to helping our admitted candidates graduate.

Specifically, our vigre program is considered innovative and effective at "fostering graduate student research at an early stage".

One area in which we excel, outside the usual pure and applied mathematical areas, is in education of mathematics teachers from primary school through high school. This is a collaboration between our excellent mathematics education department and members of the mathematics department.

A recent nationwide study identified UGA as having one of only a very few exemplary programs in math education in the nation. In particular some books for this purpose authored by Professor Beckmann in the math dept. were recognized as outstanding. Candidates interested primarily in preparing to teach mathematics would do well to look over the programs here in math and math ed.

For sincerely interested and qualified students we can usually help provide some assistance to visit campus this spring, in late February 2009.

 

[PowerIso]

1) presence of researchers working in a subject of interest.?
This was the most important factor for me. A strong research group had to be present with well known people in the field.
2) supportive grad program
This was also key. I wanted to feel that the program was behind me and that I would fit in nicely with the group of people there.
3) availability of adequate/generous student stipends.
Also played a role. Adequate was all I was looking for, generous was just a bonus.
4) appealing community/social life.
Not so much for me, I came to grad school to learn math. Good community and social life is a plus, but I also figured if I am there with other people who are interested in the same thing I am, i'll probably have a good social life regardless.
5) prestigious name/reputation of university.
Minor factor, not as important as number one on the list.
6) congenial geographic location.
Not important to me at all.
7) large diverse grad program (to maximize choice of specialty)
I didn't really think about to, maybe I should have. Thinking about it now, this probably should've carried more weight.
8) other?
One other aspect was the number of PhD that graduated from their program that had jobs five years after graduation.
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[Vid]

The VIGRE grants are pretty great. LSU got their first one this year, and already this Spring there are 5 research classes that mix undergraduates and graduates.

https://www.math.lsu.edu/dept/vigre/crews

 

[Bourbaki1123]

How much weight would solving some problems in undergrad journals such as Crux Mathematicorum and having your solutions displayed hold in admission considerations?

 

[mathwonk]

well it would be another plus, maybe a small one, but it shows ability and interest. Of course if the problems are really hard and the solutions are brilliant, it counts more.

 

[Epsilon36819]

Mathwonk,

I have a question for you. There is this graduate class given next term which is a second course in topology. The first class was given this term and I unfortunately couldn't take it, as it overlapped a core course for my degree. This first course covered the basic of topology and the fundamental group, covering spaces, simplicial complexes, singular and simplicial homology, among other things.

Now I am very tempted to take this second course without the prerequisite. I do know the basics of topology and I am willing to put lots of time and work (as well as take a lighter courseload) to make up on my own for what I don't know yet. The thing is I am really, REALLY interested in the material and the course is given by one of the best teachers in the department. I also know without a doubt that I will improve by taking this class. And I don't care what grade I get (as long as I pass, I guess...)

However, my advisor objects to this idea, saying that courses must be followed in the right order to ensure that we are properly ready.

Of course, I am not asking you what to do (you don't know me nor the course) but I would like to know, as a general rule, if you would encourage interested students to skip a few steps and put themselves in a situation where the level of difficulty is much higher for them than for anyone else in the classroom. Or would you instead suggest taking time to lay down a proper foundation, at a slower pace, risking perhaps to not be as challenged as one would like to.

Thanks in advance!

 

[mathwonk]

the general question usually has answer no. but your specific question may have answer yes. The reason is that fundamental group and so on is not really a necessary prereqisite for many later topology courses.

so the person to ask is the professor offering the spring semester course. He/she will know whether you will really be overwhelmed by not knowing the previous material. you also have the option of spending the xmas break reading a book on fundamental groups, and covering spaces, like that by massey.

 

[Epsilon36819]

Thanks mathwonk, I'll have a look at this book.

 

[mathwonk]

try this:

Algebraic Topology: An Introduction.
Massey, William S.

[30 Day Returns Policy]
Bookseller:
J. HOOD, BOOKSELLERS, ABAA/ILAB
(Baldwin City, KS, U.S.A.)
Bookseller Rating: Book Price:
US$ 15.00

 

[Vid]

I've been reading Introduction to Algebraic Topology by Wallace, and I really like it. It contains all the point set topology required.

 

[mathwonk]

great suggestion. is this about the fundamental group? i think this was the first book i read as senior that really made me understand algebraic topology for the first time! if so, it is really clear and thorough for beginners just trying to grasp the concept of homotopy.

 

[PhysicalAnomaly]

What's the difference between an undergrad journal and the typical kind? I was under the impression that the usual journals also published undergrad research.

What are living costs like in the USA? I live in aussie and 14K doesn't really sound like it's enough to live like a pauper but that's compared to our currency and living costs. Do students get much more from teaching?

 

[Vid]

Undergrad journals publish expository articles on a topic rather than just new research.

 

[mathwonk]

14K is not very much. But in Athens, Georgia life is cheaper than in many places.

Our problem is our average stipends are low, but our good stipends are high.

So I would suggest applying for our best stipends, and deciding what to do if you only get the average one.

 

[PhysicalAnomaly]

I have just been made aware that many universities require one to be able to read maths texts in German/Russian/French to do a PhD. I don't know either. *panics*

 

[SticksandStones]

Mathwonk: do you know what the current state of research into Topology is? I mean, is there still a lot of interest in the topic?

 

[mathwonk]

well with perelman's fairly recent solution of the poincare conjecture, yes, i would say topology is one of the hottest subjects.

 

[Bourbaki1123]

Mathwonk,

You seem to give quite a bit of praise to Michael Artin's book on algebra. What do you think of his father Emil's book on the subject?

 

[mathwonk]

the only books i know of by the father are "galois theory" notes from notre dame lectures, and "geometric algebra". these books are great classics, but they are not as easy to read as mike's book. mike wrote his book for sophomore students whereas emil seemed to write his books for eternity. i.e. whoever can read them is welcome, and not one word is wasted.

i myself never could really learn from e. artin's galois theory book as it was too condensed for me. he also has some algebraic geometry notes from nyu but those also leave much to be desired from my viewpoint for learning ease. But it is almost sacrilegious to criticize anything written by e. artin, who is regarded with great awe by many people.

but i regard mike's books as much more user friendly.

but as i meant to imply, i am not aware of any books by e. artin strictly on abstract algebra. of course the great book by van der waerden is based on lectures of e. artin and e. noether. Is that what you mean by e. artin's book? I like it quite well and learned a lot from it as a student.

If that is representative of e. artin's lecture style then he was a very fine teacher. Indeed I have read in his own works that he always tried to write more than usual on the board when lecturing so that the student who was not following could recover the lecture from his notes. this struck me as admirable and i long followed this practice in my own lecturing.

 

[SticksandStones]

well with perelman's fairly recent solution of the poincare conjecture, yes, i would say topology is one of the hottest subjects.

That's good to hear. I started reading Cairne's "Introductory Topology" and so far I've found it pretty fascinating. I can't wait to be able to take a class on it.

 

[mathwonk]

topology is the most fundamental branch of geometry. as such i believe it will always be one of the most fundamentally important topics.

the ideas developed in topology of ways to understand different types of connectivity, are absolutely crucial in all areas of mathematics.

the tool of cohomology, which is present in algebra, geometry, and analysis, received its greatest development within topology. Sometimes I think the greatest ideas in mathematics grew there.

that is probably unfair to analysis, but anyway.

 

[SticksandStones]

Your post went a bit over my head. :)

I really liked Abstract Algebra when I took it. It looks like group theory plays a roll in Topology, from skimming some things. Am I right in assuming this?

 

[mathwonk]

i am just saying that the ideas that were developed in the 30's, 40's and 50's within topology, like bundles, characteristic classes, and sheaves, and cohomology, grew outward and illuminated complex analysis and algebraic geometry in the 60's and 70's and are universally used now.

you are currently at the beginning, studying point set topology, but later when you study algebraic topology this will be meaningful.

 

[nomaris]

Mathwonk,

In the first page of the thread you said that a high school student should explore probability, linear algebra, calculus after having a thorough grasp of geometry and algebra. What constitutes knowing Euclidian geometry and algebra well?

 

[mathwonk]

i would say mastering harold jacobs' books on those topics are a minimum for a high schooler. if more ambitious you might search out smsg books from the 60's. say arent there numerous such recommendations in that thread? have you only read page 1?

 

[PhysicalAnomaly]

Do they really expect PhD students to learn 2 foreign languages in 3 years?

 

[Wretchosoft]

Do they really expect PhD students to learn 2 foreign languages in 3 years?

I don't see why this requirement would be intimidating. Two semesters in college is enough to teach the average student the basics of a language; with the generally higher capabilities of PhD students, I would imagine this time could be shortened. From there, it's just practice.

 

[alligatorman]

From what I've heard, the language exam is usually just to translate a mathematical paper from the language into English. I can't imagine that it's too difficult.

 

[The|M|onster]

Would Spivak's Calculus on Manifolds be a good reference text for a undergraduate course on multivariable analysis?

 

[maze]

Calculus on manifolds book is primarily useful for the exercises, which are quite good. The writing and explanation is too terse in my opinion, but some people swear by it.

 

[Vid]

I just took a course using the book and found it to be really good. Munkres Analysis on Manifolds is kind of like an expanded version of CoM and is really good as well.

 

[samspotting]

I am trying to prepare a good foundation for math. I am learning from a few sources but I will be proficient these areas from classes and books:

Real Analysis (Learned from pugh and baby rudin, and class)
Linear Algebra (Learned from Friedberg, Insel, Spence, and class)
Set Theory (Learned From Naive Set Theory)
Combinatorics (Learned from Class)

What is a good way to learn geometry? I never paid much attention to any of my high school math classes and never really got much out of it, besides the basic identities. It seems like it could be very interesting.

I was looking at Beyond Euclid's Elements, and was surprised to find Mathwonk as one of the featured reviews on amazon. Maybe he can offer some advice and input.

Is there anythink else that math majors should know before moving on? One very interesting book that caught my eye is Inequalities by hardy, littlewood, and polya. It looked intense though, is that book my level?

 

[mathwonk]

i liked calculus of several variables by wendell fleming.

as i said in my review, hartshorne's book is an excellent guide to euclid.

 

[quasar987]

IMO, "Inequalities" is a reference book, as opposed to a book you read from back to back... Say you're stuck on a problem and realize that if you had some kind of inequality then it would work... you go look in "Inequalities".

 

[mathwonk]

do you think these proof questions are too hard?

I.A i) Recently, my only guests for Thanksgivings have been turkeys.
ii) No mathematicians fail to solve crossword puzzles faithfully.
iii) The only faithful crossword puzzle solvers I know are my recent Thanksgiving guests.
Conclusion (using all the hypotheses):

IB. i) The Americans who exploited the Hawaiian natives ended up doing quite well.
ii) Some American missionaries who came to Hawaii originally to do good, started pineapple plantations.
iii) The pineapple planters in Hawaii exploited the natives’ land and labor extensively.
Conclusion(using all hypotheses):

IC. i) I consider money not spent enjoyably, to be wasted.
ii) I have had little joy out of anything lately other than comic books.
iii) An intelligent person does not waste money.
Conclusion(using all hypotheses):

ID. i) Dr. Smith has discovered the most wonderful beach.
ii) Some things are really fine, but nothing is as fine as the sand at the beach.
iii) If a person discovers something really fine, he should bury his head in it.
Conclusion(using all hypotheses):

 

[Bourbaki1123]

do you think these proof questions are too hard?

I.A i) Recently, my only guests for Thanksgivings have been turkeys.
ii) No mathematicians fail to solve crossword puzzles faithfully.
iii) The only faithful crossword puzzle solvers I know are my recent Thanksgiving guests.
Conclusion (using all the hypotheses)::

By i)&iii) the stuffing is drugged, don't eat it.


IB. i) The Americans who exploited the Hawaiian natives ended up doing quite well.
ii) Some American missionaries who came to Hawaii originally to do good, started pineapple plantations.
iii) The pineapple planters in Hawaii exploited the natives’ land and labor extensively.
Conclusion(using all hypotheses)::[/QUOTE]
Some American missionaries ended up doing quite well

IC. i) I consider money not spent enjoyably, to be wasted.
ii) I have had little joy out of anything lately other than comic books.
iii) An intelligent person does not waste money.
Conclusion(using all hypotheses)::[/QUOTE]

If I were intelligent, then I would buy comic books.

ID. i) Dr. Smith has discovered the most wonderful beach.
ii) Some things are really fine, but nothing is as fine as the sand at the beach.
iii) If a person discovers something really fine, he should bury his head in it.
Conclusion(using all hypotheses):[/QUOTE]

What if its a tar beach? Or a rock beach? If fine means the same thing in all of its uses, and is defined as to mean granulated, or ground to a very small scale , then Dr. Smith should bury his head in the sand iff the beach mentioned in i) is the beach mentioned in ii), else, we cannot say that the beach in i) even has sand, so ii) and iii) have no bearing. If, however, we take fine to mean; good, wonderful, grand, then Dr.Smith should bury his head in the beach. Now, if the beach in ii) is the same beach, or has sand as fine as the beach in ii), we conclude that Dr.Smith should indeed bury his head in the sand at the beach. Otherwise, perhaps he might be equally well off burying his head in some rocks or seaweed.

Now what if fine has two distinct meanings? Am I meant to exhaust all possibilities?

 

[mathwonk]

no one seems to notice the qualifier in B that renders it similar to a famous quote: "The American missionaries, who originally came to Hawaii to do good, ended up doing well".

and in C), wouldn't it be "...only comic books"?

I give up on D. I think the conclusion is that humor and tests do not mix, or humor and mathematicians do not. or more accurately, to cite another famous quote:

"I knew Lewis Carroll, Lewis Carroll was one of my favorite authors. ... Dr. Smith, you are no Lewis Carroll."

 

[uman]

Sorry Dr. Smith, what quote are you referencing?

Also, A) should be "all mathematicians are turkeys".