Other Should I Become a Mathematician?

[Bourbaki1123]

I think that one can often overcome the weakness of his school's math program. My school does not have the strongest math program, it is really quite weak(actually not as weak as mrb's, we covered everything in chapter 1-6 or so in David Lay's book. You must not have even covered eigenspace, nul space,collumn space, rowspace ect. if you didn't cover vector spaces.). I took the matter into my own hands and started self study in tough books before I even started college before I was really ready for them so I had to put them down,but; if you really love the subject you will not give up on a difficult aspect, you go back again and again until you can muscle through every concept and problem.(I'm talking about books by authors like Rudin, Goursat, Lang, Artin). I also make it a point to get to know my professors and ask less trivial questions that I might have encountered in my self study. I also ask for a lot of advice as far as what I should be doing, what books will prepare me well for grad school, ect.

I don't see why a weak math program would hold back a strong student. I can see how time constraints might, if one does not have the time to supplement their courses. If, on the other hand, you find a weak program too difficult to juggle with outside study, you might not be cut out for grad school, I don't know. Working might be a mitigating factor ect. There are many things which can put a hold on extra study. Hanging out with friends too much is something that might have to be sacrificed.

It seems like it would really depend on how strong the grad school is. For instance, UC Berkeley's math grad program has a very high drop out rate. This is a very difficult program that only people who can ace the Math Subject GRE and were published as an undergrad or something along those lines can do well in.

My personal circumstance is kind of similar to mrb's. I studied algebra in elementary school and became very interested in it and picked up concepts quickly, but much to my dismay, every year of advanced middle school math and basic high school math was essentially the same and I quickly became disinterested. The same thing was true of science. My 7th grade science teacher was a soccer coach or something along those lines and he would have no answers to my questions pertaining to astronomy or physics.

I did poorly in high school(not terribly, but a 2.98 gpa) and had intended to do art or music, which are two other passions of mine, but my interest in science was rekindled by my Honors physics teacher senior year(a friend told me that I should take the class). I started to self study calculus, analytic geometry, and trig, because I had not gone beyond geometry and algebra. I got a 1400 on my SAT but went to a local state university because I was wary of my math skills at that point. I nearly tested out of calc I, and could have skipped the first segment(the split it into two segments Calc I a and b) but decided not to. Instead I took the extra time I had since I knew the material in class to work on less trivial problems and study some theory. I looked at some of Apostle's book and did some problems and that really gave me a bit of an edge. Now I am a sophomore studying Rudin and Goursat on my own and I will have exhausted my school's math curriculum as of next year and will have to try to do courses at a nearby university and independent study.

The point is, if you really love math, and you have any spare time, read math, do math and talk about math to anyone you can. You don't necessarily have to abandon your social life totally, but at the same time, don't spend every waking hour hanging with your homies.

 

[PhysicalAnomaly]

The syllabus was very intimidating. But I just found their sample questions and was quite surprised - at least half the questions could easily be done with high school stuff and the most of the rest were very guessable considering that they furnish us with an explanation before they ask the question. And it's all multiple choice! I haven't had multiple choice maths questions since... I can't remember. :D

 

[mrb]

(actually not as weak as mrb's, we covered everything in chapter 1-6 or so in David Lay's book. You must not have even covered eigenspace, nul space,collumn space, rowspace ect. if you didn't cover vector spaces.).

Vector spaces were maybe not the perfect example. We also used Lay and covered through part of Chapter 5; the big problem was that this was a summer course lasting barely over a month so we got less than a week on Ch 4. A concept you work with for 3 or 4 days and then don't ever hear about again for a year tends not to be retained.

Here's another example, though: I was through Calculus 3 and Diff Eq before I ever heard of the Mean Value Theorem or the Intermediate Value Theorem (which I first read about on this forum, and then on my own from Spivak's Calculus, and finally just recently in my Analysis class).

There really is no avenue at this school for excelling. I realized last semester that my advisor was suggesting the classes she was for me based on the fact that they were easy, despite my clearly telling her I was interested in grad school and wanted to learn and my 4.0 Math GPA. We supposedly have a Math Honors program which involves undergrad research. I have been trying for 3 weeks to find a prof to be my advisor for the program. None of the profs I've asked knew there was a Math Honors program or had ever advised anyone for it. And none have been willing to advise me except possibly one, and he seems reluctant because he has admin duties and worries about his time. So I honestly don't think there really IS a Math Honors program; it's just something they put on their website and other materials because it looks good.

The point is, if you really love math, and you have any spare time, read math, do math and talk about math to anyone you can. You don't necessarily have to abandon your social life totally, but at the same time, don't spend every waking hour hanging with your homies.

I agree 100% but it took me a while to realize how much I should be studying on my own, partly because I previously had not decided with certainty to do math grad school. Social issues are, ahem, not a problem for me. I have no social life except for a gf who is generally tolerant about me spending hours and hours on math.

 

[Bourbaki1123]

Thats pretty bad. I had at least some knowledge of the IMVT and MVT in Calc 1 and definitely by calc 2. It seems necessary to give some basic results, I don't know if the teacher simply glossed over how most formulae involving it are derived? How did you guys go over the fundamental theorems of calculus and taylor series?

I guess it really isn't a big deal since you know it now and grad schools probably assume that most early calc courses are the same or generally don't care.

My school doesn't have a honors math program either.

A couple questions, directed more towards mathwonk,

Do you think that Bartle is a good Analysis text, and do you know much about Paul Dienes Text The Taylor Series (i.e. how would you rate it and why.)

 

[mrb]

Thats pretty bad. I had at least some knowledge of the IMVT and MVT in Calc 1 and definitely by calc 2. It seems necessary to give some basic results, I don't know if the teacher simply glossed over how most formulae involving it are derived? How did you guys go over the fundamental theorems of calculus and taylor series?

Nothing was ever derived. FTC and Taylor Series stuff was presented but not proved or even informally demonstrated.

 

[mathwonk]

I was responding to some posts in which people said essentially: here are my gre scores, where am i going to get in? princeton? washington? etc etc...

i am saying that gre is not a big factor in getting into the best places, that anyone struggling with the gre is simply not going to get in those places.

i hope that is useful time saving information to people thinking of applying to princeton. namely if you think the gre is hard, don't bother. princeton IS an elite school.

some students come into harvard having already read and worked through books like griffiths and harris algebraic geometry. my first advisor came to columbia having already proved the riemann singularity theorem in a rigorous way for the first time by anyone in over 100 years.

this does not mean someone who has to work to do well on gre cannot find a home where they will fit in well, but it won't be at harvard or princeton or mit or columbia.

I think I made it very clear that my post was aimed at people who want to know how to tell if they are going to get into the very best schools. i say that unless you find the gre easy, you are not going to.

when i was an undergraduate at harvard, the only people even applying to harvard grad school had already taken year long graduate courses in algebra (lang), algebraic topology (spanier), real and complex analysis (big rudin, cartan, ahlfors) as undergrads.

the rest of us looked elsewhere.
...sorry i am not familiar with bartle and dienes. bartle is a familiar name though, so probably has a good track record.

 

[PhysicalAnomaly]

Heh, same thing here. Nothing is derived and IMVT and MVT weren't taught in any of the calc units. They present stuff like Stokes's and Gauss' theorems but aren't very clear on what they actually mean much less derive it. They've neglected to mention Green's theorem or Fubini's - we just change the order of integration as we wish. But that's not really important. It'll be covered again in real analysis anyway, right?

 

[PhysicalAnomaly]

when i was an undergraduate at harvard, the only people even applying to harvard grad school had already taken year long graduate courses in algebra (lang), algebraic topology (spanier), real and complex analysis (big rudin, cartan, ahlfors) as undergrads.

Australian unis don't really offer that kind of courses. The most I can get is a year's worth of grad analysis and algebra with a mix of self-study, honours year units and maybe an exchange. Does that mean that the top universities won't be an option?

 

[mathwonk]

not necessarily. those were options at harvard so they expected the harvard undergrads to have taken them. talented people from less rich environments could be cut more slack.

but remember that was long ago. things are different now. but harvard students are still very very sophisticated and advanced.

only one thing is sure however, as a friend reminded me about applying for anything:

"if you do not apply you definitely will not get in."

 

[mrb]

But that's not really important. It'll be covered again in real analysis anyway, right?

At my school the standard undergrad real analysis is a 2 semester sequence, with single variable topics covered first and multivariable covered next. Unfortunately the single variable portion is never completed in the first semester, so the second semester is mostly spent doing what should have been done the first semester.

I don't want to drag this thread away from its purpose, but as long as I'm complaining about my math education, I want to provide this contrast:

When I thought I wanted to go into Bioinformatics, I emailed the coordinator in that department who invited me to come to his office. I met him, he told me about Bioinformatics in general, and about each of the profs and what their research was on and so forth. I contacted 2 of the profs about doing research with them; met with both of them; both offered to let me work in their labs. I chose one of them and had a rewarding semester.

On the other hand, now I want to do some math research. I have talked to four professors about it. One met with me and it went like this:

Prof: "Well, I'm a numerical guy, so you would have to be able to program to work with me, I'm sorry."
mrb: "I can program. I've been programming for years."
Prof: "Oh. Well, you would have to know C, so I guess..."
mrb: "I know C."
Prof: "Oh. Well have you had Calculus 4?"
mrb: "Yes."
Prof: "Have you taken Applied Math? [this is a course only offered every other year]"
mrb: "No."
Prof: "I'm sorry, but any work I would have for you would depend on that material, so I don't think we can do this."

He couldn't just say he couldn't do it, he had to search for some excuse. Another prof stopped responding to my emails after one reply. Another was enthusiastic and agreed to meet with me but then didn't show up and is now incommunicado. And finally there's the last one, who is still a possibility but as I mentioned above seems reluctant and since he hasn't replied to my email from a few days ago he may be going incommunicado as well.

 

[Vid]

Hmm...

I took Calculus 1-3 in high school, and we derived everything. We had a pop quiz on the formal definition of a limit for a quadratic. These wasn't an honors class this was one of the many sections of calculus taught at this school. If you didn't cover these things in a college course, something is very wrong.

 

[mrb]

Well, yes. I agree. There are some good people in our math department but something is broken at a very fundamental level.

 

[mathwonk]

I assure you there are many "good" colleges where a standard calculus class may not even include the rigorous definition of a limit.

at most schools the population is so diverse that they offer two or three or even four different calculus classes, only the most "elite" being for math majors, and hence including any theory at all.

many entering students, even those who took calculous in high school, or maybe especially those, object strongly to being asked to state a theorem, or a definition, much less actually learn to prove a theorem, since this usually never occurred in their high school course.

where did you go to school that you had a rigorous course? i think this, although desirable and excellent, is very rare.

 

[mathwonk]

mrb, it sounds to me as if the first step toward a math honors program for you would be to work through spivak's calculus book. maybe a prof at your school would supervise that, or at least sign on to give you credit for it. and try to organize a group of other like minded students and give lectures on it to them.

when i was out of grad school i taught all the way through this book in 8 weeks, to a group of returned teachers. it helped both of us. the next year i went back to grad school.

 

[mathwonk]

I interviewed some students I consider quite strong, and they thought the general gre was quite easy, but the math subject test was significantly harder.

They likened the general gre to just an sat test, but when I asked if the subject test was like an AP test, I did not get a response.

I also recall from decades ago a friend who said he did not do too well on the gre, who is now a full professor of math at a top school.

so it seems as if indeed the gre may be kind of a filter, but may not have huge relevance for actually predicting success in grad school in math.

If I feel like wasting an hour or so, I may take the sample myself and give a more informed opinion later, but to me it is kind of a waste of time, except to know what it means when I see the scores in future if I happen to be on the admissions committee.

As of now, I still do not know anyone who thinks it is highly important as a factor in measuring likelihood of success in grad school, which is ironic since it is an obstacle to admission.

Maybe Ill ask the graduate coordinator.

 

[mathwonk]

well i didn't find him but someone else who took the gre as a student said they seem designed to measure whether you know enough to be a TA in calculus. I had never thought about that aspect of readiness for grad school. he also said studying for them was useful as a way of shoring up knowledge that somehow had been omitted from his background.

 

[Bourbaki1123]

Mathwonk,
How much would you think that being published, more specifically, publishing a result in field theory showing that a closure property is not held by any finite fields, and showing that either this closure property which is superficially weaker and much easier to test for is equivalent to a commonly considered property of an infinite field or that it is a unique kind of closure that has not been considered much if at all, but I will have quite a bit of information about its properties in relation to the field, would weigh in grad admissions?

Sorry if the description is too vague.

 

[mathwonk]

I would think any kind of legitimate publication would count as something unusually good before grad school. of course there are programs, in which students are guided through relatively easy research projects by a researcher, and it results in a joint publication, and these might mean somewhat less, but even so, it is a good experience that I would think is relevant to the grad school experience.

as to the gre,it seems that my dismissal of them as almost a joke for grad school admission was based on the level of the general one, which is apparently just another sat test.

most people agree the subject test is harder and is a useful test of the topics taught in a typical undergraduate major. i still maintain however, that admission to top schools is based on something else entirely, something less cut and dried, something hopefully deeper, the opinion of teachers that an applicant has the potential to excel in the world of mathematical research.

this ability is something I feel when talking to a student, and noticing that they catch some relevant significant point more quickly than I do, or generate some original idea that impresses me.

 

[laymanhobbist]

The Story of Maths - Part 1 of 4 (6th October 2008).avi

The Story of Maths - Part 2 of 4 (13th October 2008).avi

The Story of Maths - Part 3 of 4 (20th October 2008).avi

The Story of Maths - Part 4 of 4 (27th October 2008).avithank me later

 

[Bourbaki1123]

My school was going to offer a course in mathematical problem solving(heuristics), but no one signed up but myself. The course is intended to build skill for contests like the putnam and solving problems in math journals. What would you recommend as far as good books for problem solving techniques suitable for self study?

 

[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.