Should I Become a Mathematician?

[djh101]

I want to be a mathematician. But I'm becoming a chemist instead. 8)

 

[Mépris]

James Stewart's Essential Calculus, Early Transcendentals.

Web component here: http://www.stewartcalculus.com/media/6_home.php

Generally a very disliked book, I have to say, at least by we mere undergrads. The book seems to be pared down from earlier editions to be more "concise," which actually makes it very hard to read if you're coming at it for the first time (which I was.) The earlier editions are much more readable. For one that is more mathematically literate than I was I think it's probably a fantastic book. I've just finished the three course calc sequence but I'll probably be digging into the book for years.Also, the online component is good but under-utilized. (In another attempt to pare down I guess he put stuff online.) People don't know it's there, so it doesn't get used.

You seem like you have more experience so you probably won't have a problem. What year are you, or are you a grad student?

-Dave K

Oh, don't let the link in the post above fool you! I will, hopefully, start college next year. (at twenty) I spend a lot of time reading about undergraduate study in mathematics/other quantitative fields because:

- I'm interested in the way higher education is structured in various parts of the world
- I made many poor academic decisions in the past, largely because I was unware of things. That was at the pre-college level, which is a good thing. I'd rather not have these happen during college, for the consequences will be

There is a blog which has a four year syllabus, with links to various books (with an emphasis on legally free stuff), somewhere on the internet. I believe it was Micromass linked it to me. At any rate, this is where I learned about the Terry Tao notes. If memory serves me right, they are intended to be used after one has gone through the linear algebra and calculus sequence.

I was studying algebra based physics, got incredibly bored and tried reading a few pages from there. I found a nice little result in the beginning of the file.

Consider the geometric sequence, S, below:

S = 1 + 1/2 + 1/4 + 1/8 ...

How would one go about to calculate its sum, without using the "sum to infinity formula?

Multiply by 2 on both sides.

Thus,

2S = 2 + 1 + 1/2 + 1/4 + 1/8 ...

Which is equal to: 2 + [1 + 1/2 + 1/4 ...]

Which, in turn, is equal to: 2 + S

Now, since 2S = 2 + S,

it follows that S = 2.

This is nice.

Do things such as the above fall within real analysis or number theory? Terence Tao said that "real analysis is the study of real numbers...underlying theory of calculus" (paraphrasing here) but is number theory not nearly the same, except that it covers all kinds of numbers?

 

[mathwonk]

university of chicago has one of the world's best math departments. i am not crazy about the local environment there in that part of chicago. i.e. it is right in the city and not the nicest part of the city, but that is true of some other urban campuses. the mathematicians there are incredibly good. some I have known or known of for a long time are: Nori, Drinfeld, Ginzburg, May, Nygard, Fefferman, Sally, Alperin... other younger people include Matthew Emerton, whom I have recently gotten to known through mathoverflow, and who is also very nice.

I believe the department at Chicago has long had a reputation as strong at undergraduate teaching. For a long time they were one of the few departments to continue to teach a very high powered introduction to calculus from Spivak's book, whereas other top places like Harvard discontinued it, under the (I think often false) assumption that a good grounding in beginning calculus is already known to all entering math types.

UC has a fantastic web site. read some of this:
which looks as if it describes the Spivak type class, which apparently still exists. Or ask Paul Sally.
http://www.math.uchicago.edu/undergraduate/faq.shtml

One of the best mathematicians at my department, a Sloan Fellow, and famous number theorist, Robert Rumely, went to Grinnell College for undergrad, so they should be good, and their website makes them look very engaged in student instruction. Carleton has long been well known also as good teaching college in math. I don't know the others as well. I have visited Colorado College in the summer and found it a friendly place in a nice location near Pike's Peak. The town is small but has some good restaurants.

 

[mathwonk]

Mepris, that trick for adding up an infinite geometric sequence could be called analysis but in my opinion really isn't. There is no hard work there concerning whether the sum makes sense or not, just trick for finding the sum if it does make sense. I learned that trick in the 8th grade, when I certainly did not know any analysis. Analysis is properly more concerned with defining infinite sums precisely, and proving that such tricks make sense. Carrying out such tricks is a fun game that helps magnetize people who enjoy math, but there is no real work in it.

I.e. that computation shows that IF the sum makes sense, and IF it also makes sense to multiply it term by term, THEN it must equal 2. An analysis course deals with those IF's.

 

[Sankaku]

...but is number theory not nearly the same, except that it covers all kinds of numbers?

Number theory is based around the study of the Natural numbers and, by extension, the integers. Higher-level number theory gets into other algebraic structures, but that is where it starts. With the Natural numbers, you can't always divide things the way you want. Much complexity comes out of this simple fact. They are also the quintessential countable set.

As you say, Analysis is based around the study of the Real numbers. Though the distinction seems small from the outside, it is actually huge. The real numbers are the prototypical complete ordered field and you get to grapple with the brain-bending properties of uncountable sets. Most people just accept it, but I think the Real numbers are actually the most frightening thing in all of mathematics.

 

[dkotschessaa]

Well you sound pretty conscientious for 20 Mepris, so I think you are doing alright. I'm 35 now so I'm way behind. It certainly isn't too late for you to make some good choices now.

I hope you find what's best for you, though of course I am heavily biased towards USF, and if you should come here, you would have some instant friends. (Just think, sunshine, girls in shorts all the time... oh and math.. lots of math). Here is the course flow chart: http://i47.tinypic.com/2vltump.jpg Let me know if that's not readable and I'll re-size. Looks a bit fuzzy.

-Dave K

 

[mathwonk]

Mepris, I apologize for wording my answer negatively. Yes, making sense of that computation is exactly what analysis is about.

 

[Mépris]

university of chicago has one of the world's best math departments. i am not crazy about the local environment there in that part of chicago. i.e. it is right in the city and not the nicest part of the city, but that is true of some other urban campuses. the mathematicians there are incredibly good. some I have known or known of for a long time are: Nori, Drinfeld, Ginzburg, May, Nygard, Fefferman, Sallky, Alperin... other younger people include Matthew Emerton, whom I have recently gotten to known through mathoverflow, and who is also very nice.

I believe the department at Chicago has long had a reputation as strong at undergraduate teaching. For a long time they were one of the few departments to continue to teach a very high powered introduction to calculus from Spivak's book, whereas other top places like Harvard discontinued it, under the (I think often false) assumption that a good grounding in beginning calculus is already known to all entering math types.

I don't think I will mind the location too much. One thing I appreciate with American towns is that all of them seem properly planned and everything is flat. At least, judging from what I see on TV shows and films, it looks so. I can imagine that from a bird's-eye-view, towns would seem as if they were chess boards. I am unusually fussy about such issues and it would make me happy to live some place where things are accessible and the roads are bicycle friendly. At any rate, I doubt I will have too many issues, location-wise.

http://math.uchicago.edu/~lind/161/

Yep, Spivak is indeed the prescribed text. It is interesting to note that it is merely intended to be used as a reference text. Students are expected to write a so-called "journal" in which they should each write their proofs. They call it "Inquiry Based Learning" (I think I got that right...) and it would seem that the students are expected to do the bulk of the work. (i.e, absence of spoon-feeding) Sounds like a cracking course. I will definitely try to see if I can adapt their own method when I learn from Spivak's book in the near future.

Is it not just the "higher ranked colleges" who now have multi-variable calculus as their freshman honours calculus course? My understanding is that everywhere else, where an honours variant of freshman calculus is present, the first part deals with single variables? I think of the "top schools", MIT (they use Apostol) and UChicago are the only exceptions.

Another thing. As you have pointed out before, the students who went to high school around the same time as you had access to more advanced mathematics than those students of today. Save for those participating in Olympiads or those who spend some time reading about mathematics, I doubt many have heard of that result and countless others. According to Wikipedia, the "New Math" of the 60s was created largely as a response to the threat that Soviet engineers were posing.

I'm unsure as to whether the dumbed down high school mathematics curriculum is a good or a bad thing. Only a minority will ever use such mathematics, let alone be interested in it. I think it might be a good idea to have everyone take a rigorous course (say, geometry) in mathematics and then have the next courses at varying levels of complexity and content. I cannot recall who, but a Math PhD turned coder from Stanford, had a few notes on how to change the system. He proposed three streams. One for those aiming to pursue math at university or those just interested in math. One for those going into the natural/social sciences or engineering. One which focused on more day-to-day uses of mathematics.

Number theory is based around the study of the Natural numbers and, by extension, the integers. Higher-level number theory gets into other algebraic structures, but that is where it starts. With the Natural numbers, you can't always divide things the way you want. Much complexity comes out of this simple fact. They are also the quintessential countable set.

As you say, Analysis is based around the study of the Real numbers. Though the distinction seems small from the outside, it is actually huge. The real numbers are the prototypical complete ordered field and you get to grapple with the brain-bending properties of uncountable sets. Most people just accept it, but I think the Real numbers are actually the most frightening thing in all of mathematics.

Perhaps it is because I have limited exposure to them but as of right now, my view is simply that they are fascinating, and much less scary!

The book "Challenge and Thrill of Pre-College Mathematics", which may be of interest to other prospective math majors on here, does a good job at explaining numbers. First, the set of natural numbers and the operations that can be carried out with that type of number is presented. From there, the set of integers is introduced, and the authors also explain how this new set can overcome the limitations of the previous set but also explain that new set's own limitations. They do likewise up until complex numbers and have a nice chart which shows what was "gained and lost" by "expanding" (?) the respective sets each time. A preview is available on Google Books. In fact, most of the book can be viewed.

This text and the result/computation in the previous page have made me look forward to taking an analysis course.

Well you sound pretty conscientious for 20 Mepris, so I think you are doing alright. I'm 35 now so I'm way behind. It certainly isn't too late for you to make some good choices now.

I hope you find what's best for you, though of course I am heavily biased towards USF, and if you should come here, you would have some instant friends. (Just think, sunshine, girls in shorts all the time... oh and math.. lots of math). Here is the course flow chart: http://i47.tinypic.com/2vltump.jpg Let me know if that's not readable and I'll re-size. Looks a bit fuzzy.

-Dave K

Sunshine and girls in shorts sounds awesome but then again, I might be liking the sound of it too much not inherently, but because of my new font. I'm currently running Xubuntu (a linux distrubution) and everything is in something which looks like "Consolas" or "Lucida" - not sure which.

Thank you for the flow chart. It's readable and helpful! The college I attend will depend more on the outcomes of my application, and much less on me, for getting aid (merit or need) is a massive crapshoot for international students. Nevertheless, I think I will apply to USF.

 

[mathwonk]

I went to high school in the 1950's. We studied basic algebra, euclidean plane geometry, trig, and solid geometry.

When I got to Harvard the next year, I was not sophisticated but at least I did know the basics, and I failed to succeed in a Spivak type course not because of lack of advanced preparation, but because of poor study skills as a result of how easy high school had been.

I taught several bright high schoolers here in Atlanta out of Spivak's Appendix on real numbers, and several of them succeeded at Harvard, Chicago, Yale, and Duke.

If the city is ok with you, Chicago is a great place. And because a lot of kids don't want to live there, relative to Boston or New Haven, or Berkeley or Stanford, the acceptance rate at least used to be a lot higher than those places, although the quality is comparable.

Fortunately lots and lots of schools formerly considered so-so are now quite good because of the influx of better and better faculty at all levels in US colleges and universities.

 

[Nano-Passion]

^
Sounds awesome! Post here to tell us how things pan out. What is "summer B" though? A summer class for business students?

---

Does anyone have experience with the math departments at these colleges:
- Berea College
- Carleton College
- Reed College
- UChicago
- Colorado College
-Grinnell College
- University of South Florida

These are a few places I'm considering applying for next year. I don't know much about any of them except for what is found on their website and that a number of them are in cold, bleak places. And that they're quite selective...at least, for people who're non-US citizens requiring aid!

Are you implying that it is harder for non-US citizens requiring aid to get accepted into some universities?Serious question because I've actually never heard of that conjecture.

 

[Mépris]

I went to high school in the period 1956-1960. We studied basic algebra, euclidean geometry, and trig, plane and solid.

When I got to Harvard the next year, I was not sophisticated but at least I did know the basics, and I failed to succeed in a Spivak type course not because of lack of advanced preparation, but because of poor study skills as a result of how trivially easy high school had been.

I taught several bright high schoolers here in Atlanta out of Spivak's Appendix on real numbers, and several of them succeeded at Harvard, Chicago, Yale, and Duke.

If the city is ok with you, Chicago is a great place. And because a lot of kids don't want to l,ive there, relative to Boston or New Haven, or Berkeley or Stanford, the acceptance rate at least used to be a lot higher than those places, although the quality is comparable.

Getting into a good study routine is, indeed, of utmost importance. I was bored throughout high school and got by doing the bare minimum and eventually, which worked until the last year. I half heartedly try to change things but I really did not put in the effort. Now that I've grown a little, I am more diligent.

As I said before, I don't really mind. Acceptance rates for international students at all those schools are <5% if I'm not mistaken. What do you think of Harvard's new core requirement? They don't look like they've put much thought into it and it seems to pale in comparison to what Columbia and UChicago have to offer.

Fortunately lots and lots of schools formerly considered so-so are now quite good because of the influx of better and better faculty at all levels in US colleges and universities.

Can the same be said for the large state schools in the south? Namely, Louisiana State and UofAlabama? (Huntsville and the flagship) I understand they offer merit scholarships to foreigners...

Are you implying that it is harder for non-US citizens requiring aid to get accepted into some universities?Serious question because I've actually never heard of that conjecture.

That's the way it works, yes. American colleges have to cater for American citizens, primarily. A number of colleges are need-blind and offer up to the full-need for Americans but for international applicants, they are need aware. Williams, Stanford, Columbia, Middlebury, Macalaster and I could go on...most of them are such. The remaining colleges either cannot meet the full need or can do so for a few students only (Haverford, for e.g, can do so for 3 only) or they don't offer aid to international students at all.

Academic excellence is expected. The admissions decisions is largely dependent on whether one is a "fit" and on...luck, I presume. Granted, it's a flawed system (I'm talking for everyone here, not just foreigners) but it has its advantages. I'd rather applicants be admitted that way, rather than on test scores and grades alone.

Anyway, let's not derail this thread.

http://www.its.caltech.edu/~sean/links.html
http://www.its.caltech.edu/~sean/book.html

I don't remember how I found those but here they are. These resources might be useful to some.

Is anyone here at all interested in applied mathematics? I know Chiro is a math double major but other than him, everyone seems to be in pure mathematics. Of course, I have not decided on anything yet for my exposure is too limited. I have, however, read a few things here and there, and I find the applications to biology (especially neuroscience) quite interesting.

The Courant Institute has a centre for Atmosphere Ocean Sciences, which looks pretty cool. (http://caos.cims.nyu.edu/docs/CP/59/CAOS_poster3.jpg)

 

[qspeechc]

I went to high school in the period 1956-1960. We studied basic algebra, euclidean geometry, and trig, plane and solid.

Are you saying you did spherical trigonometry in school? Or do you mean solid geometry?

 

[mathwonk]

I think I meant solid geometry. However by that time I was beginning to lose interest and did not learn much of that. High school math could be extremely boring in those days, just before the introduction of the "New math".

But just knowing Euclidean geometry and algebra gave me a big leg up over most entering college students today, even ones who have had calculus in high school. I.e. a college freshman who really knows elementary algebra and geometry but not calculus, is much better prepared for calculus than one who does not know those basic subjects well, but has had a calculus class he did not understand either, which is usual.

I myself just retired after 30+ years at a large state school in the south, and the math department there is very strong. The same can be said for many others in this category. In fact it is hard to find a school in the US anywhere with less than a strong math faculty, after decades of influx of good faculty from many countries.

A quick look at USF math web page shows professors from Japan, USSR, Poland,...
LSU looks extremely strong with professors from Stanford, Mich State, MIT, Moscow, etc...
at Univ of Alabama, they include people from Michigan, Warwick, Brown,...
this is the same pattern everywhere. As long as the US is a sound place to live economically, it seems we will never lack for talent.

The greater difference between these schools and the elite ones like Harvard is the strength of the student body, and the expectations. There are still a lot of weak students at these state schools, but not at Harvard, so the courses reflect this. Since a student learns partly, maybe largely, from his/her peers, there is a big benefit to going to a school with a strong student body, especially as an undergraduate. On the other hand, some undergraduates are overwhelmed by the atmosphere at an unforgivingly high stakes school.

Even among state schools in the south, such as UGA and Georgia Tech, the difference between the strength of the student body is greater than that of the faculty, at least in math, in my opinion. The more prestigious school has the stronger students, and in this case also GaTech is an engineering school, which should attract more strong math students.

So you can do well at a state school in the south, if you can take advantage of the strength of faculty, and not be hindered by the relatively less strong student body. Of course for grad school this matters less.

I could well be wrong too with these blanket assessments of schools, so please do your own investigating. After all I have not been to most of these schools. Maybe graduates or current students from some state schools would pitch in here with actual testimonials.

 

[eliya]

^
Sounds awesome! Post here to tell us how things pan out. What is "summer B" though? A summer class for business students?

---

Does anyone have experience with the math departments at these colleges:
- Berea College
- Carleton College
- Reed College
- UChicago
- Colorado College
-Grinnell College
- University of South Florida

These are a few places I'm considering applying for next year. I don't know much about any of them except for what is found on their website and that a number of them are in cold, bleak places. And that they're quite selective...at least, for people who're non-US citizens requiring aid!

I don't go to any of these schools, but I live in Chicago and can second mathwonk's opinions about UChicago. It's in a bad area of Chicago, but their math program is terrific.

It's safe to say that other than UChicago, none of the schools you listed are well known for math. If you want to get a good math education, then you want to look at schools with big math programs. That is, lots of math students and lots of math classes to take. Also see how many general education classes they make you take in your undergrad career. Sure, those gen eds are important, but there's nothing wrong being determined to do math, and having to take 2 gen eds every term can be annoying and set you back in math quite a bit.
Look at the current class schedules in each of those schools and see what classes are offered this year. Looking at the course catalog can be misleading. My school lists a bunch of classes in the course catalog, but many of them haven't been offered in a while. If the school has a graduate program, then it's most likely to have a broad math curriculum and more than 2 math majors. With that being said, there are some small schools that can be really surprising for math. For instance, Harvey Mudd College is a liberal arts school, but it has a strong math program. Small schools seem attractive because you think every professors is dedicated to teaching, but that's also true for big state schools. mathwonk is at UGA and he seems to be quite dedicated to teaching.

As an international student I can tell you not to expect any student loans. I believe international students aren't eligible for student loans (at least government loans). You must be truly exceptional to get a scholarship from the school you're applying to before entering. Once you entered there's a better chance for you to get some money from the school. That means that you'll probably spend at least one year paying full tuition.

I'm not trying to deter you, just trying to make you more informed.


Nano-Passion, I too heard that intl. students; chances of admittance go down if they ask for financial aid. I didn't hear it from anyone who works at a university though.

 

[mathwonk]

I will remark that I just retired from UGA in 2010 so am not teaching there any more. However the young people there and the senior people as well are very dedicated to teaching, and they are in many ways much better at it than I was. We really have a good teaching department, combined with good research.

We do separate our non honors from our honors and major oriented programs. This means we do our best in the service courses, but majors are better served by taking our honors or super honors courses, that are quite small and oriented to majors.
For example we offer the Spivak style calculus class that Chicago offers, but Harvard does not.

 

[Mariogs379]

@mathwonk,

Bit of a specific question but I thought you might be a good source of advice. Here's my background/question:

Went to ivy undergrad, did some math and was planning on majoring in it but, long story short, family circumstances intervened and I had to spend significant time away from campus/not doing school-work. So I did philosophy but have taken the following classes:

Calc II (A)
Calc III (A)
Linear Alg. (B+)
ODE's (A)
Decision Theory (pass)
Intro to Logic (A-)

Anyway, I did some mathy finance stuff for a year or so but realized it wasn't for me. I'm now going to take classes at Columbia in their post-bac program but wanted to get your advice on how best to approach this.

They have two terms so I'm taking Real Analysis I in the first term and, depending on how that goes, Real Analysis II in the second term. I'm planning on taking classes in the fall semester as a non-degree student and was thinking of taking:

Abstract Algebra
Probability
(some type of non-euclidean geometry)

Anyway, here are my questions:

1) What do you think of my tentative course selection above?

2) How much do you think talent matters as far as being able to hack it if I ended up wanting to do grad school in math?

3) I'm also having a hard time figuring out whether math is a fit for me. By that, I just mean that I really like math, I'm reading Rudin / Herstein in my free time, but I've spoken with other kids from undergrad and it's clear that they're several cuts above both ability and interest-wise. Any thoughts on how to figure this out?

Thanks in advance for your help, much appreciated,
Mariogs

 

[mathwonk]

well those are really tough questions. you are at an elite school where very little hand holding goes on, i.e. everyone assumes you know what you want, and they throw the math at you in the best form they can manage, and let it sort itself out.

There are always better people, always. I have been at all kinds of schools, and when I dropped down from ivy schools to state schools i thought well maybe now i'll be the best one here. No, there were still better people, and there always are.

So the choice has to be based on how much you enjoy what you are doing.
If you were hopelessly outclassed and had no chance, of course you should drop out, but that is not at all the case, with your record. a certain level of talent is needed as a prerequisite, but after that entry level qualifying exam, it is all about effort.

 

[Mariogs379]

Thanks, mathwonk. I guess it's just hard to know whether I'd have sufficient ability to pass quals (assuming I was interested enough to get that far).

Also, any thoughts on course selection to make myself as competitive as possible should I decide to apply to grad school? I can't apply this fall since I have to take classes to get more background, take math subject test, GRE, etc, so I'll have spring semester also if need be.

Thanks again,
Mariogs

 

[mathwonk]

it is appropriate to take real analysis, and abstract algebra. you also will want some topology. if you can read rudin and herstein you are ahead of schedule. especially be sure to work the problems. i don't like those books much myself for their explanations, but many people like them. i also studied them but i didn't learn much.

what level calculus did you have? Spivak calculus is an excellent place to learn or review calculus at the level needed for grad school. Dummitt and Foote is also a good place to review algebra, and again the problems are excellent.

 

[dkotschessaa]

So my friend and favorite math professor is willing to do some work with me in the second summer session (starting June 25th). He's given me a choice of books to work with, and we are going to spend an hour a two a week going over them.

One is "The Four Pillars of Geometry" by John Stillwell, the other is "Abel's theorem in problems and solutions" by Alekseev, based on the lectures of V.I. Arnold.

The first starts with things that I have been studying lately in my history of mathematics class. We were on greek mathematics and getting very intimate with Euclid and Archimedes, and doing straightedge and compass construction stuff. So this one looks very approachable to me.

The second is a bit intimidating at first glance, though it looks very interesting.

Any thoughts on this? I think I am leaning towards the first, but only because I am not sure I know anything about the second. The fact that I've not heard of it also gives it kind of an appeal though - like perhaps not a lot of people know about it, thus it might be a good thing to learn if I want to know about something others do not.

-Dave K

 

[mathwonk]

i would pick the one more interesting to you.

 

[Mariogs379]

Heh, when I say I can read Rudin / Herstein, I can, but it's slow going and the problems are really challenging (I think, at least). Rudin's just so slick and often he leaves out little steps that, I assume, he thinks are obvious but make for even harder reading/understanding.

When I had calculus before it was just the standard Stewart book, sounds like Spivak is higher level. How does it differ? For some reason, I got the impression that Spivak was an analysis book...?

Also rando q: Do any math people do game theory in addition to more traditional math research? I know it's often in econ departments but I really have no interest in econ aside from GT...

Thanks again for the help,
Mariogs

PS What're you doing now that you're retired?

 

[dkotschessaa]

i would pick the one more interesting to you.

It might be Abel's theorem, now that I'm looking through it in detail. I like that the book introduces so many new concepts that I'll encounter in class formally later too.

Is Abel's theorem important, special, or well known? One of the purposes of this book is to "make known this theorem." (Implying it's not) Other sources call it the "abel rufini theorem."

-Dave K

 

[Gravitational]

Hey mathwonk, I was wondering if you consider there to be greater levels of prestige amongst different areas in mathematics? Like, are there any area's of mathematics which mathematicians generally view as more important than others, or areas of mathematics which mathematicians view as more prestigious to be working/doing research in.

 

[mathwonk]

probably we all think more highly of the one we ourselves are in. i once heard of someone, maybe grothendieck, an algebraic geometer, disparaging point set topology, but if you ever met one of them, point set topologists are really impressive, and their results are very clever and difficult.

It depends on what you consider prestigious. Maybe some people will argue that a field that touches lots of other fields, and whose results thus have wide applicability, is more "important". But that is defining important as meaning broadly applicable, and someone else could define important as hard, or deep, or exotic, or practically useful, or practically useless as Hardy did.

Therefore I myself (an algebraic geometer) think of algebraic geometry as prestigious. But number theory, topology, and analysis really impress me too. Basically the fields that don't impress me as much are the ones I am most ignorant of. If you read the writings of really great mathematicians like Riemann, you will find he was interested in pretty much everything.

Also fashions change over time. If you look at the International Congress records for a list of areas in which Fields medals are given you will see they change.

It is probably not so much the field that adds prestige to the worker as the other way around. Someone who does great work in a field makes that field more prestigious. Those of us who tag along into that same field afterwards hoping to be linked with that prestige do not make as much difference as someone who helps advance another field, even one that was under the radar before they approached it.

In the 40's topology was very strong. During Grothendieck's era in the 50's and 60's algebraic geometry was king, and Atiyah and Singer gave a powerful synthesis of topology and analysis, while these and others like Bott applied differential analysis (Morse theory) to topology. With Deligne and Wiles, arithmetic geometry (a sort of combination of algebraic geometry and number theory) gained even more visibility.

Donaldson presumably applied algebraic geometry to topology. With Jones we had a beautiful combination of topology and operator theory, I believe. Since Witten, mathematical physics and its applications to algebraic geometry has been very prestigious. Lately differential) geometry applied to topology soared with the work of Perelman on Poincare's conjecture.

Maybe the pattern here is more recent work that combines several fields. And I am completely leaving out combinatorics, algebra, and logic and other subjects, although one of my own colleagues, Robert Rumely, apparently works in the interface between logic and number theory. Most of us can only work in one field, even one small part of one field, but we are wise if we try to be aware of the results and methods of other fields.

So in choosing I suggest going with what speaks to you, but with open ears and open mind toward other areas as well. You might enjoy reading the surveys of current work that appear in the reports of the ICM every 4 years, especially the work of the Fields medalists.

 

[mathwonk]

the abel ruffini theorem is the famous result that there cannot be a general formula involving only radicals and field operations, which expresses the roots of every quintic polynomial in terms of the coefficients. thus it is extremely well known, at least in the approach taken by galois. probably arnol'd means that the solution he will present, presumably due to abel, is not as well known.

 

[mathwonk]

now that I am retired I am looking a lot on the internet, but trying also to stay somewhat active on some problems that interest me on special abelian varieties. there is a conference the next couple of weeks here at UGA in honor of Robert Varley that will bring in a lot of experts on algebraic geometry, and that will be a chance to hear some stimulating talks and maybe get some ideas.

http://www.math.uga.edu/~dkrashen/agssp/varley.html

Being retired though is spoiling, and it is harder to get the energy to commute over there very often. Last summer I taught a summer course in Euclidean geometry for brilliant 8-10 year olds, but that was more than I thought I could keep up with this summer. I fantasized about teaching them elementary differential geometry, starting with spherical and then hyperbolic planes, but I don't really know the stuff myself that well, and we recruited younger and more expert instructors for this summer. these are some of my students from last year in the photo. (Yes those kids were following and doing proofs in rigorous Euclidean geometry, and constructing polyhedra such as icosahedra and dodecahedra, and apparently loving it.)

http://www.epsiloncamp.org/index.php

Its also nice having time to travel with my wife, but the tickets keep going up.

 

[mathwonk]

well to be honest i got through the first few pages, and they were great. those guys just come to grips so quickly and directly with significant results and phenomena.

But what I read jumps around a lot now that I am retired. Last week or two I spent reviewing a proposal for an introductory book giving unusual proofs of basic calculus theorems and unusual facts not everyone sees in calculus.

Here's one result he had that I didn't know: take a continuous function on [0,1] with the same values at 0 and at 1. then its graph has to have some horizontal secants, i.e. there have to be other points x1, x2 where it has the same values as well. Question how far apart do those points have to be? Can you prove there are always points at distance 1/2 apart but not always at distance 2/3?

I am also reading some topics in my specialty that I have not had time to read before, so Goursat will probably sit there awhile longer. I have dozens of books that are worth reading that I have not got to, like Bott-Tu, Mumford's books, Serre's books and papers, Grothendieck's papers, Borel Serre, Fulton, Gauss' books on number theory and on surfaces, Spivak's diff geom, Thurston's book on hyperbolic geometrry, Arbarello Cornalba Griffiths Harris, Matsumura, Weil's book on kahler varieties, Kodaira on deformation theory, Glen Bredon and Godement on sheaf theory...

 

[jbunniii]

Here's one result he had that I didn't know: take a continuous function on [0,1] with the same values at 0 and at 1. then its graph has to have some horizontal secants, i.e. there have to be other points x1, x2 where it has the same values as well. Question how far apart do those points have to be? Can you prove there are always points at distance 1/2 apart but not always at distance 2/3?

If f is continuous on [0,1] and we never have f(x) = f(x + 1/2), then the function g defined by g(x) = f(x + 1/2) - f(x) is never zero. As g is continuous, this means it must be always positive or always negative. If it's always positive, then in particular g(0) > 0 and g(1/2) > 0, which means f(1) > f(1/2) > f(0), a contradiction. Similar proof if g is always negative.

Essentially the same proof can be used for any distance of the form 1/N, where N >= 2 is an integer. My guess is that these are the only distances for which the statement is true, but I don't know how to prove it offhand.

For a counterexample for distance 2/3, just take any continuous function f such that f(0) = f(1/2) = f(1) = 0, and f(x) > 0 on (0,1/2), f(x) < 0 on (1/2,1). For example, f(x) = sin(2*pi*x). No matter which x you choose, either f(x) and f(x + 2/3) have opposite signs, or one of them is zero and the other is not.

 

[mathwonk]

that's wonderful! so you can prove that secants of length 1/n always exist. can you prove no other lengths always exist? what is an obvious function that is positive on [0,1/2) and negative on (1/2,1]?

 

[jbunniii]

that's wonderful! so you can prove that secants of length 1/n always exist. can you prove no other lengths always exist? what is an obvious function that is positive on [0,1/2) and negative on (1/2,1]?

Well, my counterexample of sin(2*pi*x) works for any d > 1/2. But it does not work for any d < 1/2, because the function is symmetric around the point x = 1/4, and so the the function has the same value at the points (1/4 - d) and (1/4 + d).

My first instinct is that if I want a counterexample for smaller d, I should use a sine function with higher frequency, because I want f(x) and f(x+d) to have opposite signs. However, this alone isn't enough, because, for example, sin(4*pi*x) is symmetric around the point x = 3/8, so the function has the same value at the points (3/8 - d) and (3/8 + d). More generally, sin(2*N*pi*x) is symmetric around the point x = (2N-1)/4N.

So this is going to require more thought.

P.S. I wonder if there exists a function that satisfies the condition for every distance of the form 1/N, but for no other distances. My guess is that this is impossible, but I'm not sure how to prove it.

 

[mathwonk]

i just tried to draw one last night with no horizontal chords of length between 1/2 and 1/3 but did not succeed. there is apparently an example though of the optimal type, with horizontal chords of no other lengths than 1/k, constructed as you surmised, from trig functions and polynomials.

 

[Gravitational]

Thanks for the response mathwonk. It's interesting to get some perspective from a real mathematician. I think in particular for me, number theory is my favourite, as it gives rise to such beautiful problems and proofs. Also, I was curious, do you have any bias towards the applied mathematical subjects, as opposed to pure mathematics?

 

[mathwonk]

i know absolutely nothing of applied math and for some reason that is beginning to make me respect it all the more.

 

[Mépris]

i know absolutely nothing of applied math and for some reason that is beginning to make me respect it all the more.

One Cal Newport (has a blog called Study Hacks - he's a CS prof) talks about a theory on this issue. Apparently, when one hears of something and thinks "Hey, that sounds impressive!", it is simply because upon hearing the aforementioned "thing", they try to mentally simulate the steps taken to achieve that "thing". If they cannot comprehend how this was achieved, they classify it as being "impressive". This applies for things that one has an idea of.

You think this reasoning could explain why your respect for (areas of) applied mathematics?

---
My teacher (he's an applied mathematician, btw!) taught me a little "trick" to integrate ln x and this came in handy in my exam last week.

Find: ∫ ln x dx.
Let y = (x)(ln x)

Since,
dy/dx = ln x + 1

It follows that,

∫ ln x dx + ∫ 1 dx = (x)(ln x)

Therefore,

∫ ln x dx = (x)(ln X) - x + C

I can't remember what exactly I had to integrate - I think it was (x)(ln x) - and all I had to do was modify the initial "trick". (Let y = (x^2)(ln x) and do as above...)

One could always go down the "integration by parts" route but for some reason, I don't like it and the method above just looks/feels nicer to me.