Should I Become a Mathematician?

[mathwonk]

since some people here solicit and take my advice i wanted to advertise my credentials.

but the bit limit seems to prohibit my displaying my 10th grade second place geometry trophy. you can take my word for it.

 

[Bourbaki1123]

since some people her solicit and take my advice i wanted to advertise my credentials.

but the bit limit seems to prohibit my displaying my 10th grade second place geometry trophy. you can take my word for it.

I feel like some sort of "Hu won first" gag should ensue but, alas, this is the wrong format.

 

[weld]

I'm firing off new questions whether you like it or not. What I really want to know is what you hear from your peers regarding employment opportunities, mathwonk. Which math subfields generally have more postdoc and tenure positions? Which generally have more applicants, that is competition for such positions? Which subfields are hard to make advances in, which have lots of low hanging fruits? Which have harder competition for gettings grants? Are there any particular subfield which are very susceptible for short-term flunctuations, resulting in a lack of job opportunities, or is this something which affects math as whole (I have the impression that it's more slow moving and stable though)? Are there any particular subfields which are harder to switch over to other subfields from? Which subfields are considered aesthetically satisfying and which are considered aesthetical unsatisfying, according to most mathematicians?

Based on criterias such as the ones I just listed, what are the overall best math subfields to work in would you say?

Also, do you know of any private industry that actually conducts high level pure math research? In academia one doesn't spend all one's time researching, one must lecture, teach and apply for grant as well. Well, how much % of an average mathematician's (On tenure track) time at a research university would you say is spent researching? Like 90%? Would you say it differs once one gets tenure?

http://en.wikipedia.org/wiki/Category:Research_institutes_in_the_United_States

That link contains a huge list of private research institutes. I don't expect anyone to read through all that of course. But I wonder, are places like "American Institute of Mathematics" and similar worth working at? Are they hard to get permanent jobs at?

Facts and figures aren't necessary if you have no such thing, but anecdotes, what you experience yourself, what you hear from peers and conventional wisdom among mathematicians are all greatly appreciated. After all it would kinda suck finding out too late that there's no jobs in one's selected subfield, now wouldn't it?

 

[Bourbaki1123]

I'm firing off new questions whether you like it or not.

lol wut?

I'd mention again that mathematical statistics people seem to be getting a lot of funding.

Which subfields are considered aesthetically satisfying and which are considered aesthetical unsatisfying, according to most mathematicians?

I would like to see a statistical breakdown of this, that said http://en.wikipedia.org/wiki/Mathematical_beauty#Beauty_and_mathematical_information_theory" could indicate that subjective beauty would depend largely on what sort of maths you've already learned. A very deep result that unifies your knowledge into a simple framework is what you will find to be the most beautiful.

Based on criterias such as the ones I just listed, what are the overall best math subfields to work in would you say?

You should probably realize that that is a non-trivial question, and you probably need to do some actual heavy duty data collection to get a decent answer. If you actually could get a solid, statistically well supported answer, you could probably publish your result!

Yes, you can gather anecdotes, but I think if you've actually got some time before grad school you could spend a fraction of your time gathering relevant data.

 

[weld]

A very interesting link you provided there Bourbaki. Thanks. I'll definitely look deeper into the pyschology underpinning math. If you have any other good sources regarding math and psychology I would be happy to see them.

Anyways, yes, I'll probably search around the web to find those much desired statistics. And I've got the time to do it as well. Part of the reason I posted here was that I hoped someone knew at least something regarding working in subfields or that someone had at least some semi-relevant and interesting knowledge at hand.

 

[mathwonk]

I don't have much of that kind of data, partly because it never interested me and partly because I am retired and do not belong to hiring committees anymore.

I just did math because I loved it more than any other subject and wanted to learn as much as possible. Then I chose algebraic geometry over my first love, algebraic and differential topology, because it was more difficult and hence more fascinating. Once I almost resigned the one permanent job I did have in favor of a temporary position at harvard because i thought it was a more mathematically exciting place to be.

I do not advise choosing a field solely for any of those practical reasons you give, because math is so hard and so competitive in every subfield that I doubt one can survive mentally unless one has a strong enjoyment of the stuff one thinks about.

You also have to enjoy teaching because the percentage of time you have in academia to think about research is MUCH less than 90%. Teaching, tutoring, advising, grading, writing notes, serving on committees, hiring, voting, writing dossiers for other people to receive awards or promotions, interviewing, preparing prelims and tests, helping students prepare for them, writing or reviewing grant proposals, revising and writing up largely finished results, ...these activities consume most of your time, especially teaching and grading.

I used to try to set aside 3-5 hours one day a week to discuss research and it frequently got cut into by other duties. Back when our teaching loads at UGA were the highest of any research university in the nation, I often noticed that research work on my computer was only updated during holidays, thanksgiving week, christmas week, spring break, summer...

My son majored in math with emphasis in numerical methods I believe and he has a good job that pays well in silicon valley. But he is very smart and very hard working and his field too is extremely competitive. He is on call essentially 24 hours a day, 7 days a week, even when on vacation, to "put out fires" at his company. And lots of his friends in the same industry have lost their jobs over the past several years.

Some people I know seem to enjoy their work at NSA, if you do not mind working for the government, say breaking codes, or making them.

 

[weld]

Thanks for the helpful reply mathwonk! :D

You also have to enjoy teaching because the percentage of time you have in academia to think about research is MUCH less than 90%. Teaching, tutoring, advising, grading, writing notes, serving on committees, hiring, voting, writing dossiers for other people to receive awards or promotions, interviewing, preparing prelims and tests, helping students prepare for them, writing or reviewing grant proposals, revising and writing up largely finished results, ...these activities consume most of your time, especially teaching and grading.

I used to try to set aside 3-5 hours one day a week to discuss research and it frequently got cut into by other duties. Back when our teaching loads at UGA were the highest of any research university in the nation, I often noticed that research work on my computer was only updated during holidays, thanksgiving week, christmas week, spring break, summer...

What? I've heard elsewhere that in math there's almost no such busywork. That you get like 89% of your time for research as a tenured prof, and 79% when on tenure track. I really would like to believe one gets away from teaching duties and other forms of busywork when in math but if reality says, otherwise, then I can't deny. You sure having way less than 90% of your time for math research is normal in most universities? Cause you said your uni at one point had the biggest teaching loads in the nation, so you sure it's not unique to UGA?

My son majored in math with emphasis in numerical methods I believe and he has a good job that pays well in silicon valley. But he is very smart and very hard working and his field too is extremely competitive. He is on call essentially 24 hours a day, 7 days a week, even when on vacation, to "put out fires" at his company. And lots of his friends in the same industry have lost their jobs over the past several years.

Interesting. How many hours/week would you say Silicon valley people put in, like 70-80/week average? Also, say I got an education in CS and got a job at some well known tech company, would the research done be as interesting as that which is done in academia or mediocre in comparison?

Some people I know seem to enjoy their work at NSA, if you do not mind working for the government, say breaking codes, or making them.

By doing this, are you doing something essentially new like when doing research? Discovering anything new? For working within the same old limited paradigms gets old quick. :(

 

[Bourbaki1123]

By doing this, are you doing something essentially new as you are when doing research? For working within the same old limited paradigms gets old quick. :(

Not only would you being doing stuff that is new, you'll have access to top secret bleeding edge math, algorithms, technology etc. The NSA actually makes you sign a legally binding contract to not work on what you had been doing there (I think it "only" binds you for a decade though) if you decide to leave.

 

[weld]

That's interesting Bourbaki. Would you say a NSA employee is essentially a researcher without busywork (teaching, applying for grants, sitting in committees etc)? Also, do you think they research things other than cryptography there? Also, what are the cances of gaining employment at a place like NSA? What do you imagine thw orkweek and job sceurity would be like? Bad job security and like 80h/week?

 

[mathwonk]

bourbaki, i guess first place probably went to jimmy fidelholtz. at least my teacher used to say we should not expect to beat him, as he had worked his way through a calculus book all by himself.

 

[Bourbaki1123]

bourbaki, i guess first place probably went to jimmy fidelholtz. at least my teacher used to say we should not expect to beat him, as he had worked his way through a calculus book all by himself.

I'm not sure if I should, but I find that to be pretty humorous. It evokes the image of an elderly schoolmarm peering down at you through her spectacles, "You ought not expect to outdo young Mr. Fidelholtz as he has worked through a calculus text all by himself, isn't that right jimmy?" :panning shot to Jimmy with a sh**-eating grin on his face:

Any idea what has become of him?

 

[mathwonk]

i googled him and he seems to be a linguist living in mexico.

 

[Bourbaki1123]

That's interesting Bourbaki. Would you say a NSA employee is essentially a researcher without busywork (teaching, applying for grants, sitting in committees etc)?

Sure, but you have to add academic freedom to the list; I doubt they'll let you research anything you want to. It has to be stuff that is critical to national security, or at least could be critical down the road. Mostly cryptology and computer security stuff as far as I know; of course I could see some game theory and computational complexity stuff as well.

Also, what are the cances of gaining employment at a place like NSA? What do you imagine thw orkweek and job sceurity would be like? Bad job security and like 80h/week?

These sorts of things I can't really give you any solid answer since I would like to know myself.

 

[deluks917]

I'm taking a graduate course in complex analysis. How hard should this class be? It appears we are skipping over things that seem important to me (admittedly I don't know what I'm talking about here). Is it a bad sign that the class is not covering the proof of Looman-menchoff and "Big" Picard. I don't know the subject yet but I purchased Narasimhan's book and he proves both of these. I just hope the class covers enough material. Its much easier to learn from a class than a textbook (which I'll have to do). Mathwonk you've taught Graduate complex do you cover these sort of results?

 

[weld]

Sure, but you have to add academic freedom to the list; I doubt they'll let you research anything you want to. It has to be stuff that is critical to national security, or at least could be critical down the road. Mostly cryptology and computer security stuff as far as I know; of course I could see some game theory and computational complexity stuff as well.


How interesting do you think cryptology is? Also, is there a "general" consensus of how interesting it is? Also, pretty much the same questions regarding game theory and complexity. Anyone one with an opinion on this, fire away.


Mathwonk, thanks for mentioning NSA. You also mentioned Silicon Valley. Do you or anyone else know of any other corporations/ organizations which does research in pure math, theoretical physics and theoretical CS, of course without busywork like teaching? If not, do any of you know of places which do applied research in math and CS, but still keep it very interesting?

I'm curious as to how interesting it really is to research at places like google, MSoft, NVIDIA, Intel, AMD, IBM, Adobe, McAfee, Apple, Mozilla, Netflix, SONY, just to name a few. Any info you can provide is interesting.



I've been reading up on research institutes. Many promising ones out there like Kavli, International Centre for Mathematical Sciences, Institut des Hautes Études Scientifiques, Institute for Computational and Experimental Research in Mathematics, Enrico Fermi Institute. But I wonder, just how good must one be to realistically have a chance of gaining permanent employment at such places?

If one can't gain permanent employment, can one survive by simply hoping back and forth between several institutes which offer short-term employments? There are those which have perma and temporary (Like Hautes Etudes), and also those which primarily focus on temporary, like Mathematical Sciences Research Institute, Institute for Pure and Applied Mathematics and Mathematical Research Institute of Oberwolfach.

 

[mathwonk]

i'm falling behind here. i do not usually cover looman menchoff (can't even remember what it says but it seems peripheral in memory) nor big picard. the main picard result is little picard and then you use it an infinite number of times plus normal families to get 'big" picard.

i don't know what level you are at in background, but i recommend starting from frederick greenleaf, then henri cartan, then lang, among the many good complex books. the most comprehensive (includes big picard) is einar hille's two volume set.

the books by konrad knopp are also quite interesting but very brief.

there are also good features about the classic of churchill, and the book by redheffer.

there are many good complex books. the most famous, by ahlfors, is one of the few i myself do not recommend, as being rather more difficult to read than average for complex books, but it does have a nice chapter on infinite products.

 

[mathwonk]

i'm sorry weld, i know little about this. i am looking for work myself.

 

[Number Nine]

How interesting do you think cryptology is? Also, is there a "general" consensus of how interesting it is?

As an amateur cryptography aficionado, I'll try to field this.
The whole issue really depends on what you're doing in cryptography. Many modern cryptosystems (RSA, ECC, etc) are designed in a such way that they really can't be "broken" in the sense that ye cryptosystems of olde were, and the most anyone really aspires to is the development of some polynomial time algorithm that will decipher the thing in an order of magnitude or two fewer billion years than the ones currently available (there's a saying in cryptography: "Crypanalysis is dead", at least as we know it).

If you want an idea of the kind of problem a cryptographically inclined mathematician might be interested in: The RSA algorithm takes advantage of the difficulty of factoring a large (200+ digits) number into its prime factors (the most efficient known algorithms (e.g. number field sieves) still run in exponential time). Another cryptosystem (ECC) takes advantage of the difficulty of calculating logarithms of points on elliptic curves. Others deal with the decomposition of groups into their generators.

Most cryptanalysis at this level takes place at the absolute deepest, darkest, most complex corners of number theory (be it elementary, algebraic, or analytic). You'll have to decide for yourself whether you find that interesting.

 

[weld]

Thanks N9! Also, mathwonk, most math professors have to teach 3-4 classes a year, right? Do you know any places where it's normal to only have to teach 1 or 2? What about other hard sciences professors, do they generally teach 3-4 as well? Are there any exceptions in hard sciences where one teaches 1-2 instead?

Also, how much does the average math professor make? If one is decent at getting grants, can one make tons then? What if one is really good at it?

I know some professors do contract work for industry sometimes, is this generally better paid than other types of work a professor can do?

 

[Bourbaki1123]

As an amateur cryptography aficionado, I'll try to field this.
The whole issue really depends on what you're doing in cryptography. Many modern cryptosystems (RSA, ECC, etc) are designed in a such way that they really can't be "broken" in the sense that ye cryptosystems of olde were, and the most anyone really aspires to is the development of some polynomial time algorithm that will decipher the thing in an order of magnitude or two fewer billion years than the ones currently available (there's a saying in cryptography: "Crypanalysis is dead", at least as we know it).

If you want an idea of the kind of problem a cryptographically inclined mathematician might be interested in: The RSA algorithm takes advantage of the difficulty of factoring a large (200+ digits) number into its prime factors (the most efficient known algorithms (e.g. number field sieves) still run in exponential time). Another cryptosystem (ECC) takes advantage of the difficulty of calculating logarithms of points on elliptic curves. Others deal with the decomposition of groups into their generators.

Most cryptanalysis at this level takes place at the absolute deepest, darkest, most complex corners of number theory (be it elementary, algebraic, or analytic). You'll have to decide for yourself whether you find that interesting.

I've done research in post-quantum or algebraic cryptanalysis, where the issue is ostensibly that an scalable quantum computer could potentially break some of these cryptosystems. While not immediately applicable to anything but toy cyphers, it promotes a lot of interesting complexity results in computational algebraic geometry and a lot of interest in algebraic geometry over finite fields. Both of those are areas that I find pretty interesting in terms of their pure mathematical/computational properties. Of course, there is also the hope that certain methods could actually successfully exploit the algebraic structure and take the problem of breaking the cryptosystem down to manageable complexity (the current methods use Grobner Basis and SAT so are NP complete, the idea is to exploit the algebraic structure of the cryptosystem to narrow down the search space).

 

[Number Nine]

I've done research in post-quantum or algebraic cryptanalysis, where the issue is ostensibly that an scalable quantum computer could potentially break some of these cryptosystems. While not immediately applicable to anything but toy cyphers, it promotes a lot of interesting complexity results in computational algebraic geometry and a lot of interest in algebraic geometry over finite fields. Both of those are areas that I find pretty interesting in terms of their pure mathematical/computational properties. Of course, there is also the hope that certain methods could actually successfully exploit the algebraic structure and take the problem of breaking the cryptosystem down to manageable complexity (the current methods use Grobner Basis and SAT so are NP complete, the idea is to exploit the algebraic structure of the cryptosystem to narrow down the search space).

The notion of taking advantage of the structure is, I think, what makes the whole business so interesting from a mathematical standpoint (a great example would be the various special number field sieves).

Weld: You can work on pretty much any level you want, from more applied areas like the actual implementation of the cryptosystem itself, information theory etc, to what we discussed above, which is essentially pure mathematics.

 

[simplicity123]

Hi you guys.

Anyway, do you need to actually know any Physics to do stuff like Quantum topology, Mirror symmetry, Quantum chaos, or Quantum group theory. As I'm interested in Physics, but more gifted at logic. If I was 20 I would switch to Mathematical Physics even through I'm not that good at it. But, I'm 22 so don't really want to start in first year to do Physics as I'm in third year now.

Also, is model theory useful if you want to go into category theory? As it looks interest, well it looks like alien writing. Like I remember picking up a model theory book and was like is this Maths?

 

[weld]

N9, that's interesting. Do you know any other places than NSA which do a lot of cryptography?Also, would you say experimental physics work is boring compared to theoretical? What's the general consensus? I have the impression that experimental is full of boring, mundane gruntwork but I might be wrong on that.

Regarding how much physicis profs makes, I saw this thread: https://www.physicsforums.com/showthread.php?t=154223

And began wondering how much do they really make? Is it really as bad as 90k after many years of experience as some say or can one end up making 200k?Do postdocs get overtimepay? What about overtime pay for profs, research profs, assistant profs, staff scientist?

 

[mathwonk]

in money terms, i know 30 somethings out there in the internet world, with an undergraduate math degree, who make 5 times what i do as a well known researcher in pure math with a phd and postdoctoral experience at harvard. if you are after money, become a salesman. deceiving people is always more lucrative than enlightening them.

 

[Sankaku]

if you are after money, become a salesman. deceiving people is always more lucrative than enlightening them.

That has to be the best line I have seen all week!

 

[kramer733]

in money terms, i know 30 somethings out there in the internet world, with an undergraduate math degree, who make 5 times what i do as a well known researcher in pure math with a phd and postdoctoral experience at harvard. if you are after money, become a salesman. deceiving people is always more lucrative than enlightening them.

What do these men do with their undergrad degree that makes them so successful?

 

[weld]

That has to be the best line I have seen all week!

Agreed.


That's pretty terrible mwonk. If you don't mind me asking, how much do you make? What's the payment range for assistant profs, full profs and research profs at your uni?

 

[lisab]

That has to be the best line I have seen all week!

Totally agree...I put it in my sig, even .

 

[mathwonk]

as a full professor i never made it to 6 figures, after 40 years in academe with some (mathematical) success. only about 8% of all professors in the state of georgia make 6 figures (mostly in medicine and engineering), which was recently attacked in the ajc as a scandal. I.e. it was considered a scandal that there WERE any such professors. now i am retired on considerably less.

but i have a home, a car, a wife and two educated children, friends, clean clothes, rosy cheeks [at least my icon], I've learned to identify good 12 dollar wine. i mean what do you want out of life?

 

[backthen]

Mathwonk,

I have got halfway through this thread. I am aware of your preference for calculus books by Spivak, by Courant, and by Courant and John. Of today's typical books, are any commendable, or are they all only typical?

At page 77, there is an exam for persons wanting to place out of first-semester calculus. One question asks for a power series solution of a differential equation. Power series? In first semester? Was that textbook material, or class material?

I graduated in '66 wanting to get into computers. But I never reviewed afterward and lost it all. I am wondering what I can do by looking up syllabi, assignments and lecture notes online. Can you point in any direction?

Thanks.

 

[TheCool]

You also have to enjoy teaching because the percentage of time you have in academia to think about research is MUCH less than 90%. Teaching, tutoring, advising, grading, writing notes, serving on committees, hiring, voting, writing dossiers for other people to receive awards or promotions, interviewing, preparing prelims and tests, helping students prepare for them, writing or reviewing grant proposals, revising and writing up largely finished results, ...these activities consume most of your time, especially teaching and grading.

I used to try to set aside 3-5 hours one day a week to discuss research and it frequently got cut into by other duties. Back when our teaching loads at UGA were the highest of any research university in the nation, I often noticed that research work on my computer was only updated during holidays, thanksgiving week, christmas week, spring break, summer...

Quick question, Mathwonk: Did the math profs at your school have freedom in choosing which textbooks were used in their courses? At some schools, professors do not have a say in that matter.

 

[mathwonk]

easy question: professors usually get to choose the textbook for advanced classes like abstract algebra or any grad class, but the committee usually chooses the calc book.\

as for textbook recommendations, those are advanced honors class recommendations for top math majors. the rest of us take normal books. the trouble is the normal books are not as well written.

if you hunt around here you will also find my suggestions for normal calc books, like cruse and granberg, thomas, thomas and finney 9th edition...

 

[weld]

http://en.wikipedia.org/wiki/Category:Mathematicians_by_field

Take a look at that link. There's tons of combinatoralists, topologists, number theorists, mathematical analysts and probability theorists, but not so much of others. What's up with that?

 

[sentient 6]

I have been wondering whether or not to attend a liberal arts college for my undergraduate, however, to be honest I don't know where to start looking. I was wondering if anybody had any recommendations for ones that are strong in math. Thanks in advance

 

[Jack21222]

I admit I haven't read all 169 pages of this thread, so I apologize if questions like this have been answered before.

I'm a physics undergrad, but the research area I'm interested is tucked away in many "applied math" departments. I'm interested in nonlinear dynamics and chaos theory, which while it has many physics applications, fits better into a math program.

However, I really haven't enjoyed math classes at the math department so far. I don't like things that get too abstract, and I hate rigorous proofs. I love the math as taught in the physics department, which is full of appeals to physical reasoning and mathematical models of physical situations.

Do applied mathematicians have to deal with rigorous abstract proofs, or is that more for the "pure" mathematicians? Do you have any suggestions for a physics major applying to applied math programs?