Other Should I Become a Mathematician?

[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?

 

[Nano-Passion]

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

I agree, can I post that on my facebook? I will quote you. ^.^

 

[mathwonk]

I am one pure mathematician who is guilty of teaching lots of math courses filled with proofs and short on applications. the truth is i taught what i knew and was interested in myself, and what was in the books we used. now that i am old and a little wiser i might teach differently but i am retired.

my logic in the old days was that understanding the ideas would enable you to apply them yourself, so i hope that is true.

 

[Jack21222]

I am one pure mathematician who is guilty of teaching lots of math courses filled with proofs and short on applications. the truth is i taught what i knew and was interested in myself, and what was in the books we used. now that i am old and a little wiser i might teach differently but i am retired.

my logic in the old days was that understanding the ideas would enable you to apply them yourself, so i hope that is true.

My problem is I struggle to understand the ideas without something real to connect it to in my mind. That's why if I pursue math, it'd be applied math.

Do you think a student who gets bored of proofs and abstractions, but is good with calculations can survive in an applied math grad program, such as this: http://www.amsc.umd.edu/programs/doctorate.html ?

 

[mathwonk]

well i can't tell from a website what the courses are like. even in our department we had professors who understood the importance of applications and emphasized them in their classes. they were recognizable by their class evaluations which emphasized this.

you see i am also becoming [too late?] more flexible in this regard. so you too should become flexible as early as possible and try to learn the pure stuff while also continually asking applied questions to provoke - inspire your professors to respond to them. even the pure guys know really a lot that they can convey if pressed. good luck.

 

[mathwonk]

have you noticed we are over 500,000 views? of course 400,000 of those are mine.

 

[Math Dad]

Greetings Mathwonk,

I am a retired engineer whose main job now is to help my 7th grade and 3rd grade kids on math. I happily find this thread, and plan to spend time to read through it. It's really nice to have a real mathematician around to provide help. Really appreciate.

A little background on myself. I have PH.D. degree on E.E. I thought math was pretty easy as a kid, until I encountered my first setback at Calculus. After a short career after undergraduate, I was getting better academically and then entering graduate school. At graduate school, I took several Math graduate course, including Algebra, Wavelet (both for my related area), and Topology (for no particular reason, just to test the mature of my math.). That was the peak of my academical life. I forget a good part of those stuffs after 15 years professional career ( which relied a lot on Fourier analysis). During that period, I can not find time to study as much as math I liked. Until recently, I have chance to study Calculus again.

My first question is do you really think "Mathematics is a branch of Physics"?

 

[Math Dad]

Professor Mathwonk,

Two more questions, thanks for the attention.

You mentioned (and I totally agree) the book "What is Mathematics" by Courant and Robbins as a good survey for math before undergraduate. Do you know any book play the same role as good survey for math before graduate? If not, anyone come close? If no, can it be done?

How do you categorize the book "Concrete Mathematics" by Graham, Knuth and Patashnik? (Similar question seems have been discussed in post #338~#340, please ignore it.)

 

[Nano-Passion]

As an electrical engineer I think physics at the college level gives people a clearly better background than math. I've met and I've worked with both categories and in most cases the math education seems narrower. I've always been impressed with physics graduates working in various companies. I cannot say the same about the math graduates.

If you want a beter standard of living go to an engineering school and specialize in EE in particular analog design. Within a few years of graduation you can be making over 120K or even more and I am not talking of California where salaries are higher.

Of course some people are purists and dream of shaping the mathematics field. Good luck with that. When we were young and naive most of us had such dreams. Nowadays the education is such that the degrees don't mean much anymore. Most of the people in industry or academia are simply parroting stuff from books and don't understand the science at a very basic level. From a handful of guys creating a treasure of knowledge 100 years ago out of nothing, the scientific field moved into a situation where hundreds or thousands of scientists with large budgets and equipment can barely make some incremental and slow progress.

Some might counter me by saying that there are so much more patents awarded and papers published nowadays. That's true, but 50 or 100 years ago a paper was published when the author had something important to say and today most of the papers are iddle chatter adding very little to what was already said. This is because the system forces increased minimum quotas on scientists while at the same time making them more compliant rather than more inquisitive and curious. Now more and more people go to school and get a degree than ever before. Is the degree type of any importance? I think it is, but less than what some people believe.

My blog: http://excelunusual.com"

Holy cow man, way to crush spirits. While you do make a sound argument.. I would say that there have been scientists throughout history that are ill-mentioned. The few that are known today are just some of the major contributions. You should notice that back then there would have been a large percentage of people not making it into the big names just as it is today. But that isn't to say that the probabilities aren't stacked high against you.. but still... nothing ventured nothing gained. Sometimes, dreams do come true. Albeit, a very small percentage.