Should I Become a Mathematician?

[Cod]

I just wanted to say that I saw this thread a long while back and picked up ,
PRINCIPLES OF MATHEMATICS - SECOND EDITION (Allendoefer ) off your recommendation of it; it's a great book, I really knew nothing before reading that book.This was years ago, but I just felt like saying thanks :)
By the way, how is the career market generally for applied mathematics in areas such as cryptography or data mining? Would you say that the jobs are particularly scarce? Competitive?

The National Security Agency (NSA) is always looking for cryptologists and data miners. Granted, you have to be able to obtain a security clearance and pass a lifestyle polygraph.

Data mining is popular in an abudance of careers. Biology / medicine is becoming a big one since more and more gens are being mapped and proteins folded.

 

[mathwonk]

Great suggestion. One of my former algebra students who then wrote a PhD thesis in algebraic geometry with Robert Varley, works at NSA. In fact I think she even was there before grad school and received support to come back to school. So that is a great job.For graduate students I just found a terrific advice discussion on the web page of Ravi Vakil at Stanford, intended for future PhD students of his, but great for anyone wanting to learn to be a mathematician.
http://math.stanford.edu/~vakil/potentialstudents.html

Notice that before aspiring to be his student you should do something I have not managed yet after retiring as a career algebraic geometer, namely work most of the exercises in Hartshorne. So for one thing that suggests it is not always necessary to know everything you "should" know, and on the other hand it suggests how much I have limited myself in what I could achieve by not preparing myself technically as far as advisable.

In that vein I have just discovered a forum called "mathoverflow", like this one only the technical question are usually at a research level. Even in the algebraic geometry section I hardly even know enough to understand many of the questions much less answer them. But it is fun and stimulating. I pretty much limit myself to questions on the classical theory of Jacobian varieties, and matters of teaching, but it makes me want to learn the more high level tools.

Notice however that in research it is not always what you know as what you can see how to do. In my experience, I have had occasions where I answered someones question merely by saying, well I do not know what this topic means, but if you can do what you say you can do, then you can also do what you want to do, because the two are analogous in the following way...

Of course then someone who actually understands the subject has to explain it to them, if they do not see what I mean. I.e. to me all discovery is about analogy of things known to things to be learned.

But just look at Ravi's publications, and see that his advice should be followed. I.e. it does no good to have a speculative research insight if you cannot then pursue it and verify it rigorously, and that requires technical power. Also even just making good conjectures often cannot occur until you have made enough technically difficult computations to generate some data.

So read and follow Ravi's advice, not mine, as he has had a lot of successful students and I have not.

 

[hunt_mat]

If you want to became an applied mathematician I would look to specialise in maths and physics. This is my outlook being an applied mathematician.

 

[radou]

Mathwonk, is the ability to think of your own problems and solve them (less or more trivial, for a start) an ability which comes with lots of work and training, or is there an individual factor of talent which can't be neglected? I'm sure there is, but I'm just interested how far you can push without special "talent" for mathematics (acutally, maybe the term isn't well defined enough either) if you work hard? Let's define "pusing far" to be to reach a PhD level.

 

[radou]

Of course, this question isn't addressed to mahwonk only, any oppinions are welcome. :)

 

[mathwonk]

I believe one can accomplish quite a lot by hard work, including get a PhD, but it is really hard. i found it the hardest thing I had ever done. My advisor suggested a problem to me and also how to attempt it. Even after that, I have always found it difficult to think of problems.

I guess you just have to practice it as much as possible. The more you listen to good talks, and read and think about good papers, the more ideas you may have.

In real life most of us are not good at everything, not even at every aspect of our own specialty. We are good at best at some one aspect of it, and we get by working very hard toward improving the other aspects, but focusing on our strength.

As an example, Dennis Rodman was a very successful pro basketball player, but when i saw him, he did not do much except rebounding. He just stayed under the basket all the time and rebounded as hard as he could. But that was all they needed from him.

 

[radou]

I am currently going through Munkres' general topology, which I find quite comprehensive, taking into account my mathematical ability, which isn't specially high, since I'm no methematician. The exercises are very useful, since they help you revise what you've learned (actually, nothing is really learned if no exercises are solved) help you gain more understanding of the subject.

What do you think would be "logical" after going through general topology? Algebraic topology is the next big part of the book, but I heard it's quite hard (a mathematician friend of mine said it was hard, and she was an excellent student).

Perhaps looking into abstract algebra would be useful first? I have Hungerford, but I don't think it's quite suitable for my level yet, so I'm thinking about first going through a more "undergraduate" text on abstract algebra, perhaps the lecture notes from our faculty of math.

As an amateur, I invest a considerable amount of my spare time into doing math, so I'd like to invest it as best as possible.

 

[mathwonk]

for abstract algebra you might try dummit and foote, as it is written so as to be an introduction starting from the beginning and going on through graduate topics. For various reasons it is not my favorite from my perspective, but it is clear and has lots of problems, which may make it a good choice for learning.

I also have free algebra notes on my website at several levels. The first level, math 4000, is based on the nice book by Theodore Shifrin, something like algebra from a geometric viewpoint.

Do you have linear algebra yet? the basic order should be something like: linear algebra, then abstract algebra including groups and rings and fields, and then algebraic topology.

But i you find a good book on algebraic topology you can understand, and that introduces the algebra needed, it may be fine to go on with it. One nice algebraic topology book is by artin and braun, but maybe hard to find.

But you should first learn some linear algebra if you have not done so yet.

There are also free, but condensed, notes on my website for math 4050 I think, which should be advanced linear algebra. And if you are willing and able to do exercises, the little 15 page book there on linear algebra might help some too.

I like Lang's books for insight and brevity, and he has an undergraduate book on linear algebra which should be an excellent introduction. The best books introducing algebraic topology for beginners are probably those of andrew wallace. anything he writes is written to be readable.

 

[radou]

Thanks for the reply, mathwonk.

I am familiar with linear algebra, since I took and passed two linear algebra courses at the faculty of mathematics (I did this while I was studying civil engineering, the study system changed in a few aspects in our country now, but back then, you could take any courses you wanted from another faculty, and go for the exams of course, but people generally didn't know about this possibility - neither did I, and if I had even earlier, I'd take a lot more math courses). We had a great professor there, his lecture notes were enough to grasp the most important parts of the subject (at least in my oppinion).

OK, I'll try to look into the basics of abstract algebra first, then.

As I mentioned above, I started to read Hungerford (let's say two years ago), and I found it a bit hard back then, although I did most of the exercises of these 4-5 chapters I went through, but I constantly felt that "something was missing" here.

Also, before going through Munkres general topology (the first part of the book I'm going through right now), I read a set of lecture notes about metric spaces and topology from our faculty here, and if I didn't do so, I'm convinced I'd find Munkres much more hard.

So basically, there always seem to be certain levels of comprehension for a subject, and choosing the right books seems crucial to learning mathematics at different stages.

 

[deluks917]

I really recommend Artin's algebra to everyone. Somewhat on topic Artin's book made most of linear algebra much clearer to me then two books spefically on linear algebra (Insel and Friedberg, Axler).

 

[mathwonk]

I agree that artin's algebra is outstanding and if you like it, it is more highly recommended (by me) than dummit and foote. actually dummit and foote contains more topics, and more exercises, but artin is a great master and will offer more insight, hence is to be preferred. maybe dummit and foote could be a supplement or read later. artin is also a "first" book, although aimed at mit undergrads.

 

[G037H3]

hey mathwonk, I have a question about Principles of Mathematics by Allendoerfer and Oakley

I ordered the Third Edition off of Amazon, is it as good as the Second (and First) Editions?

 

[mathwonk]

I'm sorry I don't know. I used it in high school in 1959-60, probably an early edition, but it shouldn't be much different. The sections on logic really helped me as I had never understood what a converse, much less a contrapositive, was. It is very useful to understand the logical structure of a mathematical statement. That book also taught me what complex numbers were rigorously, and not to be afraid of them. The parts about countable and uncountable sets were also very exciting and mind blowing. I followed up on those in some more books on set theory.

 

[mathwonk]

Here is another resource I just discovered for PhD students and people with high level math questions that may not always get answered here. There is a site called mathoverflow, where many professional mathematicians post regularly. There are so many experts that most questions, even quite advanced, get answered quickly it seems. If you have a research level or advanced graduate level math question that goes begging here, you might try it there.

 

[deluks917]

Something I've wondered about is what actually is considered "good" preparation for math graduate school. Certainly you should need undergraduate analysis and algebra (Say Rudin/Artin or D&F) and some basic topology/ODE/Complex Analysis/PDE. This sems like it would be sufficent to take the usual first year graduate courses however this is clearly not ideal. I wonder what top schools really want in terms of background knowledge.

I ask because the school websites give very conflicting answers. Harvard's prelim exam is apparently passed by a fair portion of entrants to the program day one. However to pass it requires graduate Algebraic Topology and Geometry along with Differential Geometry, Algebra and Analyis. Columbia as a list of recommended reading on their website but it didn't seem to have anything unreasonable on it. Are Columbia's expectations that much lower then Harvards? Most schools only give accounts of the minimum acceptable standard but what do you think they'd really like to see?

I don't think there is a clear answer to this question and even if there was I'm not sure it'd be worth rushing through the ciriculum so you cn finish spanier's geometry before getting to grad school. Still I think its good to know what you need to be as prepared as your fellow grad students at a good school (not nescessarily harvard).

 

[mathwonk]

If you are interested in what i wrote on admission to grad school in the past, try posts 699, and 176-186 in this thread. I probably knew more back when i was in the game of admissions than I do now. Of course it helps to know and understand as much as possible.

 

[QuantumP7]

I'm 28 years old, and am just beginning a journey into pure mathematics. I'm a physics major. I haven't gotten my degree yet (I'm severely hard-of-hearing. I wear a hearing aid in my good ear, and am hoping for a cochlear implant in the other.), but since I am out of school for this semester (money issues), and since I've decided that I want to be a theoretical physicist, I decided to get caught up on math.

I wish someone had told me a long time ago how beautiful pure mathematics is. The math I've taken (up to ordinary differential equations so far) has been geared towards science and engineering majors, so I didn't get a taste of theoretical math until about August of this year. I love it so much, that I want to get at least a master's in math. I'm working on Munkres' Topology now, and am hoping to get Artin's Algebra book soon.

So, yeah. I just wanted to put that out there. I LOVE math.

 

[Dougggggg]

I am still young into my undergraduate studies, double majoring in Physics and Mathematics Education (just in case I don't go to grad school I would like to have a job). I am only in Calculus I and we just got through discussing the Fundamental Theory of Calculus. I am by far the best in that class excluding the people that have taken Calculus in high school or at another college before. I normally excel in my math and physics classes and really enjoy both subjects. I haven't quite decided which of the 2 I want to go into but I worry that me saying I like math doesn't really matter for anything because I really haven't taken any higher math yet. Next semester I will have Intro to Abstract Math (basically an intro to proofs course) and Calc II. Have I had enough math to even think I would want to go further in math? I worry that math at the higher level will not be the same as math at this level and I may not pick it up as easy or as well. Basically my question is, do people that have good math reasoning up to Calc I typically end up doing well in higher maths?

 

[mathwonk]

let me encourage you with the following reminiscence. In the second grade i was terrified of graduating to the third grade because i had heard that third graders had to learn to swim in the deep end of the pool we had in the basement of my private school. Needless to say, at the appropriate time I managed as well as the others.

 

[Dougggggg]

let me encourage you with the following reminiscence. In the second grade i was terrified of graduating to the third grade because i had heard that third graders had to learn to swim in the deep end of the pool we had in the basement of my private school. Needless to say, at the appropriate time I managed as well as the others.

Thank you for the encouragement. Also, I remember having the fear of the deep end, granted I didn't have to do it to complete the third grade but it still freaked me out for a while. Any reading or anything you would recommend to someone at my level of education that could help me with further maths?

 

[creepypasta13]

how much linear algebra is studied by applied math phD students? Having recently completed my BS in physics and applied math, I'm considering applying to phD programs in applied math (I can't really go into pure math at this point since I missed the deadline to take the math subject GRE, and I haven't taken classes in topology or abstract algebra). My favorite math courses were proof-based linear algebra and analysis. From what I've seen of applied math, the theorem proofs involved in it are primarily related to analysis, with very little linear algebra.

Linear algebra was undoubtably my favorite class. I've also self-studied topology a little, and it seems very interesting also. If I go into applied math, will I get to do any linear algebra? Or should I just pray I can get accepted into a pure/applied math phD program where I can take more linear algebra courses?

 

[Shackleford]

Oddly enough, I just noticed and took a peak at this thread since I recently decided to swap my major and minor to math and physics, respectively. I made a few threads about it, but this is the ideal place for it. I'll start the switch after having taken Cal I-III, Linear Algebra, ODE, and Vector Analysis. I've already taken University Physics I and II and Modern Physics I. I'm currently in Modern Physics II (QM) and Intermediate Mechanics. Mechanics is kicking my butt. I just don't have the time available that the class demands, apparently. It's very difficult, time-consuming, and being taught by the associate chairman of physics who is obviously an exceptional physicist. One of the students showed the professor who taught mechanics last year our first exam, and she basically said the material is too much, or something like that. I guess I'm just making excuses for not doing so well. I'll probably end up with a C in there - I hope.

At any rate, I'm virtually readjusting my entire outlook now that it's not physics. I need to learn about the basic areas as mathwonk mentioned and figure out what interests me the most. Philosophically, I like the absoluteness of mathematics and its logic. I think I might like analysis. I remember being fascinated and impressed by my Vector Analysis professor who would derive all kinds of things off the top of his head and showed things several different enlightening ways.

 

[mathwonk]

I do not know what to suggest as the best way to learn proofs. This happens gradually as you practice them. I got my start in high school in a geometry course back when they were proof based, but I still had a lot of trouble with the language. Than as a senior we had a special course out of Principles of mathematics, which began with a chapter on sets and logic. It had a little intro to propositional calculus and truth tables, and I finally found out what a converse was, and a contrapositive, and how to negate things. Basically, in order to prove something you need to know what it would mean for it to be false. And a basic technique is proof by negation, so you need to know that A implies B if and only if notB implies notA. Then I still had trouble in a first year Spivak type proof based calculus course, but it helped more. Then later I had an abstract analysis course, where we proved set theoretic statements about measure theory, and I internalized how to negate lengthy quantified statements.And if you are getting a C from the good physicist, try for a B. It is often more instructive to get a B from a hard prof than an A from an easy one. (But a D means little from anyone.)
There are some elementary books that claim to teach you how to do proofs but I don't think any of them are much good. One of the best, and maybe hardest, general books to improve your math knowledge is "What is mathematics" by Courant and Robbins. Otherwise it probably helps just to study a proof based book on some specific topic like linear algebra, such as Halmos' Finite dimensional vector spaces.

 

[Dougggggg]

I do not know what to suggest as the best way to learn proofs. This happens gradually as you practice them. I got my start in high school in a geometry course back when they were proof based, but I still had a lot of trouble with the language. Than as a senior we had a special course out of Principles of mathematics, which began with a chapter on sets and logic. It had a little intro to propositional calculus and truth tables, and I finally found out what a converse was, and a contrapositive, and how to negate things. Basically, in order to prove something you need to know what it would mean for it to be false. And a basic technique is proof by negation, so you need to know that A implies B if and only if notB implies notA. Then I still had trouble in a first year Spivak type proof based calculus course, but it helped more. Then later I had an abstract analysis course, where we proved set theoretic statements about measure theory, and I internalized how to negate lengthy quantified statements.


And if you are getting a C from the good physicist, try for a B. It is often more instructive to get a B from a hard prof than an A from an easy one. (But a D means little from anyone.)
There are some elementary books that claim to teach you how to do proofs but I don't think any of them are much good. One of the best, and maybe hardest, general books to improve your math knowledge is "What is mathematics" by Courant and Robbins. Otherwise it probably helps just to study a proof based book on some specific topic like linear algebra, such as Halmos' Finite dimensional vector spaces.

Thank's I will check out that first book you mentioned during Christmas break.

 

[dkotschessaa]

WRT the original post, and oldie but goodie, at what point do you recommend starting to look at the works of the early mathematicians you mentioned (Gauss, Euclid). And which works? I came across Euclid's Elements in the bookstore the other day and put the thick volume down deciding I would be getting a bit ahead of myself. What I found of Gauss online was all in German.

I have one semester of calculus under my belt but it is from some time ago, so I'll be starting over this spring.

-DaveKA

 

[mathwonk]

get the green lion edition of euclid without all the commentaries. gauss is available in english but may be a little pricey. you are certainly ready to read euclid and probably gauss.

 

[dkotschessaa]

Found that, thanks. I am always interested in reading fundamental texts when exploring a subject. There is nothing quite like it.

Along the same lines, do you recommend Newton's Principia?

 

[mathwonk]

yes. one of my big regrets was selling my copy in 1977 when lightning my load to move, for a dime! I would not have let it go, but the book buyer obviously loved books and smoothed them lovingly with his hand as he stacked them up, so i thought it was going to a good home. I also highly recommend reading euler. also archimedes.

 

[dkotschessaa]

Ok, all will be added to my light summer reading list. :) You've been tremendously helpful.

-DaveKA

 

[Shackleford]

My calculus textbook says the three greatest mathematicians are Newton, Gauss, and Archimedes.

 

[andyroo]

My calculus textbook says the three greatest mathematicians are Newton, Gauss, and Archimedes.

Archimedes is probably my favorite actually!

 

[Dougggggg]

I found this https://www.amazon.com/dp/0521045959/?tag=pfamazon01-20 in my library, I was enjoying reading it so much. Only problem was that I had to leave to do homework. They had all 7 volumes. The notation is way over my head but I found where he originally showed the definition of a derivative. Granted I only realized that because of the notes at the bottom.

 

[mathwonk]

Don't expect to read it all, or even a large amount of these books, in one summer. Just read some of it, any really. Even one page of one of those greats will bless you with a special dispensation of knowledge for the rest of your life. Honest.

Once, in grad school, while studying algebraic geometry, I went to the library and struggled through a single page of Zariski. I was very frustrated because it took me all afternoon to read that page, and I felt discouraged. The next day in class however, I answered every question the professor asked, until he asked me to stop answering since I seemed to know everything.

Similarly in Archimedes there is a single sentence that says something like: a sphere is essentially a cone whose base is the surface of the sphere, and whose apex is the center of the sphere.

That one sentence, if you understand it, tells you why the volume of a sphere equals 1/3 the surface area times the radius of the sphere. Not even all mathematicians have this insight.

Just open these books and you will begin to tread on the higher ground that few people tread.

 

[QuantumP7]

Once, in grad school, while studying algebraic geometry, I went to the library and struggled through a single page of Zariski. I was very frustrated because it took me all afternoon to read that page, and I felt discouraged. The next day in class however, I answered every question the professor asked, until he asked me to stop answering since I seemed to know everything.

Which book by Zarinski do you mean?

 

[Shackleford]

Don't expect to read it all, or even a large amount of these books, in one summer. Just read some of it, any really. Even one page of one of those greats will bless you with a special dispensation of knowledge for the rest of your life. Honest.

Once, in grad school, while studying algebraic geometry, I went to the library and struggled through a single page of Zariski. I was very frustrated because it took me all afternoon to read that page, and I felt discouraged. The next day in class however, I answered every question the professor asked, until he asked me to stop answering since I seemed to know everything.

Similarly in Archimedes there is a single sentence that says something like: a sphere is essentially a cone whose base is the surface of the sphere, and whose apex is the center of the sphere.

That one sentence, if you understand it, tells you why the volume of a sphere equals 1/3 the surface area times the radius of the sphere. Not even all mathematicians have this insight.

Just open these books and you will begin to tread on the higher ground that few people tread.

Interesting. So, something like this?

https://www.amazon.com/dp/0486420841/?tag=pfamazon01-20