Should I Become a Mathematician?

[alligatorman]

I guess I'm just a little paranoid after reading bad things about the job market for Ph.Ds

 

[mathwonk]

im sort of amazed at such a calculating attitude, since when i went to school i just went wherever i could get in and for the love of the subject. pretty naive but i did ok.

yes the rep of the school helps a little, but your rep means more. look at the faculty of the top schools and ill bet most of them did not go those schools.

as i said perhaps in my thread on becoming a mathematician, when i was introduced at my present job they sort of mumbled my alma mater and trumpeted that of a similar new hire who came from a top school. but a year or two later that hire was gone and i was doing well.

how many times do i have to tell people, you succeed on what YOU can do, not what the professors at your school have done.

when someone asks you a question, you cannot get away with not answering it but saying, "well , but i did go to harvard!"

 

[muppet]

I'm currently looking at V I Arnold's Mathematical methods of classical mechanics, which according to the cover is a graduate text. One thing that really strikes me is the nature of the problems in the book. Where as an undergraduate text usually has at least twenty or thirty questions at the end of each chapter, which are generally computational in nature, Arnold will have one every other page or so, and they tend to be much more abstract:
"Cite examples where there are many extremals connecting two given points, and others where there are none at all"
is the complete wording of one that has me scratching my head a little...
Is this abstraction (which I assume is intended to get you thinking creatively, and to work out the kind of information that would be supplied in undergraduate texts) common in grad texts? I.e. should I get used to it?

 

[The|M|onster]

when someone asks you a question, you cannot get away with not answering it but saying, "well , but i did go to harvard!"

Ha ha ha!

I agree with you, Mr. Mathwonk, sir. The instructors that have had the biggest impact on my education have all come from lesser known universities. Hell, the guy who influenced my decision to declare a major in mathematics is still a graduate student. I like to believe that if someone is truly passionate about their field of study, then everyone will be able to see it and it will make them passionate about the subject as well. The majority of my teachers have exuded varying levels of passion and I think it is a joy to see them work.

One of my teachers was a 70 year old man who was absolutely awesome at what he did. He could do any problem ten times faster than anyone in the class and his tests were some of the most challenging I have ever had. Never once told the class his background, I didn't even know he was a professor. Sometime during the middle of the semester one of my classmates looked our teacher up on the school website and read his vita. Turns out he received his terminal degree from MIT. But its funny because the class started to look at him different after this kid revealed our teacher's alma mater, as if the school name validated his ability. I thought that his ability was obvious from the fact that he was a spry 70 year old man who was kicking our 18 and 19 year old asses at these problems. It scares me that the name of a college means that much to people.

 

[mathwonk]

my father never went to college and left home to work after high school. after some time he felt the need for more education and wrote the st. louis post dispatch news paper asking for a list of books which "if read and mastered:" would equate to a college degree. he wrote down for me a few books from that list, which he had read all of.

i eventually went to harvard, and on returning home found his list. i was stunned to find i had not read most of them, but was considering myself highly educated, as a "harvard man".

 

[mathwonk]

muppet i do recommend getting in synch with the tone of arnol'ds book as much as possible. he is a terrific mathematician and a wonderful expositor, and apparently a great teacher too. he has very strong and justifiable opinions about how math should be taught, and written. i have several of his books and value them highly. his opinions on math exposition are hard to argue with as well.

 

[muppet]

Don't worry mathwonk, I'm working on it... slowly!

 

[mathwonk]

i just understood jordan forms for the first time. one has a nilpotent operator T (i.e. one such that T^n = 0 for some n) on a finite diml space and one wants as simple a matrix for it as possible. define a subspace to be T cyclic, if it has a basis of form (x,Tx,T^2x,...,T^rx) for some x, where T^r+1 x = 0.

then the matrix for T in such a basis, on a T cyclic subspace is very simple. it just has 1's right below the diagonal, and 0's elsewhere.

then the theorem is that every space with a nilpotent operator T, has a basis consisting of disjoint T -cyclic sequences, i.e., the space decomposes as a direct sum of T cyclic subspaces.

but the way to look at it is this: define a subspace U as T invariant, if T(U) is contained in U. Then define a T invariant subspace U as "decomposable" if it is a direct sum of (at least two) T invariant (non zero) subspaces. Then it is almost trivial that every finite dimensional space with an operator is a direct sum of indecomposable subspaces.

then with a nilpotent operator, the main point of the jordan decomposition, is that the only indecomposable subspaces are the T cyclic ones. Thus every space decomposes into a direct sum of T - cyclic subspaces. I have not seen the theorem stated this way in my standard references.

 

[mathwonk]

some detail

the proof is also quite easy. the point is that every space is spanned by T cyclic sequences, and any T cyclic sequence is independent, but it may not be true that a union of T cyclic sequences is independent.

so the main point of the proof is to show that a union of T cyclic sequences, whose spans may overlap, can be rearranged so that they do not over lap.

e.g. if (x,Tx,...T^a-1 x; y,Ty,...,T^b-1 y) spans V, and a > b-1, we want to replace y by a vector of form y' = y - P(T)x, so that the union of T cyclic sequences
(x,Tx,...T^a-1 x; y', Ty',...,T^c-1 y') is a basis for V, where c-1 < b.

so if the dimension of V is a+c, where c-1< b, we need a polynomial P such that T^c(y-P(T)x) = 0. But since T^b y = 0, it follows (by the division algorithm) there is some c: c-1 < b, with T^c y in the span X of (x,Tx,...T^a-1 x), and such that for any other poly Q, then Q(T)y is in X iff T^c divides Q.

now since T^c y belongs to X, there is some poly Q such that Q(T)x = T^c y, and thus
0 = T^b y = T^(b-c) T^c y = T^ (b-c) Q(T) x. hence since T^a x = 0 and this is the minimum power that has this property, it must be true again by the division algorithm that T^ a divides T^(b-c) Q(T), i.e,. since b > c-1, that T^(a-b+c) divides Q(T).

but a-b >-1, so T^c divides Q(T). thus we have T^c y = Q(T) x = T^c P(T) x.
hence if we take y ' = y - P(T)x, we get T^c y' = 0.
then the sequences (x,Tx,...T^a-1 x; y' Ty',...,T^c-1 y') still span V and are independent.

'this is the key inductive step. next if V is spanned by several such sequences:

(x,Tx,...T^a-1 x; y, Ty,...,T^b y; ...; z,Tz,...T^c z), then
by induction V/X is a direct sum of T cyclic sequences, spanned by y', ...,z', and their nonzero T images.

then considering each pairwise span X + span(y', Ty',...);...; X+span(z',Tz',...) separately, we get a decomposition of V into a direct sum of X and subspaces isomorphic to the ones decomposing V/X.

 

[mathwonk]

sorry, of course this is incomprehensible. sighhhh...

 

[The|M|onster]

Wow! Give me a couple years and maybe I'll be able to figure out what you are saying. Can I get some recommendations on books concerning finite fields? It would be vastly appreciated.

 

[mathwonk]

well in a finte field, there are always p^n elements, where p is a prime, and conversely there is exactly one field with that many elements for each prime p and each n >0.

since the non zero elements form a cyclic group they all satisfy the equation X^q - X = 0, where q = p^n. moreover the solutions of that equation form a field, so the splitting field of that equation provides an example, the example, of such a field.

for such elementary introductions, see van der waerden modern algebra, and for more advanced material, see a.a.Albert, modern higher algebra.

 

[Cincinnatus]

It is not incomprehensible, it's interesting. I hadn't thought much about Jordan decomposition since my first linear algebra class.

---

What do you think about Atiyah and Macdonald's commutative algebra book. I'm going through it with a friend right now but it's incredibly slow going. We're averaging about one page per hour. Is it worth it to switch to a longer book with more exposition that might take less work to understand each individual sentence?

 

[mathwonk]

well that's up to you. the rate of understanding may be the same if there are twice as many pages and you go through them twice as fast. i myself prefer zariski and samuel, from which it appears to me that atiyah - macdonald have cribbed most of their text while omitting about 80% of the explanation.

z-s is still the classic text on the topic and has been for 50 years now.

 

[The|M|onster]

well in a finte field, there are always p^n elements, where p is a prime, and conversely there is exactly one field with that many elements for each prime p and each n >0.

since the non zero elements form a cyclic group they all satisfy the equation X^q - X = 0, where q = p^n. moreover the solutions of that equation form a field, so the splitting field of that equation provides an example, the example, of such a field.

for such elementary introductions, see van der waerden modern algebra, and for more advanced material, see a.a.Albert, modern higher algebra.

I'm familiar with the basic concepts, as I've been slowly (very slowly) teaching myself abstract algebra through artin's and hungerford's books. I've been told that finite fields have a lot of applications to cryptography, which I'm currently interested in right now. Do you know of any books that cover the theoretical underpinnings as well as applications? Or would I be better off just studying these separately?

 

[alligatorman]

mathwonk, what advice can you give someone who isn't the "quickest?"

I can understand concepts, but it takes me longer than most people. As a result, doing exercises in textbooks takes me longer than it probably should, and of course, I struggle in test situations. Often times, I gain insight when it's too late.

I guess the answer is to practice and practice and to know concepts back and forth, but I feel like it doesn't help.

Any advice?

 

[mathwonk]

if you enjoy it and eventually understand, then hang in there. speed really does not matter at all, except in coffee room situations. everyone is slow at grasping really deep ideas, and uncovering hidden facts, and proving new results.

 

[scast]

Hello everybody. I have been following these forums for a while already but I
have never really posted anything. So this is my first post.

Anyways, few days a go I found on my dad's shelf Calculus by Spivak 2nd Ed. I
thought I could read a bit on it and tackle some exercises just to open my mind
to some real mathematics before I start studying for college admission test in a
few months (here in my country to get into colleges we have to take a test
issued by the college you are applying for; it's very similar to SAT in USA but
it differs from college to college; the test for the college I want to get into
is very mathematics-based as in needing atleast 30/40 good answers in math
section).

I think I am pretty good at maths. I kinda fell in love with it a few years a
go. It has been a love-hate relationship but we always make up. I was very aware
that Spivak Calculus is very rigourous and that's pretty much why I chose it (in
my dad shelf I saw a few other calc books too). I wanted to get a real grasp of
mathematics. Anyways, I kinda went through the first chapter and I struggled a
bit, did some exercises and moved to second chapter which basically introduces
the principle of induction.

So I went to the exercise part and tried the first one. I really focused tried
very hard for a long while and I didn't get it. I kind of gave up and looked for
the answer in the back. I understood it right away but what kinda bothers me is
that I really didn't think of that solution not even for a moment. So I try
exercise number two. Same thing happens but this time I was even more far off. I
said well maybe I am not looking at this at the right way, and moved to the
second set of exercises.

In the first exercise I was able to figure it out because I saw a pattern in the
sum of the squares but I didn't really write that rigourous proof mathematicians
should write. And with a litle help of the answers in the back I also figured
the second one and tried to write a decent proof of it this time around. I came
very close to it.

So I said let's do the third one on my own. I'll write it down and put the book
away so I don't get to cheat. Heh, that was 2 days a go. I come back every now
and then to see if some new way to look at the problem has risen but no luck and
after like 30 minutes thinking I just walk away.

So after this rambling... I just want to know: should I bother? I really like
maths and would like to major it on college. I am really eager to know more and
more about it and it seems like every litle thing about it fascinates me (yet
90% of the times I don't really get what's going on). Yet again, it might not be
my true calling.

Wether or not I major on math I will end up in a career heavy on maths, probably
CS or some kind of engineering. Or physics, who knows.

(Oh and sorry if this is not the right place to post this. Maybe I should have
started a new post. If that's the case please tell me and will do that.)

 

[uman]

I'm a high school student too. The first time I tried to teach myself Calculus, I couldn't even understand the *exposition* in a Stewart type book, let alone the exercises. The second time, I worked through much of Apostol. I've decided on studying math after high school with the goal of becoming a mathematician, trying as hard as I can, and if I fail, well, at least I will have enjoyed myself. Moral of the story: Don't give up, and if you love math, continue doing it. If you realize you hate math and start loving something else, do that. At least that's my perspective.

Out of curiosity (sorry for the off-topic question), which country are you from?

 

[RocketSurgery]

I'm a high school student too. The first time I tried to teach myself Calculus, I couldn't even understand the *exposition* in a Stewart type book, let alone the exercises. The second time, I worked through much of Apostol. I've decided on studying math after high school with the goal of becoming a mathematician, trying as hard as I can, and if I fail, well, at least I will have enjoyed myself. Moral of the story: Don't give up, and if you love math, continue doing it. If you realize you hate math and start loving something else, do that. At least that's my perspective.

Out of curiosity (sorry for the off-topic question), which country are you from?

Good luck. I'm in the same boat as you. Stewart is our required book in Uni and it's supposed to be written for laypeople but for some reason it confuses me with all the little boxes everywhere in all these different colors with distracting diagrams. Apostol on the other hand has the nice clear conversational exposition with simple diagrams.

It might just be me but I have an easier time going through Apostol's exercises too. When I look at Stewarts exercises I either fall asleep from boredom or I'm so confused by what the problem is asking that I shut the book. Apostol's exercises are actually interesting and difficult but easier to solve because the text is organized so well.

 

[maze]

At this point it may be more useful to develop your problem solving skills in particular. There is a book called "how to solve it" by the great mathematician Polya, where he explains how to go about attacking these sorts of problems. It's not a textbook, there are no problems, and after the first sections there is no particular order you need to read it in. You just skip around from time to time, and try to incorporate the strategies he talks about as you go about your normal problem solving. Anyways, I'd recommend it.

 

[RocketSurgery]

At this point it may be more useful to develop your problem solving skills in particular. There is a book called "how to solve it" by the great mathematician Polya, where he explains how to go about attacking these sorts of problems. It's not a textbook, there are no problems, and after the first sections there is no particular order you need to read it in. You just skip around from time to time, and try to incorporate the strategies he talks about as you go about your normal problem solving. Anyways, I'd recommend it.

I'm not familiar with that book so I can't recommend it but I have Schaum's 3000 Problems in Calculus book and it is pretty good for problem solving practice. Most topics have over 100 questions ranging from simple calculation type to proof/theory based problems. Also for Physics Majors there is a 3000 Problems in Physics book but most of the problems aren't calculus based but there are a few challenging problems.

 

[scast]

I'm a high school student too. The first time I tried to teach myself Calculus, I couldn't even understand the *exposition* in a Stewart type book, let alone the exercises. The second time, I worked through much of Apostol. I've decided on studying math after high school with the goal of becoming a mathematician, trying as hard as I can, and if I fail, well, at least I will have enjoyed myself. Moral of the story: Don't give up, and if you love math, continue doing it. If you realize you hate math and start loving something else, do that. At least that's my perspective.

Thanks for the advice man. I like your attitude towards math. I'll try to follow it.

Out of curiosity (sorry for the off-topic question), which country are you from?

I live in Venezuela (it is not AS bad as you might think but it's pretty bad). I
might have generalized the process of getting in a college a little bit. It
might not be such a white and black process, but it is pretty much like that.

At this point it may be more useful to develop your problem solving skills in particular. There is a book called "how to solve it" by the great mathematician Polya, where he explains how to go about attacking these sorts of problems. It's not a textbook, there are no problems, and after the first sections there is no particular order you need to read it in. You just skip around from time to time, and try to incorporate the strategies he talks about as you go about your normal problem solving. Anyways, I'd recommend it.

I am definitely going to look at it. I also looked around books of this kind and
I found How To Prove It by Velleman, any thoughts on that one?

 

[mathwonk]

you are the best judge of what book helps you, but for a "professional" opinion, polya is a great timeless classic, and velleman is a hack book for college courses aimed at weak undergrads, which of course might make it about right for bright high schoolers.

 

[tgt]

These days people specialize so narrowly that a Phd might be in a very limited and small area of a field. After the Phd, they may like to continue to a postdoc position. My question is how likely is it that a recent graduate find a matching postdoc position that is in his/her area of research? Is there usually a period of unemployment due to searching a matching research position? If so how long?

 

[mathwonk]

postocs are indeed common today. ( a longer post got trashed by the browser.)

 

[mathwonk]

postdoc positions have become more common over the last 10-20 years. perhaps most students take postdocs now before moving to a permanent position.

i am not sure how many, but we have had some wonderful postdocs, several of whom would have made GREAT permanent appointments.

my own career path was a little different. With a wife and child I turned down a couple of a postdocs for a permanent job, then took leave after 2 years for a postdoc which did help a lot with my research.

 

[tgt]

But what about finding a post doc position that matches your speciality when you graduate. Do people have to wait some time before finding such a position?

 

[mobe]

Also, is there an age limit for postdocs?

 

[Jon_]

I would like to be a mathematician.But I don't think I have the skill set to be an accomplished mathematician. A while back, during the 1990's, I was studying applied math/minor physics.It was during my sophomore year,when I was talking with a friend at the time about engineering.He told me that the field had high monetary rewards and that if I chose to leave my current field, I would get a job right after graduation. Well, he was wrong.
I did change over and to put it nicely,I crashed and burned.I did not finnish school and my classmates did.I missed my boat.

I was 27 at the time.Now 20 years later,back in school trying to make it all work. I have one year left till I finish my BA in physics.It wasn't until the end of my junior year that I decided to take a minor!

What's the point in all this? Well, I chose physics because it is applied math,yet,I don't feel that my ability alone in physics is all that good.I fare much better doing math problems,modeling,etc.My rationale is "if I wanted to pursue applied math and get a physics degree at the same time" then this would be the route I'd go.

Unfortunatley,so much time has passed since the "prime of youth" that I cannot with all my abilities recall pertinant ideas I learned so long ago. Before college I always wanted to do math,but now,I think it's too late. I also lack the motivation I was had.

So back to my minor degree.The minor I chose is in geology/volcanology and it is my hope and I'm banking on it, that all I will need is a masters degree.I really don't know how much more complicated the geosciences are than mathematics but I'm hoping it's less rigourous.

For me,physics is just a vehicle to acquire the things I truly want to do.It's important to know and understand and it helps with abstract ideas as well. Had I stayed the course,I would have finished with my friends and been much happier and successful than I am today.

all that remains is to finish the BA,intership and take the GRE's and find a school that'll accept me. Hopefully,before it's too late.

 

[Werg22]

Mathwonk, say I want to get into a top notch grad school, what would you recommend me to do during my undergrad years? I am thinking something along the lines of spending the summers of my 2nd and 3rd year assisting professors in areas I'm interested in. Maybe also work part-time during my school terms. Do you think that's a good idea? Is there more stuff I could do? Also, I'm not sure professors will need the assistance of an undergrad, even if the latter offers his services for free. What is the best way to approach a professor for that sort of thing anyways?

 

[mathwonk]

I really don't know much about this. I myself teach at agood school, but not a top notch one, and we are always starved for talent. But I guess the criteria are the same everywhere, so I guess I know what to do to prepare, I just don't know how competitive the top schools are at the grad level, so I don't know how likely one is to get in by doing them.

It is not so much a matter of doing special activities or projects, or endearing yourself to professors, but just a matter of working hard to learn as much as possible, and trying to polish ones ability to do research. Although we talk otherwise in education, in recruiting we tend to behave as if math is an inherent talent rather than an acquired skill.

so we are always looking for that person who sees deeply into the subject they encounter, and who comes up with insightful comments and questions, and who finds creative approaches to problems, and who then pursues them successfully, with strong computational skills.

I guess the only part of this that you can acquire is the knowledge by sustained study with the best masters and books, and strengthened computational abilities through persistent practice at working out examples.

In my own career, I made a quantum leap by spending 2 postdoctoral years at harvard, trying to soak up as much as possible from people like david mumford and phillip griffiths, and heisuke hironaka.

i went to their lectures, asked them questions, and read the papers they referred me to (I did not read as many of those as I should have in some cases.) i volunteered to give talks and invited them to attend.

eventually i got better, and they seemed to notice it, and they helped me. but as to getting into a top grad school, i think the goal should rather be to get into the grad school that is at the right level for you, that offers courses in what you want to study and that has professors you can learn from.

for me harvard would have been a disastrous grad school, but it was an ideal postdoctoral experience. utah was perfect for me as a grad school, because it had herb clemens, the advisor who helped me find and improve my research abilities, and gave me an appropriate problem, and helped me learn to solve it.

before that my stay at brandeis helped too, by contact with brilliant and accessible professors like alan mayer, paul monsky, robert seeley, maurice auslander, david buchsbaum. i learned much more at brandeis than as an undergrad at harvard, because the professors at brandeis seemed to notice us and try to teach us. we were run over roughshod at harvard undergrad, by professors who ignored us or made us feel we were wasting their time, a really awful experience.

 

[JWHooper]

I didn't know Pikachu was good at math. :)

Talking about PokeMath!

 

[mobe]

Hi Mathwonk,

I am a first year MA student in Math at a small university. I am interested in studying algebra/algebraic geometry. I have noticed that your research area is algebraic geometry. My question for you is: what are some good universities to study algebraic geometry? I have looked at several universities such as: Michigan, Chicago but these are top ten universities. Where else can I consider? Thanks in advance.
--Mobe

(@admin: please move my question if it is not posted under the right topic. Thanks)

 

[proton]

hey mathwonk

i just finished my REU in physics this summer and was a little disapointed by it, so I'm sure that at this point I want to focus on pure math. I'll be entering my 4th yr this fall, though I plan on staying for a 5th year. So I need advice on what classes to take this fall and in the future

The only pure math classes my school is offering this fall are Honors Abstract Algebra, which I signed up for already, and Differential Geometry A. But since my school doesn't offer Diff Geom B until the spring (which comes after winter quarter, since we're on the trimester system), I figure I'll wait until the winter to take Diff Geom A. So what else should I take? Would a class in German be useful since I heard that its required for math phDs to read another language than English? or possibly take a numerical analysis class, which could prove useful should I go into industry instead of academia? or instead do undergrad research with a professor?