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The Should I Become a Mathematician? Thread

by mathwonk
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mathwonk
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Jun23-06, 03:52 PM
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Originally Posted by mathwonk
by the way pete, if you are a mathematical physicist, some posters in the thread "who wants to be a mathematician" under academic guidance, have been asking whether they should major in math or physicts to become one. what do you advise?
I had two majors in college, physics and math. Most of what I do when I'm working in physics is only mathematical so in that sense I guess you could say that I'm a mathematical physicist.

I recommend to your friend that he double major in physics and math as I did. This way if he wants to be a mathematician he can utilize his physics when he's working on mathematical problems. E.g. its nice to have solid examples of the math one is working with, especially in GR.

Pete
fournier17
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Jun23-06, 05:11 PM
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Thanks for the replys guys, this forum is so helpful.
fournier17
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Jun23-06, 05:23 PM
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[QUOTE=loop quantum gravity]you could do an undergraduate degree in combined maths & physics, and afterwards you can pursue with a phd in theoretical physics (synonymous with mathematical physics).[QUOTe]

Is theoretical physics the same as mathematical physics? If they are then thats great, more potential graduate programs to which I can apply to. However, I have heard that mathematical physics relys more on mathematics, and that theoretical physics is more physics than math. I have seen some graduate programs in mathematical physics that are in the math department of the university instead of the physics department.
matt grime
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Jun23-06, 06:01 PM
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Like many things in mathematics itself, the terms mathematical physics and theoretical physics mean different things to different people.
mathwonk
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Jul7-06, 10:33 PM
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I wrote the following letter to my graduate committee today commenting on what seems to me wrong with our current prelims. these thoughts may help inform some students as to what to look for on prelims, and what they might preferably find there.

In preparing to teach grad algebra in fall, one thing that jumps out at me is not
the correctness of the exams, but their diversity. One examiner will ask only
examples, another only creative problems, another mostly statements of theorems.
only a few examiners ask straight forward proofs of theorems.

Overall they look pretty fair, but I noticed after preparing my outline for the
8000 course that test preparation would be almost independent of the course i
will teach. I.e. to do most of the old tests, all they need is the statements
of the basic theorems and a few typical example problems. They do not need the
proofs I am striving to make clear, and often not the ideas behind them.
anybody who can calculate with sylow groups and compute small galois groups can
score well on some tests.

In my experience good research is not about applying big theorems directly, as
such applications are already obvious to all experts. It is more often applying
proof techniques to new but analogous situations after realizing those
techniques apply. So proof methods are crucial.
Also discovering what to prove involves seeing the general patterns and concepts
behind the theorems.

The balance of the exams is somewhat lopsided at times. some people insist on
asking two-three or more questions out of 9, on finite group theory and
applications of sylow and counting principles, an elementary but tricky topic i
myself essentially never use in my research. this is probably the one
ubiquitous test topic and the one i need least. I don't mind one such question
but why more?

The percentage of the test covered by the questions on one topic should not
exceed that topic's share of the syllabus itself. if there are 6 clear topic
areas on the syllabus, no one of them should take 3/9 of the test.

also computing specific galois groups is to me another unnecessary skill in my
research. It is the idea of symmetry that is important to me. When I do need
them as monodromy groups, a basic technique for computing them is
specialization, i.e. reduction mod p, or finding an action which has certain
invariance properties, which is less often taught or tested.

Here is an easy sample question that illustrates the basic idea of galois
groups: State the FTGT, and use it to explain briefly why the galois group of
X^4 - 17 over Q cannot be Sym(4). This kind of thing involves some
understanding of symmetry. One should probably resist the temptation to ask it
about 53X^4 - 379X^2 + 1129.

As of now, with the recent division of the syllabus into undergraduate and
graduate topics, more than half the previous tests cover undergraduate topics
(groups, linear algebra, canonical forms of matrices.) This makes it harder to
teach the graduate course and prepare people for the test at the same time,
unless one just writes off people with weak undergrduate background, or settles
for teaching them test skills instead of knowledge.

Thus to me it is somewhat unclear what we want the students to actually know
after taking the first algebra course. I like them to learn theorems and ideas
for making proofs, since in research they will need to prove things, often by
adapting known proof methods, but the lack of proof type question undermines
their interest in learning how to prove things.

The syllabus is now explicit on this point, but if we really want them to know
how to state and prove the basic theorems we should not only say so, but enforce
that by testing it.


Suggestions:

We might state some principles for prelims, such as:

1) include at least one question of stating a basic theorem and applying it.
I.e. a student who can state all the basic theorems should not get a zero.
2) Include at least one request for a proof of a standard result at least in a
special case.
3) include at least one request for examples or counterexamples.
4) try to mostly avoid questions which are tricky or hard to answer even for
someone who "knows" all the basic material in the topic (such as a professor who
has taught the course).

I.e. try to test knowledge of the subject, rather than unusual cleverness or
prior familiarity with the specific question.

But do ask at least one question where application of a standard theorem
requires understanding what that theorem says, e.g.: what is the determinant,
minimal polynomial, and characteristic polynomial of an n by n matrix defining a
k[X] module structure on k^n, by looking at the standard decomposition of that
module as a product of cyclic k[X] modules. or explain why the cardinality of a
finite set admitting an action by a p-group, is congruent modp to the number of
fixed points.

5) point out to students that if they cannot do a given question, partial credit
will be given for solving a similar but easier question, i.e. taking n= 2, or
assuming commutativity, or finite generation. This skill of making the problem
easier is crucial in research, when one needs to add hypotheses to make
progress.

6) after writing a question, ask yourself what it tests, i.e. what is needed to
solve it?

These are just some ideas that arise upon trying to prepare to help students
pass the prelim as well as prepare the write a thesis.
mathwonk
#150
Jul7-06, 10:39 PM
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Alg prelim 2002. Do any 6 problems including I.

I. True or false? Tell whether each statement is true or false, giving in each case a brief indication of why, e.g. by a one or two line argument citing an appropriate theorem or principle, or counterexample. Do not answer “this follows from B’s theorem” without indicating why the hypotheses of B’s theorem hold and what that theorem says in this case.

(i) A commutative ring R with identity 1 ≠ 0, always has a non trivial maximal ideal M (i.e. such that M ≠ R).

(ii) A group of order 100 has a unique subgroup of order 25.

(iii) A subgroup of a solvable group is solvable.

(iv) A square matrix over the rational numbers Q has a unique Jordan normal form.

(v) In a noetherian domain, every non unit can be expressed as a finite product of irreducible elements.

(vi) If F in K is a finite field extension, every automorphism of F extends to an automorphism of K.

(vii) A vector space V is always isomorphic to its dual space V*.

(viii) If A is a real 3 x 3 matrix such that AA^t = Id, (where A^t is the transpose of A), then there exist mutually orthogonal, non - zero, A - invariant subspaces V, W of R^3.

In the following proofs give as much detail as time allows.
II. Do either (i) or (ii):

(i) If G is a finite group with subgroups H,K such that G = HK, and K is normal, prove G is the homomorphic image of a “semi direct product” of H and K (and define that concept).

(ii) If G is a group of order pq, where p < q, are prime and p does not divide q-1, prove G is isomorphic to Z/p x Z/q.

III. If k is a field, prove there is an extension field F of k such that every irreducible polynomial over k has a root in F.

IV. Prove every ideal in the polynomial ring Z[X] is finitely generated where Z is the integers.

V. If n is a positive integer, prove the Galois group over the rational field Q, of X^n - 1, is abelian.

VI. Do both parts:
(i) State the structure theorem for finitely generated torsion modules over a pid.

(ii) Prove there is a one - one correspondence between conjugacy classes of elements of the group GL(3,Z/2) of invertible 3x3 matrices over Z/2, and the following six sequences of polynomials: (1+x, 1+x,1+x), (1+x, 1+x^2), (1+x+x^2+x^3), (1+x^3), (1+x+x^3), (1+x^2+x^3)

[omitted(iii) Give representatives for each of the 6 conjugacy classes in GL(3,Z2).]

VII. Calculate a basis that puts the matrix A :
with rows ( 8, -4) and (9, -4) in Jordan form.

VIII. Given k - vector spaces A, B and k - linear maps f:A-->A, g:B-->B, with matrices (x[ij]), (y[kl]), in terms of bases a1,...,an, and b1,...,bm, define the associated basis of AtensorB and compute the associated matrix of
ftensorg: AtensorB--->AtensorB.
mathwonk
#151
Jul8-06, 03:56 PM
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for advice on preparing for grad school, from me and others, see my posts 11 and 12 in the thread "4th year undergrad", near this one.
JasonJo
#152
Jul9-06, 05:37 PM
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how are Summer REUs regarded for graduate admissions?
mathwonk
#153
Jul9-06, 07:12 PM
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They add something, especially if the summer reu guru says you are creative and powerful.

i think my friend jeff brock (now full prof at brown) did one at amherst or williams and actually proved some theorems and got a big boost there. they are also taught by people who may be either refereeing or reviewing letters of grad school application.
mathwonk
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Jul10-06, 04:32 PM
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tiny comment, possibly superfluous to todays youth: learn to be as computer literate as possible. for example elarn to type, and learn to use TEX, and AMS TEX or LATEX.

All papers are written in TEX on computers now, usually by the author him/herself. (I even have students who refuse to read typed class notes that are not written in TEX.)

All NSF grants are submitted online. All courses have or should have webpages to support them, and even grades are submitted online.

And if you have trouble geting an academic job, there are many more openings for tech support people, and they are more essential, than are pure mathematicians.

if you want to be in the wave of the future of education, try to learn to use computers to teach effectively. i have my own doubts abut the vaue of this educationally, but it is inevitable, and can at least enhance regular classroom instruction.

if you have bad handwriting, it can at elast render it readable to project your notes on the board. long calculations, like the antiderivatives of
1/[1 + x^20] become trivial work of fractions of a second.

this can help impress on students the folly of merely learning to do such calclations, without understanding the iDEAS.
mathwonk
#155
Jul14-06, 11:34 AM
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thought for the day: students, when learning a theorem, get in the habit of trying to think up a proof by yourself, before reading one. usually if you try ahrd, you will find on reading it that you have thought of at elast the first few liens of the proof. this makes a huge difference in understanding it.
mathwonk
#156
Jul14-06, 10:44 PM
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here is another exercise: if k is any field, and c is any element of k, and p is a prime integer, prove that the equation X^p - c is either irreducible over k, or has a root in k.


hint: if it factors as g(X)h(X), with deg g = r and deg h = s, and the constant terms of g,h respectively are (1-)^r a, (-1)^s b, then show that a is a pth root of c^r and b is a pth root of c^s.

then use the fact that r,s are relatively prime to find a product of powers of them that is a pth root of c. hence X^ p -c has a root in k.
mathwonk
#157
Jul15-06, 09:27 PM
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hint: if nr+ms = 1, then (c^r)^n . (c^s)^m = c.
J77
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Jul17-06, 04:30 AM
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Quote Quote by mathwonk
All papers are written in TEX on computers now, usually by the author him/herself. (I even have students who refuse to read typed class notes that are not written in TEX.)
Adding to this, make sure you have a good (not just decent) grasp of English.

In some of the papers that I've reviewed, even the titles are ungrammatical! That's not a good start...

Being able to write a good description of your work is more important than writing down a mass of equations.
mathwonk
#159
Aug2-06, 08:55 PM
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how to get a phD; get into grad school, then pass prelims, then find a good helpful advisor, then start work as soon as possible on your thesis, [because it will take lots longer than you think it will], believe in your own intuition of what should be true and try to prove it, dont give up, because you WILL finish if you keep at it.

(secret: they really do want everyone to graduate: when they press you they are just trying to get you to extend yourself as much as possible: repeat they are NOT trying to flunk you out).

best of luck! as sylvanus p thompson put it: what one fool can do, another can.
mathwonk
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Aug2-06, 10:06 PM
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I do not know how to advise people on how to write a thesis, as I have never had a PhD student complete a thesis under me.

I am not sure why this is, but suspect it is because I was not supportive enough. When I was a student, thesis advisors sort of waited for us to produce a result, then said whether it was enough or not.

I was not too good at this and needed more help, so eventually found an advisor who proposed a specific problem and also an approach to it and then even suggested a conjectural answer and I found the solution proving his guess correct.

Along the way I needed courage and confidence however, as at one point my advisor announced that a famous mathematician had become interested in my problem. He seemed to feel that this was the kiss of death, but I cheekily responded that was fine, when i solved it I would inform the famous man of the answer. This actually occurred fortunately for me.

[I solved three problems before finding a new one. The first had already been done by Hurwitz in the 19th century and the second by deligne in the 1960s. Finally the third made progress on a problem left open by Wirtinger in 1895.]

This solution of mine was actually pretty interesting and led to some significant further work in the area by experts who extended it a lot.

Even this fairly minimal contribution is more than many students produce today, and advisors are expected apparently to essentially outline and design the thesis for them.

I.e. thesis in math is supposed to be new, interesting, non trivial, discovery, and verification of substantive results.

In many cases it consists of reproving more clearly or simply a known result, or clarifying an old solution from ancient times of an interesting tresult, or generalizing a good result to a slightly broader setting.

In mathematics, a thesis is not at all merely the recitation of the results of some experiments, whether they succeeded or not. Failed experiments are a failed thesis in math, they do not count at all, they only give the experience needed to try again more successfully.

In my thesis I partially solved a problem attempted unsuccessfully by some famous mathematicians, and discovered in the process a method that was useful in other settings, and which I used for years afterwards on other questions.


In writing a thesis I can suggest that one must take advantage of everything one has learned or heard, that one must step out on faith and belive in ones intuition, and then work very hard to substantiate the results of ones imagination.


It takes great stamina, persistence, and help from more knowledgable people, as well as some luck, to achieve something new and interesting.


But just as in other settings, even if one does not achieve the maximum result hoped for, one can still anticipate graduating. As stated (perhaps by Robin Hartshorne) to a friend of mine, the purpose of a thesis is to be the first creative work of ones life, not the last.


you CAN do more than you think, and what you can do is enough.
mathwonk
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Aug2-06, 10:13 PM
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although i have never advised a phd student, i can say what has led to my own best work: namely to read and familiarize yourself with the work of excellent people, and try to understand it as well as possible. speak on it, give a seminar on it, and it will seep into your pores and illuminate you and lead you to something further.

if things are slow, give a seminar on a paper by someone you admire. never stop working, as chern told me, maybe rest for a day or two, then go back to work.

do not be content just learning like a student, as fulton said to a friend of mine, but try to reprove significant results or extend them. at some point you will find you are going beyond what is known and on your way.
mathwonk
#162
Aug2-06, 10:15 PM
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as students we are often dependent on the simplest explanations to get on the train, but we should always aspire and try to reach the level exemplified by the masters, so do as abel said: read the masters, or prepare until one can do so.


in algebra this means to get to the point where one can read artin, van der waerden, lang, sah, jacobson. do not stop with dummitt and foote, or hungerford, rotman, herstein, or other second level texts, but do use those to get to the point one desires to reach.

(Edit: Actually of course one wants to be able to read original papers.)


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