Should I Become a Mathematician?

[mathwonk]

if you google me you will find a couple things. and there are about 6 research articles on my webpage, but it is pretty specialized.

http://www.math.uga.edu/~roy/

of more interest to you may be my class notes since several of them are quite accessible to a good high school student, such as my math 4000 notes, and maybe possibly as a project, even my 14 page linear algebra book, supplemented by A BOOK ON MATRICES, like the SMSG old 1960's high school book.

maybe even my 843 and 844 notes. they start out with groups, which could (and probably should) be learned in high school.

i.e. groups and linear algebra are really much better suited as high school topics than is calculus, because they are easier.

 

[mathwonk]

i described problems of interest to me in posts 454-457.

 

[KGZotU]

Thank you for this wonderful thread. I came across it via Google.

I'm a second year math student. It's totally irrelevant, but I can't just say that without qualifying that I'm 24. I've got a solid chance at transferring into Berkeley next year, and I've had a growing feeling that I don't really know anything about math. I've got impartial indicators that say I'm fairly intelligent...but this thread has shown me what a banal math education I've gotten.

Luckily, there's a copy of Apostol's Calculus in the library. A textbook that's been in the same edition since the 60's? That just blows my mind. I'll pull it out once the library re-opens.

I'm making my way through the entire thread, but I've got a couple of questions that I just can't sit on anymore. I understand that in some fields it's possible to find a job in a university lab as a "Research Associate" for a year or two following graduation. Are these opportunities available in pure mathematics? Would this be a good experience/prep for grad school or a waste of time?

Assuming the above is possible, that's what I've got planned following graduation. Following that, my dream for the past few months has been Cambridge's CASM, and a PhD. My final question is, supposing I'm going to spend a couple of years as a "Research Associate" following undergrad, would my junior/senior years be better spent focused on research hours or class hours?

Thanks again for this thread. It helps to know that you don't strictly need to be a genius to succeed in math; lord knows I'm no genius.

--Joe

 

[mathwonk]

I do not know of positions in pure math as research associate. there are some summer programs doing research or at least being introduced to research, called REU's.

These can be helpful at becoming introduced to new ideas and topics in mathematics that one has not met in college.

If you don't know much math presumably the last two years need to be spent learning more math. you might look at some of the posts herte on prelim exams, which show what is expected from every candidate, in the way of background knowledge.

It is hard to adivse you with so little information on what level you are at now. If you are a sophomore in college, you have a lot of subjects to learn before doing research for PhD, I would guess, real and complex analysis, topology, algebra.

there are free PhD quals algebra notes on my website, as well as elsewhere. James Milne posts many wonderful sets of notes on his site.

 

[KGZotU]

Thank you for the response. Essentially, I've been completing the transfer requirements at a California Community College. Three semesters of Calc, getting into linear algebra and differential equations. I had always heard that lower division math is quite different from upper division and beyond, but it's interesting to learn that it doesn't _have_ to be that way. I do well academically, well enough to wonder why I don't feel like I'm learning too much.

I'll be sure to add your topics to my course plan. It's nice to know the sort of breadth I'll be needing.

Thanks also for your response regarding research associateships. Sounds like it's hard to get your feet wet with pure math outside of pushing on towards the PhD.

--Joe

 

[abelian]

What does everyone here think of Mac Lane's Algebra? I'm surprised it isn't widely adopted - it develops all required machinery (including categories) from scratch, which perhaps makes for an excellent preparation to Lang. How about everyone else?

 

[mathwonk]

i think i like it, but i have not seen a copy in a long time, and do not own one. you can't go wrong with a book by someone like that. i have birkhoff and maclane which was a very good down to Earth intro, and i have maclane's homology which is very well written.

isn't maclane's algebra out of print? that would account for its lack of adoptions. chi han sah's book and nathan jacobson's books are also excellent but out of print. thank you for reminding me of maclane's book. ill take another look at it if I teach the course again.

 

[mathwonk]

by the way, i am finding myself answering questions here that are already systematically answered in the early parts of the thread, which is now too long to read in its entirety. is there some way for me to extract my basic advice posts and put them elsewhere so that can be a reference and this can be an ongoing chat room. sorry for the caps. stupid caplock key should be further from the "a" key.

 

[mathwonk]

by the way, getting your feet wet in math research is done by carefully reading a research paper. try one of mine if you like, such as the introduction to my paper with robert varley on a pfaffian structure for a prym variety, but you will get more by reading a paper by zariski, or perelman, or well, you know... a fields medalist, or a 19th century giant like riemann, or galois, or gauss's disquisitiones...

i know what, try riemann's collected works. read his thesis. its available in english now, from Kendrick Press.

http://kendrickpress.com/Riemann.htm

 

[mathwonk]

here is my latest word on archimedes computation of the volume of a sphere, revealing how he anticipated integral calculus, riemann sums, and cavalieri's principle. the moral again is to read the masters.

ok i have actually read more of archimedes and think i know how he found the volume of a sphere, or at least how he proved it. (he discovered it by setting up a lever and balancing the weights of different solids, knowing the centers of gravity of some of them, and deducing that of others.)

basic principles:
1) principle of parallel slices: two solids with equal areas for all plane slices parallel to a given plane, have equal volumes.
2) magnification principle: two pyramids with bases of equal area, have volumes in the same ratio as their heights.

these principles are proved by the method of approximation by blocks or cylinders, since solids with equal plane slices have equal approximating cylinders, and scaling the height merely scales the height of the approximating cylinders. then one proceeds as follows, first for pyramids and cones, then spheres.

step 1) right pyramids of height equal to base edge:
choose 2 opposite vertices on a cube, call them 1 and 2, and join them by a diagonal. choose a face having vertex 2 as a corner, and join every point of this face to vertex 1. this forms a right pyramid. the other two choices of faces having vertex 2 as corner, yield congruent pyramids, by rotation, and all three together make up the cube. thus the given right pyramid has volume 1/3 that of the cube, or 1/3 Bh, where B = area of base, and h = height.

step 2) using magnification principle, one extends the same formula to the case of arbitrary height in comparison to base edge, and using parallel slices one extends the same formula to pyramids which are not "right", but for which the angle to the vertex is arbitrary, since sliding a pyramid over at a new angle does not change the area of parallel slices.

step 3) approximating the base circle by polygons, hence approximating the cone by pyramids, gives the same formula for a cone, V = 1/3 Bh.

step 4) now circumscribe a cylinder about a sphere, and inscribe a double cone (vertex at center, bases at both top and bottom) in the same cylinder. then pythagoras shows that the area of a parallel slice of the cylinder has area equal to the sum of the parallel slices of the sphere and the cone.

Thus the volume of the cylinder equals the sum of the volumes of the cone and the sphere. in particular since the cone has 1/3 the volume of the cylinder, the sphere has 2/3 the volume of the circumscribing cylinder.

And that is how archimedes proved the volume of a sphere.

then by the argument above, viewing the sphere as a limit of pyramids with vertices at the center, he showed the surface area of the sphere, defined as the limit of the areas of the bases of the inscribed pyramids, was 3/R times the volume of the sphere, since that is the formula for the base area of a pyramid in terms of the volume.

I.e. the volume of a sphere is 1/3 SR where S is the surface area and R is the radius.

and that's that! hooray for archimedes, who was obviously in almost complete command of the methods of purely integral calculus.

the only thing needing to be added, was the algebraic technique of antidifferentiating the algebraic formula for the area of the parallel slices and getting an algebraic formula for the moving volumes below each slice.

so as far as i know now it had nothing to do with adding up squares of integers at all, quite opposite to my original impression.

 

[abelian]

isn't maclane's algebra out of print? that would account for its lack of adoptions. chi han sah's book and nathan jacobson's books are also excellent but out of print. thank you for reminding me of maclane's book. ill take another look at it if I teach the course again.

Not according to https://www.amazon.com/dp/0821816462/?tag=pfamazon01-20. It's affordable, too.

Out of curiosity, mathwonk, does Jacobson's Basic Algebra introduce categories from the get-go?

 

[mathwonk]

well that is a piece of luck, it has been reprinted by an inexpensive republisher, ams/dover. i'd snap it up.

no jacobson does things in a unsophisticated way in vol 1 and then more sophisticated in vol 2.

looking at maclane i see why it has fallen out of use, as it somehow combines the old fashioned feel of the original book by birkhoff and maclane with a slightly tedious use of fancy language from categories that seems unnecessary to me.

 

[KGZotU]

mathwonk,

I've got Riemann's works coming via interlibrary loan. Thanks for the heads up. Sounds like the trick might be to read it enough to get a feel for it, and maybe eventually get enough of a feel for it to be able to do it.

Also, many thanks for suggesting http://users.ictp.it/~stefanov/mylist.html quite a few pages back. Got a couple picked out that look like good introductions to the topics of higher maths. Have access to Apostol's Calculus, as mentioned earlier, and my goal is to have a fuzzy grasp of analysis by the end of summer so it doesn't kill me come fall.

A tip for anybody trying to read a pdf book: you can override the background color to something other than white. I prefer a sort of fleshy peach color, keeps my eyes from dying. You can do this, at least in reader 8, by right clicking on the document and going to display preferences -> accessibility -> replace document colors -> custom color -> page background.

I don't know about a way to extract your responses, but the archive version of this page is more imminently searchable, and it would make for easier extraction if you knew how:
who wants to be a mathematician? - archive

--Joe

Edit @ mathwonk: I realized I could pare this down to just your posts with a few rounds of search and replace in Word. I edited the html a bit to remove reliance on physics forums style sheets and give some readability. Final product is here: http://uashome.alaska.edu/~JASCHILZ/wonkmathematician.html . Of course it's yours so you can copy, modify, host elsewhere, ask me to take down, etc. Hope this helps.

 

[eastside00_99]

Mathwonk, I have two Algebro-Geometric questions. Maybe you can shed some light:

Fix a field k which may or may not have characteristic zero and may or may not be algebraically closed.

(1) If A is a CRU, then is the localisation of A at a prime ideal p, A_p, always a finitely generated k-algebra? The answer is no although I am having a problem constructing a counterexample (it appears the thing as to be really big). But, what if p is a minimal prime ideal, then I need it to be true! Do you know if it is?

(2) Hilbert Basis Theorem (HBT): I presently know two proofs of this theorem. The one given by Hilbert and the one that solidifies his idea via Buchberger's Criterion.

Now, I am working on the following theorem: Let m be a maximal ideal of k[T1,...,Tn]. There exists n polynomials P1(T1), P2(T1,T2),..., Pn(T1,...,Tn) such that for any i <= n we have m(intersect)k[T1,...,Ti] = (P1,...,Pi). I think this is pretty straight forward and can be done by induction. I just need to double check what I am thinking it is right.



Assume true for some i <n, let k_i = k[T1,...,Ti]/(P1,...,Pi). Then we have an exact sequence

0 --> (P1,...,Pi)K[T1,...,Tn] --> K[T1,...,Tn] --> k_i[T{i+1},...,Tn] -->0. The base case is trivial since k and k_i are fields (the latter is obtained by the induction hypothesis), so m(intersect)k_i[T{i+1}]=(Q) where Q is the image of some P{i+1}(T1,...,T{i+1}) in k_i[T{i+1},...,Tn]. The result easily follows.

My question is how far can one extend HBT. It seems to me if maximal ideals are finitely generated (and much more generated by n polynomials), then all other ideals will be, and this would constitute another proof of HBT. But, what about the noncommutative case? I am sure that all this has already been worked out, but what is the most general form of it?

Anyway, I am working through Algebraic Geometry an Arithmetic Cruves by Qing Liu and these are some of the first problems in the book. It is taking me forever to read this book albeit I just started a few weeks ago. Let's see, I basically average a two pages a day or 1 to 2 problems a day. Although, I am taking it pretty easy, maybe work on it for five hours a day but no less than three. Is this normal? At this rate it will probably take 1 and a half years to complete the whole book!

Further, I finished all my grad apps; if you remember I was asking for some advice a while back. I didn't apply to UGA basically because I want to get out of the south; one school I applied to almost entirely on location: what do you think of the Ph.D. program at Tufts?

 

[mathwonk]

well i don't know how to do your commutative algebra problems offhand, since i am a more complex analytic topological algebraic geometer, but they look interesting.

at tufts i know montserrat teixidor, and mauricio gutierrez, and know of zbigniew nitecki for his excellent diff eq book, and i would guess they do a fine job.

 

[eastside00_99]

Yes, I talked to Dr. Teixidor through email about the program some months. She was very ethusiastic. There are only two people at Tufts that work in Algebraic Geometry, so I said I wanted to work with them.

 

[mathwonk]

in fact could you remind me why the intersection of your maximal ideal in k[T1,...,Tn] with the subring k[T1,...Ti] is maximal?
i guess this follows from dimension theory, since the polynomial ring is cohen macaulay and you have a prime ideal of maximal height, it must be maximal?

but there must be an easier reason, since this sort of result is much more sophisticated than the hilbert basis theorem.

 

[mathwonk]

i guess by the zariski nullstellensatz the quotient of the big ring by the original maximal ideal gives a field algebraic and finitely generated over the base field, hence finite as a vector space, hence finite also as a module over the quotient of the subring by the intersection ideal. but then by the going up and down lemmas, the ring downstairs is also a field?

still this is pretty sophisticated, assuming the strongest results of basic algebra on finiteness, nullstellensatz, etc...

i sort of assumed since you were starting from the basis theorem you were assuming very little. i am probably off base here, but i am a little out of shape on this stuff not having taught commutative algebra lately, and even then generally always working over algebraically closed fields..

 

[mathwonk]

or maybe you are assuming the maximality of the intersection ideal as part of the (unstated) induction hypothesis?

 

[mathwonk]

as far as finite generation of maximal ideals implying it for all ideals i do not know if this is true, but problem 11, section 15.1 of dummit and foote second edition, outlines the proof if all prime ideals are finitely generated. i did not work the problem and do not see where the hypothesis is used hence do not see whether the same or a similar proof works for maximal ones.

 

[eastside00_99]

i guess by the zariski nullstellensatz the quotient of the big ring by the original maximal ideal gives a field algebraic and finitely generated over the base field, hence finite as a vector space, hence finite also as a module over the quotient of the subring by the intersection ideal. but then by the going up and down lemmas, the ring downstairs is also a field?

still this is pretty sophisticated, assuming the strongest results of basic algebra on finiteness, nullstellensatz, etc...

i sort of assumed since you were starting from the basis theorem you were assuming very little. i am probably off base here, but i am a little out of shape on this stuff not having taught commutative algebra lately, and even then generally always working over algebraically closed fields..

Actually, I skipped over this point very quickly, and truthfully didn't really think about it. Yes, we can use these results, but I am not sure we can go from K=k[T1,...,Tn]/m being finite over ki, to saying m(intersect)k[T1,...,Ti] is maximal. I have book on elimination theory in a box under a bunch of books. Maybe this will give a simple answer.

Sorry, about that though, I really skipped an important part of the proof. You truly are a mathematician as you saw it right away.

P.S. The starting point is Noether Normalization Lemma.

 

[mathwonk]

the point is there is an injection from k[T1,...Ti]/ pullback ideal -->k[T1,...,Tn]/maxl ideal.

hence if this extension is finite as a module, the fact the larger ring is a field implies the smaller one is too, since going down says the downstairs ring cannot have more maximal ideals than the upstairs one. look at your proof of the noether nromalization lemma, and see if it does not use this going up and down lemma. see mumford's redbook of algebraic geometry, the first section.

 

[eastside00_99]

Ah, got it, thanks. I don't have a copy of Mumfords book, but the book I am using is hinting around what you are saying--i.e. A finite as Ao-module plus injective hom Ao --> A iff A finite as Ao-algebra iff every sub-Ao-algebra of A is a field.

 

[eastside00_99]

Anyway, this is taking forever..............
.....................
..........................

 

[mathwonk]

finite as algebra does not imply finite as module, unless you also assume integral.
i.e. the analogy is with field theory where finitely generated as field and algebraic, is equivalent to finite as vector space.
here finitely generated as algebra and integral (the analog of algebraic) is equivalent to finite as module.
[or perhaps you are assuming A is a field? does that help?]
take your time, you have a lot of it. and its enjoyable and worthwhile to understand the stuff well.

 

[pivoxa15]

Mathwonk, why did you choose algebraic geometry instead of algebraic number theory?

Is commutative algebra a prereq to get into the field of algebraic number theory?

What are some supplementary books to Atiyah and Macdonald's 'intro to commutative algebra'? What do you think of that book?

 

[mathwonk]

i heard lectures from alan mayer at brandeis that were very appealing and magnetic. there were also good lectures by paul monsky in algebraic number theory, but i just loved the geometry.

well yes probably commutative algebra is useful for algebraic number theory.

atiyah macdonald is a book everyone likes. i prefer zariski and samuel, which also has far more material, and i recommend it to you as well. eisenbud is a newer book that also has much to recommend it.

 

[eastside00_99]

Yeah, eisenbud is nice as it ties Geometry into the material, but it is so long. I have an obscure book by J.T. Knight which was published as part of the London mathematical society lecture notes series in the 70s. This may be a good book for you and it very sort. As I see it, Eisenbud is more of a reference than for self-study. For instance, if you want to know something about Krull Dimension, pop open his book to chapter 11 or something, but don't read and work all the problems until you get to Krull Dimension...you will never get there.

I would say in general commutative algebra is a prerequisite for *modern* algebraic geometry which in turn is a prerequisite for algebraic number theory.

 

[pivoxa15]

i heard lectures from alan mayer at brandeis that were very appealing and magnetic. there were also good lectures by paul monsky in algebraic number theory, but i just loved the geometry.

well yes probably commutative algebra is useful for algebraic number theory.

atiyah macdonald is a book everyone likes. i prefer zariski and samuel, which also has far more material, and i recommend it to you as well. eisenbud is a newer book that also has much to recommend it.

I heard atiyah and Macdonald is very condensed. Are there books that are written especially for explaining A&M in more detail and easier to read manner?

 

[eastside00_99]

I'm going to compile a list of textbooks and papers for the aspiring Algebraic Geometer. Any suggestions?

 

[mathwonk]

nonetheless, atiyah macdonald is considered very easy to read, and i recommend it. but i especially like zariski samuel which is much more detailed and hence extremely readable.

 

[mathwonk]

well i like Shafarevich, basic algebraic geometry. there are some mistakes, but overall it is one of the most geometrically intuitive and enjoyable introduction out there.

of course Hartshorne is also very highly recommnended, but i personally recommend Shafarevich first followed by Hartshorne.

oh yes, even before those two books, there are excellent books on curves, by Walker, and by Fulton, but Fulton is out of print and very hard to find. another very nice intro to complex curves is algebraic curves by Griffiths.

so here is a list of introductory books i myself read and liked, roughly in order of difficulty: walker: algebraic plane curves. fulton: algebraic curves; shafarevich: basic algebraic geometry; mumford: algebraic geometry I: projective varieties; mumford: red book of algebraic geometry; hartshorne: algebraic geometry.

a more recent one, written from experience teaching at brown and harvard, is algebraic geometry by joe harris. it is filled with good well explained examples. griffiths and harris: principles of algebraic geometry is also useful for accessible explanations of many important topics. I especially benefited from the chapter on riemann surfaces. this book has a number of technical errors and gaps in some proofs, but one still learns a lot from it.

then maybe arnaud beauville: complex algebraic surfaces; mumford: abelian varieties; mumford:lectures on algebraic surfaces, mukai: moduli theory? and mumford on theta functions, three volumes.

many people also like miles reids introductory book: undergraduate algebraic geometry. and also his undergraduate commutative algebra. if you find atiyah macdonald too hard, try reid.

another good introductory book is algebraic curves and riemann surfaces, by rick miranda. there are also lots of free notes on the web by people such as ravi vakil, igor dolgachev, james milne, miles reid, and others.

there are also many other books written since i was a student, and hence less familiar to me. the one listed are mostly the ones i read as a student. one recent book by a colleague that is well received is invitation to arithmetic geometry by dino lorenzini. another that may be useful for making the intuitive leap from varieties to schemes is geometry of schemes by eisenbud and harris.

i myself am currently reading varchenko et al on singularities, and looijenga on isolated complete intersection singularities. there are also many good more advanced books like the one by mori and kollar on classification of varieties.

 

[mathwonk]

it just dawned on me eastside, i think no one else here ever asked me the one thing i might possibly know something about!

 

[mathwonk]

hulek also has a new introduction to alg geom. and there is the classic book by semple and roth.

 

[eastside00_99]

it just dawned on me eastside, i think no one else here ever asked me the one thing i might possibly know something about!

I do what I can. I will look into these book. I have read Fulton's "Algebraic Curves" and that's basically it for the books on that list. So, they are welcomed additions. By the way, the new Oxford Graduate Texts in Mathematics series has two books out on Algebraic Geometry--one which introduces the study of Arithemtic Geometry and the other Algebraic Groups.