Should I Become a Mathematician?

[hrc969]

Hey mathwonk, I was wondering: are you by any chance familiar with the book Differential Analysis on Complex Manifolds by R.O. Wells?

If so do you think its a good place to learn about complex manifolds?
Do you have any general comments on it?

 

[mathwonk]

well i took that course from ronnie at brandeis, before he wrote the book, and as I recall the first edition had some mathematical flaws that were corrected in later ones. It seemed like a nice book to me, but I am not an expert on books on complex manifolds.

I also read kodaira and morrow, a little computational for my taste, and for one dimensional complex manifodls, i.e. riemann surfaces, there are many good books, like griffiths and harris, springer, gunning, but wells's book is beautifully printed, a plus, and carefully written.

At a little higher level I like hirzebruch's book on topological methods in algebraic geometry. other books I have on my shelf on on complex manifolds include varietes kahleriennes by andre weil, and the very terse pamphlet of chern - complex manifolds without potential theory, and also chern's notes from recife.

books on the local theory, i.e, several complex variables, include the classical books of hormander and gunning and rossi.

its a big subject. there are some nice local theorems, like extension theorems for holomorphic functions, then some ideas that are unique to more than one variable, like holomorphic convexity and domains of holomorphy, then there are basic differential operators and their regularity proeprties, like various laplacians, and the interplay ebetween harmonic and holomorphic functions. then there is the global theory combining the complex analysis with the topology, with concepts like chern classes, and sheaves, and vector bundles, and riemann roch theorems for counting the sections of holomorphic vector bundles in terms of topological invariants involving chern classes.

there are also generalizations to indices of elliptic operators by atiyah - singer. wells is probably a good place to start. or maybe gunning on compact riemann surfaces, for an intro to the ideas but in a simpler case of one dimension.

i.e. there are two special cases of complex manifolds that are often studied, namely the riemann surfaces, and the abelian varieties, or complex tori.

these are easier than the general cases and need less machinery. gunning has the nice idea to teach the general machinery but in the easier setting of the one dimensional case. abelian varieties are quotients C^n/Lattice, so can be studied globally on C^n, using quasi periodic functions.I am not really an expert here and if you master wells's book you should know more than me.

 

[hrc969]

Thanks for all the info mathwonk. I guess I'll make sure to go through Wells' book. I am currently studying Several Complex Variables (I'm taking a reading course because we don't have an actual that I can take). I am using Steven Krantz's book, but I also have Hormander's. I have been mostly focusing on Krantz's book I'm not sure how much I should be looking at a book like Hormander's. In general do you think we should study the classics and the newer ones or just the classics or just the newer ones?

 

[hrc969]

About Grad school: I have heard (many many times) that one should get theit graduate education at a different school than where they got their undergraduate education.

I was wondering if anyone knows any particular reasons why this recommendation is made.

 

[Bitter]

Hello, I just wanted to post a link that I believe many people would enjoy. There is a professor at my math department by the name of Les Reid. He enjoys solving problems and puzzles that relate to mathematics and has a website that post some problems on various levels.
http://math.missouristate.edu/%7Eles/POTW.html

 

[mathwonk]

I am not an expert on complex analysis books, especially recdent ones. I read mostly Gunning and Rossi, Hormander, and Kodaira and Morrow, Griffiths and Harris, and some in Wells. Some of the newer books are probably very carefully written and may be more readable than some of the older ones.

But Hormander, although terse and maybe hard to read, is an excellent book. It is nice to see how brief the subject can look through the eyes of a real (or complex) master.

Wells is going to explain stuff like the sheaf theory on the complexification of the cotangent bundle of a manifold, and the Kodaira/Nakano vanishing theorem, global fancy stuff like that, also the Sobolev theory.

Hormander is going to explain also why every function which is differentiable in each variable separately is differentiable as a function of several variables simultaneously, i.e. the classical basic local theory as well. This is a useful fact. I needed it in my thesis.

I am afraid if you only read Wells you will have some of the nuts and bolts slightly hidden from you, under all the sheaf machinery. The deRham theorem proved by sheaf theory is a beautiful magic trick, but I'm not sure how enlightening it is. It eventually all boils down to the local dolbeault lemma, or poincare lemma in the real case, plus the sheaf magic to globalize it.

Magic proofs are very intoxicating to beginners. but eventually one has to get under the hood and take the pieces apart to understand them.

 

[mathwonk]

one should go to a different school for grad work just to encounter new points of view and expertise.

 

[mathwonk]

Free math books!

I just found this central site for collecting good free math books (and other things). I recommend anything by James Milne for example. His books will give you the idea of what a grad course at U. Michigan, Ann Arbor, should be like.

Milne's stuff is top notch. Sharipov is there too. Take a
look.

http://dmoz.org/Science/Math/Publications/Online_Texts/

 

[mathwonk]

Aside remark:

I hope you are not too offended if I seem to ignore your posts for long periods of time. School has started and I am having a busy week.

I only have time to log on occasionally now.

best wishes.

 

[quasar987]

Where's the link to that site you talk about?

 

[d_leet]

I just found this central site for collecting good free math books (and other things). I recommend anything by James Milne for example. His books will give you the idea of what a grad course at U. Michigan, Ann Arbor, should be like.

Mathwonk, what is the site?

 

[mathwonk]

oops, look back at my post now.

 

[mathwonk]

remarks on books:

i have noticed in other sites, maybe mostly physics, that some students are afraid old books will be outdated, and want to read newer books.

In math, the best books are the ones by the best mathematicians, regardless of their date. I.e. unlike maybe physics, math theorems are actually proved, hence mostly correct the first time, so they don't change a lot. I.e. in the words of the creationists, they are not "theories" in the speculative sense.

Hence Gauss's book on number theory is not at all outdated, for those topics which it treats. Nor is Courant's 80 year old calculus book outdated, or even Goursat's older one.

Riemann's works on topology of surfaces and complex anaklysis are indeed not complete or fully rigorous in the modern sense, but they are still highly recommended for their insight, which excels that available in most modern books.

In math, newer works on old topics tend to rewrite the discoveries of the masters, with more attention to technical detail, but without the global insight of those great masters.

all of us can only write what we ourselves understand. so newer works are primarily of interest for new topics that did not exist in olden times. Also newer books may interest young learners because they may try be easier. But this can be a drawback as well, by bowdlerizing the material.

new material is likewise best learned from the new masters who discovered it, rather than from their pupils who re - recorded it. This is an ideal, but one is encouraged to violate it whenever a given source serves ones own needs of understanding.

But eventually one benefits from returning to the source for deepest understanding.

 

[morphism]

Well said!

 

[Ki Man]

Words last as long as the peopel who listen to the, but math is forever.

 

[gmchamp2007]

Hey Mathwonk,

I just want to start out by saying that I am new to this forum. I notice that you have been giving advice on useful math texts. I wanted to know which books are good for getting a solid basis in the geometry and algebra of high school mathematics after which one can then go on to study calculus.

 

[mathwonk]

the books by harold jacobs are my favorites, Algebra, and Geometry.

the algebra ones seem to have become more expensive in the used book market:

4. Elementary Algebra Student Textbook (ISBN: 0716710471)
Harold R. Jacob
Bookseller: Adoremus Books
(Omaha, NE, U.S.A.) Price: US$ 66.27
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Quantity: 6 Shipping within U.S.A.:
US$ 3.95
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Book Description: VHPS, 1979. Book Condition: New. 100% Brand New!- In Stock at our Warehouse in Omaha, NE and ships out same day if ordered by noon CST. We provide Email Tracking and Shipment Information. We recommend Expedited Shipping for much faster delivery! Buy from us and you will keep coming back!. Bookseller Inventory # 60270

Geometry (ISBN: 0716704560)
Harold R. Jacobs
Bookseller: e-Book Traders, Inc.
(Tampa, FL, U.S.A.) Price: US$ 17.25
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Quantity: 1 Shipping within U.S.A.:
US$ 5.50
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Book Description: W.H. Freeman & Company, 1974. Hardcover. Book Condition: ACCEPTABLE. Dust Jacket Condition: ACCEPTABLE. USED Dust Jacket Ships Within 24 Hours - Satisfaction Guaranteed!. Bookseller Inventory # 121168

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2. Geometry (ISBN: 0716704560)
Harold R. Jacobs
Bookseller: OwlsBooks
(Hammond, IN, U.S.A.) Price: US$ 27.63
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US$ 3.99
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Book Description: W.H. Freeman & Company, 1974. Hardcover. Book Condition: GOOD. Dust Jacket Condition: ACCEPTABLE. USED No Dust Jacket Used items at a great price! Come to OwlsBooks for all your media needs. We have hundreds of thousands of items available today!. Bookseller Inventory # 73260added years later: For geometry: Euclid's Elements; for algebra: Euler's Elements of Algebra.

 

[mathwonk]

heres a little cheaper one found on amazon used books:
Jacobs, Elementary Algebra,
$51.99
+ $3.49shipping
LOW ITEM PRICE
New
Seller: mytextbookseller
Rating:100% positive over the past 12 months (10 ratings.) 10 lifetime ratings.
Shipping: In Stock. Ships from GA, United States Expedited shipping available. Ships from GA, United States Expedited shipping available. See

 

[hrc969]

Mathwonk, do you know anything about a problem solvers vs. theory builders thing in math? I heard that sometimes problem solvers are looked down upon (or have been in the past). Are you aware of any of this? What are the distinctions?

 

[pivoxa15]

My impression is that most good theory builders are good problem solvers but not always vice versa. Theory building is harder because it is more general - more things to deal with. This could be why pure maths is usually considered to be harder than applied maths.

 

[J77]

This could be why pure maths is usually considered to be harder than applied maths.

Have you done either to a high level?

If you had, you'd find that the distinction between the two can become very blurred.

 

[hrc969]

Have you done either to a high level?

If you had, you'd find that the distinction between the two can become very blurred.

According to one of my professors, sometimes applied mathematicians have to know MORE than pure mathmaticians. Making it harder to be an applied mathematician. (Of course this is at the PHD level or beyond).

But that's beside what I was talking about. From where I heard the problem solvers vs theory builders it was only in the context of pure mathmatics, the examples I got from each case were all pure mathmaticians.

pivoxa15, maybe you are thinking about problem solving at some level below PHD. But from the examples I got, the theory builder were not good problem solvers.

 

[mathwonk]

i think both types have been fields medal winners, but this sort of distinction is a kind of one ups manship game with no real interest to me.

just follow your dream, don't care if someone looks down on you because you are a type A mathematician instead of type B.

i.e. the purpose of building theories is to be able to solve new problems, and the solution of new problems usually requires new insights. so the two thrive on one another, and die without each other.

 

[pivoxa15]

i think both types have been fields medal winners, but this sort of distinction is silly, a kind of one ups manship game with no real interest to me.

just follow your dream, don't care if some A** H*** looks down on you because you are a type A mathematician instead of type B.

i.e. the purpose of building theories is too be able to solve new problems, and the solution of new problems usually requires new insights. so the two thrive on one another, and die without each other.

As always, very good point.

 

[mathwonk]

as an example of two fields medalists, grothendieck built the machinery of modern algebraic geometry, and deligne used it to solve the weil conjectures, which had been grothendieck's aim.

 

[bit188]

I've wanted to be a computer scientist since I was nine years old (four years ago!). This year I've had some sort of "intellectual exploration"-type thingy going on. I've screwed around in biology, chemistry, psychology, mathematics, physics, etc.

Now I'm very unsure... I started into Calc this year and it's incredible. Just... wow. It's like nothing I've done before, and I feel the same way for physics. I've decided to continue to pursue my newfound interests, but I have no idea what to do for a job. Part of me wants to be dedicated to the arts, part of me wants to be a mathematician or physicist.

On the side of textbooks, the Dover books are good for low-budget self teaching. I use them whenever they're available. ^_^

 

[mathwonk]

Just explore as many interesting things as possible at least for the next 5 years or more. Have fun!

One good cheap beginning calc. book is "calculus made easy" by silvanus p. thompson, about 100 years old now and still a classic of good writing.

I also greatly enjoyed "the universe and doctor einstein", by lincoln barnett, as a 13-15 year old. It should be very easy for you, maybe a one afternoon read, but possibly more.

 

[bit188]

Yes, trust me! I will; I have a feeling this will be a lifelong interest, no matter what my "real job" is.

I'll be starting into ODEs very soon now; however, I'm back at square one for physics (I have to go through intro. physics again with Calculus)...

It shouldn't take too long, however, because I know the concepts well enough! Then it's on to some statics and dynamics books that were recommended by a member here, and afterwards I finally move into upper-undergrad classical mechanics (I'm going on a different route than most undergrads; I'm going to tackle one subject at a time rather than many).

Oh! I don't know if these have been posted, but here are some great online math resources:

http://www.geocities.com/alex_stef/mylist.html -- Huge list of free textbooks
http://archives.math.utk.edu/visual.calculus/ -- Visual Calculus, a great way to help yourself if you get stuck. Unfortunately, only covers Calc I & II
http://tutorial.math.lamar.edu/ -- Fabulous set of lecture notes for lower-level undergrad math

 

[mathwonk]

please feel free to post questions and reactions to your exploration here for the rest of us to enjoy.

 

[quasar987]

Hey wonk, are you presently working on something? If so can you give a brief description of the problem and its implication and we're you're at regarding the "solution".

 

[bit188]

My "exploration" is going well. I think I've come to the end, as I've decided what exactly I'm sticking with (math/physics/comp sci).

I'm reading a book on proof-writing, I'll be starting my ODE book soon, and I'm waiting for my Calc-based physics text that I just got (Amazon rules!).

I'm worried about how I'll do once I hit analysis, where everything is proof-based... I'm doing very well using the book I've got, I can write most of the proofs nearly instanly, but analysis is (obviously) going to be a lot harder... and at the rate I go, considering the length of the books I read, I'll be hitting it pretty fast. Should I worry?

 

[mathwonk]

never worry. youll be fine.

 

[mathwonk]

well i have been on hiatus for a while, but my long term project was to try to show that if f:C'-->C is a 2:1 unramified map of curves, then the kernel of the induced map of jacobian varieties (the "prym variety"), plus its theta divisor, determines the map f back again, unless the curve C has a divisor of degree d that moves in a linear system of dimension r where d-2r ≤ 2.

this is the first non trivial generalization of the famous torelli theorem, now 100 years old, that the jacobian of a curve determines the curve.

my colleague robert varley and i have shown that the curve is determined by the prym variety, plus the abel map onto the theta divisor of the prym, unless the curve C has a divisor of degree d, moving in a linear system of dimension r where d = 2r.

but we do not know yet how to recover tha abel map from just the prym variety and its theta divisor.

oue hope is to imitate the proof of mark green for jacobians, using (in our formulation), deformation theory of abel maps to singular theta divisors, and extend it to the case of P^1 bundles over prym theta divisors, but we have been stalled for some time, largely from other duties, but also from technical difficulties.

We are trying to show that if you have a singular variety, and a smooth variety over it with generic fiber P^1, that under natural conditions, any deformation of the singular variety preserving the singularities, induces a deformation also of the P^1 bundle.

Our proof of Greens theorem showed that over a singular theta divisor, if the abel resolution has small exceptional locus, then the analogous statement is true.

this is part of a large program due to riemann of "moduli", i,.e. classifying geometric objects, in this case a 2:1 map of curves, by numbers, in this case the period matrix determining the prym variety and its theta divisor.edit years later: Unfortunately the time I spent over the last year or three writing these posts has apparently come at the expense of my research time.

 

[quasar987]

Sounds fun. And you succeeded in conveying a feel for what you're doing although most of the items you talk about are formally alien to me.

What about the other mathematicians? What are you doing at the moment Dr. Grime? Schmoe?

 

[mathwonk]

other progress and partial results on the problem i mentioned are: that almost all prym varieties do determine the corresponding double cover of cuvres, but we do not know precisely which ones are determined. I.e. we know most of them are but do not know what proeprty distinguishes those which are. the property stated conjecturally above is a conjecture of Donagi. It may need further modification since my colleague Izadi has at least preliminary work sugesrting more counterexamples exist.

Donagi and I also showed in 1981 that when the genus of the curve C is 6, the theorem is even generically false, and that there are in general 27 double covers with the same prym variety. there is also a link with the 27 lines lying on a hyperplane section of a cubic threefold!

this problem is now over 100 years old, having been started by riemann and pursued especially wirtinger in about 1889.