Other Should I Become a Mathematician?

[RocketSurgery]

Hey mathwonk. I've heard that a lot of mathematicians think that is good to "learn from the masters and not their students".
How do you feel about this idea? Is there even a textbook (or even a complete set of papers) in every area of math that is written by a "Master" and how do you Define master?

I'm learning Calculus from Apostol but is Apostol a master? or even Rudin for that matter?

 

[mathwonk]

well i do think apostol is a master of calculus, and i do recommend learning from masters as soon as their writings are accessible to you.

once in grad school for the heck of it i went to the libs and tried to read the famous paper on the concept of a singular point, by the master of algebraic geometry oscar zariski.

i struggled for hours to get through even a few pages and felt discouraged. but the next day in class, when the prof brought up the idea of a regular local ring, and regular sequences, i knew the answer to every question he asked instantly, so much so that ultimately he told me to shut up as i obviously knew the subject thoroughly.

that was my best day ever in class, and the only day i was ahead of the lesson.

 

[RocketSurgery]

Oh very nice example Mathwonk. I was just curious like what people mean by master and what books would be good to read from the so called masters.

From your point of view it seems you consider Apostol a master of calculus bc of how well he knows Analysis/Calculus but not necessarily because of his own contributions to the field.

Like some might consider Newton a Master and say that to understand calculus you should learn from his writings. However I don't think anyone is probably going to be better off then they would learning from Apostol or Spivak then going back to Newton's work for some enlightenment.

 

[mathwonk]

it is excellent to read Newton. for example one could learn there, well before riemanns well known definition of integration, that all monotone functions are integrable, (which one can also learn in apostol).

i myself have the book on analysis by goursat, which is also recommended.
i do not know apostol's contributions but anyone as outstanding as he must have made some.
i do know spivaks contributions to differential topology, namely the concept of the spivak normal fibre bundle, a fundamental tool in the subject.
probably apostol has some work in analytic number theory. i will check it out.

 

[mathwonk]

well he got his phd at berkeley in 1948 and is so famous since then for his book I am having trouble finding older data. his research in the past 10 years or so has been handsomely funded for projects in education of high school students.

http://www.maa.org/reviews/earlyhist.html

oh yes i believe i have commented here on some recent research by apostol on figures in solid geometry with area and volume formulas similar to those of spheres.
i.e. certain solids have the ratio of volume to area equal to something like R/3, where R is a "radius",
such as a sphere and perhaps a "bicylinder" (intersection of two perpendicular cylinders) and many others.

 

[mathwonk]

well ok, apostol is not a creator of calculus as are Newton and riemann. i recommend reading them too, for what you can get, but you will learn a lot from apostol.

there are two types of masters of a subject, those who first created it, and those later brilliant people who have indeed mastered it, and show that by the depth of their writings.

galois created galois theory, but emil artin made it accessible to modern generations, and others such as his son mike, and other modern masters like jacobson and van der waerden, and lang have given expositions some of us find useful.

it might still be useful to consult dirichlet, gauss, and legendre, for related work, but i have not much done so.

to be specific, you are invited to read my notes on my webpage, but having done so, if they are found useful, they can at best serve as an introduction to those small parts of the subject i myself understand. afterwards move on up to reading better works by more qualified persons.

e.g. even though i have criticized details in their book, dummit and foote are more accomplished algebra experts than I, as one can see from perusing their research vitae, and their book contains more than my notes.

still some features of their book cause me to feel that they are either consciously writing down to their audience, which i find troubling, or for some reason do not convey the depth one senses in artin and jacobson.

 

[RocketSurgery]

Very good point. So I'll try to make a habit of learning a subject from a good modern textbook but also look into what the creators have written afterwards to get a deeper understanding. That way I can see the point of view of the two types of masters.

 

[The|M|onster]

Thank you very much for the solutions, mathwonk. Also, would you mind recommending me some good books on Number Theory?

 

[Vid]

I really like reading Hardy's Introduction to the Theory of Numbers, and he's definitely a master.

 

[RocketSurgery]

I really like reading Hardy's Introduction to the Theory of Numbers, and he's definitely a master.

Agreed but don't forget about his coauthor, Wright.

 

[mathwonk]

forgive me, i have temporarily forgot the names of the number theory experts here, greathouse? robert ihnot? ...

lets ask them. i agree with niven and hardy by the way, but you might also check out andre weil, basic number theory (misleading title).

also borevich and shafarevich, and ...

 

[RocketSurgery]

Haha yup can't leave anyone out.

 

[mathwonk]

heres my real fave: trygve nagell. check it out.

its $120 on amazon, but here is a used one:

Introduction to Number Theory.
Nagell, Trygve.
Bookseller: Monkey See, Monkey Read LLC
(Northfield, MN, U.S.A.)
Bookseller Rating:
Price: US$ 20.00
[Convert Currency]
Quantity: 1 Shipping within U.S.A.:
US$ 3.99
[Rates & Speeds]
Book Description: John Wiley, 1951., New York:, 1951. Hardcover. ex-library with usual markings, no jacket, sound copy, text is clean. Bookseller Inventory # 4463

 

[RocketSurgery]

Seems interesting. The amazon review says Nagell is similar to Hardy/Wright. I'll see if my library has it.

 

[mathwonk]

all books have the elementary result of fermat on which integers are sums of two squares, but nagell explains which integers are sums of three squares.

stuff like that. and it is well written. i however have not read hardy and wright so it might be even better.

like i said i am a rookie at number theory. there are several people here much more expert.

 

[The|M|onster]

Cool. I am going to head to the library right after I get off of work.

 

[RocketSurgery]

Hey guys I was wondering if anyone could recommend me any books on Game Theory that I would be able to understand.

I have a good grounding in Proof based math (Set Theory, Logic, Apostol Calculus)
and I've taken an elementary Probability class.

I'd like to read Neumann's book but I don't know if it will go over my head or not.

 

[Mokae]

Usually on site like this, people rarely introduce others any books because, as you might guess, users log in with different usernames, and even the writers of the books. People care to recommend their own written books, right ? So I doubt if anyone around introducing any book to you is not the writer himself

Why don't you look up in your school library or just go straight to your school teachers to make some questions on the same problem ? I am sure they are not that selfish to not even given their students a title of an interesting book they just read or so...

 

[RocketSurgery]

Luckily I do not share your cynicism. There are many mathematicians and mathlovers on this board who recommend books all the time including Mathwonk. Unless Mathwonk is secretly Tom Apostol then I don't think we have much to worry about.

Being as you just joined this forum recently you will realize that a lot of the regulars here are very helpful people and not businessmen just trying to make a buck.

Welcome to the forums though!

 

[Shukie]

Math at university is a little too abstract for me, I like more hands-on math like in physics.

 

[RocketSurgery]

I hope that you aren't saying Calculus, Differential Equations, Partial Differential Equations etc etc at university is too abstract for physics. Did you ever do Calculus-based physics?

I ask because the Mathematics I mentioned are typically taught at university and CalculusI-III are essential for any real knowledge of physics (that is other than what Michio Kaku has told you). Or did you learn the aforementioned mathematics in High School? Now that would be awesome.

 

[cristo]

So I doubt if anyone around introducing any book to you is not the writer himself

But there is a whole forum for book suggestions/reviews here. If we could only recommend our own books, then that forum would be very empty indeed!

 

[The|M|onster]

Usually on site like this, people rarely introduce others any books because, as you might guess, users log in with different usernames, and even the writers of the books. People care to recommend their own written books, right ? So I doubt if anyone around introducing any book to you is not the writer himself

Yes, because clearly the person on the last page who recommended Hardy's Introduction to the Theory of Numbers is actually Godfrey Harold Hardy himself back from the grave to extort money out of me. Being dead for 60 + years really hits the wallet hard. Thank you oh so very much for enlightening me. Shame on you, Mr. Hardy.

 

[mathwonk]

good point, and apparently riemann, gauss, dirichlet, galois, emil artin, lang, euclid, and even archimedes, are still in the house.

and although i do at times promote my own books/notes, they are so far all available free on my webpage, where my own name also appears.

the only author of a commercially produced book, i know of who has participated here, was david bachman, author of a geometric introduction to differential forms.

but we invited him here after choosing his book for study, and at that time he made it and his updates of it available for free.

come on in mokae, this is a different world from the one you know. you might get your feet wet by actually reading some of the early posts in this thread.

 

[RocketSurgery]

Oh yea that reminds me. Does anyone have Riemann's new AIM screenname? It used to be CatcherIntheRie but I think he changed it.

 

[eastside00_99]

So, I just checked out a bunch of books from my school's library:

Basic Algebraic Geometry by Shafarevich
Algebraic Geometry by Miyanishi
Principles of Algebraic Geometry by griffiths
Commutative Algebra by Bourgaki
Geometry of Syzygies by Eisenbud

and I am currently readying Algebraic Geometry and Arithmetic Curves by Qing Liu. This should keep me busy for a while.

 

[Asphodel]

So I doubt if anyone around introducing any book to you is not the writer himself

I are mak gud books & u must reed them 2 lern 2 b l33t

 

[RocketSurgery]

I are mak gud books & u must reed them 2 lern 2 b l33t

I agree.

 

[mathwonk]

eastside, i recommend you begin with shafarevich, and work the exercises.
or maybe the chapter on riemann surfaces in griffiths.

 

[eastside00_99]

yeah, I was reading chapter 3 of Shafarevich's book where he talks about divisors. Very clearly written. It seems to be a great book for an introduction to topics. I will look at the riemann surfaces chapter in griffiths.

 

[mathwonk]

yes the main issue for algebraic varieties is what maps exist from them into projective space? such a map determines a family of subvarieties obtained by intersecting with hyperplanes. these subvarieties of codimension one, which are locally defined by one equation, are called locally principal divisors, or cartier divisors.

since the dimension condition is easier than the locally principal part, it is useful to know that on a smooth variety all codimension one subvarieties are locally principal, which is why shafarevich proves all local rings of smooth points are ufd's.

anyway, it turns out that knowing this family of cartier divisors actually determines the map to projective space in return, up to linear isomorphism of projective space, so the study of such linear familes of cartier divisors is a primary topic in algebraic geometry. the riemann roch theorem is a basic tool for this.

 

[mathwonk]

the basic fact (riemann) is that if L is the dimension of the linear family in which a given cartier divisor moves, then on a curve, L is at least as great as 1 - g + d, where d is the degree of the divisor (number of points) and g is the topological genus of the curve.

the exact number L is obtained by adding to this number, the number of linearly independent differentials vanishing on the divisor (roch).

in higher dimensions, there is a family of cohomology groups the alternating sum of whose dimensions is a computable formula in terms of topological data, such as euler characteristic, etc...(hirzebruch).

then in good cases, e.g. when the divisor has a certain very positive intersection property with other divisors, that sum collapses to give the exact number L. (kodaira)

i have notes on this topic on my website, and griffiths and harris discuss it nicely.

 

[eastside00_99]

Griffiths & harris is a little bit harder because I do not know a lot about complex analysis. Luckily I did a project on abelian varieties and so a lot of the material makes intuitive sense. Maybe you can help with the notation:

Let C be a curve defined by a cubic y^2=x^3+ax^2+bx+c=0. We then have ∫_[p,q] dx/y modulo periods is well defined. Letting t,s be generators for the first homology group of C with integer coefficients ==Z + Z, we have

a=∫_t dx/y and b=∫_s dx/y.

Apparently, these are the periods of dx/y as they are integrals over closed loops and the general periods will be an element of the lattice generated by a and b. Then to prove they are linearly independent, we assume

r=K_1a+K_2b=0 with K_i in R.

Then the conjugate of r is 0. But then it says that

$$ dx/y, \overline{dx/y} \ generate \ H^{1,0}(C) \osum H^{0,1}(C)=H_{DR}^{1}(C) $$There are two things I don't understand: what is meant by the cohomology group of 1,0 and 0,1. and why is the direct sum of these cohomology groups the first de rham cohomology group of C. It's a minor understanding but I don't get it.

The second thing I don't understand wis why this implies

k_1s+K_2 t =0 and why that is impossible if k_i are allowed to be in R.

These are minor things. But, understanding these problems would help with understanding the general casee for the jacobian variety.

 

[eastside00_99]

$ dx/y, \ \overline{dx/y} $ generate $ H^{1,0}(C) \oplus H^{0,1}(C)=H_{DR}^{1}(C) $

The last post's latex should read as the above.

 

[mathwonk]

i suggest you try reading my riemann roch notes on my webpage.

 

[mathwonk]

if you are not familiar with basic complex variables however, you should learn that before studying algebraic curves.

although very elementary algebraic curves, such as presented in miles reids undergraduate book, does not use complex variables, all advanced material in algebraic geometry is based on or motivated by complex variables theory.

 

[eastside00_99]

Well, I have tried in the past to pick up the complex analysis that I need, but my attention is never held very long. I have taken manifold theory and I know a few basic definitions and results. Is there a primer on the subject, or do I need to buckle down and learn this stuff? I was hoping either that I would pick up what I needed along the way or just ignore the stuff I don't know until next year when I take a sequence in complex analysis.

 

[mathwonk]

read henri cartan's book on complex analysis.

you cannot possibly grasp algebraic curves or algebraic geometry without a basic grounding in complex analysis.

riemann's thesis was on the topic.

 

[tgt]

Just like to know how to decide on a Phd area, let alone a Phd topic. Phd is a hard degree with 3 or 4 years so the decision is substantial. However some people may even choose a topic they know close to nothing of. What do you think? How to choose wisely?

 

[Woozya]

I wished to take an Entrance exam on mathematical faculty because I knew better mathematics, but I have changed my mind. I study in the faculty of physics.
I have a question: where in the physics it is possible to apply special branchs of algebra such as the theory ideals?

 

[MichaelL]

Just like to know how to decide on a Phd area, let alone a Phd topic. Phd is a hard degree with 3 or 4 years so the decision is substantial. However some people may even choose a topic they know close to nothing of. What do you think? How to choose wisely?

From what I understand often your PhD adviser will offer up some topics in the area you're interested it. (You can assume by this stage you've done a bunch of advanced classes, so you'll have somewhat of an idea as to what area you like)

 

[matqkks]

read henri cartan's book on complex analysis.

you cannot possibly grasp algebraic curves or algebraic geometry without a basic grounding in complex analysis.

riemann's thesis was on the topic.

Good to see you mention Henri Cartan book on complex analysis. It might be a difficult book to follow but is perhaps the most rigorous on this subject.

 

[mathwonk]

the best and the cheapest!

i myself also benefited from greenleaf's intro to complex variables.

most books had too much theory and not enough examples for me until greenleaf.

the point is to get familiar with power series. of course cartan spends the whole first part on them, but it is more proof oriented.

greenleaf also shows you how to calculate with specific ones until you feel more at home with them.

greenleaf is definitely intro however and cartan is authoritative and deep.

 

[ehrenfest]

Perhaps I should have posted this here:

https://www.physicsforums.com/showthread.php?t=233595

 

[MathematicalPhysicist]

Not sure about the best, but it does seem to be one of the cheapest.

I myself have bought markushevich's volumes, but haven't yet used it properly, I hope after I finish my undergraduate studies i will have time before graduate studies.

anyway, in my course, the lectrurer advised on the book by sardson, or something like this, but i didn't use it.

 

[tgt]

I asked a question previously which wasn't answered.
"Just like to know how to decide on a Phd area, let alone a Phd topic. Phd is a hard degree with 3 or 4 years so the decision is substantial. However some people may even choose a topic they know close to nothing of. What do you think? How to choose wisely?"

Another question based on doing a Phd is how to choose a supervisor? Is it the case that all the student need to think about is the area where he/she likes to work and need not think about whether he/she will get on with the supervisor? So it will be a bit of a gamble?

 

[mathwonk]

your question is hard. but i say, as in the nba, go with the love you feel for the topic. a phd is indeed a long hard road, so it is essential to be committed to your topic and to have a supportive advisor. i chose based on the attraction i felt for the material presented in lectures, and my ability to understand and connect with the advisor who taught the course. i still had to pass through more than one such experience before i found the maturity and commitment to carry through the job of completing a thesis.

ideally you should feel, this material is speaking to me, and this lecturer is speaking to me.

 

[tgt]

your question is hard. but i say, as in the nba, go with the love you feel for the topic. a phd is indeed a long hard road, so it is essential to be committed to your topic and to have a supportive advisor. i chose based on the attraction i felt for the material presented in lectures, and my ability to understand and connect with the advisor who taught the course. i still had to pass through more than one such experience before i found the maturity and commitment to carry through the job of completing a thesis.

ideally you should feel, this material is speaking to me, and this lecturer is speaking to me.

Is it true that Phds in other fields can be much less work? i.e I over herad a guy talking on the tram about his Phd which he only started one month ago and had already done 30,000 words. However he did have a lot of background knowledge prior to starting it. It was on the current Middle Eastern situation.

The word length is 100,000 words for a complete thesis? But how would you count the equations and symbols? Surely they would factor into the word count?

 

[mathwonk]

there is no word limit for a math thesis. some are only 30-40 pages, some are 300. riemann's entire lifes works comprise only about 400 pages.

the definition is something like "non trivial original work", and i have heard of theses where "original" could mean a new proof of an old result, not necessarily a new result never proved before.

but it is very hard to do. one trick some people have used well is to find an old result from an earlier time, and clean it up, make the proof more solid, or add something to it.

others at the opposite extreme, take a very new result, and extend it or apply the ideas to a related but different situation.

 

[tgt]

there is no word limit for a math thesis. some are only 30-40 pages, some are 300. riemann's entire lifes works comprise only about 400 pages.

the definition is something like "non trivial original work", and i have ehard of theses where "original" could mean a new proof of an old result, not necessarily a new result never proved before.

but it is very hard to do. one trick some people have used well is to find an old result from an earlier time, and clean it up, make the proof more solid, or add something to it.

others at the opposite extreme, take a very new result, and extend it or apply the ideas to a related but different situation.

So it would be a lot easier for geniuses. Didn't Grothendieck did the equivalent of 6 thesis by the time he was meant to earn his Phd. Nash's game theory was only 20 pages?