Should I Become a Mathematician?

[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:rofl:.

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.:tongue2:

 

[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.