Other Should I Become a Mathematician?

[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?