Should I Become a Mathematician?

[sponsoredwalk]

I've browsed through a lot of the drama that is this interesting thread & I've gotten a bit of confidence from people but not an answer
to what was grating on my mind so I think it's better to ask/contribute

The ultimate aim of this post is to finish Rudin's Mathematical Analysis, it's a personal journey I'll be taking over the next few months
so bear with me as you read this, I'd really appreciate some in depth input from any thoughtful reader.

I've worked through Thomas calculus (exactly like Stewart calculus) up to chapter 12 having had to go elsewhere to learn every single concept
in those first 12 chapters anyway which is the smartest thing I've ever done in my life up until now :tongue: so I quit the book &
went on amazon & found Wilfred Kaplan's Advanced Calculus which looked amazing.

I bought the book really cheap & got it like 3 weeks ago & am nearly in tears after wafting through the first chapter which is on linear algebra.
I've tried to learn linear algebra before & have quit those horrible computational style books as I absolutely despise memorizing stuff.
Admittedly Kaplan says that in his earlier single variable work he covered linear algebra more thoroughally but I have actually read the first 5 chapters
of that book, which is free online, & eventually just quit becuase of how bad it was.
I'm really stupid to have expected his advanced calc book would be any better but the allure of starting Fourier series,
functions of a complex variable & partial differential equations by the end of one single book was too strong

The thing is that I bought Serge Lang's Introduction to Linear Algebra with it as I know of Lang's reputation
& thought I'd give a slightly more theoretical book a shot.
Basically everything Lang writes is from the perspective of your inner mind & he knows how to get you to remember theorems & proofs well after you've read them.
Simple postulates have far reaching consequences!
Well, I have been toying with the idea of finishing Lang's linear algebra book then trying Kaplan's advanced calculus
post-chapter 1 but I bet the explanations will be terrible.

Because Lang's linear algebra book was making me so happy I decided to try his multivariable calculus book instead
so I went to my friends college library with him to find it.
We only got out the single variable calculus book & I've decided to go through it as a refresher then buy his multivariable book.
I'm going to sell the Kaplan book .
I've already read nearly 200 pages in 2 sittings (this is my second one :tongue:) & Lang is just brilliant.
The book isn't extremely taxing & he's clearing up so many concepts with basic ideas that are more theoretical than Thomas calc's ones for sure!
So, to close this section I would really like to hear any opinions on Lang's multivariable calculus book.
It doesn't cover as much ground as Kaplan's book but I get the feeling it will be deeper & longer lasting so I think it's a good trade off.
I've browsed PF forums & found very few multivariable calc book recommendations other than Apostol, Courant, Marsden or Stewart &
I have a plan of conquering Apostol a while after I finish Lang's book so I wonder, will Apostol be all I ever need in this field or
is the next step in multivariable calculus a solid analysis book on the topic? I really don't know


Now, I have to stress that Lang's single variable calculus book is not as difficult, by any means, as Spivak's calculus is.
I bought Spivak half a year ago when I could barely understand mathematics, being impatient, and am still shocked by it's subtlety.
I now see that it's conquerable but you need to be confident with logic, i.e. the logic of a proof, & I've never taken a course on dealing with that - but I have a plan!
I've ordered Steven R. Lay's Introduction to Analysis which takes it's first 10 chapters on this very topic!
I've tried to read some logic or proofing discussions but when they aren't applied to calculus it just doesn't click. I've looked in this book and he really shows you
how to apply logic to an analysis proof in the way that I've been looking for so I think I'll be able to pre-think actual proofs once I complete this book.

So, my idea as it stands is as follows. I'm going to finish Serge Lang's single variable calculus book in the next few days,
then as soon as I get his multivariable calculus book I'm going to work on doing that along with his linear algebra book.
Once I finish these I'm going to exclusively focus on Lay's analysis book to get used to proper proofs in a definite way.

(I may sound like I can't fathom a proof, I can but not in a sophisticated & systematic enough way to be confident,
I thought there was no theory to constructing a proof until I looked inside Lay's book so
the fact that it doesn't come out of thin air is a confidence booster)

Then, once I've really dealt with Lay's analysis book, which I know isn't that difficult from nearly every mention of the book online :tongue2:
I'm going to concurrently read both Spivak & Apostol and have confidence that I can answer the questions systematically.
Then I think I'll be able to deal with Rudin.

I'm afraid some people might say that this is overkill and it probably is but it's a personal quest & I think that if I can conquer these books then I could get anywhere in mathematics.
After failing math for nearly all 6 years of high school & having no understanding, I mean none! it's something I got to do.

Have you any tips for me, besides keeping the coffee boiled :tongue2:

 

[hunt_mat]

Have you tried "Engineering Mathematics" and "Further Engineering Mathematics" by K.A. Stroud et al?? It is the best textbook it has my fortune to read and understand.

 

[sponsoredwalk]

Have you tried "Engineering Mathematics" and "Further Engineering Mathematics" by K.A. Stroud et al?? It is the best textbook it has my fortune to read and understand.

I went through 2 engineering books to learn all of the precalculus I needed for thomas calculus (& some of the calculus) & there was so much left out of it that I can't really deal with an engineering book skipping the theory/motivation behind the concepts.
The book does look great on amazon though, I really could have used this one back then from the looks of it :tongue2:

I can't check it out but do the chapters on calculus give you a big sheet of all of the basic integrals and derivatives, their inverses & how to rederive all of them?
Does it explain least upper bounds and ε-δ limits & all that theoretically?
The first 350 pages look like they would have been useful to me but once you get past that I can't continue as it's just memorization.

The second Stroud volume looks pretty good but seeing as I don't know that much about the topics in it I'm potentially looking at repeating the past. You know what I mean, I'll go & do them but eventually just have to do it all over again because there's a wealth of material they're skipping for the sake of brevity/technique learning/memorization.

Thanks for the tip about this book though, I remember the cover from the shops & even looking in it quickly but ignoring it because another book (the one I bought) included a lot of stuf from the 2nd Stroud volume & that's why I bought it.
I can't get away with cramming anymore :tongue2:

 

[hunt_mat]

The books are "how to", so once you have an idea of what you're doing then you can apply rigour, the book I used for analysis is the Book by K.J. Binmore entitled "Mathematical Analysis: A Straightforward Approach"

 

[sponsoredwalk]

The books are "how to", so once you have an idea of what you're doing then you can apply rigour, the book I used for analysis is the Book by K.J. Binmore entitled "Mathematical Analysis: A Straightforward Approach"

Cool! How did it go? I was really going to use it, especially seeing as it has two additional books to go with it

https://www.amazon.com/dp/0521299152/?tag=pfamazon01-20

https://www.amazon.com/dp/0521299306/?tag=pfamazon01-20

but I read the first two chapters online of the logic book & it confused me so I never went and bought it :tongue2:

 

[hunt_mat]

It's a very very readable book, I taught myself Riemann integration from it, and then had a very very rigouous introduction from Prof Dima Vassiliev. Once more it has the answers in the back on the book, so it's brilliant. I was also told to read, "Yet another introduction to analysis" which is okay but not as good as Binmore.

 

[Chris11]

Hey, you should check out the following book https://www.amazon.com/dp/0387940995/?tag=pfamazon01-20

I took it out of the library while I was taking my first linear algebra class so that I could learn some more theoreitical stuff. It's really good. If you read the reviews, one person states that after reading it, you'll think that math is an art, which is perhaps the best statement I've ever heard anyone make regarding a math text.

 

[MathematicalPhysicist]

Not really, I bought a few books on integral equations and if that were the case then no one would even mention them. There are methods which don't take derivatives.

Mat

Well I know just one method which doesn't use taking derivatives, any recommendation for integral equations textbook?

 

[sponsoredwalk]

Hey, you should check out the following book https://www.amazon.com/dp/0387940995/?tag=pfamazon01-20

I took it out of the library while I was taking my first linear algebra class so that I could learn some more theoreitical stuff. It's really good. If you read the reviews, one person states that after reading it, you'll think that math is an art, which is perhaps the best statement I've ever heard anyone make regarding a math text.

A review from amazon

As a person who has a healthy interest in mathematics and has taken many classes, this is definatley one of the best! Professor Valenza taught it (he has been teaching this Linear Algebra class at CMC for ten years) and his book is essentially an excellent compilation of the lecture notes from his class. It takes a very different tack from most linear algebra texts: Usually, a linear algebra text begins by inroducing matrices and solving simultaneous equations, teaching computational methods. Prof. Valenza starts with the structure BEHIND all of that math however: Sets, Groups, and Vector Space properties. This structure is absolutely essential to knowing what's going on: My father took a (less superior) linear algebra class many years ago, and he never understood the concepts behind the mathematical manipulations; I actually sat down with him and taught him the things that I learned in Prof. Valenza's class. I really think that the knowledge in this book is invaluable to someone who wants to know what Linear Algebra is really about.

Just a few examples of the truly deep knowledge that this book communicates follows. For instance (this will ring a bell for those who have taken calculus) the "constant of integration" that must be added when doing an antiderivative is actually a property of group homomorphisms. The "absolute value" that must be introduced when taking square roots is structurally THE SAME property of group homomorphisms. Also, we all know that you can't divide by zero; it's just not allowed. But, the reason for that is ultimatley rooted in group theory; namely, the real numbers are NOT a group under multiplication. This type understanding has EVERYTHING to do with matrices and systems of equations! For instance, the fact that only square matrices can be inverted is a trivial consequence of a property of function mappings called "bijectivity." (a mapping from three- to two- dimensional space can't be bijective, for example) Many seemingly complex linear system problems can be simplified to a trivial questions by, for example, investigating the "span" of the column vectors of a matrix. There are countless problems that simply can't be understood without the kind of structural knowledge that Prof. Valenza's book gives.

Understanding the basic properties that underlie so many mathematical objects has been a true delight for me, and anyone who wants to know what is really going on "behind the scenes" with linear equations would be wise to investigate Prof. Valenza's book. It's no accident that he also wrote a book on Fourier Analysis; understanding structure is simply the key to higher math.

That is definitely bookmarked for the future! TY!

 

[mathwonk]

sponsoredwalk. Your post impressed me for your good taste in math books. I think you should always read whichever book makes things clear and enjoyable to you. Lang is great sometimes at this. I really liked his Analysis I for clarifying some ideas of several variable calc. Then his Analysis II was really high powered, almost ridiculously so, but still had the right version of some Lebesgue integration theorems I needed later on in research.

I think you will learn a lot from his one and several variable calc books, but sometimes you need a book with more mundane exercises and examples than he might have. So you might make some kind of combination of Lang and another book. I have not read Kaplan, and it is pretty old fashioned, but after getting the idea from Lang you might even learn something from Kaplan. The goal after a while is to see that all treatments are about the same subject!

One of my favorite modern several variable books is by Wendell Fleming, maybe calculus of several variables. Lynn Loomis used that in math 55 before writing his own book.

Apostol's Mathematical Analysis was thw text in that same course when Shlomo Sternberg taught it. good luck to you.

 

[Sina]

Lee's Introduction to Smooth Manifolds is a good book as an introduction to differential geometry. It tries to attach physical sense and geometric intuition to many things. It also adds some mechanics questions to the part about symplectic geometry part (under lie derivatives) which are joyous to solve. Many of the questions in the problems part are solvable but they usually help you greatly in pinning which points you haven't understood well as well as being usable lemmas themselves.

I would specially recommend this to physicists trying to learn differential geometry (with a good knowledge of real analysis of differentiation) because it attaches physical sense and rigour together. Although it seems very easily and simply written I have suspected a single derivation.

For Real Analysis I would advise Marsden as his style is also very geometric and I like the way he puts the proofs at the back because I usually attempt to prove them myself while reading the text. Hoffman Kunze's linear algebra book is very algebraic (and more like operator algebra than a geometric treatment of linear algebra) but is a good book to read about operators on finite dimensional vector spaces.

 

[ireallymetal]

If you want to be a successful mathematician do you have to be like those child prodigies or people who show aptitude in the subject very early in order for this to happen, or can you just get to that level through hard work and dedication?

I'm asking because I just took up a serious interest in mathematics this year (Sophomore in high school) and I would really like to pursue a career in pure mathematics. I have several contest books and I do a few problems every day, furthermore are there any books I can use to get a head start?

Thank you to anybody who responds.

 

[hunt_mat]

Get yourself a book on analysis, this should be acid test on whether pure maths is for you. Try "A First Course in Mathematical Analysis" by David Alexander Brannan. I would advise you to take an interest in physics too as all the interesting problems tend to come from physics these days.

 

[annoymage]

analysis only? T_T, i use herstein topic in algebra, and now i seems to like it very much. it's ok right? it can considered as acid test right?

 

[hunt_mat]

I hated algebra, group theory leaves me cold, number theory was just weird, linear algebra was quite fun. Geometry was brilliant however.

Mat

 

[annoymage]

LVL 2 courses

24 Geometry

24 Theory of Differential EquationsDISCRETE

34 Graph Theory

34 Combinatorial Mathematics

ALGEBRA

34 Linear Algebra III

34 Algebra II

34 Group Theory

34 Matrix Theory

34 Ring Theory

ANALYSIS

34 Differential Geometry

34 Complex Analysis

34 Real Analysis

34 Topology

i don't know what category is this

34 Number Theory

so first question, if i wan't to be a mathematician on algebra, should i rather choose course on the DISCRETE or ANALYSIS for my minor subject?

2. for now i really like analysis more than discrete so i want to take all ALGEBRA and ANALYSIS, and this are 2 subquestion

i) those in LVL2, am i suppose to take those? is there any strong related in ANALYSIS? if they are, I'm just taking ANALYSIS and do self study on those LVL2 myself. because i can use my extra credit hour for DISCRETE is that wise decision?

ii) i only can choose one of the DISCRETE, if i choose some of LVL2, so which should i choose? graph theory of combinatorics? which is more related on algebra?

i hope you can understand my english :P, thanks in advance

 

[hunt_mat]

I would consider the following:
Differential geometry (A wonderful subject)
Complex Analysis (this comes into so many subjects that it should be compulsary)
Differential equations can be a branch of analysis but mostly they are methods courses
Linear algebra is important as is matrix algebra.

 

[RocketSurgery]

I would consider the following:
Differential geometry (A wonderful subject)
Complex Analysis (this comes into so many subjects that it should be compulsary)
Differential equations can be a branch of analysis but mostly they are methods courses
Linear algebra is important as is matrix algebra.

I would add Tensor/Manifold/Lie theory though I guess the parts I'm thinking of are considered Differential Geometry. Seriously, allow me to make myself look ridiculous and say that Differential Geometry changed my life. They need to have it taught in undergrad more often.

 

[mathwonk]

ireallymetal: It sounds to me as if you have "the love" for the subject, which is the real necessity for success. Almost anyone, no matter how talented, will find the going difficult at some point, almost always at the PhD thesis, and also often at the beginning calculus level, so the difference is whether you enjoy the subject matter.

I would not consider anyone else's book choice as an "acid test" of whether you can do the subject. Why set yourself up for discouragement? Find a book you like. Never accept any statement as true that you must be able to read such and such a book or you are not cut out for the field. That is just not so. If that were true we would have all stopped at some point. I am still trying to really understand sheaf cohomology and I have been a professional algebraic geometer for over 30 years, writing journal articles using it and teaching courses in it.

As to whether you are late in showing ability in math, sophomore level in high school is plenty early in my opinion. At that stage I myself was just beginning to be introduced to Euclidean geometry and enjoying it. Most of us get into the subject by finding some area we like, even if we dislike others, or feel they are hard for us. But eventually many of us learn to love even the other subjects if we find a good teacher or good book that makes that subject clear and beautiful to us.

This thread is too long to read through, but I still recommend reading the first few pages where all the general advice for newbies to mathematics is given, such as how to get stimulation while in high school.

As an aside, if anyone is interested in a free copy of the classic calculus book by Courant (which is often considered harder to read than say spivak's book, so do not let it discourage you)
here is one:

http://www.e-booksdirectory.com/details.php?ebook=3553here is the more general link to a lot of free books:

http://www.e-booksdirectory.com/mathematics.php

 

[mathwonk]

Try not to exclude any subject from your exploration. At the advanced levels the subjects tend to blur together. Lie groups and their representations occur prominently in physics, and combine analysis, topology and geometry, as well as algebra and linear algebra. Cohomology which arose in topology occurs in almost all fields now, and analysis and geometry figure importantly in number theory as well.

Algebraic methods lend the benefit of symmetry and computability to whatever subjects they are used in, so are helpful in every field including geometry. As George Kempf put it, "Algebraic geometry studies the delicate balance between the geometrically plausible and the algebraically possible." I.e. without geometric intuition it is hard to predict what should be true, and without algebraic tools it is hard to prove ones conjectures or find counterexamples.

e.g. look here in Dolgachev's book "Introduction to physics" (for math grad students) at the lecture 9 on schroedinger's representation of the Heisenberg group for an interplay of physics, differential geometry, real and complex analysis and linear algebra.http://www.e-booksdirectory.com/details.php?ebook=2064

or just peruse the titles of the chapters in this related opus on "non commutative geometry" whose very title makes no sense unless you believe geometry and algebra are connected:

here are some notes for harvard course math 275, which McMullen describes as follows:

"This course will concern the interaction between:
• hyperbolic geometry in dimensions 2 and 3;
• the dynamics of iterated rational maps; and
• the theory of Riemann surfaces and their deformations."

how's that for combining geometry, analysis, and algebra?

http://math.harvard.edu/~ctm/home/text/class/harvard/275/rs/rs.pdf

 

[annoymage]

ireallymetal: It sounds to me as if you have "the love" for the subject, which is the real necessity for success. Almost anyone, no matter how talented, will find the going difficult at some point, almost always at the PhD thesis, and also often at the beginning calculus level, so the difference is whether you enjoy the subject matter.

i always remember, but i don't know who said it. The differences between work hard and play hard, And thankssssssssssss for the free book link mathwonk. :D

 

[Cod]

I know everyone around here "lives and dies" by Strang's "Linear Algbebra" textbook, but does anyone have any experiences with Otto Bretscher's "Linear Algbera with Applications"? Its the required text for my upcoming linear algebra course and planned on looking at the book this week (received in mail a few days ago).

 

[qspeechc]

Mathwonk, since you're back and posting, it seems, I'd like to share something. I know you love geometry, even Euclidean ("high school") geometry, and there is a lack of good geometry books for the high school student, so I just wanted to bring the books Geometry and Elementary Geometry from an Advanced Standpoint by Moise to your attention. Moise is one of the best expositors of mathematics that I have ever come across. I never learned geometry in high school, shockingly (I am not American), so while I was a student at university (I am a senior) I read those two beautiful books. I know you are a very busy man, but I just wanted to see what you think of them.

 

[mathwonk]

well i have the advanced standpoint one and it is marvelous. But as a US public school teacher honestly that book seems over the heads of most classes i have taught, so i have tended to ignore it over the years, but i consulted it for some proofs i needed. as you probably know i have recently taught college geometry straight from euclid himself and had good results. i think also i differed with moise as to the wisdom of using real numbers in the foundations. doesn't he do that?

the problem is that books like moise took for granted a good acquaintance with euclid and proceeded to fill logical gaps that appeared centuries later. In fact most of our students know little at all of the original euclid and hence are not at all prepared for such subtleties. for math minded people of course the matter is different.

by the way i have recently retired and hence have more time to post here, not being occupied with teaching or writing as many papers. for a while there i had to focus on my research and teaching since this is voluntary and i got no credit for spending time here that did not result in traditional publications.

think how many unpaid hours it takes to write almost 7,000 posts that do not appear on your vita. any of you planning on going into academics, probably i should warn you away from this kind of free activity, as you will not survive. i only managed because i was already old and established, and i still had trouble.

 

[EdmondDantes]

I have just completed a BS in pure mathematics in May. I did very well in all of my coursework. However, towards the end of my junior year I started to feel burnt out on the subject. I had planned on going to grad school, but during my senior year I had no drive to take the GREs or ask for letters of rec. I just wanted to get through the year.

I took the summer off and traveled to get my mind off things, hoping that all I needed was a little rest and relaxation before diving back into math. Right now I am enrolled in graduate level algebra (my favorite sub-field of math) at my university, but after two weeks I'm already sick of it. I'm doing fine on the homework sets because right now it has just been review of Group, Ring, and Field Theory. But I spend more time watching movies and reading novels than doing math.

I don't know what happened to my passion for the subject. I can't seem to get back the feeling of joy I used to get from solving a problem. I feel like I lost it in all the formalism. Is there any hope of regaining it? Or should I throw in the towel?

 

[Subdot]

mathwonk: Would you recommend Apostol's Mathematical Analysis (knowing your previous posts, you'd probably recommend as close to the 1st edition as possible) as the "best" book to get for Analysis after completing his (2nd edition) Calculus books? Is Apostol's Analysis book sufficient for a first course in Analysis like Baby Rudin or would one still have to learn some stuff that is covered in Baby Rudin after taking Apostol's Analysis? (Perhaps combining either one with Loomis and Sternberg, of course!)

by the way i have recently retired and hence have more time to post here, not being occupied with teaching or writing as many papers. for a while there i had to focus on my research and teaching since this is voluntary and i not only got no credit for doing it but was even criticized to some extent for spending time here that did not result in traditional publications.

think how many unpaid hours it takes to write almost 7,000 posts that do not appear on your vita. any of you planning on going into academics, maybe i should warn you away from this kind of free activity, as you will not survive. i only managed because i was already old and established, and i still had trouble.

Well I know that I'm glad that you posted what you did. I'll never think of American mathematics education in quite the same way again. Your posts also helped push me into being bold in mathematics. So bold that I went ahead and got Apostol's Calculus books to test my limits and work through (and still working through) instead of taking a "baby step" before getting them. Thank you very much for the time you were able to give and are giving now! =)

 

[qspeechc]

Indeed, I think most of us PFers greatly appreciate the time you have devoted to helping us students, and we are glad to see you back.

I just want to point out that Moise's Geometry is a high school geometry book, and Elementary Geometry From An Advanced Standpoint does not assume you know any geometry. From what I remember reading in the preface it is based on a Euclidean geometry course he gave to university students with no previous knowledge of geometry. He does indeed use an analysis of the reals extensively (from what I remember), but that has the benefit of allowing an easy segueway to calculus (ahem!).

 

[hunt_mat]

I would recommend Spivak, for the general introduction to analysis as it is very very good. I would recommend "Introduction to Metric and Topological Spaces" by Wilson A Sutherland as the next step in analysis for an introduction to the initially difficult topic of metric spaces.

There are lots of books on complex analysis, the two I used were, "Complex Analysis" by Priestly and "Complex Analysis: The Hitchhiker's Guide to the Plane" by Ian Stewart and David Tall.

For things like inverse and implicit function theorems, I haven't found a decent book, on it, the lecture notes I have from university are most likely the best I have come across. There was one boomk on vector analysis that was very good but I can't seem to find that.

For lebesgue integration, I would recommend the book by Alan Weir, "Lebesgue Integration and Measure". I think that covers it.

Mat

 

[hunt_mat]

I had a look the only on that covers the inverse and implicit function theorems are Return to product information
Vector Calculus (Applied Mathematics & Computing Science)
by P.R. Baxandall, H. Liebeck

 

[mathwonk]

as i have said lately, i have begun to recommend more whatever seems accessible to the reader. Apostol's mathematical analysis was the book for advanced honors calc at harvard, taught by shlomo sternberg in 1960-1961 but i did not take that course. From my acquaintance with his calc book i believe it must be wonderful.

I agree that spivak is a good intro to theoretical calc sometimes called intro to analysis. I myself did not enjoy rudin's books and do not recommend them, as they are hard to learn from for most people.

I like spivak's calculus and also spivak's calculus on manifolds, which has a nice discussion of implicit and inverse function theorems. I also recommend lang's books, such as his analysis I, and his calculus books. Dieudonne's foundations of modern analysis is also excellent, and i also liked fleming's calculus of several variables. But the point is just to get your head around the material and any source that speaks to you will do.

just try to get entrance to the subject, it does not matter a lot how you do it. Once you get inside the ideas, try some harder books with more sophisticated approaches. This is not a competition, but a collaborative endeavor.

and thank you for the kind words of appreciation.

 

[qspeechc]

As an alternative to baby Rudin I like "Real Mathematical Analysis" by Charles Chapman Pugh. Without exaggeration or levity I can say it is my favourite mathematics textbook. There are many diagrams, many problems at various levels of difficulty, far better pedagogy than Rudin, he develops intuition but is as rigorous and challenging as Rudin. It is also more modern.

 

[Subdot]

Thank you all for the advice and suggestions! It's very, very helpful. I appreciate it greatly.

 

[mathwonk]

If you are interested in Riemann surfaces, try this on for size: I just taught this, my last, and favorite course last semester. This is a survey of the whole course, given on day one.


8320 Spring 2010, day one Introduction to Riemann Surfaces

We will describe how Riemann used topology and complex analysis to study algebraic curves over the complex numbers. [The main tools and results have analogs in arithmetic, which I hope are more easily understood after seeing the original versions.]
The idea is that an algebraic curve C, say in the plane, is the image by a holomorphic map, of an abstract complex manifold, the Riemann surface X of the curve, where X has an intrinsic complex structure independent of its representation in the plane. (Although the complex structure is inherited from the plane representation, it can be described in an intrinsic way, and may be derived from many different plane representations.)
We will construct two fundamental functors of an algebraic curve, the Riemann surface X, and the Jacobian variety J(X), and natural transformations X^(d)--->J(X), the Abel maps, from the “symmetric powers” X^(d) of X, to J(X).

The Riemann surface X
The first construction is the Riemann surface of a plane curve:
{irreducible plane curves C: f(x,y)=0} ---> {compact Riemann surfaces X}

The first step is to compactify the affine curve C: f(x,y) =0 in A^2, the affine complex plane, by taking its closure in the complex projective plane P^2. Then one separates branches at points where C intersects itself, then one smooths each of those branches, to obtain a smooth compact surface X. X inherits a complex structure from the coordinate functions of the plane. If f is an irreducible polynomial, X will be connected. Then X will have a topological genus g, and a complex structure, and will be equipped with a holomorphic map ƒ:X--->C of degree one, i.e. ƒ will be an isomorphism except over points where the curve C is not smooth, e.g. where C crosses itself or has a pinch.

This analytic version X of the curve C retains algebraic information about C, e.g. the field M(X) of meromorphic functions on X is isomorphic to the field Rat(C) of rational functions on C, the quotient field k(x)[y]/(f), where k = complex number field. It turns out that two curves have isomorphic Riemann surfaces if and only if their fields of rational functions are isomorphic, if and only if the curves are equivalent under maps defined by mutually inverse pairs of rational functions. Since the map X--->C is determined by the functions (x,y) on X, which generate the field Rat(C), classifying algebraic curves up to “birational equivalence” becomes the question of classifying these function fields, and classifying pairs of generators for each field, but Riemann’s approach to this algebraic problem will be topological/analytic. We already can deduce that two curves cannot be birationally equivalent unless their Riemann surfaces have the same genus. This solves the problem that interested the Bernoullis as to why most integrals of form dx/sqrt(cubic in x) cannot be “rationalized” by rational substitutions. I.e. only curves of genus zero can be so rationalized and y^2 = (cubic in x) usually has positive genus.

The symmetric powers X^(d)
To recover C from X, we seek to encode the map ƒ:X--->C, i.e. ƒ:X--->P^2, by intrinsic geometric data on X. If the polynomial f defining C has degree d, then each line L in the plane P^2 meets C in d points, counted properly. Thus we get an unordered d tuple L.C of points on C, possibly with repetitions, hence when pulled back via ƒ, we get such a d tuple called a positive “divisor” D = ƒ^(-1)(L) of degree d on X. (D = n1p1+...nk pk, where nj are positive integers, n1+...nk = d.) Since lines L in the plane move in a linear space dual to the plane, and (if d ≥ 2) each line is spanned by the points where it meets C, we get an injection P^2*--->{unordered d tuples of points of X}, taking L to ƒ^(-1)(L).

If X^d is the d - fold Cartesian product of X, and Sym(d) is the symmetric group of permutations of d objects, and we define X^(d) = X^d/Sym(d) = the “symmetric d-fold product” of X, then the symmetric product X^(d) parametrizes unordered d tuples, and inherits a complex structure as well. Thus the map ƒ:X--->C yields a holomorphic bijection P^2*--->∏ from the projective plane to a subspace ∏ of X^(d). I.e. the map ƒ determines a complex subvariety of X^(d) isomorphic to a linear space ∏ ≈ P^2*. Now conversely, this “linear system” ∏ of divisors of degree d on X determines the map ƒ back again as follows:

Define ƒ:X--->∏* = P^2** =P^2, by setting ƒ(p) = the line in ∏ consisting of those divisors D that contain p. Then this determines the point ƒ(p) on C in P^2, because a point in the plane is determined by the lines through that point. [draw picture]

Thus the problem becomes one of determining when the product X^(d) contains a holomorphic copy of P^2, or copies of P^n for models of X in other projective spaces.

The Jacobian variety J(X) and the Abel map X^(d)--->J(X)
For this problem, Riemann introduced a second functor the “Jacobian” variety J(X) = k^g/lattice, where k^g complex g -dimensional space. J(X) is a compact g dimensional complex group, and there is a natural holomorphic map Abel:X^(d)--->J(X), defined by integrating a basis of the holomorphic differential forms on X over paths in X. Abel collapses each linear system ∏ ≈ P^n* to a point by the maximum principle, since the coordinate functions of k^g have a local maximum on the compact simply connected variety ∏. Conversely, each fiber of the Abel map is a linear system in X^(d).

Existence of linear systems ∏ on X: the Riemann - Roch theorem.
By dimension theory of holomorphic maps, every fiber of the Abel map X^(d)--->J(X) has dimension ≥ d-g. Hence every positive divisor D of degree d on X is contained in a maximal linear system |D| , where dim|D| ≥ d-g. This is called Riemann’s inequality, or the “weak” Riemann Roch theorem.

The Roch part analyzes the relation between D and the divisor of a differential form to compute dim|D| more precisely. Note if D is the divisor cut by one line in the plane of C, and E is cut by another line, then E belongs to |D|, and the difference E-D is the divisor of the meromorphic function defined by the quotient of the linear equations for the two lines. If D is a not necessarily positive divisor, we define |D| to consist of those positive divisors E such that E-D is the divisor of a meromorphic function on X. If there are no such positive divisors, |D| is empty and has “dimension” equal to -1. Then if K is the divisor of zeroes of a holomorphic differential form on X, the full Riemann Roch theorem says: dim|D| = d-g +1+dim|K-D|, where the right side = d-g when d > deg(K).

 

[mathwonk]

my apologies for the thread killer.

 

[Fragment]

Did you say it was your last course? Are you not going to teach anymore?