Book List for Undergraduates

[math...@gmail.com]

Here are some suggestion so far.

I want to organize a book list for undergraduates from all Wong's
students in US. I want to add everyone's remark into the list. So it
is more balanced. I also hope I can make this into tex files... So the
better way is that every one make into the tex files and send to me.

The following is a sample list so far. Most of the books are from
Zhang Zheng. I only add some remarks.

Please post any ideas or comments.

Best,

Yi

==============================================

The following is my recommendation and comments

GEOMETRY AND TOPOLOGY

(Yi) Mathematics is a unified object. I dont want to put any book in
any smaller category since that is meaningless. Learn math, learn
everything.

***** means five star

1. Modern Geometry ***** ***** ***** ***** ***** ***** *****

(Yi ) read as much as possible. There is a really good book written by
Postnikov, Lectures on Geometry, 1st and 2nd have chinese version.
Those are the pre-modern geometry book, best for the first year.

2. 《Topology》，James R.Munkres

(Yi) bad, Wong's point-set topology is better, you should go through
pointset top asap, dont spend too much time on this

3. Prof Wong's notes *****

4. 《Differential Geometry of Curves and Surfaces》，Manfredo P.do Carmo
*****

(Yi) First course in differential geometry, do every exercises and
examples. Learn Computation, guys!!

5.《Introduction to Smooth Manifolds》，John M.Lee

(Yi) BADDDDDDDDD. Modern geometry, or any chinese book. Try to learn
how not compute the tensor is kind of more important than the concept
itself. Lee's book is a combination of everything, but a bad
combination. Learn computation in modern geometry!!!

6. Shen Yibing's book ***

(Yi) Learn computation, guys!!! Dont think it is not necessary!!!

7.《黎曼几何初步》 伍鸿熙 北京大学出版社 *****

(Yi) Good! Read remarks every chapters before you do any serious work.

8. Wu's compact Riemann surface (chinese) *****

(Yi) Good! Read remarks every chapters before you do any serious work.

9. Algebraic Topology: An Introduction，William S.Massy, GTM? *****

(Yi) Best first course in algebraic topology forever.....

10. Algebraic Topology，Allen Hatcher，www.math.cornell.edu/~hatcher

(Yi) Best second course in algebraic topology so far....

11. Differential Forms in Algebraic topology, Bott-Tu ***** *****
***** ***** ***** ***** ***** ***** *****

(Yi) The Best forever book in algebraic topology, algebraic geometry,
complex geometry forever. If you want to be a mathematician, put the
book around your bed and read every night (for undergraduate).

12. Morse Theory, J.Milnor *****

(Yi Zhu) Best book in Riemannian geometry and topology. Morse theory
is Fundamental in mathematics.

13.《Topology from the Differentiable Viewpoint》, J.Milnor

(Yi) Good book but not good as Morse theory.

================================================================================

ANALYSIS


1. Principles of Mathematical Analysis, W. Rudin

(Yi) The Best Second course in calculus. But for the first course, I
still recommend Zorich, Mathematical analysis and 菲赫今哥尔茨 "微积分学教程". The
latter is still my favorite.

2. Complex Analysis, Ahlfors

(Yi) I really like this book although it is not possible to finish in
one semester. Gong Sheng's complex analysis book is much more
concise.

3. ODE Arnold


4. 数学物理方程，谷超豪，李大潜 高等教育出版社

(Yi) I really like this book. The book taught me that to learn
analysis, learn PDE first and to learn PDE, learn physics first. I am
still amazed by the magic idea of Fourier.

5. (Yi) Stein's three analysis books are the best. ******

==============================================================================
ALGEBRAIC STUFF.

1. Algebra, Michael Artin *****

(Yi) No doubt it is the best.

2. Introduction to Commutative algebra, Atiyah, Macdonald ******

(Yi) Strongly recommended for those interested in algebraic geometry
and number theory.

3. An Introduction to Homological Algebra, Charles A.Weibel

4. A course in Homological Algebra, P. Hilton; U.Stammbach. GTM 4

(Yi) No comments. Homological algebra is purely geometric to me. You
should learn algebraic topology, Bott-Tu, Hartshorne rather than these
books.


===========================================================================

Number Theory

1. Serre's A Course in Arithmetic

(Yi) Read it! Get some feeling for number theory.

============================================================================

It will be better if you can try to understand some basic PHYSICS.

1. 经典力学的数学方法 Arnold

2. 费曼物理学讲义1 2 3 Feynman

(Yi) Keep your eyes on it. I cannot describe it properly.....

[Pan]

First of all,

Let me introduce some classical books for algebraic topology.....Read
all of them (if possible) before u do algebraic geometry (just
kidding)

these books will provide the intuition of some morden algebraic
geometry things...

1.Hatcher Chapter 0 to Chapter 4( do most of the exercise)
this book is interesting, teach u how to think problems by
drawing....But the book do not offer u a lot of power tools.....

2.Bott and Tu, GTM 82
This book is very deep ,however, very readable, even though u are from
physics department......U can touch the modern concepts of topology
around 1940-1950s.....a lot of thing can be generalized to algebraic
geometry case....

3.<<Characteristic Class>> (Milnor)
It too good to say nothing.....Its content is an apex of 1940s'
topology!!!

4. <<topology of Fibre bundle>> (Steenord)
u can learn obstruction theory from this book, read it is enjoyable
because Steenord tell u the pictures behind every theorem...if u know
how to construct the "topological moduli" space (just infinity
projective space), u will appreciate algebraic stack.......

5.<< K-theory>> Atiyah
It is about thoery of vector bundle.....It is preparation for Atiyah-
Singer Index theorem....If u are familiar with chapter 4 in Hatcher,
that the only difficult part of K-theory is Bott
periodicity.....Milnor's Morse theory prove it, but Atiyah only use
linear algbra, amazing......The algebraic K-theory transform a lot of
idea from topological one......As usual, Atiyah's book is very
elegant....

6.<<A Concise Course for Algebraic Topology>> J.May
It is good reference and u can read it if u have the background of
previous books......It establish homotopy theory and homology theory
in a modern view which can be generalized to algebraic geometry( like
A-homotopy, model Category)....

Anyway, what u learn is just what u learn, what u do from what u learn
is the point.....So, do not take any book seriously, just enjoy it...

[Yi]

Note: This is an test email to see if Prof. Wong can receive the
email.

================================================================================

Add remark of Xuanyu,

2.Bott and Tu, GTM 82
> This book is very deep ,however, very readable, even though u are from
> physics department......U can touch the modern concepts of topology
> around 1940-1950s.....a lot of thing can be generalized to algebraic
> geometry case....

Actually, every thing can be generalized to algebraic geometry case,
which is related to intersection theory and chern classes.


Sorry for the spam!

Yi

[Yi]

====================================================
The following is the updated version, including Pan, Xuanyu's and Zhu,
Yi's suggestion.
====================================================

(Zhu) I am sorry in the new list I did not include anything about
algebraic geoemtry. I think I will write a much more detailed book
list for learning algebraic geometry (on graduate level). For those
who are interested in algebraic geometry, I only have one
recommendation Joe Harris, Algebraic geometry: an introduction. Read
every chapters and do every exercises (I am not kidding! I am very
serious about this if you want to be an algebraic geometer). Plus, GSM
55, A Scrapbook of Complex Curve Theory, C. Clemens, is also a very
good book. Try to read it.

====================================================
DIFFERENTIAL GEOMETRY AND TOPOLOGY

(Zhu) Mathematics is a unified object. I dont want to put any book in
any smaller category since it is meaningless. To learn math, learn

everything.

***** means five star

1. Modern Geometry ***** ***** ***** (Strongly recommended to after
second year)

(Zhu) Read as much as possible. This three volume book is the best
book to learn differential topology and differential geometry. First
volumes are mainly about differential geometry in higher dimensional
case, including lots of useful examples. Second volume is the standard
beginning for the differential manifold. Volume 2 is the best book for
manifold theory. It is better than Warner's and Lee's. Volume 3 deals
with differential topology and algebraic topology on manifold,
including the Morse theory.

2. Lectures on Geometry vol. 1-6, Postnikov ****** (Strongly
recommended to first year)

(Zhu) 1st and 2nd have chinese versions in Yuquan's library. Those are
the pre-modern geometry books, best for the first year. We can see how
linear algebra working in higher dimensional geometry.

3. Topology, James R.Munkres **

(Zhu) Bad. Wong's point-set topology is better. You should go through
point-set topology asap. Dont spend too much time on this. It is a
language for pure math.

4. Prof Wong's notes *****

(Zhu) Good notes! You can read it very quickly and then go to
differential geometry to use this language. Again do not spend too
much time.

5. Differential Geometry of Curves and Surfaces, Manfredo P.do Carmo
*****

(Zhu) First course in differential geometry, do every exercises and
examples. Learn the computation, guys!! The computation is the key to
differential geometry.

6. Introduction to Smooth Manifolds, John M.Lee

(Zhu) BAD. Use Modern geometry, or any chinese book. Try to learn how

not compute the tensor is kind of more important than the concept
itself. Lee's book is a combination of everything, but a bad
combination. Learn computation in modern geometry!!!

7. Shen Yibing's book ***

(Zhu) First book in Riemannian geometry. Learn how to manipulate the
tensor stuff!

8. 黎曼几何初步, 黎曼几何选讲, 伍鸿熙  *****

(Zhu) Good! Read remarks every chapters before you read it seriously.
Get some idea beore you learn.

9. Morse Theory, J.Milnor  *****

(Zhu) The Best book in Riemannian geometry and topology. Milnor can
always treat stuff in a concise and beautiful way. If you want to know
the topology of the manifold, start from the Morse theory. If you want
to get some sense in Riemannian geometry after learning the manifold
theory, Milnor's book has the best short introduction to Riemannian
geometry.

11. Topology from the Differentiable Viewpoint, J.Milnor

(Zhu) Good book but not good as Morse theory.

12. Algebraic Topology: An Introduction，William S.Massy *****

(Zhu) The best first course in algebraic topology forever..... Do
every exercises. Play some toy models.

13. Algebraic Topology，Allen Hatcher，www.math.cornell.edu/~hatcher
*****

(Pan) Hatcher Chapter 0 to Chapter 4 (do most of the exercise). This
book is interesting, teaching you how to think problems by
drawing....But the book does not offer you a lot of power tools.....

(Zhu) The best second course in algebraic topology so far.... Do as
much exercises as you can. Play lots of toy models. I like its
treatment in homology theory, but not very much for cohomology theory.
The reason is that no one use singular cohomology to do geometry. It
should emphasize the relation with de Rham's cohomology and Cech's
cohomology. The best treatment is in Bott-Tu's book. Also, I hate the
proof for the Poincare duality in the book. The better proof can be
find in Bott-Tu. My favourate prooves are using Morse theory in Modern
geometry vol 3 and Griffiths-Harris Chapter 0.4.

14. Differential Forms in Algebraic topology, Bott-Tu ***** *****
***** (strongly recommended for every one)

(Pan) This book is very deep, however, very readable, even though you
are from physics department......You can touch the modern concepts of

topology 
around 1940-1950s.....a lot of thing can be generalized to

algebraic 
geometry case.... (Remark from Yi Zhu, You can generalize
every thing in Bott-Tu to algebraic geometry.)

(Zhu) The Best book in algebraic topology, differential topology,
differential geometry, algebraic geometry, complex geometry forever.
If you want to be a mathematician, put the book in your bed and read
it every night. Key words: First chern classes for line bundls, Leray
spectral sequences, Chern Classes. This book tells you how to do math
by "clever" computations. It provides you geometric intuitions of
cohomology theory and its huge applications. You will see those ideas
transplanted into algebraic/complex geometry by Grothendieck, and
become much more essential than the algebraic topology itself.

15. Characteristic Class, Milnor *****

(Pan) It is too good to say nothing.....Its content is an apex of
1940s' topology!!!

16. Topology of Fibre Bundle, Steenord *****

(Pan) You can learn obstruction theory  from this book, read it is
enjoyable because Steenord tell you the pictures behind every

theorem...if u know how to construct the "topological moduli" space
(just infinity projective space), u will appreciate algebraic
stack.......

17. K-theory, Atiyah *****

(Pan) It is about thoery of vector bundle.....It is preparation for

Atiyah- 
Singer Index theorem....If u are familiar with chapter 4 in
Hatcher, that the only difficult part of K-theory is Bott
periodicity.....Milnor's Morse theory prove it, but Atiyah only use
linear algbra, amazing......The algebraic K-theory transform a lot of
idea from topological one......As usual, Atiyah's book is very
elegant....

18. A Concise Course for Algebraic Topology, J.May ****

(Pan) It is good reference and u can read it if u have the background

of 
previous books......It establish homotopy theory and homology
theory 
in a modern view which can be generalized to algebraic geometry
( like 
A-homotopy, model Category)....

(Zhu) To me, the book is too formal. I am willing to learn this stuff
only if I get the motivation.

===========================================================================
=====

ANALYSIS

1. Principles of Mathematical Analysis, W. Rudin *****

(Zhu) The Best Second course in calculus. But for the first course, I

still recommend Zorich, Mathematical analysis and 菲赫今哥尔茨 "微积分学教程". The
latter is still my favorite.

2. Complex Analysis, Ahlfors *****

(Zhu) I really like this book although it is not possible to finish

in 
one semester. Gong Sheng's complex analysis book is much more
concise.

3. Ordinary Differential Eqations, Arnold

4. 数学物理方程，谷超豪，李大潜 高等教育出版社 *****

(Zhu) I really like this book. The book taught me that to learn

analysis, learn PDE first and to learn PDE, learn physics first. I am
still amazed by the magic idea of Fourier.

5. Stein's three analysis books ***** ***** *****

(Zhu) The Best analysis books for undergraduates!

===============================================
ALGEBRAIC STUFF.

1. Algebra, Michael Artin *****

(Zhu) No doubt it is the best. Do every exercises!

2. Introduction to Commutative algebra, Atiyah, Macdonald ******

(Zhu) Strongly recommended for those interested in algebraic geometry
and number theory. Do every exercises!

3. An Introduction to Homological Algebra, Charles A.Weibel

4. A course in Homological Algebra, P. Hilton; U.Stammbach. GTM 4

(Zhu) No comments. Homological algebra is purely geometric to me. You

should learn algebraic topology, Bott-Tu, Hartshorne rather than
these

books.

===============================================

Number Theory

1. Serre's A Course in Arithmetic ************

(Zhu) Read it! Get some feeling for number theory.

2. Numer theory 1 Fermat's Dream *************

(Zhu) This is written by 3 Japanese number theorist. I like it very
much. The authors tells you what is going on in number theory. Lots of
deep ideas is explained, like quadratic reciprocity law and rational
points on conics, Mordell's theorem and conjeture, Kummer's trick on
Last Fermat conjecture, Zeta functions. No chinese edition yet. I will
mail the book to Prof. Wong. I hope you can see it soon.

===============================================
It will be better if you can try to understand some basic PHYSICS.
1. 经典力学的数学方法 Arnold
2. 费曼物理学讲义1 2 3 Feynman

(Zhu) Keep your eyes on it. I cannot describe it properly.....

[tt]

it seems that till now most books are about geometry and topology. In
the "motivic" point of view these are about geometry over archimedean
places (real or complex fields), which is very important. But for
balance
I want to say something about number theory. I started learning this
course pretty late. Except elementary number theory (unique
factorization of integers, congruence equations, quadratic reciprocity
etc), most of number theory uses a lot from other fields (it doesn't
have too much of its own tools). Algebraic number theory needs some
algebra and Euclidean geometry; analytic number theory needs complex
analysis and harmonic analysis. Lang's Algebraic Number theory is
pretty complete: the first part is for a first course in alg number
theory, containing basic things and terminologies one needs to know
about modern number theory; the second is about class field theory
(known as the elementary case and motivation for Langlands' program),
a beautiful interpretation of the linear characters of the Galois
groups of local / global fields in terms of "analytic" (i.e.
automorphic) things. And the third part talks about analytic
continuation and functional equation for L-functions of number fields
(most importantly, the method of Tate's thesis) as well as Weil's
explicit formula, which gives as a consequence an equivalent way of
stating RH. Many (if not most) arithmetic and geometric information of
a geometric object (e.g. alg variety) over a number field can be saved
in the form of analytic information of its zeta function, and there
are many conjectures (BSD, Stark's, Deligne... etc) telling us how to
recover the arith / geom info out of the analytic info.

For those who are interested in arithmetic geometry (roughly,
algebraic geometry + number theory), Milne's lectures notes, in my
opinion, will be very useful. So before going to Lang's book you may
want to read about Milne's notes on number theory.

For the relation between alg geom and number theory, "An Invitation to
Arithmetic Geometry" might be a good book.

[Yinbang Lin]

I am an undergraduate student in Grade Three. I would like to study arithmetic.

The first non-elementary book in arithmetic I read is Serre’s A Course in Arithmetic. I studied the book last semester. Also in last semester, I studied Stein’s Complex Analysis, in which there’re some chapters on zeta function and elliptic functions. Recently, I am reading the book Yi recommend: Number Theory 1 - Fermat’s Dream. The book contains elliptic curves, p-adic numbers, zeta function and algebra number theory. Reading the book, I get deeper understanding about what I have learned.

I wonder which book I should read next when I finish reading the book: Number Theory 1. Should I read Milne’s notes first? Can I go directly to Serre’s Local Fields? Would you mind giving me your suggestions?

By the way, what’s your name？

Yours,

林胤榜

2009/4/13 tt <sha...@gmail.com>

[Necromancer Leo]

u definitley shall start Algebraic geometry.....Try to understand the famous theorem last century.....Mordell Conjecture, Fermat last theorem and so on, they are all require deep algebraic geometry........Read part of Jeo.Harris about First part and go with atiyah's commutative algebra, finish them and start Hartshorne.....At the same time, u shall know what the class field theory is....I never do them in this way though I know that way is correct since I want to do algebraic geometry....

Xuanyu

--- 09年4月13日，周一, Yinbang Lin <yinba...@gmail.com> 写道：

> 发件人: Yinbang Lin <yinba...@gmail.com>
> 主题: Re: Book List for Undergraduates
> 收件人: cocktai...@googlegroups.com
> 日期: 2009年4月13日,周一,下午8:44
>
> #yiv1882404742 <!--
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> #yiv1882404742 I am an undergraduate student in

___________________________________________________________
好玩贺卡等你发，邮箱贺卡全新上线！
http://card.mail.cn.yahoo.com/

[Yi Zhu]

Dear Yingbang,

I think the only one you should ask are Shenghao and Xia Jie. Please follow their ideas and let us know.

Xuanyu gives a short introduction on algebraic geometry. That is good, good for algebraic geometer. But I dont think that is a proper motivation for number theory. Learn Atiyah-Macdonald, use it into algebraic number theory, using Shenghao's book or Neukirch's. Another good book I know is Marcus "Number fields" recommended by Zhang Shouwu. You should take a look. If you want to learn more like elliptic functions and modular forms, take a look at 3 books suggested by Zhou Jian (One book called elliptic function written by a germann mathematician is very very good). I think Wong has the copies. Another book I know is Shimura's Introduction to the arithmetic theory of automorphic functions. But I dont know when you should start.

About algebraic geometry, All I recommended to undergrauates are J Harris and the book of Clemens. Learn algebraic geometry, finsh them first.

Best,

Yi

2009/4/13 Yinbang Lin <yinba...@gmail.com>

[XIA Jie]

Zhu and Pan’s comments have almost covered the classical undergraduate
textbooks in every branch, and ashamed to admit, I have not read most
of them. By contrast, my comment is limited. I want to say something
on Hartshorne’s algebraic geometry.

Everyone knows it is a hard book. And it is definitely unnecessary for
undergraduate students to read it. However, prepared well and
interested in AG, you should feel free to ‘crack’ that book directly.
Neither be afraid of it nor be arrogant even if you have finished all
the exercises before graduate. In general, Hartshorne’s book is just
like J.Harris’ or Shafa’s and any other AG’s introductory textbook.
All of them are access to the world of AG but from different
approaches. It is not fair to judge that the Hartshorne’s is harder
than others. The difficulty of some exercises in J.Harris’ book can
beat Hartshorne’s.

Personally, the significance of the Hartshorne’s book is to help you
be familiar with the basic language of scheme and cohomology.
Grothendieck used the language to furnish some old stories and
problems. He did it so successful that now every Ager adopts this
language, not merely for writing paper but for talking, too. So if you
desire to enter this field, the very first mission you have to
accomplish is learning the language. Only after handling the language
well, you can further understand others, know what is happening around
and then next, you can pick up a specific problem to do your own
research.

Hence, in a sense, Hartshorne’s book is just a language. It is not
hard. However, it is a pity if you are stuffed by the abstract
language without any geometric intuition. So, complemented to
Hartshorne’s, you may need J.Harris, Shafa’s, GH and some others such
that contain more examples or cover the Hodge theory.

Back to Hartshorne’s, read the contents carefully and try to do all of
the exercises. This suggestion may seem crucial but significant. Not
only because the exercises contain many important results which may
used in the latter sections, but also due to the common sense that the
best way to learn a language is to use it! Write down your own proof
or solutions. Check and rectify them from time to time.

The prerequisite of Hartshorne’s is the commutative algebra. It is
imperative to finish AM’s book first. But do not be anxious or treat
it as an obstacle to attack Hartshorne’s if you cannot remember all
the theorems since you can capture the geometric background of the
results in AM from Hartshorne’s, which conversely help you understand
the commutative algebra. By the way, Matsumura’s book is also useful
to which Hartshorne’s refers frequently. So have these two commutative
algebra texts in hand.

I guess during the four year undergraduate life, as a math student,
one has to read and understand at least one book by heart!
Hartshorne’s can be one of the choices.

[Chen,Haojie]

Make sense!

[Shuchao Bi]

Dear all,

All the comments are awesome! But those are all about the "classical"
algebraic geometry. There is a big amazing world outside. I will list some
of these in the following (ordered by the distance to classical AG).

Toric Varieties: This subject was developed in the 1990s. There is a
classical book "Introduction to toric varieties" by William Fulton and a
uncompleted online book by David A. Cox, John B. Little, and Hal Schenck
which I like a lot. Quote W. Fulton, "toric variety have provided a
remarkably fertile testing ground for general theories".

Algebraic Statistics: This is a rather new research area, and there is
only one book on this subject by Bernd Sturmfels which is published this
year I think. Basically, algebraic statistics is about using algebraic
geometry to study statistical models. We know that, traditionally people
treat these models as manifold and use analysis to study them, on drawback
is that people don't know how to deal with singularities in the analysis
setting, but resolution of singularities in AG provides some hope. Quote
B. Sturmfels, "Algebra is as powerful as analysis in statistics".

Tropical Geometry: This is a piece-wise linear version of AG. For a
classical varieties, there is a so-called evaluation map project them to
piece-wise linear things (so called polyhedral fans or polyhedral
complexes), hope that we can get information of the original varieties
from their projections. There is no standard book on these subject yet, I
know Sturmfels are working on this, and I could email you a very draft
version of their "book" if anyone is interested.

I really liked our Cocktail Seminar back to undergraduate days at
Hangzhou, thanks to Prof. Shiu-chun Wong!

Cheers,
Shuchao

[tt]

to Yinbang:

i recommend Milne's notes on algebraic number theory; it's more fun,
with many applications. Lang's book might be a little technical to
start with (and this is Milne's viewpoint; see his notes). Serre's
Local fields is about local class field theory, and this is a more
advanced subject that you may want to reserve for a graduate course.
Nonetheless you can read it to get some idea. Milne's notes on class
field theory and Lang's algebraic number theory (part II) also deal
with (both local and global) class field theory.

On 4月13日, 上午5时44分, Yinbang Lin <yinbang....@gmail.com> wrote:
> I am an undergraduate student in Grade Three. I would like to study
> arithmetic.
>
> The first non-elementary book in arithmetic I read is Serre’s A Course in
> Arithmetic. I studied the book last semester. Also in last semester, I
> studied Stein’s Complex Analysis, in which there’re some chapters on zeta
> function and elliptic functions. Recently, I am reading the book Yi
> recommend: Number Theory 1 - Fermat’s Dream. The book contains elliptic
> curves, p-adic numbers, zeta function and algebra number theory. Reading the
> book, I get deeper understanding about what I have learned.
>
> I wonder which book I should read next when I finish reading the book:
> Number Theory 1. Should I read Milne’s notes first? Can I go directly to
> Serre’s Local Fields? Would you mind giving me your suggestions?
>
> By the way, what’s your name？
>
> Yours,
>
> 林胤榜
>

> 2009/4/13 tt <shat...@gmail.com>

[XIA Jie]

To tt
About the class field theory, I guess the proceeding <Algebraic Number
Theory> editted by Cassels & Frohlich is a good text, especially the
Tate's thesis contaned in the appendix. How do you think of it?

[tt]

yes it is. Actually when I took a course on class field theory, this
was the main reference (there was no textbook). Other than chapters 6
and 7, one can also learn some basic number theory in chapters 1-5 and
some of the research topics after chapter 7. Again this book might be
a little technical to most undergrads. Also (maybe for the experts) it
turns out that Lubin-Tate theory of formal group laws simplified the
proof of class field theory, but when the conference was held, Lubin-
Tate theory was not developed yet, therefore one can't find it in this
book (neither in Lang's ANT). This can be found in Milne's notes on
class field theory.

[Yinbang Lin]

Thank you all for your suggestions.
I will read Milne's notes first, following tt's words.
As to algebraic geometry, formerly, I planned to delay the study of AG, since I would like to spend more time on analysis and I don't have so much time. However, condering some of you graduate students are coming back and will give lectures, I decide to begin to study AG.
Thank you again.

2009/4/19 tt <sha...@gmail.com>