Re: Suggestions for undergraduate Riemann surface seminar
tian...@math.rutgers.edu
Riemann surfaces--the hyperbolic geometry approach. Just in case that
there is some interest, "Beneditti and Petronio's book: Lectures on
Hyperbolic Geometry, Chapters A and B" is a good start.
This approach (+ something else, namely, the matrix model) led Kontsevich
to the solution to Witten's conjecture on the intersection numbers of
Chern classes on the Deligne-Mumford compactification of the moduli space
of Riemann surfaces (whose statement was purely algebraic).
Any way, it is harmless to know one more approach.
Best wishes,
Tian
ps: Kontsevich, 1998 Fields Medalist.
> Hi,
>
> As a beginner, the theory of Riemann surfaces can be studied in two
> different approaches, the complex geometric side and the algebraic
> geometric
> side. If one really wants to know the theory, the final goal is that one
> could translate from either side to the other.
>
> On both sides, we need to make our hands dirty at first. From complex
> geometric side, one needs lots of complex analysis and PDE. From algebraic
> side, one needs commutative algebra. Since most of you learned the complex
> analysis, I guess the first way is more accessible.
>
> However, if you pick any book from that approach, i.e. Farkas and Kra's,
> or
> Foster's book, you will see "Let X be a compact Riemann surface Blah Blah
> Blah...." and NO example at all. "Riemann surface is a dimension one
> complex
> manifold and that is it". Could you give me a example of Riemann surface
> with genus 4 specifying the complex structure? Those books will not tell
> you
> such things.
>
> One may wonder we could choose the other way, the algebraic way. There are
> lots of examples since Riemann surfaces are actually zero locus of a
> homogeneous polynomial in complex projective space. Then if you go through
> Hartshorne or Shafarevich, you will hear words "valuation on the local
> ring.... Blah Blah Blah" with no geometric insight at all. For beginners,
> those will be definitely abstract nonsense.
>
> Today I checked every Riemann surface textbook I know in the library. And
> it
> turns out that most of them fall into one of the above "BAD" categories.
> Here "BAD" means "bad for beginners", and "not bad, pretty good once one
> knows the theory". So it is still worth to write some comments on these
> books. When one wants complete survey or further details, these books
> help.
> Please take a look at the end of the email.
>
> Since I already point out the drawbacks, as a senior seminar textbook on
> Riemann surfaces, it should have lots of examples, lots of stuff to
> compute
> concretely, lots of geometric insight and lots of fun. That is actually
> what
> Abel and Riemann did. For example, the famous Abel's theorem on elliptic
> curves can be computed and proved in a naive way, however, most of the
> books
> just want to the most general form and ignore the history.
>
> I recommend the following books.
>
> 1. A *Scrapbook* of Complex *Curve* Theory by Clemens
>
> Prof. Wong has a copy of my book in his office. The first two chapter is
> very readable. Try it! The later chapters might be hard however you can
> switch the next book if you get bored. By the way, Clemens is a very
> famous
> mathematician in algebraic geometry. He is a student of Griffiths and
> proved
> lots of amazing theorems.
>
> 2. Kirwan, Complex Algebraic *Curves*
>
> Very nice book. very readable. Try it! It should be a very good
> continuation
> after the first few chapters of Clemens.
>
> 3. Miranda, *Algebraic Curves* and Riemann Surfaces
>
> Readable and it is very popular right now, however, this should be a
> reference book since it is too thick. But when you preparing your talks,
> you
> can get something from this book.
>
> 4 and 5
> Wu Hongxi's Jin Li Man Qu Mian Yin Lun (chinese version)
> and Algebraic curves, algebraic manifolds and schemes By Danilov and
> Shokurov the first part
>
> These two books are survey books. Reading them could tell you what happen
> after Abel and Riemann. Shokurov's survey contains most of the standard
> results of Riemann surfaces. The drawback of Wu's book is the chapter on
> PDE
> going too far. The drawback on Shokurov's is no proof of anything.
>
> 6. If you are interested in the arithmetic side of Riemann surface, i.e.
> Zeta functions and Weil's conjecture
>
> Goldschmidt's book on Algebraic functions and projective curves should be
> a
> good start. GTM215 available in China.
>
> I think the above lists should be enough for a semester. Good Luck!
>
> Best,
>
> Yi
>
>
> ================================================
> BAD category #1 book lists:
>
> 1. Farkas and Kra,
>
> Complete survey for Riemann surfaces and treat it very well from analysis,
> including beautiful theorems like uniformization. Lots of complex analysis
> but a big book.
>
> 2. Foster
>
> I hated this book when I was a beginner. However, today I find it is such
> a
> small book containing so much stuff. Definitely better than Farkas and
> Kra.
>
> 3. Narasiham, Compact Riemann Surfaces (I am not sure if you can find it
> in
> Zheda's library)
>
> Smaller than Foster but containing every important results. I like it!
>
> 4. Jost, Compact Riemann Surfaces
>
> Much more analysis than previous ones. I do not think it is good for
> beginner. But the book does discuss Teichmuller theory.
>
> 5. Griffiths-Harris, Chapter 2
>
> This is the first book you should learn once you know the analytic theory
> of
> Riemann surfaces. However not good for the first study since you need 200
> pages on Hodge Kodaira's work which is much later than Riemann.
> ================================================
> BAD category #2 book lists:
>
> 1. Hartshorne, Chapter 4
> Definitely not good for first study. 200 page+ abstract nonsense. haha...
>
> 2. Shafarevich, 100 page abstract nonsense.
>
> 3. Silverman's book on elliptic curves, good but hard still not for
> beginner.
>
> 4. Serre's algebraic groups and class fields, chapter 2, hard but worth to
> read when you are a graduate student.
>
> 5. Griffiths, dai shu qu xian (algebraic curves), Chinese version
> published
> by Beida. This is a very good book but it goes really fast to
> normalization
> stuff. It should be regarded as a typical reference.
>
> =================================================
>
>
>
>
>
>
>
>
>
> 2011/2/21 Yi <math...@gmail.com>
>
>> Dear All,
>>
>> I received an email from Prof. Wong in the following.
>>
>> =====================================================
>> Dear Zhu Yi,
>>
>> A few seniors wants to study Riemann surfaces in a student seminar in
>> this
>> new semester. I'm not sure what is a good textbook for that. The first
>> that come to my mind is
>> Riemann Surfaces, by Farkas and Kra, GTM 71,
>> but I'm not sure whether that's a good choice. Do you have any
>> suggestion?
>>
>> Best wishes,
>>
>> Wong
>> =====================================================
>>
>> I will write up something soon. Please let us know your opinion.
>>
>> If you want to write anything, please do remember cc Wong's students
>> in the cc list above.
>>
>> Best,
>>
>> Yi
>>
>> --
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>>
>
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>
Yi
Yi Zhu
As a beginner, the theory of Riemann surfaces can be studied in two different approaches, the complex geometric side and the algebraic geometric side. If one really wants to know the theory, the final goal is that one could translate from either side to the other.
On both sides, we need to make our hands dirty at first. From complex geometric side, one needs lots of complex analysis and PDE. From algebraic side, one needs commutative algebra. Since most of you learned the complex analysis, I guess the first way is more accessible.
However, if you pick any book from that approach, i.e. Farkas and Kra's, or Foster's book, you will see "Let X be a compact Riemann surface Blah Blah Blah...." and NO example at all. "Riemann surface is a dimension one complex manifold and that is it". Could you give me a example of Riemann surface with genus 4 specifying the complex structure? Those books will not tell you such things.
One may wonder we could choose the other way, the algebraic way. There are lots of examples since Riemann surfaces are actually zero locus of a homogeneous polynomial in complex projective space. Then if you go through Hartshorne or Shafarevich, you will hear words "valuation on the local ring.... Blah Blah Blah" with no geometric insight at all. For beginners, those will be definitely abstract nonsense.
Today I checked every Riemann surface textbook I know in the library. And it turns out that most of them fall into one of the above "BAD" categories. Here "BAD" means "bad for beginners", and "not bad, pretty good once one knows the theory". So it is still worth to write some comments on these books. When one wants complete survey or further details, these books help. Please take a look at the end of the email.
Since I already point out the drawbacks, as a senior seminar textbook on Riemann surfaces, it should have lots of examples, lots of stuff to compute concretely, lots of geometric insight and lots of fun. That is actually what Abel and Riemann did. For example, the famous Abel's theorem on elliptic curves can be computed and proved in a naive way, however, most of the books just want to the most general form and ignore the history.
I recommend the following books.
Prof. Wong has a copy of my book in his office. The first two chapter is very readable. Try it! The later chapters might be hard however you can switch the next book if you get bored. By the way, Clemens is a very famous mathematician in algebraic geometry. He is a student of Griffiths and proved lots of amazing theorems.
Very nice book. very readable. Try it! It should be a very good continuation after the first few chapters of Clemens.
Pan Xuanyu
Try what Zhu yi suggests. Start from complex, topology, a little bit algebra (algebraic point of view is related to some algebraic number theory, especially number field, algebraic integer, finally, you will find out we can talk about Riemann surfaces and numbers in the same language, namely, valuation) , that is the best way to learn this subject.
Try examples, for instance, how to calculate the genus of a smooth projective curves of degree 4 ,5 ,6 and so on? How to calculate the elliptic integration, that is the motivation to build up the theory of Riemann Surface? Try to classify Riemann sufaces of genus 1. And understand the holomorphic maps between Riemann sufaces in the way of "covering space with branch points and Field extensions of meomorphic funtions fild"
I think most the topics I mention show up in the books Zhu yi recommended.
Enjoy the Riemann surface, it is a so beautful subject connected with complex analysis, topology, differential geometry and number theory.
Best
Xuanyu
--- 11年2月22日，周二, tian...@math.rutgers.edu <tian...@math.rutgers.edu> 写道：
> 发件人: tian...@math.rutgers.edu <tian...@math.rutgers.edu>
> 主题: Re: Suggestions for undergraduate Riemann surface seminar
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> 日期: 2011年2月22日,周二,下午2:15