Riemann Surfaces (Oxford Graduate Texts in Mathematics Book 22) by Simon Donaldson (PDF)

Description

The theory of Riemann surfaces occupies a very special place in mathematics. It is a culmination of much of traditional calculus, making surprising connections with geometry and arithmetic. It is an extremely useful part of mathematics, knowledge of which is needed by specialists in many other fields. It provides a model for a large number of more recent developments in areas including manifold topology, global analysis, algebraic geometry, Riemannian geometry, anddiverse topics in mathematical physics.This graduate text on Riemann surface theory proves the fundamental analytical results on the existence of meromorphic functions and the Uniformisation Theorem. The approach taken emphasises PDE methods, applicable more generally in global analysis. The connection with geometric topology, and in particular the role of the mapping class group, is also explained. To this end, some more sophisticated topics have been included, compared with traditional texts at this level. While the treatment isnovel, the roots of the subject in traditional calculus and complex analysis are kept well in mind.Part I sets up the interplay between complex analysis and topology, with the latter treated informally. Part II works as a rapid first course in Riemann surface theory, including elliptic curves. The core of the book is contained in Part III, where the fundamental analytical results are proved. Following this section, the remainder of the text illustrates various facets of the more advanced theory.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐I am partial to the Donaldson-style of mathematics and I very excitedly ordered a copy when I heard of it publication. This is an old subject, many treatises have been written about it, so what more can one expect from yet another monograph? I would say, that this book offers you Donaldson’s perspective on the subject, always superbly clear and efficient, informed by the author’s own extensive experience, and revealing the depths of the subject with an ease that would make Fred Astaire jealous. It has a geometric bias and it brings you from zero to where modern research begins. This is a good place to start if you want to learn about this subject that branches out into many other areas of mathematics and theoretical physics.

⭐This is an outstanding introduction to the modern study of Riemann Surfaces. By focusing on low dimensions, Donaldson bridges the gap between elementary complex variables and the deeper topics of complex manifolds, cohomology, algebraic topology, vector bundles, sheaves and so forth. I’m using it for self-study and really appreciate the extent to which the author has gone to make the topic clear and enjoyable.

⭐Amazing, I cannot recommend it highly enough.

⭐Excellent.

⭐Donaldson’s Riemann Surfaces is a lucid and modern account on Riemann surfaces; it covers a good range of topics, but most likely at the expense of being a bit sketchy at places (at least for my likings). (To put things into perspective and to reveal my bias, I have learnt most of what I know on Riemann surfaces from the Bers 1957-1958 lecture notes and the complementary lecture notes by Gunning for the more cohomological aspects, both of which are almost pedantically clear and detailed.) Donaldson’s account might not serve as the sole textbook from which one can gain working knowledge on Riemann surfaces, but it will certainly serve as an excellent and refreshing introductory text.

⭐good

⭐数論の数々の奇跡的問題解決を成し遂げたFaltingsと同時期のフィールズ賞受賞者で、トポロジーで革新的結果をもたらしたdonaldsonの新著で予約して購入したのですが、4年や院初年度の講究テキストとして最適な本の一つだと思いました。最初に正則関数、ガンマ関数から始まり、曲面の分類で復習した所で、写像類群という市販の数学書では見当たらない分野の紹介をしています。次に、リーマン面の基礎的な定義や、そこで用いられる数学的道具としてオイラー類や、ド・ラームコホモロジー、ついで楕円関数の記述があります。常微分方程式のユーリ・マニンの仕事も言及されていますが、この辺りはざっと見た程度で著者の意図までは理解していません。（後で判明したことですがガウス・マニン接続と呼ばれるもので、Deligneの仕事に関連するものです。微分方程式論が射影幾何や関数論、ちょっと高度な題材ですが数理物理と同様に抽象概念の元になるというのを知りませんでした。）リーマン面では名著のフォルスターの本がありますが、どちらが良いのかとも言えません。フォルスターの場合は抽象的で正調な立場から書かれているのに対して、ドナルドソンの場合は、双曲多様体やモジュライ、写像類群といった斬新なトピックが登場しているところで、一方でリーマン・ロッホの定理やセール双対律といった共通のトピックが出されています。まだ評価が出されていないのですが、使い道に困るという本でもなく他の本をやっている同級生や同僚が気になるとは思えません。私の場合はこの本一冊をこれから数年かけて学びたいと思っています。

⭐Not found.

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