When she was eleven years old, Sofia (sometimes called Sonja) Kovalevskaya covered the walls of her room in note sheets about differential and integral calculus by the Russian mathematician Mikhail Ostrogradski. These notes were from her father’s university years. This was how Sofia became familiar with calculus. Her first introduction to calculus was by the hand of her uncle Pyotr Krukovsky. He taught her the basics until she developed such a fascination with mathematics that she described it as “a mysterious science which opens up to its initiates a new world of wonders, inaccessible to ordinary mortals.”
This is a very nice introduction to the theory of semisimple modules and rings, central simple algebras, and the Brauer group. It starts off with some review of basic concepts such as modules, tensor products, and field extensions – much of which is in the exercises. Indeed, a lot of the material is developed in the exercises so it is very useful to go through them. Most of the exercises aren’t exceedingly difficult either, and hints are provided for the more tricky ones.
Chapter one gets the ball rolling with the theory of simple modules and rings. The Wedderburn structure theorem is proved, as well as the structure theorem for simple artinian rings. These ideas are examined further in chapter two by analysing the Jacobson radical, thereby giving another characterization of semisimple rings. Chapter three goes on to study central simple algebras. Here they prove the famous Wedderburn theorem that every finite division ring is commutative, as well as the famous Frobenius theorem on real division algebras. Chapter four finally gives the definition of the Brauer group with a discussion of group cohomology. It proves that for a Galois extension K/k the second Galois cohomology group H^2(Gal(K/k),K*) is isomorphic to the Brauer group Br(K/k). The naturality of this isomorphism is discussed later on and developed through the exercises, although they never actually define what natuarality really is.
One way I think the book could be improved would be to introduce some more abstract homological algebra. The definition of group cohomology given is in terms of n-cochains. I think it would be useful to include the notion of derived functors here, and the general definition of the cohomolgy homology of a derived functor through the use of injective and projective resolutions. Then the group cohomology could be defined using the Ext functors, and the cocycle complexes obtained by examined the particular projective resolution called the Eilenberg-MacLane bar resolution. This method would help introduce the reader to homological algebra, and at the same time give some concrete use of such abstract constructions. The authors generally shy away from categorical language though, which might appeal to some readers.
The later chapters discuss primitive rings, some representation theory of finite groups, dimension theory of rings, and the Brauer group of a commutative ring. I have not gone through this later material in detail (yet), so I cannot give comment on it.
Overall, if you want to learn some non-commutative algebra, by all means buy this book and work out as many exercises as you can. I found the exposition to be quite illuminating and overall very well written.
“This book is truly about metric spaces. … The book is packed full of material which does not often appear in comparable books. … His use of questions to increase understanding and to move on to the next topic are also to be appreciated. … this is a great book and suitable … for third-and fourth-year under-graduates and beginning graduate students.” (Marion Cohen, MathDL, January, 2007)
“The book is very readable. It includes appendixes on the necessary mathematical logic and set theory, and has a substantial number of exercises… Every concept is demonstrated via a large number of examples, starting with commonplace ones and expanding the reader’s horizon with the more abstruse ones, to give a sense of the scope of the concepts… A useful addition to any library supporting an undergraduate mathematics major.” (D. Z. Spicer, CHOICE, March, 2007)
From the Back Cover
The abstract concepts of metric ces are often perceived as difficult. This book offers a unique approach to the subject which gives readers the advantage of a new perspective familiar from the analysis of a real line. Rather than passing quickly from the definition of a metric to the more abstract concepts of convergence and continuity, the author takes the concrete notion of distance as far as possible, illustrating the text with examples and naturally arising questions. Attention to detail at this stage is designed to prepare the reader to understand the more abstract ideas with relative ease.
The book goes on to provide a thorough exposition of all the standard necessary results of the theory and, in addition, includes selected topics not normally found in introductory books, such as: the Tietze Extension Theorem; the Hausdorff metric and its completeness; and the existence of curves of minimum length. Other features include:
end-of-chapter summaries and numerous exercises to reinforce what has been learnt;
extensive cross-referencing to help the reader follow arguments;
a Cumulative Reference Chart, showing the dependencies throughout the book on a section-by-section basis as an aid to course design.
The book is designed for third- and fourth-year undergraduates and beginning graduates. Readers should have some practical knowledge of differential and integral calculus and have completed a first course in real analysis. With its many examples, careful illustrations, and full solutions to selected exercises, this book provides a gentle introduction that is ideal for self-study and an excellent preparation for applications.
نگاهی به درسهای آنالیز و توپولوژی مقطع کارشناسی ریاضی در ایران
اسماعیل اصلانی دیرانلو*
همهی ما ریاضی خواندهها کاملاً به یاد داریم که علیرغم شیرینی و جذابیت بالایی که درسهای آنالیز ریاضی 1 و توپولوژی عمومی دارند، این درسها با چه مرارتی پاس میشوند و البته در نهایت در بدو ورود به دورههای کارشناسی ارشد ریاضی محض متوجه میشویم که کم آموختهایم. کم نه از حیث «محتوا» و تعداد «فصلها» و «قضیهها» بلکه از لحاظ «عمق» و «میزان درونی شدگی» مفاهیم مهمی مانند: «جدایی پذیر» ، «فشردگی» ، «پیوستگی» و … که همه مربوط به بحث مهمی به نام «فضاهای متریک» میباشد. جان کانوی کتابی نوشته است در سه فصل و فقط با هدف جا انداختن همین مفاهیم بسیار پایه. او در این کتاب نگاه جدیدی را برای آموزش توپولوژی عمومی دورهی لیسانس معرفی مینماید:
“You can probably guess that I have long wanted to write a book on this topic, but other things took precedence. I am glad that was the case because now I think I have a better approach. I had an epiphany about halfway through my career when I realized I didn’t have to teach my students everything I had learned about the subject at hand. I learned mathematics in school that I never used again, and not just because those things were in areas in which I never did research. At least part of this, I suspect, was because some of my teachers hadn’t had this insight. Another reason is that many authors write textbooks as though they are writing a monograph directed at other faculty rather than thinking of the students as the audience. Also, mathematics refines and refreshes itself with time. Certain topics that were important at the inception of an area fade in significance, and some that are useful in various areas today must be added. Other topics are important, but only if you are one of that small percentage who specialize in a specific part of research; such things should not be taught to everyone who takes an introductory course. In addition, when a subject is developing, there is an emphasis on finding the intellectual boundaries of the concepts. Unless that viewpoint is abandoned when the subject is taught, it results in a greater prominence of pathology. An examination of early texts in any subject will reveal such an emphasis. With time, however, it is crucial to decide what should be taught in an introductory course such as the kind this book is written for. I see the purpose of a course in point set topology as giving the student a set of tools. The material is used in almost every part of mathematics.”
کتاب در سه فصل نگاشته شده است: فصل اول: «فضاهای متریک»، فصل دوم: «فضاهای توپولوژیکی»، فصل سوم: «توابع حقیقی-مقدار پیوسته». در انتهای کتاب افزونهای [appendix] شامل 5 بخش آورده شده است: «مجموعهها»، «توابع»، «اعداد حقیقی»، «لم زورن» و «مجموعههای شمارشپذیر». توصیه میکنم حتماً نگاهی به این کتاب بیندازید.
از نظر من مطابق آنچه ایشان در مقدمهی کتاب توضیح میدهد، میشود بعد از گذراندن درس «مبانی ریاضیات» در ترم اول، درس «توپولوژی عمومی» را در ترم دوم و با همین سه فصلی که جناب کانوی گفتهاند گذراند . سپس در ترم سه و چهار درس «مبانی آنالیز ریاضی» را بدون فضای متریک در حد فصل 3 تا 9 کتاب رودین ]1[ و برای تمام دانشجویان گرایشهای محض، کاربردی، علوم کامپیوتر و … ارائه داد. سپس این دانشجویان ریاضی محض هستند که در صورت تمایل و با توجه به گرایشی که قرار است در دوره ارشد انتخاب نماید، به عنوان یک درس اختیاری، «آنالیز پیشرفته» را بگذرانند: فرمهای دیفرانسیل، انتگرال لبگ و بسط مفاهیمی مانند «قضیه استوکس»، «قضیهی گرین» و …، در حد فصل 10 و 11 رودین و کل کتاب اسپیواک ]2[.در این صورت مفاهیم به صورت گام به گام فراگرفته خواهند شد و دانشجو آمادگی لازم برای فراگیری درسهای پیشرفتهی آنالیز و هندسهی دورهی ارشد را خواهد داشت.
]1[. اصول آنالیز ریاضی، والتر رودین، ترجمه دکتر علی اکبر عالمزاده، نشر علمی و فنی، 1362.
]2[. حساب دیفرانسیل و انتگرال روی خمینهها، مایکل اسپیواک، ترجمه دکتر عمید رسولیان، انتشارات روزبهان، 1385.
French President Emmanuel Macron has promised his country a revolution—and after a comfortable victory in the parliamentary elections, he is well-positioned to deliver. Macron’s brand-new centrist and reformist party, La République En Marche!, won 308 of the 577 seats in the National Assembly yesterday. Almost half of his delegates are women; most have never been active in politics.
What the upset will mean for French science is unclear. Macron has promised to raise the country’s research spending from 2.2% of gross domestic product to 3% and give universities more autonomy. He aims to make France a world leader in climate and environmental science and has promised €30 million to help attract foreign scientists using a website named “Make Our Planet Great Again.” Most French scientists were relieved that Macron defeated far-right candidate Marine Le Pen last month, but reforms in science and higher education are likely to meet resistance from leftist groups.
Science talked to one of En Marche!’s new National Assembly members, mathematician and Fields medalist Cédric Villani, 43, who won 69% of the vote in a constituency south of Paris. Villani, who heads the Henri Poincaré Institute in the capital, has won a book prize from the American Mathematical Society in 2014 and joined the prestigious Pontifical Academy of Sciences last year. Frequent media appearances over the past decade—and his trademark silk ascot and spider brooch—have made him one of France’s best-known scientists. (He also gave a TED talk explaining what’s so sexy about math.)
Q: Why did you run, and why with Macron?
A: I never recognized myself in any national political movement. But Macron’s party is enthusiastically pro-European, which has become very rare among national parties in France. It also went very much against the old political tradition of systematically attacking opponents during the presidential election; instead it promoted benevolence, pragmatism, and progress. And the party welcomed nonpoliticians with professional expertise.
Q: What do you hope to achieve in the National Assembly—in general, and for science?
A: I hope to participate in making France feel confident again—in its government, in its own abilities, and in the future. As to science, that’s a complex ecosystem, and the issues in France are well known. The efficiency of the competitive research funding agencies is one issue. How to reward researchers with significant achievements is another. How to organize the governance of universities. University entrance selection. The ratio of public and private investment in R&D. Patenting scientific discoveries and bringing products to market. And so on. There isn’t one particular topic I want to be associated with; I intend to push for the improvement of the science system as a whole.
Q: Do you have concrete measures in mind?
A: There is no simple solution. I would advocate better scientific steering of the National Research Agency. I’m in favor of awarding some researchers a special status, based on international evaluations, that comes with a reduced teaching load. On university governance, I favor relaxing the laws and making them less complicated. And universities should do a better job of informing students on the career outcomes of the degrees they offer.
But in doing this, my goal isn’t just to serve science. My goal is to serve society with scientific expertise as a tool. Currently, scientific knowledge within French political circles is close to zero. It’s important that some scientific expertise is present in the National Assembly.
I hope to participate in making France feel confident again—in its government, in its own abilities, and in the future.
Q: Part of the scientific community has yet to be convinced that Macron is really interested in science.
A: We will see. He sent a strong first signal by according science policy its own ministry, by nominating a very competent minister, Frédérique Vidal, and giving her a broad mandate. Her nomination was welcomed by everybody, including the most radical faction of the scientific community. Macron’s welcome to foreign climate scientists was important as well. He is a president who believes science is part of global politics. It is important that scientists step in and become part of the political process. Now, if there is enough money in the system, a good balance between basic and applied research, and good governance—in other words, if the system works—chances are that the scientific community will be happy.
Q: Is this the end of your career in mathematics?
A: My research essentially stopped when I became institute director in 2009 and started to get more involved with the media. Now, I will leave the directorship. Often in life when you want to gain a new experience, you need to put something aside. But the current political situation in France is so unique and extraordinary that it is more than worth it.