A Book in Metric Spaces

Metric Spaces (Springer Undergraduate Mathematics Series) 2007th Edition



From the reviews:

“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.

Link at Amazon.com


مروری بر کتاب «A Course in Point Set Topology»


  • 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.

* اسماعیل اصلانی دیرانلو، دانشجوی کارشناسی ارشد ریاضی محض، Ismael.aslani@gmail.com

Advice from Karen E. Smith for graduate students in math

“Start where you are at, and don’t compare yourself to others. Work hard, get help, and stay on the path. Sometimes you will fail. That’s OK…Do lots of calculations and examples, be curious, be solid on the basics. Also, remember to take care of yourself… Find advice and mentoring from many different people at different places in their careers and even in different careers… soothe a lot of anxiety by helping others.” Read about her life path and math in Notices of the AMS August issue at http://bit.ly/2fb9B8m (Photo of Karen E. Smith ©Eric Bronson, Michigan Photography.)




interview with Villani in Science by Elisabeth Pain

Emmanuel Macron, French Minister of Economy Presents France Project To Host the 2025 World Expo At Louis Vuitton Foundation In Paris

Emmanuel Macron (left) is “a president who believes science is part of global politics,” Cédric Villani (right) says.

Q&A: Why a top mathematician has joined Emmanuel Macron’s revolution

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.

Cédric Villani, Henri Poincaré Institute

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.

Cédric Villani Elected to French National Assembly


Cédric Villani, who won the Fields Medal in 2010, has been elected to the French National Assembly, receiving nearly 70% of the vote in a constituency south of Paris. (Photo: Sandy Huffaker.) Villani heads the Henri Poincaré Institute in Paris, and is the recipient of the Prize of the European Mathematical Society (2008), the Fermat Prize (2009), the Henri Poincaré Prize (2009), the AMS’s Joseph L. Doob Prize (2014, for Optimal Transport: Old and New), and was the 2013 AMS Josiah Willard Gibbs Lecturer. See an interview with Villani in Science by Elisabeth Pain.

some published posts about Maryam Mirzakhani


Maryam Mirzakhani is known for her work on moduli spaces of Riemann surfaces.  Some of her most cited work looks at the moduli space of a genus gg Riemann surface with nn geodesic boundary components.  In two of her papers, she computes the volume of these moduli spaces, with respect to the Weil-Petersson metric (see below).  In another, she provides a means for counting the number of simple closed geodesics of length at most LL. Mirzakhani is also known for her work on billiards (see the review of her paper with Eskin and Mohammadi below), a subject closely related to moduli space questions.  Teichmüller theory and the geometry of moduli spaces are famously deep subjects.  Making progress requires mastering large areas of analysis, dynamical systems, differential geometry, algebraic geometry, and topology.  I can only appreciate Mirzakhani’s work superficially, as I have not mastered those subjects.   Instead, some reviews of her work are reproduced below.  

 One of her biggest projects, joint work with Eskin studying the action of SL(2,R) on moduli space, is not published yet.  So there is no item in MathSciNet for it.  You can read the latest version on the arXiv.

2. Mirzakhani published three papers as an undergraduate:  MR1366852MR1386951MR1615548.  The second of these is regularly cited by combinatorists. The third paper was in the Monthly.  

3. I started writing this post back in March, when I was highlighting the work of some remarkable mathematicians.  It was delayed because describing her work is not simple: it is substantial and uses deep and difficult tools from several areas.  Her papers are quite well written, with accessible introductions.  However, the genius is in the details, which require real commitment to understand.  The video produced for the ICM where she won her Fields Medal allows her to present something of her work.  Amie Wilkinson describes Mirzakhani’s working style in this article in the NY Times.  In a recent blog post, Terry Tao comments on how Mirzakhani was able to see disparate mathematical results through the lens of the mathematics she was developing herself.

4. Thank you to Tom Ward who spotted an inequality that was reversed in the original version of this post.

Link: AMS


Terence Tao:

I am totally stunned to learn that Maryam Mirzakhani died today yesterday, aged 40, after a severe recurrence of the cancer she had been fighting for several years.  I had planned to email her some wishes for a speedy recovery after learning about the relapse yesterday; I still can’t fully believe that she didn’t make it.

My first encounter with Maryam was in 2010, when I was giving some lectures at Stanford – one on Perelman’s proof of the Poincare conjecture, and another on random matrix theory.  I remember a young woman sitting in the front who asked perceptive questions at the end of both talks; it was only afterwards that I learned that it was Mirzakhani.  (I really wish I could remember exactly what the questions were, but I vaguely recall that she managed to put a nice dynamical systems interpretation on both of the topics of my talks.)

After she won the Fields medal in 2014 (as I posted about previously on this blog), we corresponded for a while.  The Fields medal is of course one of the highest honours one can receive in mathematics, and it clearly advances one’s career enormously; but it also comes with a huge initial burst of publicity, a marked increase in the number of responsibilities to the field one is requested to take on, and a strong expectation to serve as a public role model for mathematicians.  As the first female recipient of the medal, and also the first to come from Iran, Maryam was experiencing these pressures to a far greater extent than previous medallists, while also raising a small daughter and fighting off cancer.  I gave her what advice I could on these matters (mostly that it was acceptable – and in fact necessary – to say “no” to the vast majority of requests one receives).

Given all this, it is remarkable how productive she still was mathematically in the last few years.  Perhaps her greatest recent achievement has been her “magic wand” theorem with Alex Eskin, which is basically the analogue of the famous measure classification and orbit closure theorems of Marina Ratner, in the context of moduli spaces instead of unipotent flows on homogeneous spaces.  (I discussed Ratner’s theorems in this previous post.  By an unhappy coincidence, Ratner also passed away this month, aged 78.)  Ratner’s theorems are fundamentally important to any problem to which a homogeneous dynamical system can be associated (for instance, a special case of that theorem shows up in my work with Ben Green and Tamar Ziegler on the inverse conjecture for the Gowers norms, and on linear equations in primes), as it gives a good description of the equidistribution of any orbit of that system (if it is unipotently generated); and it seems the Eskin-Mirzakhani result will play a similar role in problems associated instead to moduli spaces.  The remarkable proof of this result – which now stands at over 200 pages, after three years of revision and updating – uses almost all of the latest techniques that had been developed for homogeneous dynamics, and ingeniously adapts them to the more difficult setting of moduli spaces, in a manner that had not been dreamed of being possible only a few years earlier.

Maryam was an amazing mathematician and also a wonderful and humble human being, who was at the peak of her powers.  Today was a huge loss for Maryam’s family and friends, as well as for mathematics.

[EDIT, Jul 16: New York times obituary here.]

[EDIT, Jul 18: New Yorker memorial here.]

Link: https://terrytao.wordpress.com


New York Times:

Maryam Mirzakhani, an Iranian mathematician who was the only woman ever to win a Fields Medal, the most prestigious honor in mathematics, died on Friday. She was 40.

The cause was breast cancer, said Stanford University, where she was a professor. The university did not say where she died.

Her death is “a big loss and shock to the mathematical community worldwide,” said Peter C. Sarnak, a mathematician at Princeton University and the Institute for Advanced Study.

The Fields Medal, established in 1936, is often described as the Nobel Prizeof mathematics. But unlike the Nobels, the Fields are bestowed only on people aged 40 or younger, not just to honor their accomplishments but also to predict future mathematical triumphs. The Fields are awarded every four years, with up to four mathematicians chosen at a time.

“She was in the midst of doing fantastic work,” Dr. Sarnak said. “Not only did she solve many problems; in solving problems, she developed tools that are now the bread and butter of people working in the field.”

Continue reading the main story

Dr. Mirzakhani was one of four Fields winners in 2014, at the International Congress of Mathematicians in South Korea. Until then, all 52 recipients had been men. She was also the only Iranian ever to win the award.

President Hassan Rouhani of Iran released a statement expressing “great grief and sorrow.”

He wrote, “The unparalleled excellence of the creative scientist and humble person that echoed Iran’s name in scientific circles around the world was a turning point in introducing Iranian women and youth on their way to conquer the summits of pride and various international stages.”

Dr. Mirzakhani’s mathematics looked at the interplay of dynamics and geometry, in some ways a more complicated version of billiards, with balls bouncing from one side to another of a rectangular billiards table eternally.

A ball’s path can sometimes be a repeating pattern. A simple example is a ball that hits a side at a right angle. It would then bounce back and forth in a line forever, never moving to any other part of the table.

But if a ball bounced at an angle, its trajectory would be more intricate, often covering the entire table.

“You want to see the trajectory of the ball,” Dr. Mirzakhani explained in a video produced by the Simons Foundation and the International Mathematical Union to profile the 2014 Fields winners. “Would it cover all your billiard table? Can you find closed billiards paths? And interestingly enough, this is an open question in general.”

Link: NewYorkTimes


and at last a video the although it is not new! but I love it.

Darwin finally proven wrong by AI !


A sensational discovery in biology and genetics was made last week: Darwin’s theory of evolution was finally proven wrong using modern AI technology – a scientific discovery with consequences that cannot be overestimated at this time.

The discovery was made by Prof. Dr. E. Kaf, so far mostly only known to the scientific biology and genetic science community. Due to his recent discovery however, he has become an overnight scientific celebrity and a potential candidate for winning next year’s Nobel Prize. Dr. E. Kaf is the director of the newly established Institute of Brain Up- and Down Loads at Havrad University.

Complete Story Here