à la recherche du temps perdu

“vous êtes étranger, monsieur?” un homme avec accent sénégalais posais la question.

“non! je suis à la recherche du temps perdu, monsieur”, je lui répondis.


Photo by Jon Tyson on Unsplash


I am not the one who accept to be a looser

I have been trying to come true the biggest dream of my life for last 5 years. Now I am at almost the last step. Although it is a bit difficult but I am not giving up. As they say, “the moments before the day are the darkest moments of the night”.

Eté 2018 : vit-on une nouvelle révolution sexuelle ?

Cet article a été choisi de Radio France Culture, j’espère que vous la désirez.

Le temps passe et les affaires de harcèlement sexuels s’accumulent… Un maître bouddhiste en Chine… Une philosophe féministe, Avita Ronell, aux Etats-Unis… Tous deux contraints à la démission, cette semaine… Des prêtres et des nonnes, au Chili… Et en France, des Youtubeurs accusés d’avoir profité de leur notoriété pour obtenir des faveurs de la part d’adolescentes…
La parole ne cesse de se libérer, depuis l’affaire Weinstein, il y a près d’un an. Elle permet de prendre conscience de l’ampleur des violences sexuelles commises contre les femmes…
Si l’on a choisi de parler de ce sujet, aujourd’hui, c’est aussi à cause de quelques magazines… Des journaux qui, comme chaque année, choisissent l’été pour parler de sexe et de sexualité. Les Inrockuptibles et leur traditionnel numéro SEXE. L’Obs qui a consacré un dossier aux hommes après le mouvement #MeToo, il y a deux semaine”. Et Causeur qui titre « L’amour après Weinstein – malaise dans la libido »… On y parle, tour à tour, de désir, de domination, de victimisation, de virilité… Le débat est vif, loin d’être terminé… A tel point qu’on peut se demander si une révolution n’est pas en cours…
Bref, vivons-nous une nouvelle révolution sexuelle ?


When you accept it you deserve it.

Equality is all about having the chance to choose. if you have no choice, if you have to suffer, if you have to tolerate, if you have to keep silent, and if your words will cost you too much then how can you say this is equality? 
Any force to push you back, any force to make you to stay in an unwanted situation, any thing that stop you from growing up, is inequality and it’s against the freedom. When you accept it you deserve it. So never accept any limitation that is against your rights. There is no excuse for such thing.

Photo by Daniela Cuevas on Unsplash

Negative Probabilities

I found this great article from 2013 posts of Azimuth blog. I didn’t understand it completely but I should say that is amazing! I really enjoy it. I must read some other times.

Negative Probabilities

The physicists Dirac and Feynman, both bold when it came to new mathematical ideas, both said we should think about negative probabilities.

These days, Kolmogorov’s axioms for probabilities are used to justify formulating probability theory in terms of measure theory. Mathematically, the theory of measures that take negative or even complex values is well-developed. So, to the extent that probability theory is just measure theory, you can say a lot is known about negative probabilities.

But probability theory is not just measure theory; it adds its own distinctive ideas. To get these into the picture, we really need to ask some basic questions, like: what could it mean to say something had a negative chance of happening?

I really have no idea.

In this paper:

• Paul Dirac, The physical interpretation of quantum mechanics, Proc. Roy. Soc. London A 180 (1942), 1–39.

Dirac wrote:

Negative energies and probabilities should not be considered as nonsense. They are well-defined concepts mathematically, like a negative of money.

In fact, I think negative money could have been the origin of negative numbers. Venetian bankers started writing numbers in red to symbolize debts—hence the phrase ‘in the red’ for being in debt. So, you could say negative numbers were invented to formalize the idea of debt and make accounting easier. Bankers couldn’t really get rich if negative money didn’t exist.

A negative dollar is a dollar you owe someone. But how can you owe someone a probability? I haven’t figured this out.

Unsurprisingly, the clearest writing about negative probabilities that I’ve found is by Feynman:

• Richard P. Feynman, Negative probability, in Quantum Implications: Essays in Honour of David Bohm, eds. F. David Peat and Basil Hiley, Routledge & Kegan Paul Ltd, London, 1987, pp. 235–248.

He emphasizes that even if the final answer of a calculation must be positive, negative numbers are often allowed to appear in intermediate steps… and that this can happen with probabilities.

Let me quote some:

Some twenty years ago one problem we theoretical physicists had was that if we combined the principles of quantum mechanics and those of relativity plus certain tacit assumptions, we seemed only able to produce theories (the quantum field theories) which gave infinity for the answer to certain questions. These infinities are kept in abeyance (and now possibly eliminated altogether) by the awkward process of renormalization. In an attempt to understand all this better, and perhaps to make a theory which would give only finite answers from the start, I looked into the” tacit assumptions” to see if they could be altered.

One of the assumptions was that the probability for an event must always be a positive number. Trying to think of negative probabilities gave me cultural shock at first, but when I finally got easy with the concept I wrote myself a note so I wouldn’t forget my thoughts. I think that Prof. Bohm has just the combination of imagination and boldness to find them interesting and amusing. I am delighted to have this opportunity to publish them in such an appropriate place. I have taken the opportunity to add some further, more recent, thoughts about applications to two state systems.

Unfortunately I never did find out how to use the freedom of allowing probabilities to be negative to solve the original problem of infinities in quantum field theory!

It is usual to suppose that, since the probabilities of events must be positive, a theory which gives negative numbers for such quantities must be absurd. I should show here how negative probabilities might be interpreted. A negative number, say of apples, seems like an absurdity. A man starting a day with five apples who gives away ten and is given eight during the day has three left. I can calculate this in two steps: 5 -10 = -5 and -5 + 8 + 3. The final answer is satisfactorily positive and correct although in the intermediate steps of calculation negative numbers appear. In the real situation there must be special limitations of the time in which the various apples are received and given since he never really has a negative number, yet the use of negative numbers as an abstract calculation permits us freedom to do our mathematical calculations in any order simplifying the analysis enormously, and permitting us to disregard inessential details. The idea of negative numbers is an exceedingly fruitful mathematical invention. Today a person who balks at making a calculation in this way is considered backward or ignorant, or to have some kind of a mental block. It is the purpose of this paper to point out that we have a similar strong block against negative probabilities. By discussing a number of examples, I hope to show that they are entirely rational of course, and that their use simplifies calculation and thought in a number of applications in physics.

First let us consider a simple probability problem, and how we usually calculate things and then see what would happen if we allowed some of our normal probabilities in the calculations to be negative. Let us imagine a roulette wheel with, for simplicity, just three numbers: 1, 2, 3. Suppose however, the operator by control of a switch under the table can put the wheel into one of two conditions A, B in each of which the probability of 1, 2, 3 are different. If the wheel is in condition A, the probability of 1 is p1A = 0.3 say, of 2 is p2A = 0.6, of 3 is p3A =0.1. But if the wheel is in condition B, these probabilities are

p1B = 0.1, p2B = 0.4, p3B = 0.5

say as in the table.

Cond. A Cond. B
1 0.3 0.1
2 0.6 0.4
3 0.1 0.5

We, of course, use the table in this way: suppose the operator puts the wheel into condition A 7/10 of the time and into B the other 3/10 of the time at random. (That is the probability of condition A, pA = 0.7, and of B, pB = 0.3.) Then the probability of getting 1 is

Prob. 1 = 0.7 (0.3) + 0.3 (0.1) = 0.24,



Now, however, suppose that some of the conditional probabilities are negative, suppose the table reads so that, as we shall say, if the system is in condition B the probability of getting 1 is -0.4. This sounds absurd but we must say it this way if we wish that our way of thought and language be precisely the same whether the actual quantities pi α in our calculations are positive or negative. That is the essence of the mathematical use of negative numbers—to permit an efficiency in reasoning so that various cases can be considered together by the same line of reasoning, being assured that intermediary steps which are not readily interpreted (like -5 apples) will not lead to absurd results. Let us see what p1B = -0.4 “means” by seeing how we calculate with it.

He gives an example showing how meaningful end results can sometimes arise even if the conditional probabilities like p1B are negative or greater than 1.

It is not my intention here to contend that the final probability of a verifiable physical event can be negative. On the other hand, conditional probabilities and probabilities of imagined intermediary states may be negative in a calculation of probabilities of physical events or states. If a physical theory for calculating probabilities yields a negative probability for a given situation under certain assumed conditions, we need not conclude the theory is incorrect. Two other possibilities of interpretation exist. One is that the conditions (for example, initial conditions) may not be capable of being realized in the physical world. The other possibility is that the situation for which the probability appears to be negative is not one that can be verified directly. A combination of these two, limitation of verifiability and freedom in initial conditions, may also be a solution to the apparent difficulty.

The rest of this paper illustrates these points with a number of examples drawn from physics which are less artificial than our roulette wheel. Since the result must ultimately have a positive probability, the question may be asked, why not rearrange the calculation so that the probabilities are positive in all the intermediate states? The same question might be asked of an accountant who subtracts the total disbursements before adding the total receipts. He stands a chance of going through an intermediary negative sum. Why not rearrange the calculation? Why bother? There is nothing mathematically wrong with this method of calculating and it frees the mind to think clearly and simply in a situation otherwise quite complicated. An analysis in terms of various states or conditions may simplify a calculation at the expense of requiring negative probabilities for these states. It is not really much expense.

Our first physical example is one in which one· usually uses negative probabilities without noticing it. It is not a very profound example and is practically the same in content as our previous example. A particle diffusing in one dimension in a rod has a probability of being at x at time t of P(x,t)satisfying

\partial P(x,t)/\partial t = -\partial^2 P(x,t)/\partial x^2

Suppose at x =0 and x =\pi the rod has absorbers at both ends so that P(x,t) = 0 there. Let the probability of being at x at t = 0 be given as P(x,0) =f(x). What is P(x,t) thereafter? It is

\displaystyle{ P(x,t) = \sum_{n=1}^\infty P_n \; \sin x \;\exp(-n^2 t) }

where P_n is given by

x \displaystyle{  f(x) = \sum_{n = 1}^\infty P_n \; \sin n x }


x \displaystyle{ P_n = \frac{2}{\pi} \int_0^\pi f(x) \sin nx  \; dx }

The easiest way of analyzing this (and the way used if P(x,t) is a temperature, for example) is to say that there are certain distributions that behave in an especially simple way. If f(x) starts as \sin nx it will remain that shape, simply decreasing with time, as e^{-n^2 t} Any distribution f(x)can be thought of as a superposition of such sine waves. But f(x) cannot be \sin nx if f(x) is a probability and probabilities must always be positive. Yet the analysis is so simple this way that no one has really objected for long.

He also gives examples from quantum mechanics, but the interesting thing about the examples above is that they’re purely classical—and the second one, at least, is something physicists are quite used to.

Sometimes it’s good to temporarily put aside making sense of ideas and just see if you can develop rules to consistently work with them. For example: the square root of -1. People had to get good at using it before they understood what it really was: a rotation by a quarter turn in the plane.

Along those, lines, here’s an interesting attempt to work with negative probabilities:

• Gábor J. Székely, Half of a coin: negative probabilitiesWilmott Magazine (July 2005), 66–68.

He uses rigorous mathematics to study something that sounds absurd: ‘half a coin’. Suppose you make a bet with an ordinary fair coin, where you get 1 dollar if it comes up heads and 0 dollars if it comes up tails. Next, suppose you want this bet to be the same as making two bets involving two separate ‘half coins’. Then you can do it if a half coin has infinitely many sides numbered 0,1,2,3, etc., and you win n dollars when side number n comes up….

… and if the probability of side n coming up obeys a special formula…

and if this probability can be negative sometimes!

This seems very bizarre, but the math is solid, even if the problem of interpreting it may drive you insane.

Let’s see how it works. Consider a game G where the probability of winning n = 0, 1, 2, \dots dollars is g(n). Then we can summarize this game using a generating function:

\displaystyle{ G(z) = \sum_{n = 0}^\infty g(n) , z^n }

Now suppose you play two independent games like this, G and another one, say H, with generating function

\displaystyle{ H(z) = \sum_{n = 0}^\infty h(n) , z^n }

Then there’s a new game GH that consists of playing both games. The reason I’m writing it as GH is that its generating function is the product

\displaystyle{ G(z) H(z) = \sum_{m,n = 0}^\infty g(m) h(n) z^{m+n} }

See why? With probability g(m) h(n) you win m dollars in game Gand n dollars in game H, for a total of m + n dollars.

The game where you flip a fair coin and win 1 dollar if it lands heads up and 0 dollars if lands tails up has generating function

\displaystyle{ G(z) = \frac{1}{2}(1 + z) }

The half-coin is an imaginary game H such that playing two copies of this game is the same as playing the game G. If such a game really existed, we would have

G(z) = H(z)^2


\displaystyle{ H(z) = \sqrt{\frac{1}{2}(1 + z)} }

However, if you work out the Taylor series of this function, every even term is negative except for the zeroth term. So, this game can exist only if we allow negative probabilities.

(Experts on generating functions and combinatorics will enjoy how the coefficients of the Taylor series of H(z) involves the Catalan numbers.)

By the way, it’s worth remembering that for a long time mathematicians believed that negative numbers made no sense. As late as 1758 the British mathematician Francis Maseres claimed that negative numbers

… darken the very whole doctrines of the equations and make dark of the things which are in their nature excessively obvious and simple.

So opinions on these things can change. And since I’ve spent a lot of time working on ‘sets with fractional cardinality’, and have made lots of progress on that idea, and other strange ideas, I like to spend a little time now and then investigating other nonsensical-sounding generalizations of familiar concepts.

This paper by Mark Burgin has a nice collection of references on negative probability:

• Mark Burgin, Interpretations of negative probability.

He valiantly tries to provide a frequentist interpretation of negative probabilities. He needs ‘negative events’ to get negative frequencies of events occurring, and he gives this example:

To better understand how negative elementary events appear and how negative probability emerges, consider the following example. Let us consider the situation when an attentive person A with the high knowledge of English writes some text T. We may ask what the probability is for the word “texxt” or “wrod” to appear in his text T. Conventional probability theory gives 0 as the answer. However, we all know that there are usually misprints. So, due to such a misprint this word may appear but then it would be corrected. In terms of extended probability, a negative value (say, -0.1) of the probability for the word “texxt” to appear in his text T means that this word may appear due to a misprint but then it’ll be corrected and will not be present in the text T.

Maybe he’s saying that the misprint occurs with probability 0.1 and then it ‘de-occurs’ with the same probability, giving a total probability of

0.1 - 0.1 = 0

I’m not sure.

Here’s another paper on the subject:

• Espen Gaarder Haug, Why so negative to negative probabilities?Wilmott Magazine.

It certainly gets points for a nice title! However, like Burgin’s paper, I find it a lot less clear than what Feynman wrote.

Notice that like Székely’s paper, Haug’s originally appeared in the Wilmott Magazine. I hadn’t heard of that, but it’s about finance. So it seems that the bankers, having invented negative numbers to get us into debt, are now struggling to invent negative probabilities! In fact Haug’s article tries some applications of negative probabilities to finance.


For further discussion, with some nice remarks by the quantum physics experts Matt Leifer and Michael Nielsen, see the comments on my Google+ post on this topic. Matt Leifer casts cold water on the idea of using negative probabilities in quantum theory. On the other hand, Michael Nielsen points out some interesting features of the Wigner quasiprobability distribution, which is the best possible attempt to assign a probability density for a quantum particle to have any given position and momentum. It can be negative! But if you integrate it over all momenta, you get the probability density for the particle having any given position:


And if you integrate it over all positions, you get the probability density for the particle having any given momentum:


Reference: Azimuth Blog

Maria Droujkova: Beautiful Math is All About the People

I have chosen this article from AWM website. I hope you enjoy it. It is just a sample of biographies of women in mathematics. You can find many other articles by clicking on this link
2009 AWM Essay Contest: Grades 6 – 8 First Place

by Angela Pham

When we think of mathematics, we think of it as a solitary subject. We think of mathematicians being closed up in a small room working out complex problems. But for Maria Droujkova, working in math is almost the complete opposite.

“It’s all about people,” Maria Droujkova says. If you want to look into a career in mathematics, you should talk to math people, meet math people, go to math related events, and organize math clubs. First talk to people, then do the math. Math games, math projects, creative math, fun math, beautiful math.

Even in her young years, Maria’s mathematical education was rooted in personal relationships. Growing up in Ukraine, education was much different than in the US. At five she loved solving math problems with her mother. In early school, she found two other children who also loved math. With the help of an encouraging teacher, they found and solved math problems together. “The teacher gave us famous mathematicians’ nicknames – mine was Sonya Kovalevska.” In Maria’s adolescent years she turned toward physics, and when she was old enough for the math and science camps that her mother had her attend, she signed up for the physics division.

Maria Droujkova went to Moscow State University, a school respected throughout the Soviet Union, where she studied a branch of chaos theory. Her research led up to her thesis topic at the end of five years, which was “Bifurcations of Heart-Shaped Asymptote Polygons.” The chaos theory appealed to Maria because of the order that lay beneath complex and seemingly random events: even the smallest change can result in a whole new order. She was discovering beauty through her study of mathematics. After Maria graduated, she and her husband started looking for other countries to continue learning in. In the Soviet Union there were many restrictions on people’s activities, thoughts, and speech, even for children. As Maria began looking into math as a career, she saw that it would be too hard to accomplish where she lived. She wanted to research and discover, but Ukraine was closed to new ideas. The economy was so poor that it was almost impossible for mathematicians to work in math. “I saw a lot of my colleagues selling fruits on the streets or doing other random jobs to feed their families, and I’d rather do research.”

Because America was a good place for people who wanted to discover and the opportunities in math and science careers were wide open, Maria, 22, and her husband came to America in 1994 to enter Tulane University in New Orleans. At Tulane she got her MS degree in Applied Mathematics, which is using math in other fields. For example, math can be used to help figure out crimes, as is seen in the television show Numb3rs. Then Maria went to North Carolina State University and in 2004 got her PhD in Mathematics Education. Her dissertation topic was: “Roles of metaphor in the growth of mathematical understanding.” Metaphor in mathematical understanding is the explanation of a math principle by associating it with a familiar idea. This research must have later encouraged Maria to begin in her teaching of advanced math to young children.

What’s Maria’s favorite kind of math? Beautiful math. For example, beautiful math is in the patterns of geometric shapes. Maria uses these shapes, such as kirigami snowflakes, to teach things like multiplication. Maria’s main subject of teaching is multiplicative thinking, a central focus in her work. She strives to teach multiplication in a more creative way than just a multiplication table. She encourages students to find several different ways to learn multiplication principles rather than simply memorizing the facts.

Today, Maria works as a self-employed math education consultant near Raleigh, North Carolina. She speaks at conferences and works with other educators on math projects. Her company, Natural Math LLC, offers students fun activities, software, math clubs, and advice. Maria teaches kids and parents how to accept and love math by teaching them the beauty of it. She gives her classes different creative projects to learn complicated and advanced math that would normally be taught to older students. For example, she teaches three to six year olds about fractals. She also explores psychology and develops theories on how the human brain works when it encounters mathematical problems or patterns. Home-schooling gives her a full time job and a laboratory. “I am very happy to say that my ten year old daughter is growing up in the atmosphere where she can appreciate the beauty of mathematics. When she claps her hands excitedly upon seeing an especially elegant proof, or figuring out a tricky problem, or creating a handy representation, I can see we are doing something right.”


No Such Thing as a Math Person

She said the popular idea that boys and girls learn differently is “not supported by the neuroscientific literature.”

“Most of the things that parents and kids believe about math learning are wrong,” said Dr. Boaler, who is the co-founder of Youcubed, a website that argues for a revolution in math teaching for all children, and offers resources to teachers, students and parents. In fact, maybe what everyone needs — girls and boys both — is a different kind of math teaching, with much less emphasis on timed tests, and more attention to teaching math as a visual subject, and as a place for creativity.

“The lovely thing is when you change math education and make it more about deep conceptual understanding, the gender differences disappear,” Dr. Boaler said. “Boys and girls both do well.”

Sexism and racism, as we know, have had deep affections on our social life. One of our important issues in contemporary social life is math teaching at schools. People generally think boys learn math better than girls. The article is about to  disprove this cliche, and this idea that someone may have got a “math mind” has been disproved too. I invite you to read the article from first word to the last one:

No Such Thing as a Math Person

By Perri Klass, M.d. May 15, 2017


When I wrote about fending off math anxiety last month I learned both from the experts I interviewed and from people to whom I happened to mention the topic that math anxiety is found across all lines of gender, ethnicity and educational background. There are plenty of men and women out there, including the highly educated and the professionally aggressive (professors and corporate lawyers, say), who proudly — or shamefacedly — wave the math anxiety flag. Oh yes, that’s me, I don’t have a math brain — though the whole idea of a math brain is frowned on by those who study this topic.

There is a general assumption that women are affected more than men, and that math and math anxiety contribute to the barriers that keep women underrepresented in the STEM fields. In my own familial experiment, I have two sons and a daughter, and though everyone managed O.K. in math, the daughter was, without question, the math kid — though the very idea of “math kids” is considered part of the problem.

My daughter, who majored in math in college, feels that the key is that she attended an all-girls school from fifth grade through 12th grade, and isn’t sure she would have stayed with math if she’d gone to a coed school. I believe her — though the research literature doesn’t necessarily support her, from a statistical point of view.

That’s one of the interesting things about trying to think about girls and math: It involves questioning some of your assumptions about how children learn, and about what makes some topics harder or less accessible to lots of people. And then trying to look at research that tells you that your own perceptions and experience may not be reliable.


As far as math anxiety, “many many more girls and women than men are anxious,” said Jo Boaler, a professor of mathematics education at Stanford, “and we know anxiety holds people back — there are still messages out there that math is for boys and not for girls.” Some of the anxiety, she said, may be transmitted by elementary school teachers, who are likely to be female, and are often themselves anxious about math. “We know that girls identify with their elementary teachers,” she said, and are more likely than boys to be affected by the teacher’s math anxiety, if it is present, contributing to what she called “the cycle by which this continues.”

Sian Beilock, a professor of psychology at the University of Chicago, pointed to the pressure created by the stereotype that girls aren’t good at math. “They come in feeling pressure that could affect their performance,” she said. “That can rob people of the cognitive horsepower they would have to perform at their best.” And this can be worst for the best students, she said. “Girls who come in with the most ability to work at a high level are most impacted.”

Andrei Cimpian, an associate professor of psychology at New York University who studies children’s ideas on gender and ability, said that there is lots of variability in the distribution of men and women across fields in the sciences, social sciences and humanities. One key, he said, is whether it is perceived that in order to work in a certain field, a person needs to be brilliant, to have, even, a spark of genius. “When we surveyed academics across disciplines, in fields whose members said, yes, there is something innate, inborn, required for success,” he said, “it was particularly in those fields where we saw women underrepresented, also African-Americans.”

For math through high school, “there isn’t anything in the curriculum that any typically developing child shouldn’t be able to grasp,” he said. “The minute we start talking about who the brilliant ones are, it’s very easy to go from individual differences to group differences.”


And what about this whole idea of math brain and math kid? “Metaphors like math brain can create their own reality,” Dr. Cimpian warned. “We all fall on a certain continuum,” he said.

Dr. Boaler, the author of the book “Mathematical Mindsets,” said: “The message to all kids, girls and boys, is there’s no such thing as a math person.”

And what about my daughter’s belief that she owes a lot to being in math classes with no boys in them? Erin Pahlke, an assistant professor of psychology at Whitman College in Walla Walla, Wash., said, “often the girls in the single-sex schools have higher math achievement, better attitudes, lower levels of math anxiety, they often have better hopes for the future of wanting to take higher level math,” she said. “You say, this is incredible, single-sex schooling is the answer!”

But she was the first author of a 2014 meta-analysis representing the testing of 1.6 million students from 21 countries that found that among the high quality studies, the differences could be explained by looking at such factors as the different socioeconomic status of those choosing single-sex education, and at the pretest scores before the girls entered the single-sex schools as well as measures of school quality and resources.

Dr. Pahlke said that people tell her all the time, “my daughter or my niece went to a single-sex school and it was incredible — I would say to them, yes, I agree, you do see that, but the question is whether or not it’s due to the single-sex environment.” Instead, she said, “it’s due to being around girls who came in with higher math scores, or teacher quality differences, that’s what the research suggests.”

She said the popular idea that boys and girls learn differently is “not supported by the neuroscientific literature.”

“Most of the things that parents and kids believe about math learning are wrong,” said Dr. Boaler, who is the co-founder of Youcubed, a website that argues for a revolution in math teaching for all children, and offers resources to teachers, students and parents. In fact, maybe what everyone needs — girls and boys both — is a different kind of math teaching, with much less emphasis on timed tests, and more attention to teaching math as a visual subject, and as a place for creativity.

“The lovely thing is when you change math education and make it more about deep conceptual understanding, the gender differences disappear,” Dr. Boaler said. “Boys and girls both do well.”

Reference: https://www.nytimes.com

Photo Story: Fifth-graders Shoaa Khan, left, and Briana Berreondo work through a math problem together during class at P.S. 165 in Morningside Heights.CreditKarsten Moran for The New York Times

Where Boys Outperform Girls in Math: Rich, White and Suburban Districts

At this article you will read the result of a research studying math grades in both girls and boys depend on color of skin, income of family and the district that family live parameters. the result is interesting.

Where Boys Outperform Girls in Math: Rich, White and Suburban Districts

Blogging for the AMS

Do you like blogging? especially math blogging? even more: do you like blogging for the AMS? if so, follow the below instruction:

Blogging for the AMS

The AMS publishes a variety of blogs addressing topics related to mathematics and the profession. Our volunteer writers offer their particular perspectives as early-career mathematicians, experienced mentorsgraduate studentsexperts in their fields, and more. See a list of our current blogs here.

Would you like to propose a new AMS blog?

We appreciate your interest. Please follow these steps:

  1. Review our current set of blogs to see what our bloggers are already covering.
  2. If you think your topic is a good fit for the AMS, please submit a blog proposal including:
    • Your name, contact information, and short C.V.
    • At least two sample blog posts
    • Proposed topic
    • Motivation and background
    • Posting schedule and blog management plan
    • List of additional editors and bloggers
    • Your blog URL (if the blog already exists)
  3. Submit your proposal to AMS Blog Proposals via email.

We look forward to hearing your ideas. If you have questions about either this process or our current blogs, please email AMS Blog Proposals.

Reference: AMS.ORG

Association for Women in Mathematics (AWM)


The Association for Women in Mathematics (AWM) is a non-profit organization founded in 1971.

The purpose of the Association for Women in Mathematics is to encourage women and girls to study and to have active careers in the mathematical sciences, and to promote equal opportunity and the equal treatment of women and girls in the mathematical sciences.

AWM currently has more than 3000 members (women and men) representing a broad spectrum of the mathematical community — from the United States and around the world!

The Website of AWM: https://sites.google.com/site/awmmath/

Make Your Daughter Practice Math. She’ll Thank You Later.

“Unfortunately, thinking you’re not very good at something can be a quick path to disliking and avoiding it, even if you do have natural ability. You can begin to avoid practicing it, because to your mind, that practice is more painful than learning what comes more easily. Not practicing, in turn, transforms what started out as a mere aversion into a genuine lack of competence.” 

Nobody can deny the girls’ abilities in STEM (Science, Technology, Engineering, Math). The abilities of women have been proven several times during the history of human kind’s civilization, but we have being accepted them in all parts of the society since only few decades so far. We still have a long way to reach the equality. I invite you to read this article from NY Times:

Make Your Daughter Practice Math. She’ll Thank You Later.

The way we teach math in America hurts all students, but it may be hurting girls the most.

By Barbara Oakley
Ms. Oakley is an engineering professor and the author of a book on learning.

Aug. 7, 2018

For parents who want to encourage their daughters in STEM subjects, it’s crucial to remember this: Math is the sine qua non.

You and your daughter can have fun throwing eggs off a building and making papier-mâché volcanoes, but the only way to create a full set of options for her in STEM is to ensure she has a solid foundation in math. Math is the language of science, engineering and technology. And like any language, it is best acquired through lengthy, in-depth practice.

But for girls, this can be trickier than it looks. This is because many girls can have a special advantage over boys — an advantage that can steer them away from this all-important building block. A large body of research has revealed that boys and girls have, on average, similar abilities in math. But girls have a consistent advantage in reading and writing and are often relatively better at these than they are at math, even though their math skills are as good as the boys’. The consequence? A typical little boy can think he’s better at math than language arts. But a typical little girl can think she’s better at language arts than math. As a result, when she sits down to do math, she might be more likely to say, “I’m not that good at this!” She actually is just as good (on average) as a boy at the math — it’s just that she’s even better at language arts. Of course, it’s hard to know what’s taking place in the minds of babes. But studies revealing developmental differences between boys’ versus girls’ verbal abilities alongside developmental similarities in boys’ and girls’ math abilities — combined with studies that show that among girls, self-perceived ability affects academic performance — seem to indicate that something like the above dynamic might be going on.

Unfortunately, thinking you’re not very good at something can be a quick path to disliking and avoiding it, even if you do have natural ability. You can begin to avoid practicing it, because to your mind, that practice is more painful than learning what comes more easily. Not practicing, in turn, transforms what started out as a mere aversion into a genuine lack of competence. Unfortunately, the way math is generally taught in the United States — which often downplays practice in favor of emphasizing conceptual understanding — can make this vicious circle even worse for girls.

It’s important to realize that math is, to some extent, like playing a musical instrument. But the instrument you play is your own internal neural apparatus.

When we learn to play an instrument — say, the guitar — it’s obvious that simply understanding how a chord is constructed isn’t the equivalent of being able to play the chord. Guitar teachers know intuitively that the path to success and creativity at the guitar is to practice until the foundational patterns are deeply ingrained. The word “rote” has a bad rap in modern-day learning. But the reality is that rote practice, by which I mean routine practice that keeps the focus on what comes harder for you, plays an important role. The foundational patterns must be ingrained before you can begin to be creative.

Math is like that, too. As the researcher K. Anders Ericsson has shown, becoming an expert at anything requires the development of neural patterns that are acquired through much practice and repetition. Understanding is part of acquiring expertise, but it certainly isn’t all. But today’s “understanding-centered” approach to learning math, combined with efforts to make the subject more “fun” by avoiding drill and practice, shortchanges children of the essential process of instilling the neural patterns they need to be successful. And it may be girls that suffer most. All American students could benefit from more drilling: In the international PISA test, the United States ranks near the bottom among the 35 industrialized nations in math. But girls especially could benefit from some extra required practice, which would not only break the cycle of dislike-avoidance-further dislike, but build confidence and that sense of, “Yes, I can do this!” Practice with math can help close the gap between girls’ reading and math skills, making math seem like an equally good long-term study option. Even if she ultimately chooses a non-STEM career, today’s high-tech world will mean her quantitative skills will still come in handy.

All learning isn’t — and shouldn’t be — “fun.” Mastering the fundamentals is why we have children practice scales and chords when they’re learning to play a musical instrument, instead of just playing air guitar. It’s why we have them practice moves in dance and soccer, memorize vocabulary while learning a new language and internalize the multiplication tables. In fact, the more we try to make all learning fun, the more we do a disservice to children’s abilities to grapple with and learn difficult topics. As Robert Bjork, a leading psychologist, has shown, deep learning involves “desirable difficulties.” Some learning just plain requires effortful practice, especially in the initial stages. Practice and, yes, even some memorization are what allow the neural patterns of learning to take form.

Take it from someone who started out hating math and went on to become a professor of engineering: Do your daughter a favor — give her a little extra math practice each day, even if she finds it painful. In the long run, she’ll thank you for it. (And, by the way: the same applies to your son.)

Barbara Oakley is an engineering professor at Oakland University in Rochester, Mich., and the author of “Learning How to Learn.”

The link of the article: Make Your Daughter Practice Math. She’ll Thank You Later.

Photo from NY Times, Richie Pope

Mathematical surprises

As we know only few math discoveries are considered as shocking and unexpected results, in which open great gates instead of only windows, to the mathematician views. . Dave Richeson at “Division by Zero” has listed some of this kind of discoveries:

Mathematical surprises

I’m interested in compiling a list of “mathematical surprises.” The best possible example would be a mathematical discovery that no mathematician saw coming, but after it was discovered it changed mathematics in some fundamental way—Cantor’s discovery of the nondenumerability of the continuum is such an example. But I’ll settle for any surprise—Andrew Wiles surprised everyone with his proof of Fermat’s Last Theorem, the solution of the Monty Hall problem surprised many capable mathematicians, etc.

I’ve spent a couple days brainstorming and I’ve come up with the following list. Some are better than others, and they’re listed in no particular order. Please add your surprises in the comments below!

Paul Painlevé, a french mathematician and politician.

We have before published a post (hereOnly at France, a mathematician could be a politician too) about a french mathematician and politician, Cédric Villani, In which I am sure everybody knows him very well (his website: http://cedricvillani.org/ and Wikipedia: Cédric Villani ). He is not only a mathematician, but also a public figure trying to publicize science. He has traveled to Iran. unfortunately I couldn’t take participate in his public lecture. (you know why? because officials of IPM always publish the event news only few hours before the event!)

by the way, here and now I want to introduce you another successful french mathematician who had been successful in politics too: Paul Painlevé

From Wikipedia, the free encyclopedia: https://en.wikipedia.org/

Paul Painlevé (French: [pɔl pɛ̃ləve]; 5 December 1863 – 29 October 1933) was a French mathematician and statesman. He served twice as Prime Minister of the Third Republic: 12 September – 13 November 1917 and 17 April – 22 November 1925. His entry into politics came in 1906 after a professorship at the Sorbonne that began in 1892.

His first term as prime minister lasted only nine weeks but dealt with weighty issues, such as the Russian Revolution, the American entry into the war, the failure of the Nivelle Offensive, quelling the French Army Mutinies and relations with the British. In the 1920s as Minister of War he was a key figure in building the Maginot Line.[1] In his second term as prime minister he dealt with the outbreak of rebellion in Syria’s Jabal Druze in July 1925 which had excited public and parliamentary anxiety over the general crisis of France’s empire.[2]


Early life

Painlevé was born in Paris.[3]

Brought up within a family of skilled artisans (his father was a draughtsman) Painlevé showed early promise across the range of elementary studies and was initially attracted by either an engineering or political career. However, he finally entered the École Normale Supérieure in 1883 to study mathematics, receiving his doctorate in 1887 following a period of study at Göttingen, Germany with Felix Klein and Hermann Amandus Schwarz. Intending an academic career he became professor at Université de Lille, returning to Paris in 1892 to teach at the SorbonneÉcole Polytechnique and later at the Collège de France and the École Normale Supérieure. He was elected a member of the Académie des Sciences in 1900.[3]

He married Marguerite Petit de Villeneuve in 1901. Marguerite died during the birth of their son Jean Painlevé in the following year.[3]

Painlevé’s mathematical work on differential equations led him to encounter their application to the theory of flight and, as ever, his broad interest in engineering topics fostered an enthusiasm for the emerging field of aviation. In 1908, he became Wilbur Wright‘s first airplane passenger in France and in 1909 created the first university course in aeronautics.[3]

Mathematical work

Paul Painlevé as a young man

Some differential equations can be solved using elementary algebraic operations that involve the trigonometric and exponential functions (sometimes called elementary functions). Many interesting special functions arise as solutions of linear second order ordinary differential equations. Around the turn of the century, Painlevé, É. Picard, and B. Gambier showed that of the class of nonlinear second order ordinary differential equations with polynomial coefficients, those that possess a certain desirable technical property, shared by the linear equations (nowadays commonly referred to as the ‘Painlevé property‘) can always be transformed into one of fifty canonical forms. Of these fifty equations, just six require ‘new’ transcendental functions for their solution.[4] These new transcendental functions, solving the remaining six equations, are called the Painlevé transcendents, and interest in them has revived recently due to their appearance in modern geometry, integrable systems[5] and statistical mechanics.[6][7][8]

In 1895 he gave a series of lectures at Stockholm University on differential equations, at the end stating the Painlevé conjecture about singularities of the n-body problem.[9]

In the 1920s, Painlevé briefly turned his attention to the new theory of gravitation, general relativity, which had recently been introduced by Albert Einstein. In 1921, Painlevé proposed the Gullstrand–Painlevé coordinates for the Schwarzschild metric. The modification in the coordinate system was the first to reveal clearly that the Schwarzschild radius is a mere coordinate singularity (with however, profound global significance: it represents the event horizon of a black hole). This essential point was not generally appreciated by physicists until around 1963.[citation needed] In his diary, Harry Graf Kessler recorded that during a later visit to Berlin, Painlevé discussed pacifist international politics with Einstein, but there is no reference to discussions concerning the significance of the Schwarzschild radius.[10]

Early political career

Between 1915 and 1917, Painlevé served as French Minister for Public Instruction and Inventions. In December 1915, he requested a scientific exchange agreement between France and Britain, resulting in Anglo-French collaboration that ultimately led to the parallel development by Paul Langevin in France and Robert Boyle in Britain of the first active sonar.[11]

First period as French Prime Minister

Painlevé took his aviation interests, along with those in naval and military matters, with him when he became, in 1906, Deputy for Paris’s 5th arrondissement, the so-called Latin Quarter. By 1910, he had vacated his academic posts and World War I led to his active participation in military committees, joining Aristide Briand‘s cabinet in 1915 as Minister for Public Instruction and Inventions.[3]

On his appointment as War Minister in March 1917 he was immediately called upon to give his approval, albeit with some misgivings, to Robert Georges Nivelle‘s wildly optimistic plans for a breakthrough offensive in Champagne. Painlevé reacted to the disastrous public failure of the plan by dismissing Nivelle and controversially replacing him with Henri Philippe Pétain.[12] He was also responsible for isolating the Russian Expeditionary Force in France in the La Courtine camp, located in a remote spot on the plateau of Millevaches.[13]

On 7 September 1917, Prime Minister Alexandre Ribot lost the support of the Socialists and Painlevé was called upon to form a new government.[3]

Painlevé was a leading voice at the Rapallo conference that led to the establishment of the Supreme Allied Council, a consultative body of Allied powers that anticipated the unified Allied command finally established in the following year. He appointed Ferdinand Foch as French representative knowing that he was the natural Allied commander. On Painlevé’s return to Paris he was defeated and resigned on 13 November 1917 to be succeeded by Georges ClemenceauFoch was finally made commander-in-chief of all Allied armies on the Western and Italian fronts in March 1918.[3][12][14]

Second period as French Prime Minister

Painlevé then played little active role in politics until the election of November 1919 when he emerged as a leftist critic of the right-wing Bloc National. By the time the next election approached in May 1924 his collaboration with Édouard Herriot, a fellow member of Briand’s 1915 cabinet, had led to the formation of the Cartel des Gauches. Winning the election, Herriot became Prime Minister in June, while Painlevé became President of the Chamber of Deputies. Though Painlevé ran for President of France in 1924 he was defeated by Gaston Doumergue. Herriot’s administration publicly recognised the Soviet Union, accepted the Dawes Plan and agreed to evacuate the Ruhr. However, a financial crisis arose from the ensuing devaluation of the franc and in April 1925, Herriot fell and Painlevé became Prime Minister for a second time on 17 April. Unfortunately, he was unable to offer convincing remedies for the financial problems and was forced to resign on 21 November.[3][12][15]

Later political career

Paul Painlevé in the 1920s

Following Painlevé’s resignation, Briand formed a new government with Painlevé as Minister for War. Though Briand was defeated by Raymond Poincaré in 1926, Painlevé continued in office. Poincaré stabilised the franc with a return to the gold standard, but ultimately acceded power to Briand.[3] During his tenure as Minister of War, Painlevé was instrumental in the creation of the Maginot Line. This line of military fortifications along France’s Eastern border was largely designed by Painlevé, yet named for André Maginot, owing to Maginot’s championing of public support and funding.[citation needed] Painlevé remained in office as Minister for War until July 1929.[3]

Though he was proposed for President of France in 1932, Painlevé withdrew before the election. He became Minister of Air later that year, making proposals for an international treaty to ban the manufacture of bomber aircraft and to establish an international air force to enforce global peace. On the fall of the government in January 1933, his political career ended.[3]

Painlevé died in Paris in October of the same year.[12] On 4 November, after a eulogy by Prime Minister Albert Sarraut, he was interred in the Panthéon.[16]


Composition of governments

Painlevé’s First Government, 12 September – 16 November 1917


Painlevé’s Second Ministry, 17 April – 29 October 1925


  • 11 October 1925 – Anatole de Monzie succeeded Steeg as Minister of Justice. Yvon Delbos succeeded Monzie as Minister of Public Instruction and Fine Arts.

Painlevé’s Third Ministry, 29 October – 28 November 1925


  • Sur les lignes singulières des fonctions analytiques – 1887/On singular lines of analytic functions.
  • Mémoire sur les équations différentielles du premier ordre – 1892/Memory on first order differential equations.
  • Leçons sur la théorie analytique des équations différentielles, A. Hermann (Paris), 1897/A course on analytic theory of differential equations.
  • Leçons sur les fonctions de variables réelles et les développements en séries de polynômes – 1905/A course on real variable functions and polynomial development series.
  • Cours de mécanique et machines (Paris), 1907/A course on mechanics and machines.
  • Cours de mécanique et machines 2 (Paris), 1908/A course on mechanics and machines 2.
  • Leçons sur les fonctions définies par les équations différentielles du premier ordre, Gauthier-Villars (Paris), 1908/A course on functions defined by first order differential equations.
  • L’aéroplane, Lille, 1909/Aeroplane.
  • Cours de mécanique et machines (Paris), 1909/A course on mechanics and machines.
  • L’aviation, Paris, Felix Alcan, 1910/Aviation.
  • Les axiomes de la mécanique, examen critique ; Note sur la propagation de la lumière – 1922/Mechanics axioms, a critical study ; Notes on light spread.
  • Leçons sur la théorie analytique des équations différentielles, Hermann, Paris, 1897/A course on analytical theory of differential equations.
  • Trois mémoires de Painlevé sur la relativité (1921-1922)/Painlevé’s three memories on relativity.

See also


  1. Jump up^ Smart, Nick (1996). “The Maginot Line: An Indestructible Inheritance”. International Journal of Heritage Studies2 (4): 222–233. doi:10.1080/13527259608722177.
  2. Jump up^ Thomas, Martin (2005). “Albert Sarraut, French Colonial Development, and the Communist Threat, 1919–1930”. Journal of Modern History77 (4): 917–955. doi:10.1086/499830.
  3. Jump up to:a b c d e f g h i j k O’Connor, John J.Robertson, Edmund F.“Paul Painlevé”MacTutor History of Mathematics archiveUniversity of St Andrews.
  4. Jump up^ Painlevé, P. (1897). Leçons sur la théorie analytique des équations différentielles. Paris: Libraire Scientifique à Hermann.
  5. Jump up^ Ablowitz, M. J. and Clarkson, P.A. (1991) Solitons, nonlinear evolution equations and inverse scattering. Cambridge University Press
  6. Jump up^ Wu, T. T.; B. M. McCoy; C. A. Tracy; E. Barouch (1976). “Spin-spin correlation functions for the two-dimensional Ising model: Exact theory in the scaling region”. Physical Review B13: 316–374. Bibcode:1976PhRvB..13..316Wdoi:10.1103/PhysRevB.13.316.
  7. Jump up^ Jimbo, Michio; Tetsuji Miwa; Yasuko Môri; Mikio Sato (April 1980). “Density matrix of an impenetrable Bose gas and the fifth Painlevé transcendent”. Physica D1 (1): 80–158. Bibcode:1980PhyD….1…80Jdoi:10.1016/0167-2789(80)90006-8.
  8. Jump up^ Tracy, C. A.; H. Widom (1997). “On Exact Solutions to the Cylindrical Poisson-Boltzmann Equation with Applications to Polyelectrolytes”. Physica A244: 402–413. arXiv:cond-mat/9701067Freely accessibleBibcode:1997PhyA..244..402Tdoi:10.1016/S0378-4371(97)00229-X.
  9. Jump up^ Diacu, Florin N. (1993). “Painlevé’s Conjecture”. The Mathematical Intelligencer13 (2): 6. doi:10.1007/BF03024186.
  10. Jump up^ Harry Graf Kessler. “Berlin. 20. February 1925. Freitag” [Diary entry for Berlin 25 February 1925]. Projekt Gutenberg.
  11. Jump up^ Michael A. Ainslie Principles of Sonar Performance Modelling, Springer, 2010 ISBN 3-540-87661-8, page 13
  12. Jump up to:a b c d “Paul Painlevé” in Encyclopædia Britannica
  13. Jump up^ Cockfield, Jamie H. (1999). With snow on their boots : the tragic odyssey of the Russian Expeditionary Force in France during World War I (1st St. Martin’s Griffin ed.). New York: St. Martin’s Griffin. p. 137. ISBN 978-0312220822.
  14. Jump up^ Keegan, John (2003). The First World War. UK: Random House. p. 403. ISBN 0-7126-8040-3.
  15. Jump up^ “Édouard Herriot” in Encyclopædia Britannica
  16. Jump up^ “Painlevé To Be Buried in Pantheon Today”The New York Times. 4 November 1933. p. 13. Retrieved 29 August 2011.
  17. Jump up^ Polmar, Norman; Genda, Minoru (2006). Aircraft Carriers: A History of Carrier Aviation and Its Influence on World Events. Washington, DC: Potomac Books, Inc. p. 86. ISBN 978-1-57488-664-1.
  18. Jump up^ Schmadel, Lutz D.; International Astronomical Union (2003). Dictionary of minor planet names. Berlin; New York: Springer-Verlag. p. 84. ISBN 978-3-540-00238-3.

Further reading

  • Dutton, David (1981). “Paul Painlevé and the end of the sacred union in Wartime France”. Journal of Strategic Studies4 (1): 46–59. doi:10.1080/01402398108437065.
  • Greenhalgh, Elizabeth (2011). “Paul Painlevé and Franco-British Relations in 1917”. Contemporary British History25 (1): 5–27. doi:10.1080/13619462.2011.546094.

External links

Political offices
Preceded by
Alexandre Ribot
Prime Minister of France
Succeeded by
Georges Clemenceau
Preceded by
Édouard Herriot
Prime Minister of France
Succeeded by
Aristide Briand

Les Radio et TV Francais

Bonjour les amies et les amis

J’espère que vous allez bien aujourd’hui. J’écoute les radios francais d’améliorer mon francais. L’une d’elles est France Bleu Paris : très énergique radio que vous ne sentirez jamais que le temps passe. France Culture est l’autre radio que j’écoute beaucoup. J’adore les romans, les films, la music … donc je prends plaisir de l’écoute. Parmi les films j’ai déjà vu, j’aime spécialement la sérial « les revenants ».


If politicians were mathematicians

I found this post from 2010! still interesting and looks like an open problem.

If politicians were mathematicians

Before I start, let me get one thing over and done with: I fully admit that professional mathematicians are as capable as anyone else of making stupid collective decisions.

But I don’t want to imagine what the world would be like if it were run by mathematical researchers. I just wonder how much difference it would make if politicians understood enough mathematics to be able to understand an argument of more than one sentence. Or to put it more accurately, what would it be like if the following rules of political life were no longer accepted?

1. An argument that is slightly complicated but correct is trumped by an argument that is punchy, amusing, and wrong.

2. If option A is better than B in some respects and worse in others, then instead of weighing up the pros and cons, you decide which side you are on and then just mention the pros of the option you prefer and the cons of the other option.

3. If option A is better than B in every respect, but your party supports B, then you support B.

4. If one of your political opponents points out a flaw in your argument, then count to ten and repeat the flawed argument.

If that were the case, then one consequence would be that one could advocate new ways of doing politics and have them discussed seriously. In this post, I would like to mention a few ideas that would be dismissed as utter lunacy by any politician. But perhaps people who read this blog would be prepared to engage with them properly and weigh up the pros and cons. I’m sure there are cons — but I don’t think the ideas are utter lunacy.

I am not talking here of electoral reform, though I very much support some kind of change to the British voting system. Rather, I am talking about reform of the way that business is conducted in between elections. But before I suggest any ideas, let me discuss what I think is wrong about the current system (as it is in Britain, but I think the remarks can be generalized to many other democracies). Let us suppose that we have an ideal voting system: to please everybody, let’s suppose that it’s a first-past-the-post system that just happens to have delivered a wonderfully proportional result; and it has even delivered a good strong government, since one party has received 55% of the vote, and approximately 55% of the seats, giving it a comfortable majority.

What could possibly be wrong with that? In my view, at least two things. The way things work in Britain, the members of each political party agree that they will form a bloc and vote the same way on every issue. If you belong to a party and you think that one of your party’s policies is wrong, then you are faced with a choice. Either you vote against the party line, and become known as a rebel, jeopardizing your chances of advancement to higher office (if that is what you hope for), or you toe the line and support measures that you do not believe in.

There are of course good reasons for behaving this way. You join a party because you believe in its general principles, and the idea is that you compromise on some issues because that is the price to be paid for ensuring that the party has the political strength to make other decisions that you do believe in.

But this system can in principle lead to decisions being made that are not supported by anything like a majority of members of parliament. If the party with 55% of the vote puts forward a policy that is supported by 70% of its members (perhaps there is some committee that reflects perfectly the views of the party, and the policy is voted on in that committee) and opposed by everyone else, then it is supported by under 40% of MPs. But it is still implemented.

The second problem is one that is particularly serious in a country that has a large minority with very different interests from the majority. (This is often the case for ethnic or religious reasons, and has led to many of the worst and most persistent conflicts round the world.) Suppose that 30% of the country belongs to group A and 70% to group B. And suppose that there is a political party that represents people in group A and another political party that represents people in group B. And finally, suppose that the two groups dislike each other intensely. Then if the number of seats is roughly proportional to the number of votes, the party representing group B will have a large majority in government, which will allow it to advance the interests of group B at the expense of group A. For example, it could give all the powerful jobs to people from group B, pay for good infrastructure in regions where people in group B tend to live, and so on. This situation is sometimes referred to as the tyranny of the majority.

So far so good. Now comes the nutty bit. I would like to suggest two systems for parliamentary votes, one that would weaken the party system but without killing it off entirely, and one that would protect large minorities. Neither has the slightest chance of being adopted, because they are both too complicated to be taken seriously. But mathematicians wouldn’t find them complicated at all — hence the title of this post.

An obvious way to weaken the party system is to have secret ballots for every single parliamentary vote. That way, MPs could simply vote on every issue according to their judgment about that issue. I myself would like to see that. But it would kill off the party system almost completely. It would be criticized, with some justification, for making government virtually impossible: how could you plan ahead if every measure you proposed was in danger of being voted down? And what if somebody were to say one thing to get elected and then to vote in an entirely different way once they were elected? (Of course, entire political parties do that with their manifestos, but that’s another matter.) That would make a nonsense of representative democracy.

To meet that objection, I propose the following system. Votes are made electronically and then counted. After they are counted, the way people voted is made public. However, before that happens, each vote is changed, independently, with a certain probability such as 10% (but the precise value could be argued about, and might even vary from vote to vote, being lower for especially important votes). If you feel strongly that your party is wrong on a certain issue, then you can vote against it, and if that annoys the party whips, you can tell them that you voted for it but your vote was flipped. However, you cannot play this game too much, or the number of times your vote appears to be against the party line will be so far above 10% that it will be clear that you are not a loyal party member.

It is this last part that almost no politician would understand, since non-mathematicians have a strong aversion to the probabilistic method.

Now for a political system that would protect minorities and give power to political parties in rough proportion to the number of seats in those parties (and in particular not give 100% of the power to a party with over 50% of the seats). In this system, I shall assume that there is total loyalty within parties, so on each issue, each party can appoint a representative, who will know exactly what the party wants and will act on that knowledge. At the beginning of each year, all parties are given “credit” in proportion to their sizes. They also know how many votes there are going to be and have a good idea of how many will be very important, how many extremely minor, and so on. And then, instead of votes on different issues, there are auctions. If party A wants income tax to be raised, and party B does not, then whichever party offers to give up the most credit wins the auction and gets its way. If the issue is particularly important to both parties, the bidding may well go quite high, but if a party bids too much, then it is significantly weakened later in the year. (There may be better ways of implementing the basic idea, such as starting with a particular level of credit and continuously replenishing it at a certain rate.) And a minor party that cares deeply about one issue can save all its credit up for that issue. Perhaps it would also be permissible for two parties to get together and make a joint bid.

Note that under such a system, it would be difficult to push through a lot of controversial legislation, since your opponents would mind about it enough to force the bidding up to high levels. But if you could find compromises, then they would be cheaper, since it would not be worth your opponents’ while to spend much political capital opposing them. So the system would naturally encourage consensual politics.

To make the system more theatrical, each MP could be given a certain number of cards that was less than the number of votes to be held. If, say, there were 200 votes, then each MP could have 50 cards, each of which could be used as a vote. Then for each vote MPs would take turns: one would vote for, another against, another next for, and so on, until one side gave up. On average, a quarter of the members of each party would participate in any given vote, but they would not have to stick to the average for every single vote.

This system is a bit like a system that siblings sometimes use when their parents have died and they want to share out a number of objects of sentimental value. If they have also been left money, then they can make bids for the various items and then take those bids into account when they share out the money. It can, I am told, be a good way of avoiding acrimony.

When I thought of the above idea (I make no claim to be the first to do so, by the way), I worried that it would have the potential to lead to game playing of an undesirable kind. And since writing the last few paragraphs I have realized that indeed it does have that potential. One party could suggest an outrageously unfair piece of legislation that would be disastrous for the people represented by the other party, and then bid it up in order to force the other party to waste valuable credit ensuring that it does not pass. For instance, a party that principally represents voters in a certain region could propose a hugely expensive program of improving public transport in that region, paid for by taxpayers all round the country. Or in a country with more than one language, a party that represents speakers of one language could propose a motion to ban all use of other languages in schools.

At the time of writing, I have not come up with a good system for dealing with this problem. The difficulty I have is that the obvious ideas seem to involve having to have some measure of how “reasonable” a piece of proposed legislation is, in order to attach a cost to proposing it, whereas I was looking for a system where the cost would be determined automatically by the “market forces” that arise from the need to spend political credit.

So let me conclude, slightly limply, with the assertion that it seems wrong for a majority to be able to call all the shots, and that if one does not care about simplicity then it ought to be possible to devise a system that does not have this defect.

It is worth mentioning that in many countries with sharp ethnic or religious divisions, minorities are guaranteed ministerial posts. That is a crude way of sharing out power more fairly: I am wondering whether there are other ways.

Reference: https://gowers.wordpress.com

Math is everywhere, is it not?

What I have seen whole during my life is this fact that Math appears everywhere. You could find its footprint in every new revolutionary discover or invention in any branch of human’s thought.  Some are confused why math dose present in every aspect of our life, is it a trick? is it a misunderstanding? or it is just mathematician want to prove themselves in any possible way? in one the previous posts at this blog (math is not science) we argued why math could be more humanities rather experimental science.

Now I invite you to read this article from https://blogs.scientificamerican.com

Mathematicians Are Overselling the Idea That “Math Is Everywhere”

The mathematics that is most important to society is the province of the exceptional few—and that’s always been true

Most people never become mathematicians, but everyone has a stake in mathematics. Almost since the dawn of human civilization, societies have vested special authority in mathematical experts. The question of how and why the public should support elite mathematics remains as pertinent as ever, and in the last five centuries (especially the last two) it has been joined by the related question of what mathematics most members of the public should know.

Why does mathematics matter to society at large? Listen to mathematicians, policymakers, and educators and the answer seems unanimous: mathematics is everywhere, therefore everyone should care about it. Books and articles abound with examples of the math that their authors claim is hidden in every facet of everyday life or unlocks powerful truths and technologies that shape the fates of individuals and nations. Take math professor Jordan Ellenberg, author of the bestselling book How Not to Be Wrong, who asserts “you can find math everywhere you look.”

To be sure, numbers and measurement figure regularly in most people’s lives, but this risks conflating basic numeracy with the kind of math that most affects your life. When we talk about math in public policy, especially the public’s investment in mathematical training and research, we are not talking about simple sums and measures. For most of its history, the mathematics that makes the most difference to society has been the province of the exceptional few. Societies have valued and cultivated math not because it is everywhere and for everyone but because it is difficult and exclusive. Recognizing that math has elitism built into its historical core, rather than pretending it is hidden all around us, furnishes a more realistic understanding of how math fits into society and can help the public demand a more responsible and inclusive discipline.

In the first agricultural societies in the cradle of civilization, math connected the heavens and the earth. Priests used astronomical calculations to mark the seasons and interpret divine will, and their special command of mathematics gave them power and privilege in their societies. As early economies grew larger and more complex, merchants and craftsmen incorporated more and more basic mathematics into their work, but for them mathematics was a trick of the trade rather than a public good. For millennia, advanced math remained the concern of the well-off, as either a philosophical pastime or a means to assert special authority.

The first relatively widespread suggestions that anything beyond simple practical math ought to have a wider reach date to what historians call the Early Modern period, beginning around five centuries ago, when many of our modern social structures and institutions started to take shape. Just as Martin Luther and other early Protestants began to insist that Scripture should be available to the masses in their own languages, scientific writers like Welsh polymath Robert Recorde used the relatively new technology of the printing press to promote math for the people. Recorde’s 1543 English arithmetic textbook began with an argument that “no man can do any thing alone, and much less talk or bargain with another, but he shall still have to do with number” and that numbers’ uses were “unnumerable” (pun intended).

Far more influential and representative of this period, however, was Recorde’s contemporary John Dee, who used his mathematical reputation to gain a powerful position advising Queen Elizabeth I. Dee hewed so closely to the idea of math as a secret and privileged kind of knowledge that his detractors accused him of conjuring and other occult practices. In the seventeenth century’s Scientific Revolution, the new promoters of an experimental science that was (at least in principle) open to any observer were suspicious of mathematical arguments as inaccessible, tending to shut down diverse perspectives with a false sense of certainty. During the eighteenth-century Enlightenment, by contrast, the savants of the French Academy of Sciences parlayed their mastery of difficult mathematics into a special place of authority in public life, weighing in on philosophical debates and civic affairs alike while closing their ranks to women, minorities, and the lower social classes.

Societies across the world were transformed in the nineteenth century by wave after wave of political and economic revolution, but the French model of privileged mathematical expertise in service to the state endured. The difference was in who got to be part of that mathematical elite. Being born into the right family continued to help, but in the wake of the French Revolution successive governments also took a greater interest in primary and secondary education, and strong performance in examinations could help some students rise despite their lower birth. Political and military leaders received a uniform education in advanced mathematics at a few distinguished academies which prepared them to tackle the specialized problems of modern states, and this French model of state involvement in mass education combined with special mathematical training for the very best found imitators across Europe and even across the Atlantic. Even while basic math reached more and more people through mass education, math remained something special that set the elite apart. More people could potentially become elites, but math was definitely not for everyone.

Entering the twentieth century, the system of channeling students through elite training continued to gain importance across the Western world, but mathematics itself became less central to that training. Partly this reflected the changing priorities of government, but partly it was a matter of advanced mathematics leaving the problems of government behind. Where once Enlightenment mathematicians counted practical and technological questions alongside their more philosophical inquiries, later modern mathematicians turned increasingly to forbiddingly abstract theories without the pretense of addressing worldly matters directly.

The next turning point, which continues in many ways to define the relations between math and society today, was World War II. Fighting a war on that scale, the major combatants encountered new problems in logistics, weapons design and use, and other areas that mathematicians proved especially capable of solving. It wasn’t that the most advanced mathematics suddenly got more practical, but that states found new uses for those with advanced mathematical training and mathematicians found new ways to appeal to states for support. After the war, mathematicians won substantial support from the United States and other governments on the premise that regardless of whether their peacetime research was useful, they now had proof that highly trained mathematicians would be needed in the next war.

Some of those wartime activities continue to occupy mathematical professionals, both in and beyond the state—from security scientists and code-breakers at technology companies and the NSA to operations researchers optimizing factories and supply chains across the global economy. Postwar electronic computing offered another area where mathematicians became essential. In all of these areas, it is the special mathematical advances of an elite few that motivate the public investments mathematicians continue to receive today. It would be great if everyone were confident with numbers, could write a computer program, and evaluate statistical evidence, and these are all important aims for primary and secondary education. But we should not confuse these with the main goals and rationales of public support for mathematics, which have always been about math at the top rather than math for everyone.

Imagining math to be everywhere makes it all too easy to ignore the very real politics of who gets to be part of the mathematical elite that really count—for technology, security, and economics, for the last war and the next one. Instead, if we see that this kind of mathematics has historically been built by and for the very few, we are called to ask who gets to be part of that few and what are the responsibilities that come with their expertise. We have to recognize that elite mathematics today, while much more inclusive than it was one or five or fifty centuries ago, remains a discipline that vests special authority in those who, by virtue of gender, race, and class, are often already among our society’s most powerful. If math were really everywhere, it would already belong to everyone equally. But when it comes to accessing and supporting math, there is much work to be done. Math isn’t everywhere.



I just by accident find this link which is about STATISTICS AND POLITICS. I don’t know what the topic actually is, but I guess it’s mainly statistics of undergraduate and its applications in behavioral sciences. I put it here only I think it might somebody among my WordPress friends need it.


Statistics is used in two most common ways: as a tool to describe the world around us, and as a tool used to infer something we have limited information.

Descriptive Statistics; Basic Statistical graphs

While there are several different ways in which raw data can be presented to a reader, most commonly it is done using Line Graphs, Bar Graphs, Pie Charts and Histograms.

For example, the following graphs from Gallup Organization[1], a national

Do you consider the amount of federal income taxes you have to pay as too high, about right, or too low?

Talk about symmetry:




Most advertisements, political debates, and other public forms of argument are riddled with fallacies, that is, with persuasive but invalid forms of logic. A familiarity with the most common fallacies is important if one is to make informed decisions and generate rational opinions. Although there are a great many fallacies, the ones listed here are used most often.

Fallacies of Relevance:

Subjectivism: “I believe/want something to be true, therefore it is true.” This fallacy is often characterized by a refusal to consider evidence contrary to a cherished belief, and in some books is called “the Fallacy of the Irrefutable Hypothesis.” This is very common in conspiracy theories. Examples:

“No one with any compassion could possibly believe it is okay to torture animals.”

When the Murrah Federal Building in Oklahoma was bombed, many militia groups claimed the government knew it was going to happen since the FBI had infiltrated their organizations. When it was asked why, in that case, the authorities didn’t evacuate the building, it was replied that the government allowed it to kill so many people in order to discredit the milita movement and turn public sentiment against it. Some milita spokespeople even suggested that the FBI had done the bombing itself to achieve this purpose. These militia members have an irrefutable hypothesis.

Appeal to Ignorance: “If it can’t be proved false, it must be true.” This fallacy is often disguised by the use of very plausible language. Examples:

“Despite thousands of so called ‘sightings,’ no hard evidence for UFO’s has ever been produced. Therefore UFO’s don’t exist.”

“Thousands of studies and millions of dollars worth of research have turned up no cause for Gulf War Syndrome. Consequently, we must conclude that this so called ‘syndrome’ is not connected with having served during the Gulf War.”

This fallacy is closely related to the Fallacy of Limited Choice (or False Dichotomy), in which an argument forces a conclusion based on an artificial “either/or” statement. Examples:

“If you’re against the war, you’re not supporting our sons and daughters in uniform.”

“If you’re so smart, why aren’t you rich?”

“We must either support animal testing, or give up in the war against cancer.”

Appeal to Emotion: This fallacy attempts to evoke an emotional response to convince the listener, and is extremely common in debates over emotionally charged issues like vivisection, abortion, and capital punishment. In can involve not only verbal argument, but the use of graphic and disturbing pictures or other media, for example of animals suffering in a laboratory. Examples:

“Someone who would mutilate and kill a little child deserves to be killed – I’d throw the switch myself with a smile.”

This fallacy is closely related to the Appeal to Force, which is an attempt to get one’s point of view accepted by an explicit or implied threat. Example:

“If you don’t vote with the others, your place on this committee could be placed in jeopardy.”

Inappropriate Appeal to Authority: This is an attempt to get a position accepted by appealing to an inappropriate or unqualified authority. This is extremely common in advertising, when some well-known public figure is induced to promote a product. It can also occur when an authority is corrupt, as when, in Nazi Germany, citizens were persuaded to go along with many public policies they found abhorrent because they were dictated by the state. (This example also involved the Appeal to Force, since there was an ever-present implied threat against dissenters. This is the situation under many governments in the world today.)

Personal Attack (Ad Hominum ): A very common technique, in which one challenges one’s opponent personally, rather than his arguments. Examples:

“That’s something only a bleeding-heart liberal would believe.”

“Why shouldn’t I hunt animals if I want? You eat meat, don’t you?”

“Anyone who would support that has no feelings, no heart.”

Any time an argument impugns an opponent’s motives, veracity, or qualifications, it is an ad hominum attack, meant to shift the debate away from considering the merits of the argument itself.

Begging the Question: This occurs more often than one would suspect, and I’ve noticed that many people argue by use of this fallacy simply out of habit, and are quite surprised when their circular reasoning is pointed out to them. It generally involves a statement of the form, “this is true because it’s true.” Examples:

“Animals shouldn’t be made to suffer to further scientific knowledge, because making animals suffer is wrong.”

“Everyone in a wealthy society like ours has a right to health care, therefore health care should be made universally available.”

“Freedom of speech is an essential right in a free society, since everyone should have the right to express him or herself with complete freedom.”

Non Sequitor: This fallacy is often hard to recognize, because it so often expresses a common bias or appeals to a popular sentiment. It occurs anytime the premises and conclusion of an argument are essentially unrelated. Examples:

“A strong national defense is essential to our freedom. Therefore, we should fund production of the MX missile.”

“Sex crimes are often the result of an unrestrained libido, so castration is an appropriate punishment for such crimes.”

“Two-thirds of prison overpopulation is a result of long sentences for drug offenses. Drugs should be legalized.”

“Almost all drug addicts got started by using milder drugs like marijuana. Therefore marijuana should not be legalized.”

 False Cause (Post Hoc, Ergo Propter Hoc): The second event followed the first event; therefore the first event caused the second event. The fallaciousness of this reasoning is obvious, yet it is a remarkably easy fallacy to commit, and it is used often in politics. Examples:

“After my opponent took office, the economy plummeted. A vote for me is a vote for restoring the economic engine of this country.”

“On our watch, the crime rate has gone down. Obviously, our policies are succeeding in the war against crime.”

“After the social liberalization of the 1960’s, social ills such as teen pregnancy, urban violence, and rudeness in public discourse skyrocketed. We need to restore some traditional values.”

Complex Question: This is the famous “have you stopped beating your wife?” device, in which a conclusion is forced by disguising it in the body of a question. You should be especially suspicious of any question that demands a “yes or no” answer. Examples:

“Is my opponent prepared to renounce negative advertising?”

“Will you commit yourself to greater equity in employment by endorsing my position on affirmative action?”

“Does your procrastination cause your grades to suffer?”

Fallacies of Numbers and Statistics:

Appeal to Popularity (Numbers): This is the “everybody does it” fallacy so often used by school children, to which the wise parent invariably replies, “and if everybody jumped off a cliff, would you do that too?” Examples:

“We have the best selling (fill in the blank) in America. (So you should buy one too.)”

“So many people have seen the Loch Ness Monster – there must be something down there.”

Hasty Generalization: This involves drawing a general conclusion on the basis of anecdotal evidence. (It is similar to post hoc, ergo propter hoc.) Examples:

“I’ve never done well in my math classes. I just don’t have a ‘math brain.'”

“Two 747’s were involved in air disasters in the past year. It must not be a safe plane.”

Bias and Availability Error: This is very common in opinion polls, and boils down to the fact that how people will respond to a question often depends very much on how the question is phrased.

These examples by no means exhaust the many varieties of common fallacy, but they are representative of the most common kinds in contemporary advertising and propaganda. Becoming astute at noticing these fallacies when they are used against you is an excellent inoculation against being persuaded to accept ideas and conclusions based on flimsy or deceitful arguments.

From http://www.iun.edu

Brazil’s top math youtuber: ‘Math Maniac’

We should accept that YouTube has being become more and more important website in education. I myself has learnt French only from YouTube. YouTube has a vital role in math education and sharing advanced lecture videos. You don’t need to be at MIT to benefit those lectures! just go online and search in YouTube.

Here I am to introduce you a channel in Math: ‘Math Maniac’

Brazil’s top math youtuber: ‘Math Maniac’
Brazil’s top math youtuber: ‘Math Maniac’

With bright pink hair, it’s difficult to miss Julia Jaccoud walking down the corridors of the ICM conference, always with a camera, microphone, and tripod in hand.  She always wears her branded t-shirts too (this week’s shirt says Seja Curioso, ‘be curious’ in Portuguese).

Many international delegates will have seen the pink-haired woman at ICM, but she’s an internet icon in Brazil.  Matemaníaca (math maniac) is a two-year-old youtube channel, with more than 47,000 followers, where Jaccoud produces weekly videos that promote the learning of math.  “I wanted a fun name because math is very difficult to get people interested in. I also wanted a way to express myself,” said 24-year-old Jaccoud.  Hers is the largest math-focused youtube channel in Brazil.  While other math channels introduce math tricks found on major tests, the main objective of Matemaníaca is to get people interested in math.

Recently graduated with a degree in math education at the University of São Paulo (she missed her graduation ceremony to attend ICM), Jaccoud started producing math videos because she wanted to build on relationships developed with younger students in her student teaching assignments.  She also felt there was a bigger audience awaiting her online.  “I’m not an alien. I’m a mathematician,” Jaccoud said. “I thought if they could talk with me and be my friends online then I could introduce young people to math.”

Her Matemaníaca journey began in 2014 when she filmed a video introducing herself and her passion for math.  Her follow-up video, a tutorial on how to play Tangram, a Chinese dissection puzzle,  has been viewed more than 55,000 times. Other popular video topics include Brazilian public school math olympiad (OBMEP), math-intensive careers, and a tutorial on how to use probability to guess on exams.  He hair color evolves along with her video content.  “My channel is like art, and that is the way I communicate,” she said.

Since her youtube premiere, the 164 videos on her channel have amassed more than 1.3 million views. Most of her viewers are men aged 18-34, but she said her female audience tends to be more engaged and passionate.  Her university professors even use her videos in their classes.

At ICM 2018, Jaccoud mobilized many young volunteers with videos about the event. She has attended the international math congress every day, interviewing many of the world’s top mathematicians. Her newest video gives an overview of the World Meeting for Women in Mathematics (WM)², a satellite event held on the eve of ICM, and Jaccoud’s favorite event thus far (She dyed her hair pink to match the event’s colors).

Although Jaccoud has yet to earn a living from her online fame, she is happy to do something she loves. She launched a Padrim crowd-funding campaign this year, so her followers can directly fund her youtube channel (and so she can hire someone to edit videos). She also designs and sells her own t-shirts (the Seja Curiosa t-shirt is available here).  She plans to further specialize in math communications with a masters degree.

Read more

‘A mathematician without a dream is not a mathematician’

‘A mathematician without a dream is not a mathematician’

An article from ICM2018.ORG

Crowds of international delegates gathered around Caucher Birkar following his plenary presentation at Riocentro this afternoon. Everyone wanted a photo with the Kurdish mathematician, who was awarded a Fields Medal for his contribution to the minimum model program in algebraic geometry last Tuesday.

Chairing the packed plenary session at ICM 2018 this afternoon Jungkai Alfred Chen from the National Taiwan University said he was “fortunate to know him in person for a long time,” as Birkar was a visiting scholar at his university from 2014 – 2016. “He’s always very devoted and willing to share his mathematical ideas. I’m proud that his project was made in Taiwan.”

Birkar is known for his creative approach to math and algebraic geometry and Chen referred to the recent work of his colleague as a “huge breakthrough in birational geometry”.   Recounting some of Birkar’s early life experience as a toddler in war-torn Kurdistan, before seeking refuge in the United Kingdon, Chen said: “he has a very inspiring story, especially for those young people in difficult place, in a difficult situation.”

Algebraic geometry is about studying spaces, defined by polymonial equations (these spaces are called varieties). “Roughly speaking, birational algebraic geometry is about classifying these spaces to put them into groups, up to birational automorphism – that just means you allow a little bit more flexibility of working with these spaces, you can modify small parts of these spaces, but leaving most of it intact,” explained Birkar.  “The main goal is to show that each of these classes you can find some representative, which is simpler in a sense than others. The minimum model program is exactly this process of finding these simpler spaces.

After his return from Rio, Birkar plans to continue to focus his research on problems in birational geometry for now. “There are many interesting and very difficult problems, but maybe in the far future, I will also look at areas close to birational geometry, maybe also analytic geometry.”  He referred to a chat he had the previous day with geometry guru Michael Atiyah (89), who was sitting in the front row at the plenary session this afternoon. The 1966 Fields Medal winner told him “a mathematician without a dream is not a mathematician.”

Read more:

Behforooz: If math is not fun, what’s the point?

We have before talked about math teaching in schools here (Why math is so difficult?). At that post a fact about Iranian teachers’ habits had been explained. I myself have had this experience that how much inspiration and motivation is important. But what I have experienced in Iran schools: you have got a good situation or you haven’t got a good situation. If your total situation (family background of academic education, family income and etc.) is good enough to support you, then educational system will work for you. If you haven’t got a good situation then all doors are closed to you. nobody will help you.  Read the whole story, especially when he says:  The 77 year-old professor who has been at Utica for nearly four decades said there’s no way to make a student like mathematics.  Put simply, those that love the discipline will do best.   He used crochet as an example.

Behforooz: If math is not fun, what’s the point?

Mathematics should be fun, according to Hossein Behforooz.   Students shouldn’t be scared.  If it’s not fun, what’s the point?  “In more than half of every class we have, there is this kind of phobia. People think that math is difficult, that math is hard,” explained the mathematics professor at Utica College in New York. “Math is a way to prove something and to make it fun, not to scare students.”

Unlike some ICM  colleagues, the Iranian doesn’t agree that it’s the teacher’s role to inspire or motivate students with their personal passion.  The 77 year-old professor who has been at Utica for nearly four decades said there’s no way to make a student like mathematics.  Put simply, those that love the discipline will do best.   He used crochet as an example.  “If you do not like it, you cannot make it. But when you like it, every knot is beautiful,” he said. “It’s art, not science.”

But, Behforooz’s chosen subject area fascinates both mathematicians and non-mathematicians.   A large part of the scholarship he produced is in an area called magic squares, containing grids with special arrangements of numbers in them.   Magic squares earned their name in ancient times, due to associations with magic and the supernatural.  The earliest record of a magic square is from China (circa 2200 BC), when Emperor Yu saw a magic square in the shell of a divine tortoise. In the West, magic squares were first used in the work of Theon of Smyrna, and by Arab astrologers in 8 AD.  They were later seen in the writings of Greek mathematician Moschopoulos in the 14th century.

Magic squares retain their popularality as tools to help students solve and practice addition problems. Their arrangements are special because every diagonal, row and column add up to the same number.  Behforooz said Brazil is clearly on the right path. “I heard that there are more than 500 Math Olympiad medallists here,” he said. “That is great, that is fantastic.”

Hosting the International Congress of Mathematicians in the southern hemisphere will reinforce positive ideas around math.  It gives the message that mathematics is both accessible and fun for any student, anywhere in the world, Behforooz said.   He hopes to meet an younger cohort of mathematicians at the next ICM in Russia.

“This is my ninth time to ICM. I hope I will be alive to go to the tenth one in Russia!” he laughed. “It’s very prestigious here [in Rio]. You had the Olympics, and you had the World Cup.  The IMC was another chance to show that high achievement isn’t not just for the upper hemisphere.”


Only at France, a mathematician could be a politician too

Only at France, a mathematician could be a politician too: cederic vilani, the French mathematician who wins the 2010 fileds medal, Has Become a Crucial Political Figure in France, as https://www.bloomberg.com reported.

Read the complete article:

To hear Cedric Villani tell it, the French are better than everyone else at love, wine — and math.

A winner of the Fields Medal — the Nobel Prize equivalent for mathematics — Villani has in less than a year risen to become a key political figure in France with the ear of the tech-savvy President Emmanuel Macron. On Thursday, Villani takes center-stage when he unveils the country’s Artificial Intelligence strategy, aimed at putting his claim of France’s mathematical superiority to work in the global battle for emerging disruptive technologies.

“There is a deficit of contact between science and politics,” the 44-year-old said in an interview. “It’s part of my job to reinforce that link. It will be France’s role to lead the rest of Europe.”

Villani is an unlikely warrior in Europe’s AI battle, trying to take on China and the U.S. that are leaps ahead. The skinny scientist and lawmaker with his penchant for Gothic suits, giant frilly bow-ties favored in the late 19th century and bespoke spider-shaped brooches often draws more attention for the way he looks than for what he has to say.

Yet Macron is relying on Villani to help his modernization push by being one of the new — more optimistic — faces of France, a role the scientist has embraced with gusto. His 150-page AI report comes on top of the work he’s done on crafting a new and better way of teaching math in the country and as he prepares his next project that will involve reviewing France’s pedagogical techniques and reflects on data privacy.

New Level

Next month he’ll travel with Macron to the U.S., after having visited China with him in January. He was among the key speakers at a pre-Davos gig organized by Macron at the Versailles Palace in January to show foreign investors, including Google Chief Executive Officer Sundar Pichai, that science was now at the core of France’s ambitions.

“Villani brings France’s policies up to a whole new level of knowledge and thinking, and it seems fair that he is given even more help to do his scaling up,” said Andre Loesekrug-Pietri, an investor who launched the Joint European Disruption Initiative with major European scientific figures to accelerate investment in fundamental research. He and Villani studied together at the top Paris school of Louis-Le-Grand.

Ever since Villani won the Fields medal in 2010, the soft-spoken math whiz has endeavored to make math a part of the conversation in France and to bring more science to politics. Mathematics has taken him from Paris’s prestigious Ecole Normale Superieure, to stints at Berkeley University and Princeton University and to the helm of the French capital’s Institut Henri Poincare, the world-renowned mathematical center.

Political Life

His mathematics research fellow, Giuseppe Toscani of the University of Pavia in Italy, recalls Villani’s phenomenal ability to synthesize everything. The two men published research together in the late 1990s.

“He has the (almost unique) characteristic to be the best at anything he takes on,” Toscani said in a written response to questions. “Mathematics is one, among others. From that point of view, I am sure he will make important contributions in his new political life.”

Villani is part of Macron’s effort to change France’s political landscape, drawing into parliament people who are not professional politicians. The scientist has attempted to be more than just a new face. A fan of Marvel Comics’s Amazing Fantasy, Villani abides by the superhero’s mantra that “with great power comes great responsibility.”

France doesn’t have a Science Advisory Committee like in the U.S. The French prime minister is supposed to have a similar body, but Villani notes, “it hasn’t been used in a long, long time.”

‘Sexy’ Math

Villani, who carries a pocket watch at all times and a giant, full and often half-open backpack, is a busy man. His aides talk about their boss’s extreme multi-tasking: he writes with one hand, types with the other all while speaking on the phone.

“I must do everything at the same time, that’s the difficulty,” Villani said.

The scientist is also contributing to a much-debated government plan to revise the constitution, which has taken him into uncharted and controversial waters. For the most part, though, he’s sticking to his real passion — making France the place to be for math and science.

In a June 2016 TED Talk about why his field of study is “so sexy,” Villani joked about French people’s reputation and added more seriously that Paris has more mathematicians than any other city in the world.

“What is it that French people do better than all the others? If you take a poll, the top three answers might be: love, wine and whining… Maybe. But let me suggest a fourth one: mathematics.”

It was a bloomberg.com article; you can see the original page here

Why math is so difficult?

Is math really difficult?

You know, when I was a middle school student, our math teachers at Iran were mostly like horror movie characters: they were bad-tempered; they gave us hard problems. They were very sensitive about the homework, the cleanness of the book, and the absolute silence all during the class. Whenever we addressed to go to the board, we were supposed to know the answer of any given problem. If any other cases, unless the standards of the class, happened the punishment was very dreadful. So much stress and physical punishments. However, generally teachers were also very talented and smart, motivated and knowledgeable. The common thing that students tried to do was being silent and organized to try stay far from the punishment time. In addition, there were few math geniuses at class. The teacher, up to the end of the 2nd session, discovered them. In addition, from the third session, it was like this: everybody was silent writing what was being written on the board by teacher or those 3 – 4 geniuses. Two separated part in the class, the genius and teacher part and the stupid part (as teachers call those students). So, yes! I was in the light part of the class. I was so much better than anybody else in the math class was. Nevertheless, it is not the complete story. I started math major at high school and I was lucky enough to face with some inspiring teachers whom were kind and very patients with students. Then I changed very fast. I have had this experience in my all teaches from the first time of teaching so far: the problem is not to be dumb! The problem is not to be successful in making communication with mathematics! Math is not a bunch of pure contents, but also it is a language, it is a way of thinking and it is like three-dimensional glass in a world that people can see only two-dimensional. Anyway, I have always tried to inspire my students and to teach them the logic of the math. This is the solution. In continue, you are about to read a post from beetleypete blog which is an honest confess. We teach same book at the same rate by using same words and to different students. Different people have got different educational backgrounds, different emotional experience about math and they have different rates in learning.  So all together I am coming to this conclusion that it is our duty as math teachers to teach students how to learn math, how to understand the language and the logic of the math. I think you may feel familiar with this memory:


I have recently posted about the study of both History and Geography, so though I would continue that theme with something I was not at all good at, Maths. Short for Mathematics, and simply called ‘Math’ in the USA, most of us in Britain know this school subject as ‘Maths’.

When I started school at the age of five, I was taught simple counting. Using blocks, toys, or any other accessory, I soon learned how to count up to ten and more, along with my classmates. Then easy addition, nothing too complex for my developing mind. By the time I went to Junior School, aged seven, rote learning was still popular, and we were soon getting to grips with our ‘times tables’, to form the foundations of simple multiplication. This was 1959 of course, so no calculators, and not a thought of the computers to come. Just a teacher writing numbers on a board, and conducting our recital like a band leader.
“Once five is five.
Two fives are ten.
Three fives are fifteen,
Four fives are twenty”.
And so on.

We went as far as the number thirteen, stopping there for reasons best known to the teacher. Division was also introduced, often helped along by the use of counters or visual aids, as I learned that four into twenty makes five. Then around the age of nine, that ‘Eureka’ moment, when I suddenly got the connection between multiplication and division. We also tackled currency, as at that time we still used pounds, shillings and pence, with twelve pence to a shilling, and twenty shillings in a pound. Not that I ever had much cash, but it was good to know what change to expect when I bought something. We were also using rulers, and learning how to measure short distances.

When I was eleven, it was time to go to secondary school, and begin the exam syllabus. I had a list of things I would need just for Maths lessons; this included a set of compasses, a protractor, a triangle and a ‘proper’ ruler, with measurements down to 1/16th of an inch. The first real lesson was a double period, (why was Maths always a double?) and it hit me like a whirlwind. Algebra? Geometry? Even something called Trigonometry. I thought the teacher must be talking a foreign language, but she assured us that was all to come. Meanwhile, we were hit with some serious long division. That alone was enough to make my brain ache, and I watched my ‘working out’ get further and further down the page as I struggled with something like 295 divided by 16. By the time the first month of the new school was over, I had decided that I really didn’t like Maths, and was sure I would never be good at it.

And I was right.

Then came ‘Problems’. Things like, “If a two hundred gallon water tank has a leak of a quarter of a pint a day for ten days, then half a pint a day for twelve days, how much water will be left after twenty-two days?” I didn’t even know where to start, and my hand was soon up, informing the teacher that I didn’t have a clue. Even when she showed me how to work out the solution, I still got the answer wrong. It all got worse once we started with Algebra. “If X = ? and Y = ?, what is XY squared? ” I just laughed. There was no chance I got any of that at all. The teacher later explained that X and Y had a value and it could be anything I wanted on that occasion. X could be 2 and Y 6, for example. My reply was not well-received. “Please Miss, then why don’t you just write a 2 and 6?” I was told in no uncertain terms that I was being deliberately ‘stupid’.

But I wasn’t.

Later, we were given a complex book of numbers, called ‘Logarithms’. This baffling table introduced us to decimal points and such, but might just as well have been Sanskrit, for all my brain could take it in. I wasn’t getting any better, and had to face the next year, when it was all going to get harder. Double Maths changed to a Monday morning when I was twelve, and I began to dread the walk to school,, shuffling with the reluctance of a condemned man about to be hanged. I still had the same teacher, the formidable Mrs Widdowson, who could freeze me with one of her signature glares, and had given me a terrible entry on my end of term report the previous year. Inside, I considered I was doing alright. All the other subjects were going great. I was in the top set for English, Geography, French, History, and even Religious Education, something I had little interest in. So what if I didn’t really ‘get’ Maths? It wasn’t the end of the world, as far as I was concerned.

So, I muddled along. Bad reports, bottom section of the class, and never truly understanding anything new. I did well at everything except Maths, and that was enough for me. When it came to the final exams, I just scraped though the Maths one with a Grade Four, a ‘just passed’ result. But it wasn’t all bad. That early learning left me able to recall the times table instantly, work out money without hesitation, and even able to calculate foreign currency exchanges, on my trips abroad. These days, i see young peope reach for a mobile phone, when faced with the most basic sum to work out.

Maybe we need to go back to chanting the times tables, and using a ruler?

Photo by Jess Watters on Unsplash


ICM 2018

Dans cette page d’ams.org, vous allez trouver quelques liens sur ICM 2018:

2018 Fields Medal Media Coverage 

Mathematicians Caucher BirkarAlessio FigalliPeter Scholze, and Akshay Venkatesh were awarded Fields Medals at the 2018 International Congress of Mathematicians in Rio de Janeiro, Brazil. Read more about the mathematicians and their work. 


Continue reading “ICM 2018”

2018 Fields Medal Media Coverage

At this link of ams.org you can find a set of some links of articles and news about ICM 2018. I find it a good idea to have all these links in one single post:

2018 Fields Medal Media Coverage 
August 07, 2018

Mathematicians Caucher BirkarAlessio FigalliPeter Scholze, and Akshay Venkatesh were awarded Fields Medals at the 2018 International Congress of Mathematicians in Rio de Janeiro, Brazil. Read more about the mathematicians and their work.

“A Number Theorist Who Bridges Math and Time,” by Erica Klarreich, Quanta Magazine, August 1, 2018;

“A Master of Numbers and Shapes Who Is Rewriting Arithmetic,” by Erica Klarreich, Quanta Magazine, August 1, 2018;

“An Innovator Who Brings Order to an Infinitude of Equations,” by Kevin Hartnett, Quanta Magazine, August 1, 2018;

“A Traveler Who Finds Stability in the Natural World,” by Kevin Hartnett, Quanta Magazine, August 1, 2018;

“Fields Medals Awarded to 4 Mathematicians,” by Kenneth Chang, The New York Times, August 1, 2018;

“Five superstars win ‘math’s Nobel Prize’,” by Frankie Schembri, Science, August 1, 2018;

“Fields medal: UK refugee wins ‘biggest maths prize’,” by Paul Rincon, BBC, August 1, 2018;

“Kurdish refugee wins the Fields medal – the biggest prize in maths,” by Gilead Amit, New Scientist, August 1, 2018;

“Former refugee among winners of Fields medal – the ‘Nobel prize for maths’,” by Nicola Davis and Naaman Zhou, The Guardian, August 1, 2018;

“Maths hands out its ‘Nobel Prize’ to an Australian — here’s why you should care,”by Daniel Keane, ABC News, August 1, 2018;

“Number-theory prodigy among winners of most coveted prize in mathematics,”by Davide Castelvecchi, Nature, August 1, 2018;

“Indian-Origin Professor Wins Fields Medal, The ‘Nobel of Mathematics’,”  NDTV, August 1, 2018;

“Swiss-based mathematician wins prestigious prize,” Swiss Info, August, 1, 2018;

“Italian professor wins Fields Medal, world’s highest honor for mathematics,” by Stefania Fumo, Xinhuanet, August 2, 2018;

“Fields Medal: Aussie genius Akshay Venkatesh wins ‘Nobel Prize of mathematics’,” by Michael Slezak, ABC News, August 2, 2018;

“Stanford mathematician wins Fields Medal, ‘Nobel of math’,” by Beatrice Christofaro, The Mercury News, August 2, 2018;

“German mathematician Peter Scholze wins ‘Nobel of mathematics’,” DW, August 2, 2018;

“Barzani commends Kurdish winner of highest honor in mathematics,” by Kosar Nawzad, Kurdistan 24, August 2, 2018;

“Perth man awarded ‘Nobel Prize for mathematics’,” 9 News Sydney, August 2, 2018;

“Prestigious Mathematics Medal Stolen Minutes After It Was Awarded,” by Sasha Ingber, NPR, August 2, 2018;

“Akshay Venkatesh: What the genius mathematician did to become a Field Medal winner,” Financial Express, August 3, 2018;

“Indian Australian Mathematician Wins Fields Medal, the “Nobel of Mathematics”,”The Indian Panorama, August 2, 2018;

“Mathematician Akshay Venkatesh: Jack of all fields, master of one,” by Devangshu Datta, Business Standard, August 3, 2018;

“Akshay Venkatesh | The journey of Indian-born Australian prodigy to Fields Medal,” The Statesman, August 6, 2018;

“A Former Refugee Won The ‘Nobel Prize’ of Mathematics – And It Was Stolen Minutes Later,” by Jacinta Bowler, Science Alert, August 6, 2018;

“‘Manipulating Numbers Makes Me Feel Happy’, Says Stanford’s Akshay Venkatesh Upon Winning Fields Medal, Mathematics Highest Honor,” India West, August 6, 2018;

“Aussie Fields Medalist speaks,” Cosmos Magazine, August 7, 2018

Featured image of this post has adapted from Facebook page of ICM 2018


I just watched this video on YouTube and I really enjoyed the way he is explaining the solution step by step. More than the video by itself, the another thing that I really appreciate is how mathematicians and mathematics teachers use YouTube to communicate with a more vast world. actually mathematicians are among pioneers of bringing the new technologies into the class, although they can easily use the older methods and technologies. I hope you too enjoy the video.



Female Inventors

I just read this article from  that has been recently published in Interesting Engineering, I hope you also enjoy reading it:

Female Inventors and Their Inventions That Changed the World and Impacted the History In a Revolutionary Way

Female inventors, scientists, and engineers have discovered countless revolutionary and life-changing inventions that have caused unprecedented breakthroughs in the history of the world.

You will find whole the story here

for the second time an Iranian won a Fields Medal!

As IMPA reported at 1st august 2018:

“Researchers from Germany, India, Iran and Italy take home the 2018 Fields Medal”

the whole story:

“Four notable and promising researchers from four different countries – Germany, India, Iran, and Italy – are the winners of the most important international award in mathematics, the Fields Medal. Delivered for the first time in 1936, the medal is recognition for works of excellence and an incentive for new outstanding achievements.

Awarded every four years at the world’s largest mathematics event – the International Congress of Mathematicians (ICM) – the medal will be given this year to Peter Scholze, Akshay Venkatesh, Caucher Birkar, and Alessi Fegalli at ICM’s opening ceremony on August 1st, at Riocentro.

Founded by the Canadian mathematician John Charles Fields to celebrate outstanding achievements, the Fields Medal has already been awarded to 56 scholars of the most diverse nationalities, among them, Brazilian Fields laureate Artur Avila, an extraordinary researcher from IMPA, awarded in 2014 in South Korea. Due to its importance and prestige, the medal is often likened to a Nobel Prize of Mathematics.

The winners of the Fields medal are selected by a group of renowned specialists nominated by the Executive Committee of the International Mathematical Union (IMU), which organize the ICMs. Every four years, between two and four researchers under the age of 40 are chosen. Since 2006, a cash prize of 15 thousand Canadian dollars accompanies the medal.

Meet the winners of the Fields Medal 2018:

for the second time an Iranian won a Fields Medal!
 Fields awards ceremony 2018, showing medal winners Peter Scholze, Alessio Figalli, Caucher Birkar and Akshay Venkatesh. Photograph: Fabio Motta/Courtesy IMC 2018

Akshay Venkatesh

Conquering the greatest honor among the world’s mathematicians before the age of 40 is a notable accomplishment, although the life of Akshay Venkatesh is already marked with precocious feats. Born in New Delhi, India in 1981, and raised in Australia, at age 12 he became a medalist at the International Mathematical Olympiad. From there, he dived into world of mathematics, starting a promising career. When he began his bachelor’s degree in Mathematics and Physics at the University of Western Australia, he was a 13-year-old boy.

At 20, Venkatesh finished his PhD at Princeton University and soon became an instructor at C.L.E. Moore, at the Massachusetts Institute of Technology (MIT), a prestigious position offered to recent graduates in the area of ​​Pure Mathematics, previously occupied by prominent figures such as the American John Nash (1928-1915). Upon leaving in 2004, he became a Clay Research Fellow and was appointed associate professor at the Courant Institute of Mathematical Sciences at New York University.

He became a professor at Stanford University at the age of 27, and as of this year is a faculty member at the Institute for Advanced Study (IAS).

Venkatesh has his feet in Number Theory – an area that deals with abstract issues and had no known application until the arrival of cryptography in the late 1970s – but roves with ease through related topics, such as Theory of Representation, Ergodic Theory, and Automorphic Forms. Armed with a meticulous, investigative and creative approach to research, detecting impressive connections between diverse areas, his contributions have been fundamental to several fields of research in Mathematics. It is no wonder that his work has been recognized by several distinguished awards such as Ostrowisk (2017), Infosys (2016), SASTRA Ramanujan (2008) and Salem (2007).

Previously a guest speaker at the 2010 ICM, Venkatesh has been invited back to speak in Rio this August.

Alessio Figalli

Born in Naples, Italy on April 2, 1984, Alessio Figalli belatedly discovered an interest in science. Until high school, his only concern was playing football. The training for the International Mathematical Olympiad (IMO) awakened his interest in the subject and, upon joining the Scuola Normale Superiore di Pisa, chose Mathematics.

Figalli completed his PhD in 2007 at the École Normale Supérieure de Lyon in France, with the guidance of Fields Medal laureate Cédric Villani. He has worked at the French National Center for Scientific Research, École Polytechnique, the University of Texas in the USA and ETH Zürich in Switzerland. A specialist in calculating variations and partial differential equations, he was invited to speak at the 2014 ICM in Seoul. He has won several awards, including: Peccot-Vimont (2011), EMS (2012), Cours Peccot (2012), Stampacchia Medal (2015) and Feltrinelli (2017).

Caucher Birkar

Caucher Birkar’s dedication to the winding and multidimensional world of algebraic geometry, with its ellipses, lemniscates, Cassini ovals, among so many other forms defined by equations, granted him the Philip Leverhulme prize in 2010 for exceptional scholars whose greatest achievement is yet to come. Given the substantial contributions of Birkar to the field, that prize was a prophecy: after eight years, the Cambridge University researcher joins the select group of Fields Medal winners at the age of 40.

Birkar, who just this year received recognition for his work as one of the London Mathematical Society Prize winners, was born in 1978 in Marivan, a Kurdish province in Iran bordering Iraq with about 200,000 inhabitants. His curiosity was awakened by algebraic geometry, the same interest that, in that same region, centuries earlier, had attracted the attention of Omar Khayyam (1048-1131) and Sharaf al-Din al-Tusi (1135-1213).

After graduating in Mathematics from Tehran University, Birkar went to live in the United Kingdom, where he became a British citizen. In 2004, he completed his PhD at the University of Nottingham with the thesis “Topics in modern algebraic geometry”. Throughout his trajectory, birational geometry has stood out as his main area of interest. He has devoted himself to the fundamental aspects of key problems in modern mathematics – such as minimal models, Fano varieties, and singularities. His theories have solved long-standing conjectures.

In 2010, the year in which he was awarded by the Foundation Sciences Mathématiques de Paris, Birkar wrote, alongside Paolo Cascini (Imperial College London), Christopher Hacon (University of Utah) and James McKernan (University of California, San Diego), an article called “Existence of minimal models for varieties of general log type” that revolutionized the field. The article earned the quartet the AMS Moore Prize in 2016.

Peter Scholze

Peter Scholze was born in Dresden, Germany on December 11, 1987. Only 30 years old, he is already considered by the scientific community as one of the most influential mathematicians in the world.

In 2012, at age 24, he became a full professor at the University of Bonn, Germany. Scholze impresses his colleagues with the intellectual ability he has shown since was a teenager, when he won four medals – three gold and one silver – at the International Mathematical Olympiad (IMO).

The mathematician completed his university graduate and masters in record time – five semesters – and gained notoriety at the age of 22, when he simplified a complex mathematical proof of numbers theory from 288 to 37 pages.

A specialist in arithmetic algebraic geometry, he stands out for his ability to understand the nature of mathematical phenomena and to simplify them during presentations.

At age 16, still a student at the Heinrich-Hertz-Gymnasium – a school with a strong scientific focus – Scholze decided to study Andrew Wiles’ solution to Fermat’s Last Theorem. Faced with the complexity of the result, he realized that he was on the right track in choosing Mathematics as a profession.

He was a guest speaker at ICM 2014 in Seoul, South Korea, and will be a plenary member this year at the Rio de Janeiro Congress.

Scholze has been repeatedly recognized for his contributions to ​​arithmetic algebraic geometry. He collects major mathematics awards, such as EMS (2016), Leibniz (2016), Fermat (2015), Ostrowski (2015), Cole (2015), Clay Research 2014), SASTRA Ramanujan (2013), Prix and Cours Peccot (2012) and, finally, the Fields Medal (2018).”

Reference: IMPA

You can also find some more information from the 1st issue of the Guardian:

for the second time an Iranian won a fields medal
Fields medals 2018 winners (from L to R): Caucher Birkar, Alessio Figalli, Akshay Venkatesh, Peter Scholze. Photograph: Handout

Former refugee among winners of Fields medal – the ‘Nobel prize for maths’

An Kurdish man who came to Britain as a refugee after fleeing conflict two decades ago is one of four men who have been awarded the Fields medal, considered the equivalent of a Nobel prize for mathematics.

The winners of the prize, presented at the International Congress of the International Mathematical Union in Rio de Janeiro, have been announced as Prof Caucher Birkar, 40, from Cambridge University, Prof Akshay Venkatesh, 36, an Australian based at Princeton and Stanford in the US, Prof Alessio Figalli, 34, from ETH in Zurich and Prof Peter Scholze, 30, from Bonn University.

The Fields medal is perhaps the most famous mathematical award. It was first awarded in 1936 and since 1950 has been presented every four years to up to four mathematicians who are under 40. As well as the medal, each recipient receives prize money of 15,000 Canadian dollars (£8,750). With all the prizes this year going to men, the late Maryam Mirzakhani remains the only woman to have received the accolade.

Birkar was born in Marivan in Iran – a Kurdish city heavily affected by the Iran-Iraq war of the 1980s – and studied mathematics at the University of Tehran before coming to the UK in 2000. After a year, he was granted refugee status, became a British citizen and began a PhD.

“When I was in school it was a chaotic period, there was the war between Iran and Iraq and the economic situation was pretty bad,” said Birkar. “My parents are farmers, so I spent a huge amount of time actually doing farming. In many ways it was not the ideal place for a kid to get interested in something like mathematics.”

Birkar says it was his brother who at an early age introduced him to more advanced mathematical techniques.

Prof Ivan Fesenko of the University of Nottingham, one of Birkar’s PhD supervisors, told the Guardian how Birkar, who initially spoke very little English, came to study with him.

“The Home Office sent him to live in Nottingham while they were processing his application for asylum status,” said Fesenko. “He came to me because he was interested in research work related to my general areas.”

Birkar’s talent, says Fesenko, quickly became apparent as he began his PhD. “I thought I should give him some problem – if he solves it, then this will be his PhD. Typically a PhD lasts three or four years. I gave him a problem and he solved it in three months,” said Fesenko.

“He is very, very smart; you start to talk with him and you recognise that he can read your thoughts several steps ahead. But he never uses this to his advantage, he is very, very respectful and he gently helps people to develop further,” said Fesenko.

As with many of the winners of the Fields medal, Birkar’s research is in areas of mathematics that can seem incomprehensible to a lay audience. His citation for the award says he won the medal “for his proof of the boundedness of fano varieties and for contributions to the minimal model program.”

Prof Paolo Cascini of Imperial College London has worked with Birkar. He said that in simple terms Birkar’s work focused on classifying geometrical shapes and describing their building blocks.

Birkar said he hoped the news may “put a little smile on the lips” of the world’s 40 million Kurds.

The youngest for the four winners, Germany’s Peter Scholze, became a professor at the age of just 24, and has been described by previous award committees as “already one of the most influential mathematicians in the world.”

Among his achievements, Scholze invented the theory of perfectoid spaces – which are noted in his citation for the Fields medal, and have been described as a class of fractal structures allowing problems to be moved from one number system to another, making them easier to solve.

“Geometry is the study of space and shape,” said Kevin Buzzard of Imperial College London. “One technique that geometers have introduced is the idea of studying a complicated space by mapping a simpler space onto it. For example, a line is a simpler object than a circle. But if you imagine wrapping a line up into a spring shape and compressing the spring, you have found a way of mapping a line into a circle. Geometers might use this technique to analyse questions about circles, by turning them into perhaps more complex questions about lines.”

Perfectoid spaces, he says, turns this logic on its head. “The counterintuitive idea introduced by Scholze is that to study a geometric object, you might instead want to find a mapping from a space which is so grotesque and twisted that in some sense it cannot be twisted up any more. The result is that instead of ending up having to solve complicated questions about simple objects, you have to solve simple questions about extremely complicated objects.”

The Italian winner, Figalliworks in the field of optimal transport, which has its roots in the research of 18th-century mathematician Gaspard Monge, who studied where to send material dug from the ground for use in construction so that the transport costs are as low as possible.

Venkatesh becomes only the second Australian to win the prestigious medal, after Terence Tao in 2006.Venkatesh was recognised for his use of dynamics theory, which studies the equations of moving objects to solve problems in number theory, which is the study of whole numbers, integers and prime numbers.

Venkatesh grew up in Perth and at age 13 became the youngest person to study at the University of Western Australia. He earned first class honours in pure mathematics aged 16 before studying at Princeton.

At UWA, he went straight into second-year maths courses after he proved he could write the exam papers for all the first year subjects he had never taken.

His work also uses representation theory, which represents abstract algebra in terms of more easily-understood linear algebra, and topology theory, which studies the properties of structures that are deformed through stretching or twisting, like a Mobius strip.

Receiving his award on Wednesday, he said: “A lot of the time when you do math, you’re stuck, but at the same time there are all these moments where you feel privileged that you get to work with it.

“You have this sensation of transcendence, you feel like you’ve been part of something really meaningful.”

One of his early mentors, Prof Cheryl Praeger, who has known Venkatesh since he was 12, and supervised his honours thesis when he was 15, said he was always “extraordinary”.

“At our first meeting, I was speaking with Akshay’s mother Svetha, while Akshay was sitting at a table in my office reading my blackboard which contained fragments from a supervision of one of my PhD students.

“At Akshay’s request I explained what the problem was. He coped with quite a lot of detail and I found that he could easily grasp the essence of the research.”

Reference: The Guardian