Red diana's chapter

THIRTY AND ONE TUTORIAL TALES in and about MATHEMATICS BY Dr.Raymond N. Shekoury with Diana Shekoury Dani Shekoury Andrew Shekoury Notes ___________________________________________________ The chapter was written before the untimely death of Mariam Mirzakhani on July 14, 2017. I did not update the chapter.

• This is a chapter of the book I am writing for my grandchildren in a chatting manner.

• The word “Jeddoo “ is an Arabic word, which means grandfather • Diana the “hero” of this chapter is my granddaughter. When this chapter was written, she was 19 years old in her first year at the university.

• Paragraphs with the following fonts signify, my remarks to reader.

I am of the habit that when, continuously working for several hours, at the computer, and getting tired, I would fold my elbows right on my working table and rest my tired head on a “pillow” of scattered open books.

• Paragraphs with the following fonts signify, my talk directed to the grandchildren.

Thank you Diana, I am very proud having such a charming and intelligent granddaughter. It is not a grandfather’s day.

• Paragraphs with the following fonts signify, grandchildren’s remarks.

I am sorry, to wake you up.

It is not morning, Jeddoo. It is three PM. I just came in to tell you “Happy Father’s day”. You know I always enjoy chatting with you.

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Famous Women Mathematicians I am of the habit that when, continuously working for several hours, at the computer, and getting tired, I would fold my elbows right on my working table and rest my tired head on a “pillow” of scattered open books.

It was about ten AM. last Sunday1, when I rested my head as usual. I did not realize that I had fallen asleep at my table, until I felt that a book was being slowly drawn from underneath my elbows. Opening my eyes, I saw Diana, my eighteen year old granddaughter.

Good morning Diana I am sorry, to wake you up.

It is not morning, Jeddoo. It is three PM. I just came in to tell you “Happy Father’s day”. You know I always enjoy chatting with you.

Thank you Diana, I am very proud having such a charming and intelligent granddaughter. It is not a grandfather’s day.

Yes, a grandfather is a father.

I saw she was carefully browsing through the pages of the book that she dragged from under my head. I noticed that she turned to glance at its cover. Upon reading its title, she lifting her eyebrows, retorted with a somewhat gloomy and bewildered tone.

Men of Mathematics by E. T. Bell!

Yes, It is a book, that I greatly cherish, ever since I had read it back in 1947, when I was still at high school, and from which I learned a great deal 1

It was 19 July 2016

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of mathematics. I still refer to it, once in while, when checking historical dates. In spite of its title, the book contains a short biographical note on a woman mathematician. Do you like to read that note?

No, I prefer reading a book dedicated entirely to “Women of Mathematics”.

Do you have such a title?

No, I am sorry, I don’t have such a book.

But, do you like me, instead, to tell you about two or three prominent modern women who made significant contributions to mathematicians?

Yes, sure I love to.

I’ll start with Maryam Mirzakhani.

Maryam Mirzakhani

The year 2014 was an important turning point for women in mathematics.

It is the first year since 1936 2 , a woman, Maryam Mirzakhani was honored with the prestigious Fields Medal prize.

In fact, Maryam was not only the first woman honored by the Fields prize, but she is the first mathematician from the Middle East ever to be awarded that great prize.

2

1936 was the first year at which the Fields Medal was awarded to outstanding contributors in mathematics.

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It sounds interesting; may I ask from which country in the Middle East does Maryam come? I guess, it must be Iraq.

I wish she were. Do you intend to make your Jeddoo happy. Sorry, but, you are almost correct; you missed by only one alphabetic letter.

Therefore it must be Iran. Anyway, it is not too far away from Iraq; they are neighboring countries.

Excellent, my dear Diana; you must be good at your Geography.

I wasn’t aware that prizes were awarded for math people.

You should. Several prizes are periodically awarded to both men and women, who make outstanding mathematical achievements. For example, mentioning a few, Bocher and Cole prizes are two prizes awarded by the American Mathematical Society, and Abel prize is presented annually by the King of Norway It is worth noting that no Nobel prize is offered for mathematics.

However, Fields is the MOST prestigious prize for mathematics, it is considered to be the “Nobel” for mathematics. This prize is awarded by the International Mathematical Union, for not more than four young mathematicians under the age of forty.

How often they are awarded?

Every four years, during the meetings of the Union’s International Congress which are held every four years. Its purpose is to give recognition and support to young mathematical researchers, who have made major contributions.

Thank you, I am much intrigued by that talented woman. Please tell me something more about her, life, and her research.

Yes, I will with pleasure, provided you promise you will do the utmost to gain that prize!!

Maryam was born in 1977 in Tehran, and went to a special High school for exceptional talented students in Tehran. After finishing high school, she attended Sharif University of Technology inTehran, In 1999 she earned her Bachelor degree in mathematics.

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She, as a teenager, in 1994, received her first mathematical recognition when she won a gold medal in the international mathematical Olympiad.

She was in the first Iranian team that allowed females to participate.

I knew there are athletic olympiads. But, it never crossed my mind that there are math olympiads.

The two words seem to me self contradictory.

Sure there are international mathematical olympiads. Your math high school teachers should have informed you about them. Those international olympiads are held annually in different countries. The participants must be pre-collegiate students.

Anyway, for the moment, I am focusing on Maryam’s talents. In 1995, she won the math olympiad AGAIN, this time, with a perfect score.

Lest I forget, the Olympiad for the current year 3 will be held in Hong Kong from 6 to 16 July.

As a child she loved to read literature, especially novels. At that time she thought to become a writer. In her last year at high school she discovered the joys and challenges mathematics can offer, when learning, from her brother, the historic story about the four-year-old child prodigy Gauss, adding all the integers for 1 to hundred in a second.

Maryam mathematical journey took her, after graduation from Sheriff Technological University, to the United States. She entered the graduate school at Harvard University.

Under the supervision of her thesis advisor, and with the diversity of mathematics courses, offered at Harvard, her mathematical interests expanded to cover several mathematical fields such as hyperbolic geometry, complex analysis, topology and dynamical systems.

I am sorry to interrupt you Jeddoo. I had studied Geometry in High school, and always thought there is only one geometry. Now, I am confused by hearing about another sort of geometry; a hyperbolic one.

I am proud to have a granddaughter who asks such excellent questions.

I promise that, I’ll try my best, on some future occasion, to clarify, in simple manner, the ideas of geometries such as hyperbolic, that are different from the standard Euclidean geometry taught at schools. But, for the moment, I

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prefer to concentrate my discussion on the achievements of your femalemathematician.

I was about to say, before the interruption, that her advisor, himself a distinguished mathematician, was very much impressed by her mathematical capabilities.

She obtained the PhD degree in 2004. The title of her thesis was: Simple Geodesics on Hyperbolic Surfaces and volumes of moduli spaces of curves.

In her PhD research work, she made not only several breakthroughs in mathematics, but also gave new ingenious proofs of other known results.

The thesis was published in Annals of Mathematics, a prestigious journal.

I am tempted to question the existence of math journals.

But now I have learned, it is better to keep my mouth shut.

Sometimes, that is a good practice, but not always.

In 2004, that is ten years before the Fields Prize, Maryam’s accomplishments were recognized by being awarded the Blumenthal Prize of the American Mathematical Society. *** I like to know about her personal life. Do you know anything about her?

Please, Diana, don't ask me questions that I can't answer.

I can't imagine there are questions that Jeddoo cannot answer.

You are wrong. Sure there are. I can't answer many questions, much more than you think. Moreover, you should also know there are even questions that nobody can answer; they are unanswerable.

Anyway, I do not know her personally. The very little I know about Maryam’s personal life, is from what is published in Stanford’s daily papers.

She is married to Jan Vondrak a Czech theoretical-computer scientist, who works at IBM. The couple lives in California at Stanford, and she prefers to be called Mrs. Vondrak.

They have a five-year-old daughter named Anahita.

Maryam currently, is a professor at Stanford University.

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*** Her Field Medal citation recognizes: “Mirzakhanti outstanding contribution to dynamic and geometry of Riemann surfaces and their moduli spaces, She has made stunning advances in the theory of Riemann surfaces and their moduli and led to new frontiers in this area. Her insights have integrated methods from diverse fields such as algebraic geometry, topology and probability theory.”

I don’t understand a word of any of those mathematical terminologies.

I am well aware you don’t and never expected you do. Each of those terms is pregnant with ideas that requires a yearly university course. I am quoting the citation in order to show how Maryam was recognized.

I have a final “tangential question”. It has struck me as very strange that no Nobel prize is given for achievements in mathematics. I am wondering is there a good reason for that?

Keep wondering. I am completely in the dark about this matter. I suggest you go to the Internet to read some dozen stories about why Alfred Nobel did not include mathematics in his will. All those stories are unfounded *** Before closing, let me quote Maryam’s message to young girls like you, Diana: “I will be happy if my award the Fields Medal encourages female scientists and mathematicians. I am sure there will be many more women winning this kind of awards in coming years.”

These are nice encouraging words. *** I appreciate your patience. Diana, you should’t wait until the forthcoming Father’s Day for discussing our next woman mathematician.

No, no, If you don’t mind, I’ll drop by you tomorrow right after school.

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Next Day It was a beautiful sunny morning. However, as midday approached, clouds began to collect.

At about three PM the weather changed abruptly, rain poured mercilessly from the sky pounding the rooftops and flooding the streets.

While reading a book that I’ve recently ordered from Amazon I was enjoying the scene from the window as if it were from a movie, until I saw Diana running in the rain with an umbrella over her head.

I hurriedly ran to opened the door for her. I pulled her inside the apartment. The umbrella was heavily dripping, and her shoes were soaked with water.

My dear Diana, what made you come in this thunderstorm. I know you made a promise to come, but you don’t have to keep your promise in such a bad weather. You could have phoned me.

Now, go to the bathroom, take a towel to dry yourself.

After a while, she came into my study barefooted, her hair was still slightly wet, and smiling as if to apologize for being without shoes. I hugged assuringly and responding: That is all right, don’t worry, dear Diana. *** She reached for the only open book on my working table, glanced at its title. She gave me a questioning look and reacted: It is ‘Byron, Life and Legend’ NOT ‘The Men of Mathematics’.

Is Byron the woman mathematician you intend to talk about?

No, he is not. Hadn’t you ever heard of Lord Byron.

No, I hadn’t. Who is he?

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Ada Lovelace You should know. It is the fault of your high school English literature curricula, not yours. He was a famous British poet and was a leading figure in the ninetieth century Romantic movement. Anyway, he is the father of Ada Lovelace.

Who is Ada Lovelace?

She is the woman mathematician, we’ll be chatting about. Ada Byron Lovelace is remembered as, the mother of modern computer, and the daughter of Lord Byron. In fact she is the only legitimate child of the womanizing poet.

A woman mathematician and a daughter of a famous poet, that sounds very interesting. I like too know more about her.

Yes, it is amazing to be a woman mathematician and a daughter of a famous poet, but more amazing is the fact that the computer she programmed had not yet been built.

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She looks beautiful. But her clothes seem to be old fashioned.

Not claiming to be an expert of women’s fashions, yet I don’t agree with you. One shouldn't judge by the norms by modern standards.

When did she live?

She lived in the nineteenth century. I am sorry I forgot her birth year. Let me to look it up.

Unfortunately my “omniscience” iPad being out of charge, couldn’t be of much help to me.

Thus, referring to “The Men of Mathematics," I pointing to the book, I directed my talk to Diana.

Diana, you should remember this. It is the book that you paged through yesterday.

Yes, I do remember that. It initiated our discussion of women in mathematics.

I found the year of her birth. She was born more than 200 years ago, in 1815, in London, England, named Augustus Ada Byron. Her surname was changed after she married.

Two hundred years ago, no wonder you find her dress old fashioned.

It was a period when women were discouraged from seeking an education, especially one dealing with the sciences and mathematics.

Unfortunately, such beliefs are not dead even in the twenty - first century. My classmate told me about how a university professor discouraged her sister from pursuing a major in math on the pretext that it might hamper her chances of marriage since, he said, men don’t like women who are sharper than themselves.

He must be a very reactionary and idiot professor, who, should be ignored.

Let us go back to Ada’s early life.

In those days there were no place for girls in the United Kingdom’s universities. However, girls from wealthy, aristocratic families could be educated to a high level by private tutors. That is how Ada’s mother and Ada herself were educated.

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Her mother, Anne Isabella Milbanke was highly intelligent, had been well — educated, and was particularly enthusiastic about mathematics, and science.

Her marriage to Lord Byron was brief and unhappy. They divorced five weeks after Ada’s birth. Lord Byron was hardly an exemplary father. He never saw his daughter or his wife again after the divorce, though he mentioned his daughter in his poetry. The girl never saw her father.

This is unfortunate, for the family. educated?

How was Ada

Her mother, Lady Byron, took the responsibility of Ada’s education.

Fortunately, Lady Byron, did not share the prevailing beliefs that women should not pursue science education. She herself had enjoyed studying mathematics and science.

Moreover, Lady Byron believed a rigorous course rooted in logic and reason would enable her daughter to avoid the romantic ideals and moody nature of her father. Thus, Ada from the age of four, was privately tutored in mathematics, science, music and French.

Did the study of mathematics hinder Ada’s chances from marriage?

Diana’s remark must have been a reaction to the claim, made by that idiot professor, that women majoring in mathematics lowers their chances of marriage.

I answered: No, not at all, I’ll comeback to her marriage later on.

Many doors were open to Ada, because of her family’s prominent position.

Ada was able to converse, interact and study with a variety of talented people such as Augustus De Morgan, the mathematician and Logician, Michael Faraday, the well-known physicist, and the famous novelist, Charles Dickens.

Her life changed forever, in 1833 when she was 17 year old girl.

What happened that year? Did she fall in love with a prince?

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Yes, indeed she fell in love, but not with a prince. She fell in love with Babbage’s calculating machine that can produce mathematical tables, which he called Difference Machine.

Amazed, Diana could amazement, screamed.

not

control

her

FELL IN LOVE with a machine. She must have been in love with that Babbage, not with his machine!!

No, not with Babbage, but really with his machine.

After a short pause, I continued.

You are too young, you must know that a person can fall in love with anything, may be with his car, or with his garden, or with his pet for a few examples.

Anyway, who is Babbage?

Charles Babbage, was a British professor of mathematics at the University of Cambridge from 1828 to 1839. He is credited with inventing the first mechanical computer. Thus, he is considered to be a “father of the computer”.

He began in 1822 with what he called the “difference machine," which was designed to compute mathematical tables, and was powered by cranking a handle.

He was fond to give demonstrations of a small scale version of his rudimentary calculating machine.

How did Ada enter the scene?

Babbage, learning that the aristocratic Lady Byron and her daughter Ada were knowledgeable in mathematics, invited them to see one of his demonstrations Th young girl Ada, who seem to have inherited a poetic vision from her father and demanding love of mathematics and sciences from her mother was captivated by the idea.

She, determined to understand how the machine works. Later she sent him a message requesting copies of the machine’s blueprints.

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A professional lifelong friendship developed between Charles Babbage and the young aristocratic girl Ada Byron, Do you mean because of her love to that engine she did not marry.

No, I never said: “She did not marry”. In fact she did.

In 1835, at the age of nineteen, Ada married William, the eighth Baron King, who was made the earl of Lovelace in 1838. Thus Ada became Lady Ada Loveless.

Did the couple followed the footsteps of Ada’s parents, which ended in divorce? Or was it a happy marriage?

I don’t really know. It looks like it was a fairly stable marriage, that lasted about seventeen years, until death parted the couple.

In the four years after the marriage, she gave birth to two sons and a daughter.

After the birth of the third child, Ada was not satisfied with a role as a nurturing mother. She wanted to learn more mathematics and pursue and explore, her ideas. Fortunately, the rearing of her children was gladly supervised by her husband and her mother.

She spent more time on learning advanced mathematics from eminent mathematicians.

Did she make some breakthroughs in mathematics or settled some unsolved problem?

As far as I know, she certainly pursued learning more advanced mathematics, but did not have any contribution in the advancements of the field of mathematics. All the time she kept Babbage Engine in mind.

Diana was about to say something when my phone rang. After briefly answering the call from a friend, I forgot about Diana wanting to talk.

I continued my lecturing her.

Meanwhile, Babbage had moved on from the rudimentary difference machine to a developing a machine of a much higher level of sophistication, which he called Analytic Engine. The concept of the analytic engine is an entirely new idea, and a work of an incredible genius on 15/36

Babbage part.

It was a machine much more capable of performing sophisticated calculations than the original difference machine. In other words it was the first world’s programmable computer.

What made that analytic engine programmable?

Because, it featured all the necessary components of a modern computer.

It contained an arithmetic logic unit, control flow of loops, and separate memory.

In other words the logical structure of the Analytic Engine was essentially the same as that which have dominated computer design in our electronic era, The only essential difference is it was supposed to run using mechanical parts and powered either by hand cranking or by steam Our deep discussion turned our attentions away from thinking about the thunderstorm. However, a sudden entrance of a light beam entering the windows signified the end of the storm and revived for a few minutes our discussion about the weather unpredictability.

But soon we were attracted back to the steam or hand driven antiquated computers It is beyond my imagination to visualize a computer powered by steam.

It is beyond my imagination too.

Was such a strange steam power engine ever built?

Yes, such an engine was built in 1991, long after both Charles Babbage and Ada Lovelace, were dead.

It was built by the London Science Museum, as a historical specimen to be displayed.

I wish I would see it, if I ever visit London.

Be sure to accompany your brothers with you.

Of course I’ll. You too Jeddoo, have to come with us.

Yes, I will, on condition that I buy the tickets for the whole group.

Now, let us leave our day dreams for a more appropriate time.

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I can presuppose that Ada “broke up” with the old Machine, and fell in love with the Analytic Engine.

Yes, sure, she did.

I am curious about Ada’s “affair” with the Engine.

Would you tell me something about it.

Babbage never published anything written at all about his Analytic Engine. But he was fond to give talks about it.

The British government was not very enthusiastic about financing building the Analytic Engines.

However, his ideas were more receptive by the European governments.

He was invited by the Italian government to lecture on the Analytic Engine.

Notes of the lecture, were taken by Luigi Menabrea an Italian army engineer (who, 27 years later became a prime minster of Italy). The Italian engineer, published an article in French based on the notes he had taken.

Because of Ada’s aptitude in mathematics and her fluency in French, a British journal, asked her to translate the published article.

Ada, decided to enhance the translation by adding her own ideas in form of notes. Her notes turned out to be three times more extensive than the original article.

Diana’s facial looks and body gestures revealed that she is beginning to consider the story from a realistic and more serious point of view.

This means that Ada’s English translation of Menabrea's original article was mostly her own ideas. I am wondering did the notes contain a real addition to the functionality of the Engine?

It is true that the English translation of the article reflected her thoughts of the various potentialities of the Analytic Engine, which Babbage himself was unaware of.

I am wondering, what potentialities of the Engine, which the notes added?

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In her notes Ada broke new grounds, identifying an entirely new concept.

She realized that Babbage Engine could go beyond numbers. This was the first ever perception to a modern computer — not just a calculator — but a machine that could contribute to other areas of human endeavors. She speculated that anything content (including music, text, pictures and sound) that could be translated into digital form, can be manipulated by the machine, Thus, she foresaw the multipurpose functionality of the modern computer, while Babbage believed his engine was restricted to mathematical calculations only.

She had pointed out many of the special features of Babbage machine, such as the mill where calculation would take place, the storehouse the results were stored, the backing of the cards which allowed it to use or to reuse any card or any set of cards any number of times in solving a problem.

One should keep in mind that Ada was describing a machine that did not exist. She described how it could tackle problems that were unsolved, e.g., astronomical tables, to generate random numbers, complex numbers, complex sequences of numbers. In fact, she even wrote a computer program for computing Bernoulli numbers explaining how and where to set the calculations and read results. Today such step by step actions to be systematically performed is called an algorithm. This was an impressive work, especially when we realize that she had no machine to work out the her programs.

A replica of that historical program is depicted in Wikipedia article titled “Ada Loveless” Though realizing that the Analytic Engine did not exist at the time, yet I am curious, whether there were attempts, during the nineteenth and the twentieth centuries to construct such an engine?

The fate of Analytic Engine turned out to be an idea that remained a dream of the Victorian steam age ,destined to be fulfilled in the electronic age the twentieth century.

There were few attempts to construct such an engine in which Babbage and Ada collaborated, but those attempts seemed destined to failure, due to a variety reasons ,among which Charles Babbage ran into many financial difficulties and Ada Loveless periodic illness and her early death.

Oh poor soul what caused her death?

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Ada for quite some time, was suffering from cervical cancer. She had been in pain for several years To cope with the pain her physician gave her pain killing drugs. In 1852 Ada Lovelace died at the age of thirty - six.

Upon her request she was buried next to her father whom she had never seen. ***

Ada Loveless is to be admired for her perseverance. Despite her health issues she kept on. She, being aware that she possesses a gift and a keen love for mathematics, returned to it again and again for complex reasons — a moral compass and that it nourished her mind.

In the seventies of last century, a new high order computer language was developed by the American Department of Defense. The new language was named “Ada” in honor of Ada Loveless. Moreover, the American National Standards Institute approved “Ada” as a national all purpose language.

A movement was launched, In 2009, to establish a special day, similar to the π day, chosen to be the third Tuesday of October of every year to be dedicated for celebration, acknowledgement and honoring Ada Lovelace's work in computer science. The goal of the celebration is: “To raise the profile of women in mathematics science, technology and engineering; and to create new role models for girls and women in these fields”. *** I had promised my Mom to go shopping together. I think it is time to go home Thank you Jeddoo fo the illuminating discussion. I really loved Lovelace, and am looking forward to hear about the next woman mathematician. Who would be that woman?

Emmy Noether is the name Tomorrow I have a full day schedule at the University. I’ll come right after dinner to hear about ……

Emmy Noether Please call if there will happen a thunderstorm.

Don’t forget your shoes in the bathroom. See you tomorrow.

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Emmy Noether The most important woman in the history of mathematics Pondering after dinner, on my promised talk about Emmy Noether, I brought her picture down from the wall of my study, on which hang several pictures of famous scholars who influenced my academic life.

Planning to start my talk immediately after my granddaughter comes in. I put the picture right on my working table.

However, when she came in, she didn’t immediately notice the picture.

Thus, we started general topics:

our

conversation

on

It is nice a day. I’m happy the weather did not suddenly change into a severe storm as it did yesterday.

How many class hours did you have at school today?

Five, just imagine five consecutive hours, of English literature, calculus, European history, biology and Spanish.

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I think the Biology course was the most boring.

No, Jeddoo. it was the Calculus.

She swiftly changed the subject of our conversation, evidently feeling that I might have been hurt by her answer.

Can you guess what I had for dinner?

No, how can I? I’m sure it must be delicious: I know your Mom’s dishes are always exquisite. What was it?

It was the kubba4 .

Oh, that is wonderful. If I only knew, it was kubba, I would never have hesitated to invite myself to your dinner. Then we could have held our session about the female mathematician at your place with your folks participating.

At that moment, Diana, looked towards the Apple computer and noticed, for the first time, the picture posted there.

Oh, that must be Emmy. What did you say, Emmy what?

Emmy Noether, a name you shouldn’t forget. She is remembered as …

Impatient, just, like most girls, to express their opinions about other woman dresses, Diana, couldn’t but interrupt me: Was she dressed to meet the Pope?

Diana, did not wait for a response to her sarcastic question. Turned towards the pictures hanging on the walls, looking eager to learn about those people.

I hadn’t seen these pictures before.

4

Kubba is a well known dish in the Middle East.

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I’m sure you did see them before, probably you didn't notice them.

“Seeing” is different from “noticing”. Yesterday you were under stress and your mind probably was internally too occupied with the bad weather to notice outside items Yes, this might be true.

These must be the pictures 5 of people that inspired your admiration.

Sure they are picture of scholars, who had greatly influenced my academic career,.and won my admiration.

I can recognize only two of them, Einstein and Shakespeare.

I Pointing at Newton’s picture, asked: You must recognize this guy. Don’t you?

After a minute thinking, she asked: No, I don’t think so. Who is he?

Do you remember somebody born on the “twenty-fifth” of December, and changed the whole world?

She looked straight into my eyes me with a most bewildered stare.

You are poking fun at me. Aren’t you Jeddoo?

No, I am not making fun.

Issac Newton was born on the “twenty-fifth” of December. I’m sure, you know he changed the world by his scientific discoveries among which are the three laws of gravity, and many other laws of physics.

Now I ask: “Do you know what other important discovery he had made?”

She scratched her forehead for a few seconds. A blush painted her cheeks. Finally she declared:

5

Einstein, Bertrand Russell, David Hilbert, Kurt Gödel, WilliamShakespeare, Issac Newton, Fredrick Gauss, Mohammed AlJwahiry and Evariste Galois

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This is a difficult question. I don’t think I know.

What did Newton further discover?

The calculus The very subject that bored you.

Moving her lips as if pretending to smile, she, again intentionally diverted the conversation, by pointing at Galois’ picture and asking: Who is he? This guy seems to be a stupid kid.

Evariste Galois, a French mathematician Yes, a stupid kid he was, but also a reputed genius.

He, was able in his teens, to settle a problem unsolved for several centuries. Thereby, he laid the foundations for group theory and Galois field, two major branches of modern abstract algebra. He died at the age of twenty-one from wounds suffered in a duel.

After telling Diana few words about each of the scholars whose pictures hanged on the wall. I emphatically demanded that it is time to sit down and talk about the genius of the most important woman in the history of mathematics. ***

Emmy Noether, is not like yesterday’s Ada Lovelace a mathematics fan, but a distinguished contributor to mathematics, who is described by eminent mathematicians as the most important women in the history of mathematics. Yet, some people are ignorant who this woman was, and what were her contributions in mathematics and physics.

Diana seems to take my last remark as pointing to her personally.

Thus, she quickly responded by a disclaimer: I am one of those people, but it wouldn’t take me long to be knowledgeable about her. Thanks Jeddoo for educating me.

I disregarded her remark, and continued.

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I can only do her memory justice, if I can provide you with some understanding of her several important contributions.

First of all, you must realize how frustrating and burdensome the life of an intelligent woman born in the late nineteenth century.

Where did she live?

Well, she traveled a lot and lived in many different countries. But she was born in Bavaria (now part of Germany) back in 1882.

Being a woman, she was not allowed to attend regular college preparatory schools. Thus, she attended a ‘finishing school’ specializing in French and English, which were subjects more socially acceptable for a girl at that time.

She passed the state certificate to teach these languages at Bavarian schools for girls.

Soon after becoming a language teacher, she decided to pursue mathematics, which was then a demanding course for a woman.

Why did she change her mind?

How can I know? I can only conjecture that she discovered her potential capabilities in mathematics through studying the research work of her father Max Noether, a distinguished mathematician and a professor at Erlangen University.

The ambitious young girl applied for admission as a regular mathematics student to Erlangen university, which had at that time a reputation of being a socially progressive institution, and where her father was a prominent professor of mathematics.

Can you imagine the frustration of Fräulein Noether on learning about the refusal of her application to study the subject of her love, for no reason, except of being a women.

Yes, I do feel her frustration. But I ask couldn’t her father exert some influence on the faculty of his University?

Apparently he did. Since, she was later granted a permission to audit classes at Erlangen rather than participate fully as a regular student.

Auditing requires the permission of each professor whose lectures she wish to attend.

Did auditing count for a high university degree

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Again, I’m not sure. But it must have counted, at least in Erlangen University since, Emmy, after struggling with the administration complete in 1907 her doctorate in mathematics I guess her father supervised her PhD thesis?

No, no, I’m pretty sure it is unacceptable in all academic circles to have a close relative as an advisor.

Emmy being a woman was not allowed to teach at universities. Thus, she remained at Erlangen for seven years, during which she helped without pay, her ailing father in his teaching and supervising.

I assume she must have helped her father in grading papers?

No, You seem to be under estimating, not only the capabilities of Emmy Noether, but also her altruistic characters.

Though, I cannot categorically deny that she ever graded papers for her father. Yet, I believe it is not important whether she did or did not, the most far — reaching point, to remember, is that during those seven years, Emmy Noether actually supervised two doctoral students who were both officially registered under her father’s name. As if not enough, she during those seven years, published several papers of profound contributions.

Her research was so deep and far — reaching that attracted the attentions of eminent mathematicians.

In 1915 during the first world war, Felix Klein and David Hilbert invited Noether to teach at Gottingen University.

Who are Felix Klein and David Hilbert? And what was so special about Gottingen University?

Felix Klein and David Hilbert were the most prominent mathematicians of the first half of the twentieth century. They were both professors at the University of Gottingen. This university was, at the end of the nineteenth century and the beginning of the twentieth centuries, the most renowned center of mathematics. It was at that time, so to speak the capital of the World of Mathematics. To be invited to teach there, was considered to be a great honor.

I see that such eminent mathematicians inviting Noether to teach at such a prestigious university is an honor comparable to that of a person called by

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the President of the country to serve as secretary of some important department.

Yes, it is comparable.

I realize that her contributions are above my head, yet I would like to get some simplified ideas about her contributions, if you please Of course it pleases me.

Emmy Noether research influenced both mathematics and physics investigations. Looking at the article titled “Noether” in any encyclopedia one find terms such as the following: Noetherian group, Noetherian rings, Noetherian ideals, Noetherian modules and many more similar terms. and Noether’s theorem, Noether’s Second theorem, Lasker Noether theorem, Brauer - Noether theorem

Sprinkled all over the article. Mathematicians are more familiar with the first group of terms, while physicists with the second.

Hearing the word “ring," images of boxing rings, or a diamond rings flash in my mind. But now it seems, the term, when used in math media, is pregnant with deep mathematical notions.

Remember Diana, the doors of the wonderful world of mathematics are wide open for you, not only to peek through, but also to enter, by majoring in mathematics I took Diana’s remark as an implicit request to provide reasons making mathematicians so fond of choosing everyday simple words as terms for difficult ideas.

Not wanting to diverge from my talk about Noether, I opened my saved documents computer-file for the article titled “Abstraction” which I had written several years ago6. I printed the article and handed it to Diana saying: ‑

6

See Appendix 1.

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This is a homework. I hope, after reading it at home, the ring of integers would flash upon hearing “ring”.

Thank you, Jeddoo. I’ll read it carefully. ***

After the interruption continued my talk:

about

“ring”

I

I remember saying a little while ago, that Noether is best remembered among mathematicians for her pioneering contributions to abstract algebra, which is a field embodying the main modernization of the twentieth century’s mathematics.

She provided invaluable abstract approach methods in her lectures, and in her published papers, as well as in personal influence on her contemporaries. Because of her unique look on the topics, she was able to see relationships that traditional algebra experts could not.

She published over 40 papers in her lifetime, and her work on Ideal theory published in 1920 was revolutionary. It is said that its publication gave rise to the term “Noetherian rings” and naming several other mathematical objects as Noetherian Over the course of her career, Noether contributed to the theory of groups as result of her treatment of symmetries. This work has influenced mathematics as well as theoretical physics especially the mathematical aspect of quantum mechanics.

Physicists tend to know her work primarily through the theorem she published in 1918, later, known as Noether’s theorem.

Was Noether’s theorem in mathematics or in physics?

Well, it was in both like a thread that wove the two together.

In introductory physics textbooks, position, velocity, and acceleration are introduced, where one learns about the concept of force, Newton's laws, conservation laws of energy, momentum and angular momentum, and the laws of thermodynamics.

You know that the laws of physics, whether classical or modern, are fundamentally mathematical in nature. They are mathematical formulae. In order to express them, physicist have to utilize a system of coordinates of space and time. Since there are infinitely many coordinate systems, an important question arises: 28/36

If the coordinates are changed will the laws still hold in the new system?

I think everybody will answer “sure they do”. Nobody will claim that gravity or electromagnetic laws in New York are different from that in London.

Traffic laws can be different in New York and London.

Yes, but traffic laws are not laws of physics or chemistry.

So it is natural to ask what transformations of space and time that will keep the formulae of the of physics and chemistry unchanged.

Looking for those of transformations of space and time that preserve the desired properties, and studying them lead to the study of symmetries and hence group theory.

Emmy Noether proved in 1918 the theorem, which became known as “Noether’s theorem” that roughly states: For every symmetry there corresponds a conservation law.

The conservation laws of energy, momentum and angular momentum are perhaps the most fundamental physics laws we have.

• Conservation of mass — energy comes from time-shift symmetry: You can repeat an experiment dealing with energy of an isolated system at different times, the result comes out with the same.

• Conservation of angular momentum, comes from symmetry under rotation which when combined with the conservation of energy under the force of gravity explains the Earth’s motion around the sun.

• Conservation of linear momentum comes from transitional symmetry.

• And the list of conservation laws goes on.

Every scientist using symmetry or conservation laws in studying matter on the subatomic scale or on the cosmic scale, Noether’s theorem is present to aid.

The greatest success of Noether’s theorem came with quantum physics, and especially the particle physics revolution.

Many physicists, inspired by Noether’s theorem and the success of Einstein’s general theory of relativity, looked at geometrical descriptions and mathematical symmetries to describe the new types of particles they were discovering. ***

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Emmy Noether with such qualifications and research reputation must had been highly welcomed in Gottingen.

UnfortunateIy that wasn’t the case.In spite of Noether’s research reputation, many of the faculty at Gottingen refused to offer her a teaching position, for no reason except her gender. Even one faculty member of the history department objected: “What will our soldiers think when they return to the university and find they are to learn at the feet off a woman?”

Remember, the time was the first world war.

However David Hilbert, the great mathematician, took up her cause. He furiously addressed the administration at Gottingen: “I do not see that the sex of the candidate is an argument against her.”

“After all we are a university not a bathhouse.”

Hilbert in spite of his great defense, failed to make his case, so instead brought her on a staff more or less on permanent “guest lecturer”. Hilbert used to advertise her courses under his own name.

This means that Emmy Noether finally was taking baths at only men bathrooms!!

Thanks Diana, for this a beautiful way of describing the situation.

Did she remain guest lecturer in Gottingen throughout her academic life?

No, As a result of the reformed social attitude towards rights of women enacted after the first world war, Noether was finally granted a teaching position in Gottingen. But she was still paid only a small amount for her teaching work. She remained at that university until 1933 in spite of her underpay.

Why until 1933?

You will see the reason in a minute.

Noether, during her tenure in Gottingen made great progress in her research in abstract algebra, and applications of her theorems in physics.

She became internationally known mathematician, and was invited to give lectures at other universities and to address international meetings.

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Moreover, while at Gottingen she supervised about a dozen PhD students.

However, when in 1933, Hitler and the Nazi came in power in Germany, Jewish professors were expelled from their universities without regard to their academic stature.

Now, I understand why in 1933 she left the prestigious Gottingen university. I did not know that Emmy Noether was Jewish.

Yes, Noether was born from jewish parents. The Nazi regime discharged all professors, as long as they were born from jewish parents, regardless of their academic reputations.

Anyway, Noether had to leave her teaching job at Gottingen, and moved to the United States and taught at Bryn Mawr College in Pennsylvania until her death in 1935.

Was she ever awarded the, the prestigious prize that was awarded to Maryam Mirzakhani?

You mean the Fields Prize. No, she wasn’t, for several reasons. The most evident of which is, that she died before 1936, the year, you should remember, the first Fields Prize was awarded.

However, she was awarded the Ackermann-Teubner Memorial Prize in 1932. *** I’m wondering, about her personal life.

What kind of person was she? What were her political leanings, during the chaotic era of the beginnings of the twentieth century?

Noether having fervent feelings only to mathematics never married.

That is strange. She could have married Einstein!!

I disregarded continued:

her

clever

joke

and

She, as a person was considerate and warm, who inspired and cared deeply for her students.

During her time at Gottingen, she accumulated a small following of students known as “Noether's boys," who traveled to study with her as far as Russia. She considered her students to be like a family and was always willing to listen to their problems. She sometimes even allowed her 31/36

colleagues and students to receive credit for her original ideas in order to help them at the expense of her own.

Noether's teaching method led her students to come up with ideas of their own, and many went on to become great mathematicians themselves.

Many credited Noether for her part in teaching them to teach themselves.

Although politics was not central in her life, Noether was not aloof from political excitement. And according to some of her colleagues, she showed considerable support to the 1917 Russian Bolshevik revolution. However, in her later years, she took no part in any political matters. She always remained a pacifist, throughout all her life.

What was the cause of her death?

Noether’s death was sudden and unexpected. Strange it may seem Emmy Noether died of the same cause of Ada Lovelace. In April 1935 she had undergone a surgery to remove a uterine tumor, but she died of a postoperative infection. ***

Considering the remarkable advances made by Emmy Noether in the course of her life, she received little public recognition in her lifetime.

But she has been honored in many ways following her death. Many famous mathematicians and colleagues eulogized her. Albert Einstein published an eloquent obituary for her remembrance7.

After the second world war ended, the University of Erlangen established a co-ed gymnasium named after her, dedicated to Mathematics.

A crater on the moon is named for her.

The school at which she studied was renamed after her name. A Street in her hometown was named after her.

Our mutual chatting during these three days about three women mathematicians, impacted me in a variety of ways, it helped me to broaden my knowledge expanded my intellectual horizons and contributed to m y u n d e r s t a n d i n g o f t h e c u l t u r a l n a t u re o f mathematics.

Few people have a Jeddoo like you.

Thank you

7

see Appendix 2.

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Only a few jeddoos are as fortunate, as me to have such a remarkable

intelligent granddaughter like you.

I hope I learn the from you about other women mathematicians in the near future.

APPENDIX 1

It is true that that mathematicians are fond of using everyday simple words, like group, ring, module, field and many more, as terms loaded with deep ideas of mathematics.

Such terms originated historically by the process of abstraction on the system of numbers.

Abstraction , in general, is a thought process of stripping concrete objects from some of their properties and retaining others.

Everybody knows the system of numbers consists of a set of things called numbers endowed with the operations, of addition and multiplication, which satisfy certain known rules of arithmetic. (subtraction and division are not considered to be independent operations, they are nothing but the opposites of addition and multiplication respectively) Now imagine a mathematician subjecting the system of numbers to an abstraction process by depriving the multiplication operation of its opposite. Thus, creating a new object of thought; a system in which division has to be avoided or ignored. He feels the object is worthy to be subjected to further considerations and to be shared with his colleagues, and consequently has to be named. He chose for some reason or other to name it “ring”.

Now he or another mathematician subjected the original system of numbers to an abstraction of a different kind, by tossing the operation of multiplication away leaving the system with only a single operation, namely the addition. Again a new object was created, which is deserved to be studied and be shared with colleagues and be named.

The name “group” was chosen.

Then an analogous “game” was played, this time by stripping away the operation of addition. An object very similar to group was obtained. The adjectives “additive” and “multiplicative” are introduced to distinguish the two.

The terms “ring," “group," “additive group," “multiplicative group," and others were positively received by the math media. Hence they persisted.

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APPENDIX 2

EINSTEIN WROTE AN ELOQUENT TRIBUNE

The efforts of most human-beings are consumed in the struggle for their daily bread, but most of those who are, either through fortune or some special gift, relieved of this struggle are largely absorbed in further improving of their worldly lot. Beneath the effort directed toward the accumulation of worldly goods lies all too frequently the illusion that this is the most substantial and desirable end to be achieved.

But there is, fortunately, a minority composed of those who recognize early in their lives that the most beautiful and satisfying experiences open to humankind are not derived from the outside, but are bound up with the development of the individual's own feeling, thinking and acting.

The genuine artists, investigators and thinkers have always been persons of this kind.

However inconspicuously the life of these individuals runs its course, nonetheless the fruits of their endeavors are the most valuable contributions which one generation can make to its successors.

Within the past few days a distinguished mathematician, Professor Emmy Noether, formerly connected with the University of Gottingen and for the past two years at Bryn Mawr College, died in her fifty-third year. In the judgment of the most competent living mathematicians, Fräulein Noether was the most significant creative mathematical genius thus far produced since the higher education of women began. In the realm of algebra, in which the most gifted mathematicians have been busy for centuries, she discovered methods which proved of enormous importance in the development of the present-day younger generation of mathematicians.

Pure mathematics is, in its way, the poetry of logical ideas. One seeks the most general ideas of operation which will bring together in simple, logical and unified form the largest possible circle of formal relationships. In this effort toward logical beauty spiritual formulas are discovered necessary for the deeper penetration into the laws of nature.

Born in a Jewish family distinguished for the love of learning, Emmy Noether, who, in spite of the efforts of the great Gottingen mathematician, Hilbert, never reached the academic standing due her in her own country, nonetheless surrounded herself with a group of students and investigators at Gottingen, who have already become distinguished as teachers and investigators. Her unselfish, significant work over a period of many years was rewarded by the new rulers of Germany with a dismissal, which cost her the means of maintaining her simple life and the opportunity to carry on her mathematical studies.

Farsighted friends of science in this country were fortunately able to make such arrangements at Bryn Mawr College and at Princeton that she found in America up to the day of her death not only colleagues who esteemed her friendship but grateful pupils whose enthusiasm made her last years the happiest and perhaps the most fruitful of her entire career.

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Raymond Shekoury • Raymond Shekoury is a retired professor of mathematics at Baghdad university.

He was born in1931 in Baghdad, Iraq.

• He studied civil engineering in Baghdad University, worked as an engineer for two years. Discovering that engineering profession did not have much attraction to him. He entered the College of Science to study Mathematics.

• He was granted a scholarship to study mathematics at the university of Iowa in USA. • He obtained the PhD degree in February 1963.

• During his tenure years at Baghdad University he supervised about forty master and doctorate students.

• Among his non-teaching interests are: reading philosophy topics, classical literature, follow the world news and communicate via the Internet with his colleagues and former students.

• Currently, he lives near the families of his two sons in Nashville Tennessee.

• His e-mail address is: rshekoury@mac.com.

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