Diana's chapter

THIRTY AND ONE TUTORIAL TALES in and about

MATHEMATICS

BY

Dr.Raymond N. Shekoury

with Diana Shekoury Dani Shekoury Andrew Shekoury

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Notes This is one of the chapters of the book I’m writing for my grandchildren in a chatting manner Diana, the “hero” of his chapter, is my granddaughter. She is 19 years old, in her first year at the university when the chapter was written “Jeddoo” is an Arabic word which means grandpa

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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, lifting her eyebrows, she retorted with a somewhat gloomy and bewildered tone.

“Men of Mathematics by E. T. Bell” !

Yes, It is a book, 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 of mathematics. I still refer to it once in while, when checking historical dates. In spite of its title, the book contains a very 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.

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It was 19 July 2016

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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.

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.

Therefor 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.

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

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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.

Still, a teenager, in 1994, she received her first mathematical recognition when winning 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, about the four-year-old child 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.

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2016

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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 ask 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 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 live 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 require 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. Tomorrow I nave one class at eight A.M. I’ll drop by you tomorrow right after school.

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Monday The early morning of Monday following yesterday’s celebration of Father’s day, was beautiful and sunny. I was in my study enjoying reading a book about cosmos and science, which I recently ordered from Amazon.

All of a sudden, as the two hands of the clock approached to be at right angles, at nine A.M. clouds began to collect. In a few minutes the weather changed abruptly. Rain poured mercilessly from the sky pounding on the rooftops and flooding the streets. I spent something like half an hour or so, watching, from the window, the scene of the pouring rain, without feelings for the unlucky pedestrians, as if it were a panorama from a Hollywood movie.

All the time, I was expecting a call from Diana to cancel her appointment due to the unexpected bad weather. But, I was suddenly shocked to see her running in the rain with an umbrella over her head .

I hurriedly ran to open the door for her.

I pulled her inside the apartment. Her umbrella was heavily dripping, and her shoes were soaked with water.

My dear Diana, what made you come in this thunderstorm.

I know you promised, but you don’t have to keep your word in such a bad weather. Simply, you could have phoned me to cancel the appointment” 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, She smiled while pointing at her bare feet, as if apologizing for being without shoes.

I hugged and assured her: That is all right, don’t worry, dear Diana. Now take a seat and relax.

*** She sat in front of me at my working table.

For quiet a while, we watched in silence the rain. The only sound we heard is the rain striking the windows and rooftops, for so long a time that mesmerized us to think it is a symphony and that each drop of rain is an instrument in the orchestra, and to the question if the rain is a symphony then who is the conductor?

Diana broke the silence revealing these ideas: Can a mathematician be a conductor of the orchestra of rain and weather?

No, they can’t. Are you supposing mathematicians to be omnipotent gods?

No of course I don’t mean that.

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What I had in mind is: a group of knowledgeable mathematicians of the equations modeling the weather, and equipped with giant computers to manipulate the data obtained from the weather stations all over the globe, can’t they predict the weather.

I see what you mean. My answer is still “No," they can’t.

Back in the 1950s the first giant computer ENIAC was initially built mainly for several military purposes, among which is the control of the weather.

You realize controlling the weather is very important on a battle field.

Later on it was shown that even long distance weather forecasting is not possible.

Is that the reason why the televised weather do not forecast for more than a week in advance.

I think that’s the reason.

I know you’r eager to learn about women mathematicians. I planned to talk about Emmy Noether, an eminent mathematician. However, since our conversation has being diverted towards computers, I think it will be better if we postpone Emmy Noether talk for tomorrow.

Instead, let us consider today another woman, who is not a contributing mathematician, nevertheless, she had a demanding love for mathematics.

Historically, she is more famous for her role to computer and programming.

Have you ever heard of Byron.

No, I don’t think so.

You should.

I noticed that Diana was somewhat offended by my last remark, but she didn’t say a word I am not blaming you. It is the fault of your high school English literature curricula, not yours.

Is Byron the woman mathematician you intend to talk about?

Nnnnnnnno, yeeeeeeeees.

What do you mean by Nnnnnnnno , yeeeeeeeees?

“No” and “yes” are good answers to your question.

You see, Byron, who is better known as Lord Byron, was aa famous British poet (1788 - 1829) and a leading figure in the Romantic movement of the nineteenth century, and famous for his numerous love affairs, cannot be the woman mathematician I want to talk about. But on the other hand, that woman is the daughter of Lord Byron. She is better known as Ada Byron Lovelace, the woman who loves mathematics, She is remembered as the first programmer, the mother of modern computer and the daughter of Lord Byron. In fact she is the only legitimate child of the womanizing poet.

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A woman who loves mathematics and a daughter of a famous poet laden with scandal stories, sounds very interesting.

Certainly, I like to know more about her.

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Ada Lovelace 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.

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

You women, dress is the first thing that comes to your minds.

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’ve forgotten her birth year.

Let me to look it up.

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Unfortunately my “omniscience” iPad being out of charge, couldn’t be of much help to me. Thus, I referred “The Men of Mathematics”. Pointed to the book, i directed my talk to her Diana, You should remember this. It is the book that you paged through yesterday.

Yes, I do remember. The book that promoted the 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.

That professor must be a reactionary and idiot who, we shouldn’t care about 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.

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 mentioned his daughter in his poetry. The girl never saw her father.

This is unfortunate, for the family. How was Ada educated?

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, as I said a minute ago, 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 of high quality education in mathematics, science music and French.

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

Diana’s remark must be a kind of a reaction to the claim, made by a certain professor, that pursuing the field of in mathematics, for women 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 mathematician and Logician, Michael Faraday the well-known physicist, and 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?

Yes, indeed she fell in love, but not with a prince. She fell in love with Babbage’s calculating machine.

Diana, not able to control her amazement, screamed: FELL IN LOVE with a machine!!. She must have been in love with Babbage not with his machine!!

No, not with Babbage, but 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 his pet, just 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 by some to be a “father of the computer”.

He began in 1822 with what he called the “difference machine” 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 both knowledgeable in mathematics, and science, invited them too see one of his demonstration.

The young girl Ada, who seem to have inhered a poetic vision from her father and a demanding love for mathematics and science from her mother, was instantly captivated and fascinated by the idea.

She, determined to understand how the machine works, requested copies of the machine’s blueprints.

A professional lifelong friendship developed between Charles Babbage and the young aristocratic girl Ada Byron.

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Do you mean because of her love to that machine, 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. His wife thus became Lady Ada King, Countess of Lovelace, but generally known as Ada Lovelace.

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

I don’t really know. It looks like it was a 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 of mere 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 mother and husband.

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

She certainly pursued learning more advanced mathematics, but did not have any contribution in the advancement in that field, as far as I know.

All the time Ada never forgot about Babbage Engine. She kept his Engine in mind.

My talk was interrupted by a phone call from a friend. answering I completed my talk:

After briefly

Meanwhile, Babbage had moved on from the rudimentary difference machine to a machine of a 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 Babbage part. It was the first world’s programmable computer, a much more capable of performing sophisticated calculations than the original Difference Machine.

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 be built 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 beam of light entered the windows signified the end of the storm and revived our discussion of

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unpredictability of the weather for a few minutes. But we soon 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 powered computer 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 antiquated specimen to be displayed.

I wish I would see it when I visit London.

Be sure to accompany your brothers with you.

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

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

Let us leave our day - dreams aside for a more appropriate time and continue our discussion.

I can presuppose that Ada “broke up” with the old Difference 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 at all about his ideas of Analytic Engine. But he used to lecture about it.

The British government was not so enthusiastic about financing the construction of the Analytic Engines. However, his ideas were more receptive by the European continental governments. He was invited by the Italian government to lecture on the Analytic Engine. Notes were taken of the lecture by Luigi Menabrea, an Italian army engineer (who, by the way, became a prime minster of Italy twenty - seven years later). Luigi 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 the form of notes. The adjoint notes turned out to be three times more extensive than the original article.

Diana’s facial looks and body gestures reveal that she began to discard her initial sarcastic outlook and to view the consideration from a realistic and more serious point of view, and to appreciate the role of Ada Lovelace.

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

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It is true that the English translation of the article reflects Ada Lovelace thoughts of the various potentialities of the Analytic Engine, which Babbage himself was mostly unaware of.

What potentialities of the Engine, that were described by Ada’s added notes?

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

She realized that the Babbage engine could go far beyond numbers. This was the first ever perception to a modern computer, which is not a mere calculator, but a machine that could contribute to other fields of human enterprise. She speculated that anything,(including music, text, picture and sound) that could be translated to 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 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.

She described how the engine could tackle problems that were unsolved, e.g., astronomical tables, generate random numbers, complex numbers, complex, sequences of numbers.

One should always keep in mind that Ada was describing a machine that did NOT exist. Furthermore, her notes contained an actual step by step method for computing numbers, which mathematicians call Bernoulli numbers. She explained how and where to set calculations and read results. Today, such step by step actions to be performed is called an algorithm. This was an impressive work, especially when we realize that she had no machine to check the workability of her programs.

A replica of that particular historical program is depicted in Wikipedia under the name “Ada Lovelace�.

Though realizing that the Analytic Engine did not exist at that time, yet still I am curious, whether there were attempts, during the nineteenth and early twentieth centuries, to construct an Analytic Engine.

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

There were few attempts to construct such an Engine in which Babbage and Ada collaborated, but those attempts seemed to be destined to failure due to variety of reasons, among which Charles Babbage ran into many financial difficulties and Ada Lovelace periodic illnesses and her early death.

Oh poor soul, what caused her death?

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For quite some time, she was suffering from cervical cancer. She had been in pain for several years and was given pain killing drugs by her physicians to help her cope with it. 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 Lovelace is to be admired for her perseverance. Despite her health issues she kept on. She, knowing possessing a gift and a keen love for mathematics retuned to it again and again for complex reasons — as a moral compass and that which is most nourished her mind.

In the 1970s, a new high - order computer language was developed by the American Department of Defense. The new language was named “Ada” in honor to Ada Lovelace. Moreover, the American National Standards Institute

approved “Ada” as a national all - purpose computer language.

In 2009, a movement was launched 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 and acknowledgment and honoring Ada Lovelace’s amazing 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. *** Now it is time to go home. I had promised my Mom to go shopping together. Thank you Jeddoo for this illuminating discussion. I really loved Lovelace and am looking forward to hear about the next woman mathematician. What did you say her name, Emmy what?

Emmy Noether is the name.

Tomorrow, I have a full day schedule. dinner to learn about Emmy Noether.

I’ll come right after

Please remember to call if there happens to be a thunder storm.

Don’t forget your shoes in the bathroom!!

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Tuesday

Emmy Noether The most important woman in the history of mathematics

After dinner, while pondering on my talk about Emmy Noether, I brought down her picture from the wall of my study, where there hang several pictures of famous scholars who influenced my academic life.

Thinking to start my talk immediately the moment Diana comes in, I put the picture right on my working table. However, when she did come, didn’t immediately notice the picture. Thus, we started our conversation on general topics: It is so 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.

I think the Biology course was the most boring.

No, Jeddoo. it was the Calculus.

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

Can you guess what I had for dinner?

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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, directed her look 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 in expressing 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 know who they are.

I hadn’t seen these pictures before.

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 admiration.

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of people that inspired your

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

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 4

Kubba is a well known dish all over the Middle East.

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They are Einstein, Bertrand Russell, David Hilbert, Kurt Gödel, William Shakespeare, Issac Newton, Fredrick Gauss, Mohammad AlJwahiry and Evariste Galois.

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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. cheeks, she declared: This is a difficult question.

Newton further discover?

A blush painted her

I don’t think I know.

What did

It was the Calculus. The very subject that bored you.

Moving her lips as if pretending to smile, she 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 at his teens, to settle a problem which had 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 briefly introducing Diana, to each of the scholars, 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, seemed to take my last remark as pointing to her, so she quickly responded by a disclaimer: I am one of those people, but it wouldn’t take it long to be knowledgeable about her. Thanks Jeddoo for educating me.

I continued.

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 back in 1882.

Now, Bavaria is part of Germany.

Being a woman, she was not allowed to attend regular college preparatory schools. Thus, she attended a ‘finishing school’ specializing in French and 20

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 Again, I’m not sure.

But it must have counted, at least in Erlangen University, since, Emmy, after struggling with the administration completed 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.

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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? special about Gottingen University?

And what was so

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 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, Lasker6 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 loaded 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.

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Lasker was the world champion of chess.

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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 interrupted my talk to open the computer-file of my saved documents for the article, “Abstraction," which I had written several years ago7. I printed the article and handed it to Diana saying: This is a homework for you. I hope, after reading the article at home, the ring of integers would flash upon hearing “ring”.

Giving homework is a ritual of professors. I’ll read it carefully Thank you, Jeddoo. *** After the interruption I continued my talk: 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: 7

See Appendix 1.

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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.

*** 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:

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“What will our soldiers think when they return to the university and find they are to learn at the feet of 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.

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 has 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?

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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 her clever joke and continued: 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 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 post-operative 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 remembrance8.

8

see Appendix 2.

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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 my understanding of the cultural nature of mathematics.

Few people have a Jeddoo like you.

Thank you Only a few jeddoos are as fortunate as me to have such a remarkable intelligent granddaughter like you.

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

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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 pregnant 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 some properties of concrete objects and retaining others It is an essential part of any intellectual activity. Its importance is derived from the ability of the mind to ignore irrelevant details and forming new conceptual objects and from the use of names to reference to new object.

Abstraction is used in the arts as any object or.image from which, has been distilled from the real world.

Mathematics seems to be the best field to illustrate abstraction, its levels and its importance. In fact it is so to speak a “weapon of mass abstraction” whose purpose is to understand the world and to develop itself.

*** 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) Consider the following scenarios of caricatured history: 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 completely, 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 other s 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 in1932 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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