Tanya Khovanova's Math Blog

I was terribly shy when I was a teenager. I worked on this problem and overcame it. But when I moved to the US my shyness returned in a strange form. I was fine around Russians but shy around Americans. At first I assumed that it was a language problem.

I became friends with a Russian sexologist and psychotherapist. He pointed out that I never initiated a conversation with Americans and so I realized that my shyness had returned. He prescribed an exercise for me: I had to invite a new American guy to lunch once a week.

Why guys? Maybe because he was a sexologist or maybe because my problems with self-esteem were more pronounced when I was around men. In any case, I decided to do the exercise.

To paint the full picture I need to add some relevant details. At that time I was married, although I didn’t wear a ring, and wasn’t especially interested in other men. The reason I didn’t wear a ring was that Joseph, my husband at the time, did not himself want to wear a ring. As I love symmetry in relationships more than I love rings, I didn’t wear one either.

The men I was about to invite to lunch were mere acquaintances, because I had not yet made any American friends. So although I didn’t intend to hide it, they may not have realized that I was married.

Two things surprised me in this exercise. First, it was very easy. Most people agreed to do lunch with me.

Second, every man I invited mentioned his girlfriend. This was unexpected. From my experience with Russians, I anticipated that every man would hide his involvement with someone else, even with a wife, at least for some time. At the very least, many Russian men would try to flirt.

The Americans were different. Unclear why I had invited them out, they wanted to be upfront with me from the start, just in case I was interested in them. Since that experience, I admire the way that American men come clean.

I never invited any of these guys out twice: I just needed a supply of new men for my exercise in overcoming my shyness. I wonder if they thought I was put off by their confessions. Perhaps my loss of interest in them after the first lunch confirmed their suspicions that I was attracted to them.

The sexologist’s exercise was a success. Today I have no trouble inviting someone to lunch.

I am used to thinking that a “woman in numbers” means a female number theorist. But not anymore. I just discovered drawings by Svetlana Bogatyr. From now on the expression a “woman in numbers” will convey an additional meaning to me.

I am grateful to Svetlana for permitting me to post several of her drawings. The “Mature Woman” is on the left. “Eurydice”, “Girl in Scarf” and “Holland Woman ” are below.

Enjoy.

This year I am again on the organizing committee of the Women and Mathematics program at Princeton’s Institute for Advanced Study. Our subject is “p-adic Langlands Program.” It is a fashionable, advanced and very influential program connecting number theory and representation theory.

We invite undergraduate students, graduate students and post-docs to apply. In 2009-2010 the Institute has been running a special year in Analytic Number Theory. That has brought many number theorists to the institute already, so there will be a lot of people to talk to.

Last year I promised to hold a math party during the program. But I had to cancel it due to a scheduling conflict with George Hart’s ZomeTool Workshop. I am planning a party this year. Either way, we’ll have fun.

If you want to learn about the Langlands program, to spent time on the beautiful grounds of the Institute, to eat in one of the best cafeterias around, and to make new friends with other women interested in number theory, then please apply. The application deadline is February 20.

Israel Gelfand’s memorial is being held at Rutgers on December 6, 2009. I was invited as Gelfand’s student.

My relationship with Gelfand was complicated: sometimes it was very painful and sometimes it was very rewarding. I was planning to attend the memorial to help me forget the pain and to acknowledge the good parts.

I believe that my relationship with Gelfand was utterly unique. You see, I was married three times, and all three times to students of Gelfand.

Now that I know that I can’t make it to the memorial, I can’t stop wondering how many single male students of Gelfand will be there.

Israel Gelfand was my scientific adviser from the time I was 15. This is the story of how Gelfand helped me, when at 20 I was an undergrad at Moscow State University. At that time, I was married to Sasha (Alexander) Goncharov, who was also Gelfand’s student.

Sasha was more driven by mathematics than I. I had a lot of different interests: I wanted to hang out with friends, go to movies and read books. Sasha only wanted to do mathematics. His only other obsession was with what our colleagues (including me) were doing mathematically. So he was constantly asking me about the math problems I was thinking about.

For example, I was sitting at my side of the desk working, and he asked me to tell him about my problem. A few minutes later, I was forced to interrupt my work to go grocery shopping, because the household chores fell to me. As soon as I returned with bread and milk, Sasha excitedly told me the solution to my problem. It made me feel stupid, as if I should have solved it while I was waiting in the line for bread and milk. That feeling blocked out all the other feelings I should have been noticing, such as frustration and annoyance with Sasha.

Without his interference, I would have happily solved the problem myself. I was about to start my serious research, but I worried that I’d end up as a supplier of new problems for his papers.

You might wonder why I didn’t stop sharing my math with Sasha. But at that time, I wasn’t very in touch with my feelings and I prided myself on being a logical person. The idea that a husband and wife would discuss their work together seemed logical. Besides, even though I wasn’t particularly interested, Sasha was always ready to tell me about his math problems. It seemed important for me to be fair and to reciprocate. So I was stuck in a situation I didn’t know how to resolve.

I never confided this issue to any math colleagues. I never mentioned it to Gelfand — mostly because I was too scared of him to initiate any conversation. Besides, Gelfand delegated most of his responsibilities to others, because he was quite famous and busy. For example, all official paperwork related to his adviser role was done by Alexandre Kirillov. With me avoiding Gelfand and Gelfand being busy, we almost never spoke one-on-one.

You can understand my surprise when one day Gelfand approached Sasha and me to have a chat. He told us that we were about to start our own research, and while he permitted me to ask Sasha about what he was doing, he would not allow Sasha to interfere with my research.

Gelfand was a great judge of character. Without anyone telling him, he perceived what was going on in our marriage and gave me an excuse to stop Sasha’s prying. It was an appreciated gift.

Decades ago there was a study in Russia that claimed that a woman worked four more hours a day than a man on average. Men and women were equal in Russia and all had the same 40-hours-a-week jobs. Women were not, by and large, housewives, for they worked full-time.

So where did the additional four hours come from? They were devoted to house chores. In Russia, women did everything at home — at a time when life in Russia was much more difficult. For example, my family didn’t have a washer, or a dryer or a dish-washing machine. Plus, everything was in deficit, so to buy milk or a sweater, women had to stand in lines, sometimes for hours.

My mother was very bitter because her husband, my father, never helped her. So I always hoped that when I got married, my husband would take on some of the house chores.

When I married Andrey, he was somewhat helpful — better than the average Russian husband. Then, when I was at grad school, we had a baby named Alexey. Andrey convinced me that I had to take over all the child care because only women could get academic maternity leave. It seemed logical and I agreed.

In a year, when the leave was over, I felt that Andrey should take over some of these duties. He refused. He insisted that since I already had published a paper when I was an undergrad, and since he still didn’t have his research results for his PhD, that he had to stay focused on his work. I wasn’t strong enough to resist.

We signed up for government child care — private care didn’t exist — but we were on the waiting list for a couple of years. Almost no one in Russia — certainly not graduate students — could afford a private babysitter. I couldn’t really work on my PhD research because between caring for the house and the baby, I never had big chunks of time. The best I could do was to start preparing for my qualifying exams.

Allow me to digress from my main story for a moment to mention my gray notebook. This notebook was our baby diary. Initially I recorded important baby data — like the first time Alexey smiled. But later, as soon as Alexey turned one year old, he became very eloquent; and this notebook became my son’s quote book.

One day Andrey and I went out and my mom babysat Alexey, who was two years old. When we returned, my mother recited the following quote from Alexey:

When will Daddy be back from the university and Mommy from the store?

I don’t really remember the long hours in stores or the cooking and cleaning. I remember the quote.

Just out of curiosity I googled two phrases: “male mathematician” and “female mathematician”. The results for these phrases on May 3, 2009 were:

• male mathematician — 824
• female mathematician — 5,680

Why do you think that female mathematicians are more popular than male mathematicians? I think it is because when people hear the word mathematician, by default they picture a man, so the phrase “male mathematician” is perceived as pleonastic.

I decided to look at some of the 824 sites talking about “male mathematicians”. Many web-pages containing the words “male mathematician” are actually pages about female mathematicians, where there is a need to mention a mathematician of the opposite sex. Many other sites are dating pages where a mathematician looks for a partner, and it is wise to start with a description of the sex of the seeker.

Speaking of dating, did I ever mention that I am a female mathematician who was married three times, and all of my spouses were male mathematicians?

Until the introduction of the Abel prize, the Fields medal was the most prestigious prize in mathematics. The medal has been awarded 48 times and all of the recipients have been men. Can we conclude that women are inferior to men when it comes to very advanced mathematics? I do not think so.

The Fields medal was designed for men; it is very female-unfriendly. It is the prize for outstanding achievement made by people under age 40. Most people start their research after graduate school, meaning that people have 10-15 years to reach this outstanding achievement. If a woman wants to have children and devote some time to them, she needs to do it before she is 40. That puts her at a big disadvantage for winning the medal.

Recently the Abel prize for mathematics was introduced. This is the math equivalent of a Nobel prize and nine people have received the prize, all of them male. The Wolf prize is another famous award: 48 people have received it so far and they too have all been male.

On the grand scale of things, women have only recently had the option of having a career in mathematics. Not so long ago it was considered quite exceptional for a woman to work in mathematics. The number of female mathematicians is increasing, but as this is a new trend, they are younger people. At the same time, Abel prizes and Wolf prizes are given to highly accomplished and not-so-young people. That means the increase in the percentage of women PhDs in mathematics might affect the percentage of females getting the prize, but with a delay of several dozen years.

There are other data covering extreme math ability. I refer to the International Math Olympiad. The ability that is needed to succeed in the IMO is very different from the ability required to succeed in math research. But still they are quite similar. The IMO data is more interesting in the sense that the girls who participate are usually not yet distracted by motherhood. So in some sense, the IMO data better represents potential in women’s math ability than medals and prizes.

Each important math medal or prize is given to one person a year on average. So the IMO champion would be the equivalent of the Fields medal or the Wolf prize winner. While no girl was the clear best in any particular year, there were several years when girls tied for the best IMO score with several other kids. For example:

• In 1995, 14 students tied for the perfect score; two of them were girls. (Maryam Mirzakhani and Chenchang Zhu)
• In 1994, 22 students tied for the perfect score; two of them were girls. (Theresia Eisenkölbl and Catriona Maclean)
• In 1991, 9 students tied for the perfect score; one of them was a girl. (Evgenia Malinnikova)
• In 1990, 4 students tied for the perfect score; one of them was a girl. (Evgenia Malinnikova)
• In 1987, 22 students tied for the perfect score; one of them was a girl. (Jun Teng)
• In 1984, 8 students tied for the perfect score; one of them was a girl. (Karin Gröger)

In one of those years, a girl might have been the best, but because the problems were too easy, she didn’t have a chance to prove it. Evgenia Malinnikova was an outstanding contender who twice had a perfect score. In 1990, she was one out of four people, and she was younger than two of them, as evidenced by the fact they they were not present in 1991. Only one other person — Vincent Lafforgue — got a perfect score in 1990 and 1991. We can safely conclude that Evgenia was one of two best people in 1990, because she was not yet a high school senior.

This might be a good place to boast about my own ranking as IMO Number Two, but frankly, older rankings are not as good as modern ones. Fewer countries were participating 30 years ago, and China, currently the best team, was not yet competing.

Girls came so close to winning the IMO that there is no doubt in my mind that very soon we will see a girl champion. The Fields medal is likely to take more time.

A new study is out. Dr. Satoshi Kanazawa blogged about his new discovery. He claims that he can explain why many studies show that men have higher intelligence than women. In his posting Why men are more intelligent than women he posits that intelligence is correlated with height. Taller men on average are smarter than shorter men.

He also claims that given the same height, women are more intelligent than men. As he puts it: “Women who are 5’10” are on average more intelligent than men who are 5’10”.” According to him the only problem women have in the brains department is that they aren’t tall enough.

I couldn’t find the study itself, only his own announcement. My problem with his study is that, according to Dr. Kanazawa’s description, he used data from the National Longitudinal Study of Adolescent Health. If we are talking about adolescents we need to take into account that children grow taller and smarter with age. So any difference in intelligence among the kids of different heights may easily be explained by age. Taller boys may be older and therefore likely to perform better on intelligence tests.

As to intelligence differences between boys and girls of the same height, these girls who are 5’10” may be from higher grades than boys who are 5’10”.

Some IQ tests are designed to take age into account, but what’s totally weird is that a scientist is studying the correlation of intelligence and height, using a population that is in the middle of a physical and mental growth spurt.

As a mathematician, I’ve been around many people of great intelligence, but I’ve never perceived bright people as especially tall.

Here is the picture of the USAMO winners for 2007. These are certainly extremely smart kids and I know their heights. The sixth student on the left is my son, Sergei. I personally met all these kids and compared to their peers they are not — except for one boy — especially tall.

I bought the book “Math Doesn’t Suck: How to Survive Middle School Math Without Losing Your Mind or Breaking a Nail“ by Danica McKellar because I couldn’t resist the title. Sometimes this book reads like a fashion magazine for girls: celebrities, shopping, diet, love, shoes, boyfriends. At the same time it covers elementary math: fractions, percents and word problems.

You can apply math to anything in life. Certainly you can apply it to fashion and shoes. I liked the parallel between shoes and fractions that Danica used. She compared improper fractions to tennis shoes and mixed numbers to high heels. It is much easier to work with improper fractions, but mixed numbers are far more presentable.

Danica is trying to break the stereotype that girls are not good at math by feeding all the other stereotypes about girls. If you are a typical American girl who hates math and missed some math basics, this book is for you. If you want to discover whether the stars are on your side when you are learning math, the book even includes a math horoscope.