Tanya Khovanova's Math Blog

When I came to the US, I heard about Mensa — the high IQ society. My IQ had never been tested, so I was curious. I was told that there was a special IQ test for non-English speakers and that my fresh immigrant status and lack of English knowledge was not a problem. I signed up.

There were two tests. One test had many rows of small pictures, and I had to choose the odd one out in each row. That was awful. The test was English-free, but it wasn’t culture-free. I couldn’t identify some of the pictures at all. We didn’t have such things in Russia. I remember staring at a row of tools that could as easily have been from a kitchen utensil drawer as from a garage tool box. I didn’t have a clue what they were.

But the biggest problem was that the idea of crossing the odd object out seems very strange to me in general. What is the odd object out in this list?

Cow, hen, pig, sheep.

The standard answer is supposed to be hen, as it is the only bird. But that is not the only possible correct answer. For example, pig is the only one whose meat is not kosher. And, look, sheep has five letters while the rest have three.

Thus creative people get fewer points. That means, IQ tests actually measure how standard and narrow your mind is.

The second test asked me to continue patterns. Each page had a three-by-three square of geometric objects. The bottom right corner square, however, was empty. I had to decide how to continue the pattern already established by the other eight squares by choosing from a set of objects they provided.

This test is similar to continuing a sequence. How would you continue the sequence 1,2,3,4,5,6,7,8,9? The online database of integer sequences has 1479 different sequences containing this pattern. The next number might be:

• 10, if this is the sequence of natural numbers;
• 1, if this is the sequence of the digital sums of natural numbers;
• 11, if this the sequence of palindromes;
• 0, if this is the sequence of digital products of natural numbers;
• 13, if this is the sequence of numbers such that 2 to their powers doesn’t contain 0;
• 153, if this is the sequence of numbers that are sums of fixed powers of their digits;
• 22, if this is the sequence of numbers for which the sum of digits equals the product of digits; or
• any number you want.

Usually when you are asked to continue a pattern the assumption is that you are supposed to choose the simplest way. But sometimes it is difficult to decide what the testers think the simplest way is. Can you replace the question mark with a number in the following sequence: 31, ?, 31, 30, 31, 30, 31, … You might say that the answer is 30 as the numbers alternate; or, you might say that the answer is 28 as these are the days of the month.

Towards the end of my IQ test, the patterns were becoming more and more complicated. I could have supplied several ways to continue the pattern, but my problem was that I wasn’t sure which one was considered the simplest.

When I received my results, I barely made it to Mensa. I am glad that I am a member of the society of people who value their brains. But it bugs me that I might not have been creative enough to fail their test.

One day I got a phone call from Victor Gutenmacher, one of the members of the jury for the USSR Math Olympiad. At that time I was 15 and had won two gold medals at the Soviet Math Olympics. Victor asked me about my math education. I explained to him that although I went to a special school for gifted children, I wasn’t doing anything else. In his opinion, other kids were using more advanced mathematics for their proofs than I was. He said I was coloring everything in black and white; other kids were using calculus, while I was only using elementary math. He asked me if I would like to learn more sophisticated mathematics.

I said, “Sure.” After considering several different options, Victor suggested Israel Gelfand’s seminar at Moscow State University. He told me that this seminar might suit me because it starts slowly, picking up pace only at the end. He also told me that the seminar was like a theater. Little did I know that I would become a part of this theater for many years to come. I also didn’t know that I would meet my third husband, Joseph Bernstein, at this seminar. Joseph used to sit in the front row, and I watched his back at the seminar for more years than I later spent together with him.

The next Monday evening, I went to the seminar for the first time. Afterwards, Gelfand approached me and asked me if I had an academic adviser. I said, “No.” He asked me how old I was. I said, “Fifteen.” He told me that I was too old and that I had to choose an adviser without delay. I said, “But I do not know anyone and, besides, I need some time to think about it.” He replied, “I’ll give you two minutes.”

I paced the halls of the 14th floor of the Moscow State University for a couple of minutes, pretending to think. But really, I didn’t know about any other options. He was the only math adviser I had ever met. So I came back and asked Gelfand, “Will you be my adviser?”

He agreed and remained my adviser until I got my PhD 14 years later.

My scientific adviser Israel Gelfand was one of the greatest mathematicians in Russia. His seminar was famous.

One of the unique features Gelfand invented for his seminar was a role for a seminar participant that he called a designated listener (kontrol’nyy slushatel’ in Russian). I played this role for four years.

This is how it works. The speaker starts his lecture and Gelfand interrupts him. He then turns to me and asks if I understand what the speaker just said. If I say “no,” he says that I am a fool. If I say “yes”, he invites me to the blackboard to explain. Usually, Gelfand finds some fault in my explanation and calls me a fool anyway. As a result, whatever I do, I end up as a fool.

Ironically, I admired Gelfand for the way he conducted his seminars. I went to so many seminars where it was clear that no one understood anything. He was the only professor I knew who made sure that at least one person at his seminar — himself — understood everything.

The problem was that he convinced me that I really was a fool. I dreaded Mondays and I considered quitting mathematics. The situation changed when I started dating Andrey, my future second husband. He made a strong effort to convince me that I was not a fool; rather, Gelfand was a bully. I understood what Andrey was saying, but I wasn’t able to take it to heart. Not that I trusted my supervisor more than my lover, but I was more willing to believe that something was wrong with me than with someone else.

Andrey’s hard work wasn’t in vain. One fine day Professor G. from Western Europe was invited to give a talk at Gelfand’s seminar. During his talk Gelfand interrupted him many times, told him that he wasn’t a good lecturer, and that his results were neither interesting nor meaningful. After several hours of torture Professor G. became tearful. At that moment it hit me that Andrey was right. I am thankful to Professor G. for his tears; they opened my eyes.

The next step for Andrey was to convince me to resist Gelfand. His idea was for me to tell Gelfand, the next time he asked me if I understood: “Go f**k yourself!” (I mean the Russian equivalent).

At that time, I had never pronounced the f-word, even in my own head. But I didn’t have any other ideas. So I started preparing myself to do this. Finally one day I was ready. Gelfand interrupted the speaker and turned towards me as if he were about to ask me to be the designated listener. I looked back at him. He paused, looked at me again, and turned around. He never asked me to be the designated listener again.

In this essay I would like to explain why I am not yet a professor of mathematics.

Today at 49 I am still in search of my dream job. My gender is not the main reason that I don’t have an academic position or another job I like. My biggest problem is myself. My low self-esteem and my over-emotional reactions in the past were the things that most affected my career.

I remember how I came to Israel Gelfand’s seminar in Moscow when I was 15. He told me that I was too old to start serious mathematics, but that he would give me a chance. He said that at first I might not understand a thing at his seminar, but that every good student of his comes to understand everything in a year and a half. The year and a half passed and I wasn’t even close to understanding everything. Because of this I was devastated for a long time.

I had always had problems with my self-esteem and being a student of Gelfand just added to them. My emotional reactions, while they impacted my work in mathematics, were not exclusively related to mathematics. When my second divorce started, not only did I drop my research, I quit functioning in many other capacities for two years.

I was extremely shy in my early teenage years. By working with myself, I overcame it. When I moved to the US, my shyness came back in a strange way. I was fine with Russians, but behaved like my teenage self with Americans. For two years of my NSF postdoc at MIT, I never initiated a conversation with a non-Russian.

For the second time I overcame my shyness. Now, if you met me in person, you wouldn’t believe that I was ever shy.

I became much happier in the US, than I ever was in Russia, but still my emotions were interfering with my research. Because it was so difficult to find an academic job here, I felt tremendous pressure every time I sat down with a piece of paper to work on my research. My mind would start flying around in panic at the thought that I wouldn’t find a job, instead of thinking about quantum groups.

Over all, I think that my inability to control my emotions, together with my low self-esteem, might have impacted my career much more than the fact that I am a woman, per se. Being a mathematician is not easy; being a female mathematician is even more difficult. Still, in my own life, I know I can only blame myself.

The good news is that I have changed a lot, after many years of self-repair. This is why I have made the risky decision, at the age of 49, to try to get back to academia. And this time I have a great supporter — my new self.

This story happened at a colloquium that was conducted during the Women and Math program at Princeton last year. The room was full of women waiting for the colloquium to start. A young man appeared at the door. He looked around in complete surprise. I watched the fear fade from his face when he must have decided that he had the wrong room. He disappeared, but reappeared at the door very soon. He had obviously checked the schedule and had realized that, in fact, he had come to the right room in the first place. His face started changing colors. He was terrified. A few minutes later, he left.

I sat there thinking: women have to deal with this type of situation every week. He could afford to skip just one lecture, but if a girl wants to do math she has to be courageous almost every seminar. I mentally applauded the girls around me for being that courageous.

Wait a minute! I am a woman myself. I went to seminars where I was the only girl hundreds of times. How did I feel? Actually I think my mind never registered that I was the only girl. I never cared. The first time I really thought that the gender of people in a room might be an issue was during that colloquium last year.

I started wondering why it had never bothered me. Could it be that the Soviets did a good job of teaching me not to pay attention to people’s gender? Could be. But wondering back in time, I remembered something else too.

When I was a child I wasn’t a girly girl. I was not interested in dolls; I preferred cars. I didn’t play house or doctor; I played war. To tell the whole truth, I actually did have a doll that I loved, but I never played with it. I liked having it. The doll was a gift from my father’s second wife and it was way beyond my mother’s price range. I think I had an admiration for the quality and the beauty of this toy.

So, while appreciating the courage that the other girls might have needed to do math, I was sitting there pondering my own indifference to the gender of people at seminars and my relative comfort with large groups of men. But this comfort had its own price. I felt comfortable with the group I wasn’t a part of, while I felt different from the group I was a part of. My price of being comfortable at math seminars was loneliness.

I am interested in a career in mathematics. How hard is it to be a woman mathematician?

Let us look at some numbers from the American Mathematical Society Survey Reports for the year 2005:

• There were 40% of women among graduating math majors.
• There were 30% of women among Math PhDs granted.
• There were 11% of women among full-time tenured or tenure-track positions.

You can’t just say that women do not like math — 40% of those choosing math as a major is quite a large number, after all.

On the other hand, the downward trend of these percentages is striking. If women’s opportunities and abilities are the same as men’s, these percentages should grow with every age step, since, as we know, the percentage of women in the population increases with age due to men dying earlier.

But the numbers go down and very fast. There are many potential explanations for this, but today we’re going to look at one of them:

Women have less ability for high-level mathematics.

Was Larry Summers right when in his speech that cost him his Harvard presidency he compared math ability to height and to the propensity for criminality, and suggested that the distribution, especially standard deviation, of math ability differs for men and women?

To answer this question, I wanted to find some other data that correlates gender with math abilities. I took the results of the American Mathematical Competitions (AMC 12) for the year 2008. Among 120,000 students who participated, 43% were females. Here are some results:

• Among students scoring 72 points or higher there were 40% of girls.
• Among students scoring 98 points or higher there were 30% of girls.
• Among students scoring 134.5 points or higher there were 11% of girls.

This picture is similar to that of the academic career: the closer you climb to the top, the smaller percentage of girls you see there. Of course, winning a competition is very different from getting tenure. People who win competitions are smart and competitive — smart and competitive enough to go for money, rather than academia. On the other hand, people who are interested in mathematics often are not interested in anything else. Why would they waste their time in competitions when the Riemann Hypothesis is still waiting to be solved?

But still, both achieving tenure and winning math competitions represent mathematical ability in some sense. If Larry Summers was right and the distribution of math ability is different among males and females, then by looking around you at the percentage of females at your level, you should be able to assess how close you are to the top of the math field.

I propose the following math career predictor: Take your results in AMC 12. If among kids who did better than you, the percentage of girls is more than 11%, you do not have a chance at tenure. If the percentage of girls is more than 30%, do not waste your time working on a math PhD. If the percentage of girls is more than 40% maybe math majoring is not for you.

I hate my math career predictor. I hate it not only because it has so many flaws that it might just deserve the Ig Nobel Prize, but because it doesn’t take people’s effort into account. You really have to work very hard to be a math professor, whether you were a winner or a loser in math competitions.

You might ask why I created a math career predictor that is so flawed. My mathematician friends, those who are more honest than polite, tell me that I have no chance at getting back to academia. On the other hand, I had the second best result at the 1976 IMO, which means I have the ability. My predictor may be my only hope.

I resigned from my job half a year ago. If you remember I was planning to rebuild my career in academia and to find myself along the way.

The problem with re-entering academia is that to find a job starting September 2008, I would have had to apply in December 2007. I really didn’t want to apply before finding my new direction in mathematics and publishing some papers to rebuild my name. So I decided to apply for academic jobs in December 2008, while using the intervening year for research and to try to publish.

The problem with delaying the start of my new academic job until September 2009, is that I didn’t save enough money to cover such a long period of time. Even though I cut down my expenses significantly, I still need some additional income in the meantime. However, temporary jobs consume the time I need for research in order to go back to academia. To resolve this Catch-22 situation, I decided to choose jobs of only two kinds: first, either they pay a lot, so for a limited time of work I can buy a lot of extra time for my research; or second, they’re aligned with my goals. I wasn’t yet looking very hard for work, but nonetheless several jobs came my way.

One of the jobs I accepted was a temporary job as a math competition coach at Advanced Math and Science Academy (AMSA) Charter School. This gave me a chance to check once again how I feel about teaching. I love entertaining people with mathematics. I showed magic tricks to my students, played games with them and so on. I enjoyed myself; my students liked me. But I do not know enough tricks or games to teach 24 hours a week as a regular school teacher. I decided that I really do not want to be a school math teacher. But I love being a coach, because it takes less time and I get the best kids.

I also worked on the Organizing Committee for the Women and Mathematics Program in Princeton in May 2008. The irony is that I lived in Princeton for seven years and ignored this program for most of them. Initially I was prejudiced against such a program. I felt that I should go to lectures only for their mathematical value. The gender of the lecturer doesn’t matter.

I think I was missing the whole point of the program. I should write about this program more.

The surprising result of the last half year is that I am having a blast blogging. Wouldn’t it be fabulous to find a job in mathematical journalism, if such a profession exists.

Three years after I moved to the U.S. and had my second child here, I got an NSF postdoc position at MIT. I had had a long break from my previous research in integrable systems, so I decided to start something new for me — quantum groups. Soon afterwards I invented tetramodules and published a paper about them.

Some time later a friend of mine showed me some papers where my tetromodules had been described long before me. I felt such pain, that I couldn’t work.

I was ashamed that the quality of my paper was compromised. I was very scared that someone might think that it wasn’t original work. I felt worthless for having wasted so much energy on something that was already known.

Afterwards, I discovered that tetramodules appeared in the literature under many other names: bidimodules, Hopf bimodules, two-sided two-cosided Hopf modules, 4-modules, bicovariant bimodules. That meant that many people were reinventing them at the same time. I felt slightly better knowing that I wasn’t alone, but was in fact part of a crowd. Still, my pain impeded my ability to work and do research.

I have just read an article in the New Yorker entitled “In the Air” about some famous discoveries made simultaneously by several people. It made me realize that I am still not over that tetramodules story from 15 years ago.

Being a mathematician requires one to be emotionally strong. You need high self-esteem. Do I need to overcome my emotions or is there a niche for very emotional mathematicians? Perhaps blogging about math is such a niche.

My Number Gossip page is on hold. I am out of work and can only afford to spend time on things that can bring me my next job. As a result, numbers suffer. If you would like to support the website, consider donating to Number Gossip. With your donations, I will be able to spend more time on the website. There are many things I would like to do. Here are the three most important and fun areas that I would chose to work on.

1. Do you know that 40(forty) is the only number whose constituent letters appear in alphabetical order? I have about 1,000 unique number properties like this on my number gossip website. I also have about 600 properties in my database that need to be checked before adding them to the website. Being a perfectionist I double-check and recheck all the properties. As a result, my webpage is quite rigorous and reliable. As a downside I need a lot of time to add new unique properties to the website and 600 of such properties are waiting impatiently to join their comrades in the public view.
2. Have you heard about aspiring or untouchable numbers? What about practical or perfect numbers? There 50 famous properties like that that my website currently allows you to check for your favorite number. For example, 33 is deficient, evil, lucky and odd at the same time. There are many other properties I plan to add in the future. If you would like to know whether or not your favorite number is brilliant, fortunate or primeval, consider donating to Number Gossip.
3. Currently the limit for numbers you can input is 10,000. Some of the basic properties are very difficult to check. The expansion is not trivial and will require significant tuning of my algorithms. Even so, I would like to do that.

I’ve devoted many years to this project, and now I need some financial help. If you know a person or a company who wants to sponsor numbers, please ask them to contact me at (tanyakh at yahoo).

I did it. I handed in my resignation letter to my boss. I’m resigning effective Jan 3, 2008. If you want to know why I’m waiting until next year, I can give you several reasons.

• First, Christmas-time is usually the most enjoyable work time because no one is there. It is quiet.
• Second, I’m superstitious: I believe the way I greet the New Year determines how the New Year is going to go, so I want to be employed at the strike of the midnight clock.
• Last but not least, it appears that to get my company’s annual profit-sharing bonus, I have to be employed on December 31^st.

I am happy and sad at the same time. In four and a half years I’ve made a lot of friends and accomplished a lot professionally. Now it is my time to move forward. Where is forward? It is in the direction of a cemetery, but I would rather be doing something more meaningful to me than battle management while I am slowly crawling there.