Tanya Khovanova's Math Blog

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.

— Do you love tomatoes?
— Eating them — yes; otherwise not so much.

The word “love” expresses an emotion. But the range of emotion it can span is an enormous interval between a slight preference and a burning desire.

— Do you love tomatoes?
— I love tomatoes so much that I eat them with ketchup.

Still we can usually figure out the intensity of this emotion from the context. When someone says that he loves M.C.Escher, nobody concludes that he is a necrophiliac.

I do feel lucky that there is a special variation of the word “love” reserved to express passion. When I say that I am in love, everyone understands that I am talking about a man. You can’t say, “I am in love with my stick-shift car.” Or, maybe, you can; but I am stepping into the territory of dirty jokes:

— Anyone know any? I have lots of tomatoes, but they’re all green. A dirty joke or two might make them blush.

Why am I writing this? I do not even like tomatoes. Maybe it is because yesterday I bought some prunes and they reminded me of the tomato who went out with a prune, because he couldn’t find a date.

Anyone knows that sometimes the text is not exactly what it seems to be. There are many different simple ways to hide a secret message inside your text. So, your humble blogger decided to run an experiment. How should we go about it? I decided to hide a secret message inside this short essay. Do you see it? Do you notice that my text is artificially adjusted for some extra purpose? Everyone can feel that this text sounds different than my usual postings. Nothing should stop you from solving this puzzle now.

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.

Notation is very important in your mathematical papers. Here are the most famous rules on how mathematicians use notation.

Do not explain your notation. Do not waste your time explaining your notation. Most of them are standard anyway. Your paper will look more impressive if you plunge right into your statements. So a good paper can start like this:

Obviously, p is never divisible by 6 …

Everyone knows that p is a prime number.

Use a variety of alphabets. This way you demonstrate your superior education, while expanding your notation possibilities. Not to mention that it looks so pretty:

sin²ℵ + cos²ℵ = 1.

You also get points for drawing a parallel between alpha and alef.

Denote different things with the same letter. It is very important to maintain continuity with the papers in your references, so you should use their notation. Besides, some notation is standard:

Suppose S is an ordered set. Elements of the permutation group S act on this set: for any s in S, sS is the corresponding action.

Mathematicians secretly compete with each other. The goal is to denote as many different things as possible with the same letter in one paper. My personal record was to denote six different things with the letter G. There are two versions of this competition to maximize the number of different meanings of one letter: it can be done either on the same page or in the same formula.

Use different notation for the same thing. The ultimate achievement would be to change your notation in the middle of your sentence:

Gauss showed that the sum of integers between 0 and k inclusive is equal to n(n+1)/2.

Replace standard notation with your own. Your paper will look much more complex than it is. Besides, if someone adopts your notation, they’ll have to name it after you:

Let us use the symbol ¥ for denoting an integral.

Denote a constant with a letter. Letters look more serious than numbers. You will impress your colleagues.

We will be studying graphs in which vertices are colored in only three colors: blue, red and green. For simplicity the number of colors is denoted by k.

As a bonus, when you prove your theorem for three colors, you can confuse everyone into thinking that you proved it for any number of colors.

Do not specify constraints or limits. When you use a summation or a integral, the limits look so bulky that they distract from your real formula. Besides, it’s time-consuming and too complicated for most text editors. Look at this perfect simplicity:

∑ i² = n(n+1)(2n+1)/6.

Everyone knows that you are summing the integers between 1 and n inclusive. Oops. It could be between 0 and n. But 0² = 0 anyway, so who cares?

Be creative. You can mix up these rules or invent your own.

Let us consider a triangle with N sides. Actually, it is better to replace N by H, because in Russian the letter that looks like English H is pronounced like English N. Let us denote the base of the triangle by X. By the way, that is the Russian letter that is usually pronounced like the English letter H, so sometimes I will interchange them. The height of the triangle is, as always, denoted by H.

By following these simple rules, you will earn great respect from your readers.

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’ve got this puzzle from Nick Petry.

Captain Flint is dying. All his treasures are buried far away. He only has 99 pieces of gold with him. Filled with remorse at the last moments of his life, he decides that he only wants to take one piece of gold with him to his grave. The rest of the gold he will give to the families of two men that he had killed the day before.

Though Captain Flint is heavily drunk he notices that no matter which piece he takes for himself, he can divide the leftover 98 pieces into two piles of 49 pieces each of the same weight. Prove that all the gold pieces are of the same weight.

For an additional challenge, Sasha Shapovalov suggested the following generalization of the previous puzzle.

Captain Flint has N gold pieces and yesterday he killed not two but K men. He wants to take one piece with him to his grave and to divide the rest into equi-weighted piles, not necessarily of the same number of pieces. If he can choose any piece to take with him and is able to divide the rest, prove that N – 1 is divisible by K.

Both of these puzzles can be easily solved if the weight of every gold piece is an integer or even a rational number. If you don’t assume that the weights are rational numbers, then I do not know an elementary solution, but I do know a simple and beautiful solution using linear algebra for both puzzles.

Even pirates need linear algebra to divide their treasure. Hooray for linear algebra!

To translate from a Russian joke, borrowing money is taking someone else’s: temporary; giving back your own: forever.

This is a story about my great-uncle Fred. His name is not Fred, of course, because I don’t want to reveal which one of my thirteen great-uncles created this ingenious scheme.

My great-uncle Fred asked to borrow 100 rubles from my mother. He was notorious for not returning money, but he knew how to work my mother. He whined about being sick and urgently needing to buy pills, until my mother, who has a big heart and is an easy touch, gave up. Of course, Fred wasn’t in a hurry to return the money. But 100 rubles was a lot of money for my mother and she wasn’t planning on giving up trying to get it back. My mother started bugging her uncle with increased intensity. Finally Fred promised to return the money as a gift for mom’s upcoming birthday.

Of course, it was tacky to present the money he owed as a gift, but my mom was so glad that she would finally get her money back, that she was actually looking forward to it.

During her party, as the guests sat around the table, Fred got up to give the birthday toast. Then Fred handed my mother an envelope and said, “Congratulations on your birthday! Here is a gift for you.” Everyone applauded.

My mom felt that something in this scene was not quite right. Why was the applause so enthusiastic when he was just returning a debt? After the party my mother decided to investigate. It turned out that Fred explained to my mother’s relatives that she prefered money as a birthday gift and collected the gift money from everyone. The cash he returned as his debt in the envelope was not his. Everyone else thought he was presenting the joint gift, except for my mother, who was made to believe that he was repaying his debt.

After that my mother stopped bugging her uncle Fred. It became clear she couldn’t match his superb skills in escaping his debts.

Suppose you are a woman living in the US and you would like to be a mathematician and work in academia. Suppose also that you would like to have children and spend some time with them. Let us say that you want two or three children and you would like to be with them at home for at least their first two or three years. That is, in total you need to devote 5 or 6 years to this endeavor. When would be a good time for you to have children? American women mathematicians are commonly advised to postpone having children until tenure.

Let us look at the situation more closely. Is it a good idea to have children before you earn your PhD? There are many non-mathematical reasons not to have children too early:

• You do not know yourself well enough to be able to choose a good husband.
• You might not have met the man you love yet (see my essay The Mathematical Path to the Right Husband).
• You might not have enough money to support yourself and your children.
• If you have not yet definitely decided to become a professional mathematician, you might still be clueless that your future career contradicts your plans for children.

There are also mathematical reasons not to have children too early. My former adviser Israel Gelfand liked to tell everyone that mathematicians generate all their best ideas before the age of 23. His views may be extreme. After all, we know of mathematicians who made great discoveries later in life, but it could be that they did this using their mathematical wisdom rather than the processing power of their brains. It may well be beneficial to start your first research early in life. I am not sure about math creativity, but I swear that it is much more difficult to learn new things as I age.

In addition, you might need to relocate frequently to maintain your math career and it is more difficult to do that with children. It’s hard on the children too. Besides, having children early means that while working on your tenure, you will be distracted by your kids and their problems, leaving you less time for your research.

You might think that the closer you get to your PhD, the more the situation improves. In reality, there is a very important reason not to have children while you are in grad school. When you start working on your first topic, it is very important not to be interrupted. If you take a big break someone might finish what you started. When you come back, your topic might be resolved and you would have to start all over again.

The situation after graduate school is even worse. When you apply for jobs, employers are likely to count how many papers you have published per year after your PhD. So you need that number to be high. You can’t afford to dilute your paper count per year by a several-year break. Besides, if you have an interruption in your research it might be considered as a weakness and you might lose in comparison with other applicants. Let us break down the time between PhD and tenure into three periods: postdoc, visiting positions, and tenure track. For each period there are extra reasons not to have kids:

• Right after the PhD you begin looking for postdoc positions. All of them have formal time constraints: you are not allowed to apply for these positions if you are more than 3 (or maybe 5) years past your PhD. To have children during the postdoc period is really a bad idea.
• After the postdoc you might have several visiting positions. They usually require yearly relocations and you need to produce something new every year, so that your current curriculum vitae is different from the one you sent to the same place a year ago. At this stage of your life you are much better off without a husband — forget about children.
• When you have tenure track, you are so close to tenure that you might not want to put your dream at risk after so many years of sacrifices.

For all of these reasons, advice to wait until tenure makes sense. There is one big problem with it though: you usually get tenure in your late thirties. It might be too late for children. You might not want to risk that.

You can always compromise by having one child instead of three. Or you can suppress your desire to spend a lot of time with your children by having hired help, which means that you will miss a good deal of your child’s development, and your child will miss a lot of your love. You can compromise your academic goals by taking a more stable, but less research-oriented, technical job in industry. Or you might get lucky and marry a househusband.

In short, a math career is very kids-unfriendly — there is no right time. If you’re a woman mathematician who wants to spend time with your kids, prepare for pain and disappointment.

But here is an unconventional idea you might consider.

After you finish working on your PhD, postpone your actual defense by 5 years, and have your kids in between. This way all your PhD results will be published and no one can interfere with them. At the same time, the clock that counts your publications per year after your PhD will start 5 years later.

My idea is not a good solution — you will still have many problems — but it might be better than waiting ’till tenure. I do wish there were a better way.

Here is a logic puzzle for kids:

— John has more than a thousand books, said Pete.
— No, he has less than one thousand books, said Ann.
— Surely, he has at least one book, said Mary.

If only one statement is true, are you sure you know how many books John has?