Tanya Khovanova's Math Blog

When you talk over the phone with an adult stranger, you can generally determine if this person is male or female. From this, I conclude that the voice characteristics are often noticeably different for males and females. There are many other characteristics that have a different distribution by gender — for example, height.

My question is: “Can we find a trait such that the distributions are the same for both genders?”

Trying to find the answer, I remembered what we learned in high school about the genetics of eye color. I checked the Internet on the subject and discovered that the story is somewhat more complicated than what I studied 30 years ago, but still we can say that eye color is defined by several genes, which are located on non-sex chromosomes. That means, your eye color depends on the genes your parents have and doesn’t depend on your sex. A boy and a girl from the same parents have the same chances for any particular eye color.

Since eye coloring has nothing to do with gender, women and men are equal in the eyes of eye colors.

Does that mean that if we check the distribution of eye color for the world population, the distribution histogram will be the same for men and women? That sounds like a logical conclusion, right? I would argue that this is not necessarily the case.

Let me remind you that the distribution of eye color depends on the country. China has an unprecedented gender imbalance, with 6% more men than women in its population. As the eye color of Chinese people is mostly dark brown, this creates an extra pool for a randomly chosen man in the world to have a darker eye color than for a randomly chosen woman. If we exclude China from consideration, we can still have different distributions. For example, in Russia the life expectancy for women is 15 years longer than the life expectancy for men. Consequently, Russia has 14% less men than women, while globally the male/female sex ratio is 1.01. Therefore, eye colors common in Russia will contribute to female eye colors more than those of male.

What if we consider only one country? Let us look at the US. Immigrants to the US are mostly males. If the distribution of eye color for immigrants is different than the distribution for non-immigrants, then male immigrants contribute more to the eye color distribution than female immigrants.

There are so many factors impacting eye color distribution, that it isn’t clear whether it’s possible to find a group of people other than siblings in which the distribution of eye color would be the same for women and men.

We see that eye color distribution, which theoretically doesn’t depend on gender, when measured in a large population can produce different distributions for men and women.

Recently I wrote a theoretical essay titled “Math Career Predictor”, where I assumed that the distribution of math ability is different for men and women. In reality, there is no good way to measure math ability, hence we do not have enough data to draw a complete picture. For the purposes of this discussion let us assume that we can measure the math ability and that Nature is fair and gave girls and boys the same math ability. My example with eye color shows that if we start measuring we might still see different distributions in math ability in boys and girls.

My conclusion is that if we measure some ability and the distribution is different for boys and girls, or for any other groups for that matter, we can’t just conclude that boys and girls are different in that ability. For some distributions, like voice, we probably can prove that the difference is significant, but for other characteristics, different distribution graphs are not enough; we need to understand the bigger picture before drawing conclusions.

I got a funny book for a gift called Plato and a Platypus Walk into a Bar…: Understanding Philosophy Through Jokes. I couldn’t stop reading it. This book is an overview, and thus not very deep, but I enjoyed being reminded of philosophical concepts I’ve long since forgotten. Besides, I collect math jokes and many philosophical jokes qualify as mathy ones.

For example, self-referencing jokes:

Relativity — this term means different things to different people.

I especially liked jokes related to logic:

If a man tries to fail and succeeds, which did he do?

I knew most of the jokes, but here’s a math joke I never heard before:

Salesman: “Ma’am, this vacuum cleaner will cut your work in half.”
Customer: “Terrific! Give me two of them.”

Here is a standard logic puzzle:

A criminal is sentenced to death. He is allowed to make one last statement. If the statement is true, the criminal will be sent to the electric chair. It the statement is false, he will be hanged. Can you suggest a good piece of advice for this man?

I can offer many pieces of advice to this man. The simplest thing is to keep silent. Or he can communicate without making statements, like asking, “Can I have some crème brûlée, please?”

One can argue that the puzzle implies that it’s a favor to allow the prisoner to make a last statement, but without it he will die anyway. In this case the standard piece of advice to this man would be to create a paradoxical situation by saying, “I will be hanged.”

Another, less standard, idea is to state something that is very difficult to check. For example, to give the exact number of planets in our galaxy, or posit that P = NP. My son, Sergei, suggested saying that “Schrödinger’s cat is dead.”

But the most popular idea among my AMSA students is to say, “I am sorry.” I’m not 100% sure that they mean it as a statement that is impossible to check. Maybe they think that these words can do magic and save lives. Or maybe it could be the best thing for a criminal to say before dying.

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!