Tanya Khovanova's Math Blog

I used to be proud of my Russian math education. I am still proud of my high school one, but not so much of the one I received in college. In Soviet Russia, a student had to choose their major before applying to college. I wanted to study mathematics, and I got accepted to the best place for it in Soviet Russia: mekhmat — the math school at the Moscow State University. I used to be proud of my education there, but now I have my doubts.

I had to take, on average, four math classes per semester for five years, which totals about 40 math classes. Woo hoo! I don’t think American students could even choose to take that many. This was presumably good, but most of the courses were required, and their curriculum remained unchanged for many, many years. Obviously, the system was very rigid. The faculty members feared retaliation from the communist party and forgot how to take initiative. The bureaucracy prevented the department from adding new and exciting math to the outdated curriculum.

This post is not about my grades but about the actual subjects that we were taught then. But, in case anyone is wondering, my only B was in English; everything else was straight As.

Some of the classes listed below lasted two or more semesters, that’s why they do not sum up to the promised 40. Unfortunately, I do not remember which ones. These were the required math classes:

• Analysis
• Analytical Geometry
• Advanced Algebra
• Theoretical Mechanics
• Linear Algebra and Geometry
• Differential Equations
• Partial Differential Equations
• Functions of Complex Variables
• Probability and Statistics
• Differential Geometry and Topology
• Numerical Methods
• Introduction to Mathematical Logic
• Control Theory
• Analysis III
• Computer Science and Programming
• Programming Practice
• Physics
• History and Methodology of Mathematics
• Thesis Work

An impressive list? But guess what — I remember nothing from most of these classes. As an exception, I remember bits of Differential Equations, taught by Vladimir Arnold, a charismatic teacher. I remember Linear Algebra well, not because of my Linear Algebra class, but because I read Gelfand’s book on the subject and loved it. I remember that the Differential Geometry and Topology class was taught by Fomenko with great pictures and boring material. By the time I took Fomenko’s class, I already knew topology from an unofficial class taught by Dmitry Fuchs, which was so much better. In fact, in order to learn what I wanted, I had to take many classes unofficially, so my total is actually way above 40.

By junior year, we were finally allowed to choose some classes which would count towards our transcripts, and this is what I picked.

• Infinite-Dimensional Representations of Lie Groups
• Theory of Functions of Many Complex Variables
• Representations of Lie Groups
• Discrete Mathematics

I remember these classes much more vividly. I also wrote a graduate thesis: “Models of Representations of Generalized Clifford Algebras.” I loved working on that paper.

We had non-math classes too: everyone had to take them.

• History of the Communist Party of the USSR
• Philosophy of Marxism-Leninism
• Political Economy
• Scientific Communism
• Foundations of Scientific Atheism
• Soviet Law
• Foreign Language (English)
• Physical Education
• Foundations of Marx-Lenin’s Aesthetics

To graduate, everyone had to pass two state exams: Mathematics and Scientific Communism. Whatever the latter might mean.

Did I mention that I am no longer proud of my former Soviet college education? What a colossal waste of time!

I grew up relatively poor, but I wasn’t aware of it and didn’t care. In 7th grade, I went to a new school for children gifted in math. Looking back, I realize that most of my classmates there were privileged. My first clue about my own financial disadvantages arrived when my math teacher, Inna Victorovna, offered me several of her old dresses. I do not remember what she said to me exactly, but I remember she was tactful, so much so that I felt comfortable taking the dresses.

In an instant, I was better dressed than I had ever been. I especially loved the brown dress which I wore for my first visit to Gelfand’s seminar.

A few years passed; I went to college and married Andrey. Things got somewhat better financially, but I was still struggling. My mother-in-law, Veronika, was well-off and loved clothes. She had a habit of ordering a new dress from her tailor, every season, four times a year. In Soviet Russia, this was a lot of dresses.

One day, Veronika decided to give me some of her old dresses. Unlike my math teacher, she said something that I will never forget. She told me that she was getting rid of those dresses because they were out of fashion and made her look old. I was in my twenties at the time and didn’t want to look old either. However, I didn’t have much choice in clothes, so I wore the dresses. I hated them.

Do you know that I participated in Linguistics Olympiads in high school? They are not well-known in the US, but the Soviet Union has been running them since 1965. The first International Linguistics Olympiad was conducted in 2003, and the US joined in 2007. They are called Computational Linguistics because you are expected to discover some phenomenon in an unfamiliar language on the fly instead of knowing a lot of languages already. The problems mostly need logic and are a good fit for a person who likes mathematics. However, a feel for languages is very helpful.

I do not remember why I started attending the Olympiads, but I remember that there were two sets of problems: more difficult for senior and less difficult for non-senior years. I used to be really good at these Olympiads. When I was in 8th grade, I finished my problems before the time ran out and started the senior problems. I got two awards: first place for non-senior years and second place for senior years. In 9th grade, I got two first-place awards. I didn’t know what to do in 10th grade, which was a senior year at that time in the USSR. I couldn’t get two first-place awards, as I could no longer compete in the non-senior category. I felt ashamed that my result could only be worse than in the previous years, so I just didn’t go.

The prizes were terrific: they gave me tons of rare language books. In the picture, a guy from the jury is carrying my prizes for me. I immediately sold the books at used-books stores for a good price. Looking back, I should have gone to the Olympiad in 10th grade: my winter boots had big holes.

My friend, John Conway, had a trick to help him with tricky situations. Whenever he needed to make a non-trivial decision, he would ask himself, “What would John Conway do?” As he explained to me, he had in mind the public image he himself created. He liked the image and thought this mental trick helped him be a better, more productive, and not-to-forget, flashier person.

From time to time, I catch myself in need of a decision and ask myself, “What would John Conway do?” And he gave me the answer: I should change the question and ask myself, “What would Tanya Khovanova do?”

A familect is a portmanteau word formed by squashing together two words: family and dialect. It means words that a family uses that are not a part of a standard vocabulary. This story is about two words that my son Sergei invented, and twenty years later, my family still uses them.

As you might know, I was married three times, and I have two sons, from two different fathers. These fathers were also married several times and had other children. My two sons are half-brothers, and they also have half-siblings through their fathers. Thus a half-sister of a half-brother became a quarter-sister. Inventing this term was quite logical for a son of two mathematicians.

As you can imagine, our family tree is complicated. One day Sergei pointed out that our tree doesn’t look like a standard tree and called it a family bush.

I spend a lot of time working on my blog, and I used to think it would be nice to get some money out of it.

Years ago I got two emails from different ad agencies at the same time. They wanted to place ads at particular essays for $50 a year. I decided to give them a try.

My first correspondent wanted a link on my “Does Alcohol in Teens Lead to Adult Woes?” essay, connecting to a website offering help to alcoholics. I agreed. But when I read the actual text, I couldn’t stop laughing. The text they wanted to use was, “Many studies have already claimed that teenage alcoholism could lead to more problems later in life.” How ironic! This ad would follow my essay explaining that one of the studies is completely bogus. I rejected them.

The second agency wanted an ad accompanying the essay “Subtraction Problems, Russian Style.” I placed it. They wrote to me (and I reproduce it with all of their errors intact):

I really appreciate your efforts on this. As I checked the text link, I have seen that the text link has been label as “Sponsor ad”. Kindly omit or delete the word “Sponor ad:” or you may changed it to “Recommended site or Relevant Site” but I would love to prefer the text link be seen as natural meaning no labeled inserted on it.

They wanted me to pretend that I recommend their product. I was naive enough to think that I was selling space on my page, but what they really wanted was for me to lie that I like their product.

Before this experiment, I hoped to find some honest ads for my blog. After this experiment, I realized how much stupidity and falsehood are involved. Since then, I ignore all offers of ads that come my way. That’s why my blog is ad-free.

I grew up in the USSR, where I was clueless about the race issues in the US. I have now lived in the US for 30 years, and still feel that there are many things about race that I do not understand. As a result, I am afraid to speak about it. I am worried that I’ll say something wrong. Recent events have encouraged me to say something. This is my first piece about race.

I came back to mathematics 10 year ago and started working at MIT. I love it. With some exceptions.

Many mathematicians are introverts or snobs or gender-biased. They are not usually friendly. I often walk down a corridor and people who are coming towards don’t notice me. If I say hello, they might not even reply or raise their eyes. It could be they are thinking about their next great theorem and do not notice me. It could be that I am not faculty and therefore do not deserve their attention. It could be that as a women I am not worth of their hello.

Soon after I started working at MIT, I was reminded of one of the reasons I left academia. It was this unfriendliness. But this time was different. First, I had grown a thicker skin. Second, I was working within a group. People who were working with me were nice to me. It was enough and so I stayed.

With time I adopted the same style: passing people without saying `Hello.’ Mostly I got tired of people not replying to my hello.

One day I was passing this man who, as had happened many times before, purposefully didn’t look at me. I thought my usual thought: another introverted/snobbish/gender-biased mathematician. Then I suddenly stopped in my tracks. My logic was wrong. This guy was Black. The unfriendliness of mathematicians is surely way worse for him than for me. It could be that he is looking at the floor for the same reason I do it: he is afraid that people will ignore his greeting. I failed to think about race deep enough before this realization. What happened next should have happened years earlier.

I took the initiative and the next couple of times I saw him, I said hello. This was all it took—two hellos—to change the whole feeling between us. The guy has a great smile.

My friend Alice reminds me of me: she has two sons and she is never straight with her age. Or, maybe, she just isn’t very good with numbers.

Once I visited her family for dinner and asked her point blank, “How old are you?” Here is the rest of the conversation:

Alice: I am two times older than my younger son was 5 years ago.
Bob: My mom is 12 times older than my older brother.
Carl: My younger brother always multiplies every number he mentions by 24.
Bob: My older brother is 30 years older than me.
Carl: My mom is 8 times older than me.
Alice: My older son always multiplies every number he mentions by 2.

How old is everyone?

(Photo by Rebecca Frankel.)

When I was in grade school, one of the teachers called me Cave Lioness. She hated my unruly hair, which reminded her of a lion’s mane. This teacher was obviously very uninformed, for female lions do not have manes.

This name calling had the opposite to the desired effect. I became proud of my mane and didn’t ever want to cut it. When I grew older, I opted for convenience and started to cut my hair short—sometimes very short.

Last year I was too busy for barbers, and my hair grew more than I intended. As it turned into a mane, I remembered the story of this nickname.

Once I was giving a math lecture and my phone, which I’ve never quite understood, was on the desk in front of me. Suddenly it rang. I didn’t pick it up, as I proceeded with my lecture. The ringing stopped, while I was explaining a particularly interesting mathematical point. After a minute, my phone said, “I do not understand a word you are saying.”