Tanya Khovanova's Math Blog

As a child I felt that society expected me to grow up and be a mother. And only to be a mother. It didn’t make any sense to me, because if all that people do is reproduce, how will progress be made? So in my teens I decided that in addition to having children, I need to do something else, something to make the world a better place.

For many years I wasn’t too successful in my plan. My children were my priority. I believed that for the first three years they needed a lot of my attention, which they couldn’t necessarily get from a nanny. Later, being a single mother was so overwhelming that I didn’t have time for other missions.

As my children grew, I felt myself imposing my own ambitions onto them. It wasn’t fair to burden my children like that. I was meant to do something with my life other than bring up children. There was no substitute — I had to do it myself.

At that time, I was working at BAE Systems, but I hated my job. Its only purpose was to support my family. So as soon as my youngest son turned 16, I resigned to come back to mathematics. Though mathematics has my full attention now, my children are still my priority. I can afford to do what I love, because it is good for my children too. I am living my own ambitions, and they have the space to live theirs.

I was supposed to write about menopause. I am writing about menopause, just give me a second. I was also raised to believe that the best thing I can do for a man is to give him children. Every time I loved someone I wanted to have a child with that person, or I thought that I was supposed to want to. Now that I can’t conceive a child for any potential future husband, I feel relieved. I can’t be blamed any more for not having children. I am so happy that my current attempt at mathematics will not be interrupted any more.

I am so glad that I live in an age when women’s life expectancy is way past menopause. I might love a man again; I might marry. I am so glad that I have an excuse not to have children. I can do what I want to do. I am free.

I enjoy doing research in recreational mathematics. The biggest problem is that the probability that something I’ve done has already been done before is very high. This is partially because of the nature of this area of research. Trying to find new questions that are not buried deep in the abstract means someone else could have stumbled upon the same question. In addition, I like going in different directions: geometry, number theory, probability, combinatorics, and so on. That means I am not an expert in any of the areas. This too increases my probability of doing something that has already been done before.

This is quite unpleasant: to discover that I reinvented the wheel. And I keep reinvented different wheels on a regular basis. When I complain, my mathematician friends give me the same advice over and over:

Find your own niche!

I’ve been trying to understand what they mean, and finally I got it:

Do something no one else is interested in using methods no one else can understand.

Finding a topic that is not interesting almost guarantees that other people didn’t do it before. Developing new complicated methods helps limit the number of followers and prevents the niche from becoming crowded.

Now I know why I see so many boring math papers I can’t understand.

To hell with my own niche!

Let me start with a real story that happened a long time ago when I worked with my co-author on a math project. When I told him the idea I came up with, he dismissed it. The next day he came to me very excited with his new idea. He repeated the idea I had proposed the day before.

I said, “I suggested this exact idea to you yesterday.” He didn’t believe me. Luckily for our relationship, I knew him very well. He was not a person who lied. I remembered how preoccupied he was when I had laid out the idea to him. Now I think that he didn’t pay attention to my idea at the time, but it was lodged somewhere in his subconscious and surfaced later. I am sure that he truly believed that the idea was his. (Unfortunately, he didn’t believe me. But this is another story.)

This story made me think about the nature of collaboration and how people can’t really separate ownership of the results of a joint project. Let me tell you another story, one that I do not remember happening, although it could have.

I was working on a paper with my co-author on Sharelatex. One day an idea for a lemma came to me that we hadn’t discussed before. My co-author lives in a different time zone, so instead of discussing the idea with him, I just added Lemma 3 and its proof to our paper. I was truly proud of my own contribution.

The next day I came up with Lemma 4 while driving back home. It was another completely new direction for our research. When I came home to my laptop, I discovered my Lemma 4 neatly written and proven by my co-author. Is this lemma his? It can’t be completely his. It was a natural extension of what we discussed; he just got to a computer before me.

Did you notice in the last story that the situation is symmetric, but my hypothetical feelings are not? In a true collaboration ideas are bounced off each other. Very often people come up with the same idea at the same time, but you can’t ascribe ownership to the first person who speaks. Because of many stories like this, I made a rule for myself: never discuss in public who did what in a joint paper. Just always say “we.”

Unfortunately I keep forgetting to teach my PRIMES/RSI students that. The question usually doesn’t come up until it is too late, when one of the students in a group project during a presentation says, “I proved Theorem 5.” Oops!

I promised to explain my job situation to my readers. I currently have six part-time jobs, which I describe below in the order that I started them.

The job I have had the longest is coaching math for competitions at AMSA Charter School. I started there in the spring of 2008 and I am still teaching there every week. I post my homework assignments on my webpage for other teachers and students to use.

The next longest has been as the Head Mentor at RSI, which I started in the summer of 2009. RSI is a great program where high school students from all over the world come to MIT to do research during summer vacations. It is amazing how much an ambitious student can do during a short five-week period. My first year I saw some great research done, but I was surprised that RSI papers were generally not available online. Beginning in 2012, at my recommendation, our Director Slava Gerovitch now posts the RSI papers at the math department RSI webpage.

As good a program as it is, RSI is too short for a research project. So we decided to develop our own program called PRIMES (Program for Research in Mathematics, Engineering, and Science for High School Students). I am also the Head Mentor in this program. I provide general supervision of all the projects and review conference slides and final papers. In addition to being the Head Mentor, I mentor my own students, which is great fun. I usually have three projects, but if something happens to a project or a mentor, I take it over.

Together with Pavel Etingof, PRIMES Chief Research Advisor, and Slava Gerovitch, PRIMES Director, I wrote an article Mathematical Research in High School: The PRIMES Experience that appeared in Notices.

My job #4, since the fall of 2013, is as a Teaching Assistant for the MIT linear algebra course.

This year I took on two more jobs at MIT. I am the Head Mentor at Mathroots, a summer program for high school students from underrepresented backgrounds. I am also teaching at PRIMES STEP, the program for gifted middle-school students in the Boston area. The goal of the program is to teach students mathematical thinking, have fun, and prepare them for PRIMES.

Overall, I have reached the stage in my career when I do not have time to breath and I still do not make enough money to pay my bills. The good news: my jobs bring me satisfaction. I just hope that I will not be too tired to notice that I am satisfied.

My name is Tanya and here’s why. When my mom was pregnant, she was discussing the child’s name with my father, Gueliy Khovanov. They decided that she could choose a boy’s name while he could choose a girl’s name. She wanted me to be Andrey in honor of her half-brother who died in World War II. He wanted me to be Tanya in honor of the unrequited love of his life. In an act of incomprehensible generosity, my mother agreed to name me after this woman. My parents were not even married when I was born.

After the decision was made, I was born on my own Name Day. In Russian culture, different names are celebrated on different dates. The a holiday for Tanyas is especially big and it falls on January 25, the day I was born. This serendipity led me to be very attached to my given name.

When I was a child, I believe that my father introduced me to his love, Tanya, although I do not have any clear memory of her. I doubt they were ever actually together. I do remember how my father loved her all his life. Even today I am sure that his love for Tanya brushed off on me, being her namesake.

I especially remember one day, when I was still a kid, my father agonizing about Tanya and calling to offer her help. My mom grimaced when he offered to babysit her children. My parents were divorced by then, and my father showed no interest in babysitting his own children. Now I understand that watching Tanya’s children was the least he could do. This was happening in 1968 when the USSR invaded Czechoslovakia, and Tanya’s husband, Konstantin Babitsky, was one of the few people in the USSR who risked their lives to openly protest against it. Tanya’s life and freedom were also at risk. Offering any help was the right and only call.

Right before my father died of an illness in 1980, he asked me for one last favor: to tell Tanya that he had died. I was 20 years old and didn’t have a clue how to find her; I only knew that her name was Tanya Velikanova. She was a famous Soviet human rights activist and dissident. That made it dangerous for me and for the people I might ask for assistance.

I finally asked my father’s best friend to do it for me. He said that Tanya was in exile, but promised to pass a message to her. I am not sure he did it and I still feel guilty that I didn’t do it myself before she died.

I forgot to tell you how my dad met the love of his life. They met as teenagers at a math Olympiad, where she beat him.

My mom used to tell me “You are born to teach.” My mom was a very shy person with no desire to be a teacher. Somehow she ended up teaching chemistry all her life. She also told me that teaching was our family business. All my ancestors were teachers or priests. Given that I lived in the Soviet Union, priesthood was not an option. (Given that I’m a woman, priesthood was not an option anywhere else.)

After my postdocs, I ended up quitting academia and working for industry. As a result, I didn’t teach much. Until 2008 my only teaching experience was a course in Discrete Mathematics that I taught at Bar-Ilan University as part of my postdoc. Somehow Bar-Ilan University wanted five people to teach this same course in parallel and wanted me to teach it in Russian to attract fresh Russian immigrants. By the end of my course, I had way more people coming to my class than at the beginning, many of whom had transferred from the other parallel courses.

I had another clue that I was doing a good job. During one of my classes, the door was left open. A guy was walking down the hall, but when he heard me, he stopped. He stood there listening intently until the end of my class.

At the end of 2007 I resigned my industry job to go back to mathematics. I needed some financial support to do so, so my friends tried to arrange a math coach position for me at AMSA (Advanced Math and Sciences Academy Charter School). I had my interview in April, 2008, for a position for the following academic year. My interviewers were skeptical: they had had a series of PhDs who had proven incapable of teaching. I told them that deep down in my heart I knew that I would be a great teacher. And I made them an offer: I will start working in April and work until the end of the school year in June. Hiring me for two months was not a great risk, but gave them a basis on which to decide whether to give me a contract for the following year. I’ve been working at AMSA ever since.

My class is optional and there are no grades, but most of my students stick with me for all five years. (I teach grades 8 to 12.) My students show that they like me in many ways. I’m especially thrilled when they tell me that I am the best teacher they ever had.

Informally they call my class Tanya Time.

Some of my friends collect volumes of The Best Writing on Mathematics published by Princeton University Press. The first annual volume appeared in 2010. In 2014, one of my papers, Conway’s Wizards, was chosen to be included in the volume The Best Writing on Mathematics 2014.

All my life I have hated writing. My worst grades in high school were for writing. I couldn’t write in Russian, and I was sure I would never be able to write in English. I was mistaken. I am a better writer than I gave myself credit for being.

Writing in English is easier for me than writing in Russian because when I make a mistake, I have an excuse. As my written English gets better and better, maybe one day I’ll get the courage to try writing in Russian.

I would like to use this opportunity to thank Sue Katz, my friend, English teacher, and editor. Sue edits most of my blog essays. Originally I wrote about Conway’s wizards on my blog. When I had three essays on the subject I combined them into a paper. Sue edited the essays and the paper. I am honored to be included in this respected volume and I want to share this honor with Sue.

As I told you before I was recently diagnosed with severe sleep apnea. My doctor ordered me to use a CPAP machine&mdasha Continuous Positive Airway Pressure machine&mdashthat blows air into my nose and improves the quality of overnight breathing.

I was elated, hoping for a miracle. I haven’t had a good night’s sleep in 20 years. I was so looking forward to waking up rested.

When I went to bed, I put on my nose mask and turned on the machine. The mask was uncomfortable. It gave me headaches and I had to sleep on my back which I do not like. It also had this annoying plastic smell. It took me some time to fall asleep, but eventually I did.

To my disappointment I didn’t wake up rested; I woke up tired as usual. Still, I decided not to give up and continued trying the machine.

Several days passed and I found myself mopping the kitchen floor. I never ever mop floors. I have my cleaning crew do that. That mopping was my first positive sign. Then my relatives started telling me that my voice had changed and had become more energetic. A week later I invited my son and his family for my weekly family dinner, which I have been canceling every previous week for several months.

A month after I began sleeping with the machine, I started having night dreams, something I forgot even existed. After two months, I would wake up and my first thought would be, “What day is it today?” Prior to using the machine, my first waking thought was, “My alarm clock must be broken. It’s impossible that I have to get up now. I feel too weak to stand up.”

My machine is very smart. It records the data from my sleep and uploads it to a website, which my doctor and I can access. Instead of the 37 apnea episodes per hour I had during my sleep study, now I have 1 episode per hour. The quality of my life has changed gradually. I am not yet ready to conquer the world, but I have so much more energy. I’ve even accepted two more job offers, and now I have six part-time jobs. (I’ll tell you all about this some other time.)

There have been some side effects. For some reason I gained 10 pounds during the first two weeks of using the machine. And I feel like an elephant. Not because of my heavy frame, but because my nose mask with its pipe looks like a trunk.

But I prefer feeling like an elephant to feeling like a zombie. This indeed was a miracle, though a very slow-acting miracle.

I recently had a home sleep study. I was given a small box which I attached to my chest. I also had to attach a thingy to my finger and put small tubes into my nose. It was relatively easy. Now I have my report:

The total time in bed is 468 minutes. Overall AHI is 37 events per hour. The supine AHI is 58 events per hour. The oxygen saturation baseline is 91%. The hypoxemic burden is 58 minutes. The oxygen saturation nadir is 63%. The heart rate ranges from 76-118 beats per minute.

I didn’t have a clue what all that meant so I hit the Internet. AHI means Apnea–Hypopnea Index, and a normal score is below 5. Anything above 30 indicates severe sleep apnea. Because mine is 37, I now have my diagnosis. My 63% oxygen saturation scared me the most. Wikipedia says 65% or less means impaired mental function. I do not need mental function when I sleep, but Wikipedia also says that loss of consciousness happens at 55%. What would happen if I lose consciousness while I sleep? Can I die? Will I wake up?

Overall the sleep study was a great thing. Now I know the diagnosis and there are ways to treat it. So I am looking forward to my improved energy and health.

But there was something in this report that would bother any mathematician. As you can see apnea gets worse when people sleep on their backs. (Thanks to this study I learned a new English word: supine means lying on the back.) The apparatus that I had to attach to my chest prevented me from sleeping on my stomach, one of my favorite sleep positions.

This report doesn’t say anything about my average AHI when I am not supine. If this average is low, then the solution might be to learn to never sleep on the back. It also means that the oxygen saturation nadir number is not very meaningful. It shows how bad it can be if I am forced to sleep on my back. It doesn’t say much about my standard sleep situation.

When I next see my doctor, I hope she’ll have answers to all my questions.

I am scared of that old German gentleman. Forgot his name. Oh, yeah. Alzheimer.

I started to lose my brain processing speed a long time ago. By my estimates I am about 100 times slower than I used to be. From time to time someone gives me a puzzle I remember I solved in 30 seconds years ago, but now it takes me 30 minutes. And it is getting worse. When I moved to my new apartment recently, it took me a year to remember my own phone number.

On the positive side, I feel quite famous, according to one of the signs of success. I am often greeted by people I don’t know.

My moment of action came when I was in my basement doing my laundry and couldn’t remember how to turn on my washing machine. This is after 10 years of heavily using this damn machine. I started to look for the button to push, but there were no buttons. There were only knobs, and I couldn’t find the word “on” anywhere. After struggling with my memory and with those knobs, I pulled out one of the knobs, and the machine started.

I panicked. I am afraid of Alzheimer’s Disease. I do not want to become demented. I do not want to forget how to count or to stop recognizing my children. I do not want to become a drain on my children’s time, emotions and money.

I had complained about my memory to my doctor before, but the only thing he ever found was anemia. This time I was more insistent. I had an MRI that ruled out tumors. I had more extensive blood tests that confirmed anemia and showed a Vitamin D deficiency. But then he sent me to a neurologist who suggested a sleep study. Finally I got a diagnosis of severe sleep apnea. I am so happy now. I might not have Alzheimer’s Disease. In the worst case scenario, I might die in my sleep. In comparison, this doesn’t sound so bad.