Tanya Khovanova's Math Blog

I’m trying to lose weight. Many books explain that dieting doesn’t work, that people need to make permanent changes in their lives. This is what I have been doing for several years: changing my habits towards a healthier lifestyle.

This isn’t easy. I am a bad cook; I hate shopping; and I never have time. Those are strong limitations on developing new habits. But I’ve been a good girl and have made some real changes. Unfortunately, my aging metabolism is changing faster than I can adopt new habits. Despite my new and improved lifestyle, I am still gaining weight.

But I believe in my system. I believe that one day I will be over the tipping point and will start losing weight, and it will be permanent. Meanwhile I would like to share with you the great ideas that will work someday.

• Declare some food non-food:
  • I used to keep a kosher kitchen. It was so easy to shop. I didn’t need to go to every aisle in the store because the kosher dietary laws excluded so many foods. Now I’m not kosher anymore, but I like the idea of restricting bad foods, so I created Tanya’s own kashrut rules:
    • Soda is not drinkable.
    • Only dark chocolate deserves to be eaten.
    • Corn syrup, artificial colors and sweeteners are poison.
• Make healthy foods easily accessible:
  • I have Boston Organics fruits and vegetables delivered to me every other week. Initially they all rotted and I had to throw them out, but I am stubborn. Now I’ve learned how to make a turnip salad and how to enjoy an apple. I will soon switch to a weekly delivery.
  • When I’m in a restaurant, I have a rule that I must order vegetables. I do not have to eat them, I just have to order them. But since I do not like things wasted, I end up eating at least some of them. Now I’ve grown to like eggplants and bell peppers.
• Make unhealthy foods less accessible:
  • I buy precut frozen cakes. When I am craving sugar, I defrost one piece. A while ago I would have finished the whole cake the day I bought it, but now, after one piece, I am usually too lazy to defrost another.
  • I buy fewer sweets now. Actually I buy exactly one desert item, as opposed to the half a shopping cart I used to buy. I used to rationalize that I need deserts to serve potential guests. Then I would eat all the sweets myself. Now I’ve decided that my friends will forgive me if I don’t serve desert.
• Engage my friends:
  • Three of my girlfriends and I signed up for the gym together. Without them, I would have dropped the gym a long time ago. Natasha’s call inviting me to yoga often is the extra push that I need. Now, several years later, the habit is formed and when necessary I go alone.
  • Introduce other good habits:
    • I have a separate computer for games. I put it on top of my bookcase, so I have to stand while playing. This way I can’t play for too long, and burn extra calories at the same time.

  I have many other ideas that for different reasons haven’t yet become habits. So I am thinking about tricks to turn them into habits.

  • Start every meal with water.

  I keep forgetting to start my meals with water. Besides, I do not like plastic bottles. So now I’ve bought glass bottles with protective sleeves to carry in my car and my bag conveniently. They look so cool that I enjoy sipping from them.

  • Exercise every day.

  I never exercise in the mornings, because I want this time for mathematics. But in the evenings I am often too tired and skip my scheduled gym sessions and dance classes. I often spend the whole day inside in my pajamas. So to help me to exercise daily, my friend crocheted a small toy dog for me. Now I pretend that it’s a real dog that needs to be walked every day.

  I have many more ideas, but I gotta run now. I need to walk my dog.

More than ten years ago I went through a process of psychotherapy which, although very painful, was extremely successful. When I tell my friends about this, they are interested in knowing what can be gained through psychotherapy, so here’s my story.

I was living in Princeton, NJ, and I was very tired all the time. My primary care doctor told me that I was depressed and needed to do psychotherapy. A friend of mine recommended Dr. Ella Friedman. During my first visit Ella told me that I block my negative emotions. I protested. All my life I truly tried to be honest with myself. She insisted. I had nothing to lose because I had to solve the problem of my constant exhaustion and I had no other potential solutions. Besides, I liked her very much. So I decided to play along and started my search looking for negative emotions.

For some time I tried to convince Ella that if my best friend broke my favorite mug I wouldn’t get angry with her. Ella tried to convince me otherwise. She pushed me back in time to the source of my beliefs and feelings. After several months of therapy, I discovered that I had a strong underlying belief that for my mother to love me, I must be a good girl who is always fair. Since my friend who broke the mug didn’t do it on purpose, I wasn’t allowed to be angry with her. I repressed all my angry feelings.

It took a lot of time for Dr. Friedman to rewire me and persuade me that my negative emotions do not mean that I am a bad girl. My actions define my goodness, not my emotions. I resisted. She had already convinced me that I might have negative emotions, but I didn’t want to look at them. The power forcing me to block my emotions was the threat that my mother would withdraw her love if I wasn’t a good girl. Dr. Friedman converted me. I started to believe her and continued more vigorously searching for my hidden emotions. Finally one day I collapsed in the shower. I actually felt my blocked emotions flooding me.

Negative emotions protect us. If someone treats you badly you need to be able to recognize it and get away from the danger. Because I didn’t see my emotions I stayed in situations, like toxic relationships, that caused me great pain, without realizing it.

My psychotherapy didn’t stop then. We started working on how to understand my emotions and how to process them. Now when someone is talking to me, I listen not only with my ears, but also with my gut. Suppose someone tells me, “I am so glad to see you,” but I feel a strange tightness in my stomach. I start wondering what the tightness is about, and usually can figure it out. For the first time I was able to hear my gut and it was more illuminating than what I was hearing with my ears. All my life I processed information as text. Now the sentence “I am so glad to see you” has many different meanings.

The therapy changed my life. It feels as if I added a new sense to my palette  of senses. I feel as if I was color blind for many years and at last I can see every color. Now that I’ve learned to recognize my pain, I can do something about it. I am so much happier today than I ever was before. While my friends may not have consciously recognized the big change in me, they have stopped calling me clueless and now often come to me for advice.

Did this solve my problem of tiredness? When Ella Friedman told me that I was no longer depressed, I still felt tired. I started investigating it further. It turns out that the depression was a result of the tiredness, not the other way around. It seems that I have a sleeping disorder and an iron problem.

There is an array containing all the integers from 1 to n in some order, except that one integer is missing. Suggest an efficient algorithm for finding the missing number.

A friend gave me the problem above as I was driving him from the airport. He had just been at a job interview where they gave him two problems. This one can be solved in linear time and constant space.

But my friend was really excited by the next one:

There is an array containing all the integers from 1 to n in some order, except that one integer is missing and another is duplicated. Suggest an efficient algorithm for finding both numbers.

My friend found an algorithm that also works in linear time and constant space. However, the interviewer didn’t know that solution. The interviewer expected an algorithm that works in n log n time.

The company claims that they are looking for the smartest people in the world, and my friend had presented them with an impressive solution to the problem. Despite his excitement, I predicted that they would not hire him. Guess who was right?

I reacted like this because of my own story. Many years ago I was interviewing for a company that also wanted the smartest people in the world. At the interview, the guy gave me a list of problems, but said that he didn’t expect me to solve all of them — just a few. The problems were so difficult that he wanted to sit with me and read them together to make sure that I understood them.

The problems were Olympiad style, which is my forte. While we were reading them, I solved half of them. During the next hour I solved the rest. The interviewer was stunned. He told me of an additional problem that he and his colleagues had been trying to solve for a long time and couldn’t. He asked me to try. I solved that one as well. Guess what? I wasn’t hired. Hence, my reaction to my friend’s interview.

The good news: I still remember the problem they couldn’t solve:

A car is on a circular road that has several gas stations. The gas stations are running low on gas and the total amount of gas available at the stations and in the car is exactly enough for the car to drive around the road once. Is it true that there is a place on the road where the car can start driving, stopping to refuel at each station, so that the car completes a full circle without running out of gas? Assume that the car’s tank is large enough not to present a limitation.

Imagine a slice of buttered white bread with a heap of sugar on top. That was my favorite lunch when I was a kid. My mom was working very hard, I was the oldest sister, and this was what I would make for myself almost every day.

Later someone told me that sugar is brain food. I believed that sugar and chocolate helped me do mathematics, so my love for sugar got theoretical support. I finally figured out the source of this love when my first son was born. To teach my son to stop requesting milk at night, my mother pushed me to give him sugar-water instead. At that moment, I realized that I developed my love of sugar with my mother’s milk. Or, more precisely, instead of my mother’s milk.

Now there is more and more evidence that the love of my life is a mistake. See for example Is Sugar Toxic?. Will I ever be able to break my oldest bad habit, the one I developed before I can remember myself doing it?

Several years ago my son Sergei started a new movement: Sergeism. Followers of this philosophy seek to maximize Sergei’s happiness. Since Sergei’s happiness involves everyone being happy, becoming happy is a consequential goal of his followers.

Let me explain why this might be a perfect religion for many people, not the least myself. My parents didn’t teach me to love myself. They taught me to sacrifice myself and put other peoples’ interests ahead of my own. After reading tons of books and spending hours in therapy, I’ve learned to love myself — well, somewhat. But the truth is, I still feel guilty when I pamper myself. Sergeism eliminates this guilt. I can freely invest in my happiness as a committed member of this movement.

I became a Sergeist when I lost all hope of losing weight. I realized that my own health wasn’t a strong enough motivation. But I’m always glad to skip a cookie in tribute to Sergeism. If, like me, you put others ahead of yourself and never find the time to exercise or the will to refuse deserts, join me. Become a Sergeist and lose weight for Sergei.

My webpage and my blog generate a lot of emails. I love receiving most of the emails, but if I reply to them, I won’t have time to work on my blog. My favorite type of message is one that is full of compliments, with a note that the writer doesn’t expect a reply.

I am grateful to people who send me things I requested, like pictures of Russian plates, or some interesting number properties. I apologize that it takes me so long to reply.

The emails that I don’t enjoy reading contain amazing elementary proofs of Fermat’s last theorem, or any other theorem on the Millennium list, for that matter. I also do not like when my readers ask me for help with their homework.

Like most people, I’m already dealing with spammers who want to enlarge the body parts I do not have or to slim the ones I do have. However, if you do need to send me millions of dollars that I won in your lottery, there is no reason to waste time on email exchanges: you can process them through my “donate” button.

You are welcome to contact me, but ….

In August I visited my son Alexey Radul, who currently works at the Hamilton Institute in Maynooth, Ireland. One of the greatest Irish attractions, Broom Bridge, is located there. It’s a bridge over the railroad that connects Maynooth and Dublin. One day in 1843, while walking over the bridge, Sir William Rowan Hamilton had a revelation. He understood how the formulae for quaternions should be written. He scratched them into a stone of the bridge. Now the bridge has a plaque commemorating this event. The plaque contains his formulae. I don’t remember ever seeing a plaque with math, so naturally I rushed off to make my pilgrimage to Broom Bridge.

Quaternions have very pronounced sentimental value for me, since my first research was related to them. Let’s consider a simple graph. We can construct an algebra associated with this graph in the following way. For each vertex we have a generator of the algebra. In addition we have some relations. Each generator squared is equal to −1. If two vertices are connected the corresponding generators anti-commute, and they commute otherwise. The simplest non-commutative algebra associated with a graph corresponds to a graph with two vertices and one edge. If we call the generators i and j, then the we get the relations: i² = j² = −1, and ij = −ji. I we denote ij as k, the algebra as a vector space has dimension 4 and a basis: 1, i, j, k. These are exactly the quaternions. In my undergraduate research I studied such algebras related to Dynkin diagrams. Thirty years later I came back to them in my paper Clifford Algebras and Graphs. But I digress.

I was walking on the bridge hoping that like Hamilton I would come up with a new formula. Instead, I was looking around wondering why the Broombridge Station didn’t have a ticket office. I already had my ticket, but I was curious how other people would get theirs. I asked a girl standing on the platform where to buy tickets. She said that there is no way to buy tickets there, so she sometimes rides without a ticket. The fine for not having tickets is very high in Ireland, so I expressed my surprised. She told me that she just says that she is from the town of Broombridge if she is asked to present her ticket.

Being a Russian I started scheming: obviously people can save money by buying tickets to Broombridge and continuing without a ticket wherever they need to go. If the tickets are checked, they can claim that they are traveling from Broombridge. Clearly Ireland hasn’t been blessed with very many Russians visitors.

by Pavel Etingof and Tanya Khovanova

We worked for several years with RSI where we supervised summer math research projects by high school students. Now, we’ve started an additional program at MIT’s math department called PRIMES, where local high school students do math research during the academic year. In this essay we would like to discuss what makes a good math research project for a high school student.

A doable project. The project should not be believed to be extremely difficult to yield at least results. It is very discouraging for an aspiring mathematician not to produce anything during their first project.

An accessible beginning. The student should be able to start doing something original soon after the start of the project. After all, they don’t come to us for coursework, but for research.

Flexibility. It is extremely important to offer them a project that is adjustable; it should go in many directions with many different potential kinds of results. Since we do not know the strength of incoming students in advance, it is useful to have in mind both easier and harder versions of the project.

Motivation. It is important for the project to be well motivated, which means related to other things that have been studied and known to be interesting, to research of other people, etc. Students get more excited when they see that other people are excited about their results.

A computer component. This is not a must for a good project. But modern mathematics involves a lot of computation and young students are better at it than many older professors. Such a project gives young students the opportunity to tackle something more senior people are interested in but might not have enough computer skills to solve. In addition, through computer experiments students get exposed to abstract notions (groups, rings, Lie algebras, representations, etc.) in a more “hands-on” way than when taking standard courses, and as a result are less scared of them.

A learning component. It is always good when a project exposes students to more advanced notions.

The student should like their project. This is very difficult to accomplish when projects are chosen in advance before we meet the students. However, we try to match them to great projects by using the descriptions they give of their interests on their applications. It goes without saying that mentors should like their project too.

Having stated the desired properties of a good project, let us move on to giving examples: bad projects and good projects. We start with a bad one:

Prove that the largest power of 2 that doesn’t contain 0 is 2⁸⁶.

The project satisfies only one requirement: it contains a computer component. Otherwise, it doesn’t have an accessible beginning. It is not very flexible: if the student succeeds, the long-standing conjecture will be proven; if s/he doesn’t, there is not much value in intermediate results. The question is not very interesting. The only motivation is that it has been open for a long time. Also, there is not much to learn. Though, almost any theoretical question can be made flexible. We can start with the question above and change its direction to make it more promising and enticing.

Another bad example is a project where the research happens after the programs are written. This is bad because it is difficult to estimate the programming abilities of incoming students. It doesn’t have an accessible beginning and there is no flexibility until the programming part is finished. If the student can’t finish the programming quickly, s/he will not have time to look at the results and produce conjectures. For example, almost any project in studying social networks may fall into this category:

Study an acquaintance graph for some epic movies or fiction, for example Star Wars or The Lord of the Rings. In this graph people are vertices and two people are connected by an edge if they know each other. The project is to compare properties of such graphs to known properties of other social networks.

Though the networks in movies are much smaller than other networks that people study, the amount of programming might be substantial. This project can be a good project for a person with a flexible time frame or a person who is sure in advance that there will be enough time for him/her to look at the data.

Now on to an example of a good project. Lynnelle Ye and her mentor, Tirasan Khandhawit, chose to analyze the game of Chomp on graphs during RSI 2009.

Given a graph, on each turn a player can remove an edge or a vertex together with all adjacent edges. The player who doesn’t have a move loses. This game was previously solved for complete graphs and forest graphs, so the project was to analyze the game for other types of graphs.

It is clear how to analyze the game for any particular new graph. So that could be a starting point providing an accessible beginning. After that the next step could be to analyze other interesting sets of graphs. The flexibility is guaranteed by the fact that there are many sets of graphs that can be used. In addition, the project entails learning some graph theory and game theory. And the project has a computational component.

Lynnelle Ye successfully implemented this project and provided a complete analysis of complete n-partite graphs for arbitrary n and all bipartite graphs. She also gave partial results for odd-cycle pseudotrees. The paper is available at the arxiv. Not surprisingly, Lynelle got fourth place in the Intel Science Talent Search and second place in the Siemens Competition.

Janet Mertz encouraged me to find IMO girls and compare their careers to that of their teammates. I had always wanted to learn more about the legendary Lida Goncharova — who in 1962 was the first girl to win an IMO gold medal. So I located her, and after an interview, wrote about her. Only 14 years later, in 1976, did the next girl get a gold medal. That was me. I was ranked overall second and had 39 points out of 40.

As I did in the article about Lida, I would like to compare my math career to that of my teammates.

I got my PhD in 1988 and moved to the US in 1990. My postdoc at MIT in 1993 was followed by a postdoc at Bar-Ilan University. In 1996 I got a non-paying visiting position at Princeton University. In 1998 I gave up academia and moved to industry, accepting an offer from Bellcore. There were many reasons for that change: family, financial, geographical, medical and so on.

On the practical level, I had had two children and raising them was my first priority. But there was also a psychological element to this change: my low self-esteem. I believed that I wasn’t good enough and wouldn’t stand a chance of finding a job in academia. Looking back, I have no regrets about putting my kids first, but I do regret that I wasn’t confident enough in my abilities to persist.

I continued working in industry until I resigned in January 2008, due to my feeling that I wasn’t doing what I was meant to do: mathematics. Besides, my children were grown, giving me the freedom to leave a job I did not like and return to the work I love. Now I am a struggling freelance mathematician affiliated with MIT. Although my math blog is quite popular and I have been publishing research papers, I am not sure that I will ever be able to find an academic job because of my non-traditional curriculum vitae.

The year 1976 was very successful for the Soviet team. Out of nine gold medals our team took four. My result was the best for our team with 39 points followed by Sergey Finashin and Alexander Goncharov with 37 points and by Nikita Netsvetaev with 34 points.

Alexander Goncharov became a full professor at Brown University in 1999 and now is a full professor at Yale University. His research is in Arithmetic Algebraic Geometry, Teichmuller Theory and Integral Geometry. He has received multiple awards including the 1992 European Math Society prize. Sergey Finashin is very active in the fields of Low Dimensional Topology and Topology of Real Algebraic Varieties. He became a full professor at Middle East Technical University in Ankara, Turkey in 1998. Nikita Netsvetaev is an expert in Differential Topology. He is a professor at Saint Petersburg State University and the Head of the High Geometry Department.

Comparing my story to that of Lida, I already see a pattern emerging. Now I’m curious to hear the stories of other gold-winning women. I believe that the next gold girl, in 1984, was Karin Gröger from the German Democratic Republic. I haven’t yet managed to find her, so can my readers help?

I am starting yet another part-time job as the Head Mentor at PRIMES, a new MIT research program for high schoolers. I am very excited about this program, for it will be valuable not only to kids who want to become researchers, but also to kids who just want to see what research is like. Kids who want to learn to think in a new way will also find it highly useful.

PRIMES is in many ways similar to RSI, which it augments and complements. There are also a lot of differences. Keep in mind that I am only comparing PRIMES to the math part of RSI, with which I was working as a coordinator for two years. I do not know how RSI handles other sciences.

Different time scale. RSI lasts six weeks; PRIMES will take about a year. I already wrote about some peoples’ skepticism towards RSI in my piece called “Fast Food Research?.” PRIMES creates a more natural pace for research.

Choices. Because of the time schedule at RSI, students get their project as soon as they start. Students who realize by the end of the second week that they do not like their project are at a disadvantage: if they do not change their project, they’re stuck with something that does not inspire them or is too difficult, and if they do change their project, they won’t have enough time to do a great job. At PRIMES students will have time to talk to the mentors before starting their project, so that they can participate in choosing their project. Depending on how it goes later, they’ll have time to try several different directions. I believe that the best research comes from the heart: students who have the time and opportunity to shape their choices will be more invested in their project.

Application process. At RSI, The Center for Excellence in Education reviews the applications. Even though they usually do a superb job at sending us great students, I believe it would be an advantage if mentors were able to influence the review process, for they might find even better matches to their projects. At PRIMES, the mentors will have this opportunity to review the applications.

Geography. RSI accepts students from all over the US and from some other countries. PRIMES can only accept local students — those who live close enough to visit MIT once a week for four months. Because of this restriction, PRIMES is recruiting from a smaller pool of students than RSI. But for local students it means that it will be easier to get accepted to PRIMES than to RSI.

Coaching. At RSI, students get a lot of coaching. I think that every student is in close contact with four adults. Two of them are from the math department — mentor and coordinator (that’s me!) — and two tutors from CEE. PRIMES will have less coaching. A student will have a mentor and me, the head mentor. In addition, mentors might arrange for students to talk to the professors who originated their projects.

Immersion. RSI students are physically present. They are housed at MIT with the expectation that they completely devote their time to their research. Students at PRIMES will be integrating their research into the rest of their lives and their commitments. That will require good organizational skills and a lot of self-discipline. RSI students have discipline imposed on them by their situation — which may be an advantage to them.

Olympiads. While they are at RSI, students can’t go to IMO or other summer activities. This is why many strong Olympiad students choose not to go to RSI, or they turn down an RSI acceptance if in the meantime they have gotten on to an Olympic team. At PRIMES you can do both. It is possible to go to an Olympiad, in addition to writing a paper.

Grade. RSI students have to be juniors. There are no grade limitations for PRIMES. Thus, it is possible to go to PRIMES in one’s senior year. In this case, it may be too late to use PRIMES on college applications, but it is perfectly fine for the sake of research itself. Or it might be possible to go to PRIMES as a sophomore, and then apply for RSI the next year. This will strengthen the student’s application for RSI.

RSI is well-established and has proven itself. PRIMES is new and hopefully will offer young mathematicians additional opportunities to try research.

I think that the American system of education creates a lot of pressure for teachers to drill their students for standardized tests and multiple choice questions. This blocks creative thinking. Every program like PRIMES is very good for unleashing students’ creativity and contributing to the development of the future thinkers of American society.