Tanya Khovanova's Math Blog

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.

In the name of privacy, I have changed the names of the men I did not marry. But there is no point in changing the names of my ex-husbands, as my readers probably know their names anyway.

I received my first marriage proposal when I was 16. As a person who was unable to say “no” to anything, I accepted it. Luckily, we were not allowed to get married until I was 18, the legal marriage age in the USSR, and by that time we broke up.

To my next proposal, from Sasha, I still couldn’t say “no”, and ended up marrying him. The fact that I was hoping to divorce him before I got married at 19 shows that I should have devoted more effort in learning to say “no”. I decided to divorce him within the first year.

My next proposal came from Andrey, I said yes, with every intention of living with Andrey forever. We married when I was 22 and he divorced me when I was 29.

After I recovered from my second divorce, I had a fling with an old friend, Sam, who was visiting Moscow on his way to immigrate to Israel.

Sam proposed to me in a letter that was sent from the train he took from the USSR to Israel. At that point I realized I had a problem with saying “no”. The idea of marrying Sam seemed premature and very risky. I didn’t want to say yes. I should have said no, but Sam didn’t have a return address, so I didn’t say anything.

That same year I received a phone call from Joseph. Joseph was an old friend who lived in the US, and I hadn’t seen or heard from him for ten years. He invited me to visit him in the US and then proposed to me the day after my arrival. The idea of marrying Joseph seemed premature and very risky, but in my heart it felt absolutely right. I said yes, and I wanted to say yes.

I was very glad that I hadn’t promised anything to Sam. But I felt uncomfortable. So even before I called my mother to notify her of my marriage plans, I located Sam in Israel and called him to tell him that I had accepted a marriage proposal from Joseph. I needed to consent to marry someone else as a way of saying “no” to Sam.

After I married Joseph, I came back to Russia to do all the paperwork and pick up my son, Alexey for our move to the US. There I met Victor. I wasn’t flirting with Victor and was completely disinterested. So his proposal came as a total surprise. That was the time I realized that I had a monumental problem with saying “no”. I had to say “no” to Victor, but I couldn’t force myself to pronounce the word. Here is our dialogue as I remember it:

• Me: I can’t marry you, I am already married.
• Victor: I am sure it’s a fictitious marriage; you just want to move to the USA.
• Me: That’s not true. It’s a real marriage.
• Victor: If it were a fictitious marriage, you wouldn’t admit it. So, it’s a fictitious marriage. My proposal stands.

My sincere attempt at saying “no” didn’t work. I moved to the US to live with Joseph and I soon got pregnant. Victor was the first person on my list to notify — another rather roundabout way to reject a proposal.

The marriage lasted eight years. Sometime after I divorced Joseph, I met Evan who invited me on a couple of dates. I wasn’t sure I wanted to get involved with him. But he proposed and got my attention. I was single and available, though I had my doubts about him.

Evan mentioned that he had royal blood. So I decided to act like a princess. I gave him a puzzle:

I have two coins that together make 15 cents. One of them is not a nickel. What are my coins?

He didn’t solve it. In and of itself, that wouldn’t be a reason to reject a guy. But Evan didn’t even understand my explanation, despite the fact that he was a systems administrator. A systems administrator who doesn’t get logic is a definite turn-off.

So I said “no”! That was my first “no” and I have mathematics to thank.

Once I read a book in Russian that mentioned a study of the children of Soviet military personnel who had to move often. The conclusion was that frequent relocation is very damaging for children’s psyche. The children had to build new friendships, which they would lose the next time they had to move. After several moves they would stop making friends; later, as adults, they would be afraid of getting close to anyone.

In September 1996, my husband, my two children and I came to Princeton from Israel for my husband’s month-long visit to the Institute for Advanced Study. After the visit we were supposed to go back to Israel, but that didn’t happen. My husband returned alone and I stayed in Princeton with my children. That’s a long and sentimental story for another time.

Meanwhile, my older son Alexey started going to Princeton High School. By this time he had attended seven schools in three different countries. In light of the evidence presented in that book about the impact on children of moving, I felt very guilty. Alexey was entering 10th grade. Moving him again not only would further damage his ability to make friends, but would also screw up his college chances. He needed a stable environment leading up to college. For example, recommendation letters are better written by people who are involved with kids for several years. I was afraid to mess up his future. I promised myself not to move him again during high school, especially as Princeton High School was one of the best public schools in New Jersey.

At the same time, I got a Visiting Scholar position at Princeton University. Although it didn’t pay me any salary, through that position I received university housing, library privileges and an office. I was living on my personal savings and the monthly check my husband was sending me from Israel. My money was running out and I felt completely lost, like so many immigrants. I was new to Princeton; I didn’t have friends there; and I was struggling with English. On top of that, I had medical problems, not the least very low energy.

Ingrid Daubechies noticed me at the Princeton math department and approached me. After our conversation, she found some money for me to work on the Math Alive course she was designing. That work was a breath of fresh air. I enjoyed it tremendously, but the part-time salary was not enough. Then I received more help from Ingrid. She appreciated my work on Math Alive a lot, but realized that I needed a different solution. She sacrificed her own interests and started recommending me around. She arranged an interview for me at Telcordia, who offered me a job as a systems engineer.

The decision to accept this job was very painful, because I did not want to leave academia. However, considering that my priority was to keep Alexey in Princeton High School, I didn’t feel I had other options. I knew that I couldn’t stay much longer at Princeton University and I was aware that getting a University job often requires relocation.

Looking back, I think the reasons behind this decision were more complex than sacrificing my career for my child. If I had known more about social supports for poor families and about other possible research jobs, or if I had been more confident in my research abilities, I might not have left academics.

Alexey triumphed at Princeton High School. The school allowed him to take math courses at Princeton University. He took several, including the course in logic by John Conway and two courses in graph theory by Paul Seymour. Alexey’s multi-variable calculus professor complained to me that she couldn’t fit her grades into the required curve. If she gave Alexey 100%, the others would have to get less than 20. Luckily, it turned out that because her class was small, she didn’t need to bother about making a curve. After three years in Princeton High School, Alexey secured an impressive resume and great recommendation letters and went to MIT to pursue a double major in mathematics and computer science.

OnlineDegree.net selected the 50 Best Blogs for Math Majors, and I am pleased that Tanya Khovanova’s Math Blog is number two. Since they did not explain their criteria, I suspected that it might be according to the number of Google hits. To double check, I Googled “math blog” and once again my blog was number two.

This might be the right moment to acknowledge the others involved with my blog. First, Sue Katz, my writing teacher and editor, corrected the English in most of my posts. Now I do not “do” mistakes in English any more, I make them.

My sons, Alexey and Sergei, are a huge support. Sometimes my poor kids have to listen endlessly to my latest idea, until I am ready to write about it. And then they will even read the final piece, and continue to encourage me.

But the most important motivators are you, my readers. Your comments, your personal emails and your feedback keep me writing.

Do you know that some Russian letters are shaped exactly as some letters in the English alphabet? The shapes are the same, but the sounds of the letters are not. My Russian last name can be completely spelled using English letters: XOBAHOBA.

The adequate translation of my last name into English is Hovanova. You might ask where the first “K” came from. For many years French was considered the language of diplomacy and the USSR used French as an official language for traveling documents.

But “H” in French is silent and “Hovanova” would have been pronounced as “Ovanova.” To prevent that, Russians used “kh” for the “h” sound.

Now to my first name. I was born Tatyana, for which Tanya is a nickname. Back in Russia, Tanya is used for children and students and Tatyana for adults and teachers. As I was a student throughout my 30 years of life in Russia, I was always Tanya. When I moved to the US, I decided to keep using Tanya, which I much preferred to Tatyana.

A psychiatrist might think that I wanted to be a student forever or refused to grow up. Or I could be accused of being lazy, as Tanya is shorter. In reality, I was just trying to be considerate. Tanya is easier to write and to spell for Americans. Anyway, I already had enough problems spelling out my last name in this country.

Now that more information is getting translated from Russian into English, I keep stumbling on references to me as to Hovanova or Tatyana. For example, the IMO official website used Russian sources to come up with the names of the Russian participants. They then translated the names directly into English, instead of going through French. As a result, on their website I am Tatyana Hovanova. This is not unique to me: many Russian names on the IMO website differ from those peoples’ passport names.

By the way, if you Google my last name you will encounter other Khovanovas. Khovanova is not a particularly unusual name. Only one of the Khovanovas that came up in my search results is a close relative. Elizabeth Khovanova is my father’s second wife and a dear friend. She is also an accomplished geneticist.

Khovanova is used only for females in Russia. The male equivalent is Khovanov. Surely you have heard of my half-brother Mikhail Khovanov and his homologies.

I started my blog about two year ago. I have written about 200 entries. According to my traffic reports, these have been the top ten most popular entries:

1. The Odd One Out
2. Divisibility by 7 is a Walk on a Graph, by David Wilson
3. The Dirtiest Math Problem Ever (Rated R)
4. My IQ
5. A Miracle Equation
6. Back-to-School Funny Pictures
7. Flexible Zipcar Algorithm
8. A Very Special Ten-Digit Number
9. What Does It Take to Get Accepted by Harvard or Princeton?
10. AMC, AIME, USAMO Contradiction

My most popular category was Math Humor.

Israel Gelfand’s memorial is being held at Rutgers on December 6, 2009. I was invited as Gelfand’s student.

My relationship with Gelfand was complicated: sometimes it was very painful and sometimes it was very rewarding. I was planning to attend the memorial to help me forget the pain and to acknowledge the good parts.

I believe that my relationship with Gelfand was utterly unique. You see, I was married three times, and all three times to students of Gelfand.

Now that I know that I can’t make it to the memorial, I can’t stop wondering how many single male students of Gelfand will be there.