Love and Math

by Frenkel, Edward

  *Note that flipping a table is not a symmetry: this would turn it upside down – let’s not forget that a table has legs. If we were to consider a square or a circle (no legs attached), then flips would be bona fide symmetries. We would have to include them in the corresponding symmetry groups.

  Chapter 3

  The Fifth Problem

  Evgeny Evgenievich’s plan worked perfectly: I was “converted” to math. I was learning quickly, and the deeper I delved into math, the more my fascination grew, the more I wanted to know. This is what happens when you fall in love.

  I started meeting with Evgeny Evgenievich on a regular basis. He would give me books to read, and I would meet him once a week at the pedagogical college where he taught to discuss what I had read. Evgeny Evgenievich played soccer, ice hockey, and volleyball on a regular basis, but like many men in the Soviet Union in those days, he was a chain smoker. For a long time afterward, the smell of cigarettes was associated in my mind with doing mathematics.

  Sometimes our conversations would last well into the night. Once, the auditorium we were in was locked by the custodian who couldn’t fathom that there would be someone inside at such a late hour. And we must have been so deep into our conversation that we didn’t hear the turning of the key. Fortunately, the auditorium was on the ground floor, and we managed to escape through a window.

  The year was 1984, my senior year at high school. I had to decide which university to apply to. Moscow had many schools, but there was only one place to study pure math: Moscow State University, known by its Russian abbreviation MGU, for Moskovskiy Gosudarstvenny Universitet. Its famous Mekh-Mat, the Department of Mechanics and Mathematics, was the flagship mathematics program of the USSR.

  Entrance exams to colleges in Russia are not like the SAT that American students take. At Mekh-Mat there were four: written math, oral math, literature essay composition, and oral physics. Those who, like me, graduated from high school with highest honors (in the Soviet Union we were then given a gold medal) would be automatically accepted after getting a 5, the highest grade, at the first exam.

  I had by then progressed far beyond high school math, and so it looked like I would sail through the exams at MGU.

  But I was too optimistic. The first warning shot came in the form of a letter I received from a correspondence school with which I had studied. This school had been organized some years earlier by, among others, Israel Gelfand, the famous Soviet mathematician (we will talk much more about him later). The school intended to help those students who, like me, lived outside of major cities and did not have access to special mathematical schools. Every month, participating students would receive a brochure elucidating the material that was studied in school and going a little beyond. It also contained some problems, more difficult than those discussed at school, which a student was supposed to solve and mail back. Graders (usually undergrads of Moscow University) read those solutions and returned them, marked, to the students. I was enrolled in this school for three years, as well as in another school, which was more physics-oriented. It was a helpful resource for me, though the material was pretty close to what was studied at school (unlike the stuff I was studying privately with Evgeny Evgenievich).

  The letter I received from this correspondence school was short: “If you would like to apply to Moscow University, stop by our office, and we will be happy to give you advice,” and it gave the address on the campus of MGU and the visiting hours. Shortly after receiving the letter, I took the two-hour train ride to Moscow. The school’s office was a big room with a bunch of desks and a number of people working, typing, and correcting papers. I introduced myself, produced my little letter, and was immediately led to a diminutive young woman, in her early thirties.

  “What’s your name?” she said by way of greeting.

  “Eduard Frenkel.” (I used the Russian version of Edward in those days.)

  “And you want to apply to MGU?”

  “Yes.”

  “Which department?”

  “Mekh-Mat.”

  “I see.” She lowered her eyes and asked:

  “And what’s your nationality?”

  I said, “Russian.”

  “Really? And what are your parents’ nationalities?”

  “Well... My mother is Russian.”

  “And your father?”

  “My father is Jewish.”

  She nodded.

  This dialogue might sound surreal to you, and as I am writing it now, it sounds surreal to me too. But in the Soviet Union circa 1984 – remember Orwell?* – it was not considered bizarre to ask someone what his or her “nationality” was. In the interior passport that all Soviet citizens had to carry with them, there was in fact a special line for “nationality.” It came after (1) first name, (2) patronymic name, (3) last name, and (4) the date of birth. For this reason, it was called pyataya grafa, “the fifth line.” Nationality was also recorded in one’s birth certificate, as were the nationalities of the parents. If their nationalities were different, as in my case, then the parents had a choice of which nationality to give to their child.

  For all intents and purposes, the fifth line was a code for asking whether one was Jewish or not. (People of other nationalities, such as Tatars and Armenians, against whom there were prejudices and persecution – though not nearly at the same scale as against the Jews – were also picked up this way.) My fifth line said that I was Russian, but my last name – which was my father’s last name, and clearly sounded Jewish – gave me away.

  It is important to note that my family was not religious at all. My father was not brought up in a religious tradition, and neither was I. Religion in the Soviet Union was in fact all but non-existent in those days. Most Christian Orthodox churches were destroyed or closed. In the few existing churches, one could typically only find a few old babushkas (grandmothers), such as my maternal grandmother. She occasionally attended service at the only active church in my hometown. There were even fewer synagogues. There were none in my hometown; in Moscow, whose population was close to 10 million, officially there was only one synagogue.1 Going to a service in a church or a synagogue was dangerous: one could be spotted by special plain-clothed agents and would then get in a lot of trouble. So when someone was referred to as being Jewish, it was meant not in the sense of religion but rather in the sense of ethnicity, of “blood.”

  Even if I hadn’t been using my father’s last name, my Jewish origin would be picked up by the admissions committee anyway, because the application form specifically asked for the full names of both parents. Those full names included patronymic names; that is, the first names of the grandparents of the applicant. My father’s patronymic name is Joseph, which sounded unmistakably Jewish in the Soviet Union of that era, so this was another way to find out (if his last name hadn’t given me away). The system was set up in such a way that it would flag those who were at least one-quarter Jewish.

  Having established that by this definition I was a Jew, the woman said, “Do you know that Jews are not accepted to Moscow University?”

  “What do you mean?”

  “What I mean is that you shouldn’t even bother to apply. Don’t waste your time. They won’t let you in.”

  I didn’t know what to say.

  “Is that why you sent me this letter?”

  “Yes. I’m just trying to help you.”

  I looked around. It was clear that everyone in the office was aware of what this conversation was about, even if they weren’t listening closely. This must have already happened dozens of times, and everybody seemed used to it. They all averted their eyes, as if I were a terminally ill patient. My heart sank.

  I had encountered anti-Semitism before, but at a personal, not institutional, level. When I was in fifth grade, some of my classmates took to taunting me with evrey, evrey (“Jew, Jew”). I don’t think they had any idea what this meant (which was clear from the fact that some of them confused the word evrey, which meant “Jew,” with evropeyets
, which meant “European”) – they must have heard anti-Semitic remarks from their parents or other adults. (Unfortunately, anti-Semitism was deeply rooted in the Russian culture.) I was strong enough and lucky enough to have a couple of true friends who stood by me, so I was never actually beaten up by these bullies, but this was an unpleasant experience. I was too proud to tell the teachers or my parents, but one day a teacher overheard and intervened. As a result, those kids were immediately called to the principal, and the taunting stopped.

  My parents had heard of the discrimination against Jews in entrance exams to universities, but somehow they did not pay much attention to this. In my hometown, there weren’t many Jews to begin with, and all the purported discrimination cases my parents had heard of concerned programs in physics. A typical argument went that Jews weren’t accepted there because the studies in such a program were related to nuclear research and hence to national defense and state secrets; the government didn’t want Jews in those areas because Jews could emigrate to Israel or somewhere else. By this logic, there shouldn’t have been a reason to care about those who studied pure math. Well, apparently, someone did.

  Everything about my conversation at MGU was strange. And I am not just talking about the Kafkaesque aspect of it. It is possible to conclude that the woman I talked to simply tried to help me and other students by warning us of what’s going to happen. But could this really be the case? Remember, we are talking about 1984, when the Communist Party and the KGB still tightly controlled all aspects of life in the Soviet Union. The official policy of the state was that all nationalities were equal, and publicly suggesting otherwise would put one in danger. Yet, this woman calmly talked about this to me, a stranger she had just met, and she didn’t seem to be worried about being overheard by her colleagues.

  Besides, the exams at MGU were always scheduled one month ahead of all other schools. Therefore, students who were failed at MGU would still have a chance to apply elsewhere. Why would someone try to convince them not even to try? It sounded like some powerful forces were trying to scare me and other Jewish students away.

  But I would not be deterred. After talking about all this at great length, my parents and I felt that I had nothing to lose. We decided that I would apply to MGU anyway and just hope for the best.

  The first exam, at the beginning of July, was a written test in mathematics. It always consisted of five problems. The fifth problem was considered deadly and unsolvable. It was like the fifth element of the exam. But I was able to solve all problems, including the fifth. Aware as I was of the strong likelihood that whoever graded my exam could be biased against me and would try to find gaps in my solutions, I wrote everything out in excruciating detail. I then checked and double-checked all my arguments and calculations to make sure that I hadn’t made any mistakes. Everything looked perfect! I was in an upbeat mood on the train ride home. The next day I told Evgeny Evgenievich my solutions, and he confirmed that everything was correct. It seemed like I was off to a good start.

  My next exam was oral math. It was scheduled for July 13, which happened to be a Friday.

  I remember very clearly many details about that day. The exam was scheduled for the early afternoon, and I took the train from home with my mother that morning. I entered the room at MGU a few minutes before the exam. It was a regular classroom, and there were probably between fifteen and twenty students there and four or five examiners. At the start of the exam each of us had to draw a piece of paper from a big pile on the desk at the front of the room. Each paper had two questions written on it, and it was turned blank side up. It was like drawing a lottery ticket, so we called this piece of paper bilet, ticket. There were perhaps one hundred questions altogether, all known in advance. I didn’t really care which ticket I would draw as I knew this material inside-out. After drawing the ticket, each student had to sit down at one of the desks and prepare the answer, using only the provided blank sheets of paper.

  The questions on my ticket were: (1) a circle inscribed in a triangle and the formula for the area of the triangle using its radius; and (2) derivative of the ratio of two functions (the formula only). I was so ready for these questions, I could have answered them in my sleep.

  I sat down, wrote down a few formulas on a sheet of paper, and collected my thoughts. This must have taken me about two minutes. There was no need to prepare more; I was ready. I raised my hand. There were several examiners present in the room, and they were all waiting for the students to raise their hands, but, bizarrely, they ignored me, as if I did not exist. They looked right through me. I was sitting with my hand raised for a while: no response.

  Then, after ten minutes or so, a couple of other kids raised their hands, and as soon as they did, the examiners rushed to them. An examiner would take a seat next to a student and listen to him or her answer the questions. They were quite close to me, so I could hear them. The examiners were very polite and were mostly nodding their heads, only occasionally asking follow-up questions. Nothing out of the ordinary. When a student finished answering the questions on the ticket (after ten minutes or so), the examiner would give him or her one additional problem to solve. Those problems seemed rather simple, and most students solved them right away. And that was it!

  The first couple of students were already happily gone, having obviously earned a 5, the highest grade, and I was still sitting there. Finally, I grabbed one of the examiners passing by, a young fellow who seemed like he was a fresh Ph.D., and asked him pointedly: “Why aren’t you talking to me?” He looked away and said quietly: “Sorry, we are not allowed to talk to you.”

  An hour or so into the exam, two middle-aged men entered the room. They briskly walked up to the table at the front of the room and presented themselves to the guy who was sitting there. He nodded and pointed at me. It became clear that these were the people I’d been waiting for: my inquisitors.

  They came up to my desk and introduced themselves. One was lean and quick, the other slightly overweight and with a big mustache.

  “OK,” the lean man said – he did most of the talking – “what have we got here? What’s the first question?”

  “The circle inscribed in a triangle and...”

  He interrupted me: “What is the definition of a circle?”

  He was quite aggressive, which was in sharp contrast to how other examiners treated students. Besides, the other examiners never asked anything before the student had a chance to fully present their answer to the question on the ticket.

  I said, “A circle is the set of points on the plane equidistant from a given point.”

  This was the standard definition.

  “Wrong!” declared the man cheerfully.

  How could this possibly be wrong? He waited for a few seconds and then said, “It’s the set of all points on the plane equidistant from a given point.”

  That sounded like excessive parsing of words – the first sign of trouble ahead.

  “OK,” the man said, “What is the definition of a triangle?”

  After I gave that definition, and he thought about it, no doubt trying to see if he could do some more nit-picking, he continued: “And what’s the definition of a circle inscribed in a triangle?”

  That led us to the definition of the tangent line, then just “a line,” and that led to other things, and soon he was asking me about Euclid’s fifth postulate about the uniqueness of parallel lines, which wasn’t even part of the high school program! We were talking about issues that were not even close to the question on the ticket and far beyond what I was supposed to know.

  Every word I said was questioned. Every concept had to be defined, and if another concept was used in the definition, then I was immediately asked to define it as well.

  Needless to say, if my last name were Ivanov, I would never be asked any of these questions. In retrospect, the prudent course of action on my part would have been to protest right away and tell the examiners that they were out of line. But it’s easy to say this no
w. I was sixteen years old, and these men were some twenty-five years my senior. They were the officials administering an exam at Moscow State University, and I felt obligated to answer their questions as best I could.

  After nearly an hour-long interrogation, we moved to the second question on my ticket. By then, other students had left, and the auditorium was empty. Apparently, I was the only student in that room who required “special care.” I guess they tried to place Jewish students so that there would be no more than one or two of them in the same room.

  The second question asked me to write the formula for the derivative of the ratio of two functions. I was not asked to give any definitions or proofs. The question said specifically, the formula only. But of course, the examiners insisted that I explain to them a whole chapter of the calculus book.

  “What is the definition of derivative?”

  The standard definition I gave involved the concept of limit.

  “What is the definition of limit?” Then “What is a function?” and on and on it went again.

  The question of ethnic discrimination at the entrance exams to MGU has been the subject of numerous publications. For example, in his insightful article2 in the Notices of the American Mathematical Society, mathematician and educator Mark Saul used my story as an example. He aptly compared my exam to the Red Queen interrogating Alice in Alice in Wonderland. I knew the answers, but in this game, in which everything I said was turned against me, I couldn’t possibly win.

  In another article3 on this subject in the Notices, journalist George G. Szpiro gave this account:

  Jews – or applicants with Jewish-sounding names – were singled out at the entrance exams for special treatment.... The hurdles were raised in the oral exam. Unwanted candidates were given “killer questions” that required difficult reasoning and long computations. Some questions were impossible to solve, were stated in an ambiguous way, or had no correct answer. They were not designed to test a candidate’s skill but meant to weed out “undesirables.” The grueling, blatantly unfair interrogations often lasted five or six hours, even though by decree they should have been limited to three and a half. Even if a candidate’s answers were correct, reasons could always be found to fail him. On one occasion a candidate was failed for answering the question “what is the definition of a circle?” with “the set of points equidistant to a given point.” The correct answer, the examiner said, was “the set of all points equidistant to a given point.” On another occasion an answer to the same question was deemed incorrect because the candidate had failed to stipulate that the distance had to be nonzero. When asked about the solutions to an equation, the answer “1 and 2” was declared wrong, the correct answer being, according to an examiner, “1 or 2.” (On a different occasion, the same examiner told another student the exact opposite: “1 or 2” was considered wrong.)