When I was growing up in the Soviet Union in the 1980s, I thought math was a stale, boring subject.1 I could solve all of the problems and ace all of the exams at school, but what we discussed in class seemed pointless, irrelevant. What really excited me was Quantum Physics. I devoured all the popular books on this subject I could get my hands on. But these books didn’t go far enough in answering deeper questions about the structure of the universe, so I wasn’t fully satisfied.
As luck would have it, I got help from a family friend. I grew up in a small industrial town called Kolomna, population 150 thousand, which was about seventy miles away from Moscow, or just over two hours by train. My parents worked as engineers at a large company, making heavy machinery. One of their friends was a mathematician by the name of Evgeny Evgenievich Petrov, who was a professor at a local college preparing school teachers. A meeting was arranged.
Then in his late forties, Evgeny Evgenievich was friendly and unassuming. Bespectacled and with beard stubble, he was just like what I imagined a mathematician would look like, and yet there was something captivating in the probing gaze of his big eyes. They exuded curiosity about everything. Knowing that I was fascinated with the quantum world, he convinced me that spectacular advances in this field were all based on hardcore mathematics.
“If you really want to understand it, you have to first learn math,” he said.
At school we studied things like quadratic equations, basic Euclidean geometry, and trigonometry. I had always assumed that all mathematics somehow revolved around these subjects: perhaps problems became more complicated, but they still remained within the same general framework with which I was familiar. But what Evgeny Evgenievich showed me were the glimpses of an entirely different world, an invisible parallel universe, whose existence I hadn’t even imagined. It was love at first sight.
I started meeting with Evgeny Evgenievich on a regular basis. He would give me books to read, and, at our meetings, I would tell him what I learned and ask follow-up questions. Evgeny Evgenievich played soccer, ice hockey, and volleyball with enthusiasm, but like many men in the Soviet Union in those days, he was also a chain smoker. For a long time afterwards, the smell of cigarettes was, in my mind, associated with doing mathematics.
I was learning quickly, and the deeper I delved into math, the more my fascination grew. Sometimes our meetings 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 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 the window!
It 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, Moskovskiy Gosudarstvenny Universitet. Its famous Mekh-Mat, the Department of Mechanics and Mathematics, was the flagship mathematics program of the USSR. Since I wanted to study pure math, I had no choice but to apply there.
Unlike the U.S., there are entrance exams to colleges in Russia. At Mekh-Mat there were four: written math, oral math, an essay on literature, and oral physics. I had, by then, progressed far beyond high school math, so it looked like I would sail through these exams.
But I was too optimistic. The first warning shot came in the form of a letter I received from a correspondence school. This school was started some years earlier, and Israel Gelfand, the famous Soviet mathematician, was one of the founders. It was created in order to help those students who, like me, lived outside of major cities and did not have access to special math schools. Every month, 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 students were 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 level was still very basic, no comparison to what 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 of MGU and the visiting hours. Shortly after receiving the letter, I took the two-hour train ride to Moscow. I found the place easily. It was a big room with a bunch of desks and a number of people working, typing, correcting papers. I introduced myself and 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?”
“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.”
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 their “nationality” was. In the inner passport which every Soviet citizen had to carry with them, there was in fact a special line for “nationality,” and for this reason it was called pyataya grafa, “the fifth line.” It came after (1) first name, (2) patronymic name, (3) last name, and (4) the date of birth. 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 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, like Tatars and Armenians, against whom there were prejudices and persecution—though not nearly on 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.
Even if I hadn’t been using my father’s last name, my Jewish origin would have been 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 was Joseph, clearly Jewish, so this was another way to find out (if his last name weren’t so obviously Jewish). The system was set up in such a way that it would pick up those who were at least one-quarter Jewish and everyone of those was classified as a Jew, much like it was in Nazi Germany.
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 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 to have a couple of true friends who stood by me, so I was never actually beaten up, but this was a very unpleasant experience. I was too proud to tell the teachers or my parents, but one day a teacher was passing by. He intervened, and as the result, those kids were immediately called to the principal, and the taunting stopped.
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, like my maternal grandmother. She occasionally attended service at the only active church in my hometown, Kolomna. There were even fewer synagogues. There were none in my home town; in Moscow, whose population was close to ten million, there was only one. Going to a service in a church or a synagogue was dangerous: one could be spotted by special plain-clothed agents and 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 in the sense of ethnicity, “blood.”
My parents had heard of the discrimination against Jews at 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 of 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 the defense and state secrets; the government didn’t want Jews to be in those areas because Jews could emigrate to Israel or somewhere else. By this logic, why would anyone care about 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 real danger. And 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 they attempt to convince someone not even to try? It sounded like some powerful forces were trying to scare me and other Jewish students away. But I wouldn’t 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, which always consisted of five problems. The fifth problem was considered “deadly” and “unsolvable.” But I was able to solve all of the problems, including the fifth. Keeping in mind 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. “Hmm. . . . Friday the thirteenth,” I thought. 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 in the 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 a bilet, ticket. There were perhaps about a hundred questions altogether, all known in advance. I didn’t really care which ticket I drew, as I knew this material inside out. After drawing their ticket, each student had to sit down at one of the desks and prepare the answer (we couldn’t use any materials, just blank sheets of paper which were provided).
The questions on my ticket were: (1) With a circle inscribed in a triangle, what is the formula for the area of the triangle in terms of the radius?, and (2) What is the derivative of the ratio of two functions (formula only)? Easy questions, which I could answer 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 like this, 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 five, 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 first 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 the words. The first sign of trouble ahead.
“OK,” the man said, “What is the definition of the triangle?”
After I gave that definition, he thought about it, no doubt trying to see if he could do some more nit-picking, then 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 which 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 had been Ivanov, I would never have been 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 now. I was sixteen years old, and these men were twenty-five to thirty years my senior. They were the officials administering an exam at Moscow State University, and I felt obliged 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, the 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.
In his insightful article in the Notices of the American Mathematical Society about the discrimination at the MGU exams, the 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 article on this subject in Notices, the Swiss 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.)
After another hour and a half had gone by, one of the examiners said: “OK, we are done with the questions. Here is a problem we want you to solve.” The problem he gave me was pretty hard. The solution required the use of the so-called Sturm principle, which was not studied in school. But I knew about this from my correspondence courses, so I was able to solve it. As I was working my way through the final calculations, the examiner came back.
“Are you done yet?”
He looked at my writings and no doubt saw that my solution was correct and that I was just finishing my calculations.
“You know what,” he said, “let me give you another problem.”
Curiously, the second problem was twice as hard as the first one. I was still able to solve it, but the examiner again interrupted me halfway through.
“Not done yet?” he said, “Try this one.”
If this were a boxing match, with one of the boxers pressed in the corner, bloodied, desperately trying to hold his own against the barrage of punches falling on him (many of them below the belt, I might add), that would be the equivalent of the final, deadly, blow. The problem looked innocent enough at first glance: given a circle and two points on the plane outside the circle, construct another circle passing trough those two points and touching the first circle at one point.
But the solution is in fact quite complicated. Few of my future colleagues at Harvard and Berkeley would have been able to solve it right away. One must use “inversion,” a concept that was not studied in high school and hence could not possibly be allowed in this exam. I knew about inversion and I realized that I needed to apply it here. I started to work on the problem, but a few minutes later my interrogators came back and sat down next to me. One of them said:
“You know, I’ve just talked to the deputy chairman of the Admissions Committee and I told him about your case. He asked me why we are still wasting our time. . . . Look,” he pulled out an official-looking form with some notes scribbled on it—this was the first time I had seen it. “On the first question on your ticket, you did not give us a complete answer, you didn’t even know the definition of a circle. So we have to put a minus. On the second question, your knowledge was also shaky, but OK, we give you minus plus. Then you couldn’t completely solve the first problem, did not solve the second problem. And on the third? You haven’t solved it either. See, we have no choice but to fail you.”
I looked at my watch. More than four hours had passed by since the beginning of the exam. I was exhausted.
“Can I see my written exam?”
The other man went back to the main table and brought my exam. He put it in front of me. As I was turning the pages, I felt like I was in a surrealistic movie. All answers were correct, all solutions were correct. But there were many comments. They were all made in pencil—so that they could be easily erased, I guess—but they were all ridiculous, like someone was playing a practical joke on me. One of them still stands out in my mind: in the course of a calculation I wrote “√8 > 2.” And there was a comment next to it: “not proved.” Really? Other comments were no better. And what grade did they give me, for all five problems solved, with all correct answers? Not 5, not 4. It was a 3, the Russian equivalent of a C. They gave me a C for this? I knew it was over. There was no way I could fight this system. I said: “All right.”
One of the men asked: “Aren’t you going to appeal?”
I knew that there was an appeal board. But what would be the point? Perhaps I could raise my grade on the written exam from 3 to 4, but appealing the result of the oral exam would be more difficult: it would be their word against mine. And even if I could raise the grade—then what? There were still two more exams left at which they could easily get me. Here is what George Szpiro wrote in Notices:
And if an applicant, against all odds, managed to pass both the written and the oral test, he or she could always be failed on the required essay on Russian literature with the set phrase “the theme has not been sufficiently elaborated.” With very rare exceptions, appeals against negative decisions had no chance of success. At best they were ignored, at worst the applicant was chastised for showing “contempt for the examiners.”
A bigger question was: Did I really want to enroll in a university which did everything in its power to prevent me from being there?
I said: “No. Actually, I’d like to withdraw my application.”
Their faces lightened up. No appeal meant less hassle for them, less potential for trouble.
“Sure,” the talkative one said, “I’ll get your stuff for you right away.”
We walked out of the room and entered the elevator. The doors closed. It was just the two of us. The examiner was clearly in a good mood. He said:
“You did very well. A really impressive performance. I was wondering: did you go to a special math school?”
I grew up in a small town, we didn’t have special math schools.
“Really? Perhaps, your parents are mathematicians?”
No, they are engineers.
“Interesting. . . . It’s the first time I’ve seen such a strong student who did not go to a special math school.”
I couldn’t believe what he was saying. This man had just failed me after an unfairly administered, discriminatory, grueling five-hour exam. For all I knew, he had killed my dream of becoming a mathematician. A sixteen-year-old student, whose only fault was that he came from a Jewish family. And now this guy is giving me compliments and expecting me to open up to him?!
But what could I do? Yell at him, punch him in the face? I was just standing there, silent, stunned. He continued:
“Let me give you some advice. Apply to the Moscow Institute of Oil and Gas. They have an Applied Mathematics program, which is quite good. They take students like you there.”
The elevator doors opened and a minute later he handed me my thick application folder, with a bunch of my school trophies and prizes oddly sticking out of it.
“Good luck to you,” he said, but I was too exhausted to respond. My only wish was to get the hell out of there!
And then I was outside, on the giant staircase of the immense MGU building. I was breathing fresh summer air again and hearing the sounds of the big city coming from a distance. It was getting dark, and there was almost no one around. I immediately spotted my parents who had been waiting anxiously for me on the steps this whole time. By the look on my face, and the big folder I was holding in my hands, they knew right away what had happened inside.
Editors’ note: In the end, Frenkel enrolled in the Applied Math program at the Moscow Institute of Oil and Gas. In addition to his classes there, he read widely in mathematics and attended lectures at MGU (sneaking into the building by jumping over a fence). Later in his book Love and Math, Frenkel recounts the story of how he was rescued by two mathematicians, who mentored him privately and helped him to begin original research in mathematics.
On the strength of his early publications, which became well known in the field, he was invited to Harvard University as a Visiting Professor at the age of twenty-one, and was appointed full Professor at the University of California at Berkeley at twenty-eight.
In his book, Frenkel also describes his encounter with the president of MGU, Anatoly Logunov, the man ultimately responsible for MGU’s policy of discrimination against Jewish students. When Frenkel recounted his story to Logunov at a meeting at MIT, Logunov replied that he was outraged by the news and that he would see to it that such a case was not repeated. In expressing his feigned outrage, Logunov was able to gloss over the fact that countless other students were wrongly failed exactly as Frenkel had been. If Frenkel’s confrontation with Logunov was a personal victory, it was a painful one, leaving, as it did, the larger question of the widespread discrimination at MGU unaddressed.
1 This article is excerpted from Edward Frenkel’s forthcoming book, Love and Math.