You Failed Your

Math Test, Comrade

Einstein: Adventures

and Misadventures

of Young

Mathematicians,

or Test Your Skills

in Almost

Recreational

Mathematics

by M. Shifman

WORLD SCIENTIFIC, 2005, 232 PP., $125.00, $53.00, ISBN: 978-9-81-

256358-3, 978-9-81-256279-1

REVIEWED BY SERGEI TABACHNIKOV

his book is devoted to an episode in the history of

Soviet mathematics, namely the discrimination

against Jewish students at the entrance examinations

for admission to elite Soviet universities, in particular the

ﬂagship university of the country, Moscow State University

(MGU). The time span is about two decades, from the late

1960s to the late 1980s. Since the reader of this review

might be unfamiliar with this story, I shall describe the

historical context ﬁrst.

Before I begin, let me mention that I have ﬁrst-hand

familiarity with the events and circumstances described in

this book: I grew up in Moscow, and in 1973, I was on the

receiving end of the admissions policies at the Department

of Mathematics of MGU (Mekh-Mat) when I took the

entrance exams.

I have known many talented young people who had

similar experiences; some of them ‘‘made it’’ in spite of all

the obstacles, including one who has been honored with

the Wolf Prize in mathematics and election to the United

States National Academy of Sciences, but many abandoned

their dream of becoming a mathematician and had to

choose other career paths. The damage done to our pro-

fession and to individual lives was substantial.

Here is a very brief historical account. In the Russian

Empire, there were severe restrictions on admission to

universities for its Jewish citizens. In the late nineteenth

century, Jews were allowed to occupy at most 10% of the

university places within the pale of settlement (the western

part of the country where permanent residency was

allowed for Jews), while the numbers were 3% in the

capital cities and 5% in the rest of the empire. The deﬁni-

tion of Jewishness was purely religious then.

The revolution of 1917 repealed those restrictions, but

they were soon replaced by another kind of discrimination,

this time based on social background: admission of the

descendants of the ‘‘exploiting classes’’ (nobility and

landowners, business owners and entrepreneurs big and

small, merchants, clergy, and intelligentsia) and their family

members was severely restricted.

The next change came in the mid 1940s. Ironically, after

the victory over Nazi Germany, a rabid anti-Semitic cam-

paign began in the Soviet Union (the pejorative euphemism

used for Jews was then ‘‘rootless cosmopolitans’’).

This

campaign abruptly ended with Stalin’s death in 1953.

What followed was a short period of liberalization

known as the thaw. However, in the mid-1960s, the polit-

ical situation in the country began to change again for the

worse (the ofﬁcial name of this stage was ‘‘real socialism’’).

In particular, it brought back anti-Semitic policies that

unlike those of the Russian Empire, were unwritten and did

not ofﬁcially exist. That was the beginning of the period

described in Comrade Einstein. The end of this period

came with Gorbachev’s perestroika and the liberalization of

Soviet society in the late 1980s.

The reader may be wondering how the college admis-

sions system worked in the USSR during this period. To

start with, each Soviet citizen had an internal passport that

listed in its infamous ﬁfth item the holder’s ethnicity. One’s

birth certiﬁcate listed the ethnicity of one’s parents. At the

age of 16, every citizen received an internal passport, and at

that point, one could choose the ethnicity of their father or

their mother. (As a consequence, the children of mixed

marriages typically made the ‘‘safer’’ choice; in particular,

very few half-Jews were listed as Jews in their passports.)

When applying for university admission, students had to

submit their passports and ﬁll out a questionnaire, giving

the full names of their parents. In Russian, the full name

includes the father’s given name (patronymic); therefore,

the given names of both parents and both grandfathers

would be revealed. If any of those names, and of course the

student’s own name as well, ‘‘sounded Jewish,’’ that would

be taken into account.

The entrance exams were speciﬁc to the student’s pro-

posed major. For example, students who applied to Mekh-

Mat had to take four examinations: written mathematics,

oral mathematics, oral physics, and a written essay. In

comparison, a student applying to the History Department

of the same university had to take an oral Russian language

and Russian literature exam, an oral history exam, an oral

foreign language exam, and a written essay.

The difﬁculty of the entrance exams differed substan-

tially from university to university, and the bar was much

higher at the elite schools. The problems from past written

exams at most universities were published and were

available to the students preparing for the entrance exams.

For more on rootless cosmopolitans, see the Wikipedia article at https://en.wikipedia.org/wiki/Rootless_cosmopolitan.

Ó 2020 Springer Science+Business Media, LLC, part of Springer Nature

https://doi.org/10.1007/s00283-020-09976-y

Let me describe the inner workings of the entrance

exams to Mekh-Mat of MGU at that time.

The written mathematics exam comprised ﬁve prob-

lems. The ﬁrst three were easy, the fourth much harder, and

the ﬁfth very hard. This was done on purpose. Examina-

tions in Russia are graded on a scale of 5 at the top down to

a failing grade of 2, and very few students received even a 4

on the entrance exam, while many received a 3. As a result,

many qualiﬁed students ended up with a barely passing

total grade, which made is easier to use nonacademic cri-

teria in the admissions process. The written exam papers

were coded, so that in principle, the graders did not know

the names of the students.

Most of the elimination of the ‘‘undesirables’’ took place

during the oral exams, both in mathematics and in physics.

Those students were examined in special rooms by spe-

cially selected and specially instructed examiners. Along

with a theoretical question, a student would be given a

number of problems to solve, in a limited amount of time

(about 20 minutes per problem).

The students who were not subject to such special

treatment were given reasonably easy problems designed

to check their basic knowledge of school mathematics, but

in the special rooms (known as gas chambers), the prob-

lems given were very hard, speciﬁcally designed to fail the

students.

Some of these problems were at the level of national and

international mathematical olympiads, others were

exceedingly heavy on calculations, and still others were

puzzles with ingenious solutions that were extremely dif-

ﬁcult to ﬁnd given the short time and the hostile

atmosphere.

As a result, a student selected for special treatment

would end up with a failing grade of 2 or at best a weak 3, a

grade low enough to justify nonadmission. In addition, an

undesirable student could easily receive a 2 or 3 on the

essay, for example because ‘‘the topic was not sufﬁciently

elaborated.’’ (How can one argue with such a verdict?)

This system, crystalized in the early 1970s, created a

nearly impenetrable barrier for Jewish students and several

other categories of applicants, such as the children of

political dissidents. Later, the category of undesirables

expanded to include graduates of the best specialized high

schools for mathematics and physics, since those schools

were seen as incubators of independent thinking and

because they educated a signiﬁcant percentage of ethni-

cally undesirable students.

There were some exceptions, however. For example,

the members of the Soviet team at the International Math-

ematics Olympiad (IMO) were admitted to the universities

of their choice without entrance examinations. Some other

exceptions were made for the children of the Mekh-Mat

faculty. And the system itself, dealing with large numbers

(the incoming class at Mekh-Mat numbered around 500),

was not error-proof, and occasionally a very small number

of Jewish students managed to slip in.

Let me add that some other universities, institutes, and

colleges were unbiased in their admissions policies. For

example, in Moscow, talented students of mathematics

who had been rejected by Moscow State and other elite

universities often entered departments of applied mathe-

matics at the Institute of Oil and Gas (the legendary

‘‘Kerosinka’’ [6]), the Institute of Railroad Engineering, the

Department of Mathematics of the Pedagogical Institute

(which was my case), or one of several other institutions.

In addition to the book under review, readers interested

in learning more can consult the book [2], describing the

experience of Edward Frenkel, who was a student at

‘‘Kerosinka’’ and is now a prominent mathematician at the

University of California at Berkeley; the articles [5] and [1]

provide a selection of ‘‘killer problems’’ from the oral math

exams (in student parlance, such problems were called

‘‘cofﬁns’’).

After this long detour, without which it would have been

impossible for the reader to understand and appreciate

Comrade Einstein, let me proceed to the contents of the

book. It consists of a foreword by the editor and three

parts.

The ﬁrst part is by Ilan Vardi and comprises three essays.

The ﬁrst of them provides solutions to and analysis of 25

‘‘killer problems’’ published by A. Shen in this magazine [7]

with the names of the examiners. This essay was written

when Vardi visited the Institut des Hautes E

tudes Scienti-

ﬁques (IHES), and it was originally an IHES preprint. Vardi

acknowledges the help of a number of prominent

mathematicians.

To give the reader an idea of the level of difﬁculty of

these problems, let me describe just one, which is num-

ber 1 in Vardi’s list. This is the famous butterﬂy problem,

see Figure 1, which has a sentimental value for me, for it

was one of the ﬁve problems that I was given at my oral

exam (I failed to solve it). It is a shame that this piece of

beautiful mathematics was weaponized to achieve such

dastardly goals.

This problem has a venerable history: it was posed by

William Wallace in The Gentlemen’s Mathematical Com-

panion in 1803. The proper context for the problem is

projective and hyperbolic geometry, see, e.g., [3], but Vardi

also presents algebraic and elementary geometric solutions.

If you found this problem too easy, try another one (the

second one on the list): A quadrangle in space is tangent to

a sphere. Show that the points of tangency are coplanar.

Figure 1. The butterﬂy theorem: the chords KL and MN pass

through the midpoint C of the chord AB. Let P ¼ AB \

KN ; Q ¼ AB \ ML: Then PC ¼ QC.

THE MATHEMATICAL INTELLIGENCER

(Not more than 20 minutes, please!) And if you don’t like

geometry, you are welcome to try this one: Solve the fol-

lowing functional equation for f(x):

f ðf ðxÞÞ ¼ x

 2

(another ‘‘cofﬁn,’’ not included in Vardi’s list).

The second essay by Vardi provides solutions to the

problems offered at the 41st IMO in 2000. The author

analyzes these problems and compares them to the 25

Mekh-Mat problems. In his third essay, titled ‘‘My Role as an

Outsider,’’ Vardi describes his involvement in this project

and his moral evaluation of the events and the characters

involved. I leave to the reader to decide whether s/he

agrees with Vardi’s judgments (I do not).

The second part of the book comprises four articles:

‘‘Intellectual Genocide,’’ by B. Kanevsky and V. Senderov

(1980, Samizdat), ‘‘Science and Totalitarianism,’’ by A.

Vershik (a translation from the Russian original published

in 1998), and reprints of Vershik’s [9] and Shen’s articles [7]

published in this magazine.

‘‘Intellectual Genocide’’ provides a detailed analysis of

the 1980 entrance exams at three elite Moscow universities:

MGU, the Moscow Institute for Physics and Technology

(MFTI), and the Moscow Institute for Engineering and

Physics (MIFI). It proves, beyond a reasonable doubt, that

anti-Jewish discrimination indeed took place.

A similar analysis by Kanevsky and Senderov in 1979

yields the following picture (published in [7]). They con-

sidered the graduates of six Moscow specialized schools for

physics and mathematics and divided them into two

groups: those whose parents and grandparents were not

Jewish, and those who had at least one Jewish grandparent

(one cannot help thinking of the 1935 racist Nuremberg

Laws). This classiﬁcation is highly correlated with the rates

of admission.

The results of the students’ applications to Mekh-Mat

MGU are presented in the following table.

First group Second group

Total number of graduates 47 40

Olympiad winners

14 26

Multiple winners

411

Total olympiad prizes

26 48

Admitted

40 6

Similar data are also given for MFTI and MIFI. The

olympiads mentioned in this table do not include the IMO

(which guaranteed admission to team members without

their having to take the entrance exams).

These statistics were collected by Victor Polterovich, a

renowned mathematical economist whose son, Leonid

(today a distinguished mathematician), took the entrance

exams in 1979.

Let me comment on the number 6 at the bottom right.

This number would probably be even smaller if not for the

fact that some students and their parents already knew how

the admissions system functioned and so worked extra

hard to be prepared for whatever was thrown at them.

Valery Senderov played a prominent role in collecting

information about these biased exams and in preparing

students for them. In particular, he ran a seminar at the

famous Moscow Specialized High School for Mathematics

No. 2, teaching how to solve ‘‘killer problems.’’ At that time,

I was a math teacher at this school, and I assisted Valery

with his seminar. Two Jewish students from this class

managed to break through the nearly impenetrable wall,

and after vigorous appeals were admitted to Mekh-Mat

MGU. Today, both are prominent mathematicians.

Senderov was arrested in 1982 for ‘‘anti-Soviet activities’’

and sentenced to seven years of hard labor and a subse-

quent exile of ﬁve years. In 1987, he was released, along

with other political prisoners; he died in 2014. Kanevsky

was also arrested; he spent fourteen months in prison fol-

lowed by two years in exile. He has lived in Israel since

1987.

The third part of the book comprises four articles

devoted to the memory of Bella Subbotovskaya and the

‘‘Jewish People’s University’’ (JPU for short) that she cre-

ated. The cost of that act of deﬁance was Bella’s life. The

articles were authored by Katherine Tylevich, Dmitry

Fuchs, Andrei Zelevinsky (the last two mathematicians

taught at the underground university), and Ilya Muchnik,

also a mathematician and Bella’s ex-husband.

In 1978, Subbotovskaya began offering unofﬁcial classes

for students who were seriously interested in learning

mathematics but who were deprived of this opportunity by

the biased admissions system. At the beginning, there were

only 14 students, and the lectures took place in her small

apartment. Later, when a rumor had spread and the class

had grown, the lectures took place at various institutions of

higher education, mostly unofﬁcially and under various

pretexts.

Those meetings took place once a week, on Saturdays.

On weekdays, the students attended classes at their tech-

nical colleges. The lecturers were professional

mathematicians, some quite young and some famous.

Needless to say, no one received any money for their work.

Bella was not an instructor, but the main organizer and

administrator (if this had been a real and not a ‘‘through the

looking glass’’ university, she would have been its founding

president).

The idea was to provide the students with a fundamental

mathematical education comparable to that received by the

students of Mekh-Mat MGU, where traditionally, rigorous

mathematics was taught from day one. The courses inclu-

ded algebra, analysis, linear algebra with analytic geometry,

and differential geometry, as well as more advanced topics,

such as Lie algebras and D-modules (taught by Boris Fei-

gin) and quantum mechanics and ﬁeld theory (taught by

Michael Marinov). There were also guest speakers, for

example John Milnor, who gave a lecture during his visit to

Moscow in 1982.

Overall, this underground university boasts about 350

alumni. Fuchs and Zelevinsky estimated that their incoming

class of 1980 had about 70 students, and about 10% have

become accomplished mathematicians; this is, I believe, a

higher percentage than that of the Mekh-Mat class of the

same year.

Ó 2020 Springer Science+Business Media, LLC, part of Springer Nature

The KGB was probably monitoring JPU from the very

beginning. Clearly, this activity contradicted state policies

and was viewed as dissident and anti-Soviet. In the summer

of 1982, the authorities decided to pull the plug on it. As I

mentioned above, Senderov and Kanevsky were arrested,

along with one student (he was soon released).

Subbotovskaya was summoned to the ofﬁces of the KGB

and interrogated. She made a futile attempt to convince the

case ofﬁcer that the underground university was a worthy

undertaking, in alignment with the ‘‘party line.’’ She was

summoned again and called to be a witness against Sen-

derov; she refused.

The tragic end followed soon after: Bella was killed in a

hit-and-run accident under very suspicious circumstances.

She was 44 years old. There are serious indications that this

was a political assassination. Bella’s death was the end of

the JPU; sadly, the authorities had achieved their goal.

After the publication of the book under review, a group

of people familiar with the circumstances and involved in

these events decided to organize a conference in Bella

Subbotovskaya’s memory. In 2007, the conference ‘‘Dif-

ferent Approaches to Complexity’’ took place at the Israel

Institute of Technology; the choice of the topic reﬂected

Bella’s mathematical interests.

One day was devoted to memories of Bella Sub-

botovskaya and the JPU. The talks given on that day are

available on YouTube,

including a talk by Mikhail Shif-

man, the editor of this book. Subsequently, an article about

Bella appeared in the the Notices of the American Mathe-

matical Society [8].

Shifman maintains a website where he collects com-

ments from his book’s readers and other relevant materials,

both in English and in Russian.

The site is called ‘‘Through

the Prism of Time – Angles of Reﬂection.’’ It is essentially

another book, totaling about 160 pages.

Diverse opinions about his book and the events that

it describes are presented there. Some think that we

should forget the whole story because it happened a

long time ago a nd is no longer relevant; others argue

that the relevant documents and afﬁdavits should be

collected and published in book form or on a dedicated

website.

As Mikhail Shifman says in the preface, he is not a his-

torian of mathematics (he is a prominent physicist), and his

book is just a ﬁrst step in the right direction. I agree with

him that the events described in this book, along with other

episodes in the history of mathematics in the Soviet Union,

deserve a professional study akin to Karabel’s study of

admission to the ‘‘big three’’ American universities in the

twentieth century [4], which describes the Jewish quotas at

elite colleges (the subtitle of the book is The Hidden History

of Admission and Exclusion at Harvard, Yale, and

Princeton).

To this day, it is not known to what extent anti-Jewish

discrimination at some universities was the result of deci-

sions made at the very top, as opposed to local initiative. It

is a fact that some very prominent and inﬂuential mathe-

maticians were fervent anti-Semites, including Lev

Pontryagin and Ivan Vinogradov, who was the director of

the Steklov Mathematics Institute for about ﬁfty years.

The corrupt admissions process at Mekh-Mat matched

other policies of the department’s administration: hiring,

promotion, retention, and its systematic work toward

replacing independent freethinking employees by ‘‘tame,’’

obedient ones. These ‘‘dark two decades’’ in the history of

the department had dire consequences: one of the very top

mathematical centers in the world in the 1960s (see [10]), it

has become a pale shadow of its former glory.

It is remarkable that the biased admissions policies were

never ofﬁcially acknowledged, and as far as I know, not a

single perpetrator has admitted his guilt or expressed

regret. Some of those people are still at the helm of MGU.

I conclude with a couple of excerpts from the interview

that V. Senderov gave to Radio Liberty in 2005 (in

Russian).

Senderov was asked about the results of those admis-

sions policies. He mentioned three. First, the brain drain,

for which these policies were one of the main reasons.

Second, the personal trauma of the victims, especially

when the students came from provincial towns and were

totally unprepared for the ‘‘special treatment.’’ They exited

the entrance exams convinced that they were indeed good

for nothing and abandoned their dreams of becoming

mathematicians.

The third and most terrible result was universal

demoralization. Evil deeds were done openly, and people

had to witness them and keep silent. This ‘‘everybody

knows’’ situation led to a horrible distrust in society that

continues to this day.

Senderov also addressed the matter of quotas in

admission.

In his opinion (which I share), small quotas are not nec-

essarily bad, such as a few reserved positions for incoming

students who had served in the army. But it is crucial that

such policies be open, transparent, and small compared to the

total number of students involved. No matter whether the

intentions are good or ill, if an admissions policy is cloaked in

secrecy and lies, it is necessarily odious.

Sergei Tabachnikov

Department of Mathematics

Pennsylvania State University

University Park, PA 16802

USA

e-mail: [email protected]

Available online at https://www.youtube.com/watch?v=Xyak7yINMsI, https://www.youtube.com/watch?v=8CEN6nIN4Dk, and https://www.youtube.com/watch?v=

NDmzWMM_8wQ.

See http://homepages.spa.umn.edu/*shifman/Epilogue_12_28_2016.pdf.

Available online at https://www.svoboda.org/a/126672.html.

THE MATHEMATICAL INTELLIGENCER

REFERENCES

[1] J. Egenhoff. Math as a tool of anti-semitism. Math. Enthusiast 11

(2014), 649–664.

[2] E. Frenkel. Love and Math: The Heart of Hidden Reality. Basic

Books, 2013.

[3] I. Izmestiev. A porism for cyclic quadrilaterals, butterﬂy theorems,

and hyperbolic geometry. Amer. Math. Monthly 122 (2015), 467–

475.

[4] J. Karabel. The Chosen: The Hidden History of Admission and

Exclusion at Harvard, Yale, and Princeton. Houghton Mifﬂin Co.,

2005.

[5] T. Khovanova and A. Radul. Killer problems. Amer. Math. Monthly

119 (2012), 815–823.

[6] M. Saul. Kerosinka: an episode in the history of Soviet mathe-

matics. Notices AMS 46 (1999), 1217–1220.

[7] A. Shen. Entrance examination to the Mekh-Mat. Math. Intelli-

gencer 16:4 (1994), 6–10.

[8] G. Szpiro. Bella Abramovna Subbotovskaya and the ‘‘Jewish

People’s University.’’ Notices AMS 54 (2007), 1326–1330.

[9] A. Vershik. Admission to the mathematics departments in Russia

in the 1970s and 1980s. Math. Intelligencer 16:4 (1994), 4–5.

[10] S. Zdravkovska and P. Duren, editors. Golden Years of Moscow

Mathematics, second edition. American Mathematical Society

and London Mathematical Society, 2007.

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