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Fibonacci times Euler

November 5, 2016 2 comments

Recall the Fibonacci numbers F_n given by 1,1,2,3,5,8,13,21… There is no need to define them. You all know. Now take the Euler numbers (OEIS) E_n 1,1,1,2,5,16,61,272… This is the number of alternating permutations in S_n with the exponential generating function \sum_{n=0}^\infty E_n t^n/n! = \tan(t)+\sec(t).  Both sequences are incredibly famous. Less known are connection between them.

(1) Define the Fibonacci polytope \Phi_n to be a convex hull of  0/1 points in \Bbb R^n with no two 1 in a row. Then  \Phi_n has F_{n+1} vertices and vol(\Phi_n)=E_n/n! This is a nice exercise.

(2) F_n \cdot E_n \ge n! (by just a little). For example, F_4 \cdot E_4 = 5 \cdot 5 = 25 > 4!. This follows from the fact that

F_n \sim \frac{1}{\sqrt{5}} \, \phi^{n+1} and E_n\sim \frac{4}{\pi}\left(\frac{2}{\pi}\right)^{n} n!, where \phi=(1+\sqrt{5})/2 is the golden ratio. Thus, the product F_n \cdot E_n \sim c n! \left(\frac{2\phi}{\pi}\right)^n. Since \pi = 3.14 and 2\phi = 3.24, the inequality F_n \cdot E_n \ge n! is easy to see, but still a bit surprising that the numbers are so close.

Together with Greta Panova and Alejandro Morales we wrote a little note “Why is π < 2φ?” which gives a combinatorial proof of (2) via a direct surjection. Thus we obtain an indirect proof of the inequality in the title.  The note is not a research article; rather, it is aimed at a general audience of college students.  We will not be posting it on the arXiv, so I figure this blog is a good place to advertise it.

The note also explains that the inequality (2) also follows from Sidorenko’s theorem on complementary posets. Let me briefly mention a connection between (1) and (2) which is not mentioned in the note.  I will assume you just spent 5 min and read the note at this point.  Following Stanley, the volume of \Phi_n is equal to the volume of the chain polytope (=stable set polytope), see Two Poset Polytopes.  But the latter is exactly the polytope that Bollobás, Brightwell and Sidorenko used in their proof of the upper bound via polar duality.

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Academia is nothing like a drug cartel

November 30, 2013 6 comments

It’s been awhile since I wanted to rant. Since the last post, really. Well, I was busy. But the time has come to write several posts.

This post is about a number of recent articles lamenting the prevalence of low paid adjuncts in many universities. To sensationalize the matter, comparisons were made with drug cartels and Ponzi schemes. Allegedly, this inequality is causing poverty and even homelessness and death. I imagine reading these articles can be depressing, but it’s all a sham. Knowingly or not, the journalists are perpetuating false stereotypes of what professors really do. These journalists seem to be doing their usual lazy work and preying on reader’s compassion and profound misunderstanding of the matter.

Now, if you are reading this blog, I am assuming you know exactly what professors do (Hint: not just teaching). But if you don’t, start with this outline by my old friend Daniel Liberzon, and proceed to review any or all of these links: one, two, three, four. When you are done, we can begin to answer our main semi-serious question:

What is academia, really, if it’s not a drug cartel or a Ponzi scheme?

I can’t believe this trivial question is difficult to some people, and needs a lengthy answer, but here it is anyway.


Academia rewards industriousness and creativity

This might seem obvious – of course it does!  These are the main qualities needed to achieve success doing research. But reading the above news reports it might seem that Ph.D. is like a lottery ticket – the winnings are rare and random. What I am trying to say is that academia can be compared with other professions which involve both qualities. To make a point, take sculpture.

There are thousands of professional sculptors in the United States. The figures vary greatly, but same also holds for the number of mathematicians, so we leave it aside. The average salary of sculptors seems to be within reach from average salary in the US, definitely below that of an average person with bachelor degree. On the other hand, top sculptors are all multimillionaires. For example, recently a sculpture by Jeff Koons sold for $58.4 million. But at least it looked nice. I will never understand the success of Richard Serra, whose work is just dreadful. You can see some of his work at UCLA (picture), or at LACMA (picture).  Or take a celebrated and much despised 10 million dollar man Dale Chihuly, who shows what he calls “art” just about everywhere…  But reasonable people can disagree on this, and who am I to judge anyway?  My opinion does not matter, nor is that of almost anyone.  What’s important, is that some people with expertise value these creative works enough to pay a lot of money for them.  These sculptors’ talent is clearly recognized.

Now, should we believe on the basis of the salary disparity that the sculpture is a Ponzi scheme, with top earners basically robbing all the other sculptors of good living?  That would be preposterous, of course.  Same with most professors.  Just because the general public cannot understand and evaluate their research work and creativity, does not mean it’s not there and should not be valued accordingly.


Academia is a large apprenticeship program

Think of graduate students who are traditionally overworked and underpaid. Some make it to graduate with a Ph.D. and eventually become tenured professors. Many, perhaps most, do not. Sounds like a drug cartel to you? Nonsense! This is exactly how apprenticeships works, and how it’s been working for centuries in every guild.  In fact, some modern day guilds don’t pay anything at all.

Students enter the apprenticeship/graduate program in hopes to learn from the teacher/professor and succeed in their studies. The very best do succeed. For example, this list of Rembrant‘s pupils/assistants reads somewhat similar to this list of Hilbert‘s students. Unsurprisingly, some are world famous, while others are completely forgotten. So it’s not about cheap labor as in drug cartels – this is how apprenticeships normally work.


Academia is a big business

Think of any large corporation.  The are many levels of management: low, mid, and top-level.  Arguably, all tenured and tenure-track faculty are low level managers, chairs and other department officers (DGS, DUS, etc.) are mid-level, while deans, provosts and presidents/chancellors are top-level managers.  In the US, there is also a legal precedent supporting qualifying professors as management (e.g. professors are not allowed to unionize, in contrast with the adjunct faculty).  And deservingly so.  Professors hire TA’s, graders, adjuncts, support stuff, choose curriculum, responsible for all grades, run research labs, serve as PI’s on federal grants, and elect mid-level management.

So, why many levels?  Take UCLA.  According to 2012 annual report, we operate on 419 acres, have about 40 thousand students, 30 thousand full time employees (this includes UCLA hospitals), have $4.6 billion in operating revenue (of which tuition is only $580 million), but only about 2 thousand ladder (tenure and tenure-track) faculty.  For comparison, a beloved but highly secretive Trader Joe’s company has about $8 billion in revenue, over 20 thousand employees, and about 370 stores, each with 50+ employees and its own mid and low-level management.

Now that you are conditioned to think of universities as businesses and professors as managers, is it really all that surprising that regular employees like adjuncts get paid much less?  This works the same way as for McDonalds store managers, who get paid about 3 times as much as regular employees.


Higher echelons of academia is a research factory with a side teaching business

Note that there is a reason students want to study at research universities rather than at community colleges.  It’s because these universities offer many other more advanced classes, research oriented labs, seminars, field works, etc.  In fact, research and research oriented teaching is really the main business rather than service teaching.

Think revenue.  For example, UCLA derives 50% more revenue from research grants than from tuition.  Places like MIT are giving out so many scholarships, they are loosing money on teaching (see this breakdown).  American universities cannot quit the undergraduate education, of course, but they are making a rational decision to outsource the low level service teaching to outsiders, who can do the same work but cheaper.  For example, I took English in Moscow, ESL at a community college in Brooklyn, French at Harvard, and Hebrew at University of Minnesota.  While some instructors were better than others, there was no clear winner as experience was about the same.

So not only the adjunct salaries are low for a reason, keeping them low is critical to avoid hiring more regular faculty and preventing further tuition inflation.  The next time you read about adjuncts barely making a living wage, compare this to notorious Bangalore call centers and how much people make over there (between $100 and $250 a month).


Academia is a paradise of equality

College professors are different from drug gangsters not only in the level of violence, but also in a remarkable degree of equality between universities (but not between fields!)  Consider for example this table of average full professor salaries at the top universities.  The near $200,000 a year may seem high, but note that this is only twice that of average faculty at an average college.  Given that most of these top universities are located in the uber-expensive metropolitan areas (NYC, Boston, San Francisco, Los Angeles, etc.), the effect is even further diminished. 

Compare this with other professions.  Forget the sculptors mentioned above whose pay ratios can go into thousands, let’s take a relatively obscure profession of an opera singer (check how many do you know from this list).  Like academia and unlike sculpture, the operas are greatly subsidized by the governments and large corporations.  Still, perhaps unsurprisingly, there is a much greater inequality than in academia.  While some popular singers like Dmitri Hvorostovsky make over $3 million a year, the average salary is about $100,000 a year, giving a ratio of 30+.

In other words, given that some professors are much better than others when it comes to research (not me!), one can argue that they are being underpaid to subsidize the lackluster efforts of others.  No wonder the top academics suffer from the status-income disequilibrium.  This is the opposite of the “winner takes all” behavior argued by the journalists in an effort to explain adjuncts’ plight.


Academia is an experience

People come to universities to spend years studying, and they want to enjoy those years.  They want to hear famous authors and thinkers, learn basic skills and life changing stories, make lasting friendships, play sports and simply have fun.  Sometimes this does not work out, but we are good at what we do (colleges have been perfecting their craft for hundreds of years).  Indeed, many students take away with them a unique deeply personal experience.  Take my story.  While at Moscow University, I heard lectures by Vladimir Arnold, attended Gelfand’s Seminar, and even went to a public lecture by President Roh Tae-woo.  It was fun.  While at Harvard, I took courses of Raoul Bott and Gian-Carlo Rota (at MIT), audited courses of such non-math luminaries as Stephan Thernstrom and William Mills Todd, III, and went to public lectures by people like Tim Berners-Lee, all unforgettable.

So this is my big NO to those who want to replace tenured faculty with adjuncts, leveling the academic salaries, and commodifying the education.  This just would not work; it is akin to calls for abolition of haute cuisine in favor of more fast food.  In fact, nobody really wants to do either of these.  The inexpensive education is already readily available in the form of community colleges.  In fact, students apply in large numbers trying to get to a place like UCLA, which offers a wide range of programs and courses.  And it’s definitely not because of our celebrity adjuncts.


In conclusion

Academia is many things to many people.  There are many important reasons why the ladder faculty are paid substantially better than TA’s and adjuncts, reasons both substantive and economical.  But at no point does the academia resemble the Ponzi schemes and drug cartels, which are famous for creating the economic devastation and inequality (and, um, illegal).  If anything, the academia is the opposite, as it creates economic opportunities and evens the playing field.   And to those educational reformers who think they know better: remember, we heard it all before

College admissions I. Discrimination and lies, Jews and Harvard

December 26, 2012 1 comment

Recent reports on alleged discrimination of Asian Americans at Ivy League schools (read a discussion here and view this graph), brought a lot of disgust in me, as well as some ambivalence. Here and in the next post I will try to deconstruct these feelings.

In this post I mention my family and my own history of dealing with discrimination.  I then briefly review and make parallels with the current discussion of the issues, and make some recommendations.  In the next post, I will explain why the whole issue is overhyped and what does that say about american culture.

Russian Jews go to school

Well, this is a really long story, but when it comes to educational opportunities, things were always pretty bad.  By 1880’s most universities and gymnasiums in Imperial Russia instituted a 5 to 10% Jewish quota, which remained in effect until the Russian revolution in 1917.  Read more on the history in this book (part III), and in amazing personal memoir (in Russian).

Communists abolished Jewish discrimination replacing it with anti-religious discrimination, often having similar effect.  In the 1930s, my grandmother was expelled from college after communist officials discovered that her father (my great-grandfather) was a rabbi.  A local newspaper went all schadenfreude about her, and published an anti-clerical article “The wolf in sheep’s clothing”, apparently missing the irony of the origin of the title.

By the early 1960s, Israel became a super-enemy of USSR, and things were slowly getting hotter for the Jews.  For example, despite high exam grades, my father and few dozen Jews was denied admission to Moscow University (МГУ) on account of “lack of dorm space”.  Some scandal ensued and he was accepted a month later.   By the late 1960s, after the Six-Day War, the Mathematics Department of Мoscow University settled on 0.5% quota (about 2 Jewish students in a class of 450-500), which typically went to children of the university faculty and occasional party officials.  When I applied in 1988, I was rejected as the quota remained in effect.  In 1989, things were starting to change, and the quota was raised to about 4%.  I got in.  In the meantime, I became somewhat of an expert on “Jewish problems” (see also here and there), once even holding a seminar on them.

Curiously, the officials had supported the quota very openly, justifying it as follows:

1. We need to maintain proportion of Jews the same as in the country, so as they don’t take space from ethnically Russian students.
2. Jews are already privileged by the virtue of living in large cities, but Russians from small villages need extra help to get quality education.  Of course, Jews in Ukrainian, Lithuanian and Belorussian villages were mostly killed in the WWII as part of the Final Solution.
3. Future Russia needs an educated workforce. There is no point of preparing “cadres for Israel“.  Thus the “diploma tax“.

My little brush with discrimination in the US

In 1994, already a first year grad student at Harvard, I applied for NSF Graduate Fellowhip, which was highly selective but much less generous back then.  I mailed my proposed plan of research, letters of recommendation, transcripts, and the required GRE, both General and Subject.  I was rejected.  Since I received a more selective Hertz Foundation Fellowship (see my discussion of it here), I wasn’t too upset, but I was curious what did I do wrong.  So I filed a FOIA request, and got a reply a few weeks later.

What I learned was remarkable and made me really upset.  I discovered that the NSF reviewers rated A all my materials, both the transcript, all the letters, and plan of research.  I had a maximal GRE Subject score.  But you see, me being Russian and all, I had a mediocre to poor GRE General score on the Verbal Section.  The paperwork indicated that the committee then took weighted average of all these grades, made a list of top scorers and I didn’t make the cut.  Since I could not fathom why would I need a top GRE Verbal score for Math Ph.D., this seemed clearly discriminatory, on the basis of my native language.

So I found a lawyer (tiny Cambridge, MA is full of them).  He patiently explained to me that my Russian native language is not defining me as a member of protected class, and I have no case against NSF. He said that even politically, there is no such thing as “Russian language lobby” (despite our large numbers), and given that there was no harm done (my Hertz), I should go home and learn to be happy.  Naturally, I did.

Jews at Harvard and the geographic distribution

The story of Jews at Harvard has been described in great details at a variety of sources.  In short, Harvard instituted a 15% quota, which was later softened, substituted with geographic distribution preferences, having same effect on Jewish enrollment.   The following quote about the evolution of Harvard President James Conant (1933-1953) is revealing:

Conant’s pro-quote position in the early 1920s, his preference for more students from small towns and cities and the South and West, and his cool response to the plight of the Jewish academic refugees from Hitler suggest that he shared the mild antisemitism common to his social group and time. But his commitment to meritocracy made him more ready to accept able Jews as students and faculty.

While the quotas are both illegal and a thing of the past, the use of geographic distribution in admissions never went away.  While not discriminatory in the strict legal sense, they were created with a discriminatory intent, and still have discriminatory effect, as recent immigrants, Jews and other minorities tend to concentrate in large population centers.  Not unlike the Russian “village” arguments, this is a slight of hand, which first creates and then heavy-handedly destroys a straw man, all in an effort to deal with other issues which are kept out of sight.  We will see this in other cases as well.

All students are somebody’s children

Legacy preferences is another example of misleading practices potentially having discriminatory effect.  Universities are claiming that this creates a brand loyalty.  But that is misleading of course.  Do Ivy League schools really need to develop brand loyalty when they have 10-20 applicants per spot?  The truth, of course, is that children of alumni have money and willing to pay a full sticker price of the tuition, and the admission officers aim to have about 20% of such legacy students in each freshmen class.

In fact, the honest market based solution would be to auction this portion of the freshman class to the highest bidder, charging tuition perhaps as much as 100K per year.  This auction would raise significant funds which can pay for poor students’ scholarships and stipends, and open up these admission slots to everyone, not just children of alumni.  As it is, legacy candidates get preferences in admission and, perhaps counter intuitively, have their tuition subsidized as others may potentially be willing to pay more for their spots.  Now, I am NOT advocating for this, just showing how misguided and fundamentally unfair are the current admission policies.

Texas 10% solution

This rule was enacted in response to state losing in Hopwood v. Texas, as a novel legal way to introduce diversity in admissions.  An ultimate geographic preference, this rule fills about 75-80% of  the freshmen class at the leading Texas universities.  Note that the Fisher case is about the affirmative action for the remaining spots.

But it is exactly the kind of rule which makes wrong priorities for the students and the society.  In general, it is beneficial for the society when students have a choice which K12 school to attend.  It is undoubtedly good when they study in the most challenging environment, work hardest on the most advanced courses available.  This rule pushes students to take the easiest courses in the least challenging school, aiming to attain the highest GPA and enter the coveted 10%.  And guess what – Texas students do exactly that (this in addition to other rule troubles).

A case for honesty

As it stands, the universities are on the brink of losing another affirmative action case in the Supreme Court.  Perhaps this is not immediately apparent, but they are also on the brink of a giant PR disaster when it comes to their hidden quotas for Asian Americans.  With good intentions, the admission officers and politician keep coming up with twisted, misleading, uncomfortable and occasionally self-contradictory rationale as to why they do what they do (see above).   The problem is – with all the history, we’ve seen this all before, and nobody is buying it.  With so much public pressure, they probably have to stop and own up to their choices.

I think it is clear what many top colleges are doing.  They have a goal of a freshmen class which would have f(x)% students with property x, for many different x, which can be race, gender, wealth, political connections, geographic location, sexual orientation, sports, music, science and other achievements, etc.  So they produce all these policies like the early action, and many rationalizations aimed at reaching that goal.  One should have a lot of chutzpah to believe to know exactly the “right mix” function f, but of course they think that…

If it was up to me, I would give the universities a complete freedom to accept whoever they want without fear of lawsuits, in exchange for complete transparency.  Education is really not like housing or employment, it is fluid and highly competitive.  In exchange, make the universities publish the exact numbers of how many students with every property x have applied and got accepted.  For the sake of anonymity, delete all the names and zip codes, and publish on the web the rest of the data from their applications.  Let the future applicants, or nonprofits on their behalf, decide their chances of acceptance and make rational choice whether to spend their $75 and endless hours applying to that school.  Unfortunately, we don’t live in an ideal world, but you have to let colleges compete with each other, which is the most fair and offers the best model of education.

Finally, when it comes to Asian Americans – Harvard and the rest of the Ivies should just apologize, and starting next year accept twice as many as this year, to compensate for the real or perceived discrimination.  Otherwise, a hundred years from now, somebody might still be writing how stupid and morally twisted were these old early 21st century admission policies.

Warning:  Here I neither endorse nor reject the affirmative action, but rather advocate for some honesty, clarity and transparency.