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The problem with combinatorics textbooks

Every now and then I think about writing a graduate textbook in Combinatorics, based on some topics courses I have taught. I scan my extensive lecture notes, think about how much time it would take, and whether there is even a demand for this kind of effort. Five minutes later I would always remember that YOLO, deeply exhale and won’t think about it for a while.

What’s wrong with Combinatorics?

To illustrate the difficulty, let me begin with two quotes which contradict each other in the most illuminating way. First, from the Foreword by Richard Stanley on (his former student) Miklós Bóna’s “A Walk Through Combinatorics” textbook:

The subject of combinatorics is so vast that the author of a textbook faces a difficult decision as to what topics to include. There is no more-or-less canonical corpus as in such other subjects as number theory and complex variable theory. [here]

Second, from the Preface by Kyle Petersen (and Stanley’s academic descendant) in his elegant “Inquiry-Based Enumerative Combinatorics” textbook:

Combinatorics is a very broad subject, so the difficulty in writing about the subject is not what to include, but rather what to exclude. Which hundred problems should we choose? [here]

Now that this is all clear, you can probably insert your own joke about importance of teaching inclusion-exclusion. But I think the issue is a bit deeper than that.

I’ve been thinking about this when updating my “What is Combinatorics” quotation page (see also my old blog post on this). You can see a complete divergence of points of view on how to answer this question. Some make the definition or description to be very broad (sometimes even ridiculously broad), some relatively narrow, some are overly positive, while others are revoltingly negative. And some basically give up and say, in effect “it is what it is”. This may seem puzzling, but if you concentrate on the narrow definitions and ignore the rest, a picture emerges.

Clearly, these people are not talking about the same area. They are talking about sub-areas of Combinatorics that they know well, that they happen to learn or work on, and that they happen to like or dislike. Somebody made a choice what part of Combinatorics to teach them. They made a choice what further parts of Combinatorics to learn. These choices are increasingly country or culture dependent, and became formative in people’s mind. And they project their views of these parts of Combinatorics on the whole field.

So my point is — there is no right answer to “What is Combinatorics?“, in a sense that all these opinions are biased to some degree by personal education and experience. Combinatorics is just too broad of a category to describe. It’s a bit like asking “what is good food?” — the answers would be either broad and bland, or interesting but very culture-specific.

Courses and textbooks

How should one resolve the issue raised above? I think the answer is simple. Stop claiming that Combinatorics, or worse, Discrete Mathematics, is one subject. That’s not true and hasn’t been true for a while. I talked about this in my “Unity of Combinatorics” book review. Combinatorics is comprised of many sub-areas, see the Wikipedia article I discussed here (long ago). Just accept it.

As a consequence, you should never teach a “Combinatorics” course. Never! Especially to graduate students, but to undergraduates as well. Teach courses in any and all of these subjects: Enumerative Combinatorics, Graph Theory, Probabilistic Combinatorics, Discrete Geometry, Algebraic Combinatorics, Arithmetic Combinatorics, etc. Whether introductory or advanced versions of these courses, there is plenty of material for each such course.

Stop using these broad “a little bit about everything” combinatorics textbooks which also tend to be bulky, expensive and shallow. It just doesn’t make sense to teach both the five color theorem and the Catalan numbers (see also here) in the same course. In fact, this is a disservice to both the students and the area. Different students want to know about different aspects of Combinatorics. Thus, if you are teaching the same numbered undergraduate course every semester you can just split it into two or three, and fix different syllabi in advance. The students will sort themselves out and chose courses they are most interested in.

My own teaching

At UCLA, with the help of the Department, we split one Combinatorics course into two titled “Graph Theory” and “Enumerative Combinatorics”. They are broader, in fact, than the titles suggest — see Math 180 and Math 184 here. The former turned out to be quite a bit more popular among many applied math and non-math majors, especially those interested in CS, engineering, data science, etc., but also from social sciences. Math majors tend to know a lot of this material and flock to the latter course. I am not saying you should do the same — this is just an example of what can be done.

I remember going through a long list of undergraduate combinatorics textbooks a few years ago, and found surprisingly little choice for the enumerative/algebraic courses. Of the ones I liked, let me single out Bóna’s “Introduction to Enumerative and Analytic Combinatorics and Stanley’s “Algebraic Combinatorics“. We now use both at UCLA. There are also many good Graph Theory course textbooks of all levels, of course.

Similarly, for graduate courses, make sure you make the subject relatively narrow and clearly defined. Like a topics class, except accessible to beginning graduate students. Low entry barrier is an advantage Combinatorics has over other areas, so use it. To give examples from my own teaching, see unedited notes from my graduate courses:

Combinatorics of posets (Fall 2020)

Combinatorics and Probability on groups (Spring 2020)

Algebraic Combinatorics (Winter 2019)

Discrete and Polyhedral Geometry (Fall 2018) This is based on my book. See also videos of selected topics (in Russian).

Combinatorics of Integer Sequences (Fall 2016)

Combinatorics of Words (Fall 2014)

Tilings (Winter 2013, lecture-by-lecture refs only)

In summary

In my experience, the more specific you make the combinatorics course the more interesting it is to the students. Don’t be afraid that the course would appear be too narrow or too advanced. That’s a stigma from the past. You create a good course and the students will quickly figure it out. They do have their own FB and other chat groups, and spread the news much faster than you could imagine…

Unfortunately, there is often no good textbook to cover what you want. So you might have to work a little harder harder to scout the material from papers, monographs, etc. In the internet era this is easier than ever. In fact, many extensive lecture notes are already available on the web. Eventually, all the appropriate textbooks will be written. As I mentioned before, this is one of the very few silver linings of the pandemic…

P.S. (July 8, 2021) I should have mentioned that in addition to “a little bit about everything” textbooks, there are also “a lot about everything” doorstopper size volumes. I sort of don’t think of them as textbooks at all, more like mixtures of a reference guide, encyclopedia and teacher’s manual. Since even the thought of teaching from such books overwhelms the senses, I don’t expect them to be widely adopted.

Having said that, these voluminous textbooks can be incredibly valuable to both the students and the instructor as a source of interesting supplementary material. Let me single out an excellent recent “Combinatorial Mathematics” by Doug West written in the same clear and concise style as his earlier “Introduction to Graph Theory“. Priced modestly (for 991 pages), I recommend it as “further reading” for all combinatorics courses, even though I strongly disagree with the second sentence of the Preface, per my earlier blog post.

What if math dies?

Over the years I’ve heard a lot about the apparent complete uselessness and inapplicability of modern mathematics, about how I should always look for applications since without them all I am doing is a pointless intellectual pursuit, blah, blah, blah.  I had strangers on the plane telling me this (without prompting), first dates (never to become second dates) wondering if “any formulas changed over the last 100 years, and if not what’s the point“, relatives asking me if I ever “invented a new theorem“, etc.

For whatever reason, everyone always has an opinion about math.  Having never been accused of excessive politeness I would always abruptly change the subject or punt by saying that the point is “money in my Wells Fargo account“.  I don’t even have a Wells Fargo account (and wouldn’t want one), but what’s a small lie when you are telling a big lie, right?

Eventually, you do develop a thicker skin, I suppose.  You learn to excuse your friends as well meaning but uneducated, journalists as maliciously ignorant, and strangers as bitter over some old math learning experience (which they also feel obliged to inform you about).  However, you do expect some understanding and respect from fellow academics. “Never compare fields” Gian-Carlo Rota teaches, and it’s a good advice you expect sensible people to adhere.  Which brings me to this:

The worst idea I’ve heard in a while

In a recent interview with Glenn Loury, a controversial UPenn law professor Amy Wax proposed to reduce current mathematics graduate programs to one tenth or one fifteenth of their current size (start at 54.30, see also partial transcript).  Now, I get it.  He is a proud member of the “intellectual dark web“, while she apparently hates liberal education establishment and wants to rant about it.  And for some reason math got lumped into this discussion.  To be precise, Loury provoked Wax without offering his views, but she was happy to opine in response.  I will not quote the discussion in full, but the following single sentence is revealing and worth addressing:

If we got rid of ninety percent of the math Ph.D. programs, would we really be worse off in any material respect?  I think that’s a serious question.

She followed this up with “I am not advocating of getting rid of a hundred percent of them.”  Uhm, thanks, I guess…

The inanity of it all

One is tempted to close ranks and ridicule this by appealing to authority or common sense.  In fact, just about everyone — from Hilbert to Gowers — commented on the importance of mathematics both as an intellectual endeavor and the source of applications.  In the US, we have about 1500-2000 new math Ph.D.’s every year, and according to the AMS survey, nearly all of them find jobs within a year (over 50% in academia, some in the industry, some abroad).

In fact, our math Ph.D. programs are the envy of the world.  For example, of the top 20 schools worldwide between 12 and 15 are occupied by leading US programs depending on the ranking (see e.g. here or there for recent examples, or more elsewhere).  Think about it: math requires no capital investment or infrastructure at all, so with the advent of personal computing, internet and the arXiv, there are little or no entry barriers to the field.  Any university in the world can compete with the US schools, yet we are still on the top of the rankings.  It is bewildering then, why would you even want to kill these super successful Ph.D. programs?

More infrastructurally, if there are drastic cuts to the Ph.D. programs in the US, who would be the people that can be hired to teach mathematics by the thousands of colleges whose students want to be math majors?  The number of the US math majors is already over 40,000 a year and keep growing at over 5% a year driven in part by the higher salary offerings and lifetime income (over that of other majors).  Don’t you think that the existing healthy supply and demand in the market for college math educators already determined the number of math Ph.D.’s we need to produce?

Well, apparently Wax doesn’t need convincing in the importance of math.  “I am the last person to denigrate pure mathematics.  It is a glory of mankind…”   She just doesn’t want people doing new research.  Or something.  As in “enough already.”  Think about it and transfer this thought to other areas.  Say — no new music is necessary — Bach and Drake said it all.  Or — no new art is necessary — Monet and Warhol were so prolific, museums don’t really have space for new works.  Right…

Economics matters

Let’s ask a different question: why would you want to close Ph.D. programs when they actually make money?  Take UCLA.  We are a service department, which makes a lot of money from teaching all kinds of undergraduate math courses + research grants both federal, state and industrial.  Annually, we graduate over 600 students with different types of math/stat majors, which constitutes about 1.6% of national output, the most of all universities.

Let’s say our budget is \$25 mil (I don’t recall the figures), all paid for.  That would be out of UCLA budget of \$7.5 billion of which less than 7% are state contributions.  Now compare these with football stadiums costs which are heavily subsidized and run into hundreds of millions of dollars.  If you had to cut the budget, is math where you start?

Can’t we just ignore these people?

Well, yes we can.  I am super happy to dismiss hurried paid-by-the-word know-nothing journalists or some anonymous YouTube comments.  But Amy Wax is neither.  She is smart and very accomplished:  summa cum laude from Yale, M.D. cum laude from Harvard Medical School, J.D. from Columbia Law School where she was an editor of Columbia Law Review, argued 15 cases in the US Supreme Court, is a named professor at UPenn Law School, has dozens of published research papers in welfare, labor and family law and economics.  Yep.

One can then argue — she knows a lot of other stuff, but nothing about math.  She is clearly controversial, and others don’t say anything of that nature, so who cares.  That sounds right, but so what?  Being known as controversial is like license to tell “the truth”…  er… what they really think.  Which can include silly things based on no research into our word.  This means there are numerous other people who probably also think that way but are wise enough or polite enough not to say it.  We need to fight this perception!

And yes, sometimes these people get into positions of power and decide to implement the changes.  Two cases are worth mentioning: the University of Rochester failed attempt to close its math Ph.D. program, and the Brown University fiasco.  The latter is well explained in the “Mathematical Apocrypha Redux” (see the relevant section here) by the inimitable Steven Krantz.  Rating-wise, this was a disaster for Brown — just read the Krantz’s description.

The Rochester story is rather well documented and is a good case of study for those feeling too comfortable.  Start with this Notices article, proceed to NY Times, then to protest description, and this followup in the Notices again.  Good news, right?  Well, I know for a fact that other administrators are also making occasional (largely unsuccessful) moves to do this, but I can’t name them, I am afraid.

Predictable apocalypse

Let’s take Amy Wax’s proposal seriously, and play out what would happen if 90-93% of US graduate programs in mathematics are closed on January 1, 2020.  By law.  Say, the US Congress votes to deny all federal funds to universities if they maintain a math Ph.D. program, except for the top 15 out of about 180 graduate programs according to US News.  Let’s ignore the legal issues this poses.  Just note that there are various recent and older precedents of federal government interfering with state and private schools (sometimes for a good cause).

Let’s just try to quickly game out what would happen.  As with any post-apocalyptic fiction, I will not provide any proofs or reasoning.  But it’s all “reality based”, as two such events did happened to mathematicians in the last century, one of them deeply affecting me: the German “academic reforms” in late 1930s (see e.g. here or there), and the Russian exodus in early 1990s (see e.g. here or there, or there).  Another personally familiar story is an implosion of mathematics at Bell Labs in late 1990s.  Although notable, it’s on a much smaller scale and to my knowledge has not been written about (see the discussion here, part 6).

First, there will be huge exodus of distinguished mathematics faculty from school outside of the 15 schools.  These include members of the National Academy of Sciences, numerous ICM speakers, other award winners, etc.  Some will move overseas (Canada, Europe, Japan, China, etc.), some will retire, some leave academia.  Some will simply stop doing research given the lack of mathematical activity at the department and no reward for doing research.

Second, outside of top 15, graduate programs in other subjects notice falling applications resulting in their sliding in world ranking.  These include other physical sciences, economics and computer science.  Then biological and social sciences start suffering.  These programs start having their own exodus to top 15 school and abroad.

Third, given the sliding of graduate programs across the board, the undergraduate education goes into decline across the country.  Top US high school students start applying to school abroad. Many eventually choose to stay in these countries who welcome their stem excellence.

Fourth, the hitech, fintech and other science heavy industries move abroad closer to educated employees.  United States loses its labor market dominance and starts bleeding jobs across all industries.   The stocks and housing market dip down.

Fifth, under strong public pressure the apocalyptic law is repealed and all 180 Ph.D. programs are reinstated with both state and federal financial support.  To everyone’s surprise, nobody is moving back.  Turns out, destroying is much faster and easier than rebuilding, as both Germany and Russia discovered back in the 20th century.  From that point on, January 1, 2020 became known as the day the math died.

Final message:

Dear Amy Wax and Glenn Loury!  Please admit that you are wrong.  Or at least plead ignorance and ask for forgiveness.  I don’t know if you will ever see this post or have any interest in debating the proposition I quoted, but I am happy to do this with you.  Any time, any place, any style.  Because the future of academia is important to all of us.

You should watch combinatorics videos!

Here is my collection of links to Combinatorics videos, which I assembled over the years, and recently decided to publish.  In the past few years the number of videos just exploded.  We clearly live in a new era.  This post is about how to handle the transition.

What is this new collection?

I selected over 400 videos of lectures and seminars in Combinatorics, which I thought might be of interest to a general audience.  I tried to cover a large number of areas both within Combinatorics and related fields.  I have seen many (but not all!) of the talks, and think highly of them.  Sometimes I haven’t seen the video, but have heard this talk “live” at the same or a different venue, or read the paper, etc.  I tried to be impartial in my selection, but I am sure there is some bias towards some of my favorite speakers.

The collection includes multiple lectures by Noga Alon, Persi Diaconis, Gil Kalai, Don Knuth, László Lovász, János Pach, Vic Reiner, Paul Seymour, Richard Stanley, Terry Tao, Xavier Viennot, Avi Wigderson, Doron Zeilberger, and many many others. Occasionally the speakers were filmed giving similar talks at different institutions, so I included quick links to those as well so the viewer can choose.

Typically, these videos are from some workshops or public lecture series.  Most are hosted on the institution websites, but a few are on YouTube or Vimeo (some of these are broken into several parts).  The earliest video is from 1992 and the most recent video was made a few days ago.   Almost all videos are from the US or Canada, with a few recent additions from Europe.  I also added links to a few introductory lectures and graduate courses on the bottom of the page.

Why now?

Until a couple of years ago, the videos were made only at a few conference centers such as Banff, MSRI and IAS.  The choice was sparse and the videos were easy to find.  The opposite is true now, on both counts.  The number of recorded lectures in all areas is in tens of thousands, they are spread across the globe, and navigating is near impossible unless you know exactly what you are looking for.  In fact, there are so many videos I really struggled with the choice of which to include (and also with which of them qualify as Combinatorics).  I am not sure I can maintain the collection in the future – it’s already getting too big.  Hopefully, some new technology will come along (see below), but for now this will do.

Why Combinatorics?

That’s what I do.  I try to think of the area as broad as possible, and apologize in advance if I omitted a few things.  For the subarea division, I used as a basis my own Wikipedia entry for Combinatorics (weirdly, you can listen to it now in a robotic voice).  The content and the historical approach within sub-areas is motivated by my views here on what exactly is Combinatorics.

Why should you start watching videos now?

First, because you can.  One of the best things about being in academia is the ability (in fact, necessity) to learn.  You can’t possibly follow everything what happens in all fields of mathematics and even all areas of combinatorics.  Many conferences are specialized and the same people tend to meet a year after year, with few opportunities for outsiders to learn what’s new over there.  Well, now you can.  Just scroll down the list and (hopefully) be amazed at the number of classical works (i.e. over 5 y.o.) you never heard of, the variety of recent developments and connections to other fields.  So don’t just watch people in your area from workshops you missed for some reason.  Explore other areas!  You might be surprised to see some new ideas even on your favorite combinatorial objects.  And if you like what you see, you can follow the links to see other videos from the same workshops, or search for more videos by the same speaker…

Second, you should start watching because it’s a very different experience.  You already know this, of course.  One can pause videos, go back and forward, save the video to watch it again, or stop watching it right in the beginning.  This ability is to popular, Adam Sandler even made an awful movie about it…  On the other hand, the traditional model of lecture attendance is where you either listen intently trying to understand in real time and take notes, or are bored out your mind but can’t really leave.  It still has its advantages, but clearly is not always superior.  Let me elaborate on this below.

How to watch videos?

This might seem like a silly question, but give me a chance to suggest a few ideas…

0) Prepare for the lecture.  Make sure to have enough uninterrupted time.  Lock the door.  Turn off the cell phone.  Download and save the video (see below).  Download and save the slides.  Search for them if they are not on the lecture website (some people put them on their home pages).  Never delete anything – store the video on an external hard drive if you are running out of space.  Trust me, you never know when you might need it again, and the space is cheap anyway…

Some years ago I made a mistake by not saving Gil Kalai’s video of a talk titled “Results and Problems around Borsuk’s Conjecture”.  I found it very inspiring — it’s the only talk I referenced it in my book.  Well, apparently, in its infinite wisdom, PIMS lost the video and now only the audio is available, which is nearly useless for a blackboard talk.  What a shame!

1) Use 2 devices.  Have the video on a big screen, say, a large laptop or a TV hooked to your  laptop.  If the TV is too far, use a wireless mouse to operate a laptop from across the room or something like a Google stick to project from a far.  Then, have the slides of the talk opened on your tablet if you like taking computer notes or just like scrolling by hand gestures, or on your other laptop if you don’t.  The slides are almost universally in .pdf and most software including the Adobe Reader allows to take notes straight in the file.

Another reason to have slides opened is the inability for some camera people to understand what needs to be filmed.  This is especially severe if they just love to show the unusual academic personalities, or are used to filming humanities lectures where people read at the podium.  As a result, occasionally, you see them pointing a camera to a slide full of formulas for 2 seconds (and out of focus), and then going back for 2 minutes filming a speaker who is animatedly pointing to the screen (now invisible), explaining the math.  Ugh…

2) If the subject is familiar and you feel bored with the lengthy introduction, scroll the slides until you see something new.  This will give you a hint to where you should go forward in the video.  And if you did miss some definition you can pause the video and scroll the slides to read it.

3) If there are no slides, or you want to know some details which the speaker is purposefully omitting, pause the video and download the paper.  I do this routinely while listening to talks, but many people are too shy to do this out of misplaced fear that others might think they are not paying attention.  Well, there is no one to judge you now.

4) If you are the kind of person who likes to ask questions to clarify things, you still can.  Pause the video and search the web for the answer.  If you don’t find it, ask a colleague by skype, sms, chat, email, whatever.  If everything fails – write to the speaker.  She or he might just tell you, but don’t be surprised if they also ignore your email…

5) If you know others who might be interested in the video lecture, just make it happen.  For example, you can organize a weekly seminar where you and your graduate students watch the lectures you choose (when you have no other speakers).  If students have questions, pause the video and try to answer them.  In principle, if you have a good audience the speaker may agree to answer the questions for 5-10 min over skype, after you are done watching.  Obviously, I’ve never seen this happen (too much coordination?).  But why not try this – I bet if you ask nicely many speakers would agree to this.

6) If you already know a lot about the subject, haven’t been following it recently but want to get an update, consider binge watching.  Pick the most recent lecture series and just let it run when you do house shores or ride a subway.  When things get interesting, you will know to drop everything and start paying attention.

Why should you agree to be videotaped?

Because the audience is ready to see your talks now.  Think of this as another way of reaching out with your math to a suddenly much broader mathematical community (remember the “broad impact” section on your NSF grant proposal?).  Let me just say that there is nothing to fear – nobody is expecting you to have acting skills, or cares that you have a terrible haircut.  But if you make a little effort towards giving a good talk, your math will get across and you might make new friends.

Personally, I am extremely uncomfortable being videotaped – the mere knowledge of the camera filming makes me very nervous.  However I gradually (and grudgingly) concluded that this is now a part of the job, and I have to learn how to do this well.  Unfortunately, I am not there yet…

Yes, I realize that many traditionalists will object that “something will be missing” when you start aiming at giving good video talks at the expense of local audience.  But the world is changing if hasn’t changed already and you can’t stop the tide.  This happened before, many times.  For example, at some point all the big Hollywood studios have discovered that they can make movies simpler and make a great deal more money overseas to compensate for the loss in the US market.  They are completely hooked now, and no matter what critics say this global strategy is likely irreversible.  Of course, this leaves a room for a niche market (say, low budget art-house movies), but let’s not continue with this analogy.

How to give video lectures?

Most people do nothing special.  Just business as usual, hook up the mike and hope it doesn’t distort your voice too bad.  That’s a mistake.  Let me give a number of suggestions based mostly on watching many bad talks.  Of course, the advice for giving regular talks apply here as well.

0) Find out ahead of time if you get filmed and where the camera is.  During the lecture, don’t run around; try to stand still in full view of the camera and point to the screen with your hands.  Be animated, but without sudden moves.  Don’t use a laser pointer.  Don’t suddenly raise your voice.  Don’t appeal to the previous talks at the same workshop.  Don’t appeal to people in the audience – the camera can rarely capture what they say or do.  If you are asked a question, quickly summarize it so the viewer knows what question you are answering.  Don’t make silly off-the-cuff jokes (this is a hard one).

1) Think carefully whether you want to give a blackboard or a computer talk.  This is crucial.  If it’s a blackboard talk, make sure your handwriting is clear and most importantly BIG.  The cameras are usually in the very back and your handwriting won’t be legible otherwise.  Unless you are speaking the Fields Institute whose technology allows one to zoom into the high resolution video, nobody might be able to see what you write.  Same goes for handwritten slides unless they are very neat, done on a laptop, and the program allows you to increase their size.  Also, the blackboard management becomes a difficult issue.  You should think through what results/definitions should stay on the blackboard visible to the camera at all times and what can be safely deleted or lifted up if the blackboard allows that.

2) If it’s a computer talk, stick to your decision and make a lot of effort to have the slides look good.  Remember, people will be downloading them…  Also, make every effort NOT to answer questions on a blackboard next to the screen.  The lightning never works – the rooms are usually dimmed for a computer talk and no one ever thinks of turning the lights on just for 30 seconds when you explain something.  So make sure to include all your definition, examples, etc, in the slides.  If you don’t want to show some of them – in PowerPoint there is a way to hide them and pull them up only if someone asks to clarify something.  I often prepare the answers to some standard questions in the invisible part of my slides (such as “What happens for other root systems?” or “Do your results generalize to higher dimensions?”), sometimes to unintended comedic effect.  Anyhow, think this through.

3) Don’t give the same videotaped talk twice.  If you do give two or more talks on the same paper, make some substantial changes.  Take Rota’s advice: “Relate to your audience”…  If it’s a colloquium talk, make a broad accessible survey and include your results at the end.  Or, if it’s a workshop talk, try to make an effort to explain most proof ideas, etc.  Make sure to have long self-explanatory talk titles to indicate which talk is which.  Follow the book industry lead for creating subtitles.  For example use “My most recent solution of the Riemann hypothesis, an introduction for graduate students” or “The Pythagorean theorem: How to apply it to the Langlands Program and Quantum Field Theory”.

4) Download and host your own videos on your website alongside your slides and your relevant paper(s), or at least make clear links to them from your website.  You can’s trust anyone to keep your files.  Some would argue that re-posting them on YouTube will then suffice.  There are two issues here.  First, this is rarely legal (see below).  Second, as I mentioned above, many viewers would want to have a copy of the file.  Hopefully, in the future there will be a copyright-free arXiv-style video hosting site for academics (see my predictions below).

5) In the future, we would probably need to consider having a general rule about adding a file with errata and clarifications to your talk, especially if something you said is not exactly correct, or even just to indicate, post-factum, whether all these conjectures you mentioned have been resolved and which way.  The viewers would want to know.

For example, my student pointed out to me that in my recent Banff talk, one of my lemmas is imprecise.  Since the paper is already available, this is not a problem, but if it wasn’t this could lead to a serious confusion.

6) Watch other people’s videos.  Pay attention to what they do best.  Then watch your own videos.  I know, it’s painful.  Turn off the sound perhaps.  Still, this might help you to correct the worst errors.

7) For advanced lecturers – try to play with the format.  Of course, the videos allow you to do things you couldn’t do before (like embedding links to papers and other talks, inserting some Java demonstration clips, etc.), but I am talking about something different.  You can turn the lecture into an artistic performance, like this amazing lecture by Xavier Viennot.  Not everyone has the ability or can afford to do this, but having it recorded can make it worthwhile, perhaps.

There are some delicate legal issues when dealing with videos, with laws varying in different states in the US (and in other countries, of course).  I am not an expert on any of this and will write only as I understand them in the US.  Please add a comment on this post if you think I got any of this wrong.

1) Some YouTube videos of math lectures look like they have been shut by a phone.  I usually don’t link to those.  As I understand the law on this, anyone can film a public event for his/her own consumption.  However, you and the institution own the copyright so the YouTube posting is illegal without both of yours explicit permission (written and signed).  You can fight this by sending a “cease and desist” letter to the person who posted the video, but contacting YouTube directly might be more efficient – they have a large legal department to sort these issues.

2) You are typically asked to sign away your rights before your talk.  If an institution forgot to do this, you can ask to take your talk down for whatever reason.  However, even if you did sign the paper you can still do this – I doubt the institution will fight you on this just to avoid bad publicity.  A single email to the IT department should suffice.

3) If the file with your talk is posted, it is (obviously) legal for you to download it, but not to post it on your website or repost elsewhere such as YouTube or WordPress.  As far as I am concerned, you should go ahead and post it on your university website anyway (but not on YT or WP!).  Many authors typically post all their papers on their website even if they don’t own a copyright on them (which is the case or virtually all papers before 2000).  I am one of them.  The publishers just concluded that this is the cost of doing business – if they start going after people like us, the authors can revolt.  The same with math videos.  The institutions probably won’t have a problem with your university website posting as long as you acknowledge the source.  But involving a third party creates a host of legal problems since these internet companies are making money out of the videos they don’t own a copyright for.  Stay away from this.

4)  You can the edit the video by using numerous software, some of which is free to download.  Your can remove the outside noise, make the slides sharper, everything brighter, etc.  I wouldn’t post a heavily edited video when someone else owns a copyright, but a minor editing as above is ok I think.

5) If the institution’s website does not allow to download the video but has a streaming option (typically, the Adobe Flash or HTML5), you can still legally save it on your computer, but this depends on what software you choose.  There are plenty of software which capture the video being played on your computer and save it in a file.  These are 100% legal.  Other websites play the videos on their computers and allow you to download afterwards.  This is probably legal at the institutions, but a gray area at YouTube or Vimeo which have terms of service these companies may be violating.  Just remember – such videos can only be legal for personal consumption.  Also, the quality of such recording is typically poor – having the original file is much better.

What will happen in the future?

Yes, I will be making some predictions.  Not anything interesting like Gian-Carlo Rota’s effort I recently analyzed, but still…

1) Watching and giving video lectures will become a norm for everyone.  The ethical standards will develop that everyone gets to have the files of videos they made.  Soon enough there will be established some large well organized searchable (and not-for-profit!) math video depositories arXiv-style where you can submit your video and link to it from your website and where others can download from.  Right now companies like DropBox allow you to do this, but it’s for-profit (your have to pay extra for space), and it obviously needs a front like the arXiv.  This would quickly make my collection a thing of the past.

2) Good math videos will become a “work product”, just like papers and books.  It is just another venue to communicate your results and ideas.  People will start working harder on them.  They will become a standard item on CVs, grant applications, job promotions, etc.  More and more people will start referencing them just like I’ve done with Kalai’s talk.  Hopefully part 1) will happen soon enough so all talks get standard and stable links.

3) The video services will become ubiquitous.  First, all conference centers will acquire advanced equipment in the style of the Banff Center which is voice directed and requires no professional involvement except perhaps at the editing stage.  Yes, I am thinking of you, MFO.  A new library is great, but the talks you could have recorded there are priceless – it’s time to embrace the 21st century….

Second, more and more university rooms will be equipped with the cameras, etc.  UCLA already has a few large rooms like that (which is how we make the lamely named BruinCasts), but in time many department will have several such rooms to hold seminars.  The storage space is not an issue, but the labor cost, equipment and the broadband are.  Still, I give it a decade or two…

4) Watching and showing math videos will become a standard part of the research and graduate education.  Ignore the doomsayers who proclaim that this will supplant the traditional teaching (hopefully, not in our lifetime), but it’s clear already there are unexplored educational benefits from this.  This should be of great benefit especially to people in remote locations who don’t have access to such lectures otherwise.  Just like the Wikipedia has done before, this will even the playing field and help the talent to emerge from unlikely places.  If all goes well, maybe the mathematics will survive after all…

Happy watching everyone!

How many graduate students do we need?

In response to my previous post “Academia is nothing like a drug cartel“, a fellow blogger Adam Sheffer asks:

I was wondering what you think about a claim that I sometimes hear in this context – that one of the problems is that universities train too many Ph.D. students. That with a smaller number of math Ph.D. students the above will be less of a problem, and also that this way there will be a smaller number of people dealing with less “serious/important” topics (whatever this means exactly).

This question is certainly relevant to the “adjunct issue”.  I heard it before, but always found it somewhat confusing.  Specifically to the US, with its market based system, who exactly is supposed to decrease the number of Ph.D.’s?  The student themselves should realize how useless in the doctoral degree and stop applying?  The individual professors should refuse to accept graduate students?  The universities should do this together, in some kind of union?  The government?  All these questions are a bit different and need untangling.

I was going to write a brief reply, but after Adam asked this question I found a yet another example of lazy journalism by Slate’s “education columnist” Rebecca Schuman who argues:

It is, simply put,  irresponsible to accept so many Ph.D. students when you know graduate teaching may well be the only college teaching they ever do.

Of course, Dr. Schuman already has a Ph.D. (from our neighbor UC Irvine) — she just wants others not get one, perhaps to avoid her own fate of an adjunct at University of Missouri.  Needless to say, I cannot disagree more.  Let me explain.

Universities are not allowed to form a cartel

Let’s deal with the easy part.  If the American universities somehow conspired to limit or decrease the number of graduate students they accepts, this would be a classical example of anti-competitive behavior.  Simply put, the academia would form a cartel.  A textbook example of a cartel is OPEC which openly conspires to increase or decrease oil production in order to control world energy prices.  In the US, such activity is against the law due to to the Sherman Act of 1890, and the government/courts have been ruthless in its application (cf. European law to that effect).

One can argue that universities are non-profit institutions and by definition would not derive profit should they conspire, but the law makes no distinction on this, and this paper (co-authored by the celebrity jurist and economist Richard Posner) supports this approach.  And to those who think that only giants such as Standard Oil, AT&T or Microsoft have to worry about anti-trust, the government offers plenty of example of going after  small time cartels.  A notable recent case is Obama’s FTC going after Music Teachers National Association, who have a non-poaching of music students recommendation in their “code of ethics”.  Regardless what you think of that case, it is clear that the universities would never try to limit the number of graduate students in a similar manner.

Labor suppy and demand

As legions before her, Schuman laments that pospective grad students do not listen to “reason”:

Expecting wide-eyed, mind-loving intellectuals to embrace the eventual realities of their situations has not worked—yes, they should know better, but if they listened to reason, they wouldn’t be graduate students in the first place.  Institutions do know better, so current Ph.D. recruitment is dripping with disingenuousness.

But can you really be “wide-eyed” in the internet era?   There is certainly no shortage of articles by both journalists and academics on the “plight” of academic life – she herself links to sites which seem pretty helpful informing prospective graduate students (yes, even the link to Simpsons is helpful).   I have my own favorites: this, that, that and even that.  But all of these are misleading at best and ridiculous at worst.  When I mentioned them on MO, José Figueroa-O’Farrill called them a “parallel universe”, for a good reason.

You see, in this universe people make (mostly) rational decisions, wide-eyed or not.   The internet simply destroyed the information gap.  Faced with poor future income prospects, graduate students either choose to go elsewhere or demand better conditions at the universities.  Faced with a decreasing pool of candidates the universities make an effort to make their programs more attractive, and strive to expand the applicant pool by reaching out to underrepresented groups, foreign students, etc.  Eventually the equilibrium is reached and labor supply meets demand, as it always has.  Asking the universities (who “do know better”)  to have the equilibrium be reached at a lower point is equivalent to asking that Ph.D. programs become less attractive.  And I thought Schuman cares…

Impact of government actions

Now, when it comes to distorting of the labor market, the government is omnipotent and with a single bill can decrease the number of graduate students.  Let’s say, the Congress tomorrow enacts a law mandating a minimum wage of \$60,000 a year for all graduate students.  Of course, large universities have small armies of lawyers and accountants who would probably figure out how to artificially hike up the tuition for graduate students and include it in their income, but let’s assume that the law is written to prevent any loopholes.  What would happen next?

Obviously, the universities wouldn’t be able to afford that many graduate graduate students.  The number of them will plunge.  The universities would have to cut back on the TA/recitation/discussion sessions  and probably hire more adjuncts to compensate for the loss.   In time, this would lower the quality of education or lead to huge tuition increases, or mostly likely a little bit of both.  The top private universities who would want to maintain small classes will become completely unaffordable for the middle class.  Meanwhile the poorer state universities will commodify their education by creating huge classes with multiple choice machine testing, SAT-style, and further diminishing student-faculty interaction.  In fact, to compensate for their increasing cost to universities, graduate students will be asked to do more teaching, thus extending their time-to-degree and decreasing the graduation rates.

Most importantly, this would probably have no positive effect on decreasing competition for tenure track jobs, since the academic market is international.  In other words, a decreasing american supply will be immediately compensated with an increasing european supply aided with inflow from emerging markets (ever increasing in quantity and quality production of Ph.D.’s in Asia).   In fact, there is plenty of evidence that this would have sharply negative effect on prospects of American students, as decreased competition would result in weaker research work (see below).

In summary, who exactly would be the winners of this government action?  I can think of only one group: lazy journalists who would have many new reasons to write columns complaining about the new status quo.

Let’s go back to Schuman’s “it is [..] irresponsible to accept so many Ph.D. students” quote I mentioned above, and judge in on moral merits.  Irresponsible?  Really?  You are serious?  Is it also irresponsible to give so many football scholarships to college students if only a few of them can make it to the NFL?  Is it also irresponsible to have so many acting schools given that so few of the students become movie stars?  (see this list in my own little town).  In the previous post I already explain how graduate schools are apprenticeship programs.  Graduate schools give students a chance and an opportunity to succeed.  Some students do indeed, while others move to do something else, sometimes succeeding beyond expectations (see e.g. this humorous list).

What’s worse, Schuman implicitly assumes that the Ph.D. study can only be useful if directly applicable to obtain a professorship.  This is plainly false.  I notice from her CV that she teaches “The World of Kafka” and “Introduction to German Prose”.  Excellent classes I am sure, but how exactly the students are supposed to use this knowledge in real life?  Start writing in German or become a literary agent?   Please excuse me for being facetious – I hope my point is clear.

Does fewer students means better math?  (on average)

In effect, this is Adam’s speculation at the end of his question, as he suggested that perhaps fewer mathematics graduate students would decrease the number of  “less ‘serious/important’ topics”.  Unfortunately, the evidence suggests the opposite.  When there is less competition, this is a result of fewer rewards and consequently requires less effort to succeed.  As a result, the decrease in the number of math graduate students will lead to less research progress and increase in “less important” work, to use the above  language.

To see this clearly, think of sports.  Compare this list of Russian Major League baseball players with this list by that of Japanese.  What explains the disparity?  Are more Japanese men born with a gift to play baseball?  The answer is obvious.  Baseball is not very popular in Russia.  Even the best Russian players cannot compete in the american minor leagues.  Things are very different in Japan, where baseball is widely popular, so the talented players make every effort to succeed rather than opt for possibly more popular sport (soccer and hockey in Russian case).

So, what can be done, if anything?

To help graduate students, that is.  I feel there is a clear need to have more resources on non-academic options available for graduate student (like this talk or this article).   Institutionally, we should make it easier to cross register to other schools within the university and the nearby universities.  For example, USC graduate students can take UCLA classes, but I have never seen anyone actually doing that.  While at Harvard, I took half a dozen classes at MIT – it was easy to cross register and I got full credit.

I can’t think of anything major the universities can do.  Government can do miracles, of course…

P.S.  I realize that the wage increase argument has a “fighting straw men” feel.  However, other government actions interfering with the market are likely to bring similarly large economic distortions of the academic market, with easily predictable negative consequences.  I can think of a few more such unfortunate efforts, but the burden is not on me but on “reformers” to propose what exactly do they want the government to do.

P.P.S.  We sincerely wish Rebecca Schuman every success in her search for a tenure track appointment.  Perhaps, when she gets such a position, she can write another article with a slightly sunnier outlook.

Academia is nothing like a drug cartel

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.

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.

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).

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.

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

Admission blues: How to fix GRE Mathematics and tweak the Putnam Competition

I was thinking about the Putnam competition and the GRE Mathematics test in the context of graduate admissions.  Are they useful?  If yes, which one is more relevant?  After crunching some numbers, I concluded that while they are useful to some extend, there are problems with both.  Even worse, a number of students who fall in the gap between “very good” and “exceptional”, are ill served with either.

As I mention in my earlier post, every year the US produces around 1,600 Ph.D.’s in mathematical sciences (math, applied math, statistics) from over 100 accredited programs, of which about 900 are US citizen and permanent residents.  If you restrict to mathematics alone, the numbers drop by about 25% to about 1200. The overall 10 year completion rate is about 50%  according to the Council of Graduate Schools study, so perhaps about 3,000-3,200 students start graduate programs.

As a general rule, graduate programs in mathematics explicitly ask for the GRE Subject test scores, but are often happy to hear about the Putnam results as well.  In fact, some “how to” guides now recommend taking Putnam exam (and Putnam prep classes!) on par with the GRE test and REU programs (see e.g. here and there).  How the schools use either data is probably quite a bit different, and is the other side of our main question.

2. GRE Mathematics Subject test in numbers

The GRE Subject tests are developed and administered by ETS, which is nominally non-profit, but with about 1 billion dollars in revenue.  For a quick comparison with a for-profit, non-profit and public institutions, e.g. New York Times Corp, Harvard and UCLA, had 2.33.7 and 4.3 bln dollars in 2011 operating revenues, respectively.

From the official GRE test preparation publication:  “The questions are classified approximately as follows: calculus (50%), algebra (25%) and other topics (25%).”  This is already unfortunate, but more on that later.  Here are these “other topics”:

Introductory real analysis (sequences and series of numbers and functions, continuity, differentiability and integrability, elementary topology of R), discrete mathematics (logic, set theory, combinatorics, graph theory, and algorithms), general topology, geometry, complex variables, probability and statistics, and numerical  analysis.  The above descriptions of topics covered in the test should not be considered exhaustive […]  (emphasis mine – IP)

The GRE Guide gives .92 value for the KR20 reliability test, a solid measure suggesting the test has many questions leading to different scores between strong and weak students.  The students have 170 minutes for about 65 questions.  The scores are on the scale from 200 to 990, are rounded to nearest multiple of 10, with standard errors 31 points, and 44 for the differences.  In other words, if I understand correctly (the guide is vague on this), one should not reliably compare students with scores differing by 50 points of less.  I am doubtful most grad schools  follow that.

In the same GRE guide, ETS reports that there were about 12,800 test takers in four years (July 2008 to June 2011), roughly 3200 a year.  This loosely coincides with our graduate student data, as the students take on average one GRE Subject test.  In other words, all students with GRE scores get accepted somewhere.  So one should not be surprised to see a high correlation (but not necessarily causation) between grad school ranking and GRE Subject scores. Curiously, ETS’s own study says GRE General are a very poor predictor of success in math graduate programs, at least when it comes to GPA and graduation rate.

So how do grad schools use the GRE Math scores? That’s very much unclear. Of course, all schools gather the statistics like averages of those applied, admitted and/or accepted (reported to the dean, external department reviewers, the NRC study, the US News, etc.), but very few make it publicly available.  In a rare moment of openness, Penn State admits what amounts to not much use of GRE scores: their average scores vary widely over the years, swinging from 650 to 890, with a positive trend in recent years.  In a general MO discussion on this, Pete Clark writes that University of Georgia does  not require GRE Subject, so he looks for high GRE General scores.  UCLA is a bit evasive: “those we offer admission to have GRE subject scores in or above the 80th percentile” which according to GRE chart amounts to minimum of about 790, suggesting relevance.   MIT is blunt but imprecise: “There is no minimum GRE test score required, but if the score on the math subject GRE is not very high, evidence of excellence must be present elsewhere in the application or in the letters of recommendation.” UPenn is actually helpful: “[GRE Math score] should be at least 750, though applicants with somewhat lower scores may be admitted if the rest of their application is sufficiently strong,” and that the recent average score is 820.  This all makes a very foggy picture.

3. Putnam competition in numbers

The premise is simple: first Saturday in December, 6 hours (in two sittings) to solve 12 problems in all areas of mathematics, maximum of 10 points per problem.  Joe Gallian wrote a nice summary. The problems are difficult: the maximal score 120 is achieved only very occasionally, once in about 10-15 years.  The median score is often either 0, 1 or 2 (out of 120!), and the mean is between 5 and 10 points.  I bet it must be depressing to spend 6 hours and get no or almost no points.

The top 5 scorers are “Putnam Fellows”, another 18-20 are “in the money”, and about 50-60 get “honorable mention”.  In 2011, there were “4,440 students from 572 colleges and universities in Canada and the United States”.  The historical data shows that there is a clear correlation between doing well on Putnam and doing well in mathematics, which is even more pronounced for the top 25, and especially Putnam fellows.

Of course, the competition is not aimed at helping graduate admissions, as emphasized by the mid-March results date (way after the applications are due and the admission decisions are made).  It does not even make the scores available in any official format.  In fact, historically, it is primarily a team competition, a nerdy alternative to college athletics.  Finally, a competition is not necessarily similar to do research.  As Kedlaya said, “A contest problem is meant to be solved in the space of minutes or hours, whereas in research, one sometimes works on the same problem for days, months, occasionally even years.”

4. A bissel of analysis

(a)  GRE Math.  While useful to some extend, mostly for the middle and bottom scoring students, it is largely useless for most of the better prepared students.  Indeed, in the “upper middle range” of 75 to 90 percentile, the test scores range between 770 and 850, comprising about 500 students every year.  By the rules of GRE, many of these students cannot be even compared.  Those who can, it is unclear whether they really are better candidates for doing research and teaching in mathematics.  Indeed, the excessive emphasis on calculus, real analysis and linear algebra shows the student’s ability to memorize concepts and quickly perform routine tasks.  This does not test problem solving.  Neither do “other topics” which are heavily testing definitions of a group, ring, metric space, etc.  I bet the performance in this part strongly correlates with the quality of the undergraduate institution: better colleges offer more serious math classes, and GRE Math preparation classes, which cover these basic topics; others do not.

For the top 10%, the GRE Math scores does distinguish between them, but that’s hardly necessary.  Of the top 250-300 students over half of them are international and often come with accolades like “the best student in N years from the XYZ university.” Last year I recall even one European student described as “the best student since World War II from … country”.  Those 100-150 that are from the US, are well served with numerous REU programs both national and at their home universities, by the Budapest and Moscow semesters, Putnam, IMO and other competitions, etc.  Their GRE scores seem irrelevant in retrospect.

Now, using AMS Classification, Group I of 48 top math graduate programs graduates about 550 Ph.D’s.  All are research oriented.  I am guesstimating that they must be accepting c. 800 students in total.  So after the top 300 are accepted, how are they suppose to choose the next 500 if GRE is irrelevant?

(b) Putnam.  Even though a majority receive only single digit score, there is a clear benefit for the top programs to know who the winners are.  The top 25 individuals, clearly possess excellent problem solving abilities, which is useful in a number of areas of mathematics.  The are multiple problems with this.  First, it would be nice to have the list of winners available by December.  Second, it would be nice if Putnam is offered overseas.  But even for the US/Canada based students, as it stands, the senior’s performance is not counted in admissions due calendar issues.  Since students often are encouraged to take their junior year abroad, the best performance they can include in their applications is from their sophomore year, which is often inferior to their senior year performance.  So with exception of the truly top students, Putnam results are not used in the admissions.

5. A modest proposal, Russian style

(a) GRE Math.  Split the GRE Mathematics into two parts.  Keep Calculus/Linear Algebra in the first half, more or less in the same multiple choice form as you have now.  It is clearly helpful for middle and bottom tier students and programs.  For the second part, make it a no-hard-math-required problem solving style.  Make many relatively simple problems, much much simpler than IMO problems, more like Moscow olympiads for the freshman-sophomore HS years (8-9th year out of 11).  This would allow relatively unbiased testing of problem solving, extremely useful to mathematics programs.  Both scores would need to be reported (kind of like 4 GRE General scores).

As revenue figures suggest, ETS is essentially a large utility company which does not want to rock the boat.  But it has made changes before, and this particular change would be relatively painless and have the added advantage that no “other fields” need to be argued about – all students will know exactly what is the scope of the test.

(b) Putnam.  Ugh.  It’s true that “if it ain’t broke, don’t fix it“, so I don’t want to propose major changes.  Just three minor tweaks, which will not change the core competition, but hopefully will make it more democratic and helpful for graduate admissions.

* First, move the competition to late September, so the scores can be revealed before Jan 1.  I really don’t see what exactly is hard about that.  Perhaps, some Putnam prep classes will have to be moved to the Spring.  So what?

* Second, open it for international students.  I know, I know, time difference, language issues, etc.  Whatever, keep it on the US time and only in English, as it is now.  If the overseas students want to participate, they might have to do this at night perhaps (simply allow unlimited tea, coffee and Red Bull).  This is still better than not giving them an opportunity at all.  Another issue is trust (in foreign faculty supervisors).  For that, use the technology.  Reveal the problem on some website for all at once.  Videotape what’s happening in all rooms where the competition is taking place.  Have all solutions uploaded as .pdf files to the main server within minutes after the end of the competition (they should still be graded locally, with top scores re-graded at a central location).  While some of this might be an obstacle for some universities in poor countries, the majority of foreign universities already have all the necessary technology to make this happen.

Third, and most controversially, at least for the US/Canadian students allow an easy “parallel track”.  That is, come up with substantially easier problems which can be administered at the same time in parallel.  The students should be given a choice – either real problems which are hard, or easier problems which do not count.  This would be good for students’ morale as a means to prevent the annual 40% of 0 scores, and the scores can be useful for admission.  I am modelling this based on the widely successful Tournament of Towns, which has two levels and two tracks (harder and easier), see this problem archive.

P.S.  Full disclosure: I took GRE Math in 1994 and received maximal score available at that time. I recall finishing early, but missing a couple of problems possibly due to some English language difficulties.  I did not participate in the Putnam – was busy in Moscow.  More recently, I also participated in graduate admissions, but everywhere above made sure I use only open sources and no “inside information”.