## The power of negative thinking, part I. Pattern avoidance

In my latest paper with Scott Garrabrant we disprove the *Noonan-Zeilberger Conjecture*. Let me informally explain what we did and why people should try to disprove conjectures more often. This post is the first in a series. Part II will appear shortly.

#### What did we do?

Let *F ⊂ S _{k}* be a finite set of permutations and let

*C*(

_{n}*F*) denote the number of permutations

*σ ∈ S*avoiding the set of patterns

_{n}*F*. The

*Noonan-Zeilbeger conjecture*(1996), states that the sequence {

*C*(

_{n}*F*)} is always

*P-recursive*. We disprove this conjecture. Roughly, we show that every Turing machine T

*can be simulated by a set of patterns*

*F,*so that the number

*a*of paths of length n accepted by by T is equal to

_{n }*C*(

_{n}*F*) mod 2. I am oversimplifying things quite a bit, but that’s the gist.

What is left is to show how to construct a machine T such that {*a _{n}*} is not equal (mod 2) to

**any**P-recursive sequence. We have done this in our previous paper, where give a negative answer to a question by Kontsevich. There, we constructed a set of 19 generators of

*GL(4,Z)*, such that the probability of return sequence is not P-recursive.

When all things are put together, we obtain a set *F* of about 30,000 permutations in *S _{80}* for which {

*C*(

_{n}*F*)} is non-P-recursive. Yes, the construction is huge, but so what? What’s a few thousand permutations between friends? In fact, perhaps a single pattern (1324) is already non-P-recursive. Let me explain the reasoning behind what we did and why our result is much stronger than it might seem.

#### Why we did what we did

First, a very brief history of the NZ-conjecture (see Kirtaev’s book for a comprehensive history of the subject and vast references). Traditionally, pattern avoidance dealt with exact and asymptotic counting of pattern avoiding permutations for small sets of patterns. The subject was initiated by MacMahon (1915) and Knuth (1968) who showed that we get Catalan numbers for patterns of length 3. The resulting combinatorics is often so beautiful or at least plentiful, it’s hard to imagine how can it not be, thus the NZ-conjecture. It was clearly very strong, but resisted all challenges until now. Wilf reports that Richard Stanley disbelieved it (Richard confirmed this to me recently as well), but hundreds of papers seemed to confirm its validity in numerous special cases.

Curiously, the case of the (1324) pattern proved difficult early on. It remains unresolved whether *C _{n}*(1324) is P-recursive or not. This pattern broke Doron Zeilberger’s belief in the conjecture, and he proclaimed that it’s probably non-P-recursive and thus NZ-conjecture is probably false. When I visited Doron last September he told me he no longer has strong belief in either direction and encouraged me to work on the problem. I took a train back to Manhattan looking over New Jersey’s famously scenic Amtrack route. Somewhere near Pulaski Skyway I called Scott to drop everything, that we should start working on this problem.

You see, when it comes to pattern avoidance, things move from best to good to bad to awful. When they are bad, they are so bad, it can be really hard to prove that they are bad. But why bother – we can try to figure out something awful. The set of patterns that we constructed in our paper is so awful, that proving it is awful ain’t so bad.

#### Why is our result much stronger than it seems?

That’s because the proof extends to other results. Essentially, we are saying that everything bad you can do with Turing machines, you can do with pattern avoidance (mod 2). For example, why is (1324) so hard to analyze? That’s because it’s even hard to compute both theoretically and experimentally – the existing algorithms are recursive and exponential in *n*. Until our work, the existing hope for disproving the NZ-conjecture hinged on finding an appropriately bad set of patterns such that computing {*C _{n}*(

*F*)} is easy. Something like this sequence which has a nice recurrence, but is provably non-P-recursive. Maybe. But in our paper, we can do worse, a lot worse…

We can make a finite set of patterns *F,* such that computing {*C _{n}*(

*F*) mod 2} is “provably” non-polynomial (Th 1.4). Well, we use quotes because of the complexity theory assumptions we must have. The conclusion is much stronger than non-P-recursiveness, since every P-recursive sequence has a trivial polynomial in

*n*algorithm computing it. But wait, it gets worse!

We prove that for two sets of patterns *F* and *G*, the problem* “C _{n}*(

*F*) =

*C*(

_{n}*G*) mod 2 for all n” is undecidable (Th 1.3). This is already a disaster, which takes time to sink in. But then it gets even worse! Take a look at our Corollary 8.1. It says that there are two sets of patterns

*F*and

*G*, such that you can never prove nor disprove that

*C*(

_{n}*F*) =

*C*(

_{n}*G*) mod 2. Now that’s what I call truly awful.

#### What gives?

Well, the original intuition behind the NZ-conjecture was clearly wrong. Many nice examples is not a good enough evidence. But the conjecture was so plausible! Where did the intuition fail? Well, I went to re-read Polya’s classic “*Mathematics and Plausible Reasoning*“, and it all seemed reasonable. That is both Polya’s arguments and the NZ-conjecture (if you don’t feel like reading the whole book, at least read Barry Mazur’s interesting and short followup).

Now think about Polya’s arguments from the point of view of complexity and computability theory. Again, it sounds very “plausible” that large enough sets of patterns behave badly. Why wouldn’t they? Well, it’s complicated. Consider this example. If someone asks you if every 3-connected planar cubic graph has a Hamiltonian cycle, this sounds plausible (this is Tait’s conjecture). All small examples confirm this. Planar cubic graphs do have very special structure. But if you think about the fact that even for planar graphs, Hamiltonicity is NP-complete, it doesn’t sound plausible anymore. The fact that Tutte found a counterexample is no longer surprising. In fact, the decision problem was recently proved to be NP-complete in this case. But then again, if you require 4-connectivity, then *every* planar graph has a Hamiltonian cycle. Confused enough?

Back to the patterns. Same story here. When you look at many small cases, everything is P-recursive (or yet to be determined). But compare this with Jacob Fox’s theorem that for a random single pattern π, the sequence {*C _{n}*(π)} grows much faster than originally expected (cf. Arratia’s Conjecture). This suggests that small examples are not representative of complexity of the problem. Time to think about disproving ALL conjectures based on that evidence.

If there is a moral in this story, it’s that what’s “plausible” is really hard to judge. The more you know, the better you get. Pay attention to small crumbs of evidence. And think negative!

#### What’s wrong with being negative?

Well, conjectures tend to be optimistic – they are wishful thinking by definition. Who would want to conjecture that for some large enough *a,b,c* and *n,* there exist a solution of *a*^{n} + *b*^{n} = *c*^{n}? However, being so positive has a drawback – sometimes you get things badly wrong. In fact, even polynomial Diophantine equations can be as complicated as one wishes. Unfortunately, there is a strong bias in Mathematics against counterexamples. For example, only two of the Clay Millennium Problems automatically pay $1 million for a counterexample. That’s a pity. I understand why they do this, just disagree with the reasoning. If anything, we should encourage thinking in the direction where there is not enough research, not in the direction where people are already super motivated to resolve the problem.

In general, it is always a good idea to keep an open mind. Forget all this “power of positive thinking“, it’s not for math. If you think a conjecture might be false, ignore everybody and just go for disproof. Even if it’s one of those famous unsolved conjectures in mathematics. If you don’t end up disproving the conjecture, you might have a bit of trouble publishing computational evidence. There are some journals who do that, but not that many. Hopefully, this will change soon…

#### Happy ending

When we were working on our paper, I wrote to Doron Zeilberger if he ever offered a reward for the NZ-conjecture, and for the disproof or proof only? He replied with an unusual award, for the proof and disproof in equal measure. When we finished the paper I emailed to Doron. And he paid. Nice… 🙂

## What do math journals do? What will become of them in the future?

Recently, there has been plenty of discussions on math journals, their prices, behavior, technology and future. I have been rather reluctant to join the discussion in part due to my own connection to *Elsevier*, in part because things in Combinatorics are more complicated than in other areas of mathematics (see below), but also because I couldn’t reconcile several somewhat conflicting thoughts that I had. Should all existing editorial boards revolt and all journals be electronic? Or perhaps should we move to “pay-for-publishing” model? Or even “crowd source refereeing”? Well, now that the issue a bit cooled down, I think I figured out exactly what should happen to math journals. Be patient – a long explanation is coming below.

#### Quick test questions

I would like to argue that the debate over the second question is the general misunderstanding of the first question in the title. In fact, I am pretty sure most mathematicians are quite a bit confused on this, for a good reason. If you think this is easy, quick, answer the following three questions:

1) Published paper has a technical mistake invalidating the main result. Is this a fault of author, referee(s), handling editor, managing editor(s), a publisher, or all of the above? If the reader find such mistake, who she/he is to contact?

2) Published paper proves special case of a known result published 20 years earlier in an obscure paper. Same question. Would the answer change if the author lists the paper in the references?

3) Published paper is written in a really poor English. Sections are disorganized and the introduction is misleading. Same question.

Now that you gave your answers, ask a colleague. Don’t be surprised to hear a different point of view. Or at least don’t be surprised when you hear mine.

#### What do referees do?

In theory, a lot. In practice, that depends. There are few official journal guides to referees, but there are several well meaning guides (see also here, here, here, here §4.10, and a nice discussion by Don Knuth §15). However, as any editor can tell you, you never know what exactly did the referee do. Some reply within 5 min, some after 2 years. Some write one negative sentence, some 20 detailed pages, some give an advice in the style “yeah, not a bad paper, cites me twice, why not publish it”, while others a brushoff “not sure who this person is, and this problem is indeed strongly related to what I and my collaborators do, but of course our problems are much more interesting/important – rejection would be best”. The anonymity is so relaxing, doing a poor job is just too tempting. The whole system hinges on the shame, sense of responsibility, and personal relationship with the editor.

A slightly better questions is “What do *good* referees do?” The answer is – they don’t just help the editor make acceptance/rejection decision. They help the authors. They add some background the authors don’t know, look for missing references, improve on the proofs, critique the exposition and even notation. They do their best, kind of what ideal advisors do for their graduate students, who just wrote an early draft of their first ever math paper.

In summary, you can’t blame the referees for anything. They do what they can and as much work as they want. To make a lame comparison, the referees are like wind and the editors are a bit like sailors. While the wind is free, it often changes direction, sometimes completely disappears, and in general quite unreliable. But sometimes it can really take you very far. Of course, crowd sourcing refereeing is like democracy in the army – bad even in theory, and never tried in practice.

#### First interlude: refereeing war stories

I recall a curious story by Herb Wilf, on how Don Knuth submitted a paper under assumed name with an obscure college address, in order to get full refereeing treatment (the paper was accepted and eventually published under Knuth’s real name). I tried this once, to unexpected outcome (let me not name the journal and the stupendous effort I made to create a fake identity). The referee wrote that the paper was correct, rather interesting but “not quite good enough” for their allegedly excellent journal. The editor was very sympathetic if a bit condescending, asking me not to lose hope, work on my papers harder and submit them again. So I tried submitting to a competing but equal in statue journal, this time under my own name. The paper was accepted in a matter of weeks. You can judge for yourself the moral of this story.

A combinatorialist I know (who shall remain anonymous) had the following story with *Duke J. Math*. A year and a half after submission, the paper was rejected with three (!) reports mostly describing typos. The authors were dismayed and consulted a CS colleague. That colleague noticed that the three reports were in .pdf but made by cropping from longer files. Turns out, if the cropping is made straightforwardly, the cropped portions are still hidden in the files. Using some hacking software the top portions of the reports were uncovered. The authors discovered that they are extremely positive, giving great praise of the paper. Now the authors believe that the editor despised combinatorics (or their branch of combinatorics) and was fishing for a bad report. After three tries, he gave up and sent them cropped reports lest they think somebody else considers their paper worthy of publishing in the grand old Duke (cf. what Zeilberger wrote about *Duke*).

Another one of my stories is with the *Journal of AMS*. A year after submission, one of my papers was rejected with the following remarkable referee report which I quote here in full:

The results are probably well known. The authors should consult with experts.

Needless to say, the results were new, and the paper was quickly published elsewhere. As they say, “resistance is futile“.

#### What do associate/handling editors do?

Three little things, really. They choose referees, read their reports and make the decisions. But they are responsible for everything. And I mean for *everything*, both 1), 2) and 3). If the referee wrote a poorly researched report, they should recognize this and ignore it, request another one. They should ensure they have more than one opinion on the paper, all of them highly informed and from good people. If it seems the authors are not aware of the literature and referee(s) are not helping, they should ensure this is fixed. It the paper is not well written, the editors should ask the authors to rewrite it (or else). At *Discrete Mathematics*, we use this page by Doug West, as a default style to math grammar. And if the reader finds a mistake, he/she should first contact the editor. Contacting the author(s) is also a good idea, but sometimes the anonymity is helpful – the editor can be trusted to bring bad news and if possible, request a correction.

B.H. Neumann described here how he thinks the journal should operate. I wish his views held widely today. The book by Krantz, §5.5, is a good outline of the ideal editorial experience, and this paper outlines how to select referees. However, this discussion (esp. Rick Durrett’s “rambling”) is more revealing. Now, the reason most people are confused as to who is responsible for 1), 2) and 3), is the fact that while some journals have serious proactive editors, others do not, or their work is largely invisible.

#### What do managing editors and publishers do?

In theory, managing editors hire associate editors, provide logistical support, distribute paper load, etc. In practice they also serve as handling editors for a large number of papers. The publishers… You know what the publishers do. Most importantly, they either pay editors or they don’t. They either charge libraries a lot, or they don’t. Publishing is a business, after all…

#### Who wants free universal electronic publishing?

Good mathematicians. Great mathematicians. Mathematicians who write well and see no benefit in their papers being refereed. Mathematicians who have many students and wish the publishing process was speedier and less cumbersome, so their students can get good jobs. Mathematicians who do not value the editorial work and are annoyed when the paper they want to read is “by subscription only” and thus unavailable. In general, these are people who see having to publish as an obstacle, not as a benefit.

#### Who does not want free universal electronic publishing?

Publishers (of course), libraries, university administrators. These are people and establishments who see value in existing order and don’t want it destroyed. Also: mediocre mathematicians, bad mathematicians, mathematicians from poor countries, mathematicians who don’t have access to good libraries (perhaps, paradoxically). In general, people who need help with their papers. People who don’t want a quick brush-off “not good enough” or “probably well known”, but those who need advice on the references, on their English, on how the papers are structured and presented, and on what to do next.

#### So, who is right?

Everyone. For some mathematicians, having all journals to be electronic with virtually no cost is an overall benefit. But at the very least, “pro status quo” crowd have a case, in my view. I don’t mean that *Elsevier* pricing policy is reasonable, I am talking about the big picture here. In a long run, I think of journals as non-profit NGO‘s, some kind of nerdy versions of Nobel Peace Prize winning Médecins Sans Frontières. While I imagine that in the future many excellent top level journals will be electronic and free, I also think many mid-level journals in specific areas will be run by non-profit publishers, not free at all, and will employ a number of editorial and technical stuff to help the authors, both of papers they accept and reject. This is a public service we should strive to perform, both for the sake of those math papers, and for development of mathematics in other countries.

Right now, the number of mathematicians in the world is already rather large and growing. Free journals can do only so much. Without high quality editors paid by the publishers, with a large influx of papers from the developing world, there is a chance we might loose the traditional high standards for published second tier papers. And I really don’t want to think of a mathematics world once the peer review system is broken. That’s why I am not in the “free publishing camp” – in an effort to save money, we might loose something much more valuable – the system which gives foundation and justification of our work.

#### Second interlude: journals vis-à-vis combinatorics

I already wrote about the fate of combinatorics papers in the *Annals, *especially when comparison with Number Theory. My view was gloomy but mildly optimistic. In fact, since that post was written couple more combinatorics papers has been accepted. Good. But let me give you a quiz. Here are two comparable highly selective journals – *Duke J. Math.* and *Composito Math. *In the past 10 years *Composito *published exactly **one** (!) paper in Combinatorics (defined as primary MSC=05), of the 631 total. In the same period, *Duke* published **8** combinatorics papers of 681 total.

**Q:** Which of the two (*Composito* or *Duke*) treats combinatorics papers better?

**A: ***Composito*, of course.

The reasoning is simple. Forget the anecdotal evidence in the previous interlude. Just look at the “aim and scope” of the journals vs. these numbers. Here is what *Compsito* website says with a refreshing honesty:

By tradition, the journal published by the foundation focuses on papers in the main stream of pure mathematics. This includes the fields of algebra, number theory, topology, algebraic and analytic geometry and (geometric) analysis. Papers on other topics are welcome if they are of interest not only to specialists.

Translation: combinatorics papers are not welcome (as are papers in many other fields). I think this is totally fair. Nothing wrong with that. Clearly, there are journals which publish mostly in combinatorics, and where papers in none of these fields would be welcome. In fact there is a good historical reason for that. Compare this with what *Duke* says on its website:

Published by Duke University Press since its inception in 1935, the

Duke Mathematical Journalis one of the world’s leading mathematical journals. Without specializing in a small number of subject areas, it emphasizes the most active and influential areas of current mathematics.

See the difference? They don’t name their favorite areas! How are the authors supposed to guess which are these? Clearly, Combinatorics with its puny 1% proportion of Duke papers is not a subject area that Duke “emphasizes”. Compare it with 104 papers in Number Theory (16%) and 124 papers in Algebraic Geometry (20%) over the same period. Should we conclude that in the past 10 years, Combinatorics was not “the most active and influential”, or perhaps not “mathematics” at all? (yes, some people think so) I have my own answer to this question, and I bet so do you…

Note also, that things used to be different at *Duke*. For example, exactly 40 years earlier, in the period 1963-1973, *Duke* published 47 papers in combinatorics out of 972 total, even though the area was only in its first stages of development. How come? The reason is simple: Leonard Carlitz was Managing Editor at the time, and he welcomed papers from a number of prominent combinatorialists active during that time, such as Andrews, Gould, Moon, Riordan, Stanley, Subbarao, etc., as well as a many of his own papers.

#### So, ideally, what will happen to math journals?

That’s actually easy. Here are my few recommendations and predictions.

1) We should stop with all these geography based journals. That’s enough. I understand the temptation for each country, or university, or geographical entity to have its own math journal, but nowadays this is counterproductive and a cause for humor. This parochial patriotism is perhaps useful in sports (or not), but is nonsense in mathematics. New journals should emphasize new/rapidly growing areas of mathematics underserved by current journals, not new locales where printing presses are available.

2) Existing for profit publishers should realize that with the growth of arXiv and free online competitors, their business model is unsustainable. Eventually all these journals will reorganize into a non-profit institutions or foundations. This does not mean that the journals will become electronic or free. While some probably will, others will remain expensive, have many paid employees (including editors), and will continue to provide services to the authors, all supported by library subscriptions. These extra services are their raison d’être, and will need to be broadly advertised. The authors would learn not to be surprised of a quick one line report from free journals, and expect a serious effort from “expensive journals”.

3) The journals will need to rethink their structure and scope, and try to develop their unique culture and identity. If you have two similar looking free electronic journals, which do not add anything to the papers other than their .sty file, the difference is only the editorial board and history of published papers. This is not enough. All journals, except for the very top few, will have to start limiting their scope to emphasize the areas of their strength, and be honest and clear in advertising these areas. Alternatively, other journals will need to reorganize and split their editorial board into clearly defined fields. Think *Proc. LMS,* *Trans. AMS, *or a brand new* Sigma*, which basically operate as dozens of independent journals, with one to three handling editors in each. While highly efficient, in a long run this strategy is also unsustainable as it leads to general confusion and divergence in the quality of these sub-journals.

4) Even among the top mathematicians, there is plenty of confusion on the quality of existing mathematics journals, some of which go back many decades. See e.g. a section of Tim Gowers’s post about his views of the quality of various Combinatorics journals, since then helpfully updated and corrected. But at least those of us who have been in the area for a while, have the memory of the fortune of previously submitted papers, whether our own, or our students, or colleagues. A circumstantial evidence is better than nothing. For the newcomers or outsiders, such distinctions between journals are a mystery. The occasional rankings (impact factor or this, whatever this is) are more confusing than helpful.

What needs to happen is a new system of awards recognizing achievements of individual journals and/or editors, in their efforts to improve the quality of the journals, attracting top papers in the field, arranging fast refereeing, etc. Think a mixture of Pulitzer Prize and J.D. Power and Associates awards – these would be a great help to understand the quality of the journals. For example, the editors of the *Annals* clearly hustled to referee within a month in this case (even if motivated by PR purposes). It’s an amazing speed for a technical 50+ page paper, and this effort deserves recognition.

**Full disclosure:** Of the journals I singled out, I have published once in both *JAMS* and *Duke. *Neither paper is in Combinatorics, but both are in Discrete Mathematics, when understood broadly.

## What’s the Matter with Hertz Foundation?

Imagine you have plenty of money and dozens of volunteers. You decide to award one or two fellowships a year to the best of the best of the best in math sciences. Easy, right? Then how do you repeatedly fail at this, without anyone notice? Let me tell you how. It’s an interesting story, so bear with me.

**A small warning**. Although it may seem I am criticizing Hertz Foundation, my intention is to show its weakness so it can improve.

#### What *is* the Hertz Foundation?

Yesterday I wrote a recommendation letter to the Hertz Foundation. Although a Fellow myself, I never particularly cared for the foundation, mostly because it changed so little in my life (I received it only for two out of five years of eligibility). But I became rather curious as to what usually happens to Hertz Fellows. I compiled the data, and found the results quite disheartening. While perhaps excellent in other fields, I came to believe that Hertz does barely a mediocre job awarding fellowships in mathematics. And now that I think about it, this was all completely predictable.

First, a bit of history. John Hertz was the Yellow Cab founder and car rental entrepreneur (thus the namesake company), and he left a lot of money dedicated for education in “applied physical sciences”, now understood to include applied mathematics. What exactly* is* “applied mathematics” is rather contentious, so the foundation wisely decided that “it is up to each fellowship applicant to advocate to us his or her specific field of interest as an ‘applied physical science’.”

In practice, according to the website, about 600 applicants in all areas of science and engineering apply for a fellowship. Applications are allowed only either in the senior year of college or 1st year of grad school. The fellowships are generous and include both the stipend and the tuition; between 15 and 20 students are awarded every year. Only US citizen and permanent residents are eligible, and the fellowship can be used only in one of the 47 “tenable schools” (more on this below). The Foundation sorts the applications, and volunteers interview some of them in the first round. In the second round, pretty much only one person interviews all that advanced, and the decision is made. Historically, only one or two fellowships in mathematical sciences are awarded each year (this includes pure math, applied math, and occasionally theoretical CS or statistics).

#### Forty years of Math Hertz Fellowships in numbers

The Hertz Foundation website has a data on all past fellows. I compiled the data in Hertz-list which spanned 40 years (1971-2010), listed by the year the fellowship ended, which usually but not always coincided with graduation. There were 67 awardees in mathematics, which makes it about 1.7 fellowships a year. The Foundation states that it awarded “over 1000 fellowships” so I guess about 5-6% went into maths (perhaps, fewer in recent years). Here is who gets them.

**1)** **Which schools are awarded?** Well, only 44 US graduate programs are allowed to administer the fellowships. The reasons (other than logistical) are unclear to me. Of those programs that are “in”, you have University of Rochester (which nearly lost its graduate program), and UC Santa Cruz (where rumors say a similar move had been considered). Those which are “out” include graduate programs at Brown, UPenn, Rutgers, UNC Chapel Hill, etc. The real distribution is much more skewed, of course. Here is a complete list of awards per institution:

MIT – 14

Harvard, Princeton – 8

Caltech, NYU – 7

Berkeley, Stanford – 5

UCLA – 3

CMU, Cornell, U Chicago – 2

GA Tech, JHU, RPI, Rice – 1

In summary, only 15 universities had at least one award (34%), and just 7 universities were awarded 54 fellowships (*i.e.* 16% of universities received 81% of all fellowships). There is nothing wrong with this *per se*, just a variation on the 80-20 rule you might argue. But wait! Hertz Foundation is a non-profit institution and the fellowship itself comes with a “moral commitment“. Even if you need to interfere with “free marketplace” of acceptance decisions (see P.S. below), wouldn’t it be in the spirit of John Hertz’s original goal, to make a special effort to distribute the awards more widely? For example, Simons Foundation is not shy about awarding fellowship to institutions many of which are not even on Hertz’s list.

**2)** ** Where are they now?** After two hours of googling, I located almost all former fellows and determined their current affiliations (see the Hertz-list). I found that of the 67 fellows:

University mathematicians – 27 (40%)

Of these, work at Hertz eligible universities – 14, or about 21% of the total (excluding 3 overseas)

At least 10 who did not receive a Ph.D. – 15%

At least 13 are in non-academic research – 19% (probably more)

At least 8 in Software Development and Finance – 12% (probably more)

Now, there is certainly nothing wrong with directing corporate research, writing software, selling derivatives, designing museum exhibits, and even playing symphony orchestra or heading real estate company, as some of the awardees do now. Many of these are highly desirable vocations. But really, was this what Hertz had in mind when dedicating the money? In the foundation’s language, “benefit us all” they don’t.

I should mention that the list of Hertz Fellows in Mathematics does include a number of great academic success stories, but that’s not actually surprising. Every US cohort has dozens of excellent mathematicians. But the 60% drop out rate from academia is very unfortunate, only 21% working in “tenable universities” is dismaying, and the 15% drop out rate from graduate programs is simply miserable. Couldn’t they have done better?

#### A bit of analysis

Every year, US universities award over 1,600 Ph.D.’s in mathematical sciences, of which over a half go to US citizen (more if you include permanent residents, but stats is not easily available). So they are choosing 1.7 out of over 800 eligible students. Ok, because of their “tenable schools” restriction this is probably more like 300-400. Therefore, less than half of one percent of potential applicants are awarded! For comparison, Harvard college acceptance rate is 10 times that.

Let me repeat: in mathematics, Hertz fellows drop out from their Ph.D. programs at a rate of 15%. If you look into the raw 2006 NRC data for graduation rates, you will see that many of the top universities have over 90% graduation rate in the math programs (say, Harvard has over 91%). Does that mean that Harvard on average does a better job selecting 10-15 grad students every year, while Hertz can’t choose one?

Yes, I think it does. And the gap is further considering that Hertz has virtually no competition (NSF Fellowships are less generous in every respect). You see, people at Harvard (or Princeton, MIT, UCLA, etc.) who read graduate applications, know what they are doing. They are professionals who are looking for the most talented mathematicians from a large pool of applicants. They know which letters need to be taken seriously, and which with a grain of salt. They know which undergraduate research experience is solid and which is worthless. They just know how things are done.

Now, a vast majority of Hertz interviewers are themselves former fellows, and thus about 95% of them have no idea about the mathematics research (they just assume it’s no different from the research they are accustomed to). Nor does the one final interviewer, who is an applied physicist. As a result, they are to some extend, flipping coins and rolling dies, in hope things will work out. You can’t really blame them – they simply don’t know *how to choose*. I still remember my own two interviews. Both interviewers were nice, professional, highly experienced and well intentioned, but looking back I can see that neither had much experience with mathematical research.

You can also see this lack of understanding of mathematics culture is creeping up in other activities of the foundation, such as the thesis prize award (where are mathematicians?), etc. Of course a private foundation can award anyone it pleases, but it seems to me it would do much more good if only some special care is applied.

**A** modest** proposal**

There is of course, a radical way to change the review of mathematics applicants – subcontract it to the AMS (or IMA, MSRI, IPAM – all have the required infrastructure). For a modest fee, the AMS will organize a panel of mathematicians who will review and rank the applicants without interviewing them. The panel will be taking into consideration only students’ research potential, not the university prestige, etc. The Hertz people can then interview the top ranked and make a decision at the last stage, but the first round will be by far superior to current methods. Even the NSA trusts AMS, so shouldn’t you?

Hertz might even save some money it currently spends on travel and lodging reimbursements. The 13% operating budget is about average, but there is some room for improvement. Subcontracting will probably lead to an increase in applications, as AMS really knows how to advertise to its members (I bet you currently receive only about 40 mathematics applications, out of a potential 400+ pool). To summarize: *really*, Hertz Foundation, think about doing that!

**P.S.** It is not surprising that the 7 top universities get a large number of the fellowships. One might be tempted to assume that clueless interviewers are perhaps somewhat biased towards famous school names in the hope that these schools already made a good decision accepting these applicants, but this is not the whole story. The described bias can only work for the 1st year grad applicants, but for undergraduate applicants a different process seems to hold. Once a graduate school learns that an applicant received Hertz Fellowship (or NSF for that matter), it has every incentive to accept the student, as the tuition and the stipend are paid by the outside sources now.

**P.P.S.** Of course, mathematicians’ review can also fail. Even the super prestigious AIM Fellowship has at least one recipient who left academia for bigger and better things.