T-Duality, Mirror Symmetry, Dedekind-Weil-Langlands .Rosetta Stone, Ising Model, Automorphy, Galois, Automorphic Representations

If we look at current work on Mirror Symmetry in the context of quantum field theory and string theory, and in the context of mathematics, there are a number of suggestive analogies with the solutions to the two-dimensional Ising model. There are very many approaches, and they reveal different aspects of the Ising system.

1. If T-Duality is taken as (m,n) and R becoming (n,m) and 1/R, it may be useful to look at duality in the Ising lattice: (K,K') and k go to (K'*, K*) and 1/k, connecting high and low temperature lattices, rotating the lattice pi/2.  k=1/sinh2Ksinh2K' and K'* is defined by sinh2K'sinh2K'*=1.

2. The Dedekind-Weil-Langlands triplet of Riemannian (analysis, automorphy), Italian (function theory), and German (arithmetic number theory) nicely classifies the various ways of solving the Ising problem: Yang-Baxter, functional equations and automorphy, and adding up by means of transfer matrices or graph counting. As for the Langlands program connecting Galois groups to automorphic representations, perhaps think in terms of the physicists' partition function and its also being automorphic, and that the transfer matrix might well be a group representation (I am unsure of just what), its characters or traces, being the actual partition function (the thermodynamic free energy is the -1/kBT ln Partition-Function = 1/kBT ln tr Transfer-Matrix = 1/kBT ln Lamda-max, where Lambda is the eigenvalue of the Tranfer-Matrix. Note as well that the Ising pseudo-Hamiltonian, in Onsager, A+B/k, where if k is small (low T, B dominates and is a potential energy, and if k is large A dominates and is a kinetic energy, much as we might have for the harmonic oscillator, as is the case in the T-duality system.

3. None of this denies the value of the electromagnetic duality that inspires much of Mirror Symmetry, but provides a rather simpler model.

4. The Big Questions are:
Why is there regularity here, here being crystalline solids, quadratic reciprocity, in the congruences and the distribution of primes, in the number of solutions to an equation modulo p, How does counting lead to scaling? In effect, what makes harmonic analysis so powerful.

How does the regularity encoded in the Galois group, permutations essentially, become the regularity encoded in automorphic forms? We do know that in the asymptotic realm, sums of reasonably nice random variables have asymptotic form (the Gaussian) and scale nicely at the square root of N.

We know, by the way, that the counting of eigenvalues or the spectrum of a differential operator gives information about areas, volumes, kinetic energy sums. Weyl asymptotics

And we know that particles and fields are intimately connected.
Why are group representations so powerful? For physicists, they allow calculation of numbers!

There are two sorts of non-answers to these questions. Kant would say that we cannot have knowledge of the ultimate source, but we surely can go deeper and deeper. Husserl would say that what we are learning are various aspects of a phenomenon, an identity in a manifold presentation of profiles, when we have many way of getting at something. In any case, questions about why mathematics and physics are so useful to each other, demand historical answers, not in the end something mysterious.

Having many layers and aspects will often lead us to understand something and see it at self-evident, as described by Gian-Carlo Rota in his discussions of mathematics and beauty and evidence.

Two-Years' Work/Article

1. I believe that the popularity of the three-article equivalent in economics has to do with the nature of the research most economists do.  The articles do not need to be published, but the advisor needs to be sure they are "publishable." In that field, there is a sense that they know this, and also that they can say if the article would be in an A, B, or whatever journal.

2. Planning is much more diverse a field. Public policy, too. It would be good if all articles that were submitted to journals represented at least one person-year of full-time work. I am not sure that is the case now. Maybe it should be two person-years.  

(Note that in some fields, such as biology and medicine, it would seem that there are many more articles published/person, often they are smaller contributions, and the number of authors is likely more than three. On the other hand, in mathematics, the articles are fewer, they are more extensive, and have one or two authors, and this is for the strongest people.)

3. My point here is that research in our field would benefit from more depth, more long term case-studies, more ... In other words, the problem with the three-article dissertation is that it encourages a practice by our colleagues (junior and tenured) that does not produce the work that will transform knowledge and practice.  

4. The issue is not whether the dissertation is a book or an article, or even its actual length. It's a matter of learning how to do serious deep research. What it also means is that getting a PhD would be a matter of total devotion to one's research project for perhaps two or three years. It's not a part-time job, or one that is balanced by doing lots of other stuff. 

Now this goal may be impossible, given the level of support we can provide, the limited teaching and research assistantships we have, etc. I went to graduate school in a time, the 1960s, when students were supported throughout their studies, but this was in a field where there was such support. And support was much weaker in earlier decades, and perhaps subsequently.

It may be that we are giving people training in doing research, and only later do they have a chance to do serious work.

Grant Giving is a matter of allying yourself with your best bets.

From the point of view of applicants for research grants, they are competing to be one of the chosen. But from the point of view of the foundation or grants officer or investor, your problem is to choose to give your resources to:
1. Recipients who will deliver on what the promise. And what they promise is OK.
2. Long-shots that may pay off, or may not. 
3. People whose work has proved valuable and you are trying to have them be part of your stable of grantees. You want their work allied with your program or foundation. Not a matter of agreeing with it, so much as sharing in its prestige and influence.

Surely you want reliable recipients, and you may want to take some risks that you know are risky. But what you want as well is to support those who are likely to be very strong, and there you are competing with other funders--and it's not just a matter of dollars, it may be a matter of how you encourage them. Similar problems come up in venture capital, where #1 and #2 are nicely addressed by finance, but #3 is a very different sort.

Understanding something.

I am revising my 2003 book (Doing Mathematics) for a second edition. There was lots of stuff I did not understand then, and I still don't understand, not to speak of the new stuff I don't understand. But...

An article I must have read 5 times, finally made lots of sense, at least 4/5 of it.

I discovered a nice way of putting much of it together. Right now it is in a three or five column chart, and I know that some of the stuff belongs in different columns. But I can fix it.

I finally figured out a way of describing something in everyday terms. I looked at the first edition, and I more or less said the same thing--but now I can be clearer, I hope.

There were various places where I had elided over difficulties. I did not realize I was eliding, for I did not appreciate the deeper issues. Now I can insert better explanations and details.

[Note that I am willing to believe that all my changes will not be much noticed by most if not all readers. But scholarly authors are usually an authority on what they are saying. Still, they might well be found out by a reviewer or a competitor. But often not!]

There are some bloopers in the first edition, but fewer than I was anticipating. I suspect that the prospect of going over something you wrote a dozen years ago makes one both vigilant and apprehensive. It's always better than I remembered, and surely there are places where the elisions are now realized, the mistakes now standing out.

My experience is that I understand something because I eventually find a perspective on that something that mirrors something I understand already. Not exactly, but close enough. What I need to do is to keep thinking about what I do not understand (better, a few things I do not understand), do lots of reading, and perhaps I will encounter an exposition that opens my eyes. I'd like to ask a real expert, but I am not sure how to do so: namely, what I want are the keys to the kingdom, and those are supposed to be earned by advanced degrees and having a member of the club show you the key closet. Club members may be pleased to be of help, but they don't know that they might know what I need. One day, I might be able to ask the right questions, and they respond readily and without pretense, and let me know what's up. But asking the right question only is clear after you know the secret answer.

Put differently, when I reread stuff I have written, I am often surprised by the quality of my understanding, it's that good. Not all the time, but often enough. I forget all the work I did then so that I could figure out what was going on.

As far as I can tell, this act of understanding something cuts across the disciplines, the sciences, the humanities, social sciences, the professional schools. Every field is esoteric to outsiders. Some are more forbidding, but it all depends on where you feel most vulnerable.

Building at the margin on the positive-sum outcomes you are already generating...

One description of some of MIT's Judith Tendler's recent fieldwork concerning regulation (it is a commonplace that regulation never works, is inefficient, etc., although the evidence is rather more complex) in Brazil says, These grim expectations in themselves often contribute to regulatory inaction. This research project, therefore, asked how could regulatory actors and others build at the margin on the positive-sum outcomes they were already generating, by observing the  patterns running across them.  

The big point here is that there are some places where things are working better than most others. Look at them and learn from them. 

In personal life, despite all the nonsense, you actually do some good stuff. Pay attention to that good stuff and build on it.  

Book Fields and Salary Increments

In so far as we are asked annually about our research, those who write big books, or even more modest books, don't fit. In book fields, this may not be a problem. The idea that someone would work for 5-7 years (or twice that in some cases) on a project, and then have a book come out does not quite fit with the idea of two research articles/year in many fields. Or monitoring research productivity.  Yes, I know people should publish articles about their project as they go along. But what you want to encourage is deep difficult work. Now of course we know of promised books that never appear, and we know of people who produce books rabbit-like. I don't know what to do with the barren or the fructiferous. But for most people, the way the university works, the book writers have a difficult time.

As far as I recall, in promotion and tenure and UCAPT, the book fields are well recognized and two articles/year is never brought up. (Book fields might be defined by what scholars do at peer or better institutions.)

In fields where books or articles are considered, then the one book every 5-7 or more years person might well not get raises when the book appears to compensate for what they did not receive in previous years when their two article/year colleagues were rewarded (but I don’t know if this happens).

The other problem is that all universities now publicize “findings,” and much of the humanities and some of the social sciences, do not have findings or discoveries. I do believe that there is a very useful kind of university publicizing: scholars and scientists, and what they are doing, with a focus on the ideas and problems rather than on the person, per se.

As for part time faculty and a dual labor market, I suspect that there was a short period when there were fewer part-time etc. faculty, maybe 1955-1975, but before or after we are back to the not-so-good old days.  [In Judith Shklar’s address about the life of a scholar, she reminds us of her experience (although she tells us it gave her time to write and bring up her children—no meetings, committees!—and that she felt she was not majorized by her colleagues in terms of their excellence).]

Things I Have Not Done, But Ought To Have Done

1. Choose an advisor who wants to work with you.

2. If you are not made for a research career--something you discover as you are doing your dissertation work--acknowledge that, and find another path. Finish the degree if that is not too onerous. Better to have at least one PhD.

3. Publish your dissertation research. Now!

4. If you have a book in process, get the book out, now, not five years from now. The same for articles.

5. If you have a success (say an article published in a top journal), publish more in that area immediately.

6. Wherever you work, you are not above it all. Be enthusiastic, surely in public.

7. You can change your position when you have something more suitable. You may have to build up your portfolio, where you are now, if someone is to be further interested in you elsewhere.

8.  Your colleagues will be suspicious if you publish outside the field, even if you publish well and successfully in the field. You are not being 100% loyal. If you plan to publish outside the field, you may have to use a pseudonym.
 
9. There is likely to be a number of smartasses or grumps in any audience. You must engage them substantively. 

10. What you want is a job, not a grant. If you have a job, you want a grant. Grants without real jobs will not be well received.

11. Produce doctoral students and make sure they get good jobs.

12. It's nice if somebody up there likes you, but it's better if the folks down here like you.

Evaluating Public Policies and Programs: Marketing, Finance, and Going to Look

1. What to look for: There is always the idea that if you can't measure it, you are not doing it right. In much of public policy, you try for various measures, but you know as well that they are proxies, and they may not be reliable or even accurate. So what do you do? What you want to do is to do fieldwork, in effect it is sophisticated theory-informed journalism. You interview people, you use some survey instruments (but this often is unhelpful). You find out what is happening. In other words, whether it be in health, education, housing, you have to find out what is happening as a consequence of your policies, and then you might be able to figure out how to measure it. If you don't know what to look for, you are out of business. Some of the time you are doing exploratory research, and you will see surprises. And it's not a matter of quantitative stuff, per se. It's phenomena you had not expected.

Reliability. People worry about objectivity or about bias. But looking at the world is more accurate than at some data set whose connection with the world is dubious. The problem is not measureable, it is to figure out what is going on and then you might want to measure it. Preconceived ideas about what is going on are typically artifacts of fantasies.

a. What have we done in the past? 

b. Sociological or economist fieldworkers to go out and tell you what is going on.

c. I assume there is a historical literature. 

A wide range of indicators and methods.

Examples.  a. Robert Sampson (Chicago, now Harvard) just did a book on neighborhood effects in cities, and surely there were dicey neighborhoods. But they drove up and down the streets and photographed the facades of buildings. Sampson is a sociologist, not a public policy person, but his work is central to public policy.

b.  Judith Tendler, an economist at MIT, a former student of Albert Hirschman, thinks like an economist and acts like a social fieldwork researcher. (Michael Piore, of the Economics Dept at MIT, now retired, is similar.) What she is looking for are places where things work. The idea is that social scientists are excellent at figuring out what won't work, they are skeptics at heart. But you can be committed to reliable research, but focus on what works or best practices. Of course, it's not obvious they can be replicated, but at least you are focusing your efforts. Rather than the so called "gold standard" one hears about in medical research, here you are more like a market researcher trying to find out how to tune the product so that it is more likely you have what people want.

In sum. You don't want to act like evaluation researchers, nor like conventional social scientists. You want to act much more inquiringly, trying to see what happens. Once you can tell others what happens then the agency can help you and themselves figure out if this is what they want. Put differently, people are much more inventive than our hypotheses allow. We have to find out what they do with what we put out. 

2. Finance theory offers a set of ideas about risk and uncertainty: probability, variances, fat tails, value at risk, correlation of actors leading to surprising events. The world is not a random walk, or a geometric random walk, but that might be a good place to begin. In any case, be a Bayesian, do some simulations, and get your decision theory and statistician friends to help you think about potential policies and likely glitches. It's not about quantitative stuff, it's about organizing your thinking about a future that is surely uncertain, but you can get a feel for how to bound that uncertainty, and how to insure so that awful things are less likely, consequences are more modest, and losses are compensated.  Much of this insurance is self-insurance, about risk management, about not being too stupid. And you can buy flexibility in the future through real options. Still, there will be residual uncertainties and you can be vigilant for surprises and be imaginative about those uncertainties--this is not a solution, but it gives you an active stance.

Portfolio notions encourage you to have a range of actions, so that you are less vulnerable to a single line of action not working out.

3. Marketing is having/selling what people want. So you have to do lots of consumer research, test marketing, pilots, so that your new products are less likely to flop. You may have limited resources, you may have limited time, but focus groups, competitive analysis (Red Teaming), and backup plans can make it less likely to do not have the capacity to respond to glitches. 

4. Creativity  You train people, you give them resources, and you hope that some will invent. You can make the resources you give people be like venture capital.  I usually run for the fallout shelter when people talk to me about "creativity," the creative class, and other such. In my experience, maybe 10% of faculty or researchers are "creative." But lots are systematic, interesting, careful, disciplined, and that matters a lot. Maybe they are all "creative" at Google and Goldman Sachs and Pixar and Harvard, and at MIT, Stanford, and ...,

Gian-Carlo Rota on Academic Life

Gian-Carlo Rota

MIT, April 20 , 1996 on the occasion of the Rotafest

Allow me to begin by allaying one of your worries. I will not spend the next half hour thanking you for participating in this conference, or for your taking time away from work to travel to Cambridge.

And to allay another of your probable worries, let me add that you are not about to be subjected to a recollection of past events similar to the ones I've been publishing for some years, with a straight face and an occasional embellishment of reality.

Having discarded these two choices for this talk, I was left without a title. Luckily I remembered an MIT colloquium that took place in the late fifties; it was one of the first I attended at MIT. The speaker was Eugenio Calabi. Sitting in the front row of the audience were Norbert Wiener, asleep as usual until the time came to applaud, and Dirk Struik who had been one of Calabi's teachers when Calabi was an undergraduate at MIT in the forties. The subject of the lecture was beyond my competence. After the first five minutes I was completely lost. At the end of the lecture, an arcane dialogue took place between the speaker and some members of the audience, Ambrose and Singer if I remember correctly. There followed a period of tense silence. Professor Struik broke the ice. He raised his hand and said: "Give us something to take home!" Calabi obliged, and in the next five minutes he explained in beautiful simple terms the gist of his lecture. Everybody filed out with a feeling of satisfaction.

Dirk Struik was right: a speaker should try to give his audience something they can take home. But what? I have been collecting some random bits of advice that I keep repeating to myself, do's and don'ts of which I have been and will always be guilty. Some of you have been exposed to one or more of these tidbits. Collecting these items and presenting them in one speech may be one of the less obnoxious among options of equal presumptuousness. The advice we give others is the advice that we ourselves need. Since it is too late for me to learn these lessons, I will discharge my unfulfilled duty by dishing them out to you. They will be stated in order of increasing controversiality. 

1. Lecturing
2. Blackboard Technique
3. Publish the same results several times.
4. You are more likely to be remembered by your expository work.
5. Every mathematician has only a few tricks.
6. Do not worry about your mistakes.
7. Use the Feynmann method.
8. Give lavish acknowledgments.
9. Write informative introductions
10. Be prepared for old age.

 1 Lecturing    top

The following four requirements of a good lecture do not seem to be altogether obvious, judging from the mathematics lectures I have been listening to for the past forty-six years.

a. Every lecture should make only one main point The German philosopher G. W. F. Hegel wrote that any philosopher who uses the word "and" too often cannot be a good philosopher. I think he was right, at least insofar as lecturing goes. Every lecture should state one main point and repeat it over and over, like a theme with variations. An audience is like a herd of cows, moving slowly in the direction they are being driven towards. If we make one point, we have a good chance that the audience will take the right direction; if we make several points, then the cows will scatter all over the field. The audience will lose interest and everyone will go back to the thoughts they interrupted in order to come to our lecture.

b. Never run overtime Running overtime is the one unforgivable error a lecturer can make. After fifty minutes (one microcentury as von Neumann used to say) everybody's attention will turn elsewhere even if we are trying to prove the Riemann hypothesis. One minute overtime can destroy the best of lectures.

c. Relate to your audience As you enter the lecture hall, try to spot someone in the audience with whose work you have some familiarity. Quickly rearrange your presentation so as to manage to mention some of that person's work. In this way, you will guarantee that at least one person will follow with rapt attention, and you will make a friend to boot.

Everyone in the audience has come to listen to your lecture with the secret hope of hearing their work mentioned.

d. Give them something to take home It is not easy to follow Professor Struik's advice. It is easier to state what features of a lecture the audience will always remember, and the answer is not pretty. I often meet, in airports, in the street and occasionally in embarrassing situations, MIT alumni who have taken one or more courses from me. Most of the time they admit that they have forgotten the subject of the course, and all the mathematics I thought I had taught them. However, they will gladly recall some joke, some anecdote, some quirk, some side remark, or some mistake I made.

 

 2 Blackboard Technique    top

Two points.

a. Make sure the blackboard is spotless It is particularly important to erase those distracting whirls that are left when we run the eraser over the blackboard in a non uniform fashion.

By starting with a spotless blackboard, you will subtly convey the impression that the lecture they are about to hear is equally spotless.

b. Start writing on the top left hand corner What we write on the blackboard should correspond to what we want an attentive listener to take down in his notebook. It is preferable to write slowly and in a large handwriting, with no abbreviations. Those members of the audience who are taking notes are doing us a favor, and it is up to us to help them with their copying. When slides are used instead of the blackboard, the speaker should spend some time explaining each slide, preferably by adding sentences that are inessential, repetitive or superfluous, so as to allow any member of the audience time to copy our slide. We all fall prey to the illusion that a listener will find the time to read the copy of the slides we hand them after the lecture. This is wishful thinking.

3 Publish the same result several times    top

After getting my degree, I worked for a few years in functional analysis. I bought a copy of Frederick Riesz' Collected Papers as soon as the big thick heavy oversize volume was published. However, as I began to leaf through, I could not help but notice that the pages were extra thick, almost like cardboard. Strangely, each of Riesz' publications had been reset in exceptionally large type. I was fond of Riesz' papers, which were invariably beautifully written and gave the reader a feeling of definitiveness.

As I looked through his Collected Papers however, another picture emerged. The editors had gone out of their way to publish every little scrap Riesz had ever published. It was clear that Riesz' publications were few. What is more surprising is that the papers had been published several times. Riesz would publish the first rough version of an idea in some obscure Hungarian journal. A few years later, he would send a series of notes to the French Academy's Comptes Rendus in which the same material was further elaborated. A few more years would pass, and he would publish the definitive paper, either in French or in English. Adam Koranyi, who took courses with Frederick Riesz, told me that Riesz would lecture on the same subject year after year, while meditating on the definitive version to be written. No wonder the final version was perfect.

Riesz' example is worth following. The mathematical community is split into small groups, each one with its own customs, notation and terminology. It may soon be indispensable to present the same result in several versions, each one accessible to a specific group; the price one might have to pay otherwise is to have our work rediscovered by someone who uses a different language and notation, and who will rightly claim it as his own.

4 You are more likely to be remembered by your expository work    top

Let us look at two examples, beginning with Hilbert. When we think of Hilbert, we think of a few of his great theorems, like his basis theorem. But Hilbert's name is more often remembered for his work in number theory, his Zahlbericht, his book Foundations of Geometry and for his text on integral equations. The term "Hilbertspace" was introduced by Stone and von Neumann in recognition of Hilbert's textbook on integral equations, in which the word "spectrum" was first defined at least twenty years before the discovery of quantum mechanics. Hilbert's textbook on integral equations is in large part expository, leaning on the work of Hellinger and several other mathematicians whose names are now forgotten.

Similarly, Hilbert's Foundations of Geometry, the book that made Hilbert's name a household word among mathematicians, contains little original work, and reaps the harvest of the work of several geometers, such as Kohn, Schur (not the Schur you have heard of), Wiener (another Wiener), Pasch, Pieri and several other Italians.

Again, Hilbert's Zahlbericht, a fundamental contribution that revolutionized the field of number theory, was originally a survey that Hilbert was commissioned to write for publication in the Bulletin ofthe German Mathematical Society.

William Feller is another example. Feller is remembered as the author of the most successful treatise on probability ever written. Few probabilists of our day are able to cite more than a couple of Feller's research papers; most mathematicians are not even aware that Feller had a previous life in convex geometry.

Allow me to digress with a personal reminiscence. I sometimes publish in a branch of philosophy called phenomenology. After publishing my first paper in this subject, I felt deeply hurt when, at a meeting of the Society for Phenomenology and Existential Philosophy, I was rudely told in no uncertain terms that everything I wrote in my paper was well known. This scenario occurred more than once, and I was eventually forced to reconsider my publishing standards in phenomenology.

It so happens that the fundamental treatises of phenomenology are written in thick, heavy philosophical German. Tradition demands that no examples ever be given of what one is talking about. One day I decided, not without serious misgivings, to publish a paper that was essentially an updating of some paragraphs from a book by Edmund Husserl, with a few examples added. While I was waiting for the worst at the next meeting of the Society for Phenomenology and Existential Philosophy, a prominent phenomenologist rushed towards me with a smile on his face. He was full of praise for my paper, and he strongly encouraged me to further develop the novel and original ideas presented in it.

5 Every mathematician has only a few tricks    top

A long time ago an older and well known number theorist made some disparaging remarks about Paul Erdos' work. You admire contributions to mathematics as much as I do, and I felt annoyed when the older mathematician flatly and definitively stated that all of Erdos' work could be reduced to a few tricks which Erdos repeatedly relied on in his proofs. What the number theorist did not realize is that other mathematicians, even the very best, also rely on a few tricks which they use over and over. Take Hilbert. The second volume of Hilbert's collected papers contains Hilbert's papers in invariant theory. I have made a point of reading some of these papers with care. It is sad to note that some of Hilbert's beautiful results have been completely forgotten. But on reading the proofs of Hilbert's striking and deep theorems in invariant theory, it was surprising to verify that Hilbert's proofs relied on the same few tricks. Even Hilbert had only a few tricks!

6 Do not worry about your mistakes    top

Once more let me begin with Hilbert. When the Germans were planning to publish Hilbert's collected papers and to present him with a set on the occasion of one of his later birthdays, they realized that they could not publish the papers in their original versions because they were full of errors, some of them quite serious. Thereupon they hired a young unemployed mathematician, Olga Taussky-Todd, to go over Hilbert's papers and correct all mistakes. Olga labored for three years; it turned out that all mistake scould be corrected without any major changes in the statement of the theorems. There was one exception, a paper Hilbert wrote in his old age, which could not be fixed; it was a purported proof of the continuum hypothesis, you will find it in a volume of the Mathematische Annalen of the early thirties. At last, on Hilbert's birthday, a freshly printed set of Hilbert's collected papers was presented to the Geheimrat. Hilbert leafed through them carefully and did not notice anything.

Now let us shift to the other end of the spectrum, and allow me to relate another personal anecdote. In the summer of 1979, while attending a philosophy meeting in Pittsburgh, I was struck with a case of detached retinas. Thanks to Joni's prompt intervention, I managed to be operated on in the nick of time and my eyesight was saved.

On the morning after the operation, while I was lying on a hospital bed with my eyes bandaged, Joni dropped in to visit. Since I was to remain in that Pittsburgh hospital for at least a week, we decided to write a paper. Joni fished a manuscript out of my suitcase, and I mentioned to her that the text had a few mistakes which she could help me fix.

There followed twenty minutes of silence while she went through the draft. "Why, it is all wrong!" she finally remarked in her youthful voice. She was right. Every statement in the manuscript had something wrong. Nevertheless, after laboring for a while, she managed to correct every mistake, and the paper was eventually published.

There are two kinds of mistakes. There are fatal mistakes that destroy a theory; but there are also contingent ones, which are useful in testing the stability of a theory.

7 Use the Feynman method    top

Richard Feynman was fond of giving the following advice on how to be a genius. You have to keep a dozen of your favorite problems constantly present in your mind, although by and large they will lay in a dormant state. Every time you hear or read a new trick or a new result, test it against each of your twelve problems to see whether it helps. Every once in a while there will be a hit, and people will say: "How did he do it? He must be a genius!"

8 Give lavish acknowledgments    top

I have always felt miffed after reading a paper in which I felt I was not being given proper credit, and it is safe to conjecture that the same happens to everyone else. One day, I tried an experiment. After writing a rather long paper, I began to draft a thorough bibliography. On the spur of the moment, I decided to cite a few papers which had nothing whatsoever to do with the content of my paper, to see what might happen.

Somewhat to my surprise, I received letters from two of the authors whose papers I believed were irrelevant to my article. Both letters were written in an emotionally charged tone. Each of the authors warmly congratulated me for being the first to acknowledge their contribution to the field.

9 Write informative introductions    top

Nowadays, reading a mathematics paper from top to bottom is a rare event. If we wish our paper to be read, we had better provide our prospective readers with strong motivation to do so. A lengthy introduction, summarizing the history of the subject, giving everybody his due, and perhaps enticingly outlining the content of the paper in a discursive manner, will go some of the way towards getting us a couple of readers.

As the editor of the journal Advances in Mathematics, I have often sent submitted papers back to the authors with the recommendation that they lengthen their introduction. On occasion I received by return mail a message from the author, stating that the same paper had been previously rejected by Annals of Mathematics because the introduction was already too long.

10 Be prepared for old age    top

My late friend Stan Ulam used to remark that his life was sharply divided into two halves. In the first half, he was always the youngest person in the group; in the second half, he was always the oldest. There was no transitional period.

I now realize how right he was. The etiquette of old age does not seem to have been written up, and we have to learn it the hard way. It depends on a basic realization, which takes time to adjust to. You must realize that, after reaching a certain age, you are no longer viewed as a person. You become an institution, and you are treated the way institutions are treated. You are expected to behave like a piece of period furniture, an architectural landmark, or an incunabulum.

It matters little whether you keep publishing or not. If your papers are no good, they will say, "What did you expect? He is a fixture!" and if an occasional paper of yours is found to be interesting, they will say, "What did you expect? He has been working at this all his life!" The only sensible response is to enjoy playing your newly-found role as an institution.