(Sigue de la entrada Aprenda un poco de inglés con… Gian-Carlo Rota (1/11))

1 Lecturing

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.

A continuación publicamos, en once entradas semanales, la conferencia de Gian-Carlo Rota en el Rotafest, organizado en su honor por su 64 aniversario.

Ten Lessons I wish I had been Taught

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.

How to make a marriage stable

by Marianne Freiberger in +plus magazine

We’ve always got our finger on the pulse here at Plus! This year’s Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel has been awarded for work closely related to something we covered in an article back in August. The Prize was announced this morning and the laureates are Alvin E. Roth of Harvard University and Harvard Business School and Lloyd S. Shapley of the University of California, Los Angeles.

The work they have been honoured for concerns matching problems: how do you best allocate students to universities, doctors to hospitals, or kidneys to transplant patients? Lloyd Shaply started investigating such problems in the 1950s, and substantially developed an area called cooperative game theory in the process. In 1962 he published a short paper together with the mathematician David Gale on pairwise matchings. They phrased their problem in terms of marriages: suppose you have a group of men and a group of women and you want to marry them off in a way that keeps everyone as happy as possible. A central concept here is that the matching should be stable: there should be no two people who prefer each other to the partners they actually got.

Shaply and Gale described a straight-forward algorithm for doing the matching and showed that it always leads to a stable outcome (this is the algorithm we looked at in our article Mixing doubles). They also showed that the algorithm leads to very different outcomes for the two groups, depending on how it is applied. If the women do the proposing and the men decide who to accept and who to reject, then the algorithm is optimal for the women: no other stable matching is better (from the women’s point of view) than the one given by the algorithm. The same is true for the men if they do the proposing.

Shaply and Gale’s work was theoretical, but in the 1980s Alvin Roth made the connection to real world applications. He investigated the National Resident Matching Program (NRMP), which had been introduced in the US to allocate medical graduates to hospitals and seemed to work rather well. Roth showed that the algorithm used by the program was closely related to that of Shaply and Gale and he conjectured that the reason why it worked was because it produced stable matches. He went off to investigate similar medical matching problems in the UK and found that stability really was the secret to success: algorithms that produced stable matches worked, while others didn’t.

Ironically the NRMP ran into problems caused by real-life marriages: as the number of female medical students grew there were more student couples, many of whom wanted to stay in the same area when they applied for internships in hospitals. The NRMP algorithm did not cope well with this demand. It had also been criticised because it favoured the hospitals, who in this case did the “proposing”, over the students. In 1995 Roth, together with Elliott Peranson, was drafted in to improve the algorithm.

Roth also noticed that the original algorithm could be manipulated by those who receive the “proposals”, in this case the students, by lying about their true preferences. He worked out exactly how such manipulation would function and benefit the student and made sure that the revised algorithm couldn’t be tampered with in this way. Computer experiments have shown that, in practice, the algorithm is equally robust when it comes to manipulation by the hospitals. The new method was implemented in 1997 and has worked well ever since.

Shapley and Gale’s work has delivered new insights into a whole range of problems, from matching kidneys to patients to internet auctions. “Even though these two researchers worked independently of one another, the combination of Shapley’s basic theory and Roth’s empirical investigations, experiments and practical design has generated a flourishing field of research and improved the performance of many markets,” says the Nobel Prize press release. “This year’s prize is awarded for an outstanding example of economic engineering.”

You can find out more in the public information document on the Nobel Prize website and in the previous article here on Plus.

Nota: Quien quiera ver el artículo original sobre emparejamientos, puede visitar este enlace. ¿Donde está publicado?  En el American Mathematical Monthly. Dejamos para las mentes inquietas buscar en qué tercio del JCR está.