"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."
###Lesson 1: Every lecture should only one main point. "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, [...]"
###Lesson 2: Never run overtime. "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."
###Lesson 3: Relate to your audience. "Everyone in the audience has come to listen to your lecture with the secret hope of hearing their work mentioned."
###Lesson 4: Give them something to take home. Most alumni of Rota have forgotten the technical contents of his lectures. "However, they will gladly recall some joke, some anecdote, some quirk, some side remark, or some mistake I made."
"By starting with a spotless blackboard you will subtly convey the impression that the lecture they are about to hear is equally spotless."
"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."
Two examples are given: David Hilbert and William Feller.
"Even Hilbert had only a few tricks!"
Rota describes that some friends of Hilbert wanted to compile a collection of his papers for his birthday. They had to employ a young mathematician who would correct the errors in his papers over the course of three years. "it turned out that all the mistakes could be corrected without any major changes in the statement of the theorems." A purported proof of the continuum hypothesis by Hilbert was the only exception to this.
"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!"
See Lesson 3.
"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 motviation to do so."
"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 furniture, an architectural landmark, or an incunabulum. [...] The only sensible response is to enjoy playing your newly found role as an institution."