*It beats getting stoned, and it'll keep you off the streets.*

— Serge Lang

When we wish to communicate we use a language of some kind and use one or more of the following, not necessarily related, skills: reading, writing, speaking, listening. The last of these is by far the most difficult and least efficient. It is the only one in which you have no control over the speed. It's true that a good lecturer will not only make noises, but will also draw symbols, words and pictures, and perhaps display them, in not too much quantity, and with not too much speed, on an overhead projector. A good lecturer will also interact with the audience, though such interaction is limited for all but the smallest audiences.

Editors and publishers are often reluctant to put out anything that's informal. So when you write you use quite a different style from when you talk. Or do you? Not if you're Serge Lang:

I have kept some of the style of original talks, when students are questioned or interrupt freely. ... Halmos once characterized this style as "vulgar" , and ... later ... the present talk [Prime Numbers] was turned down by the editor [of the *Monthly*] because of the spoken style. Well I like the spoken style, and I find it effective.

So, although it's a book, it's not for reading! It's for teachers, or better still, students, to present to (other) students. As Serge Lang says, they will find some talks easy, others difficult or impossible.

As you might expect, each "talk" contains much more than can be attempted in a 50-minute presentation. You'll want to know about the assumed background.

**Prime Numbers**: practically none.
**The abc Conjecture**: surprisingly little.
**Global Integration of Locally Integrable Vector Fields**: no knowledge of complex analysis is assumed. Some basic knowledge about partial derivatives is taken for granted. Fundamental use is made of rectangular paths.
**Approximation Theorems of Analysis**: hard to say. Nothing specific, but a considerable level of general mathematical sophistication. Some sort of an analysis course. As Lang says, the material provides opportunities for a whole series of talks.
**Bruhat-Tits Spaces**: "The notions and auxiliary theorems which we shall use are all standard from basic undergraduate courses in linear algebra and analysis."
**Harmonic and Symmetric Polynomials**: "Courses in linear algebra or more general algebra at the undergraduate level provide foundations."

There's a bibliography for each talk. Here are brief surveys of the topics covered and some snippets to whet your appetite and give you the flavor of the book.

**Prime Numbers.** This is the least unpredictable of the talks, but even here there are surprises. After the prime number theorem the Riemann hypothesis is stated as

which draws a comment from the audience that that's not the way it's usually stated. Read Lang's reply.

Concerning the infinitude of twin primes:

It is a famous problem of mathematics to answer this question. If you answer it, you'll make it in the history books.

Concerning the sufficiency of conditions for the Bateman-Horn conjecture:

... two professors in the audience said yes. Then I pointed to a student, and asked: "What do you say?" The student said "sure".
[after discussing the example T^{3} - T - 3]

So it's not so sure, is it? Don't do what you did, saying "sure" based on what someone else says. Use your own brains. If you don't know, or want to think about it, say so. You won't do it again?

**The abc Conjecture.** Lang is first concerned with the Mason-Stothers proof of the conjecture for polynomials. Then the connexion is made between the integer version and Fermat's last theorem. Finally,

I had occasion to talk on the phone with a high school senior, Noah Snyder,... I told him about the abc conjecture, and the fact that it was proved for polynomials. ...he found his own proof which is simpler than the one I learned from Mason.

An appendix gives Noah Snyder's proof.

**Global Integration of Locally Integrable Vector Fields.** A presenter of this talk may wish to find some other title. The motivation is Cauchy's residue theorem from complex analysis, but Lang "is basically concerned with real analysis in the plane." I found this the least exciting of the talks.

**Approximation Theorems of Analysis.** Lang starts with a very general approximation theorem for Dirac sequences. Then, as applications and examples, he discusses the Weierstrass approximation theorem, Fourier series, harmonic functions, Poisson families and solutions of the heat equation. We meet the Landau sequence, the Dirichlet kernel, Cesaro summation, the Poisson approximation theorem, theta series and the convolution product, the Poisson summation formula, the functional equation for the zeta-function and elliptic functions: numerous talks — perhaps a whole course.

I like to say, for instance, that the heat kernel which we'll discuss later is the big bang of the mathematical universe. You can find it all over the place.

**Bruhat-Tits Spaces.**

There should be some application of the Bruhat-Tits fixed point theorem in economics. It is very likely that if you find one, you will get a Nobel prize.

We start with Apollonius's theorem (though Lang doesn't call it that), define a Bruhat-Tits space, prove Serre's theorem that a subset has a circumcentre, leading to the Bruhat-Tits fixed point theorem. The other sections concern the space of positive definite matrices and the metric increasing property of the exponential map. Untypically, Lang appends some historical notes, because

The presentation of the above material essentially follows a path which is the reverse of the historical path.
... Klingenberg asserts that von Mangoldt essentially proved what is called today the Cartan-Hadamard theorem for surfaces, 15 years before Hadamard did so. Actually, von Mangoldt refers to previous papers by others before him, Hadamard refers to von Mangoldt, and Cartan refers to Hadamard. I am unable to read the original articles.

Intriguing! Are they physically, or psychologically, inaccessible?

**Harmonic and Symmetric Polynomials.**

... the ring of polynomials and two important subsets, the harmonic polynomials and the symmetric polynomials, provide a fertile ground to illustrate concepts from linear algebra, not to speak of the calculus involved with partial derivatives. When dealing with polynomials, however, the analysis side of things is mostly illusory. Differentiation of polynomials is a purely algebraic notion, ...

Ask any engineering student for a definition of derivative and he'll tell you it's nx^{n-1}.

The book provides a good template, or set of templates, for teachers to present other topics outside the often over-rigid confines of specific course syllabuses, and to get across the message that mathematics is one subject, not to be separated into "branches" and not to be classified as "pure" or "applied".

Richard K.Guy (rkg@cpsc.ucalgary.ca) is Faculty and Emeritus Professor of Mathematics at The University of Calgary.