One of the odd perks of getting old is that one can look back to the
distant past and wonder with cold detachment just who the hell we
As a kid I was obsessed with cryptography and secret messages. I even
halfheartedly tried to create a language that only my friends and I
would understand. I wanted to build a robot. I also wanted to live in
a desert island and made endless plans in my head on how I would run
my life on it. Perhaps nothing of this is too surprising in a restless
kid, reader of Jules Verne (“La isla misteriosa/The mysterious
island”, “Viaje al centro de la Tierra/Voyage to the center of the
So how did I get from there to becoming a researcher in pure
mathematics? Not along a straight line for sure.
I started engineering at the Universidad de Buenos Aires, aiming for
electronics, the robot thing. I found myself not caring about the
specific answer to any exercise but was fascinated instead by the
solution process. In a basic physics class the lecturer casually
mentioned the exponential, even the cosine, of a matrix. What?
In the vaulted, vast, inspiring space of the Engineering central
library (Paseo Colón!) I discovered a book that described the “dual
numbers”. Just imagine this grotesque incarnation of the complex
numbers: you add to the reals a quantity whose square is zero! How
amazing, how beautiful.
I got a TI-99 programmable calculator, like all my classmates. We
carried it around hanging from our belts looking like some kind of
geeky cowboys. I spent hours playing with the built-in functions for
matrices. Computing, for example, the limiting value of the inverse
of x – A for x near an eigenvalue of A. You could program the
calculator using at most 50 steps. What a triumph to realize two
steps of your code could be combined to gain an extra line. It warped
my programming style for years.
In the hall of that imposing building, at a booth of the university
publishing house EUDEBA, I bought Kelley’s book “General Topology”,
convinced by its preface that the only requirement was a knowledge of
the real numbers. I couldn’t read past the first page.
About then I decided to switch to a degree in math, in the second year
of University; didn’t have any idea of what I was doing, didn’t know
you could make a living doing mathematics.
In my second year of high school in Buenos Aires, mathematics meant
solving problems about triangles using a slide rule or a log table. The
teacher would bring to class a huge slide rule he would hang from the
top of the blackboard and start sliding away. That year I devised an
encryption scheme involving permutations of the five vowels:
a,e,i,o,u. I kept some notes I wrote then on this scheme, written in
the standardized three-holed sheets (hojas de carpeta Rivadavia!) we
had for all our high-school work. I never actually ever used the
The all-male group marooned on the desert island is rescued at the
very end of “The mysterious island” by the cryptic Captain Nemo, the
main character of another Jules Verne book, with his
submarine. Anachronistically apparently but a fabulous coup de theatre
that impressed me immensely nevertheless.
I look at the stuff in these notes on permutations now, more than 45
years later, and don’t really understand any of it. But it sure looks