# The early days:

1957 – 1963

The years 1957 to 1966 were a golden period at the School of Sciences of the University of Buenos Aires, Argentina. A number of prestigious scientists had returned after the fall of Perón and were enthusiastically reconstructing the infrastructure of Argentine science.

In mathematics there was great activity, with many foreign visitors being sponsored by the Latin-American School of Mathematics, financed by Unesco and ably headed by Alberto Gonzalez Domínguez. Besides our local world known mathematicians, that included Luis Santaló, Eduardo Zarantonello, and Mischa Cotlar, there were some very prominent expatriates, such as Alberto Calderón, which were pivotal in creating the strong Argentine analysis school that is still well regarded today.

A long list of very important mathematicians visited for extended periods, giving lectures, guiding research, and creating long lasting relationships. I can mention: Jean-Pierre Kahane, Anthony Zygmund, Mary and Guido Weiss, Alexander Ostrowski, Lothar Collatz, Stefan Vagi, and Ian Mikusinski, among others. I was lucky to interact with several of them.

I completed my undergraduate studies in 1961, obtaining a Licenciatura degree in Mathematics. I recall Luis Santaló, Oscar Varsavsky, Gregorio Klimovsky, Pedro Zadunaisky and Manuel Sadosky as my main and most influential teachers.

We were a small group of students in Mathematics and most of them have had outstanding carriers. I recall among others Horacio Porta, Nestor Riviere, Ricardo Nirenberg, Hector Fattorini, Carlos Segovia, and Corita Sadosky.

In the summer of 1960, Porta, Riviere and I went to spend a month at the Centro Atómico Bariloche, to study Algebra of Logic with Antonio Monteiro. That was a very interesting experience and included some adventures back packing in the Lage Region.

In 1961, a revolutionary event came to pass, thanks to the hard work of Manuel Sadosky and Rolando Victor García. The first large computer was purchased and the Instituto de Cálculo was created in the School of Sciences.

The computer was a British made Ferranti Mercury, very large and very expensive, although puny for today’s standards.

That was when I switched from pure to numerical mathematics. At the invitation of Sadosky I was one of the first employees of the Instituto de Cálculo and initiated work to understand the new machine, its programming language Autocode (under the ineffable Miss Poppelwell, who had worked with Alan Turing in Manchester) and the mysteries of numerical mathematics.

Just a few months before I had written my first and only paper in pure mathematics, with the help and guidance of Anthony Zygmund:

This paper extended some theorems of Kintchine on generalized derivatives of measurable functions.

Around this time I also had the chance of becoming Alexander Ostrowski’s assistant, to whom I introduced to Muller‘s method, which was used in the Mercury to calculate roots of polynomial equations. I could not understand the proof of convergence given in Muller’s paper. Professor Ostrowski concurred with me that the proof was incorrect and then proceeded to write a correct proof and extend the method to the approximation of polynomial equations by polynomials of lesser degree.

In 1962 Pedro Zadunaisky arrived from the Goddard Space Center in Maryland. Zadunaisky was an astronomer with strong leanings towards the numerical aspects of data fitting and ordinary differential equations. I was assigned to work with him, both as his assistant in the numerical analysis courses he lectured on, and also in his research. The Astronomical Union had commissioned him to study a large set of calculations that the Italian-Argentine astronomer Bobbone had performed using worldwide observations of the 1908 passage of Halley’s comet. These observations and orbit calculations were collected in a number of handwritten notebooks. Our task was to verify the accuracy of those calculations using our electronic computer. Zadunaisky had also as a hidden agenda the calculation of the orbit with great and guaranteed accuracy, in order to study some anomalous behavior of comets, which could not be ascribed to gravitational effects. This were very small perturbations and thus one needed to make sure that numerical errors were not tainting the results.

We proceeded then to attack this fairly large computing and data processing task that involved collating and clustering the observations, calculating the orbit parameters by the method that the astronomers called differential corrections, which was none other than the Gauss-Newton method for nonlinear least squares, and finally integrating the orbit. We did all that, taking into account the ephemerides of the Moon, Sun, and the various planets, which unfortunately came from different tables and were in different coordinate systems. That involved a lot of manipulation with our paper tape input and very restricted memory, but we finally managed to reproduce all of Bobbone’s results that were error free.

As part of this work I wrote one of the first proofs of the convergence of the Gauss-Newton method and I presented the results at the 1965 IFIP meeting in New York: