Page 11 - Textos de Matemática Vol. 39

EDUARDO MARQUES DE SA´
AND THE INTERLACING INEQUALITIES
GRACIANO DE OLIVEIRA
I first met Eduardo Marques de S´a when he was a student, in 1972, just after my return from Brazil. Since I was very interested in continuing my math- ematical activities, I looked for students who shared my interest. After some efforts, I found Jos´e Perdiga˜o Dias da Silva, Maria Alice Ina´cio and Eduardo Marques de Sa´. I didn’t know them before, but we started working. Later on, others joined us.
That year they had to submit a dissertation for their first degree. This obligation contributed strongly for them adhering to my plans. I began with a series of lectures on multilinear algebra, a subject I was then working on and about which I was preparing a monograph, later published by the Gulbenkian Foundation. I noticed that my presence did not please the Mathematics De- partment, and so we moved to the Ateneu de Coimbra, whose executive did not hesitate to place a room at our disposition, where we improvised a blackboard.
In this way I met Marques de Sa´ at the beginning of his career. And I boast of having contributed to attract him to Linear Algebra. He had started his studies as an Engineering student, but did not resist the calling of Mathe- matics. The subject of his dissertation, van der Waerden’s conjecture on doubly stochastic matrices, was suggested by me. After getting his degree, he thought of going into Analysis, but he eventually came back to Linear Algebra, I sup- pose because of his excitement with his discovery of a result which came to be known as the “S´a-Thompson interlacing inequalities”. As far as I remember, I was the first person to mention to him the problems which led to that now famous result, as I was working on that at the time.
The Sa´-Thompson interlacing inequalities, in my view, are one of the best results obtained in the last decades in the field of Linear Algebra. This is confirmed by the fact that Marques de Sa´’s PhD thesis received the Householder Prize.
It is such an important result, and has such potential, that it is worth describing it. It is a good opportunity to attract young researchers, who often have the discouraging feeling that everything has already been done. The result is relatively easy to state, and even to understand by those not working in the field. The proof, on the contrary, is very hard, and I am convinced that the number of mathematicians who have ever studied and understood it is very small. It is a very deep result, with relations to several interlacing phenomena
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