Interview with Joseph Ayoub

Joseph Ayoub, Professor of Mathematics at the University of Zurich, is the first holder of the “Alexzandria Figueroa and Robert Penner” Chair. He is interested in the cohomology of algebraic varieties and the theory of motives.

How did your interest in mathematics start?

I’ve always been very interested in maths. In my early teens, I had good grades in all subjects but maths was always a special interest of mine: in my spare time, I enjoyed solving maths problems. When I ran out of them, I made new ones up. I was particularly keen on plane geometry but I also liked calculating things and solving equations. During breaks, I often disappeared into the library to look through the Encyclopaedia Universalis in search of maths articles. This is how I became familiar with a number of modern concepts such as the classification of finite simple groups.
I was able to access bits of “advanced mathematics” at a very young age, when I found some papers in the storage room of our small apartment in Beyrouth. They were notes of the lectures on general topology which my father – a maths professor – had followed at the university. My mother, who was a librarian at the science faculty, knew someone who helped me lay my hands on a copy of Differential Geometry and Symmetric Spaces by Helgason. I remember having spent most of the summer holidays compulsively going through that book. I ended up reading it from start to finish and feeling I had understood everything!
In 1998, straight after my baccalaureate, I was lucky enough to be admitted to Lycée Louis‑le‑Grand in Paris. That’s when I understood that you could earn a living from mathematical research, which was a real revelation for me. It was my maths teacher, Hervé Gianella, who made me realise this and who encouraged me to take the École normale supérieure entrance examination. I had previously seen myself as becoming an engineer with a “proper” job and an “eccentric” hobby: reading maths books.

What is your connection with IHES?

The first time I heard about IHES was in connection with Alexandre Grothendieck. His name is inextricably linked with that of IHES. In a way, I first discovered IHES with the élément de Géométrie Algébrique and the “Séminaire de Géométrie Algébrique”, which were largely prepared and drafted at IHES. It was much later that I came to IHES, and that was for a conference in honour of Luc Illusie.
I am very grateful to the scientific council for having chosen me as the first holder of Alexzandria Figueroa and Robert Penner Chair. It is a great honour of course and I am already looking forward to the time I will be spending at IHES. I don’t yet know what impact my visits will have on my work but I will try to extract the maximum benefit from them.

How would you summarise your main contributions?

For a long time, I worked on a particular and crucial conjecture in motive theory called the “conservativity conjecture.” The conjecture is very easy to state and offers a bridge, or rather a return path to two different kinds of objects. One is a motive, which is a very rich algebraic geometric object, the other is its realisation which is a topological object with no additional structure.
The conservativity structure turned out to be very difficult. Nonetheless, I devised a strategy to demonstrate it. Even if I haven’t managed to make it work yet, I consider this unfinished business to be my most important contribution.

What inspired you so much to pursue your research and what do you find most exciting in what you do?

What I love most in mathematics is the coherence that emanates from a well‑constructed theory. Once the right point of view has been identified, the right definition, the right context, what follows is more or less inevitable and the result is very coherent. I think I really value that coherence. Luckily, there is no shortage of well-constructed theories in algebraic geometry, which is probably one of Grothendieck’s legacies.
I also like the writing stage. In fact, I think doing and writing maths are activities which cannot be separated. It’s only when I write an article that I really understand the demonstration of a problem and the cogs and wheels in a theory. Unfortunately, the big questions I’ve addressed so far have turned out to be very tough. This is naturally the source of some disappointment but I am an optimist. What inspires me to carry on is definitely the hope of seeing the solution to these great questions one day. Another source of hope and inspiration is to have been witness to spectacular progress on other topics and in other mathematics fields.