Interview with Alexander Goncharov - IHES
Alexander Goncharov IHES

Interview with Alexander Goncharov

A Professor of Mathematics at Yale University since 2010, Alexander Goncharov is the first holder of the Gretchen and Barry Mazur chair. He is interested in different fields of mathematics and mathematical physics, such as the theory of motives, Hodge theory, representation theory, higher Teichmüller theory and its quantification.

How did your interest in mathematics start?

When I was a kid, I was fascinated by astronomy and nuclear physics. At the time, even in my small town in Ukraine, one could find some great popular science books on these subjects. For example, I remember a book titled Entertaining Nuclear Physics, by Mukhin, explaining the subject in a serious, yet entertaining way. I read it many times, as I had done with The Three Musketeers.
A little later, my interests shifted to mathematics: I enjoyed solving problems and reading the Kvant magazine (a popular science magazine in physics and mathematics).
In 1976 I was admitted to Moscow University. The first Monday in September I attended Israel Gelfand’s seminar, which became the place where I grew up mathematically. I learned a great deal from D. Fuchs, J. Bernstein, S. Gindikin, Y. Manin and A. Beilinson. Moscow was a fantastic place to learn mathematics. But, for a young mathematician, it was not easy to survive the inevitable collision with officialdom. I was helped by Israel Gelfand and Simon Gindikin.

What are your research interests?

In mathematics, I like being at the cross-roads between different fields. Since the mid 1980s I have been studying integrals of algebraic geometric origin using methods of arithmetic geometry, often conjectural. This allows predictions to be made on the integrals without calculating them – I call this “arithmetic analysis.” Studying such integrals is an old enterprise, which significantly motivated the development of algebraic geometry. Entirely new insights brought Grothendieck’s idea of motives and, even more importantly, Beilinson’s conjectures on mixed motives. Using these ideas as guiding principles, one can predict values of integrals by performing easy algebraic calculations, and that is at the very heart of what I do: I use integrals to get further insights into the theory of motives and I apply the motivic philosophy to the study of integrals.
How has your relationship with the Institute evolved over the years?
I visited IHES for the first time in June 1990, just after the USSR opened its borders. Since then, my main motivation to come to IHES has been Maxim Kontsevich – we met almost forty years ago, and have been discussing mathematics ever since.
Yet, at IHES one gets the chance to meet many new people, which makes life delightfully unpredictable. For example, in 1996 I met Dirk Kreimer, and learned about amazing computations he and David Broadhurst were doing in quantum field theory. I suggested that one should apply techniques of arithmetic analysis in perturbative calculations of Feynman integrals. In particular one should upgrade correlation functions to their motivic avatars – motivic correlation functions – which lie in the motivic Galois Hopf algebra. That led to new insights and raised new type of questions.
The many contacts with physicists I have had since then can be traced back to these discussions at IHES.
Since January 2019, I am the first holder of the Gretchen and Barry Mazur Chair at IHES and that is a great honor for me. I am delighted to be here and especially grateful for the chance to give a series of lectures at IHES.

What do you find most exciting in what you do?

I get excited when sensing a mystery in mathematics. For example, the motivic symmetries show up only a posteriori. Yet in the real Hodge realisation one can make them visible by writing a single Feynman integral. It seems therefore that mathematical ideas underlying the quantum field theory paradigm will play an essential role in our description of the motivic world.