Yuri Ivanovich Manin (16 February 1937-07 January 2023)

There are few great mathematicians who were also great human beings. Yuri Manin is surely one of them. I say this, because great mathematicians like Gauss or Cauchy had not at all pleasant attitude toward a young mathematician like Abel, who was only seeking for survival and a little bit of recognition of his work. In my Ph.D thesis, there was the notion of Abelian integrals. They satisfy Picard-Fuchs differential equations and a more algebro-geometric framework for this statement was formulated by Grothendieck under the name "Gauss-Manin connection''. This was the first place I encountered Manin's name.
My first physical encounter with him was in Max Planck Institute for Mathematics (MPIM) at Bonn in the winter of 2001. I can imagine that my application to MPIM was accepted as I had made a good advertisement in my research statement that I am going to do this and that with Abelian integrals and Gauss-Manin connection. I saw him in the library trying to figure out why the printer was not working. I was so excited to see him. I directly went to him and asked a question on Gauss-Manin connection and a part of his 1963 article which stamped the name. "It was a long time ago and I do not remember details of this old article", he said. Now, I understand the situation. If a student of mine ask a question of an article of mine, not 40 years ago, but even 2 years ago, I have to read my own article for a day in order to recall the mathematics I was doing at that time, and then, I will be able to reply the student.
Despite this first encounter, conversation with Manin during my first post-doc at MPIM, enabled me to broaden my mathematics beyond holomorphic foliations that I learned at IMPA. At that time he had a collaboration with Claus Hertling on Frobenius manifolds arising from versal deformation of singularities, and I learned a lot about this topic in conversations with him and Claus in cafe and cookie time at MPIM. I was never able to get close to him and his mathematics, as I was still learning new territories in mathematics. I remember that I spent almost all of my stay at Bonn trying to learn Algebraic Geometry a la Grothendieck (schemes, motives and all type of strange staff), therefore, it was hard for me as a fresh Ph.D man to start a mathematical discussion with him in a way to produce some new ideas for collaboration. Therefore, I do not think I marked a place in his brain as he was nice with all visitors and discussing mathematics with almost all of them.
In 2002 I wrote my first paper on modular forms jointly with D. Mayer and J. Hilgert. Depite the fact that one expert in this topic told my coauthors that the content of our article was already done by him and it was somewhere in his shelves at his office, Manin had a very positive reaction to a seminar given by one of my coauthors on our article. I can imagine that later this has been effective in getting the article accepted. I have to say that until 2009 I got one rejection and one acceptance of my articles submitted with Manin as Editor. Even though I was upset with the rejection, it was understandably justified by a report that I have to do more in the article. Many years later in a submission to the same journal, but a different editor, I got again a rejection saying that the level of my article does not match the high level of the Journal!
In 2014 I visited once again MPIM and at that time he was organizing the seminar "Algebra, Geometry, and Physics". I was already working on a generalization of modular forms for Calabi-Yau varieties and I gave a talk in his seminar. After the talk we discussed a little bit about the preliminary of my talk which is a very simple relation between Eisenstein series and Gauss-Manin connection of families of elliptic curves. He was also surprised why this simple observation about these already classical objects was not noticed earlier. This conversation was in particular useful for me, as in this classical topic I wrote few papers, and I had always a nightmare that maybe what I am writing was already discovered some hundred years ago. Maybe the time of Abel and Eisenstein, as it deals with so classical objects such as Eisenstein series.
Manin was one of the great leaders in trying to combine Number theory, which Gauss once said queen of Mathematics, and Physics. Number theory is well-known for many problems, such as Fermat's last theorem, which for centuries persists as pure mathematical thoughts, without any connection to the outside world and Physics. The final scene in these notes, also justifies the first scene in 2001 near the MPIM printer.
For my GADEPs seminar, which started during the pandemic, I started to arrange for interviews, as I thought it is more useful for young mathematicians to hear about a piece of history of Mathematics directly from the creators. The few interviews that I made had more success than the specialized talks. In 2021 I wrote an email to Manin and asked him for a possible interview. The answer was: "Unfortunately, I cannot accept it, in particular, because I dislike more and more impromptu talks, when I have no time to think about questions, to collect materials and/or check facts etc.".
It is hard for me to digest the whole work of a mathematician like Manin who until last moments of his life produced significant Mathematics. Still, when I am searching in the internet on a topic, I land on an article of Manin that I have never read, and for sure until the end of my life I will have surprises like this. Few of Manin's work was directly affected my research. Manin's work on the Mordell conjecture for curves over function field for sure coined the name "Gauss-Manin connection" which is one of the central objects in my research. Actually, for me the main ingredient of Hodge theory is not Hodge decomposition but rather the Gauss-Manin connection. Manin's work on modular forms, and in particular periods of modular forms, has produced a fundamental notion called Manin's symbol. Long before Andrew Wiles prove that all elliptic curves are modular, Manin's symbols have been a fundamental computational tool in order to produce more and more modular elliptic curves. His work on Painlevé VI equation has been a fundamental paper for me to get an overview of the topic and write my own contributions. He also wrote few papers on iterated integrals for modular forms, this being far from my interest on iterated integrals for holomorphic foliation, for sure broaden the appearance of these type of integrals in mathematics and physics. At last but not least, his book on cubic forms still after many decades, is the main source for researchers who wants to learn the basics of the theory.


Hossein Movasati
January 14, 2023.