Yakov Eliashberg: Unfolding insights from a life in mathematics
January 26, 2025 | Bharti Dharapuram
Yakov Eliashberg is the Herald L. and Caroline L. Ritch Professor of Mathematics at Stanford University, whose research spans the fields of symplectic and contact topology. As the inaugural speaker of The TNQ Distinguished Lectures in Mathematics, he delivered a lecture on ‘The Strange and Wonderful World of Symplectic Geometry’ at the Indian Institute of Technology Madras on 26 September 2024. As a part of the TNQ Numbers & Shapes program, he also taught a five-day workshop on symplectic topology, hosted at The Institute of Mathematical Sciences, Chennai between 23-27 September 2024. During his visit, Bharti Dharapuram met with him for a conversation about symplectic geometry, mathematics and his journey.
Mathematics and mathematicians
Author’s note: Symplectic geometry finds its origins in Hamiltonian mechanics. Here, the motion of a particle is described by a curve in a coordinate system known as phase space. Every point in a phase space is defined by position and momentum, combinations of which describe all possible states any particle can assume. Given an initial state, a particle’s motion is described by a curve in the phase space such that the area of the parallelogram enclosed by its position and velocity vectors in a specific coordinate direction remains constant over time. In this case, the property of area preservation emerges from Hamilton’s equations of motion, which constrains the paths a particle can take from the set of all possible trajectories. This property of area preservation in specific coordinate directions is known as symplectic structure. Mathematicians study properties of symplectic structures and transformations which preserve it.
Symplectic geometry, put simply
Symplectic geometry is a geometric structure discovered in attempts to formulate equations of mechanics. In the 19th century, great polymaths (Lagrange, Jacobi, Hamilton) were trying to derive better formulations for Newton’s laws of mechanics. At the time, two approaches - the Lagrangian and Hamiltonian formalisms, came about. These ways of stating mechanical problems and writing equations still exist today and depending on the purpose, one might be better than the other. In some sense, the approach of Hamilton is more geometric.
It describes a mechanical system by what is called its phase space. A point in the phase space determines not only the position of the system but also its velocity or momentum. You can think of the development of a dynamical system as a curve in this space. Every point in this space gives you the system’s initial position, and the Hamiltonian function, which is the energy of the system expressed through velocity or momentum, gives you its dynamics. Early work showed that this flow of the system given by the Hamiltonian function preserves many interesting geometric features. For instance, it preserves volume, but Poincaré observed that it preserves much more. It preserves something called symplectic structure, which is a geometry based on area rather than distance. This is in two-dimensions, but symplectic structure can be generalised to higher dimensions.
Poincaré was interested in celestial mechanics and probably was the first to realise that mechanical equations for most real-life problems are not integrable, in the sense that you cannot just write down a formula and find a solution. Even with powerful computers you cannot answer many of these interesting questions. Even if you compute a numerical trajectory, you don't know if the trajectory follows periodic motion and if the periodic orbit is stable or not. For instance, it is still an open problem if the motion of the solar system is periodic - but we are kind of safe for the next billion years!
What mathematicians do
There is, I think, a misconception among the general public about what mathematicians do. I don't know how it is in India, but in the United States when you read an article about some great mathematical discovery, the newspapers say "this mathematician solved a puzzle”. It is made to look like some kind of entertainment for adults. There are many interesting and famous classical problems we would like to solve and what do mathematicians do in order to solve these? We create new worlds. For example, to study symplectic geometry we created the symplectic world. When we come to this world, we do not know what is possible and are exploring. We go there, we try to see what is possible and sometimes discover that there is a prohibition, there is a law of nature which says that a certain construction is not possible. We then try to change our approach to see what is possible, subject to newly discovered constraints. We act in this mathematical world exactly the way an explorer would in the real world.
I think it is extremely rare [that mathematicians work in complete isolation]. Mathematics is collective work, which is very important. Even if there are great people who appear to achieve results in isolation, most of them wouldn't be able to do it without a lot of background work done by the whole mathematical community. I think most reasonable people can become good mathematicians, no doubt. Of course, nobody is guaranteed to make a great discovery, even if they are a genius. But everybody can do some useful and important work, and there is enough important work for everybody. Of course, it is also important to have some fantastic people who appear a few times in a century.
Problems and solutions
There are good questions that you can ask and there are not so good questions. In my view, finding good questions is at least as important as solving them. This, of course, requires experience. Usually, there is a problem and in many cases, the goal of solving it is very far. In science, there is an extremely important thing called arguing by analogy. It is important not to be very narrow in your field but to have a broader understanding in mathematics, and to know many subjects, so you can bring ideas from one to another.
[Imagine] you have a problem and a preliminary idea of what you expect will happen and you try to prove it. In many cases what you are trying to do may be wrong, but it is not obvious. It looks like you are coming very close to finding the solution but somehow something always goes wrong. Sometimes persistence does pay dividends but in most cases, it is a sign that maybe things are not right. It is important to be open-minded, maybe you can completely change your mind to see if the opposite route would work.
When I was a student in Leningrad university, my advisor was Vladimir Abramovich Rokhlin, who was a very good mathematician. I learned a lot from him but also from a senior student, Mikhail Gromov, who was a fantastic mathematician. He greatly influenced all my views of mathematics. There was a problem related to symplectic geometry [Arnold’s conjecture, generalizing Poincaré's last geometric theorem] that we were both thinking about. It was completely unclear what kind of outcome it could have. It could be A , or B, the complete opposite. In the beginning, both of us thought that B was the answer, so for a while, I tried to prove B, but there were always some mistakes. At one moment I realised that probably just the opposite was true. Therefore, I started to think about it and thought about it for many years before finding the solution. It was an effort spread over at least six to seven years.
Drawing Indian students to geometry
In general, I think [mathematics] research in India has traditionally been restricted to certain areas. For example, number theory, probably thanks to Ramanujan, is extremely popular. There are fantastic number theorists in India. When I first came here around ten years ago—it was my first time in Chennai—I met with Dishant Pancholi, and MS Raghunathan, a great mathematician whose work is very close to geometry. We discussed that it would be great to do something here to develop geometry and topology in India, and I am happy to see that it is happening now.
Reflections on his past
Author’s note: Yakov Eliashberg was born in Saint Petersburg (Leningrad at the time), and as a young child took serious interest in playing the violin, dreaming of becoming a professional musician. Alongside this, he was inspired to learn mathematics from the system of olympiads and math circles that were prevalent in the Soviet Union, mentored by some great teachers. At a critical juncture in school that shaped his future, he chose to study science and mathematics over professional violin. He went on to finish his doctorate from Leningrad University and found a position at the Syktyvkar State University where he taught mathematics for a few years. With a rising wave of antisemitism, he quit his job to try to emigrate in the footsteps of family and friends who left abroad. When he was denied permission, he worked as a substitute teacher and computer programmer for many years before he was allowed to emigrate to the US, where he resumed mathematics research. Since 1989, he has held a position as a professor of mathematics at Stanford University.
Playing the violin
I continued playing the violin for a while in the US but stopped at some point. It is difficult for everyone to understand, but I had the following very strange problem. I always had an extremely good pitch, what you would call a perfect pitch. But sometimes a blessing can become a problem - because this sense of a perfect pitch at some point shifted by almost a single tone in my ears. Whatever I heard had this shift, which made it extremely difficult for me to play in ensembles. I had to correct for it while playing, and for me, it was like playing something wrong.
Math circles in the Soviet Union
The whole system of olympiads and math circles was extremely well-organised and there were a lot of really great people participating there. When we were at university, many students went to schools and organised mathematics circles. It was loosely organised but, I think, it was extremely important. I was running this circle for a couple of years when I was at university and I know at least two or three extremely good mathematicians who came from this school. But there were extremes in this movement. For instance, the system had some people who were oriented towards training students to win prizes, which can be unpleasant. I was luckily not participating in this. A prize is fine as a stimulus, but when it is the only goal it becomes like a professional sport and defeats the purpose.
To be reborn as a mathematician
I was essentially out of mathematics for maybe seven to eight years in the Soviet Union. I needed to do something to earn money for living. I was trying to do mathematics but was too tired with the other things I was doing, which were also mentally taxing. I was invited as a speaker at the International Congress of Mathematicians, Berkeley [August 1986] and, of course, I was not allowed to go. But after this meeting, a Berkeley mathematician - Alan Weinstein, one of the founding fathers of symplectic geometry, organized a special year-long program on symplectic geometry at the Mathematical Sciences Research Institute (MSRI). He sent me an invitation to the Soviet Union. I considered it a joke because there seemed to be no chances for me to go there. But suddenly, I was allowed to go. I had an invitation, I got the permission and I had the opportunity to spend a year at this research institute and do mathematics. That, I think, was extremely important. I was very lucky when I emigrated and came to the States. But I was prepared that it may be difficult for me [to return to doing mathematics]. I really didn't know whether I would be able to restart and effectively function as a mathematician. I am happy that I managed.
The author thanks Dishant Pancholi and Sushmita Venugopalan for discussions related to symplectic geometry that greatly helped in writing the introduction.
