Discovering New Frontiers

Conversation with Samuel Grushevsky
Interview by Luis Álvarez-Gaumé
Edited by Maria Shtilmark

Samuel Grushevsky
Photo: Joshua Klein

The Simons Center for Geometry and Physics welcomes Samuel Grushevsky, Professor of Mathematics at Stony Brook University, as the second Deputy Director of the SCGP. To ensure a smooth transition, in 2021-2022 Prof. Grushevsky will work in parallel with Prof. Abanov, who will then step down from his position as Deputy Director. The SCGP Director Luis Álvarez-Gaumé sits down with Professor
Grushevsky, recently elected Fellow of the American Mathematical Society for his contributions to algebraic geometry and Teichmuller dynamics and service to the mathematical community. Grushevsky talks about his career, his view of the Simons Center’s unique traditions, and his vision of new directions of bringing mathematics and physics together.

Luis Álvarez-Gaumé: Sam, it is a pleasure to welcome you as the new Deputy Director of the Simons Center. Before we enter into the SCGP activities, perhaps you could briefly outline for us your professional career: where did you study, when did you come to the U.S., and what made you choose Stony Brook University as your long-term destination?

Samuel Grushevsky: I came to the U.S. in the middle of my undergraduate studies. Before that I was in Moscow, during what was the time of great enhancement of mathematical activities in Russia, due to the opening of the borders and new freedoms. New people were doing new kinds of mathematics, and I was privileged to study with many of them at two universities at once: Moscow State University and the Independent University of Moscow. I did not finish my education at either. In fact, I have a document from the Independent University saying that to get a degree I need a course in partial differential equations. I never took that course, but I have taught a course on partial differential equations since then, so perhaps I’m now qualified for a degree.

I then transferred to Harvard and got my BA in math and physics there. This time I was lucky to have wonderful instructors in both math and physics, taking various advanced math and theoretical physics courses. I wanted to do more mathematics, so I stayed at Harvard and got my PhD in math. Then I went to Princeton, first as an instructor, then as an Assistant Professor in Mathematics, and, finally, I arrived at Stony Brook.

This was in 2009, exactly when the Simons Center for Geometry and Physics was founded, and its creation was in fact one of the main reasons why I decided to come to Stony Brook. The university had a great reputation in geometry, going back to the times of Jim Simons and C.N. Yang, and with all the activities that have been happening since. But I believed the Simons Center would provide many excellent opportunities to see more mathematics and more physics and to interact with many experts worldwide, and this is what happened. So, I am still at Stony Brook, but now not just at the mathematics department.

LAG: What are your major and minor areas of expertise, or how would you describe them?

SG: This is a question that mathematicians, and not only mathematicians, like to ask when they meet, and I always find it hard to answer. To my mind, many of the partitions between fields in mathematics are somewhat porous and abstract, and it is not always clear what the distinctions really are. Sometimes when I talk to people, I say I am a geometer, sometimes I say I am a number theorist. Other times it’s mathematical physics—it really depends.

To my mind, you can describe your work, and what labels get attached to it is sort of secondary. The best way to describe my work is to say that I work on Riemann surfaces, or algebraic curves. These are some of the most fundamental objects in mathematics and physics. There is a quote, apocryphally attributed to André Weil, that says “God created integers and Riemann surfaces and everything else was created by the devil.” This somehow says that Riemann surfaces are really fundamental to a lot of mathematics and a lot of physics as well, and my interests mostly stem from that, but have gone in many different directions throughout geometry.

LAG: You mentioned your degree in mathematics and also in physics, and the next question is kind of natural. Have you had much interaction professionally with physicists? In what sense do you find those interactions positive and enriching, if at all?

SG: That is a great question. When I was an undergraduate, I was a precocious one, studying very advanced math and physics, and I have done homeworks on string theory, which I can still look at, though that doesn’t necessarily mean I can understand them well now. But I wanted to focus on mathematics and went in that direction.

And almost by chance, about seven years after my PhD I accidentally went back to do something related to physics, based on interactions with physicist Duong Phong (who is also a mathematician, with separate physics and mathematics interests) at a conference, and eventually some of my work that was related to modular forms turned out to be related to string theory and to superstring theory. I have written some papers and discussed physics with some people, and then I co-organized a workshop at the Simons Center with mathematicians and physicists, which was devoted to moduli of super Riemann surfaces1. 

At the time, the foundations of the theory of super Riemann surfaces were largely lacking: some development had happened in 1980s, yet not much happened since and we were hoping there would be enough impetus to try to work on this again. We organized a conference, including an influential lecture series given by Pierre Deligne and by Edward Witten. After that nothing happened for a while, but recently this field has been picking up pace, and I am happy to say that next year there will be a program at the SCGP2 devoted to this subject that I’ll be co-organizing with experts who have recently done wonderful things in this area. So yes, it has been very influential.

LAG: Now, a more philosophical question. Since its beginnings, most of the mathematical research carried out at the SCGP has hovered above more or less conventional, or traditional, geometry. Yakov
Eliashberg, the chairman of the SCGP Board of Trustees, said that from a modern perspective, geometry is a point of view on mathematics. Do you feel that the Center’s research should extend into areas like perfectoid spaces and the p-adic realm, so far from physical applications, or should it remain in a more conventional territory?3 

SG: To my mind, again, the labels in mathematics are sometimes assigned semi-arbitrarily, so what constitutes geometry is indeed a point of view, perhaps more than the description of a field. And more and more work really unites and connects different areas of mathematics, some with connections to physics, some without. In this sense, hopefully, the Simons Center will evolve and extend its reputation into other areas which are not maybe so present now. Whether or not this would include the work of Peter Scholze and everything around it is too early to say, but certainly any kind of development in mathematics is something that the Simons Center could be related to. Whether or not this is specifically the direction that we would like to go in at this point is not clear. Stony Brook has an excellent reputation in many areas of geometry, but doesn’t at the moment have expertise in anything related to p-adics. Perhaps building in that direction would require more time, starting with some other excellent hires, which I hope will come at some point.

LAG: I think it will be nice to increase the mathematics activity at the Simons Center. Do you share that view, and if so, what do you think is the best way to enhance that activity?

SG: Well, I am a mathematician, despite my great admiration and some interactions with physicists, so of course I can always wish for more and more mathematics happening. So yes, it would be great to have more activity in mathematics at the Simons Center, and I think we are doing well in that regard. For the next academic year our program is very well balanced, with respect to mathematics, physics, and programs and workshops that unite mathematics and physics. We have been successful at attracting stellar mathematicians and physicists to our Scientific Advisory Committee, and we have recently hired John Pardon, which of course is going to greatly increase the level of activity at the Simons Center. I very much hope and I am sure that we will have more and more mathematical activity.

LAG: Throughout the world there are special institutes, like Oberwolfach, the MSRI (in Berkeley), and others. Do you believe the mathematics community is ready to accept a new competitor in that domain? Could the SCGP become another center of reference? How to achieve it?

SG: I don’t think there is any competition. I don’t believe that mathematics research centers, or science research centers, are competing with each other. We are all colleagues really, trying to do different aspects of the same great thing, which is advancing research and having educational, training, and mentoring activities throughout the world. In this sense, these are not our competitors, but our colleagues, and we will continue to work to strengthen our relationships with them. And certainly, there is room for the Simons Center in this scene. Mathematics and physics have been expanding greatly, as witnessed by the explosion of the number of journals, papers and all sorts of activities, and there is certainly room for more.

Whether or not the Simons Center will find itself at the forefront I think is not really the right question—we already are. We are well-recognized and well-established in what we do. We are certainly one of the peer institutions of the places like MSRI or Oberwolfach you named, and others. We will of course need to continue working to do things even better —because everybody should be striving for perfection, ourselves included. But the Simons Center is also different from any other place I know of, in two aspects.

One is the existing tradition of bringing math and physics together, focusing on activity closer to the interface between the two disciplines, not to the exclusion of excellent activities in pure mathematics with no connections to physics, at least no connections to physics quite yet, and similarly for activities purely in physics. Second, we are a rather unique institution, in having both permanent faculty and postdocs, like maybe the Institute for Advanced Study does, and also hosting a large collection of workshops and programs, like MSRI or Oberwolfach do. We do a mix of activities that is different from any other place I know. And the fact that we do all this allows different activities to benefit from each other. For example, our workshops and programs greatly benefit from the Center’s excellent faculty and postdocs and, furthermore, from the proximity of Stony Brook’s excellent physics and mathematics departments, the Yang Institute for Theoretical Physics (YITP), and the Institute for Mathematical Science (IMS). We are special, and I think we are doing very well. We certainly are a peer of the best research institutions in the world in this regard.

LAG: Finally, what are the most pressing issues you see at the Center that need to be addressed and/or corrected?

SG: The last two years have been truly special, and not in a good way, due to COVID-19 with all its complications, including cancellations, remote work, and lack of meetings. I think the present times will allow us to reinvent and revamp our activities. We should strive to optimize our use of resources to achieve the maximum level of scientific activity that we can, while maintaining our ability to host successful workshops and programs and to recruit excellent postdoctoral scholars. One thing I think we should work on is further broadening our reach within the mathematics and physics communities, beyond the core fields that are extremely well represented at SCGP. For example, while our reputation in fields related to subjects covered during the Simons Summer (Physics) Workshops is stellar and needs maybe no further enhancement, there could be further mathematics and physics communities where we haven’t done any particular activity recently, and we could try to expand in those directions and try to advertise our opportunities to people in these disciplines.

At some point, there was a Fields medalist visiting Stony Brook who said: “I thought I was coming to this quiet place, where people sit and do excellent work, [because that’s the level of work that people at Stony Brook have done-SG] but I came here and now I see how much is really going on!” I hope we can communicate this message to the world: that we are not just a quiet place where a lot of great work is being done. We are also a very active place where a lot of great work is being done, with lots of great visitors, and lots of great activities. I think this will help reestablish and reconfirm the SCGP as one of the preeminent centers of research excellence in physics and geometry worldwide, post-pandemic.

LAG: Thank you very much and, again, welcome to the Simons Center!

SG: Thank you, Luis. It is an honor to become a part of the SCGP.

3 For more on this subject see page 8 of SCGP News Vol XVIII