After 9 years of service to Friends of IHES as a member of its Board of Directors, Nikita Nekrasov, Permanent Professor in Physics at IHES from 2000 to 2013 and now a Professor at the Simons Center for Geometry and Physics at Stony Brook University, joins its Honorary Committee.

Find out more about his relationship with IHES and Friends of IHES in the interview below.

Nikita, you have not only been a member of Friends of IHES, but you also worked at the Institute itself as a Permanent Professor in Physics for nearly 15 years. Can you tell us a bit more about your history with IHES?

I joined IHES as a Permanent Professor in Physics in 2000.

Of course, I knew about IHES before that. For me, it was a place where great scientific work was done, and where great scientists were allowed to completely focus on their research, free of any teaching or administrative duties. It also had a magical flavor, being located in a forest near Paris.

I was raised in Moscow, in the Soviet Union. Nevertheless, my family was what one may call francophiles. My grandmother started to teach me French when I was 5 years old. At home, we had French books, maps of France and some of its cities. I studied French in elementary school. I even thought I would be a historian specializing on France when I grow up. Then puberty hit, and my interests shifted towards physics and mathematics. English language took a more prominent role as a means of communication. I became a graduate student at Princeton, then Junior Fellow at the Harvard Society of Fellows, and finally returned to Princeton as a Dicke Fellow. It was there that I received a letter from Jean-Pierre Bourguignon with the offer of a permanent professorship at IHES. Some of my closest collaborators envied me to the point that they tried to convince me I couldn’t have possibly been offered that position. Surely, IHES offered me some sort of junior half-time gig! Yet, other colleagues warned me that IHES would put me under enormous pressure to produce groundbreaking results. Despite all these premonitions, I viewed my family legacy as a good omen. More importantly, I saw IHES as the place where I would find experts with whom I would be able to solve the problem I had been working on for the previous five years!

I was 26 years old when IHES offered me the position, an important number in string theory! In the end, I spent 26/2 years at IHES, and I did solve (to the extent really good problems can actually be solved, some say they can only be approached), the problem I wanted to solve. I got to know extremely bright people, learned a lot, and got very much inspired by the place. The environment stimulated me enormously, and I wrote some of my best papers while being there. So, everything worked out in the best possible way.

What exactly were you working on during your time at IHES?

The reason I was offered a position at IHES was probably my 1998 paper with Albert S. Schwarz, who, at the age of 90, is still very active and regularly visits IHES. Our paper on Instantons on noncommutative ℝ⁴ and (2,0)-superconformal six-dimensional theory interpreted certain mathematical constructions, developed in the late 1980s and early 1990s by algebraic geometers (such as David Gieseker, Hiraku Nakajima, and others), in the light of non-commutative geometry (proposed by Alain Connes), in a way that was important to physicists.

Mathematicians often study structures, be they geometric, algebraic, combinatorial… Often, these structures have parameters, usually called moduli, i.e., they come in families. You may have singularities in these families, and you may have to deal with them. One way to do this is to compactify the moduli space. Doing so, you might get new geometric structures, and you need to find an interpretation of the compactification, such that the limiting objects also belong to the structure you were exploring. One example of such a construction, which came from physics in the early 1970s, is the moduli spaces of instantons (discovered by A.A. Belavin, A.M. Polyakov, A.S. Schwarz and Yu.S. Tyupkin), which was studied a lot by mathematicians in the 1980s. Instantons are special solutions of equations arising in Yang-Mills theory that provide an approximation to correlation functions in gauge theory, a theory that describes elementary particles and their interactions. Unexpectedly, mathematicians, most notably Simon Donaldson, were able to use instantons to produce new invariants of smooth structures on manifolds.

In my research, I tried to extract useful information from the moduli space of instantons to answer questions in quantum field theory. What we did with Albert was to recast a compactification of the moduli space of instantons in a way that would be useful for physicists.

At the time, IHES visitors were quite involved in the revival of interest in noncommutative geometry in physics, in the light of developments in string theory having to do with dualities. Indeed, people realized that some specific examples of non-commutative geometries can be realized in string theory. The conclusions of our paper with Albert inspired Nathan Seiberg and Edward Witten to write String Theory and Noncommutative Geometry, one of their most cited papers. It was an exciting time for the subject, and I was supposed to continue this research at IHES.

However, I was actually interested in something for which my paper with Albert was only a stepping stone: learning how to integrate quantities of interest to physicists on the moduli space of instantons. My work with Albert helped me to recast this integration procedure in a new useful way. In 2002, I then wrote a paper (see below) in which I was able to produce exact results in special kinds of quantum gauge theory, helping me to rigorously prove a 1994 conjecture by Seiberg and Witten on effective field theory (this was the problem I was interested in!).

This conjecture was about one of the main problems in theoretical physics, which is to understand how the microscopic description of a physical system translates into the description of the system at low energies, i.e., the macroscopic picture. One of the ways to formulate this translation is effective field theory. In 1994, Seiberg and Witten, in Monopole Condensation, and Confinement In N=2 Supersymmetric Yang-Mills Theory proposed a candidate for such a theory, guided by symmetry arguments. For several years no one could derive the theory rigorously from first principles.

In my 2002 paper Seiberg-Witten Prepotential from Instanton Counting, I formulated a strategy to derive this theory rigorously. As sometimes happens in mathematics, if you cannot solve the problem head on, you generalize it. This is what I did, reducing the infinite dimensional path integral to an infinite sum. I saw that this sum not only contains the Seiberg-Witten geometry, but also knows about the Gromov-Witten theory of special Calabi-Yau manifolds, two dimensional conformal field theories, and other exciting things. It was in evaluating this sum that I got stuck. This is where the IHES magic kicked in. At the Bures-sur-Yvette train station, I met Andrei Okounkov, a remarkable mathematician, who was just passing by IHES. We started to discuss what we were working on, and I told him about my problem. I needed to take the sum over a family of Young diagrams, and Andrei told me that this was his bread and butter, that he was basically summing over Young diagrams from dusk to dawn.

We thus started to look at the problem together, and he recognized that what I wanted to do was actually a generalization of the limit shape problem in asymptotic representation theory, which was solved simultaneously in the late 1970s by mathematicians Anatoly Vershik and Sergei Kerov. This insight allowed us to solve the problem, and that led to an enormous amount of activity. Even today, this area of research in physics is very active, and I am going to lecture about the material at this year’s Les Houches Summer School.

Now that’s just one example of the beauty of IHES, very much advocated for by Dennis Sullivan: you get to meet people with great expertise, and often, this yields unexpected connections to the problem you are working on. My work with Andrei is a good example of the synergies that IHES may provide to its scientists.

How did you start to get involved with Friends of IHES?

My involvement with Friends of IHES started right when I joined the Institute. Jean-Pierre would always ask me to present what is done at IHES to the public, both in France (IHES sometimes hosts open house events for the general public) and in the United States. I remember giving several presentations about the Institute at the French Consulate, and at the Centre culturel français (Villa Albertine) in New York. More generally, I was involved in spreading information about the Institute in the United States. That’s also how I met Jim and Marilyn Simons, around 2003. It was a very sad time for the Simons family because they had just lost their son. Nevertheless, to my surprise and awe, they never canceled a single event in support of IHES. I find Simons’ commitment truly remarkable. I got to know them better, and we became good friends, raising money for IHES. I also got involved in their efforts to raise the level of physics and mathematics at Stony Brook, notably through the Simons lectures, Simons summer workshops and, eventually, via the creation of the Simons Center for Geometry and Physics.

My connection to Friends of IHES is deeply linked to my friendship with Jim and Marilyn. Leaving IHES was a difficult decision for me. I was torn because I wanted to do what is best for IHES, for Jim and Marilyn’s project, and of course for myself. In the end, I decided to move, but it was always clear to me that I wanted to keep in touch with IHES.

The Institute really means a lot to me, and I consider it my family. Continuing to be involved with Friends of IHES after my departure was the simplest and most natural thing for me to do.

Is there a Friends of IHES event for which you have particularly fond memories?

I have a lot of fond memories with Friends of IHES, but one of the most extraordinary events I remember was the 2015 Gala at the Pierre Hotel in New York where, with the help of some of my friends in the film industry, I was able to get the performing artist Marina Abramović to attend. Everybody was surprised and amazed that she was there in person, and we had great fun. It was fascinating to see scientists interact with an artist.

The memory of that gala is also an emotional moment for me because it was the last time I saw Joe Polchinski, a pioneer in string theory, the inventor of D-branes (whose mathematical counterparts were studied by Maxim Kontsevich at IHES), whom I also invited to attend. Just before the gala Joe, Peter van Nieuwenhuizen and I took a stroll around the area, went up to Madison Avenue, where he jokingly put his head in between the jaws of the Tyrannosaurus Rex featured, at the time, in the atrium of the IBM building. He was his sparkling self later that evening in the gala, entertaining guests with stories of high energy physics. Sadly, shortly after, he was diagnosed with brain tumor which eventually took his life three years later.

Now that you are stepping down from the Friends of IHES Board of Directors, is there anything that you believe is important to develop for the organization in the future?

The main thing I would advocate for is to continue to diversify the portfolio of Friends of IHES members and donors. One direction I think is important to pursue is to make the Institute and Friends of IHES better known in the Silicon Valley, and more generally on the West Coast.

Historically, Friends of IHES has been very active on the East Coast, particularly in New York, and its benefactors often work in finance. I think that the tech industry is not adequately represented in Friends of IHES for now. There is definitely potential there because these people often have a scientific background, and their companies heavily rely on innovations building upon cutting-edge research in mathematics and physics.

Photo credit: © Alina Artemeva