BEAUTIFUL EQUATIONS

Six Hours with Nikita Nekrasov: Mathematical Physicist and String Theorist

A discussion of black holes, consciousness, and long durational collaboration

Professor Nikita Nekrasov stands at the front of his classroom scratching out figures on a large chalkboard. He writes so quickly that chalk dust flies from the board, somehow avoiding his crisp dress-shirt and slacks.

The subject of the lecture: “Instantons on Non-Commutative Spaces.”

The rain outside is barely audible over the sound of equation-scratching and Nekrasov’s calm, Moscow-inflected tone. In the first ten minutes, I recognize only the terms "plumber," "plug the hole," "eliminate the variable," and the letters M, K, N, and Z. Many of the symbols on the board also read like a foreign alphabet.

After about twenty minutes of work on what appears to be a single problem, Nekrasov notes that two variables in the equation are like “beasts from different zoos.”

An appropriate metaphor. Theoretical physics aims to tame the wild beasts of the universe that are arguably the most difficult to understand: particles so small that it is impossible to observe them experimentally.

To test the phenomena that Nekrasov studies, one would need to produce so much energy that a black hole would be created in the process. No light could escape this warp in space-time and therefore, no experimental evidence would ever be retrieved.

Nekrasov buys lunch. After our meal, we take a sluggish elevator to the fifth floor ("It has to think about where it’s going," he remarks). His office is pristine but for a giant, chaotic pile of papers on his desk. His shelves are lined with books in Russian, French, and English. A large coffee table displays a potted orchid and two large books of photography.

In physics circles, Nekrasov is known for his work on supersymmetric gauge theory and string theory. He was awarded the Jacques Herbrand Prize of French Academy of Sciences in 2004, the Hermann Weyl Prize in 2004, and the Compositio Prize in 2009. These awards are not on display. The wall opposite Nekrasov's desk consists solely of an enormous chalkboard. There is a smaller one hanging outside his office door and countless more lining the halls of the Simons Center for Geometry and Physics. I comment about this abundance of writing surfaces.

“That’s nothing,” he replies. “At a university I visited in Vienna, they have three chalkboards in each bathroom.”

He laughs. “You never know where you’ll get an idea.”

Once he’s seated at his desk, we begin.

SIENA ORISTAGLIO: Can you start by stating your name, where you’re from, and what you do, by way of introduction?

 

NIKITA NEKRASOV: This is already complicated. [Laughs.] My name is Nikita Nekrasov in English, Nikita Nekrasov in Russian, and Nikita Nekrasov in French. I was born in Moscow, Russia, studied at Moscow Institute of Physics and Technology for five or six years, and then came to the United States to study at Princeton University, where I got my Ph.D. I worked at Harvard, Princeton, and then was hired by the Institut des Hautes Etudes Scientifiques France near Paris, where I worked for 13 years. Now I’m back in the United States, here at the Simons Center for Geometry and Physics, as a professor of physics.

 

SIENA ORISTAGLIO: Throughout this time, you studied physics and mathematics?

 

NIKITA NEKRASOV: Yes. I position myself as theoretical physicist, but my research is somewhere on the boundary between physics and mathematics. Where it is exactly is hard to define. Sometimes I say that hardcore physicists call me a mathematician and hardcore mathematicians call me a physicist, so I am nowhere. Just like with the countries. [Laughs.]

 

Nikita Nekrasov at The Simons Center for Geometry and Physics at Stony Brook University Photo by Maria Sprowls

Nikita Nekrasov at The Simons Center for Geometry and Physics at Stony Brook University

Photo by Maria Sprowls

SIENA ORISTAGLIO: Were either of your parents in those fields?

 

NIKITA NEKRASOV: Both my mother and my father were doing research that was related to mathematics but it was mainly focused on economics. They were interested in mathematical methods of economics — models that nowadays are very fashionable — but they were doing this in the Soviet Union 50 years ago, when everything in economics was supposed to be according to a plan with no predictions and no models, so they didn’t have a chance to fulfill their mathematical inclinations. Somehow they probably passed these inclinations down to me.

 

SIENA ORISTAGLIO: Can you briefly describe the lecture that you just gave?

 

NIKITA NEKRASOV: I was lecturing my students to prepare them for the research that they will be undertaking. The general domain of these lectures is gauge theory, which is a branch of theoretical physics that describes interactions of elementary particles. Generally, we look at the forces that keep the atoms and molecules together, that keep electrons and nuclei together inside the atom, and that keep protons and neutrons together inside the nuclei. All of these forces are described by what physicists call gauge theory, and it has a sophisticated mathematical apparatus, which is mostly what I was lecturing about.

The first gauge theory was invented by James Clerk Maxwell, who wrote equations describing electromagnetic forces in the 19th century. Many years later, maybe in the 60s of the 20th century, people realized that Maxwell equations have generalizations that could possibly describe other forces. The major force in this discovery was the Chinese-born American physicist C.N. Yang, who was one of the founders of this physics center.

 

It turns out that at the same time physicists were discussing the ways to describe various forces of nature, mathematicians were building the theory that described the same thing but in a different language.

 

C.N. Yang discovered forces, but he also was a force of nature of his own. [Laughs.] He was one of the great enthusiasts of interactions between physicists and mathematicians. Another such enthusiast was Jim Simons, who was a mathematician here, and together they held joint seminars at which physicists and mathematicians would discuss their fields. It turns out that at the same time physicists were discussing the ways to describe various forces of nature, mathematicians were building the theory that described the same thing but in a different language. Now, this is called gauge theory and is understood by both physicists and mathematicians.

 

SIENA ORISTAGLIO:  I also noticed the term “non-commutative space” in the title of the lecture. What does this mean?

 

NIKITA NEKRASOV:  I work with all kinds of spaces in my research: flat spaces, such as Euclidean, or non-Euclidean (Minkowski), and curved spaces. A non-commutative space is a kind of space that is sort of exotic and also, for the moment, is just a theoretical concept. This is a space where you cannot even define the notion of point. It’s a space where points are fuzzy. They have a kind of vague notion of being close to each other, but not exactly there. [Brings his fingertips together at a single point.]

These are concepts that are potentially useful for describing nature but they may end up being purely theoretical. Up to now, we don’t know the true nature of space-time because if we try to look at things that occur at very small distances, it costs energy. Take the Large Hadron Collider in Geneva, Switzerland, for example. When physicists there explore very small distances, the smallest distances currently possible, they have to pay for the electricity bill, which is enormous, really.

Interesting distances from the point of view of theoretical physics are even smaller, twenty order of magnitudes smaller. If you make a microscope — or, in the case of physics, an accelerator — which shoots particles with high enough energy that they can explore distances this small, you would create a black hole. This is a curved region of space-time from which even light cannot escape. This is what would happen if you tried to explore things experimentally at this very small scale. The moment you look in, the door closes on you, so you don’t know what’s there.

 

If you make a microscope — or, in the case of physics, an accelerator — which shoots particles with high enough energy that they can explore distances this small, you would create a black hole.

 

That’s what makes exploring space-time at small distances very complicated, theoretically — you create a paradox. You need a lot of energy to “zoom in,” but once you do so, this energy destroys what you wanted to explore by creating a black hole at that place. This leaves only theoretical research as a possibility.

 

SIENA ORISTAGLIO: Does it ever bother you that you cannot test your research experimentally?

 

NIKITA NEKRASOV: I have been studying a similar area of physics since my PhD. I understand things much better now than 20 years ago, but I cannot say that I drastically changed my field of research. It’s still quantum field theory, gauge theory, string theory, this whole area of physics. It’s complicated because in a sense you are asking fundamental questions about nature but there are no experimental tools to confirm or disprove your theories. Part of the challenge is to find some predictions or indirect consequences of the theories that would be testable and experimentally verifiable or falsifiable, but, no, I am not particularly interested in that, actually. I find my main driving force is actually the mathematical beauty of the equations I am writing. There are other people who are more interested in connecting theoretical research to experimental physics, but this is my way.

 

I find my main driving force is actually the mathematical beauty of the equations I am writing.

 

SIENA ORISTAGLIO: Are there any particularly long durational aspects of your own professional work?

 

NIKITA NEKRASOV:  There are different kinds of researchers in physics and mathematics — or really, in any field. Some people like to find a new problem, work on it for a couple of years, maybe have a solution or be fed up with it, and then switch to something else. This is actually considered to be healthy, especially in America. Some people dig deeper and deeper and deeper into something and take pleasure in learning more and more about their subject. I’m somewhere in between. There are problems that I work on for many, many years, and in a sense, all of my life is a kind of preparatory work on that problem. Then there are some smaller problems that I work on, solve, and then move onto something else. The fact that I am still interested in the things I was interested in 20 years ago means that for me, my whole physics research is a long durational work, and everything I do is a preparation for the next work, or for deeper work.

 

SIENA ORISTAGLIO: Can you talk about the importance of long durational work in physics research, generally?

 

NIKITA NEKRASOV: One of the predictions of Einstein’s theory of relativity was that the geometry of space-time — these days, the fashionable word is the “fabric” of space-time — can have ripples, it can fluctuate in changes which propagate in time and space. These are called gravitational waves and the prediction of Einstein’s theory of relativity is that — just like the motion of charged particles in your radio antenna emits radio waves, which are electromagnetic waves — the particular motion of any massive object emits gravitational waves. The difference is the intensity of gravitational waves is much weaker. Gravitational interaction is relatively weak. So it’s very hard to observe them directly, these gravitational waves.

People were looking for sources of this radiation, so the bright idea was to look at extremely rapidly rotating stars, called pulsars. These are neutron stars that are very compact, very small, and massive at the same time, which makes them very fast rotators. People found these pulsars in the sky by looking for radio signals of a specific shape. One experiment followed one of these pulsars over the course of 30 years and they observed that the rate, the angular velocity, of that pulsar decreased in time, which meant that somehow that star was losing energy. Since it was not interacting with anything else, there was no other way to lose this energy besides through emitting these gravitational waves.

 

The fact that I am still interested in the things I was interested in 20 years ago means that for me, my whole physics research is a long durational work, and everything I do is a preparation for the next work, or for deeper work.

 

People made the calculation, which was quite a non-trivial calculation. Actually, Thibault Damour, one of my colleagues at the institute in France that I worked at, worked on these calculations as a graduate student. The predicted energy loss was found to be in very good agreement with what physicists observed, so this was one of the last major confirmations of Einstein’s theory of relativity.

It took 30 years because the effect is so small, so at every moment of time, the energy loss is really tiny. In fact, it was very lucky that it only took 30 years. I mean, 30 years seems long, but it’s still within the lifetime of a person — it could have taken 300 years, which would have required much more devotion and many more resources. So that’s the experiment of Joseph Taylor, Joel Weisberg, and others that confirmed the existence of gravitational waves, indirectly.

 

SIENA ORISTAGLIO: You described that something that you love about your work is the beauty of some mathematical equations. How do you conceptualize beauty with respect to mathematics?

 

NIKITA NEKRASOV: Well, it’s something you learn to appreciate. It’s like with wine or food. First, you are simply hungry and you would eat anything, and then you learn to differentiate between different kinds of foods. It’s the same with physics and math. First, you are just happy that you computed this integral or that your equation makes sense. And then you start to appreciate that, well, this equation is simple but beautiful because it explains many things in concise form. A short equation that has a lot of content is beautiful. A long equation that has less content is ugly. It’s something along these lines — simplicity but complexity at the same time. A beautiful equation should also leave some room for imagination and contemplation. It shouldn’t be too obvious. If it is obvious, it is not beautiful, and it is not interesting.

 

A short equation that has a lot of content is beautiful. A long equation that has less content is ugly ... A beautiful equation should also leave some room for imagination and contemplation.

 

The most difficult thing in physics is to formulate a theory in simple words, without too much machinery or introduction. That’s difficult. That’s mastery. The fact that I have to lecture my students for many months in order to prepare them for their research is actually a bad sign, in a sense, but that’s the state of the art. I wish it were simpler, but I’m not there yet. I have another 20 years left to simplify. [Laughs.] I hope.

 

Nikita Nekrasov Lectures at The Simons Center for Geometry and Physics at Stony Brook University

Photo by Maria Sprowls


SIENA ORISTAGLIO: Can you talk about how you came to be introduced to MAI? Are there any things that excite you about our mission?

 

NIKITA NEKRASOV:  I met Marina Abramovic at the filming of the Dau Project, which was a film by Ilya Khrzhanovsky, originally centered around the life of Russian physicist Lev Landau. Part of that film project involved building a film set, which was called the Institute, which represented a research institute where people worked and lived. It was an interesting place to try to live in and to do what I normally do in a different environment for many reasons — partly for historical reasons but also for personal reasons. It was a way to explore myself and learn about the way I interact with interesting and difficult circumstances. Marina was one of the participants of the project, which is how we met. Since then, I’ve been interested in all kinds of platforms where scientists and artists can interact. I find this inspirational for my research and hopefully for the artists involved as well.

Just as before, when there was a great potential in the interactions between physicists and mathematicians, I believe there is a great potential in interactions between scientists and artists of all kinds. It remains to be seen how to make these interactions sensible. Maybe we will not see it in the immediate future, maybe it will take time, but it is my intuition — I have a feeling — that it’s there and should be explored. The fact that Marina is building an institute sounds like a great step in that direction.

 

Just as before, when there was a great potential in the interactions between physicists and mathematicians, I believe there is a great potential in interactions between scientists and artists of all kinds.

 

SIENA ORISTAGLIO: I notice that you have this Egon Schiele book here, and that you have some interest in art, generally speaking. Are there particular works of immaterial art that have impacted you on a personal or a professional level?

 

NIKITA NEKRASOV: Well, any good opera or film impacts me. I am quite sensitive, actually, to all forms of art. I can’t really tell you that there was a particular piece of art which I was in contact with that led me to some discovery, but there is a general state of mind which art puts me into that helps me to expand the boundaries of the domain in which I am currently thinking.

I like paintings of the Russian avante-garde, or of French impressionism, or, yes, Egon Schiele here. I like the way my wife paints, for example. I find all this inspiring. I like opera, and I’m very happy to live not far from the city so that from time to time we can go to the Met and listen to various great performers. In particular, some of them are Russian, like the great singer Anna Netrebko. I actually found some of the works of Marina quite inspiring, like “The Artist Is Present” (2010). Troubling, some of them are troubling, which is also a good thing — like the performance with the onion.

 

SIENA ORISTAGLIO: Art certainly explores the unknown in a way that is more abstract than science. But it's true to say that your field is also exploring the unknown...

 

NIKITA NEKRASOV: Yes. Yes and no. You see, it’s not like we are smoking pot and coming up with some crazy ideas. What we do is part of science. In science, we have rules. Two plus two is equal to four. You cannot change that. You can change the meaning of “four” but you cannot say that, well, today, I feel good — it will be five. In a sense, we are constrained by these rules. The fact that the rules are hard to explain means that sometimes people think we do not know what we are talking about, because it’s very hard to communicate this in a way that people will accept. But if you say that you want me to think in terms that are even more abstract than science, I am willing to try, but I might not know how to begin.

Nikita Nekrasov at The Simons Center for Geometry and Physics at Stony Brook University

Photo by Maria Sprowls

This is where the long durational part is important. In order to appreciate science, there is no other way than to suffer through long periods of time to absorb it. If you see an artist’s impression of what science is and you spend five minutes looking at the picture and the artist maybe spent two days drawing the picture, this sacrifice of time was not there. The artist didn’t appreciate the whole depth of what the physics community had to go through to come up with even this seemingly simple statement of Einstein — that gravitational force is not a Newtonian force, it’s really just the curvature of space-time.

This is a very simple statement, but it compresses centuries and centuries of research, experiments, and observations. If an artist just draws a curved line for you and says, "Well, this is what gravitational force is," it might even be an accurate impression of what it’s like to move in a gravitational field, but you would not appreciate it properly because you didn’t live with it for a long time. You have to live with it for a long time.

 

SIENA ORISTAGLIO: This is something that we’re talking about a lot at MAI — and this is my experience working at the intersection of arts and sciences. Often times, if you have a short collaboration between people who come from different backgrounds, you end up with very reductive results and content. But if you can imagine a very long durational collaboration between an artist and a scientist where both have the time to experience the depth of what the other person is working on and the historical context for this work, this is where we see real potential for meaningful collaboration. We encourage the audience to spend time with the work as well.


NIKITA NEKRASOV: Yes. This is why I think MAI is a very interesting initiative, and I am very hopeful.

 

SIENA ORISTAGLIO: Speaking of which, would you be interested in collaborating with artists on a long durational project in the future?

 

NIKITA NEKRASOV:  Sure, yes. I like artists, I like arts, and I like to collaborate with people. [Laughs.] As I said, it remains to be seen whether there is anything useful or reasonable that can come from these collaborations, but I’m willing to try it. Of course.

 

SIENA ORISTAGLIO: I saw you did some drawings on the board during the lectures. Do you visualize in your mind when you do research?

 

NIKITA NEKRASOV: Yes, I do both. I use equations lots, so when I do calculations I work with symbols and numbers. But if I have an equation and there is some consequence of this equation, I also like to draw a picture that helps to understand what the equation means. Sometimes pictures help because I can see structures that would be hard to see in the equations.

 

I like to think of consciousness as a new form of complexity.

 

SIENA ORISTAGLIO: You mentioned earlier that art puts you in a certain state of mind. I recently read some studies claiming that consciousness is being looked at as a new state of matter, and that physicists are beginning to theorize this. Can you speak to this topic?

 

NIKITA NEKRASOV: I am not an expert in this field, but the question of consciousness interests me, actually, as a physicist. I like to think of consciousness as a new form of complexity. After all, our brains are built out of elementary blocks, which we can understand as simple physical devices, but somehow the way that they are arranged is such that the physical system that they form operates at a different level than physical systems that we normally discuss and understand.

I like to think of this as some kind of phase transition, which may be similar to the phase transition between ice and water, water and vapor. Clearly, there is something non-trivial happening here. I don’t think anybody knows the adequate language to describe consciousness. “State of matter” in the physics sense of the word is probably the wrong term, but maybe the right term is not far from that.

 

SIENA ORISTAGLIO: Another article I read recently said that there was a structure in quantum physics that was found to be crystal-shaped. Can you talk about this?

 

NIKITA NEKRASOV:  Yes, this is actually part of my research. In a sense, this is the place where I think in pictures. Let’s go back to this discussion about how it is very hard to explore space-time at small distances because you create black holes. Because of that difficulty, British physicist Steven Hawking came up with the concept that he called space-time foam — a long time ago, maybe in the 70s, early 80s. He said that even though, macroscopically, at our scales, space-time looks flat — meaning that lines are straight and parallel lines don’t intersect — at small scales, space-time looks like a foam. It has bubbles that are created and then disappear like a boiling soup. These bubbles are pieces of space-time and they are curved, so they are like little black holes.

People have tried to describe this bubbling space-time, but for a long there were no adequate descriptions because this is precisely where quantum physics gets into contradiction with the Einstein’s general theory of relativity. Then, there was a proposal for a theory that would allow Quantum Physics and general relativity to co-exist. This was string theory. String theory posits that instead of elementary particles, which are point-like, you have elementary strings, which are one-dimensional objects, like little rings. There are some mathematical apparatuses that come out of this assumption that allow us to describe things that are at the same time quantum and gravitating.

Unfortunately, for many years, it was not possible to apply string theory to describe Hawking’s space-time foam, this boiling space-time, because people understood string theory mainly as a way to better describe gravitational waves. So, sometime ago, myself and my collaborators at Harvard, here at Stony Brook, at Princeton, and at Columbia University, came up with a mathematical model of quantum space-time that looks like a piece of three-dimensional crystal.

 

Nekrasov pauses. “I’ll need to explain this with a chalkboard. Do you mind?”

I don’t mind.


Forthcoming from IMMATERIAL:

Professor Nikita Nekrasov Explains Crystal Melting Theory (Video)


Photograph by Maria Sprowls

Photograph by Maria Sprowls

Nikita Nekrasov was a visiting professor at the Simons Center in 2009 - 2010 and again from 2011 to the present. He earned his Ph.D. at Princeton University in 1996 under the supervision of David Gross. His dissertation was on Four Dimensional Holomorphic Theories. Nekrasov was a postdoc in physics at Harvard University, a Junior Fellow at the Harvard Society of Fellows in 1996-1999, and the Dicke Fellow at Princeton University in 1999-2000, before becoming a permanent professor at the Institut des Hautes Études Scientifiques. He is a leading mathematical physicist, and the world expert in non-pertubative calculations in supersymmetric gauge theory. Among his best-known achievements is the proof of the Seiberg-Witten solution of Donaldson theory in 2002, which had been one of the major unsolved problems in the Quantum Field theory since 1994. He is also well known for his work on non-commutative geometry, topological string theory and ADHM construction, for which he received the Hermann Weyl Prize in 2004. While at the Center, Nikita has co-organized many programs and workshops including the Fall 2012 program ‘Integrability in Modern Theoretical and Mathematical Physics’, along with Samson Shatashvili, and the workshops ‘Gauge Theory Angle at Intergrability’ in 2012 and Branes and Bethe Ansatz in Supersymmetric Gauge Theories’ in 2011.