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Nikita Nekrasov: Why Physicists Say We Don't Understand Quantum Field Theory
July 14, 2025
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Some people will say they of course understand everything about quantum field theory, but we don't understand it as the complete structure. To really understand superposition as a quantum, you have to be a quantum entity.
Professor Nikita Nekrasov unified previously disconnected fields, instanton moduli spaces, random partitions, integrable systems, and quantum strings. What was required was so novel that when he was presenting at conferences, other physicists couldn't tell if his results contradicted or confirmed theirs. His mentor David Gross advised him, keep your poker face. That gamble paid off. It turns out four dimensions aren't arbitrary.
I'm Kurt Jaimungal and I interview researchers regarding their theories of reality with technical depth. Today we explore how non-commutative geometry cures quantum singularities, why understanding superposition requires becoming quantum yourself, and Nikita's speculative connection between R4's exotic smooth structures and the chemistry undergirding life itself. From falling asleep during Witten's seminal lecture to late-night residue calculations with Greg Moore, this is mathematical physics
at its most human and revolutionary. Why don't we understand quantum field theory? So it probably depends on who you ask. So some people will say they of course understand everything about quantum field theory. Practitioners who use it, they will tell you that they understand because they know how to use it and they know how to do some calculations which will then can be
may be compared to experiment, and more often than not, the comparison is favorable. But we don't, maybe as theoretical physicists with some kind of mathematical ambitions, we don't understand it as the complete structure built out of axioms, let's say.
So you want an axiomatic quantum field theory that reproduces all the successes of the standard model? Something which, well, we would like to have some basic principles, like maybe not axioms, but some basic principles from which we can build structure from bottom to top, bottom up.
We have many, many examples of what we think quantum field theory is. So many examples of quantum field theories. And there are many overlaps, but there are also missing kind of areas. We don't know whether gravity is part of quantum field theory or it's something else which requires a different approach. And so there are many, many questions like that.
So there is, of course, a very, very practical, practical approach, which works with many, many theories which, which are involved in standard model, which you can just simulate them on a computer to do lattice lattice field theory. Right. There are various various solities with that, because not all structures which we know there are in
in smooth fields and fields defined on smooth manifolds. Manifolds may have different structures like spin structure. All those things are kind of hard to represent faithfully on the lattice. But with some room for error, people do that. But that still looks like a bypass, not a hole.
the whole thing. And we know examples of quantum field theories which probably don't have lattice description. And so the work which we do, some of us do, which we're actually trying to chip away at the unknown in connecting us to those theories by coming up with
Maybe mathematical conjectures which sort of follow from the fact that these theories exist and then checking them maybe by more standard means and then gaining more confidence that the reason for those conjectures to be there probably is valid. Now we're going to build up to gauge origami, but before we do, I want to get to another one of your constructions. So Dirac said to Feynman, infamously, do you have an equation
And Feynman could have said to Dirac, well, do you have a diagram? And now I'm thinking you could have chimed in and said, well, do any of you have a partition function? Well, they, of course they had lots of partition functions. Yes, yes. But anyhow, you have one named after you. So let's get to that story. I believe this was around 2002 where you cracked the Cyberg-Witten instanton counting puzzle.
Which was when 1990 in the mid 1990s, 1994. Yeah, right. And it was there where you devise the Microsoft partition function. So tell us about what that partition function is and what it was like coming up with that solution. Was it a lightning bolt? Was it an incremental climb? What did that moment feel like?
Yeah, so this is a story which is a kind of worth a book because it was it's a story worth of many sleepless nights, but also there's a book called The Count of Monte Cristo. So yours would be The Count of Instant Hans. Right. Right. Thank you for observing the analogy. Right. You have to dig underground tunnel from from some predicament you are to actually to get to the
You know, dry land and some people say there is also vengeance involved, but in my story, it's all pure love. Are you sure? Well, at least that's my inner work. So my inner work is to realize that all we do in science is joy. Even though it feels like suffering along the way, but
The goal is to recognize beauty and maybe add a little bit of beauty and use it later, just because beautiful things are good to have around and sometimes they're useful. So the story started, I don't know when it started actually, maybe started in 1992. Witten wrote a paper which was called Two-dimensional Gauge Theory Revisited.
and in that paper he proposed to use a technique from rather abstract mathematics at the time called a covariant localization to compute the path integral so it's fine an integral over trajectories or field configurations in example of our
rather simple but nevertheless interesting gauge theory, two-dimensional Young-Mills theory. So our world is described to some approximation by four-dimensional Young-Mills theory, quantum Young-Mills theory. But if you imagine a world in which there is only one space dimension and one time dimension, then you could study the simplified version of Young-Mills theory
And that was a theory which was actually solved by several tools. But one tool which was kind of interesting was by Sasha Migdal, Alexander Migdal, who actually first defined it using lattice approximation. And then he found out that it was an interesting theory in which you can define it on any kind of lattice. You can make the lattice finer or
or economical. So use as little number of edges as possible. Yes. Given the topology of space time and the the partition function will be the same. So it was interesting theory, which was almost topological. We would not call it now almost topological. It only dependent on the area of space time. So space time here, I'm using a little bit of a jargon.
The theory is interesting when space-time is not physical space-time when you have time and space, but what we sometimes call Euclidean space-time. So it's a manifold in which the notion of a distance between points is similar to the notion of distance between points in space. There is an ordinary mining metric.
How disconnected to the space time where the notion of distance involves events. And so some some distances can have negative square, for example. That's a that's a story which we can discuss, but let's not get there. OK, anyway, so we then observed that this this theory, the reason why McDowell was able to solve it and get a very interesting answer.
So we can observe that it had to do with with certain topological structures present in the final dimensional space, which you can associate to any two dimensional surface. We call them Riemann surface. So it's like a surface of a well, your audience knows probably. So Riemann surfaces have if they are intimately, they have only one one
number which characterizes the topology of the number of handles. Young-Mills theory deals with connections, with ways to transport things from one point to another in some bundle over the base space.
The lowest energy configurations, lowest action configurations in Yanin's theory are the connections which have zero curvature, which means that if you try to, if you transport things around small loops, the result doesn't depend on which loop you choose, as long as the loops are small. But if the loops are not small, if they wind around handles, so they cannot be contracted to a point, then those transports can be non-trivial.
This fact actually was observed in physics in the famous our own bomb experiment. So it's right. It's a famous, famous story which showed to the world and to physicists also that the connection, the vector potential actually has a meaning, has a physical meaning, not just the curvature, which if you only if you if you're familiar with Maxwell equations, you can
uh, write them only using the curvature, this fuel strength. And so for a while, people thought that the vector potential is not really meaningful because first of all, it's not universally defined. You can make the so-called gauge information and change it. But the fact that they transport, especially the transport observed by quantum quantum particle depends on the, on the path, which can be non-contractable. Uh, well, it was a big deal. And so,
On the two-dimensional human surface, because you have several handles, you can have those transports and then they can have pretty much arbitrary values. And there is a space which parameterizes them, which is called the modular space of light connections. These days, this space is, of course, very popular in the stories involving geometric wavelengths program. But anyway, so if you're
Connection, if your gauge theory is defined with compact gauge group, then it's some complex space, although with singularities, because you have symmetries and you divide by symmetries, and sometimes symmetries act not freely, so it's a singular space. But you can define its volume. It has a natural, well, it turns out it actually could be used, viewed as a phase space. It has a symplectic structure.
which was actually found by, I guess, at here in both in the early eighties. And so there is a natural number which you can associate to every human surface, which is the volume of that modular space computed in the Louisville measure. Right. Right. And so we can compute this number by a very clever, clever trick using localization applied to two dimensional theory. Got beautiful answers involving functions. So some some of us now call them Riemann written data functions because they
In 1992, I was just beginning my studies. I was very much impressed by the paper. In fact, my friend and later collaborator, Andrei Losyev, suggested I study that paper and present it at the seminar.
No, I come from Moscow, so I was attending some string theory seminars there, and the tradition there was that you study some paper, and this was the beginning of archive. So it was the first, maybe second year of archive. So the papers from the West started coming in not by
And at that point you were an undergraduate?
I was an undergraduate at the time, so it was my probably third year after high school. Right, right. So, but I mean, I already knew what I wanted to do. So I mean, I was lucky to, you know, to get interested in mathematical physics in what I call mathematical physics early on.
And so. Somehow I got into this crowd of people, group of people discussing whatever was interesting. So I was very much impressed by this paper and I thought, OK, maybe one can use this localization to compute interesting things in other theories. And I remember we were discussing with my
other friend and also collaborator later, Sasha Gorsky. And we ask ourselves, why couldn't we apply this supersymmetric technique to quantum mechanics, especially quantum integrable systems? Later on, the spectrum was analyzed by Haldane, Duncan Haldane, who got Nobel Prize for many, many things. And so the way
The spectrum was organized, is now called Haldane statistics. So these particles, because of this particular interaction, they kind of behave as neither bosons nor fermions, but something that can be in between. And so this coupling constant can be an interpolating parameter. Anyway,
Gorsky and I found that this model of particles moving on the circle and repelling each other with the potential which is proportional to inverse square of the distance, chord distance, is actually part of the two-dimensional Agnus theory. So quantum mechanics is equivalent to two-dimensional theory. So it's one of the interesting examples of deceiving dimension that
This is part of the structure which I mentioned when I say that we don't quite understand quantum field theory, because when we explain to engineers that quantum field theory is the continuous limit of something you define in the lattice, of course an engineer will think that the lattice is embedded into space-time of some dimensionality, so that's the dimensionality of the theory.
But it turns out that you can define a theory in one number of dimensions, but then by looking at special observables, so by restricting the set of observables you're allowed to. Yes, to look at. You will find you will not be able to distinguish it from the theory in the lower dimensions and vice versa. So you so we think we live in three plus one dimensional space time.
But it might be just that we don't have access to observables which can probe, you know, high dimensional space time. And eventually it said that your work implies that particles live across multiple dimensions at once. Maybe particles is not the correct word here, though. When I was talking about particles in the collider model, that was more of a
Just in a mental picture, like beads, like some small objects. So in particular, in this case, the way this relationship between quantum mechanics and field theory was working was that the role of those particles was played by the eigenvalues of the holonomies or the transport
of the two-dimensional gauge field around the circle of space. We found that those points in the circle behave, so time continues to be the time, so those points in the circle behave as particles in the Collogero-Sutherland model. The mathematicians call it the Collogero-Moser-Sutherland model because Jürgen Moser, the Swiss mathematician,
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So two years later, okay, so let's just keep it. So let's just remember that there is some relation between quantum many body systems and gauge theories with some supersymmetry. Okay. And that in gauge theory, certain things can be calculated exactly if you can recognize some symmetry and can use the mathematical tool of equivalent localization.
Got it. So two years later, it was 1994, and I was traveling, one can say backpacking through Europe, and I found myself in Paris. When it just happened, the International Congress of Mathematical Physics took place. And so there was an interesting talk. There were two talks by Witten there.
who presented his work with zyberg on zyberg-witten solution of n equals two gauge theory and on his work with kumrunwafa on the strong coupling tests of s-duality yes so that's s-duality of n equals four superagnos theory i spoke to kumrun on this podcast i'll place the link on screen and in the description as well right so uh
I should admit that, I mean, okay, so it was 1994. I was just out of my, uh, I just finished, uh, basically my undergraduate studies. I finished the, um, military bootcamp and I just went to Europe and I was going to graduate school later. It was a very hot summer. And at Wittenstock, uh, on Zabrik Witten theory, I fell asleep.
So it was, I mean, I, I remember, I mean, I wasn't there, but I remember you telling the story. Yes, I was asleep. I must be ashamed. I am ashamed. I could not really comprehend the significance of what was going on, but I got, I was impressed with the formula he wrote for certain quantity, which, which is called pre-potential of, of, of, of theory. So just to recall the abracadabra solution was the ansatz.
for what is called low energy effective action of some gauge theory they said that so this the quantity which they they wanted to compute the one which determines this low energy physics can be expanded in power series in instant dots so what are the instant dots so this gauge theory
being monobillion gauge theory. There are configurations of gauge fields in one could say in the vacuum where they fluctuate in a way which changes the topology of the gauge bundle in which they are defined. So let me maybe unpack that. The thing which
Not often appreciated is that the fact that our world is described by gauge fields means that, you know, maybe conceptually or maybe physically there are extra dimensions to the world we live in. Because what these gauge fields are, they describe the transport of things over our space-time, but which happen
In a kind of additional space, which is fibered over our space time. And this space could be a group. So typically one takes as a basic object, one takes the principal bundle. So the fiber of this vibration is just a group manifold, or it could be a space on which the group acts.
So, for example, fermions of the standard model, they are all sections of some bundles which are associated with some principal bundle. Now, one can take the totality of those fibers of a spacetime and look at the space, the resulting total space. So, it will be a manifold of dimensionality 4 plus the dimension of the group.
So if the group is SU2, the total space will be seven dimensional. If the group is SU3, the total space would be 12 dimensional and so on. Some people like group E8, so the total space would be 152 dimensional. At any rate, just given the fact that your base space is your four dimensional space time and the fiber is the standard group,
doesn't specify uniquely what would be the total space. So there are different ways of fibering this group over the base manifold. So the ways to parametrize, to enumerate the topology of those total spaces is the subject of the study in homotopic topology. And so there are
special, not people, special classes which are responsible for those classifications called characteristic classes found by Pontjagin and Chern. And so in doing gauge theory, we're actually doing a little bit of quantum gravity in the sense that we are summing over different topologies of this total space.
So, when I say summing, I mean it in quantum mechanical sense. So, if you think about Feynman path integral, which is the kind of integral over paths over possible trajectories of evolution of your system, engaged theory in the evolution of the system involves the choice of the total bundle, total space of a principal G bundle.
And so we're summing over the topologies of those of those bundles. Let me see if I can break this down for the audience so far. Okay, and you tell me if I'm incorrect at any point. So space time, we ordinarily think of it like x, y and z. And then well, there's also a time. So you
Get X and Y the plane that most people know about by doing X cross Y. So that's something that people are familiar with. You take R, R1, and then you cross it to get R2. And then you're wondering, well, what is this fiber business that people are talking about? Locally, meaning if you look closely, you can think of a fiber as another cross. So we're just crossing with a gauge group.
So your example is already in your example, we can give an example of a non-trivial bundle. Sure. So if your y variable is compact, it's actually not aligned by the circle. And now I'm adding x, sorry, x was first, let's say y was first. Sure, sure, sure.
So I want to fiber the line over a circle. And so one way I can just, you know, take a product, direct product. So I will get the space which I will get will be a cylinder. I think I even have a scene drawn on my blackboard somewhere, but anyway. So it's a cylinder. It's a tree. We would call it as it's a, it's a trivial bundle. So the cylinder, so the base is a circle and the fiber is a, it's a line.
But now let's, there's another option, which is to say that as the, as I go around the circle, my line flips orientation. And so it comes back as the same line, but it's inverted. So I used the action of the group Z two, the two element group, which acts on the real line by multiplying X by my minus one. So X goes to negative X.
And so that gives you another space, which is known as Möbius strip. And that's a non-trivial bundle. Right. But you can also fiber the circle over a sphere in a non-trivial way. So imagine, so let's take. Actually, let's take our two donuts. It's probably a good place to start. Now imagine, so I would think of the.
So donut is a product of a dimensional disk times a circle. So now what I want to do. And these two donuts currently are disconnected. Yeah. So they had separate two separated separate donuts. Unfortunately, I don't have donuts with me. Tim Hortons is what they would say here in Toronto. Right. Yes. Yes. Also coffee. Now,
I want to take the boundary of those donuts, of those two donuts. So those would be products of S1 times S1. Now I want to identify the first S1 so they will be the same S1 and glue them together. But the second S1, the fiber, I want to identify with a twist in such a way that as I go around the first S1 the second one makes the full rotation.
It's almost like this example with the Möbius strip, but now the rotation is 360 degrees and it's a rotation of a circle, not the flip of the line. So if you make a little mental exercise, glue these things together, you will realize that the resulting space is now a three-dimensional sphere. So one way to maybe imagine this is to
actually embed one so take one donut and then realize that the complement to this donut in the three dimensional space is actually topologically also a donut if you add the point of infinity and so the union of those two donuts is the complete three dimensional space with the point of infinity which is a three dimensional sphere and it turns out that if you are studying the parallel transport
On this bundle with non-trivial non-trivial topological class, the minimal Yang-Mills energy configuration will be not flat, so the curvature will be non-zero, but it will have a very interesting property. It will be self-dual or anti-self-dual. So this is something which is specific to four dimensions. So if you
List the coordinates of a four-dimensional spacetime, like x, y, z, and t. Suppose we orient them, so I say x, y, z, and t. Now, if you have a two-form, so a two-form is something which likes to take in two planes or two bi-vectors and produce numbers. So, for example, I take x, y.
But if I remember I had x, y, z, and t, so if I took x, y, then the remaining is z, t. And if I took x, z, the remaining would be y, t, and so on. So there is this duality between two forms, which is given by the orientation of the four-dimensional space. And so the minimal energy configurations turned out to be such for which the curvature
is cell dual. So if you apply duality to it in that sense, you get itself or maybe negative of that, depending on the sign of the topological class. So these are called instantons in the technical sense. But this is not the most general definition, of course, of what instanton is. And the significance of those solutions in quantum field theory is that
These are configurations for which most of the action in both literal and mathematical sense happens in a kind of compact region of space time. And here I should make a disclaimer that the space time I'm talking about is not the real space time. It's the Euclidean space time. Which incidentally is real.
Which is incidentally real, but but it's also imaginary. So in this real real manifold, what we perceive as a time direction is actually imaginary. Yes, yes, yes. I don't like when people actually present it this way. I like to think of it as. As a part of the structure in which we study, we study quantum fields on
Many faults endowed with complex metrics in general and so the metric can be Real with different signature so You know square of the Distance and time direction could be positive or negative or it could be positive in all directions. So we call that Euclidean signature metric But in general
We could and it should also study the metrics, which are just complex. So the metrics elements are complex. The values of the metric are complex. And but that's maybe a slight digression. I'm sorry, this is a bit technical. That's fine. Anyway, so Zadrikin-Witten wrote a formula and they said that to really understand this formula from the first principles, one should derive it by doing honest path integral
in supersymmetric gauge theory and evaluating those contributions to the effective action coming from those peculiar fluctuations of gauge fields, which happen in compact regions of Euclidean spacetime. And the moduli space of instantons, is that a connected space? So it is connected for each topological class.
So if you fix the topological charge, it is connected. But for different topological charges, it is there, you know, disconnected distinct spaces. So a priori, a priori, mathematically, you would have said, OK, I mean, I have just the infinite number of integrals to compute. Why would they combine into something nice? So. I would say that.
You never approach an interesting problem directly. It's very hard to study things face-on. You need some kind of preparation work, and maybe if you're lucky, you will get to the point where you hit on the problem you're actually interested in, or you dare to be interested in.
So at this point, you were exploring, hoping that you were going to get to the solution, but allowing yourself to not get to the solution because it was interesting what you were exploring anyhow. Right. So the goal was so that at this point, I mean, I had to get familiar to familiarize myself with instantons with modular spaces of instantons, what they are, what they look like, what what do people know about them? And so one,
The thing which became popular at the time was the study of quib of varieties. But they were not defined as the modular space of solutions of some partial differential equations. Baffelin-Witten mapped this property to the very non-trivial property of the four-dimensional gauge theory, which inversed the gauge coupling. So somehow they related the coupling of gauge theory to the geometry of some at the time abstract
two-dimensional torus elliptic curve. So that was part of the one of the hints towards the BPS safety correspondence. That some structures found in the studies of four-dimensional gauge theory study structures involving supersymmetry and the cohomology of some supercharges are related to the structures found in the in two-dimensional conformal field theory.
Right, so just to linger on this point, so many people have heard of ADS-CFT and that involves differential geometry and harmonic analysis and string worldsheets and so on, but there's also something else called BPS-CFT which you helped popularize and articulate. Well, BPS-CFT was just maybe a joke term which I coined at the time because it sounds like ADS-CFT, but of course it's
Just like, just as ADS-CFT is a kind of a string duality, it's a kind of open-close string duality in my mind. Even though some people claim it's just holography in its purest form and doesn't require string theory, I don't think it's the case. I think it's... Oh, you think it's string-dependent, it's not just string-inspired. I mean, it's kind of a, it's one of these things which makes string theory great.
It gives you a mechanism for holographic duality. ADS-CFT in that sense is a projection of some 10-dimensional entity. There are versions of ADS-CFT which use the same theory. It turns out that the integrals which one had to compute to get the Zebrick-Witten solution were independent of that parameter.
So you can hope that if you send it to zero, you got the same answer as if you don't. So this is one of the tricks which I learned essentially from Whitten. Whitten used in his studies of supersymmetry breaking a quality which is now called Whitten index, which is the
also a partition function. It's a kind of a partition function of a quantum system with supersymmetry, which differs from the conventional partition function where you average exponential negative Hamiltonian divide by temperature. You also insert the operator which weighs bosonic and fermionic states in your Hebel space with different sides.
And so we can show that in many favorable circumstances, this partition function is independent of the temperature. And so you can compute it in different limits when temperature goes to zero or temperature goes to infinity. And when temperature goes to zero, this partition function receives contributions from ground states only from the vacuum. And when the temperature goes to infinity,
It is usually the regime where things are computable. And so one of the consequences of this of this method is the physics proof of I.T.S. Singer index theorem, which was used, for example, by Luis Andres Gomez, my current director.
So if they can prove that there is a parameter in your Hamiltonian or in your action or in your system to which the observable you want to compute is insensitive, then you vary this parameter as much as you can and try to find the regime in which things become computable. And so in my story, one of these parameters was this parameter of non-commutativity.
Unfortunately, this trick cured one half of the problem. Namely, it dealt with this singularity of the modular space of instantons, which corresponds to small distances in spacetime, so things which happen at short distances.
But since I'm interested in working over R4, so we want to understand physics on the four-dimensional flat space, I like mathematicians who use instant and modular spaces for things like invariance of four manifolds, which is the content of Donaldson theory.
for them the interesting spaces are complex spaces even though sometimes with very clever clever tricks they draw conclusions even for r4 and famously found that that there are unlike all r dimensions so as a smooth manifold there is only one r1 there is only one r2 only one r3 only one r5 only one r6 there's only one rn for any n except four
And four dimensions, there are exotic Euclidean spaces, which is, which is quite controversial. Just a moment. Do you think that fact is connected to the R4 of our space? The fact that our space time is R4? Could be. Yeah, because it has to do, it has to do with the fact, with the fact that two dimensional surfaces meet in four dimensions and they don't meet in, in, in higher dimensions generically. And, uh,
So the fact that certain things meet often, like probabilistically, so if you throw random walkers, it's probably crucial for life. So let me just put it this way. OK. Life needs interaction and propagation.
So the exotic R4 is our world. The R4 is our world because there's an exotic R4. And if there wasn't, then we wouldn't have enough interactions for us to be here. Right. So, okay, I would say, I don't know what the causality is, but I would say that the reason exotic R4 exists is probably
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important in things like having chemistry and things like that. So there is certain topology involved in having complex structures, in having biochemistry, depends on certain topological features of our space and space time.
Some of these things are crucial in the construction of exotic R4. So these things might not be unrelated. But since it's discussing things for which you have only one sample, it's hard to draw conclusions. Because I haven't heard anyone connect chemistry to R4's exotic property. Unless this is just a speculation of yours, you're not sure.
No, maybe we could probably elaborate. I mean, I'm not strong in chemistry, so if I venture in this direction, I'll probably say something silly. I would say that the fact that four-dimensionality, well, what do we know? We know that four dimensions are special in interacting
Propagating vector fields so gauge gauge theory as quantum theory Is well defined in four dimensions and has very interesting properties, which which is And so it is not well defined in five or higher dimensions and it's less interesting in in few dimensions and
People size of three dimensions because it still maintains some has some creative features, but it's there are certain things you get almost for free because you get mass scale for free and so on. In four dimensions, you things happen just, you know, they're kind of on the boundary between things becoming trivial or ugly. So it's a four dimensions is interesting border case.
And the fact that the possibility of exotic smooth structures in four dimensions uses the same kind of coincidences, which is just all I'm trying to say about that. Okay, so getting back to the story. So back to the story. So in physics, I'm interested. So in physics, I need to for physics applications, I need to compute meaningful integrals over
the modular space of instantons on non-compact, literally non-compact infinite R4. And this modular space is again non-compact because now those instantons, which are like pseudo particles, like events in spacetime, they can run away to infinity. So you can have a sequence of events which happen here and then on Saturn and then on
in the Andromeda and then go all the way to infinity. So there's no limit to that. So again, I'm in trouble because I cannot integrate by parts. I cannot, I mean, I'm not sure my integral is convergent. Life is difficult. So I need to invent another trick to somehow cure that possible divergence or non-compactness at least. Now, if we're doing something on R4,
R4 has a beautiful symmetry. It's a symmetry of rotation. Well, it has also symmetry of translations, but translations act without fixed points. So if we use translations, it will kill everything. So it's not the right symmetry in this case. The right symmetry to use is symmetry of rotations. And so the group of rotations of Euclidean four-dimensional space is the group SO4 and its maximal torus, maximal Abelian subalgebra,
Subgroup is two-dimensional and it's generated by rotations into orthogonal planes. Now, of course, if you just do some mathematical construction, which is completely artificial, first of all, you might make mistakes, but also you lack intuition. So it would be hard to know what is the right way to proceed. What are you doing? Meaning you lack physical intuition?
Right. So this mathematical construction was nice, but I wanted to have some kind of physical understanding where these parameters come from and what would be the physical realization. Is there a physical meaning for such a deformation? Remember, I was doing this in the late 90s and at that time, even though the non-community field theories were already popular and non-community,
information breaks Lorentz symmetry, because you see, if I tell you that x and y don't commute, the fact that x and y commute in one way and z and t commute in another way breaks the symmetry between x, y and z, which is part of the Lorentz symmetry. So people started discussing quantum field theories, which were not Poincare invariant, but people were reluctant. So this partition function scale, like a partition function of a non-ideal gas where the volume
The role of the volume of the space in which the gas was confined was played by the inverse product of these parameters epsilon 1, epsilon 2. And the leading term, the free energy, so it's the energy, free energy per unit volume, was its arbitrary potential. So first I discovered that experimentally, just by expanding the term by term,
Because I felt like these partition functions were good for something because they were so beautiful. They fit nicely to each other, one instant on, two instant on, so on. And then when I saw that the one case people computed, the one instant on term people computed, and sometimes took it to instant terms, people computed by very hard work, matched with what I got by relatively simple work, then I was convinced it must work.
Of course, I was not very confident in myself, so I had to go up to five instantons and get some help from numerics and got some errors along the way. So I had to. Is that the one about the story where you were working for two weeks on a laptop and then you were in a hotel room and it worked out? That's a different one. That's a different story. By that time, I was much more confident.
So that's when I published it. I made a conjecture in this paper that this partition function, which was a deformation of, I mean, it gave more than Zybrokrutin pre-potential. It gave something else because it had two more parameters. And so the conjecture was that it gives the amplitudes of topological string. And the reason for that we can discuss. And also that it is given by some
It was the moment both of joy and frustration because
It felt like I should be able to prove it, that it's not just conjecture, but just to prove that indeed the curves that Zabrik and Witten proposed and conjectured and people then extend their conjectures to other gauge groups and types of theories with various metric content should be provable. But I was too close and I couldn't do it.
Uh, it took me another year and another happy, happy meeting, just a fortunate meeting with the right person at the right time to be able to, to prove at least one of some of these conjectures. So that was the meeting with Andrej Okunkov, uh, just on the train station in the, in France, in Bureaux-sur-Rivette. And, um, so I knew Andrej from before, but we've,
basically just played volleyball together when when I was in Princeton as a student he was at Princeton as a postdoc and we just played volleyball but we never really discussed physics or math I knew that he was an expert in combinatorics and maybe he knew something about partitions and so I just stumbled upon him on the train station and he said what what are you doing I said what do I have this
Difficult problem when I have a sum over sets of infinite set of partitions Which I believe has this hidden structure there with curves emerging, but I don't know how to prove it. I mean, I don't have It feels like it should be something to have something to do with conformal field theory to dimensions, but I'm not sure exactly what and he said well, but You are in luck because I'm computing the partition sums
from dusk till dawn and then from dawn till dusk. And he also knew of a problem about random partitions for which the answer is given by some kind of a curve. So he knew about some emergent geometry in the problem involving random partitions. And that was the famous limit shape of Vershik, Kerov and Logachev.
which mathematicians studied for completely, you know, different reasons also in late 70s. And it was a different crowd of mathematicians interested in different problems. And, but fortunately I met Andre and he said, well, but this is a similar problem. So maybe, maybe there is a similar solution. And again, it turned out that the problem which mathematicians solved was for the trivial case, but
With some ingenuity, and Andrei had a lot of ingenuity at the time with this, so he used a lot of interesting complex analysis ideas to transform the problem of computing my asymptotics to the problem of finding a limit shape. Limit shape is kind of the most probable geometric shape in the ensemble of random geometries you are given.
We found that indeed it was the family of zebra Putin curves that govern the asymptotics of my partition function. Wonderful. So you and volleyball. Yes, because.
You went to Paris for volleyball and you weren't doing much math there, but then I think in 98 or so you with Maxime Koncevich, I think you went there for volleyball, but then it ended up being a fortuitous lunch, something like that, if I recall. You don't need to get into that story, but it's so... Well, volleyball is important for mathematicians for some reason. So I like, I mean, I'm real amateur in volleyball.
Sometimes I manage to get a good serve, sometimes I don't. But at IHS, which I visited in 1998 and where I worked later for many years, there was this tradition that in the summer, in the compound where visitors live, people would play volleyball. It used to be
The main driving force behind that used to be Professor Kirillov, Alexander Kirillov, the inventor of geometric quantization, who also was the advisor of Andrey Okunkov. And so that's why we played volleyball, because Kirillov invited us to play volleyball.
Okay, so from Moscow to Princeton to Paris to New York, you've worked in these different cultures. Would you say that the culture of academia is different there, or at least the one that you interacted with? Or would you say there's more similarities than dissimilarities?
Yeah, it's a... Actually, I mean, I wouldn't say... I don't think they're so different. I wouldn't... Right now, I mean, after having spent 30 years in doing physics and math in between,
I think it's pretty much universal. I think that's what makes science beautiful. It's kind of universal. It's independent of the nationality. Yes. I mean, in Moscow, I had kind of a double, I led maybe double life, maybe triple life. I had two advisors who
One was my advisor in particle physics. Another was my advisor in, I guess, singularity theory. Both of them knew that I'm doing something else. I'm actually, I mean, I had some, my main interest was in string theory, in kind of modern mathematical physics. But part of their kind of training, maybe part of the coming of age kind of thing,
You have to work on traditional problems for a while. So I was computing radiative corrections to standard model parameters to get bounds on the mass of top quark. It was before top quark was discovered. So that was part of my work, kind of part of a duty, so to speak.
So I worked with my advisor was Lev Okun, who was basically a phenomenologist, an expert in weak interactions. He wrote several very good textbooks on electroweak theory, on physics of elementary particles. But apart from one paper which I wrote with him on those bounds, we didn't work much
But he kind of protected me from, I mean, I guess he was my protector in many respects. I mean, we could get into that life in Moscow. So it was Soviet Union and just post just post Soviet Union. Just it was very different from what it is now. So so the the expectations of our young fees from young thesis were different than the
There were different obstacles and different challenges. My math advisor was Vladimir Arnold. He didn't know, I mean, I was just going to his lectures in Moscow State University, even though I studied at a different university, and he didn't know I was not a student there. So I just attended his seminar, his lectures, and he gave me, I mean, every year he would give a list of problems to the participants of his seminar.
and if you wanted you could work on them and so I worked on one of these problems and it was very useful for me because I learned about topology about many many things but he actually gave me some advice which I guess I used and I mean he gave me lots of advices but one of them was to you know always try
Find all the parameters in your problem and always try to take them to extremes. That's the advice which you can take outside outside the science as well. So your physics supervisor was David Gross in Princeton. Yes, when I got to Princeton, my advisor was David Gross. But again, when I was a graduate student, David was
Kind of going through a complicated phase in his personal life and he basically let me do whatever I wanted, which was the best case scenario for a good student. Good students should not ask problems from their advisors. They should find problems and maybe if they have some difficulty in solving them, they should come to the advisor and ask for advice.
So David gave me when I was a student, he gave me some practical devices, but we started working together already later when I already had the job actually. So which was actually the best, also the best time because I got to work with happy and confident David. So he already was in Santa Barbara and I got him interested in non-commutative geometry.
So we found interesting solutions of gauge theory of non-commutative space. We found non-commutative monopoles and found that they actually carry the physical string attached to them. So the Dirac string of magnetic monopole of Dirac, the string of Dirac monopole, which is kind of imaginary object on a non-commutative space becomes physical.
He always told me to believe in myself and to be confident and that don't get intimidated by other intimidating people.
So if people tell you that you should not work on something because it's wrong or because it's morally wrong or because it's, uh, whatever reasons, if you feel that it's important, work on that. So was there something in particular that you were insecure about that you were coming to him saying, I'm not sure if I should work on this. And then it was that that moment that he gave you that advice. Well, it was, uh, I think, uh,
It was before one of the string conferences. Incidentally, I think it was a string conference in Paris. I think I was competing, working on similar topics with a group of prominent physicists. So you felt in competition with them, like you felt there was a rivalry? Yeah.
He had some stories when
He had a
result, which he could confirm with the models and examples, which showed precisely the opposite. And so it was a very happy coincidence. So it made a good show in the sense that people make strong claims and then one claim after another. And then there's some discussion. And of course, David was right. So it's his story.
So he told me that, you know, you should just aim for be yourself, believe in yourself and present what you have. And maybe the other party doesn't have as strong a result as you think. And he was right. So I think to do. So I've had another one of your collaborators on Edward Frankel and I'll post that podcast on screen and in the description. He was on several times, actually, and it's coming up again to talk about the geometric Langlands. So
What is it about Edward Frankel that you admire? You all both seem to get along both mathematically, but also as people, so you can speak about both. Well, we are, well, what I admire about Edward, so he's, first of all, he's charming. He's very intelligent, very knowledgeable, but also
Well, I would say that for a person of his charm, he actually has a big heart, which is very rare. Sometimes people kind of use the charm to the advantage and eventually they become cynical. Edward is the opposite.
He's evolving, he's learning and he's interested in things I'm interested in. So it's always, I mean, I always learn something from him and it's always not just a pleasure, it's a challenge and a pleasure to be in communication with him.
So mathematically now, what is it that you admire? So you admire his heart, sure, at a personal level. Yeah. So he combines, he combines kind of a deep knowledge and understanding of very abstract things. And also, but he also knows how to connect them to things which kind of I understand as a physicist. So he can, he can speak to me
Yes. Okay. So earlier you also mentioned physical intuition and I want to touch on this because while physical intuition can take you so far, do you find that it hinders at some point? Do you find that? Well, it helps in general. And if it depends, then what does it depend on?
Well, on my path, I was not always guided by physical intuition. I would say I'm guided either by mathematical intuition or by physical intuition. So there is always some kind of some sense of beauty in both worlds. And they sometimes complement each other. Sometimes they are compatible.
Well, of course, historical famous examples where they are in contradiction and then mathematical intuition, for example, for Dirac was was more important physical intuition and it led to a change in our physical understanding, like with the concept of antiparticles, with positron and things like that. But
Of course, physical intuition sometimes could be misleading because we are limited by our experience and the models we study. So again, the good example would be the invention of quantum mechanics. You have to really step outside, step out of your mind box to embrace the quantum intuition.
You cannot understand it using everyday experience. We get used to it, but to truly understand quantum mechanics, I don't know what kind of meditation you need for that. I haven't found it. I have a very good friend
film director Ilya Khrzhanovsky who wanted to make sort of a quantum film which where you could kind of experience quantum reality through interesting through you know multitude of universes you live in but well I don't know it's some people say that
He succeeded to some extent. Some people say he made a step. But so we keep talking about this with him, whether maybe animation could be good media for for kind of explaining what quantum world is. So here's a question. I was just at dinner last night with some people.
And it's commonly said that when we measure, we observe something real and we don't see a superposition. And in part, that's the measurement problem. And then I just asked, well, what would it look like if you observed a superposition and the people were pausing it? Because it's not exactly that you see something here and here, but it's half transparent. No, we observe effects of the position because we, I think the
Issues that the devices we measure with a kind of classical devices. So and and the way the brain will analyze this is a classical brain. So to really understand superposition as a quantum, you have to be quantum entity to to better superposition. So Greg Moore, which I want to talk about as well, he described something called physical math. And that's in contrast with mathematical physics.
Now, I know that you didn't come up with that distinction, but do you see yourself as more of a physical mathematician or more mathematical physicist? I think of myself. So I think of myself more. So I'm more physical than a mathematician, but I put it this way. So just just that. But I'm equally interested in mathematics as I'm interested in physics. And for me, mathematical beauty is as important as physical beauty.
physical beauty, not physical beauty, physical beauty. Physical beauty is also important, but it's different. Yes. Okay. Well, okay. Let's talk about mathematical beauty. Physical beauty is something for another time, but give a taste for what it is to to find something mathematically beautiful for people who haven't experienced that.
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Well, of course, there are kind of professionals who come up with the with examples of mathematical constructions for for layman, which are kind of beautiful. For me, mathematical beauty is what unites distinct parts of mathematics. For example, the fact that. You can think of prime numbers. Prime integers.
as of points on some space. So they, for me, this is a beautiful concept. In what sense? Well, it's in the sense that integers, they are analogous to the ring of polynomials. So the ring of integers, integers you can add, you can multiply, you cannot always divide, but some integers are divisible by other integers. So just like polynomials,
You can multiply them. Some polynomials are divisible by other polynomials. So if the polynomial vanishes at a point, let's say polynomial of x vanishes at point a, it means that it's divisible by x minus a. So in that sense, prime numbers is what the other integers might be divisible by. So these are the analogs of those minimal polynomials, x minus a.
Okay, you have a lecture titled natural language, geometry and physics. Yes. What I want to know is what's the relationship between human language and then the mathematical language that is used to describe the universe. So I was I'm very much impressed that I'm amazed by the
The fact that human language, so natural language in the human sense, evolves, so it's not a static object, it's a kind of a dynamical system, and as such, I was proposing to study it as a physical system like a growing crystal or melting crystal.
In my work on Instant Dots, which I reviewed for you, the concept of emerging geometry appears. The problem from gauge theory became a problem of enumerating some combinatorial structures, partitions, which you can visualize as young diagrams, or you can visualize as some kind of
The enumeration of those objects, you can also view as a search for what's called equilibrium distribution in the ensemble of dynamically changing shapes. So for one young diagram, you can go to another one by changing a few wiggles. And so you can assign the probability for those wiggles
What if those wiggles were analogous to the words or the sentences in the natural language? And so the way the language changes in time could be
may be assigned some transition amplitudes, transition probabilities, and maybe there is some emergent geometry which emerges as the most probable configuration. So maybe the natural language, maybe to the natural language, certain geometric shape is associated, just like to these random partitions in my incident calculus, the Zyberputin curve, the geometric object was associated. So language is much more complicated than
So, I was proposing in this lecture to embark on the statistical analysis of the language, not in the way it's done in the large language models,
but in doing some kind of time, time sequence analysis of language. So to study how it changes from, from, from, you know, century to century, interesting, maybe from year to year, but also the, the pun in this title was that the geometry is the language of nature, of physics. So that, so that's when there's a natural language in that sense that also physics is so,
Have you followed up this work? Not quite. Not yet. Not yet. But in the back of your mind, are you developing conjectures or you've just abandoned it and it was a fun activity at the time? No, no, no. As I told you, this is a very hard problem. So you don't attack it head on. So I'm developing models which will kind of surround this problem and then I will attack when the time comes.
So I'm developing other models for simpler systems, simpler than languages. So speaking of young diagrams, and I don't want to get us into more ground that'll take us quite some time to get to, but there is gauge origami. And if I understand that correctly, that has to do with counting young diagram configurations associated with tetrahedra, whose edges are colored by vector spaces. So
So we should fast forward to a few years later. So 2003, with Andrea Kunikoff, we understood how to evaluate asymptotics of the partition function. But then it was understood that the whole partition function is a very meaningful and interesting object because it contains information about strings, black holes,
local collibial also thinking about this particular function people came up with the notion of the so-called refined topological string so there so this whole thing was worth studying and so one way to approach this problem is to say maybe the the zabir-quintin curve which is the
the object which captures the limit shape, the asymptotic of the partition function can be somehow quantized or deformed and had to be deformed by two parameters, by two deformations to capture the structure of the whole partition function. So this led to the notion of QQ characters, which is maybe not a very
Maybe not the most fortunate name, but, well, it has a reason. I mean, this name didn't come from nowhere. So it's a notion. It's not observable. So it's a way to measure the partitions. So it's a device. So the device with expectation value, first of all,
contains information about the the shapes of the random young diagrams and so on but on the other hand has some analytic properties allowing to maybe compute it or to write some differential equation on it so i realize that one can define some kind of gauge theory problem it's best done in the context of non-commutative gauge theory where gauge fields leave
the tricky space-time which has different parts. So it's four-dimensional, but it's not like four small dimensions. It's four dimensions, four dimensions with coordinates x, y, z, t, and then four dimensions with coordinates, I don't know, u, v, w, s. Sure.
So then once I understood that it works and it has good properties and it actually is behind this observable, which I call Q-character for the reason, which maybe I will explain later. I asked myself, this is where the mathematical intuition kicks in. So if you have two transverse four planes inside the eight-dimensional space,
Why not adding other four-dimensional planes? And so the total number of four-dimensional planes you could have is six. And so you can label, they would be in correspondence with the edges of a tetrahedron. And the axis of the four complex dimensional space would be in correspondence either with vertices
or with the faces of a tetrahedron, depending on how you want to think about it. So the tetrahedron is just an object which helps in, you know, just keeping track of all the moving parts of this construction. But it's really just a gauge theory which lives on a particular singular spacetime. And the reason it's interesting to study is because
From the point of view of the observer who lives on just only one of six sheets of this complicated structure, you are just exploring all possible local or semi-local observables of gauge series. You have observables inserted at the point. You have observables which are inserted along a two-dimensional plane or another two-dimensional plane.
and it's constrained to be six because of the rotational symmetry. Of course, if you don't have any symmetry in question, you can, of course, place things any way you like. It will be kind of, you know, unruly. But if you insist on things being invariant under rotations in now 10 dimensional space time, you will be bound by placing things at along the coordinate plates.
And so this is what this gauge origami picture is emerging. It's called gauge origami because it looks like you have your folding paper different different folds. Right. Is this at all related to the positive geometries of NEMA or the amplitude Hedron? Not as far as I know, but in my experience, things are
Okay, what's something that you've learned as a lesson from collaborating with Edward Witten?
And also, I'd like to get to Greg Moore as well, but afterward. So, I guess with Edward, I share kind of affinity to rigor and
He told me on many occasions that this desire to be rigorous is sometimes very constraining. So sometimes it really stops you from doing something because you cannot proceed in a rigorous manner. But I found that one can be rigorous in a different field. So our stumbling blocks
We're kind of complicating each other. So when whenever. So we were trying to in our work, we were trying to combine something I learned with something he learned. So he was working with Kapustin on on. This approach to geomagic Langlands, I was working on my partition functions. I was certain it was the same thing. So these these these were
The domain, the realm where we were discussing, the subject of our study is the same subject. It was certain that there was a way to relate our viewpoints.
but I had my stumbling blocks and he had his stumbling blocks and it just turned out that somehow they were at the right places and so I could advance where he couldn't and he could advance where I couldn't and so I guess my the main lesson from Edward which unfortunately I could still to this day I cannot quite embrace is that one should
And by getting stuck in the problem, you mean that if you're not making progress, you still continue to think about it. Whereas with Edward, he will transition.
Probably, yes. I mean, it's not that I'm... I'm continuing to think about the problem, not because I'm kind of stupid, that I don't understand, that it's at least nowhere. It's because I feel like there is a progress to be made, or I just cannot not do this. It's one of these things when you do what you cannot not to do. Like it's an obsession? Like an obsession, yes. Like an obsession.
Okay, so what did you learn from Greg, Greg Moore? Persevere. So if you persevere. The opposite. Okay, got it. So Greg is close to be in that sense. Yes, you can get obsessed with things and it's okay. Yeah. Do you have a specific example in mind with you working together with him? Yes, we had
So remember, I told you that before attacking the Zyberquint problem, one had to learn how to compute certain integrals. And so we were playing with integrals over like hypercal and manifolds, hypercal equations, using all kinds of a covariant localization, all kinds of tricks. And at some point, so this was my
What is called Miraculous Year? So it is 1998 when they met Albert Schwartz and Santa Barbara. Also in Santa Barbara, I met at some point Greg Moore and Saf Seti. I don't know if you know Saf Seti. He's a professor at Chicago. So Saf was presenting his work with Stern, I think, on the computation of Witten index
for the quantum mechanics of the zero brains. So the zero brains are the interesting particles which are defined as Z-brains. So they are particles in type 2-A string theory. But secretly, the secret life is that they are also black holes and also graviton, gravitons of 11-dimensional supergravity. And if, so the main conjecture of Witten of 1995 was that
The strong coupling limit of type 2A string theory is 11-dimensional end theory, which contains 11-dimensional supergravity in its low energy limit. And for this conjecture to be consistent, D0-branes should form bound states. So it should have a bound state with certain properties for every number of D0-branes. And so it's a question about
Supersymmetric quantum mechanics. This is a question about conventional mathematical physics, I should say. And so Saf was discussing the case of 2D zero brains. So just two particles. And he showed that to compute with an index, one had to compute certain integral over matrices, supersymmetric matrices.
And Greg and I, we were in the audience. And I remember after this talk, I told them, but I told Greg that this is an example of it's one of the examples of the integrals we should be able to compute because this is the space he's integrating over happens to be, among other things, also happy color space. And so we quickly set up to set out to do the calculation using our tricks and
I quickly confirmed that for two particles it produced the answer that SAF was giving, but for three particles one had, it was already kind of complicated, multiple residue thing, so it looked hopeless. But Greg, with his perseverance, he set out, I don't know how many hours he spent, but he
carefully analyzed all these residues and it came out to give the answer which which we wanted. So then of course once you know it works then you can be clever about it and find more more scientific ways of proving proving what you want to prove but you really need this confirmation and so
This was just one example of what Greg, with his ability to do two-de-four calculations, Persevered and Pustas. And so we were able to prove the conjecture of Michael Green and Gadparli, which, thanks to the work of Satya Stern, actually implies
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other than saving m theory. No, I would like, of course, I would like to. This is my so I'm pushing my physics hat. I'd like to to contrast some of my calculations, some of my predictions with actual experiment. And so
Right now it seems that the most promising venue for high energy physicists to make predictable predictions is condensed metaphysics. So with my current student, Yugov, we just published a paper where we applied a clever mathematical technique from Young-Mills theory to the physics of two-dimensional graphene. So maybe when people
Now speaking of your student, what advice do you have for researchers young and old? Be passionate about what you do. That's the best advice.
Many years ago, like 20 years, 20 something years ago, I gave an interview to my former university, to my undergraduate school. And so they asked me, what do you what advice do you give to what you give to current students? And my advice at the time was don't overburden. So don't don't burn yourself. So don't don't don't burn out. Don't burn out. Right.
But I think these days the attitude of modern students is different.
My advice is go the opposite way. Be passionate about what you do. Burn yourself, whatever you want. Burn yourself. So are you suggesting that, I don't know how long ago that was, a decade or two decades ago? Two decades ago. Yeah, okay. Are you suggesting that back then the problem was over passion and it would be to their detriment and you're saying now there's maybe some apathy?
I don't want to put it that negatively. I think people now know a little bit better how to take care of themselves. In my time, it was a really typical thing that people would commit suicide because they were not successful, because they would overwork themselves. I think people are now much more aware of those things.
So personally speaking, how do you stay strong during the bad times? Well, I guess I was lucky not to be in too dark times, but
I do get stressed a lot. Physical exercise is one way to deal with it. I exercise a lot. There are lots of ways to take care of yourself, which are important. Breath work, meditation, yoga, gym.
be in nature, bicycle, skydiving, see other people travel, see the world, spend time with the native people, just all kinds of things. Well, speaking of spending time,
Thank you for spending so much time with me. You're welcome. It was a pleasure. It's three hours at this point, over three hours now. I appreciate it. Thank you. Thank you.
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▶ View Full JSON Data (Word-Level Timestamps)
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"text": " The Economist covers math, physics, philosophy, and AI in a manner that shows how different countries perceive developments and how they impact markets. They recently published a piece on China's new neutrino detector. They cover extending life via mitochondrial transplants, creating an entirely new field of medicine. But it's also not just science they analyze."
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"text": " Where senior editors argue through the news with world leaders and policy makers in twice weekly long format shows. Basically an extremely high quality podcast. Whether it's scientific innovation or shifting global politics, The Economist provides comprehensive coverage beyond headlines. As a total listener, you get a special discount. Head over to economist.com slash to subscribe. That's economist.com slash to for your discount."
},
{
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"text": " Some people will say they of course understand everything about quantum field theory, but we don't understand it as the complete structure. To really understand superposition as a quantum, you have to be a quantum entity."
},
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"text": " Professor Nikita Nekrasov unified previously disconnected fields, instanton moduli spaces, random partitions, integrable systems, and quantum strings. What was required was so novel that when he was presenting at conferences, other physicists couldn't tell if his results contradicted or confirmed theirs. His mentor David Gross advised him, keep your poker face. That gamble paid off. It turns out four dimensions aren't arbitrary."
},
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"text": " I'm Kurt Jaimungal and I interview researchers regarding their theories of reality with technical depth. Today we explore how non-commutative geometry cures quantum singularities, why understanding superposition requires becoming quantum yourself, and Nikita's speculative connection between R4's exotic smooth structures and the chemistry undergirding life itself. From falling asleep during Witten's seminal lecture to late-night residue calculations with Greg Moore, this is mathematical physics"
},
{
"end_time": 164.889,
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"text": " at its most human and revolutionary. Why don't we understand quantum field theory? So it probably depends on who you ask. So some people will say they of course understand everything about quantum field theory. Practitioners who use it, they will tell you that they understand because they know how to use it and they know how to do some calculations which will then can be"
},
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"text": " may be compared to experiment, and more often than not, the comparison is favorable. But we don't, maybe as theoretical physicists with some kind of mathematical ambitions, we don't understand it as the complete structure built out of axioms, let's say."
},
{
"end_time": 218.456,
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"start_time": 193.097,
"text": " So you want an axiomatic quantum field theory that reproduces all the successes of the standard model? Something which, well, we would like to have some basic principles, like maybe not axioms, but some basic principles from which we can build structure from bottom to top, bottom up."
},
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"end_time": 247.432,
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"text": " We have many, many examples of what we think quantum field theory is. So many examples of quantum field theories. And there are many overlaps, but there are also missing kind of areas. We don't know whether gravity is part of quantum field theory or it's something else which requires a different approach. And so there are many, many questions like that."
},
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"text": " So there is, of course, a very, very practical, practical approach, which works with many, many theories which, which are involved in standard model, which you can just simulate them on a computer to do lattice lattice field theory. Right. There are various various solities with that, because not all structures which we know there are in"
},
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"text": " in smooth fields and fields defined on smooth manifolds. Manifolds may have different structures like spin structure. All those things are kind of hard to represent faithfully on the lattice. But with some room for error, people do that. But that still looks like a bypass, not a hole."
},
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"text": " the whole thing. And we know examples of quantum field theories which probably don't have lattice description. And so the work which we do, some of us do, which we're actually trying to chip away at the unknown in connecting us to those theories by coming up with"
},
{
"end_time": 355.282,
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"text": " Maybe mathematical conjectures which sort of follow from the fact that these theories exist and then checking them maybe by more standard means and then gaining more confidence that the reason for those conjectures to be there probably is valid. Now we're going to build up to gauge origami, but before we do, I want to get to another one of your constructions. So Dirac said to Feynman, infamously, do you have an equation"
},
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"index": 14,
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"text": " And Feynman could have said to Dirac, well, do you have a diagram? And now I'm thinking you could have chimed in and said, well, do any of you have a partition function? Well, they, of course they had lots of partition functions. Yes, yes. But anyhow, you have one named after you. So let's get to that story. I believe this was around 2002 where you cracked the Cyberg-Witten instanton counting puzzle."
},
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"text": " Which was when 1990 in the mid 1990s, 1994. Yeah, right. And it was there where you devise the Microsoft partition function. So tell us about what that partition function is and what it was like coming up with that solution. Was it a lightning bolt? Was it an incremental climb? What did that moment feel like?"
},
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"text": " Yeah, so this is a story which is a kind of worth a book because it was it's a story worth of many sleepless nights, but also there's a book called The Count of Monte Cristo. So yours would be The Count of Instant Hans. Right. Right. Thank you for observing the analogy. Right. You have to dig underground tunnel from from some predicament you are to actually to get to the"
},
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"end_time": 461.903,
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"start_time": 433.012,
"text": " You know, dry land and some people say there is also vengeance involved, but in my story, it's all pure love. Are you sure? Well, at least that's my inner work. So my inner work is to realize that all we do in science is joy. Even though it feels like suffering along the way, but"
},
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"text": " The goal is to recognize beauty and maybe add a little bit of beauty and use it later, just because beautiful things are good to have around and sometimes they're useful. So the story started, I don't know when it started actually, maybe started in 1992. Witten wrote a paper which was called Two-dimensional Gauge Theory Revisited."
},
{
"end_time": 517.159,
"index": 19,
"start_time": 492.739,
"text": " and in that paper he proposed to use a technique from rather abstract mathematics at the time called a covariant localization to compute the path integral so it's fine an integral over trajectories or field configurations in example of our"
},
{
"end_time": 545.128,
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"start_time": 517.79,
"text": " rather simple but nevertheless interesting gauge theory, two-dimensional Young-Mills theory. So our world is described to some approximation by four-dimensional Young-Mills theory, quantum Young-Mills theory. But if you imagine a world in which there is only one space dimension and one time dimension, then you could study the simplified version of Young-Mills theory"
},
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"text": " And that was a theory which was actually solved by several tools. But one tool which was kind of interesting was by Sasha Migdal, Alexander Migdal, who actually first defined it using lattice approximation. And then he found out that it was an interesting theory in which you can define it on any kind of lattice. You can make the lattice finer or"
},
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"end_time": 605.759,
"index": 22,
"start_time": 576.237,
"text": " or economical. So use as little number of edges as possible. Yes. Given the topology of space time and the the partition function will be the same. So it was interesting theory, which was almost topological. We would not call it now almost topological. It only dependent on the area of space time. So space time here, I'm using a little bit of a jargon."
},
{
"end_time": 631.698,
"index": 23,
"start_time": 606.442,
"text": " The theory is interesting when space-time is not physical space-time when you have time and space, but what we sometimes call Euclidean space-time. So it's a manifold in which the notion of a distance between points is similar to the notion of distance between points in space. There is an ordinary mining metric."
},
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"end_time": 660.128,
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"start_time": 632.056,
"text": " How disconnected to the space time where the notion of distance involves events. And so some some distances can have negative square, for example. That's a that's a story which we can discuss, but let's not get there. OK, anyway, so we then observed that this this theory, the reason why McDowell was able to solve it and get a very interesting answer."
},
{
"end_time": 688.114,
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"start_time": 660.674,
"text": " So we can observe that it had to do with with certain topological structures present in the final dimensional space, which you can associate to any two dimensional surface. We call them Riemann surface. So it's like a surface of a well, your audience knows probably. So Riemann surfaces have if they are intimately, they have only one one"
},
{
"end_time": 709.497,
"index": 26,
"start_time": 688.626,
"text": " number which characterizes the topology of the number of handles. Young-Mills theory deals with connections, with ways to transport things from one point to another in some bundle over the base space."
},
{
"end_time": 738.37,
"index": 27,
"start_time": 709.991,
"text": " The lowest energy configurations, lowest action configurations in Yanin's theory are the connections which have zero curvature, which means that if you try to, if you transport things around small loops, the result doesn't depend on which loop you choose, as long as the loops are small. But if the loops are not small, if they wind around handles, so they cannot be contracted to a point, then those transports can be non-trivial."
},
{
"end_time": 766.288,
"index": 28,
"start_time": 738.882,
"text": " This fact actually was observed in physics in the famous our own bomb experiment. So it's right. It's a famous, famous story which showed to the world and to physicists also that the connection, the vector potential actually has a meaning, has a physical meaning, not just the curvature, which if you only if you if you're familiar with Maxwell equations, you can"
},
{
"end_time": 796.015,
"index": 29,
"start_time": 766.988,
"text": " uh, write them only using the curvature, this fuel strength. And so for a while, people thought that the vector potential is not really meaningful because first of all, it's not universally defined. You can make the so-called gauge information and change it. But the fact that they transport, especially the transport observed by quantum quantum particle depends on the, on the path, which can be non-contractable. Uh, well, it was a big deal. And so,"
},
{
"end_time": 825.213,
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"start_time": 796.34,
"text": " On the two-dimensional human surface, because you have several handles, you can have those transports and then they can have pretty much arbitrary values. And there is a space which parameterizes them, which is called the modular space of light connections. These days, this space is, of course, very popular in the stories involving geometric wavelengths program. But anyway, so if you're"
},
{
"end_time": 855.111,
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"start_time": 825.964,
"text": " Connection, if your gauge theory is defined with compact gauge group, then it's some complex space, although with singularities, because you have symmetries and you divide by symmetries, and sometimes symmetries act not freely, so it's a singular space. But you can define its volume. It has a natural, well, it turns out it actually could be used, viewed as a phase space. It has a symplectic structure."
},
{
"end_time": 885.486,
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"start_time": 855.691,
"text": " which was actually found by, I guess, at here in both in the early eighties. And so there is a natural number which you can associate to every human surface, which is the volume of that modular space computed in the Louisville measure. Right. Right. And so we can compute this number by a very clever, clever trick using localization applied to two dimensional theory. Got beautiful answers involving functions. So some some of us now call them Riemann written data functions because they"
},
{
"end_time": 912.551,
"index": 33,
"start_time": 885.998,
"text": " In 1992, I was just beginning my studies. I was very much impressed by the paper. In fact, my friend and later collaborator, Andrei Losyev, suggested I study that paper and present it at the seminar."
},
{
"end_time": 934.991,
"index": 34,
"start_time": 913.2,
"text": " No, I come from Moscow, so I was attending some string theory seminars there, and the tradition there was that you study some paper, and this was the beginning of archive. So it was the first, maybe second year of archive. So the papers from the West started coming in not by"
},
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"end_time": 960.794,
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"start_time": 935.674,
"text": " And at that point you were an undergraduate?"
},
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"end_time": 991.715,
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"start_time": 962.244,
"text": " I was an undergraduate at the time, so it was my probably third year after high school. Right, right. So, but I mean, I already knew what I wanted to do. So I mean, I was lucky to, you know, to get interested in mathematical physics in what I call mathematical physics early on."
},
{
"end_time": 1013.916,
"index": 37,
"start_time": 992.073,
"text": " And so. Somehow I got into this crowd of people, group of people discussing whatever was interesting. So I was very much impressed by this paper and I thought, OK, maybe one can use this localization to compute interesting things in other theories. And I remember we were discussing with my"
},
{
"end_time": 1041.084,
"index": 38,
"start_time": 1014.48,
"text": " other friend and also collaborator later, Sasha Gorsky. And we ask ourselves, why couldn't we apply this supersymmetric technique to quantum mechanics, especially quantum integrable systems? Later on, the spectrum was analyzed by Haldane, Duncan Haldane, who got Nobel Prize for many, many things. And so the way"
},
{
"end_time": 1062.927,
"index": 39,
"start_time": 1041.886,
"text": " The spectrum was organized, is now called Haldane statistics. So these particles, because of this particular interaction, they kind of behave as neither bosons nor fermions, but something that can be in between. And so this coupling constant can be an interpolating parameter. Anyway,"
},
{
"end_time": 1093.08,
"index": 40,
"start_time": 1064.565,
"text": " Gorsky and I found that this model of particles moving on the circle and repelling each other with the potential which is proportional to inverse square of the distance, chord distance, is actually part of the two-dimensional Agnus theory. So quantum mechanics is equivalent to two-dimensional theory. So it's one of the interesting examples of deceiving dimension that"
},
{
"end_time": 1121.22,
"index": 41,
"start_time": 1093.524,
"text": " This is part of the structure which I mentioned when I say that we don't quite understand quantum field theory, because when we explain to engineers that quantum field theory is the continuous limit of something you define in the lattice, of course an engineer will think that the lattice is embedded into space-time of some dimensionality, so that's the dimensionality of the theory."
},
{
"end_time": 1148.916,
"index": 42,
"start_time": 1122.142,
"text": " But it turns out that you can define a theory in one number of dimensions, but then by looking at special observables, so by restricting the set of observables you're allowed to. Yes, to look at. You will find you will not be able to distinguish it from the theory in the lower dimensions and vice versa. So you so we think we live in three plus one dimensional space time."
},
{
"end_time": 1173.746,
"index": 43,
"start_time": 1149.309,
"text": " But it might be just that we don't have access to observables which can probe, you know, high dimensional space time. And eventually it said that your work implies that particles live across multiple dimensions at once. Maybe particles is not the correct word here, though. When I was talking about particles in the collider model, that was more of a"
},
{
"end_time": 1202.739,
"index": 44,
"start_time": 1174.872,
"text": " Just in a mental picture, like beads, like some small objects. So in particular, in this case, the way this relationship between quantum mechanics and field theory was working was that the role of those particles was played by the eigenvalues of the holonomies or the transport"
},
{
"end_time": 1227.159,
"index": 45,
"start_time": 1203.285,
"text": " of the two-dimensional gauge field around the circle of space. We found that those points in the circle behave, so time continues to be the time, so those points in the circle behave as particles in the Collogero-Sutherland model. The mathematicians call it the Collogero-Moser-Sutherland model because Jürgen Moser, the Swiss mathematician,"
},
{
"end_time": 1249.548,
"index": 46,
"start_time": 1227.671,
"text": " Just a moment. Don't go anywhere. Hey, I see you inching away."
},
{
"end_time": 1273.404,
"index": 47,
"start_time": 1250.043,
"text": " Don't be like the economy, instead read the economist. I thought all the economist was was something that CEOs read to stay up to date on world trends. And that's true, but that's not only true. What I found more than useful for myself personally is their coverage of math, physics, philosophy, and AI, especially how something is perceived by other countries and how it may impact markets."
},
{
"end_time": 1297.295,
"index": 48,
"start_time": 1273.404,
"text": " For instance, the Economist had an interview with some of the people behind DeepSeek the week DeepSeek was launched. No one else had that. Another example is the Economist has this fantastic article on the recent dark energy data, which surpasses even scientific Americans coverage, in my opinion. They also have the chart of everything. It's like the chart version of this channel. It's something which is a pleasure to scroll through and learn from."
},
{
"end_time": 1315.179,
"index": 49,
"start_time": 1297.295,
"text": " Links to all of these will be in the description of course. Now the economist's commitment to rigorous journalism means that you get a clear picture of the world's most significant developments. I am personally interested in the more scientific ones, like this one on extending life via mitochondrial transplants, which creates actually a new field of medicine."
},
{
"end_time": 1339.548,
"index": 50,
"start_time": 1315.179,
"text": " Something that would make michael levin proud the economist also covers culture finance and economics business international affairs britain europe the middle east africa china asia americas and of course the u.s.a. whether it's the latest in scientific innovation or the shifting landscape of global politics the economist provides comprehensive coverage and it goes far beyond just headlines."
},
{
"end_time": 1364.121,
"index": 51,
"start_time": 1339.548,
"text": " Look, if you're passionate about expanding your knowledge and gaining a new understanding, a deeper one of the forces that shape our world, then I highly recommend subscribing to The Economist. I subscribe to them and it's an investment into my, into your intellectual growth. It's one that you won't regret. As a listener of this podcast, you'll get a special 20% off discount. Now you can enjoy The Economist and all it has to offer."
},
{
"end_time": 1381.101,
"index": 52,
"start_time": 1364.394,
"text": " Thanks for tuning in, and now let's get back to the exploration of the mysteries of our universe."
},
{
"end_time": 1409.019,
"index": 53,
"start_time": 1382.449,
"text": " Hola, Miami! When's the last time you've been in Burlington? We've updated, organized, and added fresh fashion. See for yourself Friday, November 14th to Sunday, November 16th at our Big Deal event. You can enter for a chance to win free wawa gas for a year, plus more surprises in your Burlington. Miami, that means so many ways and days to save. Burlington. Deals. Brands. Wow! No purchase necessary. Visit BigDealEvent.com for more details."
},
{
"end_time": 1440.452,
"index": 54,
"start_time": 1410.555,
"text": " So two years later, okay, so let's just keep it. So let's just remember that there is some relation between quantum many body systems and gauge theories with some supersymmetry. Okay. And that in gauge theory, certain things can be calculated exactly if you can recognize some symmetry and can use the mathematical tool of equivalent localization."
},
{
"end_time": 1469.77,
"index": 55,
"start_time": 1441.544,
"text": " Got it. So two years later, it was 1994, and I was traveling, one can say backpacking through Europe, and I found myself in Paris. When it just happened, the International Congress of Mathematical Physics took place. And so there was an interesting talk. There were two talks by Witten there."
},
{
"end_time": 1494.309,
"index": 56,
"start_time": 1470.213,
"text": " who presented his work with zyberg on zyberg-witten solution of n equals two gauge theory and on his work with kumrunwafa on the strong coupling tests of s-duality yes so that's s-duality of n equals four superagnos theory i spoke to kumrun on this podcast i'll place the link on screen and in the description as well right so uh"
},
{
"end_time": 1522.551,
"index": 57,
"start_time": 1494.838,
"text": " I should admit that, I mean, okay, so it was 1994. I was just out of my, uh, I just finished, uh, basically my undergraduate studies. I finished the, um, military bootcamp and I just went to Europe and I was going to graduate school later. It was a very hot summer. And at Wittenstock, uh, on Zabrik Witten theory, I fell asleep."
},
{
"end_time": 1553.609,
"index": 58,
"start_time": 1524.309,
"text": " So it was, I mean, I, I remember, I mean, I wasn't there, but I remember you telling the story. Yes, I was asleep. I must be ashamed. I am ashamed. I could not really comprehend the significance of what was going on, but I got, I was impressed with the formula he wrote for certain quantity, which, which is called pre-potential of, of, of, of theory. So just to recall the abracadabra solution was the ansatz."
},
{
"end_time": 1581.135,
"index": 59,
"start_time": 1553.814,
"text": " for what is called low energy effective action of some gauge theory they said that so this the quantity which they they wanted to compute the one which determines this low energy physics can be expanded in power series in instant dots so what are the instant dots so this gauge theory"
},
{
"end_time": 1611.135,
"index": 60,
"start_time": 1582.688,
"text": " being monobillion gauge theory. There are configurations of gauge fields in one could say in the vacuum where they fluctuate in a way which changes the topology of the gauge bundle in which they are defined. So let me maybe unpack that. The thing which"
},
{
"end_time": 1636.988,
"index": 61,
"start_time": 1611.664,
"text": " Not often appreciated is that the fact that our world is described by gauge fields means that, you know, maybe conceptually or maybe physically there are extra dimensions to the world we live in. Because what these gauge fields are, they describe the transport of things over our space-time, but which happen"
},
{
"end_time": 1662.671,
"index": 62,
"start_time": 1637.841,
"text": " In a kind of additional space, which is fibered over our space time. And this space could be a group. So typically one takes as a basic object, one takes the principal bundle. So the fiber of this vibration is just a group manifold, or it could be a space on which the group acts."
},
{
"end_time": 1688.814,
"index": 63,
"start_time": 1663.063,
"text": " So, for example, fermions of the standard model, they are all sections of some bundles which are associated with some principal bundle. Now, one can take the totality of those fibers of a spacetime and look at the space, the resulting total space. So, it will be a manifold of dimensionality 4 plus the dimension of the group."
},
{
"end_time": 1715.589,
"index": 64,
"start_time": 1689.377,
"text": " So if the group is SU2, the total space will be seven dimensional. If the group is SU3, the total space would be 12 dimensional and so on. Some people like group E8, so the total space would be 152 dimensional. At any rate, just given the fact that your base space is your four dimensional space time and the fiber is the standard group,"
},
{
"end_time": 1742.398,
"index": 65,
"start_time": 1716.544,
"text": " doesn't specify uniquely what would be the total space. So there are different ways of fibering this group over the base manifold. So the ways to parametrize, to enumerate the topology of those total spaces is the subject of the study in homotopic topology. And so there are"
},
{
"end_time": 1768.695,
"index": 66,
"start_time": 1743.046,
"text": " special, not people, special classes which are responsible for those classifications called characteristic classes found by Pontjagin and Chern. And so in doing gauge theory, we're actually doing a little bit of quantum gravity in the sense that we are summing over different topologies of this total space."
},
{
"end_time": 1793.609,
"index": 67,
"start_time": 1769.258,
"text": " So, when I say summing, I mean it in quantum mechanical sense. So, if you think about Feynman path integral, which is the kind of integral over paths over possible trajectories of evolution of your system, engaged theory in the evolution of the system involves the choice of the total bundle, total space of a principal G bundle."
},
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"index": 68,
"start_time": 1794.804,
"text": " And so we're summing over the topologies of those of those bundles. Let me see if I can break this down for the audience so far. Okay, and you tell me if I'm incorrect at any point. So space time, we ordinarily think of it like x, y and z. And then well, there's also a time. So you"
},
{
"end_time": 1835.52,
"index": 69,
"start_time": 1812.978,
"text": " Get X and Y the plane that most people know about by doing X cross Y. So that's something that people are familiar with. You take R, R1, and then you cross it to get R2. And then you're wondering, well, what is this fiber business that people are talking about? Locally, meaning if you look closely, you can think of a fiber as another cross. So we're just crossing with a gauge group."
},
{
"end_time": 1858.439,
"index": 70,
"start_time": 1836.049,
"text": " So your example is already in your example, we can give an example of a non-trivial bundle. Sure. So if your y variable is compact, it's actually not aligned by the circle. And now I'm adding x, sorry, x was first, let's say y was first. Sure, sure, sure."
},
{
"end_time": 1883.848,
"index": 71,
"start_time": 1859.531,
"text": " So I want to fiber the line over a circle. And so one way I can just, you know, take a product, direct product. So I will get the space which I will get will be a cylinder. I think I even have a scene drawn on my blackboard somewhere, but anyway. So it's a cylinder. It's a tree. We would call it as it's a, it's a trivial bundle. So the cylinder, so the base is a circle and the fiber is a, it's a line."
},
{
"end_time": 1911.203,
"index": 72,
"start_time": 1884.923,
"text": " But now let's, there's another option, which is to say that as the, as I go around the circle, my line flips orientation. And so it comes back as the same line, but it's inverted. So I used the action of the group Z two, the two element group, which acts on the real line by multiplying X by my minus one. So X goes to negative X."
},
{
"end_time": 1939.445,
"index": 73,
"start_time": 1912.841,
"text": " And so that gives you another space, which is known as Möbius strip. And that's a non-trivial bundle. Right. But you can also fiber the circle over a sphere in a non-trivial way. So imagine, so let's take. Actually, let's take our two donuts. It's probably a good place to start. Now imagine, so I would think of the."
},
{
"end_time": 1966.51,
"index": 74,
"start_time": 1939.77,
"text": " So donut is a product of a dimensional disk times a circle. So now what I want to do. And these two donuts currently are disconnected. Yeah. So they had separate two separated separate donuts. Unfortunately, I don't have donuts with me. Tim Hortons is what they would say here in Toronto. Right. Yes. Yes. Also coffee. Now,"
},
{
"end_time": 1996.63,
"index": 75,
"start_time": 1967.773,
"text": " I want to take the boundary of those donuts, of those two donuts. So those would be products of S1 times S1. Now I want to identify the first S1 so they will be the same S1 and glue them together. But the second S1, the fiber, I want to identify with a twist in such a way that as I go around the first S1 the second one makes the full rotation."
},
{
"end_time": 2024.326,
"index": 76,
"start_time": 1997.585,
"text": " It's almost like this example with the Möbius strip, but now the rotation is 360 degrees and it's a rotation of a circle, not the flip of the line. So if you make a little mental exercise, glue these things together, you will realize that the resulting space is now a three-dimensional sphere. So one way to maybe imagine this is to"
},
{
"end_time": 2051.783,
"index": 77,
"start_time": 2024.889,
"text": " actually embed one so take one donut and then realize that the complement to this donut in the three dimensional space is actually topologically also a donut if you add the point of infinity and so the union of those two donuts is the complete three dimensional space with the point of infinity which is a three dimensional sphere and it turns out that if you are studying the parallel transport"
},
{
"end_time": 2078.353,
"index": 78,
"start_time": 2052.534,
"text": " On this bundle with non-trivial non-trivial topological class, the minimal Yang-Mills energy configuration will be not flat, so the curvature will be non-zero, but it will have a very interesting property. It will be self-dual or anti-self-dual. So this is something which is specific to four dimensions. So if you"
},
{
"end_time": 2103.916,
"index": 79,
"start_time": 2078.695,
"text": " List the coordinates of a four-dimensional spacetime, like x, y, z, and t. Suppose we orient them, so I say x, y, z, and t. Now, if you have a two-form, so a two-form is something which likes to take in two planes or two bi-vectors and produce numbers. So, for example, I take x, y."
},
{
"end_time": 2130.52,
"index": 80,
"start_time": 2104.377,
"text": " But if I remember I had x, y, z, and t, so if I took x, y, then the remaining is z, t. And if I took x, z, the remaining would be y, t, and so on. So there is this duality between two forms, which is given by the orientation of the four-dimensional space. And so the minimal energy configurations turned out to be such for which the curvature"
},
{
"end_time": 2159.394,
"index": 81,
"start_time": 2131.596,
"text": " is cell dual. So if you apply duality to it in that sense, you get itself or maybe negative of that, depending on the sign of the topological class. So these are called instantons in the technical sense. But this is not the most general definition, of course, of what instanton is. And the significance of those solutions in quantum field theory is that"
},
{
"end_time": 2187.278,
"index": 82,
"start_time": 2160.128,
"text": " These are configurations for which most of the action in both literal and mathematical sense happens in a kind of compact region of space time. And here I should make a disclaimer that the space time I'm talking about is not the real space time. It's the Euclidean space time. Which incidentally is real."
},
{
"end_time": 2217.261,
"index": 83,
"start_time": 2188.148,
"text": " Which is incidentally real, but but it's also imaginary. So in this real real manifold, what we perceive as a time direction is actually imaginary. Yes, yes, yes. I don't like when people actually present it this way. I like to think of it as. As a part of the structure in which we study, we study quantum fields on"
},
{
"end_time": 2245.196,
"index": 84,
"start_time": 2218.148,
"text": " Many faults endowed with complex metrics in general and so the metric can be Real with different signature so You know square of the Distance and time direction could be positive or negative or it could be positive in all directions. So we call that Euclidean signature metric But in general"
},
{
"end_time": 2274.019,
"index": 85,
"start_time": 2245.589,
"text": " We could and it should also study the metrics, which are just complex. So the metrics elements are complex. The values of the metric are complex. And but that's maybe a slight digression. I'm sorry, this is a bit technical. That's fine. Anyway, so Zadrikin-Witten wrote a formula and they said that to really understand this formula from the first principles, one should derive it by doing honest path integral"
},
{
"end_time": 2302.807,
"index": 86,
"start_time": 2274.633,
"text": " in supersymmetric gauge theory and evaluating those contributions to the effective action coming from those peculiar fluctuations of gauge fields, which happen in compact regions of Euclidean spacetime. And the moduli space of instantons, is that a connected space? So it is connected for each topological class."
},
{
"end_time": 2329.019,
"index": 87,
"start_time": 2303.319,
"text": " So if you fix the topological charge, it is connected. But for different topological charges, it is there, you know, disconnected distinct spaces. So a priori, a priori, mathematically, you would have said, OK, I mean, I have just the infinite number of integrals to compute. Why would they combine into something nice? So. I would say that."
},
{
"end_time": 2354.701,
"index": 88,
"start_time": 2330.555,
"text": " You never approach an interesting problem directly. It's very hard to study things face-on. You need some kind of preparation work, and maybe if you're lucky, you will get to the point where you hit on the problem you're actually interested in, or you dare to be interested in."
},
{
"end_time": 2383.097,
"index": 89,
"start_time": 2356.254,
"text": " So at this point, you were exploring, hoping that you were going to get to the solution, but allowing yourself to not get to the solution because it was interesting what you were exploring anyhow. Right. So the goal was so that at this point, I mean, I had to get familiar to familiarize myself with instantons with modular spaces of instantons, what they are, what they look like, what what do people know about them? And so one,"
},
{
"end_time": 2413.66,
"index": 90,
"start_time": 2384.36,
"text": " The thing which became popular at the time was the study of quib of varieties. But they were not defined as the modular space of solutions of some partial differential equations. Baffelin-Witten mapped this property to the very non-trivial property of the four-dimensional gauge theory, which inversed the gauge coupling. So somehow they related the coupling of gauge theory to the geometry of some at the time abstract"
},
{
"end_time": 2440.469,
"index": 91,
"start_time": 2414.445,
"text": " two-dimensional torus elliptic curve. So that was part of the one of the hints towards the BPS safety correspondence. That some structures found in the studies of four-dimensional gauge theory study structures involving supersymmetry and the cohomology of some supercharges are related to the structures found in the in two-dimensional conformal field theory."
},
{
"end_time": 2469.974,
"index": 92,
"start_time": 2441.34,
"text": " Right, so just to linger on this point, so many people have heard of ADS-CFT and that involves differential geometry and harmonic analysis and string worldsheets and so on, but there's also something else called BPS-CFT which you helped popularize and articulate. Well, BPS-CFT was just maybe a joke term which I coined at the time because it sounds like ADS-CFT, but of course it's"
},
{
"end_time": 2500.299,
"index": 93,
"start_time": 2470.52,
"text": " Just like, just as ADS-CFT is a kind of a string duality, it's a kind of open-close string duality in my mind. Even though some people claim it's just holography in its purest form and doesn't require string theory, I don't think it's the case. I think it's... Oh, you think it's string-dependent, it's not just string-inspired. I mean, it's kind of a, it's one of these things which makes string theory great."
},
{
"end_time": 2530.486,
"index": 94,
"start_time": 2500.794,
"text": " It gives you a mechanism for holographic duality. ADS-CFT in that sense is a projection of some 10-dimensional entity. There are versions of ADS-CFT which use the same theory. It turns out that the integrals which one had to compute to get the Zebrick-Witten solution were independent of that parameter."
},
{
"end_time": 2555.998,
"index": 95,
"start_time": 2530.794,
"text": " So you can hope that if you send it to zero, you got the same answer as if you don't. So this is one of the tricks which I learned essentially from Whitten. Whitten used in his studies of supersymmetry breaking a quality which is now called Whitten index, which is the"
},
{
"end_time": 2585.247,
"index": 96,
"start_time": 2556.783,
"text": " also a partition function. It's a kind of a partition function of a quantum system with supersymmetry, which differs from the conventional partition function where you average exponential negative Hamiltonian divide by temperature. You also insert the operator which weighs bosonic and fermionic states in your Hebel space with different sides."
},
{
"end_time": 2615.691,
"index": 97,
"start_time": 2586.937,
"text": " And so we can show that in many favorable circumstances, this partition function is independent of the temperature. And so you can compute it in different limits when temperature goes to zero or temperature goes to infinity. And when temperature goes to zero, this partition function receives contributions from ground states only from the vacuum. And when the temperature goes to infinity,"
},
{
"end_time": 2639.309,
"index": 98,
"start_time": 2616.51,
"text": " It is usually the regime where things are computable. And so one of the consequences of this of this method is the physics proof of I.T.S. Singer index theorem, which was used, for example, by Luis Andres Gomez, my current director."
},
{
"end_time": 2668.012,
"index": 99,
"start_time": 2639.872,
"text": " So if they can prove that there is a parameter in your Hamiltonian or in your action or in your system to which the observable you want to compute is insensitive, then you vary this parameter as much as you can and try to find the regime in which things become computable. And so in my story, one of these parameters was this parameter of non-commutativity."
},
{
"end_time": 2692.944,
"index": 100,
"start_time": 2669.77,
"text": " Unfortunately, this trick cured one half of the problem. Namely, it dealt with this singularity of the modular space of instantons, which corresponds to small distances in spacetime, so things which happen at short distances."
},
{
"end_time": 2717.227,
"index": 101,
"start_time": 2694.94,
"text": " But since I'm interested in working over R4, so we want to understand physics on the four-dimensional flat space, I like mathematicians who use instant and modular spaces for things like invariance of four manifolds, which is the content of Donaldson theory."
},
{
"end_time": 2747.517,
"index": 102,
"start_time": 2718.49,
"text": " for them the interesting spaces are complex spaces even though sometimes with very clever clever tricks they draw conclusions even for r4 and famously found that that there are unlike all r dimensions so as a smooth manifold there is only one r1 there is only one r2 only one r3 only one r5 only one r6 there's only one rn for any n except four"
},
{
"end_time": 2776.476,
"index": 103,
"start_time": 2748.046,
"text": " And four dimensions, there are exotic Euclidean spaces, which is, which is quite controversial. Just a moment. Do you think that fact is connected to the R4 of our space? The fact that our space time is R4? Could be. Yeah, because it has to do, it has to do with the fact, with the fact that two dimensional surfaces meet in four dimensions and they don't meet in, in, in higher dimensions generically. And, uh,"
},
{
"end_time": 2796.937,
"index": 104,
"start_time": 2776.732,
"text": " So the fact that certain things meet often, like probabilistically, so if you throw random walkers, it's probably crucial for life. So let me just put it this way. OK. Life needs interaction and propagation."
},
{
"end_time": 2822.363,
"index": 105,
"start_time": 2797.381,
"text": " So the exotic R4 is our world. The R4 is our world because there's an exotic R4. And if there wasn't, then we wouldn't have enough interactions for us to be here. Right. So, okay, I would say, I don't know what the causality is, but I would say that the reason exotic R4 exists is probably"
},
{
"end_time": 2854.087,
"index": 106,
"start_time": 2825.333,
"text": " This episode is brought to you by State Farm. Listening to this podcast? Smart move. Being financially savvy? Smart move. Another smart move? Having State Farm help you create a competitive price when you choose to bundle home and auto. Bundling. Just another way to save with a personal price plan. Like a good neighbor, State Farm is there. Prices are based on rating plans that vary by state. Coverage options are selected by the customer. Availability, amount of discounts and savings, and eligibility vary by state."
},
{
"end_time": 2880.384,
"index": 107,
"start_time": 2855.555,
"text": " important in things like having chemistry and things like that. So there is certain topology involved in having complex structures, in having biochemistry, depends on certain topological features of our space and space time."
},
{
"end_time": 2911.254,
"index": 108,
"start_time": 2881.766,
"text": " Some of these things are crucial in the construction of exotic R4. So these things might not be unrelated. But since it's discussing things for which you have only one sample, it's hard to draw conclusions. Because I haven't heard anyone connect chemistry to R4's exotic property. Unless this is just a speculation of yours, you're not sure."
},
{
"end_time": 2939.855,
"index": 109,
"start_time": 2912.756,
"text": " No, maybe we could probably elaborate. I mean, I'm not strong in chemistry, so if I venture in this direction, I'll probably say something silly. I would say that the fact that four-dimensionality, well, what do we know? We know that four dimensions are special in interacting"
},
{
"end_time": 2958.882,
"index": 110,
"start_time": 2940.759,
"text": " Propagating vector fields so gauge gauge theory as quantum theory Is well defined in four dimensions and has very interesting properties, which which is And so it is not well defined in five or higher dimensions and it's less interesting in in few dimensions and"
},
{
"end_time": 2984.241,
"index": 111,
"start_time": 2959.343,
"text": " People size of three dimensions because it still maintains some has some creative features, but it's there are certain things you get almost for free because you get mass scale for free and so on. In four dimensions, you things happen just, you know, they're kind of on the boundary between things becoming trivial or ugly. So it's a four dimensions is interesting border case."
},
{
"end_time": 3011.937,
"index": 112,
"start_time": 2984.957,
"text": " And the fact that the possibility of exotic smooth structures in four dimensions uses the same kind of coincidences, which is just all I'm trying to say about that. Okay, so getting back to the story. So back to the story. So in physics, I'm interested. So in physics, I need to for physics applications, I need to compute meaningful integrals over"
},
{
"end_time": 3041.254,
"index": 113,
"start_time": 3012.551,
"text": " the modular space of instantons on non-compact, literally non-compact infinite R4. And this modular space is again non-compact because now those instantons, which are like pseudo particles, like events in spacetime, they can run away to infinity. So you can have a sequence of events which happen here and then on Saturn and then on"
},
{
"end_time": 3069.667,
"index": 114,
"start_time": 3042.21,
"text": " in the Andromeda and then go all the way to infinity. So there's no limit to that. So again, I'm in trouble because I cannot integrate by parts. I cannot, I mean, I'm not sure my integral is convergent. Life is difficult. So I need to invent another trick to somehow cure that possible divergence or non-compactness at least. Now, if we're doing something on R4,"
},
{
"end_time": 3100.06,
"index": 115,
"start_time": 3070.145,
"text": " R4 has a beautiful symmetry. It's a symmetry of rotation. Well, it has also symmetry of translations, but translations act without fixed points. So if we use translations, it will kill everything. So it's not the right symmetry in this case. The right symmetry to use is symmetry of rotations. And so the group of rotations of Euclidean four-dimensional space is the group SO4 and its maximal torus, maximal Abelian subalgebra,"
},
{
"end_time": 3130.299,
"index": 116,
"start_time": 3100.316,
"text": " Subgroup is two-dimensional and it's generated by rotations into orthogonal planes. Now, of course, if you just do some mathematical construction, which is completely artificial, first of all, you might make mistakes, but also you lack intuition. So it would be hard to know what is the right way to proceed. What are you doing? Meaning you lack physical intuition?"
},
{
"end_time": 3159.77,
"index": 117,
"start_time": 3130.555,
"text": " Right. So this mathematical construction was nice, but I wanted to have some kind of physical understanding where these parameters come from and what would be the physical realization. Is there a physical meaning for such a deformation? Remember, I was doing this in the late 90s and at that time, even though the non-community field theories were already popular and non-community,"
},
{
"end_time": 3189.991,
"index": 118,
"start_time": 3160.043,
"text": " information breaks Lorentz symmetry, because you see, if I tell you that x and y don't commute, the fact that x and y commute in one way and z and t commute in another way breaks the symmetry between x, y and z, which is part of the Lorentz symmetry. So people started discussing quantum field theories, which were not Poincare invariant, but people were reluctant. So this partition function scale, like a partition function of a non-ideal gas where the volume"
},
{
"end_time": 3212.619,
"index": 119,
"start_time": 3190.299,
"text": " The role of the volume of the space in which the gas was confined was played by the inverse product of these parameters epsilon 1, epsilon 2. And the leading term, the free energy, so it's the energy, free energy per unit volume, was its arbitrary potential. So first I discovered that experimentally, just by expanding the term by term,"
},
{
"end_time": 3239.991,
"index": 120,
"start_time": 3213.797,
"text": " Because I felt like these partition functions were good for something because they were so beautiful. They fit nicely to each other, one instant on, two instant on, so on. And then when I saw that the one case people computed, the one instant on term people computed, and sometimes took it to instant terms, people computed by very hard work, matched with what I got by relatively simple work, then I was convinced it must work."
},
{
"end_time": 3270.282,
"index": 121,
"start_time": 3240.35,
"text": " Of course, I was not very confident in myself, so I had to go up to five instantons and get some help from numerics and got some errors along the way. So I had to. Is that the one about the story where you were working for two weeks on a laptop and then you were in a hotel room and it worked out? That's a different one. That's a different story. By that time, I was much more confident."
},
{
"end_time": 3299.428,
"index": 122,
"start_time": 3271.032,
"text": " So that's when I published it. I made a conjecture in this paper that this partition function, which was a deformation of, I mean, it gave more than Zybrokrutin pre-potential. It gave something else because it had two more parameters. And so the conjecture was that it gives the amplitudes of topological string. And the reason for that we can discuss. And also that it is given by some"
},
{
"end_time": 3328.899,
"index": 123,
"start_time": 3300.35,
"text": " It was the moment both of joy and frustration because"
},
{
"end_time": 3359.565,
"index": 124,
"start_time": 3329.889,
"text": " It felt like I should be able to prove it, that it's not just conjecture, but just to prove that indeed the curves that Zabrik and Witten proposed and conjectured and people then extend their conjectures to other gauge groups and types of theories with various metric content should be provable. But I was too close and I couldn't do it."
},
{
"end_time": 3389.326,
"index": 125,
"start_time": 3360.213,
"text": " Uh, it took me another year and another happy, happy meeting, just a fortunate meeting with the right person at the right time to be able to, to prove at least one of some of these conjectures. So that was the meeting with Andrej Okunkov, uh, just on the train station in the, in France, in Bureaux-sur-Rivette. And, um, so I knew Andrej from before, but we've,"
},
{
"end_time": 3416.101,
"index": 126,
"start_time": 3389.991,
"text": " basically just played volleyball together when when I was in Princeton as a student he was at Princeton as a postdoc and we just played volleyball but we never really discussed physics or math I knew that he was an expert in combinatorics and maybe he knew something about partitions and so I just stumbled upon him on the train station and he said what what are you doing I said what do I have this"
},
{
"end_time": 3445.486,
"index": 127,
"start_time": 3416.732,
"text": " Difficult problem when I have a sum over sets of infinite set of partitions Which I believe has this hidden structure there with curves emerging, but I don't know how to prove it. I mean, I don't have It feels like it should be something to have something to do with conformal field theory to dimensions, but I'm not sure exactly what and he said well, but You are in luck because I'm computing the partition sums"
},
{
"end_time": 3474.872,
"index": 128,
"start_time": 3445.93,
"text": " from dusk till dawn and then from dawn till dusk. And he also knew of a problem about random partitions for which the answer is given by some kind of a curve. So he knew about some emergent geometry in the problem involving random partitions. And that was the famous limit shape of Vershik, Kerov and Logachev."
},
{
"end_time": 3504.036,
"index": 129,
"start_time": 3475.691,
"text": " which mathematicians studied for completely, you know, different reasons also in late 70s. And it was a different crowd of mathematicians interested in different problems. And, but fortunately I met Andre and he said, well, but this is a similar problem. So maybe, maybe there is a similar solution. And again, it turned out that the problem which mathematicians solved was for the trivial case, but"
},
{
"end_time": 3532.466,
"index": 130,
"start_time": 3504.821,
"text": " With some ingenuity, and Andrei had a lot of ingenuity at the time with this, so he used a lot of interesting complex analysis ideas to transform the problem of computing my asymptotics to the problem of finding a limit shape. Limit shape is kind of the most probable geometric shape in the ensemble of random geometries you are given."
},
{
"end_time": 3547.517,
"index": 131,
"start_time": 3533.336,
"text": " We found that indeed it was the family of zebra Putin curves that govern the asymptotics of my partition function. Wonderful. So you and volleyball. Yes, because."
},
{
"end_time": 3575.776,
"index": 132,
"start_time": 3547.927,
"text": " You went to Paris for volleyball and you weren't doing much math there, but then I think in 98 or so you with Maxime Koncevich, I think you went there for volleyball, but then it ended up being a fortuitous lunch, something like that, if I recall. You don't need to get into that story, but it's so... Well, volleyball is important for mathematicians for some reason. So I like, I mean, I'm real amateur in volleyball."
},
{
"end_time": 3599.838,
"index": 133,
"start_time": 3576.408,
"text": " Sometimes I manage to get a good serve, sometimes I don't. But at IHS, which I visited in 1998 and where I worked later for many years, there was this tradition that in the summer, in the compound where visitors live, people would play volleyball. It used to be"
},
{
"end_time": 3620.435,
"index": 134,
"start_time": 3600.384,
"text": " The main driving force behind that used to be Professor Kirillov, Alexander Kirillov, the inventor of geometric quantization, who also was the advisor of Andrey Okunkov. And so that's why we played volleyball, because Kirillov invited us to play volleyball."
},
{
"end_time": 3648.968,
"index": 135,
"start_time": 3621.118,
"text": " Okay, so from Moscow to Princeton to Paris to New York, you've worked in these different cultures. Would you say that the culture of academia is different there, or at least the one that you interacted with? Or would you say there's more similarities than dissimilarities?"
},
{
"end_time": 3674.172,
"index": 136,
"start_time": 3650.282,
"text": " Yeah, it's a... Actually, I mean, I wouldn't say... I don't think they're so different. I wouldn't... Right now, I mean, after having spent 30 years in doing physics and math in between,"
},
{
"end_time": 3704.309,
"index": 137,
"start_time": 3676.015,
"text": " I think it's pretty much universal. I think that's what makes science beautiful. It's kind of universal. It's independent of the nationality. Yes. I mean, in Moscow, I had kind of a double, I led maybe double life, maybe triple life. I had two advisors who"
},
{
"end_time": 3735.572,
"index": 138,
"start_time": 3706.135,
"text": " One was my advisor in particle physics. Another was my advisor in, I guess, singularity theory. Both of them knew that I'm doing something else. I'm actually, I mean, I had some, my main interest was in string theory, in kind of modern mathematical physics. But part of their kind of training, maybe part of the coming of age kind of thing,"
},
{
"end_time": 3762.858,
"index": 139,
"start_time": 3736.357,
"text": " You have to work on traditional problems for a while. So I was computing radiative corrections to standard model parameters to get bounds on the mass of top quark. It was before top quark was discovered. So that was part of my work, kind of part of a duty, so to speak."
},
{
"end_time": 3790.265,
"index": 140,
"start_time": 3763.865,
"text": " So I worked with my advisor was Lev Okun, who was basically a phenomenologist, an expert in weak interactions. He wrote several very good textbooks on electroweak theory, on physics of elementary particles. But apart from one paper which I wrote with him on those bounds, we didn't work much"
},
{
"end_time": 3819.104,
"index": 141,
"start_time": 3791.032,
"text": " But he kind of protected me from, I mean, I guess he was my protector in many respects. I mean, we could get into that life in Moscow. So it was Soviet Union and just post just post Soviet Union. Just it was very different from what it is now. So so the the expectations of our young fees from young thesis were different than the"
},
{
"end_time": 3848.063,
"index": 142,
"start_time": 3819.872,
"text": " There were different obstacles and different challenges. My math advisor was Vladimir Arnold. He didn't know, I mean, I was just going to his lectures in Moscow State University, even though I studied at a different university, and he didn't know I was not a student there. So I just attended his seminar, his lectures, and he gave me, I mean, every year he would give a list of problems to the participants of his seminar."
},
{
"end_time": 3875.23,
"index": 143,
"start_time": 3848.422,
"text": " and if you wanted you could work on them and so I worked on one of these problems and it was very useful for me because I learned about topology about many many things but he actually gave me some advice which I guess I used and I mean he gave me lots of advices but one of them was to you know always try"
},
{
"end_time": 3903.541,
"index": 144,
"start_time": 3876.067,
"text": " Find all the parameters in your problem and always try to take them to extremes. That's the advice which you can take outside outside the science as well. So your physics supervisor was David Gross in Princeton. Yes, when I got to Princeton, my advisor was David Gross. But again, when I was a graduate student, David was"
},
{
"end_time": 3931.152,
"index": 145,
"start_time": 3904.206,
"text": " Kind of going through a complicated phase in his personal life and he basically let me do whatever I wanted, which was the best case scenario for a good student. Good students should not ask problems from their advisors. They should find problems and maybe if they have some difficulty in solving them, they should come to the advisor and ask for advice."
},
{
"end_time": 3961.647,
"index": 146,
"start_time": 3932.193,
"text": " So David gave me when I was a student, he gave me some practical devices, but we started working together already later when I already had the job actually. So which was actually the best, also the best time because I got to work with happy and confident David. So he already was in Santa Barbara and I got him interested in non-commutative geometry."
},
{
"end_time": 3989.172,
"index": 147,
"start_time": 3962.261,
"text": " So we found interesting solutions of gauge theory of non-commutative space. We found non-commutative monopoles and found that they actually carry the physical string attached to them. So the Dirac string of magnetic monopole of Dirac, the string of Dirac monopole, which is kind of imaginary object on a non-commutative space becomes physical."
},
{
"end_time": 4019.411,
"index": 148,
"start_time": 3990.077,
"text": " He always told me to believe in myself and to be confident and that don't get intimidated by other intimidating people."
},
{
"end_time": 4048.643,
"index": 149,
"start_time": 4019.735,
"text": " So if people tell you that you should not work on something because it's wrong or because it's morally wrong or because it's, uh, whatever reasons, if you feel that it's important, work on that. So was there something in particular that you were insecure about that you were coming to him saying, I'm not sure if I should work on this. And then it was that that moment that he gave you that advice. Well, it was, uh, I think, uh,"
},
{
"end_time": 4080.52,
"index": 150,
"start_time": 4051.391,
"text": " It was before one of the string conferences. Incidentally, I think it was a string conference in Paris. I think I was competing, working on similar topics with a group of prominent physicists. So you felt in competition with them, like you felt there was a rivalry? Yeah."
},
{
"end_time": 4111.903,
"index": 151,
"start_time": 4082.346,
"text": " He had some stories when"
},
{
"end_time": 4138.609,
"index": 152,
"start_time": 4112.415,
"text": " He had a"
},
{
"end_time": 4162.688,
"index": 153,
"start_time": 4139.735,
"text": " result, which he could confirm with the models and examples, which showed precisely the opposite. And so it was a very happy coincidence. So it made a good show in the sense that people make strong claims and then one claim after another. And then there's some discussion. And of course, David was right. So it's his story."
},
{
"end_time": 4192.722,
"index": 154,
"start_time": 4163.524,
"text": " So he told me that, you know, you should just aim for be yourself, believe in yourself and present what you have. And maybe the other party doesn't have as strong a result as you think. And he was right. So I think to do. So I've had another one of your collaborators on Edward Frankel and I'll post that podcast on screen and in the description. He was on several times, actually, and it's coming up again to talk about the geometric Langlands. So"
},
{
"end_time": 4220.316,
"index": 155,
"start_time": 4193.643,
"text": " What is it about Edward Frankel that you admire? You all both seem to get along both mathematically, but also as people, so you can speak about both. Well, we are, well, what I admire about Edward, so he's, first of all, he's charming. He's very intelligent, very knowledgeable, but also"
},
{
"end_time": 4251.886,
"index": 156,
"start_time": 4225.145,
"text": " Well, I would say that for a person of his charm, he actually has a big heart, which is very rare. Sometimes people kind of use the charm to the advantage and eventually they become cynical. Edward is the opposite."
},
{
"end_time": 4275.708,
"index": 157,
"start_time": 4252.585,
"text": " He's evolving, he's learning and he's interested in things I'm interested in. So it's always, I mean, I always learn something from him and it's always not just a pleasure, it's a challenge and a pleasure to be in communication with him."
},
{
"end_time": 4306.169,
"index": 158,
"start_time": 4276.493,
"text": " So mathematically now, what is it that you admire? So you admire his heart, sure, at a personal level. Yeah. So he combines, he combines kind of a deep knowledge and understanding of very abstract things. And also, but he also knows how to connect them to things which kind of I understand as a physicist. So he can, he can speak to me"
},
{
"end_time": 4330.623,
"index": 159,
"start_time": 4308.814,
"text": " Yes. Okay. So earlier you also mentioned physical intuition and I want to touch on this because while physical intuition can take you so far, do you find that it hinders at some point? Do you find that? Well, it helps in general. And if it depends, then what does it depend on?"
},
{
"end_time": 4367.142,
"index": 160,
"start_time": 4338.592,
"text": " Well, on my path, I was not always guided by physical intuition. I would say I'm guided either by mathematical intuition or by physical intuition. So there is always some kind of some sense of beauty in both worlds. And they sometimes complement each other. Sometimes they are compatible."
},
{
"end_time": 4396.135,
"index": 161,
"start_time": 4367.91,
"text": " Well, of course, historical famous examples where they are in contradiction and then mathematical intuition, for example, for Dirac was was more important physical intuition and it led to a change in our physical understanding, like with the concept of antiparticles, with positron and things like that. But"
},
{
"end_time": 4425.828,
"index": 162,
"start_time": 4396.698,
"text": " Of course, physical intuition sometimes could be misleading because we are limited by our experience and the models we study. So again, the good example would be the invention of quantum mechanics. You have to really step outside, step out of your mind box to embrace the quantum intuition."
},
{
"end_time": 4454.275,
"index": 163,
"start_time": 4426.937,
"text": " You cannot understand it using everyday experience. We get used to it, but to truly understand quantum mechanics, I don't know what kind of meditation you need for that. I haven't found it. I have a very good friend"
},
{
"end_time": 4481.527,
"index": 164,
"start_time": 4454.718,
"text": " film director Ilya Khrzhanovsky who wanted to make sort of a quantum film which where you could kind of experience quantum reality through interesting through you know multitude of universes you live in but well I don't know it's some people say that"
},
{
"end_time": 4503.183,
"index": 165,
"start_time": 4481.869,
"text": " He succeeded to some extent. Some people say he made a step. But so we keep talking about this with him, whether maybe animation could be good media for for kind of explaining what quantum world is. So here's a question. I was just at dinner last night with some people."
},
{
"end_time": 4530.572,
"index": 166,
"start_time": 4503.66,
"text": " And it's commonly said that when we measure, we observe something real and we don't see a superposition. And in part, that's the measurement problem. And then I just asked, well, what would it look like if you observed a superposition and the people were pausing it? Because it's not exactly that you see something here and here, but it's half transparent. No, we observe effects of the position because we, I think the"
},
{
"end_time": 4560.043,
"index": 167,
"start_time": 4531.169,
"text": " Issues that the devices we measure with a kind of classical devices. So and and the way the brain will analyze this is a classical brain. So to really understand superposition as a quantum, you have to be quantum entity to to better superposition. So Greg Moore, which I want to talk about as well, he described something called physical math. And that's in contrast with mathematical physics."
},
{
"end_time": 4589.889,
"index": 168,
"start_time": 4560.401,
"text": " Now, I know that you didn't come up with that distinction, but do you see yourself as more of a physical mathematician or more mathematical physicist? I think of myself. So I think of myself more. So I'm more physical than a mathematician, but I put it this way. So just just that. But I'm equally interested in mathematics as I'm interested in physics. And for me, mathematical beauty is as important as physical beauty."
},
{
"end_time": 4613.268,
"index": 169,
"start_time": 4590.367,
"text": " physical beauty, not physical beauty, physical beauty. Physical beauty is also important, but it's different. Yes. Okay. Well, okay. Let's talk about mathematical beauty. Physical beauty is something for another time, but give a taste for what it is to to find something mathematically beautiful for people who haven't experienced that."
},
{
"end_time": 4644.087,
"index": 170,
"start_time": 4618.37,
"text": " Ford BlueCruise hands-free highway driving takes the work out of being behind the wheel, allowing you to relax and reconnect while also staying in control. Enjoy the drive in BlueCruise enabled vehicles like the F-150, Explorer and Mustang Mach-E. Available feature on equipped vehicles. Terms apply. Does not replace safe driving. See Ford.com slash BlueCruise for more details."
},
{
"end_time": 4709.991,
"index": 171,
"start_time": 4680.026,
"text": " Well, of course, there are kind of professionals who come up with the with examples of mathematical constructions for for layman, which are kind of beautiful. For me, mathematical beauty is what unites distinct parts of mathematics. For example, the fact that. You can think of prime numbers. Prime integers."
},
{
"end_time": 4738.404,
"index": 172,
"start_time": 4710.23,
"text": " as of points on some space. So they, for me, this is a beautiful concept. In what sense? Well, it's in the sense that integers, they are analogous to the ring of polynomials. So the ring of integers, integers you can add, you can multiply, you cannot always divide, but some integers are divisible by other integers. So just like polynomials,"
},
{
"end_time": 4766.305,
"index": 173,
"start_time": 4738.78,
"text": " You can multiply them. Some polynomials are divisible by other polynomials. So if the polynomial vanishes at a point, let's say polynomial of x vanishes at point a, it means that it's divisible by x minus a. So in that sense, prime numbers is what the other integers might be divisible by. So these are the analogs of those minimal polynomials, x minus a."
},
{
"end_time": 4798.49,
"index": 174,
"start_time": 4768.916,
"text": " Okay, you have a lecture titled natural language, geometry and physics. Yes. What I want to know is what's the relationship between human language and then the mathematical language that is used to describe the universe. So I was I'm very much impressed that I'm amazed by the"
},
{
"end_time": 4828.541,
"index": 175,
"start_time": 4800.452,
"text": " The fact that human language, so natural language in the human sense, evolves, so it's not a static object, it's a kind of a dynamical system, and as such, I was proposing to study it as a physical system like a growing crystal or melting crystal."
},
{
"end_time": 4858.302,
"index": 176,
"start_time": 4830.93,
"text": " In my work on Instant Dots, which I reviewed for you, the concept of emerging geometry appears. The problem from gauge theory became a problem of enumerating some combinatorial structures, partitions, which you can visualize as young diagrams, or you can visualize as some kind of"
},
{
"end_time": 4887.449,
"index": 177,
"start_time": 4859.121,
"text": " The enumeration of those objects, you can also view as a search for what's called equilibrium distribution in the ensemble of dynamically changing shapes. So for one young diagram, you can go to another one by changing a few wiggles. And so you can assign the probability for those wiggles"
},
{
"end_time": 4915.282,
"index": 178,
"start_time": 4888.422,
"text": " What if those wiggles were analogous to the words or the sentences in the natural language? And so the way the language changes in time could be"
},
{
"end_time": 4944.667,
"index": 179,
"start_time": 4915.896,
"text": " may be assigned some transition amplitudes, transition probabilities, and maybe there is some emergent geometry which emerges as the most probable configuration. So maybe the natural language, maybe to the natural language, certain geometric shape is associated, just like to these random partitions in my incident calculus, the Zyberputin curve, the geometric object was associated. So language is much more complicated than"
},
{
"end_time": 4974.462,
"index": 180,
"start_time": 4944.855,
"text": " So, I was proposing in this lecture to embark on the statistical analysis of the language, not in the way it's done in the large language models,"
},
{
"end_time": 5003.336,
"index": 181,
"start_time": 4975.179,
"text": " but in doing some kind of time, time sequence analysis of language. So to study how it changes from, from, from, you know, century to century, interesting, maybe from year to year, but also the, the pun in this title was that the geometry is the language of nature, of physics. So that, so that's when there's a natural language in that sense that also physics is so,"
},
{
"end_time": 5032.108,
"index": 182,
"start_time": 5004.138,
"text": " Have you followed up this work? Not quite. Not yet. Not yet. But in the back of your mind, are you developing conjectures or you've just abandoned it and it was a fun activity at the time? No, no, no. As I told you, this is a very hard problem. So you don't attack it head on. So I'm developing models which will kind of surround this problem and then I will attack when the time comes."
},
{
"end_time": 5061.732,
"index": 183,
"start_time": 5032.517,
"text": " So I'm developing other models for simpler systems, simpler than languages. So speaking of young diagrams, and I don't want to get us into more ground that'll take us quite some time to get to, but there is gauge origami. And if I understand that correctly, that has to do with counting young diagram configurations associated with tetrahedra, whose edges are colored by vector spaces. So"
},
{
"end_time": 5092.841,
"index": 184,
"start_time": 5063.114,
"text": " So we should fast forward to a few years later. So 2003, with Andrea Kunikoff, we understood how to evaluate asymptotics of the partition function. But then it was understood that the whole partition function is a very meaningful and interesting object because it contains information about strings, black holes,"
},
{
"end_time": 5119.735,
"index": 185,
"start_time": 5094.172,
"text": " local collibial also thinking about this particular function people came up with the notion of the so-called refined topological string so there so this whole thing was worth studying and so one way to approach this problem is to say maybe the the zabir-quintin curve which is the"
},
{
"end_time": 5148.473,
"index": 186,
"start_time": 5120.247,
"text": " the object which captures the limit shape, the asymptotic of the partition function can be somehow quantized or deformed and had to be deformed by two parameters, by two deformations to capture the structure of the whole partition function. So this led to the notion of QQ characters, which is maybe not a very"
},
{
"end_time": 5175.708,
"index": 187,
"start_time": 5149.224,
"text": " Maybe not the most fortunate name, but, well, it has a reason. I mean, this name didn't come from nowhere. So it's a notion. It's not observable. So it's a way to measure the partitions. So it's a device. So the device with expectation value, first of all,"
},
{
"end_time": 5205.657,
"index": 188,
"start_time": 5177.602,
"text": " contains information about the the shapes of the random young diagrams and so on but on the other hand has some analytic properties allowing to maybe compute it or to write some differential equation on it so i realize that one can define some kind of gauge theory problem it's best done in the context of non-commutative gauge theory where gauge fields leave"
},
{
"end_time": 5230.93,
"index": 189,
"start_time": 5206.169,
"text": " the tricky space-time which has different parts. So it's four-dimensional, but it's not like four small dimensions. It's four dimensions, four dimensions with coordinates x, y, z, t, and then four dimensions with coordinates, I don't know, u, v, w, s. Sure."
},
{
"end_time": 5261.527,
"index": 190,
"start_time": 5233.166,
"text": " So then once I understood that it works and it has good properties and it actually is behind this observable, which I call Q-character for the reason, which maybe I will explain later. I asked myself, this is where the mathematical intuition kicks in. So if you have two transverse four planes inside the eight-dimensional space,"
},
{
"end_time": 5289.258,
"index": 191,
"start_time": 5262.193,
"text": " Why not adding other four-dimensional planes? And so the total number of four-dimensional planes you could have is six. And so you can label, they would be in correspondence with the edges of a tetrahedron. And the axis of the four complex dimensional space would be in correspondence either with vertices"
},
{
"end_time": 5316.118,
"index": 192,
"start_time": 5289.753,
"text": " or with the faces of a tetrahedron, depending on how you want to think about it. So the tetrahedron is just an object which helps in, you know, just keeping track of all the moving parts of this construction. But it's really just a gauge theory which lives on a particular singular spacetime. And the reason it's interesting to study is because"
},
{
"end_time": 5342.961,
"index": 193,
"start_time": 5316.561,
"text": " From the point of view of the observer who lives on just only one of six sheets of this complicated structure, you are just exploring all possible local or semi-local observables of gauge series. You have observables inserted at the point. You have observables which are inserted along a two-dimensional plane or another two-dimensional plane."
},
{
"end_time": 5374.428,
"index": 194,
"start_time": 5344.428,
"text": " and it's constrained to be six because of the rotational symmetry. Of course, if you don't have any symmetry in question, you can, of course, place things any way you like. It will be kind of, you know, unruly. But if you insist on things being invariant under rotations in now 10 dimensional space time, you will be bound by placing things at along the coordinate plates."
},
{
"end_time": 5403.387,
"index": 195,
"start_time": 5375.896,
"text": " And so this is what this gauge origami picture is emerging. It's called gauge origami because it looks like you have your folding paper different different folds. Right. Is this at all related to the positive geometries of NEMA or the amplitude Hedron? Not as far as I know, but in my experience, things are"
},
{
"end_time": 5433.933,
"index": 196,
"start_time": 5404.701,
"text": " Okay, what's something that you've learned as a lesson from collaborating with Edward Witten?"
},
{
"end_time": 5463.029,
"index": 197,
"start_time": 5434.94,
"text": " And also, I'd like to get to Greg Moore as well, but afterward. So, I guess with Edward, I share kind of affinity to rigor and"
},
{
"end_time": 5493.029,
"index": 198,
"start_time": 5463.695,
"text": " He told me on many occasions that this desire to be rigorous is sometimes very constraining. So sometimes it really stops you from doing something because you cannot proceed in a rigorous manner. But I found that one can be rigorous in a different field. So our stumbling blocks"
},
{
"end_time": 5520.657,
"index": 199,
"start_time": 5493.558,
"text": " We're kind of complicating each other. So when whenever. So we were trying to in our work, we were trying to combine something I learned with something he learned. So he was working with Kapustin on on. This approach to geomagic Langlands, I was working on my partition functions. I was certain it was the same thing. So these these these were"
},
{
"end_time": 5546.852,
"index": 200,
"start_time": 5521.169,
"text": " The domain, the realm where we were discussing, the subject of our study is the same subject. It was certain that there was a way to relate our viewpoints."
},
{
"end_time": 5573.592,
"index": 201,
"start_time": 5547.637,
"text": " but I had my stumbling blocks and he had his stumbling blocks and it just turned out that somehow they were at the right places and so I could advance where he couldn't and he could advance where I couldn't and so I guess my the main lesson from Edward which unfortunately I could still to this day I cannot quite embrace is that one should"
},
{
"end_time": 5603.08,
"index": 202,
"start_time": 5575.128,
"text": " And by getting stuck in the problem, you mean that if you're not making progress, you still continue to think about it. Whereas with Edward, he will transition."
},
{
"end_time": 5631.988,
"index": 203,
"start_time": 5603.763,
"text": " Probably, yes. I mean, it's not that I'm... I'm continuing to think about the problem, not because I'm kind of stupid, that I don't understand, that it's at least nowhere. It's because I feel like there is a progress to be made, or I just cannot not do this. It's one of these things when you do what you cannot not to do. Like it's an obsession? Like an obsession, yes. Like an obsession."
},
{
"end_time": 5661.067,
"index": 204,
"start_time": 5632.773,
"text": " Okay, so what did you learn from Greg, Greg Moore? Persevere. So if you persevere. The opposite. Okay, got it. So Greg is close to be in that sense. Yes, you can get obsessed with things and it's okay. Yeah. Do you have a specific example in mind with you working together with him? Yes, we had"
},
{
"end_time": 5690.913,
"index": 205,
"start_time": 5663.063,
"text": " So remember, I told you that before attacking the Zyberquint problem, one had to learn how to compute certain integrals. And so we were playing with integrals over like hypercal and manifolds, hypercal equations, using all kinds of a covariant localization, all kinds of tricks. And at some point, so this was my"
},
{
"end_time": 5720.64,
"index": 206,
"start_time": 5692.534,
"text": " What is called Miraculous Year? So it is 1998 when they met Albert Schwartz and Santa Barbara. Also in Santa Barbara, I met at some point Greg Moore and Saf Seti. I don't know if you know Saf Seti. He's a professor at Chicago. So Saf was presenting his work with Stern, I think, on the computation of Witten index"
},
{
"end_time": 5750.06,
"index": 207,
"start_time": 5721.067,
"text": " for the quantum mechanics of the zero brains. So the zero brains are the interesting particles which are defined as Z-brains. So they are particles in type 2-A string theory. But secretly, the secret life is that they are also black holes and also graviton, gravitons of 11-dimensional supergravity. And if, so the main conjecture of Witten of 1995 was that"
},
{
"end_time": 5780.981,
"index": 208,
"start_time": 5750.981,
"text": " The strong coupling limit of type 2A string theory is 11-dimensional end theory, which contains 11-dimensional supergravity in its low energy limit. And for this conjecture to be consistent, D0-branes should form bound states. So it should have a bound state with certain properties for every number of D0-branes. And so it's a question about"
},
{
"end_time": 5806.51,
"index": 209,
"start_time": 5781.374,
"text": " Supersymmetric quantum mechanics. This is a question about conventional mathematical physics, I should say. And so Saf was discussing the case of 2D zero brains. So just two particles. And he showed that to compute with an index, one had to compute certain integral over matrices, supersymmetric matrices."
},
{
"end_time": 5836.442,
"index": 210,
"start_time": 5807.142,
"text": " And Greg and I, we were in the audience. And I remember after this talk, I told them, but I told Greg that this is an example of it's one of the examples of the integrals we should be able to compute because this is the space he's integrating over happens to be, among other things, also happy color space. And so we quickly set up to set out to do the calculation using our tricks and"
},
{
"end_time": 5863.422,
"index": 211,
"start_time": 5837.637,
"text": " I quickly confirmed that for two particles it produced the answer that SAF was giving, but for three particles one had, it was already kind of complicated, multiple residue thing, so it looked hopeless. But Greg, with his perseverance, he set out, I don't know how many hours he spent, but he"
},
{
"end_time": 5890.111,
"index": 212,
"start_time": 5863.916,
"text": " carefully analyzed all these residues and it came out to give the answer which which we wanted. So then of course once you know it works then you can be clever about it and find more more scientific ways of proving proving what you want to prove but you really need this confirmation and so"
},
{
"end_time": 5919.445,
"index": 213,
"start_time": 5891.015,
"text": " This was just one example of what Greg, with his ability to do two-de-four calculations, Persevered and Pustas. And so we were able to prove the conjecture of Michael Green and Gadparli, which, thanks to the work of Satya Stern, actually implies"
},
{
"end_time": 5949.053,
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"start_time": 5920.06,
"text": " Hi, everyone. Hope you're enjoying today's episode. If you're hungry for deeper dives into physics, AI, consciousness, philosophy, along with my personal reflections, you'll find it all on my sub stack. Subscribers get first access to new episodes, new posts as well, behind the scenes insights and the chance to be a part of a thriving community of like minded pilgrimers."
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"text": " By joining you'll directly be supporting my work and helping keep these conversations at the cutting edge so click the link on screen here hit subscribe and let's keep pushing the boundaries of knowledge together thank you and enjoy the show just so you know if you're listening it's c u r t j i m u n g a l dot org kurt jaimangal dot org. What do you want to be known for."
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"text": " other than saving m theory. No, I would like, of course, I would like to. This is my so I'm pushing my physics hat. I'd like to to contrast some of my calculations, some of my predictions with actual experiment. And so"
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{
"end_time": 6038.524,
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"text": " Right now it seems that the most promising venue for high energy physicists to make predictable predictions is condensed metaphysics. So with my current student, Yugov, we just published a paper where we applied a clever mathematical technique from Young-Mills theory to the physics of two-dimensional graphene. So maybe when people"
},
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"text": " Now speaking of your student, what advice do you have for researchers young and old? Be passionate about what you do. That's the best advice."
},
{
"end_time": 6096.169,
"index": 219,
"start_time": 6068.968,
"text": " Many years ago, like 20 years, 20 something years ago, I gave an interview to my former university, to my undergraduate school. And so they asked me, what do you what advice do you give to what you give to current students? And my advice at the time was don't overburden. So don't don't burn yourself. So don't don't don't burn out. Don't burn out. Right."
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"end_time": 6104.224,
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"text": " But I think these days the attitude of modern students is different."
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"text": " My advice is go the opposite way. Be passionate about what you do. Burn yourself, whatever you want. Burn yourself. So are you suggesting that, I don't know how long ago that was, a decade or two decades ago? Two decades ago. Yeah, okay. Are you suggesting that back then the problem was over passion and it would be to their detriment and you're saying now there's maybe some apathy?"
},
{
"end_time": 6162.483,
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"text": " I don't want to put it that negatively. I think people now know a little bit better how to take care of themselves. In my time, it was a really typical thing that people would commit suicide because they were not successful, because they would overwork themselves. I think people are now much more aware of those things."
},
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"end_time": 6191.8,
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"start_time": 6163.404,
"text": " So personally speaking, how do you stay strong during the bad times? Well, I guess I was lucky not to be in too dark times, but"
},
{
"end_time": 6221.374,
"index": 224,
"start_time": 6193.319,
"text": " I do get stressed a lot. Physical exercise is one way to deal with it. I exercise a lot. There are lots of ways to take care of yourself, which are important. Breath work, meditation, yoga, gym."
},
{
"end_time": 6250.23,
"index": 225,
"start_time": 6222.978,
"text": " be in nature, bicycle, skydiving, see other people travel, see the world, spend time with the native people, just all kinds of things. Well, speaking of spending time,"
},
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"end_time": 6277.346,
"index": 226,
"start_time": 6251.135,
"text": " Thank you for spending so much time with me. You're welcome. It was a pleasure. It's three hours at this point, over three hours now. I appreciate it. Thank you. Thank you."
},
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"text": " Some of the top perks are that every week you get brand new episodes ahead of time. You also get bonus written content exclusively for our members. That's C-U-R-T-J-A-I-M-U-N-G-A-L dot org. You can also just search my name and the word sub stack on Google. Since I started that sub stack,"
},
{
"end_time": 6312.039,
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"text": " it's somehow already became number two in the science category now sub stack for those who are unfamiliar is like a newsletter one that's beautifully formatted there's zero spam"
},
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"end_time": 6340.316,
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"text": " This is the best place to follow the content of this channel that isn't anywhere else. It's not on YouTube. It's not on Patreon. It's exclusive to the Substack. It's free. There are ways for you to support me on Substack if you want and you'll get special bonuses if you do. Several people ask me like, hey Kurt, you've spoken to so many people in the field of theoretical physics, of philosophy, of consciousness. What are your thoughts, man? Well,"
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"text": " While I remain impartial in interviews, this substack is a way to peer into my present deliberations on these topics. And it's the perfect way to support me directly. KurtJaymungle.org or search KurtJaymungle substack on Google. Oh, and I've received several messages, emails and comments from professors and researchers saying that they recommend theories of everything to their students. That's fantastic."
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"text": " If you're a professor or a lecturer or what have you and there's a particular standout episode that students can benefit from, or your friends, please do share. And of course, a huge thank you to our advertising sponsor, The Economist. Visit economist.com slash totoe to get a massive discount on their annual subscription. I subscribe to The Economist and you'll love it as well."
},
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"text": " Toe is actually the only podcast that they currently partner with. So it's a huge honor for me. And for you, you're getting an exclusive discount. That's economist.com slash toe. And finally, you should know this podcast is on iTunes. It's on Spotify. It's on all the audio platforms. All you have to do is type in theories of everything and you'll find it."
},
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"text": " I know my last name is complicated, so maybe you don't want to type in Jymungle, but you can type in theories of everything and you'll find it. Personally, I gain from rewatching lectures and podcasts. I also read in the comment that toe listeners also gain from replaying. So how about instead you re-listen on one of those platforms like iTunes, Spotify, Google podcasts, whatever podcast catcher you use. I'm there with you. Thank you for listening."
}
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}No transcript available.