So, I grew up at a very little town in Ukraine, but in 1976 I moved to Moscow.

I was very lucky to escape the entrance examination, so I got to Moscow State University. And I grew up mathematically there on Gelfand Seminar. And at that time it was an absolutely great place to learn mathematics and many

other things. So it was lots of mathematics, lots of culture, and there were many other people around. And so I learned a lot on Gelfand Seminar, of course, but also from Joseph Bernstein, Simon Gindikin, Yuri Manin, and a little later, Alexander Bellinson.

Then in 82 I graduated from Moscow State University. And I should admit that it was absolutely a great place to learn mathematics and study mathematics, but for young mathematicians it was a little bit difficult to escape the

system, the official system. And so somebody had to help you. And so in my case, I got great help from Gelfand and Gindikins, who, with the help of Gelfand, I moved to the Institute of Crystallography where I became the greatest student in 1982. And in the very beginning, like January 1985, things got better in the country.

So Gelfand organized his new laboratory, mathematical laboratory. And so he took me there in the very beginning of January as a first member. And so this was great. But actually one accidental thing happened.

So on my way out of Institute of Crystallography, where I was a graduate student officially, I was a director, he said, okay, you're leaving, but can you at least do a little bit? Can you maybe write a little report for my article in Encyclopedia on some things related to electron microscopy and reconstruction of biological objects?

That was the subject his laboratory was working on. So I said, okay. He gave me some papers to look at them, didn't understand anything, but understood what the problem was. So the problem was that in biology, you have some big molecule like ribosome and you cannot,

you can make only one picture of this object because it's destroyed. And so, but what they do, they put many, many objects, put them randomly and take one picture, but then you have to reconstruct to do tomography by reconstructing this object, having many projections from different angles, but you have no idea what the angles are.

So I need to write something. And so I wrote, I realized that you can actually reconstruct the angles between these projections, unknown projections, if you know the projections. And so I wrote this to, this is little note, but then somewhere like middle of January,

I got a call, first and last call ever, I got from the director. He said, where did he get this? And I said, sorry, I didn't understand the papers, I just wrote my, my thoughts about this. And he said, so you found this on your own? I said, yes. He said, oh, don't you understand that's important? Write a paper.

And so after this, so for about, so I wrote a paper, we wrote a paper with Einstein, and for about two years, I was heavily involved in this. I wrote a number of papers on the subject, but this ended up in a very funny way. So later, like in 88, I was approached by some guy,

applied mathematician, who said, you know, I'm working for military, and we know that you are interested in the reconstruction objects from random projections. We're also interested in this, we have lots of benefits, so why don't you work with us? And then I hear that my voice saying that, you know, I no longer work on the subject.

And so this was the end of the discussion. And later I realized that if one works with military, at least in Soviet Union, so you lose your freedom to do mathematics, and this was the last thing I wanted to do. So luckily I skipped, and I never had any problems after that with my freedom in mathematics.

In general mathematics, I prefer to work on crossroads where different subjects, mathematical subjects meet, so that I have more freedom where to move and how to move.

And so in more concrete terms from like mid of 80s, I was very interested in the problem of understanding of properties of integrals of algebraic, or algebraic-geometric nature, by using some methods of arithmetic algebraic geometry.

And so this allows to make conclusions about integrals without calculating them, by using, as I said, some conjectures in arithmetic algebraic geometry, due to Bellinson mostly,

which are not available, but have a huge predictable power. And so I just want to explain a little bit what I'm talking about, because it sounds very general. So first of all, this problem of understanding of integrals motivated algebraic geometry always, from the time of Euler.

And in 19th century, the attempt to understand integrals, a billion integrals, led to creation of a theory of curves on the Jacobians. In 20th century, it was the Hodge theory, and mixed Hodge theory, Hodge, Griffiths, Deling.

But then things changed somewhere in around 1992, when Bellinson came up with his conjectures on mixed motifs and their properties, relating them to special values of all functions, the extensions of them. And so what I was doing after was trying to understand what implications this has for this problem of calculating of integrals.

So here's an example. So if you consider some rational number q, then if you're interested in numbers like logarithm of qi, and some of them with some integral coefficients, okay, this is just logarithm of the product of qi.

So that's a rational number. And so basically, the point is that this number is defined modulo 2 pi i z. And so if you're interested in this number's modulo 2 pi i z, that all you need to know is this number.

It's very simple. One way it's obvious, other way it's a statement from transcendental theory. But what if you take more complicated functions, the simplest of them is the dialog algorithm, which you can write as a series generalizing as a logarithm,

which are convergent when absent value of z less than 1. And you wanted to study the question, so you wanted to understand, again, some of the values of the dialog algorithms at rational numbers. And you wanted to know this modulo something because dialog can be analytically extended,

but has monodromy, so it has to be modulo 2 pi logarithm of some non-zero rational number. So, okay, so you wanted to know this number and all about this number, for example, when it is zero.

And so the claim you should make is that this is zero, modulo 2 pi log q, if and only if the following algebraic statement holds, that if you consider sum of this ni, it's an integer, times 1 minus qi, tensor qi, and this is zero in the abelian group q star, tensor q star over z.

And so this is a free group with a basis p tensor q up to little 2 torsion with n q primes. So it's very easy to handle this question, this is just, you can handle this for any collection of numbers immediately.

But this question looks difficult and transcendental, and this theory of mixed motives, arithmetic theory of mixed motives, imply that actually the question was, when it's zero, modulo this little freedom, is equivalent to this algebraic statement. So that's what I mean by making statements about integrals using arithmetic algebraic geometry.

And I call this arithmetic analysis because you make statements about analytic, make statements of analytic nature, but you use arithmetic basically. So, and in general, you can describe this as a starting of mixed state,

sorry, mixed motives and their periods, and more generally, motivic Hopf algebra. And the general idea about this is that whenever you have any integral of algebraic geometric nature, it produces some element in the certain Hopf algebra. This is this motivic Hopf algebra, and this kind of motivic avatar of this integral.

And the main benefit which you get after getting to this more sophisticated level is that now you live in a Hopf algebra, so you can apply the coproduct for this element, and then this gets simpler. And so what's written here is just the first instance of this

kind of line of thought where you apply, what's written here is a coproduct of the motivic element which corresponds to this element. And it's much simpler, it's actually just some algebraic expression, and it keeps all the information about the number. And so that's the point. So I first arrived in Chayes in June of 1990,

so almost 30 years ago, 29. And it's very clear why, because the borders of the Soviet Union just opened up. So this is the first chance I had. And ACHES is the place where all kind of

new ideas in arithmetic algebraic geometry were coming to us, to Moscow, like 70s, 80s, 60s. And so I obviously wanted to see the place and Paris. So that's why I came when I had the first

chance. And so why I came into ACHES all this, you know, years after? So the main magnet for me in ACHES is Maxim Koncevich, whom I know very well from 1980. And a discussion with Maxim was and is one of the main sources of joy in my mathematical life, so I wanted to keep

them going. But this is not the only reason, because there are other people, first of all, you meet new people when you come and you don't know whom. And secondly, there are people who work here permanently. And it's also very interesting to discuss with them. So I just

want to give you one example, how this worked out. So in something like 1996, I came to ACHES and I met Dirk Reimer, who was working at that time in ACHES. And he was telling me about this amazing computations he and David Brodhurst were doing, calculating multi-loop contributions

to Feynman integrals, this kind of fine dimensional Feynman integrals. And they discovered that they get multiple zeta numbers and multiple Euler sums, some little generalization of them. And this was very exciting. And on the other hand, I worked on the subject. So the main idea was whenever you see integrals, they are integrals, you want to put them to the

motivic framework. And so I was saying that one needs to, even in general, so if you have any Feynman integral, one should take the corresponding correlation functions and make motivic correlation functions, meaning putting them, putting their motivic avatars, not them,

into some huge Hopf algebra, motivic Hopf algebra. And then you have a great benefit because now you can apply the coproduct. So motivic Hopf algebra is, very roughly speaking, is an algebra of functions on a group. And so you can use a group law. But this is actually

very vague analogy because this motivic Hopf algebra is a Hopf algebra in the category of Grothendieck's pure motif. So it's not exactly leaving the vector space. But so I was saying that, okay, so it's not clear, it wasn't clear to me what the general question is about calculation of this correlation functions. But if you put them to this motivic Hopf algebra,

we can start asking different questions, which we didn't see before, like, what is the coproduct of those motivic correlation functions? Do all these motivic correlation functions, are they closed on the coproduct? If they are, what kind of quotients of motivic Hopf algebra you get? And so that's maybe one of the ways to try to

handle the general problem. And so this all was kind of reaction on talking with Dirk. And later on, all my interaction, most of my interactions with physics community was somehow inspired by this discussion. So it continued over many, many years. But this was,

these discussions in 1996 were very crucial, first of all, to formulating this kind of approach. And secondly, to getting contacts with physicists after that. So, and many other contacts with other people, of course. First of all, it's a great honor to me to

be first holder of Gretchen Barry Mazer chair. Regarding the question, what would it bring to me? What I think about this for me is the main benefit is, of course, it's the possibility to

come to HES, but the main thing is that I have the ability to give recorded lectures. And so I already give, finished last week, the lecture series on quantum geometry of modeling spaces and representation theory, which was about my joint work with Volodya Fok and Lin Kui Shen. And I hope to be another lecture series in future on quantum field

theory. So this is the main benefit. Surely, so there are points of contact. So I was lucky to be in Boston somewhere like 94, 93, 94, 95. And that time a little later, I had a lot of

discussions with Barry about the following things. So I was studying the action of this motivic Galois group on the motivic fundamental group of C star, punch rate, and P torsion points.

And I bumped into some strange connection with a modeler curve at that time, modeler curves. And so we were discussing with Barry, so why these things happen. So basically what happens is that the simplest subquotient of the image, non-abelian subquotient of the image of this motivic algebra turns out to coincide with the chain complex of the modeler curve

of level P. And so Y1 of P. And so Y, it was a big mystery, it is a big mystery. And so given lectures two weeks ago at the conference, I just wanted to give Barry an update on what happened after this. So this relation with modeler manifolds of high rank GL3, GL4, and the

role of major symbols, major model of symbols plays there. The generalization of classical model of symbols to this group GLN. Actually, so as I said, I grew up in a very small town,

there was basically nothing to do there. And so I was sitting at home and reading books available. And originally, at Soviet Union at that time, there was a number of books on interesting subjects like astronomy, or even nuclear physics. And so I remember buying one of them, it was like 1969, by Muhin, it was called Entertaining Nuclear Physics. And I read it through

many times, I read it like three musketeers. And the point is that it was a very serious book, but also very entertaining. So it explains the subject seriously and without, you know, glossing things over. But on the other hand, it can be read by a kid. And so,

but then I realized that I actually prefer to do mathematics, start solving problems. And so I shifted to mathematics. But this kind of access to books, which are serious, but still available, I think this was crucial. So another example was,

it was already in mathematics, I remember reading a paper in a magazine, Quant, when I was in high school, written by Gindiki on the golden serum, it's about proving of Gauss, Gauss reciprocity law. Again, it was, it was a proof, but it was written in a way, a high school, you know, student can understand. So this was a very good interested in math.

I get excited when I see a mystery in mathematics. And so to give a kind of concrete example, how this motivates, the study just wants to give an example of this mystery.

So we were talking about this motivic symmetries, motivic Galois group, motivic Hop algebra, it's kind of idea of motivic symmetries. But unlike Galois symmetries, they do not come in a direct way. So they come through Grothendieck's kind of dream of

motifs, and then his idea of having Tanachian formalism, which brings you, so if you have this category of mixed motifs, you get back the group of symmetry, but you don't see directly, first of all, it's not a primary object, it's kind of secondary after you see the category itself. And in Galois theory, it's just the opposite. The Galois group acts on everything,

then you get the whole theory. And so I was motivated and actually studying the idea that actually the ideas and somehow the paradigm of quantum field theory should actually play a very

considerable role in our understanding of how this symmetries come out in a more natural way. And so in particular, so example is that if you consider a real mixed structure, for example, and simplest object is on the fundamental group of let's say of a curve. So you get lots of

numbers, periods, but you can organize them as infinite connection of numbers, you can organize them into correlation functions of just one Feynman integral. And then you can understand that actually these correlation functions, they provide you explicitly the action of the real

Hodge-Galois group, the Tanachian-Galois group of the category of mixed, real mixed Hodge structures explicitly. And so now this kind of Feynman integral allows you to produce this action of the Hodge symmetries explicitly without constructing first the mixed Hodge structure.

It's the other way, you get mixed Hodge structure from this construction. And so, as I said originally, I was interested in applications of the arithmetic algebraic geometry to analysis or to physics, but then the things turned out in the other way. And so it seems that it's actually physical ideas should play important role in understanding

where does this symmetries and structures coming from, because many of them are still conjectural and we have no clue, for example, Ballyson's conjectures, which underline all this line of thought about mixed motives. So is there relation to special values of L-functions?

We have absolutely no idea why this should be true and why this should happen, but they're extremely important. So that's an example.