 So, I grew up at a very little town on 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. 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 1982 I graduated from Moscow State University. And I should admit that it was absolutely a great place to learn mathematics and study mathematics. But it was, 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 Gindikin, with the help of Gelfand I moved to the Institute of Crystallography, where I became a graded 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 graded 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 I understood what the problem was. So the problem was that in biology you have some big molecule like ribosome. And you can make only one picture of this object because it's destroyed. 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 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 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 Wenstein. And for about two years, I was heavily involved in this. Wrote a number of papers on this object, 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 reconstruction of 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 this 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, 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 theory of curves from the Ajakobians. In the 20th century, it was the Hoch theory and mixed Hoch theory, Hoch, Griffiths, Deline. Since changing somewhere in around 1992, when Bellinson came up with his conjectures on mixed motives and the properties relating them to special values of 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 Q i and some of them with some integral coefficients, OK, this is just a logarithm of the product of Q i. So that's a rational number. And so basically, the point is that these numbers defined model 2 pi i z. And so if you're interested in this number's model of 2 pi i z, then 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 a dialogarithm, which you can write as a series generalizing the logarithm, which are convergent when epsilon value of z less than 1. And you wanted to start as a question. So you wanted to understand, again, some of the values of the dialogarithms at rational numbers. And you wanted to know this modula, something because dialogarithm can be analytically extended, but has monodramies. It has to be modula 2 pi logarithm of some nonzero rational number. So you wanted to know this number and all about this number, for example, when it is 0. And so the claimation makes that this is 0, modula 2 pi log q, if and only if the following algebraic statement holds. But if you consider the sum of this n i, it's an integer times 1 minus q i tensor q i. And this is 0 in the billion group q star tensor q star over z. And so this is a free group with the basis p tensor q up to little to torsion with pin q r 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 whether when it's zero modula, 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 on a little nature, but you use arithmetic basically. So and in general, you can describe this as a starting of mixed motive and period, and more generally motivic HOF algebra. And the general idea about this is that whenever you have any integral of algebraic geometric nature it produces some element in a certain HOF 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 things get 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 the 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 CHS in June of 1990, so almost 30 years ago, 29, and very clear why, because the borders of the Soviet Union just opened up, so this is the first chance I had, and CHS 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 in Paris, so that's why I came when I had the first chance. And so why I came into CHS all these years after? So the main magnet for me in CHS is Maxim Konsevich, whom I know very well from 1980, and a discussion with Maxim Woznan 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 CHS and I met Dier Kreimer, who was working at that time in the CHS, and he was telling me about these amazing computations that he and David Broadhurst were doing, calculating multi-loop contributions to Feynman integrals, these kind of find-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 this subject, so the main idea was whenever you see integral, and 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 their motivic avatars, not them, into some huge HOF algebra, motivic HOF algebra, and then you have a great benefit because now you can apply the co-product, so motivic HOF 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 a very vague analogy because this motivic HOF algebra is a HOF algebra in the category of Grotten-Expure Motives, so it's not exactly leaving a vector space, but so I was saying that, okay, so 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 HOF algebra, we can start asking different questions, which we didn't see before, like what is the co-product of those motivic correlation functions, do all these motivic correlation functions as they're close to the co-product, if they are what kind of quotients of motivic HOF 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 these discussions, so it continued over many, many years, but 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. 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 Berry-Mazer Chair. Regarding the question, what it brings to me, what I think about this, for me the main benefit is, of course it's a possibility to come to HHS, but the main thing is that I have ability to give recorded lectures, and so I already gave, finished last week the lecture series on quantum geometry of modular spaces and representation theory, which was about my joint work with Volodya Falk and Link Vishen, 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 1993-1994-1995, and that time a little later had a lot of discussions with Berry about the following things. So, I was studying the action of this Motivic-Gullo group on the Motivic-Fundamental group of C-Star, Pancheret and Pete Orson points, and I bumped into some strange connection with the modular curve at that time, modular curves, and so we were discussing with Berry why these things happen. So, basically what happens is that the simplest sub-quotient of the image, non-Abelian sub-quotient of the image of this Motivically algebra comes out to coincide with the chain complex of the modular 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 a Berry update on what happened after this, so this relation with modular manifolds of high-rank GL3, GL4, and the role of major symbols, major modular symbols play there, the generalization of classical modular symbols to this group GLM. 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 the time there was a number of books on interesting subjects like astronomy or the 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 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 shift 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 magazine Quant, and when I was in high school, written by Gendikin on the Golden Serum, it's about proving of Gauss-Rissuprosti law. Again, 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 interest in maths. 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 Gaullo group, motivic hop algebra. It's kind of idea of motivic symmetries, but unlike Gaullo symmetries, they do not come in a direct way. So they come through Grotendig's kind of dream of motives and then his idea of having Tonakin formalism, which brings you, so if you have this category of mixed motives, you get back the group of symmetry, but you don't see it directly. First of all, it's not a primary object, it's kind of secondary after you see the category itself, and in Gaullo's theory, just the opposite. The Gaullo group acts on everything, then you get the whole theory. And so I was motivated in 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 these symmetries come out in a more natural way. And so in particular, so our example is that if you consider a real mixed-hot structure, for example, the 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 phyman integral, and then you can understand that actually these correlation functions, they provide you explicitly the action of the real-hot Gaullo group, the Tonakin-Gaullo group of the category of mixed real-mixed-hot structures explicitly. And so now this kind of phyman integral allows you to produce the section of the hot symmetries explicitly without constructing first the mixed-hot structure. It's the other way. You get mixed-hot structure from this construction. And so as I said, so originally I was interested in applications of the charismatic algebraic geometry to analysis or to physics, but then the things turned out in the other way. So it seems that it's actually physical ideas should play important role in understanding what are these symmetries and structures coming from, because many of them are still conjectural and we have no clue, for example, Balanson's conjectures, which underline all this line of thought about mixed motives. So it's a relation to special values of health 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.