 I am very happy that Ambassador of India, Madam Reena Sandu, she kindly agreed to attend this ceremony and as you, I am sure all of you have heard the name of Ramanujan, you probably seen his movie, The Man Who Knew Infinity. He was quite, he has made such far-seeing contributions to mathematics. I mean many of the works that he did 100 years ago, things like how to partition, the number of ways to partition an integer was one of his very favorite problems as well as things like what is known as the hardy Ramanujan expansion and also he introduced some very interesting new functions called the mock theta functions and quite surprisingly 100 years later these ideas are finding their way in remote areas of physics like quantum field theory, in string theory, even in my own work in quantum black holes suddenly you encounter Ramanujan's work. So that is sort of kind of attest to the power and depth of his ideas that 100 years later people find them useful in very different areas. So the Ramanujan Prize was basically created to honor, so therefore Ramanujan is one of the, you would say the greatest, one of the greatest mathematical geniuses to come from the developing world and the Ramanujan Prize was instituted to recognize the contributions of young mathematicians from developing countries and the prize is usually given to a mathematician under 45 and it is in collaboration with the International Mathematical Union, ICTP and the Department of Science and Technology from India and these three organizations have their members on the selection committee and each year they select the brilliant or promising mathematician from the developing world and this year we are very happy that Dr. Pham has been the recipient of this, is going to be the recipient of this year's medal. So with these opening remarks, I will request Mother Sandhu to say a few words. You should be able to. Thank you. Good afternoon everyone, it's a great pleasure to be in Trieste today, it's a beautiful sunny day, my first visit to Trieste and first time at the ICTP, it's a wonderful occasion as well, the Ramanujan Prize, I think Dr. Dabholkar has already explained the relevance and the importance of this award and that it is in the memory of one of the greatest mathematical geniuses of our times. I want to congratulate the winner of the award this year, that is Dr. Huang Heap Pham from Vietnam, from the Institute of Mathematics at the Academy of Sciences and Technology of Vietnam in Hanoi. So it's a great pleasure to be here when he's receiving the award, he's very young and I hope that his work and his contribution will inspire other young mathematicians to also follow in the steps of bringing their knowledge and their genius to all of us and which benefits in fact all humanity. So congratulations very much on winning the award. Dr. Dabholkar has shared some very interesting facts and figures about India's collaboration with the ICTP over the years and I'm very happy that ICTP which has been contributing so much to the growth of scientific expertise in developing countries that India has also been able to participate, both contribute as well as be a recipient itself of the knowledge base and the facilities and the capacity building that is provided by this important institute. According to the figure he has shared, India is at the top of the list of countries for visiting scientists to ICTP in 2018. Last year 328 Indians participated in the ICTP programs. Since 1970 more than 10,000 visitors from India have been at the ICTP and in the last decade 24% of the Indian women have been women. Indian visitors have been women and since 1983, 16 of the prestigious ICTP prize winners have been from India. So it's you know it's a kind of has been a very I should say a nurturing and a very helpful collaboration where scientists from India have benefited greatly from this collaboration. I understand that many of them have gone on to receive many more awards and hold important positions in academic and scientific institutions. I have actually received a message from the Secretary Department of Science and Technology for Dr. Pham which they've asked me to read here a message of congratulations from India. So I shall read it for you. I congratulate Professor Hoang Khe Pham Institute of Mathematics Vietnam Academy of Science and Technology for being awarded the prestigious Ramanujan prize for the year 2019. The award is a recognition of your work and contribution in the field of complex analysis and in particular the pluripotential theory which has been important which have important applications in algebraic and complex color geometry. The award also recognizes your important organizational role in advancement of mathematics in your country. It is also a great honor that the award has been instituted in the name of S Ramanujan, the great Indian mathematician. I'm sure that your contribution will contribute to enlarge the legacy of the great Ramanujan. I'm also sure that this recognition will motivate you further to expand your research with more noteworthy outcomes in the future. It will also inspire researchers and young mathematicians not only in your country but in the entire developing world to undertake research in mathematical sciences. I wish you all the best for your future endeavors from Mr. Ashutosh Sharma, Secretary Department of Science and Technology. Thank you very much for having me here and wish you all the best. Thank you. Okay, so now, yeah, please. So now I request Professor Brach from, I'm sorry. So I am Carlos Henig. I'm the president of the IME. It's good to be here. And it's my pleasure to congratulate Juan Herzpan from the Institute of Mathematics of the Vietnamese Academy of Sciences for his being awarded the Ramanujan prize for his impressive work in complex geometry. The Ramanujan prize is awarded by DST, ICTP, and the IMU. Thank you very much. Yeah, sorry. That was a message from the International Mathematical Union Representative. Now I request Professor Filippo Bracci. Actually, unfortunately, these terminals here are not working. So it's probably better we sit there. Just today, just five minutes ago, they stopped working. So apparently. Okay. So he will talk about pluripotential theory. So first of all, I would like to thank very much ICTP. I'm very honored to be here to give this special lecture on pluripotential theory. I also congratulate with Dr. Fan for forgetting this prize. And I, well, I have to say as a mathematician that the name of Ramanujan is one of those names which reminds of magic and beautiful in mathematics and also show, I mean, mathematician, when they start working, they get accustomed to name of all over the world. And we, I mean, we don't have really frontiers or barrier in mathematics. We try to cooperate all together just for the sake of knowledge and to go in advance on difficult problems. So I'm very happy also for being here, for the Ramanujan prize. I myself have collaborators from India, from the Indian Institute of Technology of Science was, in fact, one was visiting me a couple of months ago. So I'm very happy for this. Okay. Let's come to the talk. First of all, I have to apologize because probably for mathematician, the talk would be too easy for no mathematician, but probably that would be too difficult or even boring. So I'm sorry, but I try to do my best to calibrate. So a potential theory. I want to give at least to those who are no mathematician, not mathematician, an idea of why we use this strange word or why people are considering this object. The most simple things that you can have is just a line. And the function like this is linear. So linear is something very easy. You would like that everything would be linear, but that's not in reality. And then, well, you say, okay, linear is not enough everywhere, essentially. And you can have in physical phenomena, in reality, you have a phenomena which are very far from being linear. In fact, most of the nowadays phenomena are even fractals. So it's something very, very complicated. But that's too much. That's too complicated. We don't have the appropriate technology to understand. So you can say, well, instead of linear, you might consider something which is some kind of sublinear. So it's what the mathematician called a convex function. What's the main property of a convex function? Well, if you take any two points in this figure and you consider the line joining these two points, this line here is always above this one. And that's essentially the main philosophy which is underneath the pleurisobarmonic function and pluripotential theory. So it's something which is complicated enough to include or to be good to determine or to study many phenomena that come from physics or from natural science. But still, they are not too complicated. So we can somehow work on that. Yeah. So I gave some definitions, mathematical definitions. So I start from the very basic object is a upper semi-continuous function is a function from a certain domain in Rn on the real line such that all sublevel sets are open. So you probably know, I mean, just by heart that when you talk about continuous function means that you can draw a picture without taking out your hand from the blackboard. And here upper semi-continuous means that, okay, it's not continuous but almost. It tend. It would like to pretend to be continuous although it is not. And the main one I think the very beautiful result and one of the, probably the guiding result of all the theory is this that we call maximum principle. This means that if you have this upper semi-continuous function, so that's just the technical matter. And suppose that you have this function defined on some domain which satisfies this inequality. So this inequality here means essentially that if you take a ball and you consider this function on the ball. So here you can imagine a graph above this ball here and you measure the area or the volume of this object. Then it is always greater than the value in the center when you divide by the volume of the ball itself. And if this happened, then the function cannot have maximum inside. You can think more or less about this case here. You can have this picture and this picture essentially contain all my talk. So if you are in this situation you see that the maximum is rich here. It cannot be rich here. Okay. Yeah, then people define this special class of function to be subharmonic. So subharmonic are function that aside this kind of strange regularity, they have this sub mean property. And if you have a function which is both subharmonic and minus the function is also subharmonic, then this is called harmonic. You can think of here. So you have this function here and if you take the negative of this function, then you get something which is concave, so in the other side. But if you have a function which is convex and when you take minus is still stay convex, then you see that the only possibility is to be the line. So somehow harmonic function are better subharmonic function. In fact, in mathematical term, they are analytic, which means that they have a lot of regularity. They can expand in series, Taylor series. And well, there is this symbol. As a mathematician, I am forced to use symbol, which is called the Laplacian. So the Laplacian is just this object, which means that you take the function. Well, the function we assume for the moment. As I said, if it's harmonic, you can prove that it's analytic. So in particular, you take the second derivative with respect to all the variable. You take the sum and you call this operator is the Laplace operator. And then the function is harmonic, if and only if you satisfy this equation. Now, subharmonic function and harmonic function, they are very, very important in applications. For instance, electric and gravitational potentials, diffusion equation for heat and fluid dynamics and flows, wave propagation, quantum mechanics, they are all based on equation, which are based essentially on this operator here, the Laplacian. So if you want to apply any kind of mathematics, most of the time you will find yourself dealing with this guy. Okay, and this is what usually mathematicians call potential theory. Yeah, now just to give an idea of this, so a function u is subharmonic. And then it has the, well, this is another submin property that instead of taking, dealing with volume, you deal with the surface of the ball. But that's a technical part. But the most interesting is exactly that it acts like the linear function. So if you have a subharmonic function, if you, and if you take any other function is subharmonic, whenever if you take any harmonic function, which has the same value or a bigger value on the boundary, then the subharmonic functions stay behind. So it's like linear and convex. That's exactly the same, except that here is a little bit more complicated. So there is this idea of subharmonic, the word sub, which means they under harmonic function and satisfy this maximum principle. Okay, this probably is too technical. You can prove that if you have a subharmonic function, this is locally integrable. And they satisfy the same equation as before, but this time with greater or equal than zero. And conversely, you can prove that that's equation is essentially enough. Okay, I will skip probably these two technical parts. Yeah, and then here is the same as we said before, that if you have something linear, it means that they stay above any other linear function. And here is exactly the same. If you have a subharmonic function, then it is harmonic, if and only if it stay above any other subharmonic function, which is bigger on the boundary. And then this is what I was saying that harmonic function are maximal subharmonic function. Okay, that's the way I mean, probably you heard this name directly problem, which is another kind of differential equation that one tried to solve to construct certain objects. Okay, now I will be a little bit more technical just to understand from a mathematical point of view how to move from the real world, because what I said now is essentially real to the complex world. You know that in here I wrote in CN, but just consider the plane or the real number and the complex number. So you know that complex number can be multiplied by i. If you consider a complex number as a sum of a real plus i imaginary part, so a couple of number, that means that you stay in a plane. And then the multiplication by i becomes what we call a linear operator that here I define by j, it's j here, which is just a multiplication. So you multiply the original complex number by i. Now you look in terms of real variable and that's exactly what you would do if you consider these complex numbers as a couple of real numbers and multiply by this operator here. Okay, yeah and then I mean in complex analysis the main object to study are those so-called holomorphic functions. So this term comes from Latin and means same form essentially, which are because holomorphic functions at least those in dimension one are those who preserve from an infinitesimal point of view the form. So if you have like a circle it cannot be the form, it's a stay continue to be a circle. And here you can use this operator j to say analytically what means that the function is holomorphic. Okay, now I need to introduce these two operators which I call d and d bar and they are defined formally in this way. So essentially this d u means that you take the differential of the function or a known two technical term, you try to approximate your function by using if you have like this curve here at this point you try to approximate with something linear and that's essentially the meaning of this d u in the formula. And now you take this operator j because this I mean it's a real object and you multiply by i and so you do the same here but you put plus instead of minus. Okay that's of course there is a reason why one does this because that gives provides a so called C linear and C anti-linear decomposition of the differential of u which means well C linear means that if you multiply first by a complex number then this complex number can gets out from the differential and C anti-linear is the same for let's say conjugation. Okay and then you can define also this operator d c which is here is the analytical formula. And again you can play a little bit with these terms and you see that the function if you write in terms of real and complex part real imaginary part this comes holomorphic if and only if they satisfy this relation. So for the moment it's just plain with some strange objects and now here comes an important part which is the complexation of a matrix which means essentially that you take instead of considering just the real differentiation you differentiate with respect to z and z bar and this is called the complexation and now the operator the delplacian that we saw before this becomes four times the trace of this complex matrix but that's an Hermitian matrix something more. Okay another object is the so called levy form and the levy form is technically is zero to tensor and this is defined in this way so you take the complexation of a function and then you define this quadratic form in this way. So what you see now is that if you restrict the function to a line a complex line and you compute the Laplacian you get the levy form applied to the vector of this direction and this suggests the the following definition that you say that the function is pluriarmonic if whenever you restrict to a slice the function is harmonic so harmonic was the exactly the same definition as before so now we are just try to move a wave hence we are in the space and so we can look at all the figure or we can slice up and see what happens on the slice and that's what we do in this context and then by the previous formula you see that the function is pluriarmonic if and only if the levy form is identically vanishing. Okay so in particular well the pluriarmonic function are analytic and the the levy form vanishing identically and they are also harmonic by the same reason because if the levy form is identically zero then the trace is identically zero but the conference is not true and in fact one can characterize the pluriarmonic function by saying that it's a function which is locally the real part or the imaginary part of a lumbar function so even if you didn't get anything about what I said the main point here is that the potential theory the Laplacian and so on is good to work with the real problem but when you are in the complex word and many even at least theoretical problems are modeling using the complex language the complex function so there the Laplacian is not good so we need something else and that's something else is exactly this definition of pluri so when you heard about pluripotential pluriarmonic plurisobarmonic means that it's about the same as here except that you are in the complex word that's the the philosophy yeah because the point is that the the trace of the asian is not invariant if you change coordinate holomorphically it's not invariant it changes so if you have a function which is harmonic and you change coordinate then you take the function it's no longer harmonic in that coordinates but if you have an harmonic function which stay harmonic and the holomorphic change of coordinate then it's pluriarmonic so it's something which is really well related to the complex word okay and now I mean since we know that we have the notion of harmonic subharmonic we have the notion of pluriarmonic and we can have the notion of pluri subharmonic so a ps age or plurisobarmonic function is a function which when you restrict to these lies as before now is no longer harmonic but it's subharmonic or minus infinity but that's a technical detail okay and then if the function is regular enough you can use the the same operator as before the levy form and you see that a condition and a necessary and sufficient condition for a function to be pluri plurisobarmonic is this one that the levy form is positive semi definite okay okay this is a technical definition and again I mean plurisobarmonic function are a class of subharmonic function but they are not the same because of the same reason and here is essentially the same property subharmonic function which is plurisobarmon is plurisobarmonic if it is so under any change of holomorphic coordinates and this which might be important for for Dr. Fahm next talk is that the this allowed to define plurisobarmonic function on complex manifold so complex manifold you can when you when you heard the word manifold means that you have the ambient space which is our space or complex fine space and you patch together part of this in a certain way so you construct objects which cannot be in general embedded in our world but they still exist and then you want to understand what are the objects living on there and then I mean you have to be able to define them and because of this observation you can really do okay then you have that in some manifold for instance if the manifold is compact there are no global plurisobarmonic function because of the maximum principle is where compact means that you have a hand somewhere so in this picture if you try to go at infinity but then you know that you can glue then you lose convexity somewhere and that happens more or less here and okay that's another technical point that somehow even if it's not too regular you can manage to have a regular plurisobarmonic function and okay here is another characterization using the so-called currents but probably I'm asking too much to understand this but it's this ddcu is a one-one form if u is regular which is constructed by the previous d and dc and this is going to be a positive one-one form and that's very important because this part here is related to the so-called keller manifold and so on where you really use going to study this this object and curvature and so on okay yeah then we have a maximal function as we said before the harmonic function are maximal in the class of subarmonic function that's unfortunately it's not true in general for plurisobarmonic function if you have a plurisobarmonic function which is maximal is not in general pluriharmonic and and this is what the sadulla have defined as a maximal plurisobarmonic function which is a plurisobarmonic function which has the property of being above any other plurisobarmonic function for which it is above on the boundary and as I said that they are not in general pluriharmonic but what you you might see is that if you have a a proper olomorphic embedding in at each point in such a way that the restriction of your function is harmonic then this is maximal so somehow harmonicity is included in some sense and then there is this operator which is the so-called complex monjampere operator and that's became very famous after the calabic injection and yaw results which is essentially this operator here so you remember maybe you don't but did this form a one-one form a ddcu which this is you which is a if you make a computation it's just a determinant essentially of this complex asian that we defined before and that's the so-called complex monjampere operator and you see that if it is plurisobarmonic then the determinant so in particular this operator here is always greater or equal than zero and if it is maximal then this operator is zero so you might try to solve in an analytic way this problem by considering not the plassium but this complex monjampere operator okay and yeah and then as I said by really by definition if you have this complex Hermitian format you take the determinant this is zero if and only if there is a one again vector which is one again vector related to one zero again value and so in particular this means that at every point you have such condition you have a a privileged direction and in fact if you have a kind of foliation which is an object meaning that at each point you you can prescribe a certain direction in such a way that your function restricts to this foliation is harmonic then the function is maximal and the converse is also true I mean if you have a maximal plurisobarmonic function now with some regularity then there exists the monjampere foliation which means that you can foliate you can divide your manifold in leaves in such a way that the restriction of your function on these leaves are harmonic and this is called the monjampere foliation as I said as I said there was probably you heard this directly problem which means that you study the you look for an harmonic function so you study Laplacian equal to zero with some prescribed boundary data and this comes this kind of problem comes in many physical phenomena as I said here you can consider the same kind of problem but now considering the complex monjampere operator so you look essentially for a maximal plurisobarmonic function which has a prescribed boundary data and in fact you can solve this problem in well at least in reasonable case and you have the so-called Perron-Bremmerman function which means the following it's very natural you consider all plurisobarmonic function which has the property of being less when you go to the boundary than the data that you have on the boundary so here for instance just consider this case you want to you give some value this value and this value and you want to find a function possibly which is linear and attains these two value here and of course you can do it's just this straight line but now the idea is well I don't know if I can do but I know that there are a lot of these convex functions which stay here behind and then at each point I take the maximum as possible and then you see that if you do this picture here you just go to the line and here you do the same essentially but using this kind of objects so this pluripotential theory has been used Klemek introduced the so-called pluricomplex green function green function is something that if you study mathematics but even problem of physics the comes naturally and here is the pluri version of this green function so you do exactly the same game as before you consider a domain you consider all negative plurisobarmonic function but now you assume that to have a logarithmic singularity inside so you have something which now as a prescribed singularity and you see that it has some good property that I'm just going to skip briefly but mainly what is important is that you see you want to the main object of study morally would be a lomorphic function but these are too complicated and too rigid so you pass to study this plurisobarmonic function it's exactly like here you want somehow to approximate or using linear function but they are too rigid and then you you're allowed to use convex and here you do the same but now you see that this kind of function did this pluricomplex green function as good property because if you have a lomorphic map then it is almost preserved it is reduced so this gives information in fact okay and then if you have a domain bounded then you can really find such a solution such a function which is also maximal and satisfy this complex monjampere equation and then if you have some other condition like this called hyper convexity then still again you can you can prove some more regularity condition and they might in fact show that there exist and it is also a unique solution to this kind of generalized directly problem for the complex monjampere equation I will well I want just to I won't spend too much time here because I think it's too technical there is this this function here however is related also to a special metric which is called the Kobayashi distance which is very natural in it's not the Euclidean so to go from here to here probably I don't have to go straight but maybe I have to turn and go there and that's still the the shortest way so it's another distance but it's very natural and this function here is pretty much related to to this distance okay and then I just wanted to mention that you can do the same game at the boundary and you can define a pluricomplex Poisson function which I defined some 10 years ago maybe more with Giorgio Patrizzi and Stefano Trapani where now you assume that your function you consider the family of pluricobarmonic function negative which has a kind of a simple pole when you go non-tangentially to the boundary and in the unit disk this is really the Poisson kernel they probably you heard this name okay and and again it has many good property the this kind of function it has you still have a solution which is essentially unique although we don't know exactly if it is unique of a kind of generalized directly problem and again it has very special property if you look to the level set they turns out to be the so-called aurasphere and and it allows to get a reproducing formula for pluriarmonic function in fact even for pluricobarmonic function but here I just mentioned this one so essentially means that you now you have your boundary data and you want to reconstruct the entire function and you can do if you have reproducing formula and this kind of kernel this kind of solution allows to reconstruct the entire function by starting from boundary data problem okay and that's I think this was my last slice I hope I was not bored and thank you very much for your attention so I request Professor Arezzo to now give a presentation about the work of this you prefer while while we solve the technical problems first of all let me let me thanks a lot Filippo Bracci good friend of ICTP and for giving a very gentle and clear introduction to some of the most technical parts of mathematics that I I know so and we will see and also I really also on behalf of the mathematics section as a subset of the whole of ICTP but I am really very grateful to the ambassador your excellency the ambassador to be here for us that the the the name of Sri Vaza Ramanujan is ubiquitous in mathematics as we all know but actually it was not planned but just by accident we just finished a special section a set of lectures by Professor Don Zaghiere who works with us for a few months per year on the on the Rogers Ramanujan identities and I mean four beautiful lectures I mean his work is still is still with us strange enough I will mention another Ramanujan which is less famous but as a geometry I would say almost has has genius with the same level of geniality unfortunately with the similar sad story personal story anyway my role now is just to introduce you thanks Filippo for introducing the general area of research of the winner of this year I will now I introduce the person so I give you some informations about the work of Professor Pham so few bio data I hope I'm not violating any privacy but I put so he is well below the our limit age for the Ramanujan prize is very young also for among among young scientists he's very young I mean and he has studied bachelor and master in Hanoi got his PhD in in Sweden he has taken few postdoctoral positions in Europe in Italy and France before going back to to Vietnam where he's now professor at the Institute of Mathematics at the Vietnam is Academy of Science sorry for the question mark I just solved the problem since 2018 but actually I have to apologize for either he has been promoted I knew him as the deputy director my fault he's actually the the director of the UNESCO Center in Hanoi so he's a good collaborators a good collaborator of ICTP also in many other in in other directions and in fact we are working together as as both as UNESCO team okay he also got some important recognitions in Vietnam for for his work coming I mean of course I don't expect you to read this page it's just a small sample of his mathematical production I counted 38 could be more sorry even at least greater than or equal to 38 papers in international journals and these are just a sample you can see some of the most distinguished journals appearing here in the mathematical community and of course you will start seeing the word pluripotential monjampere and that's why Philippo has been kind enough to explain to us a bit what they mean because that's where he has been working on the main research teams I would say are probably related to three three areas existence and regularity of solutions of complex monjampere equations. Philippo told us a little bit what what is this problem about the more generally properties of singularities of plurisubharmonic function because the key point is that plurisubharmonic functions tend to be very singular I mean he of course Philippo could not dig too much into the technicalities but I mean the point is that they tend to go to minus infinity a bit too often than we hope and when they do we want to understand what does it mean and what do they measure the rate of singularity in this language means going to minus infinity essentially so what does it mean for something like that but I'm a geometer so in fact that's why I asked a good friend the complex analyst to explain the first two because I'm I'm in love with the third line because that's really where pluripotential theory which is I think branch of complex analysis comes to speak with geometry okay now it's a very strange history and I I hope I mean I hope I mean this this ceremony is also live streamed outside ICTP I hope I'm not offending because I cannot trace and I think nobody it's not really known where this log LCT stands for log canonical threshold this this number was born for sure in many papers more or less at the same time at the end of the 60s and the 70s somebody claims it was a Tia the first one to to to define it but it was essentially contained with small variations in paper by Le Long by Skoda by Siu and so on the part is taken a plurisubarmonic function Phi you want to know which is the least the the the the big well the least because it's a minus otherwise if you put the minus in front of the exponent is the the soup of the exponents you have to put when you exponentiate it to to make this function integrable so remember we are dealing with a problem of going to minus infinity so that's why as a log the log is our is the thing that we have in mind so we exponentiate it in order to make it treatable and then we we work like that now for any function you define this number this is called a complex integrability exponent but and at the main situation where you actually see in geometry appearing this is when actually this function is not just any function but is actually the norm squared of the section of a bundle okay let me just say it like that so it appears in geometry because when your function is not just a generic function but it's really the section of the holomorphic section of a line I mean the norm of a holomorphic section of a line bundle then you're really speaking about the log canonical threshold of the pair of the complex manifold and the the zero locus of the section which is what we call a divisor anyway these two numbers appeared essentially at the same time in a number of papers and the point I just want to make is this sorry for the technological I could not draw it in a in a more fancy way so I did it by hand but the point is these numbers as I said defined more or less 50 years ago now they are still a big mystery I mean in the sense that they have been used and they have helped to solve problems in all branches of mathematics I would say the problem they appeared I mean even the most surprising one if you if you now think this is just complex analysis for example igusa in a very famous paper of 1977 found essentially this number has the the rate of growth of solutions over z mod a prime to some power of a polynomial with integer coefficients okay so now purely number theory collar 20 years later in variants of singularities and burns and polynomials I don't know where you would put it in the geography of mathematics whether topology or again number theory or the intersection of the two more classical and actually probably what made the everything very famous was essentially these numbers appearing in the famous classification of algebraic surfaces in characteristic p we all know that codire and requests and severely classified algebraic surfaces in characteristic zero that's the famous codire and request classification but then 30 years later more or less a month for them bombieri gave the classification in characteristic p and actually again this number played a closure rule in their work based on an idea which goes back essentially to the birth of these numbers of the log canonical threshold by skovsky manning then through the works of puklikov then essentially alessio courti much more recently applied again the theory related to these numbers to the problem of studying the rationality of phano varieties that if you are an algebraic geometry you probably consider one of the central problems in algebraic geometry rationality means whether a manifold an algebraic variety is actually biomorphic to pn by rational to p to the complex project space okay so that's a kind of state to algebraic geometry like the poincare conjecture states to topology I mean you really want to know I mean is is this manifold trivial or not okay algebraic geometers call it to be rational okay then as I said the other ramanujan the other famous ramanujan cp ramanujan in the beginning of the 70s actually again use these numbers to open a line which actually was very fertile in the years to come lads as well I mean many many people worked on that to realize that these are actually good estimates on these numbers could provide good vanishing theorems which in geometry are very good vanishing means vanishing for comology I mean if every time a group is zero you you you open a bottle I mean you are happy because something is easy finally okay and then probably well again another fundamental this is actually direct application of the study of these numbers was given by Tian right after his PhD thesis basically the Tian's theorem is probably the easiest here to quote almost correctly in the sense that if you can prove that this number when you take the anti canonical device that is sufficiently big then actually that means that your space is actually Einstein that's Tian's theorem so there's an actually precise bound that if you can exceed that then your space is an Einstein space so for theoretical of positive curvature I mean that's where the problem really stays so as you can see I mean this is just if you want just psychology but as I said the the work of this year's winner is very technical and goes really to the heart to the center of this picture now the problem if you prove a great theorem at the center presumably you can imagine some kind of earthquakes in the surrounding no and in fact I would like just to to finish this my little presentation of fam's work just by quoting what is probably the most at least I mean for a geometers for sure for a geometry this is really a great theorem that he proved in collaboration with demae because you see you can really look down to the bottom and the idea here you take a you want to have an estimate on the complex integrability exponent of any plurisubharmonic function I mean here it's normalized just to be the origin the bad point and you see well this is telling us I mean what demae and fam are telling us is that this is greater or equal to this rather simple expression but the point is that not this expression is simple the point is that this expression is actually these numbers are actually defined here and now of course we don't have the time to to go into details but I mean these are comological numbers related to the function phi so basically the geometry is completely dominating the problem and it in principle computable plus the fact that I mean this is the end of probably 40 years of work by many many people I mean there were estimates of various type the point is that they also prove that this is sharp so it's as good as it gets okay there are you can construct easily examples where this is an equality and not an inequality okay so I think now you put for example this kind of theorems back in the previous page and you can imagine that this says corollary is in all in all directions and I think if I I mean there's one one way to come to to congratulate again doctors from for for the prize and to convince you that this is absolutely very very well deserved and and my congratulations okay so now actually this is the moment of the award I was I was jumping to the to your lecture but so now if you can come here and we should stand there I we do the unboxing as now it's very popular on I didn't I didn't remove the if the statue I give the this or you want to do I give this and you give that okay so this is the prize and you want some help okay so now we are very happy to leave the winners on the stage for special lecture Magistral is for on good afternoon everyone thank you for nice introduction before very interesting my talk I will say some words I have a watch a movie about the india race madam decent in the Vasa Zaman Zubai let me say I am impressed by his own life and his great contribution to mathematics therefore I'm very happy and proud to receive the prize let me thank the ICTP, DST and IMU for organizing the prize and so many times all people here and many times to all former advisor and all my collaborators and all my colleagues and finally I would like to express my deep gratitude to my parents who will support and encourage me to pursue my career now I will talk about about the security of two solid funds in corporate analysis and the energy is this talk we declare for how to combine the techniques of corporate mobile operator and deep by equation with mental and the statement to obtain some of these data in the theory of security and the data application in corporate analysis and the energy and finally we present a question and a question in this talk base first I will give a short very short introduction to corporate analysis and the energy and we know corporate analysis in one variable is the price of mathematical analysis that investigates the function of corporate numbers it was developed by many mathematicians such as Cossie, Erlen, Raoult and Midtown Nebler Zaman were shot in the 18th century it is useful in many parts of mathematics including as a play they are multi number theory under the textbook in the dynamic system of the model of physics and I will say some words about the corporate analysis of several variables corporate analysis of several variables is the start study of function fz1.zn which z1.zn is in domain in CN it was first introduced by and studied by Carton, Chlores, Hustop, Zemez, Stens, Oka and Thuran in the 19th century it is connected to other fields of mathematics including as a play they are multi function analysis, partial differential equation, commutatives, as a play, module theory and dynamical system. Now I will introduce a corporate manifold and corporate vector bundle a corporate manifold is a topological space which some label hood of every boy is a homomorphic to another subset of CN such as the transition between the three open sets as given by Horowitz Hansen and a corporate vector bundle on corporate manifold is a vector bundle Zin vector bundle whose fibers are corporate vector space and transition function as homomorphic function and we talk about the space of Horowitz Hansen and let E be a corporate vector bundle over a corporate manifold we denote HZOXE by the set of own Horowitz Hansen of E over H a fundamental problem in corporate analysis and Zia-Merci is to contract Horowitz Hansen in HZOXE suck that decide the condition here we have a remark here if it if you achieve your bundle of Zinc card then HZOXE is a space of Horowitz mass from H2CK that I will give a short introduction to Horowitz Hansen and analysis survive now there will be a corporate manifold of dimension n and a function f from omega to c is said to be a homomorphic on omega if for every boy ZZO in omega then f equal power series on the neighborhood of ZZO and would defy omega defy by own of Horowitz Hansen on omega and all ZZO denote by own Horowitz Hansen groups as a boy ZZO then as we know the Zinc all ZZO is a unique factorization and return and we are other way as soon as omega is corporate manifold of dimension n and a subsets A of omega is said a line it takes if for every boy ZZO in omega we have a intersergent U equal exactly Z in U suck that f1z equal dot dot fmz equal ZO where f1 fm is homomorphic on U and the subset A of Cn is said to be other place if there exists polynomial f1 fm suck that A equals Z in Cn suck that f1z equal dot dot fmz equal ZO a subset R of positive space Cpn is said to be other place if there exists homomorphic polynomial f1 fm suck that A equals Z1z n plus 1 suck that f1 z n plus 1 equal fm z1 z n plus 1 equal ZO by its show told up we know that every a line it takes the superiority in positive space Cpn is as a as a place now I will introduce about blue sonic function thank you very much for nice lecture also Philip Pulaski he talked very clear about this function and we test a per semicolon function phi for omega to minor phi t plus e phi t the phi on the domain domain omission it's going to to be sub pollution if phi z next z number it's average of the value phi z plus c w which would see a present i r then we defy the b ss omega be developed by the set of blue sonic function and b ss minus to be a own blue sonic function negative on omega here we have a some some example we have a first own convert function as blue sonic and all blue sonic are sub harmony of course we know that every convert function as continuous even nipsy but blue sonic is not continuous as second example we have we take f1 fm be our whole which function on omega and it takes phi z equal noga more than f1z 2 by 2 plus dot dot plus f1z 2 by 2 then we have a f5 the function phi is blue sonic and we have a the set z in omega success phi z equal minor e phi t equal z in omega success f1z equal fmz equal zero then we have this this this set is a variety in omega but in previous hotel we call this set is a blue process here we talked about the we have some collection between the idea is a ring original with blue sonic function we take idea i is a ring original and for every basic f1 fm of e i and we set phi z equal 1 divided by 2 noga f1z 2 by 2 plus fmz 2 by 2 that phi is blue sonic function on you and we take z z1 z k be another another basic of the idea as such says cj equal 1 divided by 2 noga f1z 2 by 2 plus z kz 2 by 2 we have a different blue sonic function phi and psi however phi minus psi is bounded all neighborhood of u of z o it's means that phi and psi have the same singularity at z o then we can use the function phi to to understand the identity now i will give a short introduction about the current and interest in theory the difference form has much of difference in geometry influenced by linear as a plan it was introduced by kaktang in 1899 we did did not the this is a key omega with a sense of different form of dj k i whose coefficient will not smooth function with support in omega as a currency of the degree of degree key or dimension and malice key in the sense of girl drama it's corporate continuous need the functional on-space of corporate supported differential form then in a geometric setting they can represent integration over sub-manifold generalizing the design measure as a boy many design of iteration 3 is still applied to colors here we give some notation in the previous in protest hotel is about the no negative current and the zin zin current here we have a some rematch we take the t be close and no negative current of degree 1 1 then t equal ddc phi which phi is a blue sonic function on omega and it takes a a is a close analytic surprise subrarity of poor dimension and malice p tends a eclipsed pp current of integration associated to a and is defined by this formula is entered on the zin rate zin rate of a and it's easy to say to see that the current is non-negative as top total implies the current is close but the long has sold us the current is locally 59 miles near the security point and this for all these kinds can see be stood as consequence of circular emu attention teller we talked about the way product of close non-negative current and application for further maintenance works of the web product of close non-negative current and application to iteration 2 theory and corporate analysis and as a player the emoji we refer the leader to the articles of urban second and the primary he has some paper so then the definition of the corporate monster up there operator and the primary monster pair operator landlord numbers and iteration theory as a pretty new medical criterion for every ample night bundle and for recent works on the west product of close non-negative current on corporate economy for an application to iteration theory and corporate dynamics in higher dimension we refer the leader to the difficult and article we have some article regularization of current and HOP and gamma g of colors and iteration theory and dynamics of holy room line maps and super pressure of positive color current iteration theory and animation now we talk about the difference of corporate operator for smooth prusonic function the class classical potential theory plays in important zone these studies the last need the operator and a couple analysis is test the operator we consider the corporate monster pair no need operator defined by follow dc2n equal and the determinants of the corporate has some of my chicken company has some and it's the notebook measure and dc2n is a zadon negative measure but if the function is not smooth we defy the corporate monster pair by if we as soon as we assist a measure in you it's called the corporate monster a pair operator of fine if for every the open set d in omega for every smooth sequence sequence of smooth prusonic function phi z decreased to phi then dc phi z to the n converged to new which need as z to go up infinity we denote we need or dc phi to n by the corporate monster a pair operator of fine in one dimension the dc v phi equal not like a operator is a really fine for every some of the function but is a kind of higher dimension we have many counter example to show that the corporate motion pair is not defined for example on the board we take the function phi equal no gamut zero one it's the corporate moon pair the corporate motion pair of this function is not identified by meaning by this meaning he talked about some history of the division of corporate monster pair operator in 1982 where fort and tenor proved that the corporate motion pair operator is identified for class about this prusonic function and later that means so that the corporate motion pair operator is identified for class with bounded value near the boundary and it should be noted that this is an instinctual for current and it's many applications in interest in theory and corporate dynamic system and in his urban section paper he introduced and studied a general class of prusonic function on which the corporate motion pair is well defined he defined by class e omega and he also introduced some class prusonic function e b omega and f b omega with finite energy which are useful tools to solve some pressure in corporate analysis of several very able and in the case of two dimensions we have e omega e square prusonic function interesting they kept log one two is the sub-level and we talked some some correct motion per equation on domain is some design in 1976 the corporate motion pair equation on the main scene was first studied by father manton goes back fort and tenor and later it has intensive developments by a series of paper mathematics suck ass with one second so we call it a nozzle numbers perhaps left this and also myself we have some and here we talked about corporate motion pair equation on copper cannon manifold if all we talked in local case in global context the corporate motion pair equation appeared naturally in problem of existence unique color magic is the same class we see form is a any given two form that is presenting the first check glass on copper cannon manifold the solution of copy conjecture equivalent to the solution of motion per equation of copper cannon manifold by city hall in 1976 the revolution night the complex difference of geometry and to the disease is and has many consequences and it's been an open of its its presentation for current style since the corporate motion per equation have been useful tune in present-day the corporate geometry as a application to construct kind of ice-time magic on copper cannon manifold and new magical characterization of the kind of cone of copper cannon manifold we see some paper of philippe acl vixen get america in 2009 that may and be high prune into and for synto synto in 78 and gatian is 80 87 here i talk about the synto already done he he proved us that is omega p a compact cannon manifold of dimension n assume that f is smooth function such that e turn over f omega 2 n equal e turn over f omega 2 n then there exists a unique smooth omega prune some link function phi on each such that dz phi plus omega 2 n equal f omega 2 n as sub phi on x equals a zone now i will talk about the deeper equation written to estimate in a basic document domain the zone of debugation written estimate domain was invented by hobbinder and we also see a lot better me in 97 this omega p be a special to converge domain in synt as such that phi is presently on omega and az is a continuous map from omega to a set of relative Hermitian magic such that dd bar phi got Sunday and the volume of z in omega the determinant az equals a zone equals a zone that forever was a zone one form in n2 f be such that the bar f equals a zone and such that this condition then there is exist a function f u in n log 2 such that we buy you equal f and such that condition it's a very nice form estimate here you can use this and they have a second version when you touch the omega is the domain is a subset of the bone that we have the same with the same condition f e 1 1 z zone one form such that debug d bar f equals a zone and such that co-descent then there exists a solution function u such that we buy you equal f and such that this co-descent but uh i will talk about the two app uh extension term of osuwa and tachyrocy a useful consequence of sunny by western with the two estimates is uh n2 extension term of osuwa and tachyrocy is was a main tool in the product of many uh fundamental designs in cobalt analysis and they are muchy we test uh omega be about the basic basic convergence domain in sin f phi is blue solid function and and its complex subspace then for every function whole which function f on omega it is an n then there exists a whole which function f on whole domain omega with e turn over omega moon f 2 but 2 is minus 2 uh minus 5 the big measure then some constant d brand on omega e turn e turn uh on omega it is an n f 2 but 2 is 2 minus 5 and the big measure on n the number the c omega is only depend on the diameter of omega this the the number c omega is not good it uh osuwa and tachyrocy paper i think the c omega is uh 1700 but uh z blocky he finds the best the best constant c omega here we have we have omega in c m minus 1 time t is domain and we that we have phi is blue solid function on omega and f is a whole which function on n it is an omega then we can uh find the function f to f on the sasa condition with some the best constant here now i will talk about the singularity of blue solid function in last five is blue solid function and the opposite side of the singularity with the research or study the behavior of the function phi on neighborhood on neighborhood of phi equal minus e phi t that we use some tune here corporate analysis of several uh variable method and the biogasic method to estimate and the two functions of osuwa and tachyrocy corporate browser operator and invariant of singularity of blue solid function of boy s le long number and no color code transfer we talked about the le long number it was introduced by le long it's the measure of the singularity of this function we touched the le long number of blue solid function phi at the boy z equal by definition it means that we compare phi with log loga module of w minus z at the boy z is the cosine le long number of z it's introduced uh start by le long here we have some uh property of le long number uh first uh we have we take the f be a whole like function then the le long number of loga module f of z equal multiplicity of f at the boy z and we take the ideate i at the z for basic f 1 f m of the ideate i which says phi z equal 1 divided by 2 loga f 1 z 2 by 2 plus dot dot dot plus f m z 2 by 2 then le long number phi at the boy is minimum multiplicity of f 1 at the boy multiplicity f m at the boy then now i uh talk about some a very beautiful design of the in thong siew by saying that the function from z to le long number phi of z is the upper semi continuous with respect uh along the text this this is the topology this group of literature by heavy and king in uh 1972 and is proved by in thong siew the paper as a simple proof of siew theorem was found that found by the bidenby who devised an official estimation of arbitrary proof of the function by one which add analytics securities we talk about the theorem let's find the proof of the function on about the lumian omega in cn then the upper never sets z in omega such that the le long number phi z greater than c as analytics sub-variety of omega now we talk about local code has transfer up to solid function as professor colonial is also he talked about history of local code has there are many versions of definition of this uh in uh corporate analysis and you know so in uh i can pick them with him for example uh i find very uh i find the end invariant and many different definition but here uh you talk about the definition by the uh the bay encoder we take five be a blue something function on omega uh of cn then look the local code transfer of phi at the boy z as o in omega is a positive z number c phi z as o equals to the moon c glass and z o such that e speed to power to minus two c phi is n one on a neighborhood of z as o is a invariant number of phi uh then uh i introduce a what the local code transfer up to solid function for every non-negatives they don't measure new or the label hood of uh z as o we introduce the weighted local code transfer of phi with the west wheel z as o by c mu phi z as o is suitable c glass and z as o such that e to power minus two c phi is n n one will or the label hood of z as o it means that we if the phi have a big uh big infinity then the c mu phi is smaller is smaller now i will introduce the martin martiner i that's it unless it be a complex manifold and not then introduce a i phi with a c program of homic function i have such that it turns over you uh moon f to power two is uh minus two phi less than e phi t this uh shift is closed idea shift over east is a 35 z equals c divided by two look look look at moon f one z to power two plus uh dot dot plus uh f m z to power two where f m f m i hold function on x then i find shift of worm or hole function f such that condition is uh come from uh commutative agro plan and we uh also introduce uh weighted martiner i that's it for every non-negative they don't measure new on complex manifold we denote by in new the shift of worm or hole function f such that it turns over you uh moon f to power two d mu less than e phi t on some label hood of mu of z in x now i need to talk about the notation and uh some design basic design and some motivation now i will talk about some recent design that pair the at the me and cola blower of uh fundamental design on effective version of semi continuous theorem of local core threshold of blue solid function f following we take the uh a sequence of blue solid function phi z suck up phi z convert to phi in n one look and we assume that it turns over d e to uh to uh minus two c phi no big measure less than e phi t for some d co-park in omega then for every key is co-park in d c prime less than z z let us see then exist z is over as an one suck that's suitable z glass and z so it turns over key e uh to power minus two c prime phi z the big measure less than e phi t this is on is very deep and have many uh consequences in the couple and it and they are basically suck as co-constructed kind of like magic on follow manifolds then in uh from 14 uh i use n the two it turns into them uh also quite like we'll see for on the global basic uh uh martin uh i don't see to prove the effective person of semi continuous theorem forward that's not going to cause a cell it proves us it takes the sequence of blue solid function phi z suck that's phi z less than phi and phi z convert to phi in n one look then we assume that integral uh is uh minus two c phi moving up to but to less than infinity we have a moving up to but to here for some d is co-park in omega therefore every key is co-park in d then exist z is over as an one suck that's suitable z glass and z so it turns over key e to power minus two c phi z moving up up to but to the big measure less than e phi t just uh uh here we have a remark the problem with our condition phi z less than phi is not true in the case we with uh mu z one no big measure we see the mass one three in uh farm 14 then we have a some consequences of uh oh these are uh the main color theorem deeply uh directly applies the following consequence we net uh it's uh about the same uh it's about continuous the my dad and my dad I don't see it that's phi z be a sequence of blue solid function on omega and suck that is a convert to phi in n one look we assume that the element one is in this uh shift then for every c prime less than zero less than c there exists the less than one suck that one is i c prime phi z for all type z glass and z so and something mainly than is farm 14 and farm 17 apply the following consequence the next phi z is uh uh sequence of blue solid function on omega and phi be a blue solid function on omega be suck that phi the convert to phi in n one look that's the to the following statement as hope first if we assume that phi z less than phi for every z less than one therefore every omega prime is compact in omega then exists the less than one suck that e phi equal e phi z equal e phi on omega prime for all the z glass and z z so and if we assume that c one z n is i i phi z so we don't need the assumption with z movies then as it then there exists the less than one suck that z one z n is e phi z for every z less than z at all and so i talked about the relationship between the local control and motion per mass with uh let me add my paper and myself found the integrated between local control and motion per mass it let's find e omega then c phi glass than some z from one to n z minus one phi divided by e z phi where e z phi is the interest number is equal it turns over the so d z phi to the z d z phi local moon z to the n minus z and we uh now we consider the the ck phi you call soup c phi s for all key dimension linear subspace s to z that's uh we set ck title phi you call soup c phi for all smooth summoning for of p dimension s to z then uh we found 17 the also found the inequality between with the local control that's the and system local control ck phi we proved that for every phi is personally faster than ck uh title phi equal ck phi you can say c phi with wooden is this method when we also the find the inequality between the system local control ck phi ck phi and motion per mass it we told them in farm 18 it takes phi in e omega then c phi glass and c m minus one phi plus m minus one to the m minus one c m minus one phi to the m minus one e n phi i've uh and finally i have asked a question before future study in first question about the semi continuity for what the local control what condition one new is necessary and sufficient such that if e turn over omega problem e to power minus two c phi d new net z plus e phi t or solid shall open set up omega problem with omega then assist some data is positive number that's that for every side is presently function as a norm and one norm sin minus phi that's on data then sin u sin e plus sin u sin d z over c and also if sin convert to phi in n one omega problem then the function e to power minus two c problem side convert to e to power two c problem five in n one dot degree for all c problem never see and i self uh ask a conjecture the next five e omega then c phi glass c m minus one five plus e n minus one five divide e n five and here is now some uh self reference and i finish my lecture thank you for attending and listening any questions if not i would like to make just two final very final concluding remarks first of all an event like this cannot be organized without the help of many people and i would like to thank them in particular the director's office the public information office the media office of ictp mrs mabilo i mean there's really a teamwork of everybody many many people to have this an event like this to happen and i really would like to thank them publicly i hope i haven't forgotten i am sure i have forgotten somebody but you know and uh secondly actually uh so of course we are happy to have the ramanujan price will exist in this shape we are happy to announce it now for the next five years at least because we sign just sign the agreement with the uh indian government and the international mathematical union to continue in this way and actually just i think yesterday we opened the call for nominations for the 2020 so we we move on to the next event and of course we would be very happy to have you again here next year around this time it's usually earlier but okay so we thank and congratulate dr farm again would you like to say a few words yes i'm also wasting a little bit of your time because there is a refreshment outside the door but i think i take responsibility i think we said that five now they are running to do it as quickly as possible it's ready it's ready so we thank also the refreshment office and so there is a light refreshment outside the door for everybody thank you