 Okay. So good afternoon everybody. It's a it's a great pleasure to have all of you here for this wonderful ceremony of the Ramanujan Prize, awarding the Ramanujan Prize for the year 2022, which is a result of a very beautiful collaboration between the Department of Science, Technology, ICTP and the International Mathematical Union. Let me begin by welcoming our distinguished guests who are here, Professor Ulrike Tilman. She's the Vice President of the International Mathematical Union, one of our partners in this prize. An important delegation from Senegal. I'm very happy to welcome them here. They have taken the time to visit us at ICTP, led by his Excellency Dr. Papa Abdullah Esek, Ambassador of Senegal to Italy. Professor Mamadu Sai, Director of Strategies and Planning of Research, who is also a former ICTP associate and Professor Dheeraf Sek, Technical Counselor at the Ministry and a friend of ICTP who has collaborated on many scientific activities. So we are very happy to have this very distinguished people visiting at ICTP. And I'm pleased that this year's Ramanujan Prize is awarded to Professor Mohammed Mustafa Fahl. This is the first time a Senegalese scientist receives this prestigious award, and I'm happy that his family is here too. So far we've had four awards from India, four from Brazil, four from China, two from Argentina, one from Vietnam, one from Gabon, and one from Mexico. This 2022 Ramanujan Prize Selection Committee had many imminent mathematicians on the selection panel. Professor Ji Rangarajan from the Indian Institute of Science Bengaluru in India, Professor Toro from University of Washington and MSRI Berkeley, in the U.S., Professor Nang from Equal Normal Superior, Libriville in Gabon, Professor Ayesha Wad, Penn State University, who is also a member of ICTP Scientific Council, and Professor Lothar Goche, who chaired this committee and his mathematician professor at ICTP. And the prize citation I will read as follows. The prize is in recognition for his impressive results on existence and non-existence of solutions of linear and non-linear partial differential equations inspired by geometry and mathematical physics, especially for his remarkable study of solutions to fractional Schrodinger equations and non-local mean curvature problems. The committee also recognizes his outstanding contributions to the advancement of mathematics in Africa through research and public engagement. The prize is the result of, as I said, is a collaboration between ICTP, DST, and IMU, and we hope that this collaboration will continue for a long time. I would like to especially thank the Department of Science and Technology and Government of India for their generous support for the Ramanujan Prize every year. Mustafa is the president of the African Institute for Mathematical Sciences and has an endowed chair of mathematics and its application at Ames Senegal, African Institute for Mathematical Sciences. He has an old important connection at ICTP. He was a diploma student in Maths for the academic year 2004-2005 and then was also an ICTP associate. I would like to take this opportunity also to thank the Simons Foundation who awarded Dr. Fal the ICTP Simons Association in 2014, a prestigious fellowship which allows for extended visits for conducting mathematical research both for Dr. Fal and his students. So congratulations once again to Mustafa. And today is a special day for ICTP and for Senegal because the president of Senegal himself, his Excellency Dr. Makke Sal, decided to send his personal greetings in a video recorded messages. So this is also I think congratulations again to Mustafa and so we will start with video recorded messages from the president of Senegal, his Excellency Dr. Sal. Thank you. I congratulate you warmly, Mr. Atis Tabalkar, Director of the International Center for Theoretical Physics for this excellent initiative that will soon take place in 20 years. Senegal is honored to see this prestigious distinction given to our compatriot Dr. Mohamadou Mustafa Fal, President of the African Institute of Mathematic Science. I am even more proud that Mr. Fal is a pure product of Senegalese school where he obtained his degree in Mathematics before frequenting the International Center for Physics Theoretical Physics and did his thesis at the International School for Advanced Studies. His current role as President of the African Institute of Mathematic Science in the world that our country has been supporting since 2011 is a beautiful stage of a career that I wish him a long and successful career. I congratulate all my congratulations to our distinguished Aurea and give her my pride and celle of the Senegalese nation. His performance confirms the persistence of the choice that we have made to promote more the teaching of scientific and technological disciplines in our educational system, especially by the creation of the high school of excellence, the opening of the preparatory classes at the Grand School, the modernization of the laboratories of our university and university institutes, and the high school price for the scientific and technical research of the KAMESK, the African Council and Malgash for the Higher Education. In the history of human societies, science, technology and innovation have always carried the progress to be the economic and social well-being of the people. This is why I want to give a tribute to the efforts that your Centre has made to create the emulation between young mathematicians in development by distinguishing the best of the best. All my wishes for success are with you. So it is my pleasure now to ask our partners to give welcome remarks. So we're expecting some from the DST, but apparently there is a problem in connecting. So I will pass the floor first to the Ambassador of India to Italy, her Excellency Dr. Nina Malhotas, and as a message for congratulating and for being with us today. Honourable Secretary, Department of Science and Technology, Government of India, Director of the Abdu Salam International Centre for Theoretical Physics, Shri Atish Dabholkar, President of the International Mathematical Union, Carlos Kenick, Professors, Faculty Members, Researchers and Practitioners of Mathematics. I'm delighted to join you all today. This year this event is taking place in the backdrop of many significant developments in India-Ikli relationship. We recently had an immensely successful visit of PM Georgia Maloney to India. Our two countries are also celebrating the 75th anniversary of the establishment of diplomatic relations this year, and India is holding the G20 presidency. I extend my heartfelt congratulations to Professor Mustafa Fawad, Professor and President of the African Institute for Mathematical Sciences in Senegal for being awarded the prestigious Ramarajan Prize for the year 2022. The award is a recognition of his outstanding work and contribution in the theory of partial differential equations and also reflects your important organizational role in advancement of mathematics in your country. I'm sure that this well-deserved award will motivate you further to expand your research and inspire other researchers and young mathematicians not only in your country, but in the entire developing world. I feel proud and stated that this award not only showcases India's historic contribution to the field of mathematics but also highlights and perpetuates the legacy of Srinivas Ramarajan, a brilliant Indian mathematician. I'm intrigued by the fact that the wealth of ideas he created a century ago are being researched upon and finding applications in diverse contracts till today. I would like to take this opportunity to thank ICTB Italy, IMU Germany and DST India for their contribution to the legacy of the great Ramarajan. I wish all the very best to the event and look forward to mutually enriching association in the future. Thank you. Thank you again for being with us and sending her message. I would like now to give the floor to Professor Richard Tillman. Professor Tillman, she is actually a fellow of the Royal Society and chairman of one of the most important institutes in mathematics in the world. She is the director of the Isaac Newton Institute for Mathematical Sciences but today she's here also in her capacity as vice president of the International Mathematical Union which is of course a crucial partner in this initiative of the Ramarajan Prize. So I'm happy to leave Professor Tillman the floor. Thank you very much. It's a pleasure to be here. Let me just say something about mathematics itself. The physicist Wigner famously commented on the unreasonable effectiveness of mathematics when 60 years ago he received the Nobel Prize. His discovery and application of fundamental symmetries principles, in other words group theory, led to his groundbreaking contributions in particle physics. He also points out at the same time another mathematical theory, the theory of complex Hilbert spaces which in turn became fundamental to the formulation of quantum mechanics and nowadays quantum information. Physics is not the only area of application where mathematics has proven to be essential. Modern cryptography, think public key encryption, is based on number theory, the very subject that Ramanuchan studied and that his mentor G.H. Hardy claimed to have no practical use whatsoever in his famous essay a mathematician's apology. The power of mathematical abstraction is indeed unreasonable. To give just one concrete example in the work of an Oxford colleague, the same mathematical analysis used to design dust filters in vacuum cleaners has been used to develop filters to remove arsenic from ground waters in the Ganges Delta, impacting hundreds of thousands of people and saving lives. In the opening ceremony of the 2002 International Congress of Mathematics in the great hall of people in the presence of the then president, Jiao Ziming, a doubling of the financial investment into mathematics was announced to accelerate the development of science and technology. Twenty years later, the success of the Chinese strategy is evident. As mathematics is underlying nearly all of science and technology, it may not be so easy to quantify the effect of this investment. But let me just quote a study by the global consulting company Deloitte, just as a proxy. It estimated the impact of mathematics on the UK economy as more than 15% of GDP. So mathematics interpreted in the broadest way provides a cost effective because we don't need big laboratories and hence democratic route into high tech and prosperity for many countries. The investment in teaching and research mathematics is in particular an investment in people. It is enabling, it is giving people the power to think and the tools and confidence to attack the most taxing problems, whatever shape. I'm very honored to represent the International Mathematical Union here today as we celebrate the achievements of one of the world's brightest mathematical minds. The IMU was founded in 1920 at the inaugural International Congress of Mathematicians held in Strasbourg, France. It was created to promote international cooperation in mathematics and to support the development of mathematical research and education worldwide. Over the past 100 years, the IMU has indeed played a vital role in advancing mathematics, not least through its very active commission for developing countries. Most visibly, it organizes every four years the International Congress of Mathematicians, its flagship event. The next ICM will be in the US, but earlier this century, it has been held in Brazil and in India and I very much hope that it will soon also be possible to hold an ICM in Africa. Today, we have every reason to be hopeful for the future of mathematics in Africa and I just want to extend my warmest congratulations to Mohamed and his family. Thanks a lot and I would like to ask his Excellency Dr. Papa Abdul-Alyasek, Ambassador of Senegal to Italy, to join us in welcoming this award. Contestably, I am delighted to see a representative of his Excellency, Professor Musa Balde, Minister of Higher Education, Research and Innovation of the Government of the Republic of Senegal. His Excellency, Professor Musa Balde, had well wanted to be among us, but he was held in Dacar for government obligations. He asked me to send you his life back, not to be among us today. Dear friends, there is a strong correlation between the promotion of scientific research in a country and its economic development level. That is why the Government of Senegal considers a strong research generating knowledge and technology in harmony with our deepest aspirations, built without a doubt, a cynical condition to ensure our economic social emergence. That is why this will is illustrated by the installation of a research-based Makisal prize. A vast program of modernization of our laboratories. Today, Senegal has the best calculator, the most efficient in our space, which is absolutely remarkable because for those who do research, to have good precision, you really need a tool such as a super calculator. There is also the creation of scientific research. The President talked about it. We focus on mathematical, informatic and technical excellence, but also on new universities. To diversify the training effort, we have to ensure employment training. In reality, the Government of Senegal has made a clear choice. It is about a densification, a fortification, and a massification of our research and teaching device. This is precisely what will allow us to create the Uduan conditions, to call ourselves a plural expression of excellence, a plural expression of excellence, consisting of liberating the potential genius and creators of Senegalese and Senegalese. Because, in reality, progress is not the case. Progressing is doing things other than better, with technological innovation as an achievement, that no one considers as the first, the first matter to stop our economy. So, strength of all this, the Government of Senegal, by my voice, is simply and deeply rejoicing. This high distinction is dedicated to Prof. Mohamed Moussaafal, a former president of the Republic of Senegalese, a young and brilliant mathematician, deeply attached to excellence in every field, in every time and in every place. Prof. Fahl is currently conducting with Brio the African Institute of Mathematics. We are sure and convinced, Prof. that this price will not be the upper part of your performance. Prof. Fahl will, incontestably, benefit and contribute more to the domestication of mathematics. And yes, the domestication of mathematics is a global challenge, more mathematics, more development, so we must domesticate mathematics so that it is the case of all of us. Prof. Fahl, our pride is infinite, but our expectations for you are also immense, because we expect always more of superior services, because the world is governed by the good news. I would like, naturally, to associate with my demonstrations with your father, Honorable Khalil Gibray-Mofal, eminent deputy of the National Assembly of Senegal, eminent economic operator, a man of development who has contributed a lot, and who still contributes to the Senegal's standing, prosperity, and the divinity of sharing. Grand has been my joy to learn that you are the father of the man we celebrate today. Also, I would like to associate with my demonstrations with your response, which is also mathematical, as I said. Like me, learning mathematics has not yet led to mathematics. Congratulations also to the national scientific community, because it is one of my products. We are really proud. We are proud because we are used to saying that mathematics are inaccessible, that it is extremely complicated, but to have a young university professor who has just been promoted constitutes the entity of this assertion. Mathematics must be studied by all, because we have to say that it is a word. To avoid making this word a sentence, I may have to conclude. Congratulations to the professor. He is the director of the International Faculty of Physics. You have made us a presentation that shows that this center is an excellent center, that it can consistently contribute to strengthening the scientific achievement in our spaces thanks to what we could call a personalization of responsibility to come out of an aggregation of skills to better understand the world in view of my explanation. Congratulations for the work you are doing. Thank you and congratulations to the Indian government. You must never forget what they do more. Thank you and congratulations to all those who work and fight for a better world, a world of peace, a world of sustainable progress, and a world of social success. Science can be a major strategic input in the construction of this new world. I would like to thank you for your kind attention. Thank you very much. It is always good to practice our French, but just to help us, we should actually, we should definitely improve on that side, but fortunately we have our colleague Corine, she will read the translation in English. Okay, so His Excellency allowed me to make a short summary of what he said. So first of all, sort of presented the regrets of the Minister of Education that really wanted to be with us, but was unfortunately unable to attend today, but His Excellency's warm congratulations and thanks to ICDP and to the Government of India and the Association of International Mathematic Union and for sort of rewarding Professor Pfau. He reinforced in his address actually the importance of what the President of Senegal just presented in his talk, which was research and education is really key for the development of any country, and particularly in Senegal, and that Senegal embarked in really investing in excellence in research and education, starting at the level of the LISSE, which is this sort of, when you are about 18, up to building new universities and new tools for research, like the calculator, if I'm not mistaken, you translate calculator by calculator probably, and he really sort of emphasized the fact that this is really very important for any nation in its growth to learn mathematics, although it might be from a very first contents with mathematics, it looks difficult and maybe not the kind of discipline that you would envisage or embrace, but it's really important to encourage young people and through the example of Professor Pfau and his success to sort of path the way, and he associated, I wanted to sort of thank and congratulate Professor Pfau, of course, his family, his father, who is really a renowned and very important figure in Senegal, his wife as well, and we know that wives are very important for the career, behind the scenes careers, and that's my personal touch, sorry, that's not his excellency. And in conclusion, he wanted to congratulate Professor Atish Davalkar, the center of STP, that for its mission, since the very beginning, which is really making a better world through science, and he thanks you all, and I hope I was not mistaken in sort of translating what you had in mind. Thank you so much. So, now we open the second part of today's ceremony, and I think we are all quite curious to understand a bit better. Some of the contribution of Mohamed Mustafa Pfau to mathematics, I must say, I myself had to study some of Mustafa's works, and he is definitely one of the great experts in partial differential equations, but we deny to applications also to geometric problems, and for example, his kind of non-local Alexander of Theon, and I think it's one of the beauty of the past few years in the crossroad of analysis and geometry, actually something for which he has already received important honours around the world. I mean, just to mention one, he has been invited speaker at the International Congress in 2018. So, in order to understand a bit more about Pfau's work, we have asked one of the greatest experts in analysis to help us. Unfortunately, he had to leave, but before he managed to leave a message explaining to us mathematically Pfau's work, I think it should appear. So, Professor Luigi Ambrosio from the Scola Normale Superiore has given this short presentation of Pfau's work. Good morning. First of all, let me congratulate with the Professor Foll for this important achievement, for this recognition for his work. This is an important prize associated to the name of an outstanding mathematician, and while it is also sponsored by important institution, the ACTP, the International Mathematical Union, the Department of Science and Technology of the Indian Government. So, congratulations. And it is a real pressure for me to have this occasion to outline in a few minutes the career and the scientific production of Professor Foll. So, let me start to share the screen. Professor Foll is presently a professor of mathematics at the African Institute for Mathematical Sciences in Senegal and also president of the institute. His career has been fast. After his bachelor studies in his country, he came to ACTP where he got a diploma in 2005 and immediately after he got a PhD position at CISA where he got the PhD thesis under the supervision of Michael Lee, Andrea Malkiovi in 2009. Then he had several post-doc positions, which also were important to establish contacts with scientific contacts which are still going on now. And this professor now has had also a professor position at ACTP for four years. Among his main achievements I would like to stress in particular the invitation as a speaker at the ICM in Rio de Janeiro. And by the way, also it is evident for his ACP, his capacity also to raise, to train young students in mathematics. He had already several PhD students and also his important engagement in the development of mathematics in his country and in Africa. What are the topics covered by Professor Foll in his career so far? Let me say that eventually I have chosen since my time is limited to spend more words about two papers which I consider particularly striking and interesting, but we started this scientific work at the time of the PhD thesis in CISA on the theory of isoperimetric sets with a small volume. So trying to analyze the asymptotic behavior of surfaces with a large micrometer associated also to variational problems with a small volume constraint. Then we also studied several hard type problems, problems where singularity appears either inside the domain let's say at the origin or on the boundary of the domain. He studied problems dealing with the before creation of expansions of eigenvalues, in particular Neumann eigenvalues. And I will say the main research interest of him in the last few years has been the theory of fractional differential operators. This is actually the topic of the two papers I will tell you later on. All these things of course with the variety of tools coming from functional analysis in Hilbert Manifold's of course the many tools from partial differential equations and in particular to stress Morse theory and harmonic analysis. To go on, let me start indeed from this paper. I want to tell you something about this paper published in this important journal more or less 10 years ago. This paper was written with Rico Beltinoce and this with this semi-linear fractional PD and of course the semi-linearity comes from this non-linearity in the right hand side and the fractional character comes from the fact that we are considering here the alpha power minus the Laplacian with alpha street less than 1. As many of you know, the left hand side in this PD let's say the linear part is naturally associated with this so-called norm which consists of the Gallardo semi-norm which is basically at least formally the L2 norm of the alpha fractional derivative and this is all this semi-norm by adding for instance the L2 norm we turn it into a norm which defines a so-called space and a fractional so-called space. The exponent PD non-linearity stays between 1 and 2 alpha star minus 1 where in this fractional theory 2 alpha star plays the role of the classical so-called even bending theorem. Indeed, you see that for alpha equal 1 you recover 2n over n minus 2 which is the exponent of the classical theory and this is associated to this so-called embedding. So what is the relation between this embedding and our PD? Well, it comes from the fact that this PD can be solved can be also attacked with variational methods by looking at the best constancy in this inequality and because P plus 1 is smaller than 2 alpha star there is enough compactness to gain ground states and so the infimum in this kind of Rayleigh quotient provides to you the inverse of the best constant in this inequality and the connection with the PD star comes from the fact that if you consider a ground state normalised on the unit sphere of the LP plus 1 norm then the Euler Lagrange equation appears and displays the role of a Lagrange multiplier and by scaling since this side is linear this side is not linear because P is larger than 1 so by scaling you can recover by multiplying by a positive and independent of x constant you can recover a solution to the problem star and it is well known that the course of variational solutions have special properties and one of the main results of this paper is obtained by a kind of perturbative analysis by analysing the linearized operator and suite of branches of eigenvalues in this paper Valdinochi and Professor Foll have been able to prove that for alpha sufficiently close to 1 uniqueness of the ground state persists and also the symmetrization gives you that it is radial this paper generated quite some interest indeed shortly later in a paper which appeared on Akta Frank, Lenzman and Sylvester were able to extend this result to orange subalpha between 0 and 1 the second paper I want to tell you somehow related is a paper appeared on Krell in 2018 and to put this paper into context let me remind that in 2010 in a similar paper Caffarelli, Roccageoff and Savin introduced a notion of fractional perimeter actually here P should depend on alpha so the P-alpha perimeter which as you see takes into account the interaction between points inside and points outside the domain weighted by this singular kernel and this paper initiated the study of the regularity of minimizes really along the lines of the classical theory meaning you may try to understand the structure of cones or better solutions the dimension of the singular set and so on and most results but not all of them have a perturbative nature meaning that if you know that some property is true for classical surfaces, minimal surfaces then by a perturbative analysis you can try to show that they persist also like indeed in the previous paper I mentioned to you for alpha sufficiently close to 1 this is also justified by the fact that if in this formula you multiply P by a constant by a factor proportional to 1 minus alpha indeed one can show that the alpha perimeter converges what we call gamma converges to the classical perimeter but what is even more important is that as in the classical theory a fractional notion of mean curvature emerged for locally minimizing and even stationary solutions you can write it as a principal value and also maybe by integration by parts if it is sufficiently smooth you can write it as a boundary integral which in a sense makes more clear the geometric character of this expression which is the so-called fractional mean curvature and then again we can try to follow the analogy with the classical theory and what Cabret, Sole Morales and Pett together with Foll they were enabled in this paper to obtain one of the first classification or rigidity results for constant fractional mean curvature surfaces extending a classical theorem for the classical mean curvature surfaces due to the rational mathematician Alexandro precisely the result is this that if he is a bounded open set with sufficiently regular boundary sufficiently regular with respect to the parameter alpha which appears in the fractional perimeter under this assumption if the mean curvature h is constant on the boundary then e modulo negligible set must be a ball so there is a rigidity result which actually then can be generalized also in several other ways using viscosity notions of mean curvature relaxing and also the regularity assumption but this has really been one of the first results for fractional constant fractional mean curvature of the theorem so I hope that this short presentation has given to you a glimpse of the range of activity of Professor Foll and let me congratulate with him again thank you for the attention thanks a lot to Professor Ambrosio for being with us and explaining to us some of the beauty of Foll's work so I think now we are all convinced and now I think it's time that Foll finally receives this prize we have been talking enough so I think I invite Professor Tillman and Atish Dalbukhara over there to pass the award to Professor Foll I think you can go in front here just to comment what's going on the prize consists of a statue of Ramanujan a certificate and a cash prize supported by the Government of India now with the statue, the certificate and the check we ask you to give us a talk about it to explain to us some of your work and I think ok you will come from... backwards and you press this from the pointer would you like to do this one? no it's ok I can use this one so thank you very much Claudio of course it is a great honour to be here so I didn't really prepare a speech because I know this place so much I spent so much time and so many years in this place here so it's very hard for me to write a speech and express all that I have in my heart so I start Professor Atish Dabholkar the International Centre for the Vertical Physics Director Professor Eric Tillman Vice President of the International Mathematical Union His Excellency Dr. Papa Abdulasek Ambassador of Senegal and the delegation sent by the Minister of Higher Education and Research in Senegal I thank also Professor Ambrosio for his great presentation so it is a great honour of course to be here and I would like to start really to thank his Excellents President Makisal for his constant support of the promotion of science in Senegal his testimony, his video shows that in fact he is quite active and strong supporter of mathematics and in general science in Senegal my greetings also go to the Minister of Higher Education and research who really wanted to be here I know really well that who could not so I have also many people to thank so I start first with my family because without whom I would not be here in particular my father, my mother and our big family so we have really we really have so my wife also who is here supporting me day and night so research is very jealous as you all know my research collaborators who are not here and except one Professor Muzina who is here who taught me hard inequalities this is one of also the research field I studied and the one that Professor Ambrosio was mentioning my teachers when I am talking about my teachers it's not only those from Senegal but also those from ICTP here and almost everywhere and also very big family so which is also EMS African Institute for Mathematical Sciences which is a big family founded by Professor Nil Turok and which provides one of the best places in Africa to do research and in fact there are even some alumni who are here see them here yeah so this is just to say that we are in fact trying our best to train so I would not say the young scientist because I think I'm in the young scientist because of the price but the trend so the young Africans also to be part of the future I would say so it is of course an honor to receive this price because I would say of course a kind of marriage because Ramonujan is a number theorist and my wife is a number theorist so he was an autodidact person which of course I was not unfortunately or fortunately because it's because of that probably I was here in ICTP so where I really met great people, great scientists who really make a big change in my research career, in fact when I left the university Gaston-Berger of Saint Louis in Senegal so I left there as applied mathematician and when I came here I met new people I meet new people who are specialized in different fields and great teachers and it's from there that I decided to study geometric analysis so it's not in general common I think people in general they start with pure mathematics and go to applied maths but I did the opposite but I didn't regret it because at least I can train people who are doing applied science so thank you very much I hope I didn't forget anyone it was almost a mistake I really want also to thank not only the director of ICTP but also those people who are really working in ICTP who also made our stay when we were here very young almost 10 years ago to come for the first time in Europe so for some people it's not it's very, it's nothing but for us it was not that easy to come for the first time so the first time I left Senegal in 2004 I arrived here in Europe it was quite cold a cold that I never knew before in fact I didn't even know which type of clothes that I should wear to go to sport there is something that I loved very much but we really had many people in here in ICTP who was really also supporting us so the support is not only science but also in the social part so thank you very much and thank you very much also to CISA where I did my PhD at the School of International the Superior Advanced Study so let me just start my presentation which should be very quick and I'm sorry because for those who really who are not mathematicians so my father I hope we are going to discuss this later my thoughts will be about these type of problems so over determined boundary problems so it is a subject that I start studying with my PhD students who was from Cameroon so he is now doing a post-doc in Frankfurt with my also collaborator WSW and so we so when he came to Senegal and he wanted to work with me so I gave him this type of problems and so since then we found a lot of interest and we keep working on this and in particular in the recent I would say since last year I would say so we decided also to tackle one of the to try to tackle one of the most difficult conjectures in this type of problems which is known as the Schiffer conjecture and so this is around my presentation and yeah so I am going to show what is going on almost and exactly what we have on this subject yeah so so there are many type of over determined problems coming from many different branch of mathematics but so those that I really would like to talk about will be those which are very specific type and of course driven by the Laplace operators or the diffusion operator so it is a very interesting field of research so which of course appears also in fluid mechanics in capillarity theory in elasticity and also in electrostatic problems they appear also in shape optimization so as many people know professor general expert also in this type of shape optimization so directly problems and no one eigenvalues so in general so when you differentiate these eigenvalues so you end up with an additional thing that comes on the eigenfunctions and when you prescribe that then you end up with an over determined problem so they are also very interesting as application because in general so whenever you solve this type of problems then you have a solution also with explicit solutions to the stationary also incompressible equations of course in the in the 2D case so with some very good information on the pressure of the pressure of the fluid on the boundary so the problems they look like this and they look quite simple but still so there are many things that are going on but it's very short equation so we just say so Laplacian of u equal a function in an open set and at the same time you want to prescribe 2 things at the same time so you have to prescribe the value of the function and the value of the norm of the gradient so but in most papers so you see people they write the noman boundary condition but the 2 are the same because u is constant at the boundary so so the problem is a free boundary problem because so what I am writing here is that we want to find a domain we want to find a function u we want to find functions f so that something happen so and this thing that is happening is that we you have to find all the solution and therefore omega is part of the problems and this makes in fact the problem is well difficult to solve so in general you don't even have solutions you cannot use direct amount of calculation because you prescribe too many things and therefore in general what people needs to do is to construct solutions construct the domain construct the functions f in such a way that these things happen because as I said before whenever you solve this type of problems you have immediate problems in different fields in physics or so and so this is exactly what I said so in general also since this problem cannot be solved immediately so if they have a solution then that domain omega which as a solution has to be special so and so let me just give in the next 2 slides some few examples of where these type of problems they come where they appear so the most famous one is what is known as sometimes people call it the serans problem or it is also sometimes call it the torsion problem because they appear in also solid mechanics and also maybe if you look at only in the simplest problem so where u of x is of course in general so for probabilist in this equation u of x is the first exit time of the Brownian motion in a domain omega so in this situation so u is proportional to the flow velocity of a viscous incompressible fluid moving in a parallel tip this cross section that is named after omega and the fact that omega is special is that whenever you have the omega you are saying that you want to find the domain in such a way that the tangential stress of the fluid is constant along the pipe so as it was proved in 1971 that if of course omega is bounded and sufficiently smooth then the only domain in which such type of fluids can have this property must be a ball so there is another applications or maybe another problem that is very well studied also is the case where you have that is 0 so in this case you have an harmonic function but of course in that you must ask also the domain to be unbounded otherwise you don't have anything and in this case so u has this very beautiful shape so I am saying here r is the normalized Newton potential because you need to normalize if the boundary of omega is unbounded it is a single layer so electric potential so and this to by when you charge when we charge the boundary of the domain by my constant function I would say and here the fact that omega is special would be that omega is such that so the electric field so nabla u is uniformly proportional to the integral of the normal so the charge density is in equilibrium in a sense such type of domains also are called exceptional domains and have strong connections also is the theory of minimal surfaces we are going to see that later but there is also some specialty which is sometimes called rigidity is that if the boundary of omega is compact then so omega must be again so the boundary of omega must be a sphere so in general this problem one study of a sphere so this exterior of a ball so as was proved by Raichel which was solving a conjecture by Grober so the most studied case is so in general this directly problem so as I was saying so in some cases the problem is completely known due to an interesting argument of coming from geometry that was applied by Saren which is the moving plane argument in order to show that whenever you have such problems and like f is beautifully enough omega is bounded and smooth then the domain must be a ball and Weinberger also well obtained also another interesting way to prove this using only base on the maximum principle and using the p function so kind of p function so it looks like when you look at the p function it looks so the pressure so when you want to transform this problem to solve the solution of the earlier equation so there was also a very big people also who studied this problem so which is in 1997 when omega is unbounded and for certain types of domain so they were able also to classify this type of domain so using this argument that is also known as the sliding methods but so the sliding method is not very far away from the moving plane method but it is very convenient when you work also with unbounded domains and also they made some interesting conjecture that maybe so when omega is unbounded so the bound of omega is connected so you must so they propose that you know you must have specific shapes of the domain which was disproved by Sigbaldi who was able also to construct some undeloid surfaces so when f is lambda u so there are also several works that comes just later on which are based on bifurcation argument so which is due to Ross, Ruiz and Sigbaldi and other collaborators and some results that is obtained also by Ruiz so Ruiz was usually here in ICTP and the time that he was doing his post-doc here in CISA so he also proved that in fact the positivity of you is important in the science result and it's very strange that this result was not known since 1971 in fact I really didn't know that and so this is a recent paper so in the case where f equals 0 so the argument so the electrostatic problem that I was talking about in this case so there is this work of Helen, Auschwitz and Packard so who were able in two dimensions also to prove that in fact there are domains which looks the boundary of domains looks like catamines so in which you can have a solution to these problems so this was very interesting so they were interested in more geometric problems and so I think in 2014 made a certain correspondence between the solutions the boundary of the solution to this problem in two dimensions and also the theory of minimal surfaces so after that so there are generalizations so Liu, Wang and Wei also obtained a dimensional result of this result of Helen, Auschwitz and Packard and with my collaborators also we were able to find to prove also the existence of completely also new exceptional domains we call it in a sense so we are talking about so since I was talking about this what I call the name this Dirichlet problem so the most there is the most probably difficult I would say which still leave many open problems is in fact the Neumann problem and so let me just start so with this long standing conjecture so which is named after Schieffer so one can see this conjecture so in this in this kind of book of Yao so the problem 80 so which states that so any time you have the Neumann function so for which the function is constant on the boundary so then omega has to be a ball the very simple conjecture but which is really open since many many years and so there are many people working on this but there are very very very few results direction so this problem is also linked with another very open problem which is called so the Pompey problem so which states that so when you have the integral of a function so which vanishes in a domain omega which is bounded and connected after changing the domains by region motion then the function must be zero so this is of course a very strange thing that these two problems are linked but in fact they are linked in a sense whenever so you have a domain omega so so with this property so which satisfies the Pompey problem so the conjecture is false and this verse and this is a very interesting proof probably that is based probably I would say on complex analysis so in a sense there is a characterization of functions of domains that satisfy the Pompey problem by looking at so the zero of the Fourier transform of the characteristic function of the domain and once you have that then you can link this property also with the Fourier transform of the shift of course the shift of the solution to this normal problem so you plus one will solve this and then you can take the Fourier transform of this equation it is it has a strong link between this and the Fourier transform of the zeros of the Fourier transform of the function in fact the proof is this type of ideas so there are a few results I would say of course there are many results in the Pompey problem many people are working on this but the problem is still open and also the shifter contract is still open but there are partial results and those so the one that I probably would say would be probably the result of Bernstein so in 1980 which states that if you have TORM so for fixed domain omega so you have infinitely many eigenvalues and infinitely many eigenfunctions then the conjecture is a TORM so there is also a result of Deng in 2012 so which extend result of Avelis which Avelis in Italian it will be called Avelis right but in French it would be Avelis but I would say Avelis since I am in Italy so he also were able also to prove this conjecture so for some small values of the parameters and in Riemannian manifolds also there are some results by Bernstein and Young so I would say in not on the sphere but mainly on hyperbolic spaces also which generalize the result of Bernstein and in higher dimensions and there are also stability results so that if the domain satisfy this problem and is very close to a ball then in fact it is a ball so which means this result so we cannot expect a better version of the ball to show where to disprove the conjecture so there was a paper by Cheklova so who was really interested in these problems on different forms and he was able also to prove that in fact so when you have certain domains which homogenous boundary or as a parametric boundary then you can have of course solutions so in a sense for the simplicity so if you are on a sphere so if you take the spherical cap or an annulus so you can build solutions to this conjecture to this problem so in a sense you can have solutions which have this problem but it's not nothing to do with the conjecture but it's just example of domains which support this problem and so there was also is a walk of swarms so who started the problem on the two sphere so she first was no Cheklova's walk was mainly in dimension on the sphere but in sphere with dimension larger than three so and he also conjectured that only balls and annulus solve the shifter problem on the two sphere so in a sense it's likely so you can have only homogenous homogenous domains or maybe shift or maybe annulus domains maybe isoparametric domains would solve the shifter problems on Riemannian manifolds and in fact so there is this generalized shifter problem so on Riemannian manifold that if you would have a solution to the shifter problem is it true that the bound of omega is homogenous or isoparametric and so in a recent work so we in fact disprove this conjecture so this is just recent so in 2003 that in fact this is not true so which somehow disprove the shifter conjecture when you are on the flat cylinder and also when you are on the two sphere and in this case so we were able to prove that in fact the domain cannot be a geodesic ball and it cannot be also an annulus so the idea is that in fact so we use bifurcation arguments so which is so here quite different from what people used to do in the Dirichlet problem because in a sense so when we started with initial perturbation so we start with this annulus domain we want to pair to but however it was probably very difficult to do what people used to do so in a sense if you want to find to have new result probably you need new things new tools and so we transformed the problem in such a way that you know the perturbation of the boundary stays in the unknown function u because you have two unknowns you have a perturbation of the boundary with h and you have a solution that you want to find so you have to find a domain and a solution so once we did that in fact we had the serious trouble because we had the problem of loss of regularity and of course we tried to solve this problem for like six months we tried to use you know this Nash-Mozier implicit function theorem but in fact we luckily found a very very key very interesting argument that was able that allowed us to solve the problem so in the case of the two spheres so we had an argument that we developed on the two-cylinder in order also to develop the results on the two spheres so in a sense here so I put many bombs but the idea is that in fact when you have this type of result in the cylinder so which is so you have something which is a band which is periodic and so if you wanted to put it on a sphere so you want to squeeze a little bit so that you develop many many bombs and in this case you can squeeze on this other loss on this sphere so this is this is the recent result that I wanted to present so yeah so this is my last slide so I hope I didn't take too much time about this thank you very much I just want to ask the audience if there are any questions or comments I would like to congratulate Professor Mustafa Falco his nice talk and since he invited me to talk a little bit about shape optimization problem and I am very glad to see that this is a very great result the last one you can find it in the archive and I would like to ask you some question because these two types of problem are I am thinking right now about these two types of conjecture Pompeo and Chiefer problem here you have there is a mu a positive mu such that you disprove the is it possible to do some characterization related to the eigenvalues problem here and for the other question is when you change instead of taking the gradient the absolute the norm of gradient equals zero you take a constant and you change for the boundary condition the drichler one you put zero and you authorize that mu may be an eigenvalue what happened is it possible to go ahead with your outstanding idea you have developed in the last paper yes so you mean if I change the the drichler then I put you is zero and then the gradient of you is constant yes in fact we could but there is a first result I think a problem you know of sick Baldy in which maybe he is asking you to be positive and so my student in us so he went to the presentation in the experts of this in Granada and they was asking him about this generalization so this you can change the shine and things like that and we already we were already able to do it so we already write the draft but we didn't find it too much now to publish it because for now because we already have this the which correspond to the Schiffer concept yes thank you Mustafa for the nice presentation and congratulations for the price which is really honoring all of us from Senegal so my question is so I see that you use a lot like a metric you know like you talk about does it click complete and stuff like this so are you just using a Euclidean metric or would you be able to generalize some result or is it interesting to talk about different Riemannian metric for these type of problems so here we just consider for example for the two sphere is the metric the Euclidean metric the metric on the sphere I didn't write maybe the G which says that I'm saying maybe is it interesting to put like a different Riemannian metric on the space and study so we need also a lot of symmetries so in a sense yes but probably you can make you can maybe perturb maybe this metric so you may have this result probably for a generic metric I would say a generic sensor but for the cylinder it is really for the flat metric yes thank you thank you very much Mustafa so happy for your last paper I was wondering if the assumption n equals 2 is a technical assumption in your opinion or is something more no in fact we could generalize it into any dimension it's just we wanted to combine the two results because we first wrote the paper on the cylinder and later we met this conjecture of we could do it quickly and then we added it but in higher dimension you can do just that you need to ask axially symmetry in certain directions in order to resolve the problem it's very possible thank you any other question I think also not strictly mathematical after all the price is also to point to a role model to make a sense of if our students want to ask have some curiosities for ok well so if not I think I speak here but stay here with me ok so we came to the end of this afternoon and I'd like just to add my personal congratulations I mean I've been discussing mathematics with Professor Fal a few times I managed to visit Ames Senegal once and the hope is that this is again as often happens with prices that it's also a boost for to do more and we are here also to discuss how and when and I really want to thank all the authorities for being with us I want to thank DST I want to thank the International Mathematical Union with Professor Tilma and his excellency the ambassador of Popsenegal in Italy and the ambassador of India in Italy for having been with us today thanks a lot I mean the party continues on the terrace and I hope I mean people will be curious to get a bit more personally outside so thanks to everybody and see you next year applause