We are live now, I don't still see it, not now, because it's ready to start, because the coloquium starts with the slide, yes, and after you will see, okay, tell me, tell me when I can start. Good afternoon, I'm Claudia Rizzo and has a head of the Mathematics section at ICTP, it is my great pleasure to welcome you all at this Mathematical Fest for the Ramanujan Prize Award. Now the Ramanujan Prize has been established in 2005 and since 2014 is the output of a beautiful collaboration between the Department of Science and Technology of the Government of India, the International Mathematical Union and ICTP. So now we are very happy today that we have with us the highest representatives from these institutions. So let me introduce to you immediately Professor Ashutosh Sharma, Secretary of the Department of Science and Technology of the Government of India and Professor Carlos Koenig from President of the International Mathematical Union. We are also very honored to have today three distinguished ambassadors. So your Her Excellency, daughter Nina Malhotra, Ambassador of India to Italy, His Excellency Mr. Vishal Sharma, Ambassador of India to UNESCO and His Excellency Mr. Santiago Murao, Ambassador of Brazil to UNESCO. We are really very grateful for their presence and it's an honor. So now let me just introduce all the people you will see in this panel. So given that we do not do like others' institution, we don't try to keep the secret of who the winner is. So let me welcome immediately and my partner, hello Karolini. Claudio you should repeat, I think you were frozen for a while. Difficulties with the song? Yes, when frozen. And who is speaking now? It is Claudio Arezzo still, but he got frozen. If you know what he was going to say, perhaps you can come back. Okay, maybe. I don't know what he was talking about. He's back, okay. Is he back? Claudio are you back? Well, I'm sure he was. Thank you. Okay, maybe. He was congratulating Dr. Carolina Malhotra and indeed Haiti, congratulations from all of us. It is such a compelling gathering today here with several ambassadors, one mathematician, two mathematicians, including you. So it is indeed a very, very exceptional extraordinary moment. And all of us are so happy that mathematics is getting the highlight that it truly deserves. And now Claudio is going to come back. Okay. If Claudio is not back, maybe I can take over here. So first of all, I would like to also say that a number of prize winners of the Ramanujan prize. Dr. Araho, if you want to speak, you need to unmute yourself. No. Can you hear me? Hello. Can you hear me? Are they? Can you hear me? Hello. Are they able to hear me? Do you see? You need to come to Euclid. Can you? Hello. First of all, is Lothar, are you on online? No. Your regular three-dimensional space. What happened?  And perhaps, I know most of you must have read about it in the... They cannot hear me. Much earlier. But for the benefit of those who have not, I would like to thank you. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. Thanks a lot to everyone who have not. Let me put it forth in a very simple language because of my ignorance. It's like this. The sums of the angles of a triangle in Euclidean geometry cannot exceed 180 degrees. But the sums of the angles of a triangle in non-euclid geometry may exceed 180 degrees. But unable to see... But this is the beauty of mathematics. And this is the beauty of geometry. But thankfully, this geometry is non-differential. At least there is some constant in it. Once we go into higher spaces, we come into what is called differential, seen the moving installer, you know, this magic of the magic of the universe is put forth in the mathematical language. Okay. Hello. I think Claudio is here. Can you hear me or not? Yes. Yes. Okay. Very good. So I think Claudio has joined me in the auditorium. There was some problem in the network in his office. So please, Claudio, please take over. Yes. So, sorry. I was congratulating Professor Haral Karolina Haralco for getting the prize of 2020. And I was also introducing and thanking Professor Kautcher Birker from Cambridge University and 2018 Fields Medalist for joining us today and giving a special lecture later this afternoon. Okay. So let me give you a quick introduction to the, very quick introduction to the prize. So the prizes are awarded annually to researchers from a developing country who is less than 45 years of age and who has conducted outstanding research in a developing country. So we are also happy to have with us, even though you won't see them on the panel, but many of the past winners and actually a brief video with the history of the prize will be shown during the break. Now the selection for the winner, later also the chairman of the selection committee will take the floor, but let me just say that the winner is selected by a joint committee by ICTP IMU and DST. And we are very grateful to DST also for providing the prize money coming with the prize. Now let me also take this opportunity because I don't know exactly how many, but I'm sure I'm speaking to a very large audience that actually today we opened a new call for the next year's winner for 2021. So nominations are welcome and actually important for us. Okay. So I think I'm done. I want to be quick and I'm very, very happy and honored to pass the floor to Professor Ashutosh Sharma, Secretary of the Department of Science and Technology of the Government of India. And thanking again DST for this collaboration and for funding the prize. Professor Sharma, I pass you the floor. Thank you very much. Indeed, good afternoon to all who have joined us. There are some who have joined us from India, but I've already told them good afternoon many times. So I think that in Brazil it is morning now or yeah, in Europe, Europe is still afternoon. So great. So let's congratulate Dr. Carolina Arajo and she is from the Institute for Pure and Applied Mathematics in Rio, Brazil, in recognition of her work and contribution in the field of algebraic geometry, in particular, in bi-rational geometry and the theory of external raise. This is so fantastic, it's all Greek to me, but I'm sure that this is something very great. And that's why we, you know, mathematicians recognize you for your work. Dr. Carolina is also the first non-Indian woman mathematician to receive this prize and second woman mathematician after Sujata Ramdurai in 2006 to receive this mathematics prize. It's a great honor that award has been instituted in the name of Ramanujan, the great Indian mathematician who was indeed he did all his mathematics when he was young. So it also makes a correlation with this award. Of course, he's a mathematician who dreamt mathematics all the time, even while awake and certainly a very singular personality in the entire history of mathematics.  And I totally, totally urge you to get hold of that book, The Man Who Knew Infinity, as well as a movie by that name is so truly fantastic, inspiring, totally inspiring. So your contributions will enhance the legacy of Ramanujan, we are so pleased about it. And I note that the award, of course, is bestowed a woman mathematician. And this would actually encourage bring in role model for all women who want to excel in mathematics. I know that culturally, I know in India and elsewhere, culturally, women are not greatly encouraged to do mathematics, to do theoretical sciences, to do engineering. And what we need is the leadership, the role models to make that possible. In fact, there's a new program is called Vigyan Jyoti, Vigyan is science Jyoti is the, in fact, the ray, the external ray, and that program Vigyan Jyoti would actually select top 50,000 girls from Indian schools and make it totally possible for them by cultural transformation by training to get into top schools of engineering and mathematics. So this is going to make you very proud. And Dr. Araho, if you see us again, after five years, I would present to you at least 35 women, percent, 35 percent women in all the top engineering and mathematics schools in India. So that's going to, and I want your participation in the success of this program that we totally invite you to be in India. I mean, right now remotely, and then once we get out of COVID-19 times to travel in India to talk to our mathematicians, especially talk to women mathematicians and inspire them by your own example. So I would conclude by saying that this recognition will suddenly would motivate you, Dr. Araho, but also further expand your research horizons and will inspire the young mathematicians in the entire world, and especially what we call the developing world. I don't particularly like this world, but okay, it basically means some part of the world and to undertake research in mathematics will be so inspirational, so motivational for them. I wish you all the very best for your future activities in mathematics, as well as bringing more people in this fold. Of course, I understand a little bit personal note and I kind of mathematician myself, what is called applied mathematics. Of course, I know that there's a packing order in mathematics like everywhere else. So you have a strength theory, pure mathematics, applied mathematics, but I must say applied mathematics is also really beautiful. At the time of this COVID-19, we set up a group of top mathematicians in the country to make a very good model for the spread of pandemic and they work really very hard day and night. All these kind of people, these fellows of Royal Society and clay awardees and stuff like that. And even though they are pure mathematicians, they took it on them to make a very good model of pandemic spread and in fact it was very useful for planning. So I have to tell you that if we can inspire people to say how can we, when the situation demands it, can we bring mathematics for the service of people and be such a compelling example because a lot of planning could be done based on that model and its predictions. So thank you very much and all the best going ahead in your career and in your mathematics. Thank you. Thank you very much. Thank you very much. I am now happy to welcome Professor Carlos König, President of the International Mathematical Union, which is the largest association of mathematicians around the world and with whom we are very happy to have various type of collaborations on a number of programs. So, Benvenuto Carlos and I leave you the floor. To be here today representing the International Mathematical Union on this very happy occasion. The Ramanujan Prize celebrates outstanding young mathematicians in developing countries and for the International Mathematical Union supporting mathematics in the developing world is one of its core missions. On this occasion, I wish to congratulate Carolina Araujo from IMPA in Rio de Janeiro on receiving the Ramanujan Prize. This prize was awarded for her deep contributions to algebraic geometry, in particular for her works in bi-rational geometry, algebraic foliations, and final varieties. Carolina is also honored for the key role that she plays promoting women in mathematics all over the world. I'm very proud to say that Carolina is the Vice Chair of CWM, the IMU's Committee for Women in Mathematics. Carolina was also the lead organizer of WM Squared, the first world meeting for women in mathematics which was held in Rio de Janeiro in 2018 just before the International Congress of Mathematicians. Her tireless work in reducing the gender gap in mathematics all over the world is deeply appreciated by all of us. Thank you Carolina and congratulations once again. Thank you. Thank you very much and now as the final and third party involved in the in the in the in the prize let me pass the word to my boss, the ICTP director, Professor Atish Dabh Kokalar for few words. Okay thank you thank you all for coming to this virtually to this important ceremony. I just wanted to say a few words and I'm also very glad that the three ambassadors could come in addition to the Secretary of DST and the President of the IMU. I just want to say that ICTP is one of those rare institutions which really has played a great role in bringing the world together through science through excellence in science. We believe in science without borders and we have really played a very important role in overcoming barriers of gender, geography, ethnicity and economics. We have something like 6,000 visitors every year scientific visitors okay before COVID time this was the case and we promote with in collaborations with organizations like DST and IMU prizes like Ramanujan one of the prestigious prizes or the Dirac medal and so on. I'm also very happy that the three ambassadors are here the Ambassador of India to Italy and the Ambassadors of Brazil and India to UNESCO because ICTP is actually a category one UNESCO Institute so in that way we contribute to the the mission of UNESCO of promoting science in the developing world and we are governed by a tripartite agreement with the Atomic Energy Agency the Italian government and UNESCO and as a special guest today I'm very happy that many of the past Ramanujan prize winners were contacted and I'm very glad that they were able they are able to attend and also Professor Birkar is a fields medalist he will be the special he will give a special colloquium after this. I hope that I will next year I think next year I will be able to we will be able to invite you and see you in person here in the beautiful city of Trieste for the next Ramanujan prize. Ramanujan of course is a very inspirational figure for many of us coming from India and his very deep work in number theory has had such wide ranging applications including my own work for example in the theory of quantum black holes somehow Ramanujan makes his appearance surprisingly so it's a it's a it's wonderful that actually Dr Carolina Rauho is the winner she has a long association with ICTP and I'm very happy to see a distinguished mathematician such as herself from Brazil a woman mathematician getting this prize so congratulations Carolina and I give the floor to so these were the three institutions handling the the the prize but now we are very honored to have the three distinguished guests and I'm very honored to pass the floor to her excellency Dr Nina Mahlotra ambassador of India to Italy distinguished guests ladies and gentlemen let me just begin by saying that I'm very honored and delighted to join you all for today's Ramanujan prize award ceremony the award which is instituted to honor and encourage high level of research a field of mathematical science also reflects long-standing association between ICTP Italy and India I would like to take this opportunity to congratulate this year's winner professor Carolina Rauho of Brazil for her outstanding work in algebraic geometry I'm confident that her recognition would encourage young people to strive for excellence in mathematical research as an Indian and as a woman I feel proud in stating that this award not only showcases India's historic contribution to the field of mathematics but also highlights and perpetuates the legacy of Srinivas Ramanujan I'm very glad that a woman has once again won this award the wealth of ideas as Srinivas Ramanujan created a century ago are being researched upon and finding applications in diverse contexts to acknowledge the indelible mark this India's great scientist and mathematician left on mathematics and to imbibe his vision for the subject the government of India celebrates December 22nd his birth anniversary every year as national mathematics day friends research development and innovation capabilities are fundamental for social and economic progress in a developing country for more than 50 years ICTP has been contributing immensely to advance scientific expertise in the developing world we have very strong association with ICTP with a large number of scientists and mathematicians from India visiting ICTP and participating in in its programs developing countries have abundant talent and large pool of researchers who could be tomorrow's scientists, technocrats and innovators the need is to nurture acknowledge and support our scientific community and integrating their expertise into developmental activities I'm very happy to note that ICTP has been supporting growth of excellence in the developing world and has emerged as a crucial link between north and south we commend the valuable contributions of this institute once again I extend my best wishes for this annual ceremony unfortunately I couldn't join today in person but due to COVID situation but I definitely hope to be physically present next year to participate in this award ceremony and I look forward to our mutually enriching association in future thank you thank you very much and now it is my honor to pass the floor to his excellency Mr Santiago Murao for ambassador of Brazil to UNESCO thank you thank you very much my colleagues ambassador Vishal ambassador Marotra and Dr. Tenning Dr. Sharma and of course Carolina Araujo it is my pleasure for being here today this uh Ramanujan prize ceremony first of all because today we celebrate science and this pandemic has shown us how important science is for our societies no contemporary challenge from COVID-19 pandemic to global warming can be solved without sound science secondly because today we award a young woman mathematician and brazil attaches great importance to gender equality in science Carolina Araujo Araujo I have the honor to congratulate you for the prize for your important work on algebraic geometry and also for your leadership as vice president of the committee for women in mathematics at the international mathematics union Brazil actively supports gender equality in science from programs such as innovative women and women and science program and in partnership with UN women brazil supports women in science and innovation as a result of those efforts brazil is among the three top countries in terms of gender equality in science and innovation according to 2020 Elsevier report thirdly brazil is honored to see the work of a researcher from brazilian institute for pure and applying mathematics recognize internationally today this institute is connected to the ministry of science technology innovation and plays an important role in promoting research in mathematics training young scientists and disseminated mathematical knowledge for all these reasons it is a pleasure for me to be here representing brazil once again I congratulate Dr Carolina Araujo for receiving this this prize and I also take this opportunity to warmly thank UNESCO the ICTP the international mathematical union and the government of India for supporting this prize and the promotion of science in developing countries thank you thank you very much and I now pass the floor to his excellency Mr Ishaal Sharma ambassador of India to UNESCO thank you good afternoon ladies and gentlemen good afternoon excellency Malotra excellency morale and doctors Carlos professor Sharma and fellow mathematicians and Dr Carolina I'm honored to participate in this award ceremony instituted in the name of Ramanujan who's a great mathematician a mathematician does not belong to a particular country Carolina may be a brazilian but because she's a mathematician she belongs to entire humanity she belongs she does not even belong to humanity she belongs to the universe because mathematics is the language of the universe the idea of binary numbers zero and one are essentially from an indian philosophical perspective jad and chetan alive and dead or the states of to be or not to be everything in this universe is in a way alive or it is inactive without awareness or with awareness if there is one unifying factor not just for humans but for the universe itself then that factor is mathematics such a unity was evident in the interactions we read between Mr. Hardy and Srinivas Ramanujan Ramanujan and Hardy came from different countries different cultures had different outlooks but their passion for mathematics united them Ramanujan held that an equation has a meaning if it is expressed as a thought of God because I'm from India for me physics and philosophy mathematics and philosophy go hand in hand together in fact my favorite subject physics I will say is a mathematical interpretation of God's desires perhaps even God is a mathematician after all when you see the harmony in the universe Ramanujan in his letter wrote to Hardy I am poor but if you are convinced that there is anything of value in my writing I would like to have my theorem published Ramanujan had introduced himself as a clerk in the accounts department who worked at the Negro annual salary of 20 pounds but in Hardy's words he was a mathematician of the highest quality a man of altogether exceptional originality and power in fact Ramanujan bettered the value of pi even more accurate than the Greeks Ramanujan gave a ratio that is even better than 22 divided by 7 it underlines the rise in the etra of this award for young researchers a young genius from Tamil Nadu in pre-independence India would have been so easily overlooked by the world but thank god that humanity did not overlook Ramanujan the man who understood zero as well as infinity and I'm so happy that you have kept your this memory of his alive and today the world is not overlooking great mathematicians but it is honoring them I applaud the former laureates of this prestigious prize who are among the attendees and I once again congratulate Dr Carolina for her exceptional talent and for getting this prize thank you so much well thank you very much for all all the beautiful words I heard I mean and now I I we start presenting the the winner so as we said now already many times we are speaking about professor Carolina Arrajo from the Institute of Mathematica Purae Aplicata in Sao Paulo Brazil which by way it's an institution with whom we had long long-term connections and we really want to continue collaborating in the in the future so now I ask my colleague professor Lothar Goetje hoping that from his office things are working a bit better otherwise he has to run the three floors in less than one minute as I did anyway professor Goetje served as chairman of the selection committee and he can introduce to us the professor Haravu's work thank you thank you very much so okay so I've so thank you very much so on the also on behalf of the other members of the selection committee which were Alicia Dickenstein, Filibert Nang, Kapil Paranjapil and Van Buu it's a pleasure to introduce professor Arrajo so I've written here some of her data of her CV so she did her bachelor in 1998 at the Pontifica and Universidade do Rio de Janeiro then in 2004 she did her PhD with Jano Scola at Princeton University working on the variety of tensions to rational curves since 2004 she is at first she was a postdoctoral fellow there and then she became researcher and has risen to the ranks so now she's a full researcher so she has had many short and longer time research visits in many places of the world in particular she was for instance in the MSRI the Mitter Glifler Institute in Grenoble and since 2015 she is also an ICP Simons associate and has been here many times so she as was mentioned she is a very active both an organization of conferences and that I can also very active for the benefit of women in mathematics so as was said she is since 2015 a member of the committee of women in mathematics of the international mathematical union and since 2019 its vice chair and she was a member of the organizing committee of the International Congress of Mathematicians in 2018 in Rio de Janeiro and she also as Professor Kienic mentioned was the lead organizer of the world meeting for mathematics for women in mathematics which was a satellite event to this ICM 2018 she also has received a number of awards I just mentioned the Loyal Award for Women in Science Brazil and she was also an invited speaker at the International Congress of Mathematicians 2018 talking about positivity and algebraic intercapidity of holomorphic for the issues so I can briefly mention a few things about her research so as mentioned her field is algebraic geometry in particular by rational geometry and she applies the theory of extreme race more theory to to the study of algebraic varieties in particular to classification for instance of found varieties and she also has a long ongoing work on foliations of varieties so can maybe say two words more so so algebraic geometry is a study of solution spaces to holonomic equations say in projective space these are called varieties and by rational classification means that you classify them in a slightly closer way and and this is the by rational classification of higher dimensional varieties is one of the main directions of modern algebraic geometry where a lot of important work has been done in recent years and possibly the main tool in this work is the theory of extreme race so final varieties which are the special what she has done more work on in this context and in some sense the simplest class of algebraic varieties which in this case just means this are the only ones where there's any hope of achieving a classification and so Carolina Ujo has made important contributions to this but an example which maybe strikes in particular is relatively early paper on the characterization of projective spaces and hyper quadrics so and so these are characterized in terms of the properties of the tension bundle of these varieties and this can be seen as a very far reaching generalization of the famous result of Mori who proved the hard shunt conjecture that the only varieties with ample tension bundle are projective spaces so as Simon's associate professor Araujo has been here many times and has given quite a number of talks here which I always like very much he's a wonderful speaker and he's very clear talks and is able to make difficult subjects easy and I remember particularly well however her lecture on final varieties at the school in Simat in Mexico in 2017 where she actually was not present but did it remotely like she's going to do here so I expect she will do her as wonderful a job of it now as she did then so I'm looking very much forward to her lecture now thank you very much thank you very much so now we can you can finally present like the actual price to Carolina what she will get in some way and then so I don't know if exactly what you are seeing now but so the ICTP director will now present the price which is constituted by of course a prize money as I said supported by DST but also a nice statue of Ramanujan which and a nice certificate which is of course not as big as that one but on a smaller scale is the one that you see next to the to the ICTP director okay so I'm very happy to present this to professor Carolina Araujo congratulations Carolina okay so congratulations and I hope to see you here in DST sometime soon and you can just reach out like that and grab the the trophy we can do it virtually yes very soon we'll have augmented reality thank you congratulations okay well thank you everybody in some sense now we entered another part of of of today's program in the sense that finally we hear from Carolina herself about algebraic geometry and in particular she will talk about algebraic varieties with positive tangent bundles and I just add my personal and warmest congratulations to Carolina for this award Carolina you are free to speak okay thank you very much Claudia I would like to start by thanking um thank you all of you for this very supportive words I am uh it is my great honor to receive the Ramanujan prize in 2020 and I would like to thank ICTP the Department of Science and Technology of the Government of India and INU for funding this remarkable prize that recognizes and celebrates the work of mathematicians in developing country and also highlights the importance of basic science at challenging times like this science and fact-based policies are indispensable the Ramanujan's life and his striking mathematics are proof that mathematical talent is a human quality it transcends boundaries it has neither race nor gender however differently than Ramanujan I was not self-taught I have benefited from long-term national programs of investment in science in Brazil especially through SEMPK, CAPIS and FAPERGE my initiation to research fellowship brought me into science and I cannot stress enough the importance of these programs in for president science I'm also very glad that my efforts to promote women in mathematics have been acknowledged I'm very proud to serve the Committee for Women in Mathematics in mathematics from the IMU and this prize is a great incentive and adds to the certainty that this struggle for diversity and inclusion in science is the right one so before I move to mathematics I would like to to thank some people so first of all I would like to thank my family for the support that they always gave me I would like to thank professors from whom I learned so much mathematics in particular my undergraduate advisor Professor Ricardo Saerri and my PhD advisor Professor Janusz Polar I really I really honed to them a lot of what I what I know today I would also like to thank my colleagues at INPA for the constant support and and also the women mathematicians that have been with me in this journey they are really important for me to really feel that I am part of this community in in all senses so I would love to be at ICTP today I have been to ICTP many times through the Simon's associate program and they have been fundamental for my research so I really appreciate also this this partnership but well this is what we have today so we have to be here online and on the other hand I see that there are many friends that are joining this this ceremony I see many of my students many of my colleagues and and and this is actually something that I really appreciate too so what now I will share my screen and and my goal with this lecture is to explain to to many to my colleagues the mathematics that I do or at least some of it and and how I view it so most of my friends and colleagues that are that are here are not algebraic geometry so I will really try to to at least give them some taste of what of my what my research is so now here this is this is the the title of my talk algebraic varieties with positive tangent models so let me first okay so so let me just start by the general setup so we have so we start with x some complex projected manifold and and we look at its tangent bundle so first of all what is this so if you if you look at your your if you given your variety x for every point it has a tangent space attached to it and the tangent bundle is just a way to assemble all these tangent spaces together in something that locally looks like a product so it looks like it's just a vibration like this and and there are many ways of defining what it means to be what is positivity in the tangent bundle and very roughly speaking positivity can be measured in terms of how many sections the vector bundle has how many homomorphic sections so let us look at this picture just to get some intuition so what does a section of the vector bundle mean in this case homomorphic section well so to each point of x we attach a point in its corresponding tangent space so this is a section of this vector bundle in this case is nothing but a vector field on the variety x so understanding if the tangent bundle has many sections can be can can have this geometric interpretation so let us look at an example that is probably known to to most of you when x is a smooth projective curve or in other words a Riemann surface so in this case topology tells us everything so there is this very well known trichotomy between these three classes of Riemann surfaces that is given in terms of the genus and if you think about vector fields homomorphic vector fields are in this case just this just smooth vector fields continuous vector fields then the the well-known Harry Ball theorem says that the any vector field on the sphere must have zeros in fact it always has two zeros if counted with multiplicity on the other hand one can easily construct a non-vanishing vector field on on the torus and for genus greater equal than two then a compact Riemann surface of genus g cannot admit any any global vector field so this is this is sections of the tangent bundle for for this classes of for smooth projective curves and more generally if you have any meromorphic vector field on your smooth projective curve or Riemann surface then you can count a number of zeros and poles taking into account multiplicity and this gives a very meaningful number so if for any meromorphic vector field if you add the number of zeros counted with multiplicity and then subtract the number of poles you always get two minus two g so this is really controlled by the by the topology also in this case this has deep connections with with the curvature so the let me also remind you that in this case the average Gaussian curvature of the Riemann surface x again is it's two pi times this number two minus two g so this trichotomy between Riemann surfaces can be phrased in terms of positive curvature zero curvature and and negative curvature so this is the intuition that we we want to take to higher dimensions and in higher dimensions there are many ways and different ways of measuring positivity in the tangent bundle so let me just state for instance Mori's theorem from 1979 which solves what was known as the Frankel's conjecture that says that the complex projective space cpn is the unique compact scalar manifold with positive sectional curvature so this is measuring positivity in terms of of curvature in fact this was a this was a consequence of a priori stronger result which was already mentioned by by Lothar that which is that the solution to the hard conjecture so in in algebraic geometry we have this notion of positivity which is called ampleness of a vector bundle and Mori proved that that the complex projective space was the unique protective manifold having ample tangent bundle so so i would like to explain this notion of ampleness and i will not explain everything but i would like at least to explain what is what it means for for a line bundle to be ample okay so let's instead of looking at vector bundles in more generally let us look at line bundles so these are bundles by by lines and if you um so if you have again a complex projective manifold and a complex line bundle holomorphic line bundle on x then we can define a integer which is called the degree of this line bundle and this is search that for every meromorphic section of this line bundle again if you add the number of zeros and subtract the number of poles of this meromorphic section then this number is constant and this is what is called the degree of the line bundle and so in for um for services we can define the sorry for line bundles we can define the notion of ampleness by saying that line bundle is ample if this degree is positive so um let me just remark that in the case of the complex projective line so this is the Riemann sphere then the degree completely determines the line bundle so this is the notation that we use for the OD for the isomorphism class of the line bundles that has the property that any meromorphic section has um number of zeros mind their number of poles equals to D so in in for for higher genus this is no longer the case the degree is just the first discrete invariant of the line bundle and then and then there but there may be many line bundles with the same degree that are not isomorphic so this is a very important feature of the projective line and and we are going to come back to that later on okay so for instance when we look at the tension bundle of the projective line this is as we saw this is just isomorphic to O2 so any vector field has two zeros if you count them with multiplicity okay so now in in in generally in a more general situation when we want to define line bundles now for projective manifolds of higher dimensions so we fix a complex projective manifold and we fix a complex line bundle on x um then from what we have just seen if you fix a smooth projective curve on x then you can supplesate a number by the degree by restricting first the line bundle to the curve and now we know how to compute its degree so for every curve we get a a number and this is usually denoted by l.c the intersection number of l and c okay and now i won't know i will not uh so in in general if you if you give me a projective variety and a line bundle there's no reason why these intersection numbers are always have the same sign in general for some curves this will be positive some others this will be negative others zero um and the notion of ampleness asks in a very strong sense that this number is always positive so i will not define to you what ampleness means but let me just give you this important consequence of being ample so if a line bundle is ample then its intersection number with any curve is positive the converse is not true so ampleness is a much finer a notion so in fact if you if you understand all the ample line bundles in in a projective variety then you understand all the possible baggings of this variety in projective spaces so this is more than this numerical condition however this this positivity can be if you have if you if you're not familiar with algebraic geometry then you can for this talk you can take this as a definition and in fact uh there are uh there there are numerical characterizations of ampleness so um so climate's ampleness criterion tells us that a slightly stronger condition of positivity of this type is enough to actually to actually recover ampleness so for this talk you can think of this as a as a definition and let's look at the what and so what what about the tangent of the the tangent bundle of that so the tangent bundle is the vector bundle in general at rank the rank is equal to the dimension so in general this is not a a line bundle but we can cook up line bundle from it so we can take the determinant of this vector bundle that is the the top wedge power so this this determinant now is a line bundle on x and so we can consider the intersection with curves and so this this line bundle that we construct this way is usually denoted by minus kx and the jargon is that this is the first term class of the tangent bundle so this is the line bundle a line bundle associated to that we construct from the tangent bundle it's called the anti-canonical line bundle on x and we say that a variety is funnel if it's anti-canonical class is ample so we can think of this as this positive so as i said in general it is not true that the canon in general it's not true that the canonical class will have a definite sign but let me tell you that this this this sign is actually measuring something about the positivity of the of the variety so these this intersection number of the anti-canonical with the given curve is is measuring up to up to the average of reaching curvature of x along c so if we remember our classification of projective curves or Riemann spheres then they can be restated in this way in terms of the canonical class and so and the minimal model program which is a very important area in framework in developments in algebraic geometry its consequence is that every so the philosophy that it tries to prove and make precise is that every projective manifold is built up from varieties with anti-canonical class ample zero or or negative so this so this so from the point of view of the minimal model program it is very important to to study funnel manifolds fun of and also they they're they're these are they they include many of the very natural projective varieties that appear such as projective spaces with monions hyper surfaces of low degree and so on and let me just so this is as these three classes they have very different geometries sort of say so let me just focus on one very very relevant property of funnel manifolds so this is that funnel manifolds contain many rational curves these are rational curves are just embeddings of the Riemann sphere the cp1 so to make this precise the this famous theorem of Kampana-Colomio-Kamori says that funnel manifolds are rationally connected so through any two points of x we can trace there is a rational curve through these two points and on the other hand projective manifolds where the the anti-canonical class is negative or canonical class is non non-negative they do not contain any rational curve through a general point so this is a very special feature of this variety and that becomes an important tool in the study of funnel manifolds so there's this theory of rational curves on algebraic varieties that that that is one of the most important tools that i have been using in my research and so let me explain to you first why this theory is so suitable to understand funnel manifolds so there is this theorem by Birkhoff Crotentik that says that there that p1 has this very special feature that any vector bundle on cp1 on the projective line can be can be written as a sum of flying bundles so we have a locally every vector bundle can admit such splitting but in p1 this is always a global splitting and understanding this splitting tells us a lot about the variety so let me just give you a concrete example just to fix the idea so let's take x to be the the product of pitch p1 and let me take as a rational curve a line l in a fiber of the second projection so the thing we really restrict our tangent bundle to l then as i said it splits as a sum of line bundles and then we do start seeing some positivity in this splitting and this positivity in fact defines a a fact in this tangent bundle if i fix a point x in my variety this decomposition in fact defines a stratification of the tangent space of x into linear subspaces so for instance this O2 direction will correspond on x precisely to the to the tangent direction of the curve at at the line at the point x and this is because O2 is precisely the tangent bundle of p1 as we have already seen and the part in red the two-dimensional part will correspond to a tangent the tangent the tangent space to the plane to the pitch to containing the line and so and and they correspond also to the direction where we can deform the curve l keep in the point s fixed and if i move it away from this fiber then it will not it will no longer contain x so this so so there's some tools that i have developed have to do with understanding how this splitting varies when i vary a curve from from a family and this and understanding this infinitesimal variation sometimes it's enough to recover the whole the whole space x and and let me just give you an application of this so i return to the mori mori's theorem or a hard turn point chapter that that that shows the pn is the unique protective manifold with ample tangent bond so we have been looking at line bundles and one may ask whether one can ask something something something weaker or or in a sense the last also positive but just to check if you can contain at least the tension bundle contains at least a ample line bundle so this so let me fix notation these have a protective manifold we we consider a line bond an ample line bundle on it and then we'll prove that if this line bundle is contained in the tangent bundle then again this is this is the your the ambit variety must be cpn and and let me now discuss the theorem that that lot also mentioned which is the characterization of protective spaces and hyper quadrics which i proved with the stephan dwell and shander kovach in 2008 so we consider a generalization of these results so now the type of positivity that we assume is that some tensor power of the line bundle leaves inside some wedge power of the tangent bundle so this is much weaker than actually the previous assumptions and then in this case again we show that either x is isomorphic to the projective space or p is the dimension of x and x is a quadric hyper surface okay so now i have a few minutes left and so this was actually the the starting point for much risk a lot of research that investigates positivity of the tangent bundle and in a long long-term collaboration with stephan dwell we've been looking at what we called a funnel foliation so let me let me say a few words about holomorphic foliation so the theory of holomorphic foliation starts with the study of algebraic differential equations on the complex plane c2 so for instance one can look at these two differential algebraic differential equations and they can be easily solved but when you solve you notice that even though both equations are algebraic in the first case the solutions are algebraic they're just respond to the lines through the origin while the in the second case the solutions are transcendental so a fundamental question that we can address is when are the solutions of an algebraic differential equation in fact algebraic so now i would like to turn this problem into a problem of a geometric problem that has to do with with positivity and curvature so if we look at this the solutions in each of these cases the solutions of the differential equations then we see that they partition the plane into into this sub varieties which we call leaves and so if we for every point we look at the tangent space at this point then we get a line bundle contained in the tangent bundle of the plane and then you can extend it to the complex projective plane and then if you check which line bundle that is then in the first case this is the the correspondence of 01 so this is ample while in the second case this line bundle is just a trivial line bundle so what we see here is that is a at least in this simple example a correlation between algebraicity of the of the solutions and ampleness of this this line bundle that we construct inside of our tangent bundle okay so more generally a homomorphic filiation is a partition of my have a complex projective variety and a filiation on it is just a partition of this space in in sub varieties of smaller dimensions and well and locally they look like a product except maybe at some points like the ones that i mark there where you don't have it so those are called the singularities of the filiation and there you do not have the structure of the product so if for every point you consider the tangent space of the leaf through this point you construct a sub bundle of the tangent bundle of of the variety and the fundamental question now becomes so this tangent bundle is always algebraic this is a global objects on x however the leaves themselves may not be algebraic they may be transcendental and the fundamental question is when are the leaves of f algebraic and so the way that we still now in in order to answer or to shed some light into this the answers to these questions we consider we can define the anti-canonical class of the filiation so this is defined as i did before this is i take the the determinant of tx and this gives me a line bundle on x and and we define a funnel filiation for the filiation is funnel if it's anti-canonical class is ample just like we did when we defined funnel filiations and and and the and what our work has shown is that the algebraicity of the leaves are very much connected with positivity of the of the filiation so let me just state a state an example of a result that we proved so we can define an index that measures the positivity of this anti-canonical class of the funnel filiation so this is measuring positivity of the filiation and we also define an invariant integer that is that that measures how how algebraic the leaves are so this is what we call the algebraic rank and this is a just a type of theorem that we prove is that the algebraic rank that means us the much algebraic city that you have in the leaves is bounded from below by the positivity of the anti-canonical class so these are some of the ideas that i have been exploring in my research i thank you very much for your attention thank you very much carolina and finally thanks again thanks again and congratulations for also for this beautiful lecture i'm afraid i would never like to collect questions but let's see if if there are questions then my colleague Lothar from can read them up okay until now there are no questions just from here actually i can see many many many people congratulating and of course and okay so in that case carolina thanks again but the the the program is not over and we are going to ask you also as you know another another bit of work but now we take we take a break and we reconvene on this channel at the court a half past half past four so x 16 30 the time so in 20 minutes essentially in 18 minutes for the math colloquium by professor cultures birker okay so and we will show you a little video in the break about the history of the of the price itself so see you soon thank thank you to everybody and congratulations again to carolina well so here we are again and let's continue this celebration of the ramanujan price with extending to the broader area of algebraic geometry so it is my great pleasure to introduce professor culture birker from cambridge university and he's one of the leading experts in algebraic geometry and by rational geometry so let me mention in particular some of his mathematical achievements to start with so among his accomplishments certainly in collaboration with kashini hekon and mcternan he has set the long-standing problem of the final generation of canonical rings and in his studies of linear systems of phano varieties he has solved deep conjecture asked by shokurov ser and tian among others he has also solved the alexa bodies of bodies of conjecture actually passed into literatures like the bub conjecture on the boundedness of epsilon log terminal phano varieties and this is just to mention the most outstanding contribution he has given to this field so for his mathematical achievements he has been awarded the number of important prizes in which include he has been induced in the he's a fellow of the royal society he's he got the level room and prize he got the price of the foundation of science the mathematics the party he got the ams more prize and the 2018 fields medal in mathematics he is also an honorary doctorate of the salahudin university now this is as far as his mathematical contribution to algebraic geometry but let me say that birker's personal history deserves also to be recalled briefly here because it is my opinion this is a beautiful living inspiration for the whole ictp community and for others so he was born in the kurdistan province of iran and he was raised during the iran iraq war then after and the undergraduate studies at the university of tehran he has been a refugee in the uk and eventually entered nothing comes graduate school so i cannot think of a better mathematical proof of the fact that scientific curiosity and talent are indeed universal characteristic of human beings but they need also to be fed and nurtured also in the most complex situations and i think this is a beautiful message to ictp for our mission for for the future so let me just mention one technical thing everybody of course is welcome to ask a question but please write the question in the questions and answer section of your zoom channel and it will be read at the end and ask the directory to professor birker kautcher can you hear me yes i can perfect so now i leave you the floor and looking forward to your colloquium thank you thank you claudio thank you very much for the kind introduction first of all let me congratulate carolina on getting the ramanujan prize and secondly i would like to thank ictp for the invitation to participate in this event so i know a little bit about the background of ictp and i think they are on a great mission and i really admire the work is probably one of the most maybe the most accessible center of excellence for theoretical physics and mathematics in the world so it's a kind of place that people in the developing country especially people who have no support at all can go can visit spend some time or maybe even get a degree and i really admire this this mission um i have been to ictp only once actually and that was a long time ago when i was still a phd student but i'm happy to be here at least virtually today okay i will talk about algebraic geometry a little bit in broad terms and also its connection with a couple of different topics so my intention is not to go deep into any of this topic but just to be very broad and very general and just to give some example basically of how algebraic geometry is connected with some other fields and also just start with a very basic from scratch what algebraic geometry is at least for people who are not familiar with this field okay one way to characterize algebraic geometry is that is the study of solutions of systems of polynomial equations so that was already mentioned before today so in general in mathematics the kind of functions that you study in a subject almost determines the subject in a way and in algebraic geometry the kind of functions that we study are polynomial functions so essentially everything is constructed out of polynomial functions this has an algebraic side and also geometric side but everything pretty much involves polynomial equations and so you can construct a whole theory a whole branch of mathematics out of these polynomial equations and the geometric and algebraic structures associated to them the subject is of course very central to mathematics and also deeply related to many other parts of mathematics but also to outside mathematics for example mathematical physics computer science and so on in recent years actually it has found even more applications in down-to-earth subjects by the time in real-world kind of applications now the field the subject starts with a number system so you need to pick a number system before you actually start doing algebraic geometry in technical terms this means that choosing a field but if you don't know what is a field you don't have to worry about it because we will mostly work with some very familiar examples and this includes the field of rational numbers real numbers complex numbers and also sometimes finite fields so these are at least for today these are the most important examples of fields so the field is basically a set of some sort of numbers where you can add things together and do multiplication and so on we start with local geometry a local version of algebraic geometry and later on we will discard also a global kind of compact version of the same things so now that we have picked our field k we look at all the polynomial equations over this k and we put all of them together into one object which i denote by k t1 to tm these are all the polynomials in variables independent variables t1 to tm with coefficients in the chosen field k and this is basically the algebraic side of the theory then we want also to go to the geometric side of the theory and we first define the ambient space the n-dimensional affine space which is denoted by an k uh this is just the set of all the points a1 to an n where the coordinates of these points come from the chosen field so this is just k to the n now if you if you work over the real numbers r then we have just like r to the n and if you take n to be 2 for example then we have r2 which is just the plane or r3 which is just the usual three-dimensional space so now that we have this ambient space then we can define subsets inside the ambient space and these subsets are the primary the primary objects in algebraic geometry and affine variety is then defined as the common solutions of a collection of polynomials in the ring that we fixed already so we have f1 to fr to all polynomial equations we look at all the points where these equations vanish all at the same time so they can be considered as points inside this an and that's why we think of this as something geometric uh because an itself is a sort of geometric object space and then subsets which will be just collections of points we can also think of it as geometric objects now everything about this x this variety x is actually encoded in this in this ring which is just a polynomial ring divided by the ideal generated by this polynomial f1 to fr so here we can see the algebraic geometry clearly again x here is something geometric and this ring here is something algebraic but there is a dictionary the kind of the geometric properties that we are interested in about this x these properties can be translated into algebraic properties statements about this ring so that's why there is this dictionary the duality of algebra and geometry obviously that's where the name comes from so algebraic george now the goal of algebraic geometry in general is to understand the shape of these kind of spaces x so that can be local and that can be also a global shape of these spaces we can look at the the the very simplest case where we have only one variable let's say t this is a very classical basically ancient kind of algebraic geometry so if you have a polynomial with one in one variable then we want to find its solutions its solutions will give you the the algebraic variety associated to this equation so for example if we take k to be the complex numbers and f this polynomial to be a quadratic polynomial like this t square plus bt plus c then it has solutions in c and the solutions can be calculated by some formula and this goes back to something like four thousand years ago to the Babylonian mathematics but people of course try to generalize the statement that instead of working with degree two equations if you work with degree like three or four or five or a million can you always write down the solutions of this polynomial in some nice form like this or not that as every mathematician knows led to gawa theory to the theory of fields and gawa theory says that yes you can always do that if the degree of f is at most four but not otherwise and as also all mathematicians know this gawa theory which you can see that the other kind of elementary algebraic geometry is actually one of the most beautiful examples of modern mathematics by modern i mean mathematics in the last one or two centuries so gawa theory goes back to more than 200 years ago so you see that even in this simple case algebraic geometry tells something very non-trivial and it's really exciting so this statement that i said here of gawa here it usually occupies a whole undergraduate course in mathematics so you can imagine now if you go to more variables and if you look at high degree equations that the geometry that you get will be much more complicated in fact let's look at some very simple example just to illustrate what we do suppose that we look at this equation t1 square plus t2 square plus one it has rational coefficient obviously so it's defined over q you can also see very clearly that it has no solutions over q and also not over r but it has solutions over complex numbers and also over some finite fields so to understand this polynomial really at some time working with one specific field maybe doesn't show you the whole picture but when you extend the field go to a larger field you can see more and that's what happens in this case when you consider the polynomial over the complex numbers there you basically see everything if we now look at this equation t1 to the m plus t2 to the m minus one where m is at least three then it is known that it has only trivial solutions over q meaning that any solution with will have either the first coordinate zero or the second coordinate zero of course you also know that it is just fair mass zero so this illustrates that even working with some of the most simple equations can lead to a very difficult and deep problems in mathematics but let's consider something simpler some equation like t1 square plus t2 square minus one as we all know from high school this defines a circle in a square in the two-dimensional affine space especially if you take k to be the real numbers you can see that much more clearly but over different fields we we get different shapes at least the visually we get different things so over the real numbers we just have a really just a circle what we know this equation defines but over the complex numbers it's a sphere without two points and over finite fields we will have a bunch of points so you see that if you work with the equations purely only the equations life can get quite complicated but at least if you work with the geometry also you can use we can use our intuition our geometric intuition to get information or at least get some idea of how to study this space and these equations the thing is that because human brain is very is very powerful visually that's just how we have evolved and we can take advantage of this power to study algebraic geometry let me give you a couple of other examples again very simple equations where in the first example I have an equation t2 square minus c1 to the 3 defined over the complex numbers it gives us this red curve here I have another simple equation which gives me a second red curve now these are the kind of thing we have all seen actually stimulating in high school but in high school we looked at these kind of objects these curves from different perspective from different points of view but here in algebraic geometry we look at the properties of these curves which are different from what we usually do in high school so for example one of the things we can see here is that at the origin both shapes here have something strange going on to be technically to be precise there is a singularity at this origin I mean visually we can already see it just by looking at the picture but we can also see this mathematically we can explain why this happens the reason this happened is that if you look at the partial derivatives of our equation in each case both partial derivatives vanish exactly at the origin and that basically characterizes the property of being singular something that is intuitive we turn it into something which is precise and algebraic now the singularity theory itself is quite interesting because if we study it deep enough it leads to interesting things for example we can resolve our singular curve the example from the previous page we can somehow remove or resolve this singularity by an operation which is called a blow up in algebraic geometry intuitively what it does it replaces that point that the origin will align and in this way it also modifies the red curve the red curve becomes something like this where here the curve intersects itself and creates a singularity but up still we see that there is no intersection anymore and the curve becomes completely smooth without any singular points so if you know what is a blow up if you write down the equation then you can see workout this thing very easily why this is the case so this example is quite simple but what is amazing is that there is a general statement for algebraic variety in general over the complex numbers or actually over any algebraically close field of characteristic zero which says that is a result of Hironaka from the 60s he says that we can always resolve singularities of every algebraic variety by some kind of birational transformations basically this kind of blow ups because you may not be able to do it by one blow up but a whole sequence of them and so the whole process can get very complicated but the thing is that in the end you can resolve all the singularities of the variety in this way so these are the kind of things that we didn't see in high school but we see it in algebraic geometry and in fact if you go deeper into studying singularities you see that its behavior in a way is very much related and somehow similar to global behavior of algebraic varieties so I will define global compact algebraic varieties in maybe in the next slide but this connection played a fundamental role in modern algebraic geometry especially birational geometry and it's really very nice but let me also remark that for fields of past the characteristic for example if you take the finite field and its algebraic closure then varieties defined on this kind of finite fields they are expected also to have resolution of singularities but this remains to be a major open problem in algebraic geometry so for most of the talk I will work over complex numbers so far I talked about local geometry if we make a comparison with with differential geometry many falls at each point locally they are defined by some patch and if you go to a different point then the patches will basically be the same there is no difference between locally between any two points but that's not the case in algebraic geometry even locally between two different points even on the same algebraic variety you can have very different kind of geometries but if we glue together this affine variety in a way we can define also compact or projective varieties or we can do it a bit more direct by going into projective spaces so first we worked in the ambient affine spaces but now we work in ambient projective spaces which are defined formally as follows the n-dimensional projective space is the set of points with n plus one coordinates n so i0 to an the coordinates all come from the chosen field and we want at least one coordinate to be non-zero and this is all defined up to the following equivalence that if you have any point with a set of coordinate then if you multiply all the coordinate by some fixed non-zero number the coordinate at the point will not change although the coordinates will change so in other words coordinates here are not unique you can have different representations for the same point but it's not difficult to work with these kind of spaces and in fact it turns out that pn is simply an with a copy of pn minus one added at the infinity so in other words this pn is a kind of compactification of the usual am what did all come from projective geometry where people wanted to have something compact so that you can see everything that that there is nothing missing so for instance if you work in a two let's say if you work in the usual two-dimensional space over the real number then if you have parallel lines they will not intersect so that's one of the axioms of classical geometry but if you go to the projective space then that is not the case anymore so the the lines will intersect eventually at some point which is at the infinity we can't see it in a2 but we will see it in p2 now we can define then projective varieties inside pn they will be similarly as in the affine case there are solution common solutions of a bunch of polynomial equations and the only difference is that now we have to look at homogeneous polynomials homogeneous means that all the degrees in each polynomial all the terms have the same degree we have to do that because points in this projective space are defined up to this equivalence and if the polynomial is not homogeneous it would not make sense to say that a polynomial vanishes at the point or not but if it is homogeneous then it makes sense although even in this case the value of the polynomial is actually not well defined but to say that whether it vanishes somewhere or not that is well defined um projective varieties are actually compactification of affine varieties and even more interestingly every projective variety can be covered by finitely many affine varieties so this is similar to differential geometry but the difference is that as i say locally different patches will not necessarily be isomorphic okay so let's look at the simplest case but let's look at the simplest case and simple non-trivial case and that is the case of looking at x varieties inside p2 over the complex numbers and this is defined by one irreducible homogeneous polynomial we are working over in p2 so we need three variables and these f let's say have degree d so we have that one polynomial and i assume that this x has no singularities so singularity here let's understand it just intuitively as in the examples where nothing strange happens at any point then topologically it's actually very simple very easy to understand this kind of x x will look like a donut as we see in this shape in this picture here is so there are different cases depending on the degree there is a number called genus we can calculate it just by this formula it's a half of d minus one times d minus two and that's a genus this number happens to be the number of holes in this space so here we just have a sphere there is no hole so g will be zero and this corresponds to equations of degree one and two because if degree is one or two we have zero here but if degree is three the g becomes one and we get one hole or we get three and so on so the g which is calculated purely algebraically by this formula also has a topological interpretation here so for example if we take f to be something of degree three and put three in this equation in this formula we see that g is equal to one and this is the kind of topological shape we get and the curve in this case the algebraic variety is called an elliptic curve and people in non-bacteria and computer science know that elliptic curves are also very important in their subjects not only in algebraic geometry I will actually come back to this point a bit later so at least topologically you see that this kind of x can be understood very easily but algebra actually is still not that simple because for example just in degree three there are infinitely many different even have infinitely many different algebraic varieties like elliptic curves with the same topology but their geometry are different it's not the same now we come to special varieties on every non-singular or even some mildly singular variety x we can define curvature in some way let's not go into that detail and we say x is a final variety if the curvature is everywhere positive is colloquial if curvature is zero everywhere and canonically polarized if curvature is everywhere negative for those who are familiar with algebraic geometry this is just saying final just means that minus the canonical bundle is ample colloquial means that the canonical bundle is trivial and canonically polarized means canonical bundle is positive it is ample so at least intuitively we can look at these things in terms of curvature and now these are very special classes of algebraic varieties in algebraic geometry but they also happen to be extremely important in many other areas in particular in differential geometry in arithmetic geometry in mirror symmetry mathematical physics and so on even in applications like algebraic statistics and so on this kind of spaces just come up very naturally in many different areas but in algebraic geometry they are really fundamental as we will see in a in a couple of minutes just as examples if we take x inside p and define just by one single polynomial let's say of degree r then this kind of x will always be belong to one of these classes and we can determine which class by simply by looking at the degree in fact x will be always final if the degree is less than n plus one it's colloquial if the degree is exactly n plus one and canonically polarized if the degree is more than n plus one so for example if we look at the case when n is equal to two then in that case if degree is one or two we get a line which is final we will get p one for degree three we get exactly elliptic curves which are colloquial in dimension one and for the degree more than three we get canonically polarized which topologically there are those with their number of holes more than one now these special classes are very important in algebraic geometry because of the following at least conjectural scheme which is usually discussed in bi-rational geometry bi-rational geometry itself tries to classify algebraic varieties up to bi-rational transformations bi-rational transformations are similar to meromorphic transformations in analytic geometry now there is a major conjecture in algebraic geometry especially bi-rational geometry which says that every algebraic variety x is bi-rational so in other words you can modify just some small part of the variety and leave the rest intact so x is bi-rational or meromorphic to a projected variety x with possibly some manageable kind of singularities so it may not be smooth such that one of the following things should happen either this new variety y it admits a funnel vibration or it admits a calabia of vibration or it is already canonically polarized so this is an incredibly powerful statement it basically says that if you pick any algebraic variety x you can let's say x is smooth and projective but then in fact there is a more precise version of the conjecture we say that you can transform this x in a step by step program where every step is very clear what it should be so that in the end you end up with y and if you apply a similar process to y itself to the base for example of the vibration and so on the whole statement would say that you can somehow reconstruct this x by only using fauna varieties calabia varieties and canonically polarized varieties so in other words if you consider x to be like a house then the fauna or calabia and this canonically polarized will be like the bricks and mortar and and so on i don't know the wood or whatever so that's amazing everything somehow is constructed out of these three type of spaces this conjecture is known in full generality up to dimension three and we know a lot of cases in higher dimensions but otherwise it's still still fully wide open now if we know this conjecture if we have already proved it then what we can do next is try to classify this kind of y and usually in technical terms that means to construct modular spaces, parameter spaces for these kind of objects and that's also an important and active area of algebraic geometry so that's just a few words about biorational geometry and the general classification theory of algebraic varieties and now i will discuss a couple of other topics and somehow i try to connect them to for example the biorational geometry to this classification theory just to illustrate some to you that there are strong connections between algebraic geometry and other fields and these connections are quite exciting let's go into group theory suppose that cn is the group of field automorphisms of ct1 to tn that fix c so this is a purely algebraic definition which is understandable to any and the graduate student who is familiar with the notion of fields and automorphism so c itself is the number the complex number field so it is a field and t1 to tn are independent variables and if you take all the rational functions rational means polynomial divided by polynomials you get a new field which is denoted usually this way ct1 to tn this is also a field which is much bigger than c now you can look at automorphisms like maps from this field to itself algebraic maps which fixes c so that means that it doesn't change coefficient but it may change polynomials and this is a group it's again a purely algebraic stretch however this is related to geometry in a very nice way in fact cn is isomorphic as an algebraic object to the setup all biorational maps all meromorphic maps from pn to pn so remember that meromorphic or biorational means that you have a map which is defined by polynomials and it may not be defined at every point but actually it's defined on a big part of this space and you can also have an inverse for this map which that means biorational or meromorphic so this is a geometric characterization geometric definition but here we have a purely algebraic definition and so it's clear to see what the group structure here is how you multiply elements inside c and by this algebraic definition but in the geometric definition here to multiply two elements you just compose maps so an element is just a biorational map from pn to pn another map will be another biorational map if you compose them that will give you the product inside this group okay now let's just look at one simple example of an element of this group which is a biorational map from p2 to p2 this is defined by simply taking a point a0 a1 and a2 to the product of a1 a2 and then a0 a2 and a0 a1 so you see that these are all defined by polynomial functions which in this case happen to be very simple you can easily also find the inverse of this map so this happened to be a biorational map and therefore it is an element of c2 of this two-dimensional group in fact cn contains pgl n plus 1c here by this i mean the set of n plus one times n plus one invertible matrices but it's projective version that means that you need to divide it by so if you have a matrix and you multiply by one non-zero number would pretend to be the same a matrices so you make it projective so it is basically some set of matrices and this is naturally contained inside cn why the reason for that is because the elements of this pgl n plus one c give exactly isomorphisms from pn to pn so if you are given any matrix of these four invertible matrices that will define if you apply it to coordinates of points in pn it will give you a new point in pn and since the matrix is invertible this map will also have an inverse and the reason to make it projective is because if you take a matrix and multiply it by a fixed non-zero number the map will be the same so that's why you have to make it pgl rather than gf so this is a set of group of matrices which is very well known and classical let's see you go back to 19th century at least okay so so far we haven't seen anything very exciting apart from the fact that we have something algebraic you can characterize it in terms of geometry but now i want to tell you a few things which may be more exciting first of all c1 is just pgl2c so in dimension one we just have matrices and that's the end of the story but in dimension two for c2 we know that this group is generated by pgl3 and the phi that i just defined a few minutes ago so i mean this phi in this example uh so if you put together this particular simple phi and put together also all the those birational math or isomorphism which come from these matrices they all together generate the whole group c2 but this happened to be something special in dimension two because in starting from dimension three that's not the case even if you take some simple similar maps phi together in matrices you cannot generate the whole group so people try to understand these kind of groups from a different point of view and ser also started these groups not a long time ago and he asked whether this cn is german which is just a group theoretic statement is meaning is that the question is that whether there is a natural number h such that for any finite subgroup g of cn uh there is a normal abelian subgroup h of this group g of index at most h so in other words all the finite subgroups of this cn the question asks whether they are very close to being abelian so the non abelian part is somehow bounded by this number h so that's a very nice question ser himself he proved that this is the case in dimension two and then not long after that Prohorov and Sramov they started this question of ser using birational of the right geometry so using the connection that this group cn is interpreted in terms of birational from pn to pn they try to use geometry here to write geometry to approach this question and to understand these groups in general and there is some very nice works in this direction what they prove is that if you have some boundedness statements about final varieties something that I proved some years later they say that if you have that then you can show that these groups are always Jordan for every n so since I proved that bound in the statement a few years ago then this becomes an unconditional statement so it gives a positive answer to ser's question so you see that there is this very nice connection between algebraic geometry on one side and group theory on the other side in fact maybe I spend a few minutes here to go very quickly over the the sketch of the proof of this statement how you can use algebraic geometry to prove something which is potentially just a purely algebraic state so I'm going to be quick so that I can talk about other topics well take a finite subgroup g of cn and bioregional geometry as you write geometry state that there is a bioregional map from some other projective variety y to pn so that so that on these new varieties which is bioregional to pn we have a morally final vibration so we have a vibration or a vibration structure on y to some variety set so let's say if you work with p3 then this y will be fibrous over something two-dimensional or three or one-dimensional or zero-dimensional so the fibers will be most of the fibers will be final varieties and these are called morally fibrous spaces but it also has an action of the group g that we started with so in other words this vibration is compatible with the action of g the g x on y and also this action is compatible with this vibration so we have a very nice picture here in fact to arrive at y itself this involves some against some nice algebraic geometry and is relied on some highly trivial facts from algebraic geometry so the existence of this y is not so simple as it may look so as I said the g x on y and considering the fibers and considering also the base we get an exact sequence we get a sequence of groups where it kind of gives you a decomposition of this g of group g into two different groups h and e so by the composition I just mean that we have this exact sequence where this h now which is a subgroup of g it acts on the fibers f of this vibration but ex on z so in other words you said you kind of divide this g into two parts in this sense where one part is acting on the fibers another part of g somehow comes from the base now if dimension of z is positive if it is not a point then we can use induction to understand this group g and prove that Jordan property for it but it may happen that the z is a point so that this dimension is zero in that case so the vibration is just a trivial vibration it just collapses y to a point and in that case you can't do induction because y and f are the same which is let's say n dimensional and but what happens is that f itself a y itself here is a final variety with some mild kind of group singularities and what happens in this case is that if you put all such y together independent of their group g they all somehow form above this family there is a finiteness statement here of course that doesn't mean that such y belong to a finite step but it's just that a finite in some algebraic geometric sense can be infinitely many of them but they all together can be put in a finite dimensional family so this one of the things that I proved a few years ago and then you can use this property to show that we can embed this g into the automorphism group of pd for some d this is not the same as n some maybe larger number d which in this case this automorphism group is just pgl d plus one for some d so in other words we can embed this g into pgl of some higher degrees but so that this d does not depend on the g that we started with d only depends on n so there is a uniform kind of embedding here and the thing is that there is a classical theorem result of Jordan which says that these kinds of pgl are actually Jordan in other words any finite subgroup here will have an abelian subgroup with bounded index so that that group is the theorem so you see that this involves all kind of facts from algebraic geometry so this is a really nice connection between algebra and algebraic geometry now the next topic I would like to talk about is quite different this is about connections with computer science in computer science cryptography is a very important topic of course not only for computer science but for economics and for so many other things so in general this means that we would like to have secure digital communications we want to send messages from a computer to another computer so that even if someone on the way reads all this information being sent they cannot make sense of it although they have all the numbers now elliptic curves which i talked about earlier these are algebraic varieties defined by the grid three equations inside p2 these are elliptic curves and they play an important role in cryptography there are different versions of cryptography but elliptic curves provide one other version and in fact is providing a nice version because in the sense that you need less computer power to run it although you get the same kind of security compared to other systems but you need less computational power so that's an advantage here now i'm not going to tell you how exactly it worked but i'm going to tell you how algebraic geometry is used what's the main point so what we do here is suppose that x is an elliptic curve over a field k and in practice this k will be a finite field now x is a smooth with a non-singular projective algebraic variety of dimension one and it has genus one in general for this kind of algebraic variety for the elliptic curves the points of x carry an additional structure which we usually don't see in other algebraic varieties what happens is that the set of points of x itself it forms an algebraic group it becomes an algebraic structure we can make sense of this structure just by looking at some pictures now if there's a red curve here is the elliptic curve then we can add points on this elliptic curve in the following way that if let's say we have p and we have q if we add them together and also add r then we will get zero inside this group structure so basically this tells you how to add things together and in this picture here if we draw a vertical line then that let's say intersected it in two points like in this picture here then that means that this sum of these points will be equal to zero so in reality there is also a point at the infinity which is hidden here in the picture at that point we'll play as the the trivial element of the group so there is a non-trivial structure on this space now assume that we define x over some finite field fp and suppose that we are given points x and y in this elliptic curve such that x is n times y so n times y here means that you sum y with itself n times in the group structure of this elliptic curve for sum n now the whole point the reason that this works that you can use elliptic curve in cryptography is that if you know x and y only so if you give x and y to a computer then it can be almost impossible to find the number and just from the numbers from the points x and y so that's the whole point but there are other systems cryptography systems which for example rely on the fact that if you have two let's say prime numbers and if you take their product but give the product to a computer then it becomes almost impossible to find the two prime numbers so this is similar to that situation where instead of prime numbers here you take points on some elliptic curve so given x and y it can be it can take like billions of years for even the most powerful computer in the world to find n and this is the whole point of using elliptic curves in public key cryptography now a billion varieties which are higher dimensional versions of elliptic curves they can also be used to define cryptography to do this kind of public key cryptography but on the other hand people have recently been trying to use these groups cn these are the group of biorectional transformations of pn in cryptography in fact some people told me so i'm not an expert in this area but some people told me that you know that there is a whole discussion about quantum computing if it becomes a reality in the sense that if they get really powerful then these cryptography systems for example using prime numbers or elliptic curves would not work anymore because those quantum computers can break all the codes very easily however one of the candidates that can resist that kind of quantum computing is in fact this kind of biorectional transformation so these are important only not only in the right geometry but also in computer science which of course also implies that it's important for many other things like economics and in general for security okay that was computer science i will also spend a few minutes i don't have too much time i will spend a few minutes to talk about a completely different topic now arithmetic geometry and again to draw some connections between algebraic geometry and this kind of sort of arithmetic geometry let's say number theory now arithmetic geometry basically unifies some parts of algebraic geometry with some parts of actually a big chunk of number theory the point is that i told you that in the beginning that to do algebraic geometry we need to start with a field some kind of number system but in fact modern algebraic geometry especially starting with growth and it tells you that you can start with any commutative ring and do algebraic geometry in a similar way especially you can do algebraic geometry by taking the ring of integers so the integers don't form a field but they form a ring and something a bit weaker however you can still construct algebraic geometry over a spec of zero or just say over this kind of structure over the integers and that immediately kind of creates a bridge between number theory and algebraic geometry so there are other ways that connects algebraic geometry and number theory for example when you start the varieties over fields which are not algebraically close for example just the rational numbers or function fields of some other algebraic varieties now in general if you are given a variety x over some field especially when the field is not algebraically close like you then it's very hard to determine whether this x has any point or not we already maybe saw some examples for example x inside p2 defined by this equation it's very simple looking equations now to ask whether x is empty or to determine its point is just firm as last zero so even in this kind of it looks very simple at least the statement okay the other example maybe i skipped this because i don't have much time let me just get this example i want to tell suppose that we have a curve x inside p2 defined over the rational numbers and suppose that this x has at least one point then there are several things that can happen one is that if the genus of the curve so this is just the number of holes in x when we draw x over the complex numbers which is also the same as saying that x is a final variety then in that this case x have in fact infinitely many points and we can there is a way of parameterizing all the points but if the genus is equal to one which is exactly when x is calabi-al in this case it's just an elliptic curve then x is a finitely generated abelian group this is a famous result of it's called model val theorem but if genus is more than equal to two that's exactly when x is canonically polarized and in this case x is finite the set of points is finite and this is a deep result of faultings now there is a conjectural extension of this result to higher dimensions so let me skip a few things and maybe go to the to the conjecture yes so this is the conjectural statement which says that suppose we have x which is a non-singular projective variety defined over a number field for example just the rational numbers q then the statement says that then if x is final or if it is calabi-al then there is a finite extension of this field such that the set of points after this finite extension become dense among the whole here of course you need to make sense of points in the sense of scheme theory but just intuitively just saying that the number of points becomes huge you will have a lot of points almost everywhere inside x but on the other hand if x is canonically polarized then the set of points over k is not going to be dense even if you extend your field by any finite extension the number of points will not be dense they will be like concentrated inside maybe some sub-ride so this is really amazing you see that this kind of space is final and calabi-al and canonically polarized they are interesting from many different point of views whether that's algebraic geometry or differential geometry or arithmetic geometry or even applications outside mathematics or as we saw group theory and so on so i think i'm going to stop here thank you for your attention thank you very much Kavser for a busy full talk let's see i can see from here there are a few questions okay so can you read them yeah okay so first i have two questions from fozy Higa so the first one is is there a known analog in different geometry in g in many folds for the conjecture regarding the composition of varieties to funny calabi-al or canonically polarized varieties whether there is some kind of version people have tried for example to use something which is called richy flow to run a similar program in at least in some cases in differential geometry for example there people also have tried to look at calabi-al and many folds and run a similar kind of program so in general this variety these special classes where the curvature takes a particular shape particular like positive negative or zero it just happened to be fundamental in in all these areas so in differential geometry in some cases yes it's expected to have something similar so the second question by fozy Higa is also about same thing but now are the classes of varieties well um yeah hypersurface i can read the other questions oh you can read the questions oh okay so maybe then i don't have other classes of varieties what is the composition is known i say hypersurface well hypersurface it always happened to be it is yes because any hypersurface will automatically belong to one of these classes it's simply because it's defined by one equation so that's yes varieties of more it would be property like projective varieties of a billion varieties a billion varieties are calabi-al always so that they always belong to that class of calabi-als property like projective varieties well i mean all the variety we looked at in that context where projective varieties so yeah there are there are special cases so i mean for example there are for instance tori varieties these are special classes of algebi varieties which are related to to combinatorics then in that case we we know if you are given a new tori variety you can always decompose it into yeah exactly as in the conjecture but in that case tori varieties are very special so what you will always get will be a final vibration for example you cannot get i mean there are of course there are also cases where you have for instance elliptic vibration and so on but when you run this program for tori varieties you end up with the final vibration or more refined spaces yeah so there are certain cases and another case is these are called varieties of general time now this might be difficult to define what is the variety of general type if you have any variety of general type in fact then we know that the statement is is true in this case in the sense that such a variety is always birational to economically polarized varieties and varieties of general type their names suggest that in fact most varieties are of general time so the difficulties are in the cases where you have like a smaller class of variety where we don't know how to prove this statement but i mean from a statistical point of view most varieties actually are of general time so we already know the statement in that case then another question is said is there an explicit formula for the smallest d such that cn in beds in pgl pgl d plus or so what we did we did not really embed cn into any pgl what we did was we we picked a finite subgroup of cn and then according to that finite subgroup we constructed some birational model of of pn and then i say that in one particular case when that birational model is the final variety then you can embed the group into some pgl d plus one so only those kind of groups were embedded into pgl d not the whole cn and so in general you can i mean we know that there is an if there is some bound d although we haven't tried to calculate exactly what d should be but probably at least in lower dimension maybe you can do that and there should be some way of maybe getting an ineffective bound but at least for the statement we did not need an effective bound okay did all the questions that i can see thank you in this case i'm i'm just giving a few seconds just to check if the answers are giving space for new questions or okay so it is my pleasure to to to thank again everybody i mean i hope we i mean of course we missed a lot the possibility of being all together here in trieste but i hope even with this with the help of technology we managed to conceive a little bit our philosophy and staying together and especially to to send to carolina our warmest congratulations and our also love for the subject and algebraic geometry so i just really have to thank everybody now we know there is a tradition of ictp that math colloquia and especially this one for on the occasion of the ramanujan prize every colloquia is followed by closed meeting between the speaker the winner and the diploma students so i ask all of them to move to their reserved channel and i thank everybody for taking parts to this mathematical fest i really hope we can see you all here in trieste next year goodbye thank you all i'm moving out to the other zoom channel to be with the diploma students but i just wanted to thank you again thank you for everything thank you with carolina and kasha okay


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