 I would like to, yeah. So, yeah. Professor Karestan, would you like to take the questions during the talk or after the talk? I could take some questions during the talk and some afters. I can stop maybe every fight. Okay, so I'll keep track of it. Yeah. We had a- Are we live on air? To, yeah. So, yeah. Okay, so welcome everybody. So, it's actually a pleasure to have Professor Chandrasekhar Karya as a speaker of the second talk of the day. And as a formal introduction to the speaker, Professor Chandrasekhar Karya is a professor of mathematics at the University of California, Los Angeles. In 2005, he made a major advance in the field of graduate presentations and number theory by proving the level one study conjecture and later approved of the full conjecture with Jean-Pierre Linton Berger. Professor Karya is the winner of the Insta Young Scientist Award, which he won in 1999. He won the Pharma Prize in 2007, the Infosys Prize in 2010 and the Cold Prize in 2011. He has been on the Mathematical Sciences as a jury for the Infosys Prize from 2015, serving as the jury chair from 2020. Now, in 2012, he became a fellow of the American Mathematical Society and was elected as a fellow of the Royal Society. And I'm glad to let you know that he started his career as a fellow at TIFR. So it's really an honor to have him here as a speaker on an occasion of TIFR. Thank you very much. So it's over to you Professor Karya. Yeah, thank you so much for the kind introduction. So I would like to share my screen. Let me see if I can do that. Yeah. Is it visible or should I make it full screen? Yeah. Okay. You have to enlarge it. You have to enlarge it further. It's fine, yeah. Maybe a little bit more. A little bit more. Okay, I think this is the best I can do at the moment. It's visible, right? Okay, that's fine. Yeah, that's fine. I could try something, okay. Yeah, but this is what it comes to. Okay, all right. I think it's kind of, we can make do with this, I guess. Yeah. All right, thanks so much. It's a great pleasure to be present for this occasion. As was said, I started my career after my PhD in TIFR where I worked for roughly 10 years. And in fact, maybe the work I'm best known for which is a proof of SARS conjecture. I thought about it for 10 years in my time at TIFR. So in some sense that illustrates one thing about mathematics is that it's a marathon race that everything takes a long time and persistence. One hopes that it'll pay off sometime. So I'm sure that many of you students have been very persistent and have done fantastically well as I see from looking at the program at the Olympiads of various sorts, math, physics, chemistry, astronomy, maybe not physics this year. We really congratulate you for that. And I hope some of you will be inspired to do basic science later. It has great rewards because questions are hard. The subject is beautiful and it's also powerful. So even if you kind of think about something carefully and have a small idea, the subject amplifies the power of your idea. So I really encourage you to think about a career in research and basic sciences. I also really enjoyed the previous talk by Professor Yamuna Krishnan. And in fact, funnily enough, she mentioned these sort of symmetries, the icosahedral symmetry, the icosahedron and it's the group of symmetries of that and the tetrahedron and the octahedron. And in fact, the symmetries of these objects which are groups of finite order like 60 and 12 and 24, these groups of symmetries show up in a big way. In the topic I'm going to talk about, not that I'm going to mention it explicitly in my talk, but it somehow gets us into certain technicalities. But the proof of Fermat's last theorem essentially uses tetrahedral and octahedral symmetries. So it's kind of interesting coincidence. Okay, so the title of my talk is Fermat's Last Theorem. So many of you know that this is a theorem which is a misnomer because Fermat, this very great French mathematician from the 17th century, formulated a thought. He kind of posed this question or said he had a proof of this statement which no one found for 300 years, more than 300 years. So this is what I'm going to talk about today. Yeah, the other thing I wanted to mention is that this is of course, it's called National Mathematics Day in India with a birthday of Ramanujan. And again, Ramanujan actually plays a big role in the proof of Fermat's Last Theorem in terms of the work he did, especially in a 1916 paper called on some arithmetical functions. He kind of made some interesting observations and stated some conjectures. And the work on those conjectures actually in some ways there's a direct link between that work and the proof of Fermat's Last Theorem. So when you, again, Ramanujan is, this is number 1729 associated with the lore of Ramanujan because as the story goes, G.H. Hardy visited Ramanujan in a sanatorium at which the Ramanujan was recovering. And to cheer Ramanujan up, who did not appear to be in the greatest of spirits, Hardy said, oh, I came by a cab and it had a boring kind of number, 1729. And then Hardy at Ramanujan famously quipped that, no, it's not boring because it's the smallest number which can be expressed as a sum of two cubes in two different ways, right? So there's a sum number 1729, which is a part of the Ramanujan lore, but actually there's another number, which is 691, right? And I leave it, and it's actually a prime number. And the number 691 also Ramanujan made some remarkable observations about it. And those observations again have a very deep link to the proof of Fermat's Last Theorem. So maybe with that I shall start on the lecture proper. Okay, so Fermat's Last Theorem is a statement about integers, right? So it's a statement that an nth power of an integer can never be the sum of two nth powers as long as that n is bigger than two. So a cube cannot be the cube of an integer, cannot be the sum of two cubes, right? And that's one of them is zero, of course. So this was a statement that no nth power can be the sum of two nth powers in a non-trivial way as long as that power n is bigger than two, right? So this was a problem which Fermat kind of thought about in the 17th century and claimed that he had a solution and for 350 years no one found a solution, right? So this is this kind of very sort of old problem. And again, now these problems in number theory sometimes they have this air of being rather random, right? Maybe why should one care about knowing the solution to this problem? But I think one of the hallmarks of a question in number theory is that though the question might seem kind of very elementary in some sense, it often leads to the creation of sort of whole theories to try and answer that question, right? So in some sense those theories in the end become more interesting than the answering the question, though the fact that one is able to answer the question is a kind of litmus test of how far we have come in mathematics, right? So this was one question which Fermat thought about in the 17th century. So let me point out that this is a huge feature of number theory, right? Number theory somehow has the ability to pose questions. The questions seem very beguilingly simple and then you're kind of enticed to think about them. But actually to answer it takes enormous effort. So let me start with the quote of Barry Maze that one of the great number theorists of the present times. So if he says the field number theory produces without effort innumerable problems which have a sweet innocent air about them like tempting flowers. And yet the quest for the solution of these problems have been known to lead to creation from nothing ab initio of theories which spread the light on all of mathematics and has been known to go to mathematicians on to achieve major unifications in their science. So I won't read all of it. And number theory swarms with bugs waiting to bite the tempted flower lovers who once bitten are inspired to excesses of effort. So there's a very poetic way of describing this kind of feature of number theory that produces very seemingly innocent looking questions which maybe even a 10 year old can understand but to answer it, to answer those questions sometimes requires like maybe hundreds of years of collective efforts of mathematicians across the globe. So Fermat's last theorem is a classic illustration of this sort of problem in number theory. So number theory is about integers of things we've known since we were in nursery school, right? Numbers one, two, three. So sometimes people might ask and people do ask me what is there to know about numbers one, two, three? We know that, right? But the thing is that in fact, questions about the integers in mathematics are sometimes the hardest to answer. Questions about real numbers, complex numbers in some sense are easier in the sense you have to first answer them before you can answer questions about integers. So number theory again has this feature of dealing with the basic things like numbers, one, two, three, four, but then the questions about them tend to be very interesting and complicated. Yeah, are there any questions at the moment? I can't really see them, but I see that there are, or I can take them later. No, no, that's not it. Okay, so we have, so number theory is about integers and questions about integers. So one of the basic things about integers is when you look at the multiplicative structure, the numbers which distinguish themselves are the prime numbers. These are kind of the atomic elements, these are the atomic building blocks of integers because every number is a product of prime numbers in an essentially unique way, right? So the primes are the numbers which have this property that there are no interesting devices. They have nothing devised to emphasize themselves on one. Then one of the oldest subjects, one of the oldest theorems in mathematics or number theory is Euclid's theorem that the number of primes is infinite, but the list of primes never stops, right? And primes are the fundamental building blocks of the integers and every integer can be written uniquely as a product of primes, right? So somehow primes are kind of a distinguished sort of, these are the stars in the firmament of numbers and the key property of a prime number which is gonna play some role in thinking about Fermat's last theorem is that if it divides the product of two integers, then it divides one of them, right? So primes have a naively defined are those integers which have no interesting divisors, but the key property which kind of they have is that they have this property that if they divide the product of two things, they divide one of, at least one of the two integers involved, right? So, but this very natural looking kind of thing, if you try to extend it to number systems beyond the integer that actually becomes false in some sense that plays a role in Fermat's last theorem. All right, so again, so now coming back to Fermat's equation, so when I was looking at this equation, X to the n plus Y to the n equal to Z to the n, right? And this equation, let's call it the equation F sub n and in 1637, Fermat wrote in the margin of a book he was reading, Diffantus is arithmetic, that he had a truly marvelous proof that there were no solutions of this in integers X, Y, Z with X, Y, Z not equal to zero and with n bigger than two. And then he said that this margin of this book was too small to fit his proof into, right? So we all hope that we all wish that the margin was bigger and he had had more space to write his proof because no one could find a proof for the next 300 years or more, right? So there was, and on the other hand, Fermat had a lot of credibility as a mathematician because he had made claims before and not really told people how to prove these things, because he was giving hints and almost all the time he was proven to be correct, right? So because of his stature, this was some of a statement which people thought he probably knew how to prove and then there was this massive hunt to find the solutions, to find his proof because it was a picture of Fermat. He worked in Toulouse in the Southwest of France and at that time in the 6th, 17th century, one could not make a living as a mathematician. So he worked, his day job was, I think, as a jurist and as a parliamentarian and then he pursued number theory and other mathematical disciplines as a hobby, right? So that's the picture of, so when I went to Toulouse in 2008 or something, I did go to the city center and then there's a statue of Fermat out there, right? So he's remembered in the city. And so this is the statement because at that time, just as nowadays English for Godabad has become kind of the lingua franca of my scientific communication. Most people write their papers in English. At that time, Latin was used and this is the statement of Fermat's last theorem in Latin, right? In the original form in which it was stated. Okay, so I myself don't know much Latin, so I'll just leave it there. And this is the picture of the book he was reading at the time when he kind of jotted down in the margin of this book, that he had this statement and he had a marvelous proof of it, which the margin was too small to contain. All right, so Fermat had this habit, right? Because at that time there was not, I mean, I don't think there were people publishing papers and journals, people corresponded to their friends by mail, I don't know, email, they just posted letters. Some of Fermat's correspondents were this Jesuit priest, Mara Mercen. Then there was this very famous mathematician philosopher, Blaise Pascal. And he wrote to these people, this posing this problem. And in fact, maybe, particular cases of this question of his were N equal to four to show that a fourth power cannot be the sum of two fourth powers and a cube cannot be the sum of the cubes. And, but the thing is that Fermat wrote this in the margin of his book and maybe mentioned this question to a few of his friends, but for the, after that, he never mentioned his truly marvelous proof, right? So there's somehow the feeling that maybe Fermat had this proof or at least he thought he had this proof, but maybe he did not revisit the question or he was not sure of his proof or whatever. But for whatever reason, he did not mention, he did not come back to his claim of this marvelous demonstration in the next 30 years before he died. So this is the translation of the Latin, right? Again, it's the same statement that you cannot separate a cube into two cubes or a fourth power into two fourth powers. And in general, any power higher than the second into two like powers. And I would discover a truly marvelous proof of this and the statement of Fermat led mathematicians in some sense on a wild boost chase, right? But it was not a wild boost chase. It was very productive because it generated a lot of interesting new mathematics. Okay. So besides his fundamental work in number theory, he also made important contributions to what is now called calculus of variations, geometry, probability. In fact, the correspondence between Pascal and Fermat is the starting point of probability and also optics that Fermat's the least principle, least action of them. So it was a very sort of broad mathematician. So now the attempt to prove Fermat's last theorem led to the development of whole branches of number theory, right? Because so even within number theory, it has no real application. The Fermat's last statement, even if you knew the answer, it doesn't tell you something, it doesn't tell you something very useful even within the subject. But because such a great man as Fermat made this claim and because the question is beguilingly simple, I think maybe someone even in the sixth or seventh grade can understand this question. It led mathematicians to kind of break the head against it, right? Because somehow it was challenging in terms of being something very sort of easy to understand but extremely hard to prove. And though of course some people did not care to work on it, for example, Gauss famously refused to work on it. He said one could ask a thousand other such questions and one need not answer every question one can one thinks about, right? But many other people actually did think very seriously about it. Lots of great mathematicians in the 19th century in the 20th century thought about it, one of them being Kumar and the German mathematician Kumar who lived in the 19th century and he made tremendously deep contributions which kind of revolutionized number theory algebra but motivated by trying to at least find ways of attacking Fermat's last theorem. So now one can ask why is the question different for squares, right? And if you think of the question x squared plus y squared equal to z squared, in that case, one is asking in some sense for rational points on the curve x squared plus y squared equal to one, right? You can just divide by z squared. And then you're somehow looking at this picture, you're looking at the unit circle, circle of radius one and you're asking about rational points for points x, y with x, y both rational numbers which lie on this circle. And why are these sort of available plentifully and in some sense that is because the circle is a particularly simple sort of object. In some sense it is, it can be parameterized in, has a rational parameterization, right? So for example, if you fix the point minus one zero on the x-axis and you pass lines through it, then it'll intersect the circle in one other point, right? And that other point will, as long as the slope of the line is rational, the other point of intersection will also be rational and you can work out what the point of intersection is, right? And then you get this parameterization for the circle given by the x coordinate being one minus m squared upon one plus m squared and the y coordinate being two m upon one plus m squared. So I think we mathematicians really suffer from having to talk about Zoom because our mode of operation still, we are very old fashioned and want to talk on the blackboard. So it's much nicer to give a talk on a blackboard, but okay, but technology being what it is and the time being what they are, we shall, this is the best form of communication right now. Okay, so I encourage you for those of you not seen this, of course, to work it out for yourselves, why is this parameterization and how justify the details of an argument which I've just sketched here? Any questions? I think I don't see any, okay. No, all right. So this in some sense explains the fact that squares can be written as a sum of squares because the geometry of the circle is very simple, right? And in a technical sense, it has a rational parameterization which makes it have plentifully, which makes it have lots of rational points, once it has one rational point and one rational point is minus one zero, right? So this is a, so in fact, one can ask, Fermat's last time seems a rather random question and you can ask variants of this question, right? So Euler, another great mathematician just after Fermat, he asked, he tried to find the right sort of context in which to ask Fermat type questions, right? Questions. So you ask, can you just sort of generalize Fermat's statement to say the sum of certain number of nth powers can never be an nth power, right? So you looked at this equation, x1 to the n plus dot, dot, dot xk to the n, so the sum of k nth powers, can it be the sum, can it also be an nth power, right? As, again, to avoid degenerate cases like the circle in the case of sums of squares, you really need to say that the number of variables on the left hand side of the k, the number of variables should be less than the degree, right? If that is not the case, then in some sense it has this rational type parametrization. And then again, he asked, okay, so is it true that this also has no interesting solutions as long as n is bigger than k, right? So for example, if in Fermat's case, k was equal to two and then Fermat's assertion was for n bigger than two. So you can ask a variant of this question, can the sum of k nth powers equal to an nth power as long as n is bigger than k? Now, again, this was actually, so this again seems a very plausible question. And again, it was around for hundreds of years, where it turned out actually to be false. So L keys, I know I'm L keys, I think maybe, in the 1990s, found actually this kind of solution for this equation when the number of, so he showed that the sum of three fourth powers can indeed sometimes be a fourth power and the numbers involved are pretty big, as you can see, right? So he didn't just search for a solution and found some solution, but he somehow did some geometry associated with this equation, right? You can think of this equation and think of all the points which satisfy this equation that let's say with all these variables, real numbers or complex numbers and you can plot them, right? And then you can try to do some geometry of this associated variety or shape. And so he did something very clever. And so, and found these solutions. So sometimes questions which look very plausible can turn out in fact to be false after centuries of people trying to wonder about that. But Fermat's last theorem was not falsified for a long time, okay? And in fact, essentially proved as you know, so now how do I move to the next slide? So this is a picture of Euler, right? So he lived in the 18th century. And so the Fermat actually did prove by a very ingenious method of descent. He's very famous for his method of infinite descent that his equation for the fourth powers is actually his assertion is indeed true. And that also he perhaps gave some hints of the proof and the proof was kind of recovered. And so after this, then if you think about it a little, the only thing to solve Fermat's last theorem, you just need to answer this question for nth powers when n is the odd prime number P. That's somehow the focus then of, the focus of the question shifts to showing the Fermat equation has no solutions when F sub P, the Fermat equation when P is a prime, right? So somehow that's what we are gonna focus on, okay? So the idea of descent, again, I'm giving you a sort of very loose formulation of it, is his idea of descent is somehow some procedure. If you want to, if somehow if you suspect an equation has no solutions and integers to find a procedure so that you keep on cutting down the size of the solutions. If there is a solution, you try to show that there's a solution of smaller size. And as you know, an infinite sequence of natural numbers cannot keep decreasing, right? They're bounded below by, let's say one, if you're looking at natural numbers. So this is kind of the basic logical logic of some of his arguments. And so this infinite descent did indeed work to answer this question for the fourth powers. So you found an ingenious way that if there existed some solution, then you could find a solution of smaller size and you can then iterate this construction. And that's how we solved the fourth power version of his question. But in some sense, one of the reasons why people believe that Fermat did not actually have a solution is because these methods, like infinite descent and so on, they don't really work that well when you look at powers beyond the fourth power. So some of there's a natural limit to what, to which cases of Fermat's question, Fermat could have attacked and that natural limit somehow seems to be n equal to four. So some of all the methods he had at his disposal, some of them he invented kind of stopped working beyond the fourth power because the geometry of the corresponding equation, x to the n plus y to the n equal to z to the n, that becomes very much more complicated for higher and higher values of n. And the tools he had at his disposal in the 17th century would not address, would not be able to really robustly address these questions for small, for values of n bigger than the fourth power, bigger than four. All right, any questions? I don't see any, is that right? No, no. There's, no, well, there is a couple of questions. So sir, can you please explain again why the fourth power example wasn't applicable? And there's another one, does the generalization for odd primes work because if it works for n equal to four, it can be extended to x power four and plus y power four and equal to c power four. Yeah, I mean the, okay, I've not explained to you how to form as proof of x to the four plus y to the four equal to z to the four has no solution that integers x, y, z with x, y, z not equal to zero. But there's a method of infinite descent that kind of uses, it uses a, in fact, this parameterization I talked about for the case x squared plus y squared equal to z squared. And there's some sort of procedure which I'm not really going into it. And the fact that you would reduce everything to the case of odd primes prime numbers p is because every number can be factorized as a product of primes. So if you have x to the n plus y to the n equal to z to the n, and if p divides n, right, some of you can pull out the p in the exponent, you can write it as x to the n upon p to the p, et cetera. So yeah, so it's something you can think about as an exercise, yeah, that you, it's enough to prove Fermat's, to answer Fermat's question in totality, it is enough to answer it for the fourth power because for squares, in fact, it is false. In fact, the sum of squares can easily be squares, but for, it's enough to answer the question for fourth powers and then for odd primes p, yeah. Okay, there are two more questions, if you want to take them now. So do we have counter examples for Euler's conjecture for k equal to greater than three also? I don't know really. I think, yeah, what happens is if n is bigger than k, if the exponent is bigger than k, then the variety, then the corresponding set of solutions, the loci, the complex loci of solutions has no rational parametrization. So one expects there's a slogan in number theory that topology dictates arithmetic. So if the locus of the equations you're studying, the corresponding geometric object has a complicated topology, then one expects that the rational solutions are very few. So that's somehow the philosophy of why, if you look at this equation, this Euler type equation, and as long as the exponent is bigger than the number of variables, then one expects very few solutions, but there might still be some sporadic solutions, yeah. But I don't know the exact status of this question currently. Okay, so there's one more. What is the maximum known value of n minus k for which the generalized version of Fermat's last theorem is disproved? Okay, so again, I don't know. Maybe we'll have to Google that, but yeah, I don't know. Let's move on. Okay, I think I'll move on. So, okay, so Fermat was, I mean, of course, number theory has been in a state of continuous development for thousands of, maybe a couple of thousand years, but Fermat still in the 17th century was at an early stage of the development of number theory. So he had a few tools at his disposal. Some of those he invented like the method of infinite descent. The other standard tool he had was to, if you wanna show an equation has no solutions, sometimes you can kind of rule out solutions by doing something much simpler, which is by looking at that equation in some sense from the viewpoint of some small number, right? So this is the idea of using congruences. So we have a very simple illustration of this basic idea. Suppose you look at this equation, x squared plus y squared equal to one million and three. And suppose you need to show, suppose you want to show it has no solutions and integers x and y. Now, what you can do is now one million and three looks like a very large number, at least to me, right? So rather than looking at this number by itself, you can look at it with respect to four, right? You can look at the remainder leaves when you divide it by four. So when you divide one million three by four, the remainder is just three, right? So then the question is, if you have a number which leaves a remainder of three when divided by four, can it be the sum of two squares? Now, actually, the answer turns out to be no, because when you look at the square of an integer, then when you divide it by four, it either leaves a remainder of zero or one. Now that's again, a simple exercise. And therefore, if you sum zero plus one, right? Or one plus one, you never get three. So therefore, if you look at this number, module four, this is the technical term. If you look at this term, if you look at this equation with respect to remainders, all these quantities leave when divided by four, you easily rule out a solution, right? Because if there existed a solution, there would also exist a solution which would work with respect to four, module of four. And just module of four, you show that it has no solutions. And hence you rule out. Okay, okay, okay, okay. Yeah, okay. I take it that was not a question, right? Okay. No, no, that was not a question. I don't know where it came from. Sorry about that. Yeah, no, no problem. Okay, so that was another tool in his arsenal that he could use congruences. Though this notion of congruences is formalized later by Gauss. So, okay, so these were tools and some of the tools he had, as I said, the guests nowadays when when people look back at what really Fermat had in mind when he said he had a solution. Because Fermat was a great mathematician, right? He was not trying to be, trying to just kind of pull a fast one by saying he had a solution. He really thought he had a solution. But the point, but the methods he knew somehow were inadequate when one looked at the question for powers beyond the fourth power. I'll come to another reason why people think he might have been mistaken in thinking he had a solution. But again, because he was, I mean, one of the features of being a good mathematician is that even your mistakes can, you have to make good mistakes, right? So in the sense that the mistakes also can generate interesting new mathematics. And certainly Fermat made a fantastic mistake if indeed he did make a mistake in claiming he had a solution because it stimulated much further work. And because of the fact that it was Fermat who had claimed that he had a solution. And as Fermat was a great mathematician, people were very motivated to try and find, recover the solution, right? So it was a great, it was making mistakes also. It's important to make fruitful mistakes. All right, so as I said, one of the other things that may have led Fermat to think he had a solution is that this property of numbers, integers, that every number can be uniquely factored into prime numbers, which is kind of something intuitive and which one can prove. Or the integer actually goes false when you look at more complex number systems. Like instead of looking at the integer Z, you can look at the integers with some version of integers which are linear combinations of integers with square root of minus five, right? So these are numbers of the form A plus B times square root of minus five, where A, B are integers. So if you look at this number system, then in fact, then this number system does not enjoy the property of integers that every number can be uniquely factored into primes. First of all, I want to define what it means to be prime carefully. But for example, here's a numerical example of why numbers like six can be written as two times three and also can be written as one plus root minus five times one minus root minus five, right? And both, and this factorization that they're not related to each other in any simple way. So somehow this tells you that you can factor numbers in this larger world of numbers Z root minus five in ways which are not unique, right? So somehow this uniqueness of factorization property which holds for integers goes wrong when you work in larger number systems. And in fact, if this uniqueness of factorization was uniformly true for all kinds of number systems, then in fact, you could prove Fermat's last theorem rather easily. So one other natural guess for why Fermat thought he had a solution to his problem was that he might have rather naturally assumed that this uniqueness of factorization into primes which works for integers works for slightly larger number systems. So that's another reason why he might have thought he had a solution. And in fact, this is false and hence, yeah. So actually, so when you try to think about Fermat's last theorem, the one natural way to do it is you factorize the left hand side. So the X to the P plus Y to the P equals Z to the P, right? We're looking at Fermat's equation for the exponent P and just so for example, if you look at X squared as Y squared, just to give an example, you can factorize the left hand side as X plus IY into X minus IY, right? So you can generalize that factorization for higher powers, for higher powers for P bigger than two. And you kind of factorize it, the left hand side of the equation into linear factors, but the kind of the coefficients turn out to be something about P through the unity. So now if you assume that number systems like generated over Z by P through the unity, right? These kind of number systems, linear combinations of powers of Zeta P to the I with coefficients and integers, if this number system had uniqueness of factorization property, then it would be easy to prove Fermat's last theorem, right? But in fact, it is known to be basically false for almost all prime speed. So this uniqueness of factorization fails. So one natural way to approach Fermat's last theorem runs into a major roadblock, right? Which Fermat may not have anticipated. So just conjecture because we don't exactly know what proof he had in mind. So in fact, Kumar, this mathematician I mentioned earlier gave a beautiful sort of checkable criterion for Fermat's last theorem. So he gave an explicit series of numbers. Now again, I'm not gonna emphasize the exact nature of these numbers that are given by this generating series. So if P does not divide the numerators of a few of these Bernoulli numbers, these are called Bernoulli numbers, then Fermat's last theorem is true because then somehow this numbering Zeta P behaves as if, at least for the purpose of Fermat's equation, behaves as if it has unique factorization. And hence you can kind of carry out this naive idea, which again, I'm not explaining what the naive idea is, but there's a natural way of attacking Fermat's last theorem which is to use uniqueness of factorization of these number systems. So Kumar gave some criterion for when you could carry out the naive idea. But unfortunately that even now one does not know whether his criterion will be valid in the sense will be applicable for infinitely many primacy. So though it's a beautiful criterion and it's theoretically of great importance, it's used for Fermat's last theorem is still limited. So until why is maybe I'm gonna come, I'm giving away the punchline a little bit, but until why is solved the problem in 1994, Fermat's last theorem had been checked for lots of values of P up to several million, right? But again, in mathematics, the proof is the holy grail. So it can't just be a plausible statement. You really wanna prove it for all primes P, right? So now one great thing which happened, which actually led to the solution of Fermat's last theorem was that someone had the idea of Frey, German mathematician Gerard Frey, I think sometime in the 1980s, had this idea that instead of looking at this very, very complicated equation, X to the P plus Y to the P equals Z to the P, when P is large, this is very, very complicated. He somehow had the idea of reducing this Fermat's last theorem to some rather exotic property about elliptic curves, right? Elliptic curves are much simpler equations. They are equations in degree three, right? So what is an elliptic curve? An elliptic curve is an equation of the form Y squared equal to XQ plus AX plus B, right? With A and B rational numbers. And then there's some technical condition of the right-hand side not having multiple rules, right? So we're looking at such equations with the coefficients A and B in the rational numbers. And so these equations are of course much simpler than the Fermat's equation X to the P plus Y to the P equal to Z to the P because when P is large, that is very complicated. Well, this is just a cubic equation, though it is still very mysterious from the point of view of number theory. Lots of very famous conjectures are around about these kind of curves, which are called elliptic curves. Right? So if you plot the locus of these, the solutions of this equation, for example, for these, some of these particular equations with particular values of A and B, you get these sort of simple pictures, right? So there's some of the great shift which happened was that P, that the Fermat's last theorem was related to some much simpler, a priori things, things which look easier, which are elliptic curves. So these are equations of a fixed degree. They're the Y squared equal to some cubic polynomial, right, rather than some high degree equations. So that was one, so and one of the great things about these elliptic curves is that they have a group law on them. So I want, so there's a way, just as in this rational parameterization of I talked about, starting with this one minus one zero, you can pass the lines and look at the other points of intersection. In this case, if you pass a line through a rational point P, it'll intersect the elliptic curve in two other points, right? And somehow, okay, because you can turn this into some group law. Anyway, so the main point about elliptic curves is that they are cubic equations and they have this miraculous property of we start, that you can start with two points on this curve and produce a third point, right? In some systematic way. This is called the chord and tangent process. So, but anyway, but the main thing you can take away from this is that Fermat's last theorem is now got related to studying kind of equations which look much simpler because they're just cubic equations, right? So maybe I will not, for the sake of time, I don't know how much time I have. When I'm looking at these slides, I cannot look at the time. So you have time. Yeah, yeah, 30 minutes. 30 minutes, okay. About 30 minutes. So yeah, so, okay, so we're looking at, so I'm making some kind of lateral shift. I'm not yet explaining what is the connection with Fermat, but it turned out, right? That the solution of Fermat's last theorem turned out to be related to elliptic curves which are equations of a totally different sort, right? They're cubic equations, y squared equal to cubic. And they're very special kind of equations because it turns out that the locals of solutions of this have a group law on them, which roughly means that starting with two points, you can systematically produce a third point. By joining the two lines, by joining the two points with the line and looking at the third point of intersection and then reflecting it in the x-axis. Anyhow, so, but these equations have this property that they have low degree, right? They're just of degree three. Well, Fermat's equation, Fp is of degree P. When P is large, that becomes larger and larger degree and of greater and greater complexity. Okay, so people study rational points of these elliptical, right? There's a very famous conjecture of Birch and Swinney-Dyer which tells, which in some sense, analyzes the structure of these rational points though no one knows how to prove this conjecture. So again, by a method of dissent, people, I have understood what the rational solution, so E of q means looking at the values x and y which satisfy this y squared equal to cubic equation with x and y rational numbers. So you can look at this rational points and they form a group, right? And that group has a particular structure, right? So anyway, so the elliptic curves are totally different kind of beasts rather compared to Fermat's equation and they're much studied. And there's some sort of basic theorems about them which were actually proved by a model. And it turned out that the solution of Fermat's equation was intimately related to properties of elliptic curves. Right, so this is some sort of describing the rational points in a particular elliptic curve. Okay, I'm not gonna use this much. Okay, so now let me get back to Fermat's idea, to Fermat's equation. So the amazing idea of Gerard Frye and I think it was preceded by some hint of that idea in the 1970s by a French mathematician, hello Gourache. They had this, so this is a totally kind of brilliant idea of unexpected that reduced the study of Fermat's equation, Fp, so this is an equation x to the p plus y to the p equal to z to the p. Now their idea reduced studying this equation to the study of elliptic curves. Now why is this idea so brilliant? Because if you think about Fermat's last theorem, it is asking you can divide by z to the p and essentially you're led to thinking about the equation x to the p plus y to the p is equal to one, right? And instead of looking at the solutions and integers, you're looking at solutions in rational numbers x and y. Now instead of looking at rational numbers, if you look at solutions just over complex numbers over real numbers, then you can think of this as some kind of space, right? You get a solution, you get some locus and then you can look at the complexity of the space, the topology of the space. And then this solution of this equation turns out to be topologically like a doughnut with these many holes, right? Roughly p squared upon two holes, right? So in some sense, this is an example of a Riemann surface and you can think of its top complexity and it turns out to be a very, very complex kind of surface, right? So it's a two-dimensional kind of structure, two-dimensional manifold, and it has a growing complexity with the primes p and the complexity is roughly like p squared upon two or p minus one into p minus two upon two, right? So the complexity of the Fermi equation grows fast with p. But on the other hand, the elliptic curve is some equation of fixed degree, cubic basically, a degree three. So the amazing idea of Fry and Hedda-Gurash was that the rational points and fp curves of growing genus and hence growing topological complexity as the prime p increases are actually controlled by properties of elliptic curves that actually correspond to a doughnut with one hole, right? So somehow, okay, you have to kind of imagine what I mean by this. And sometimes when you plot the points, right? Some of you will get some space which looks like a doughnut with several holes. And in the case of Fermi's equation, they're p minus one into p minus two upon two holes. Well, elliptic curves just have just one hole in them, right? So somehow you've done some dramatic thing where rather than thinking of Fermi's equation which is very complicated, you're thinking about elliptic curves which topologically are much less complicated but they're still very subtle, right? So and the equivalence of the way to solve Fermi's theorem using elliptic curves is still rather subtle, okay? So the other great progress before why is the solution in Fermi's equation of Fermi's last theorem was that in 1983, Gert Falking, the German mathematician proved that this equation x to the p plus y to the p equal to one for any prime p bigger than three has only finitely many rational points, right? So Fermi is saying that this equation has no rational points. So what for any prime p? What Falking's did as a particular case of his much more general theorem where he dealt with all curves of genus bigger than one, he showed that this equation, x to the p plus y to the p equal to one has no solutions with x, y rational numbers as long as p is bigger than three. Sorry, he showed that it has only finitely many rational points, rational solutions. Fermi was asserting that it has no solution but at least he showed that it has at most finitely many. Right, but that is not good enough because one is trying to show that it has no solutions. Right, so it's still left the possibility that there existed some sporadic solutions only finitely many, but there could be some solutions. Right, so, but it was a dramatic piece of work. It was a, it again was a brilliant proof for which Falking's got the Fields Medal. And again, it harness the entire power of mathematics which has been developed in the last couple of hundred years especially in this revolution and algebraic geometry brought about by Rothending in the 1960s. So it used a lot of very sophisticated mathematics but to again answer some very simple questions, simple looking questions about numbers. Any, any question at this point? No question, but let me just tell you that you can speak for another 10 minutes then after that we'll take questions. All right, okay. Thank you. Okay, so this is a picture of Gert Falking's. Okay, it was one of the great number theorists of our times. And so what was, what was the, okay, to describe precisely the idea of Frye would be rather technical, right? But like, what was the idea? Just at the level of logic or just at the level of, yeah, at the somewhat superficial level I can describe what the idea was. So that the idea is to argue by contradiction, right? So somehow that's one tool mathematicians have. I mean, the logical tool, if you wanna show something is true, I mean, assume the contrary and try to derive a contradiction. So instead of, so suppose you start with a solution to Fermat's equation, right? A to the P plus B to the P equal to C to the P where A, B, C are all non-zero integers and P is a prime bigger than three or something, right? And suppose you start with a solution and you kind of massage it in some way so that you make A and A, B and C have no common factors and a few, a couple of more properties. And then you write down the, some elliptic curve, right? So you just write down, you just write down and that was Frye's great idea that you just kind of wrote down this elliptic curve. Y squared equal to X. So you make the roots of the cubic equation on the right-hand side, these p-th powers which are occurring in a hypothetical counter-example to Fermat's last theorem, right? Counter-example to Fermat's last theorem for exponent P and you create out of this, this elliptic curve. And of course, the one is trying to, one is going to try and show, get a contradiction by showing that this elliptic curve should not exist, right? But how does one show it? Because we're starting with the solution, you've written down an elliptic curve and then actually Jean-Pierre has said another great French, another great mathematician of the 20, of our modern times. He actually made Frye's idea precise, saying that if indeed Fermat's last theorem had a counter-example for exponent P, this elliptic curve should at least conjecturally, he made some conjecture, which were very precise. This elliptic curve should have properties which should be so contradictory that it should not exist, right? So Viles proved that this elliptic curve came from another world of modular forms and he proved a very celebrated conjecture of Shimura Tanyama. And so what happened was, I mean, it's like a very, very complicated piece of work for Viles' paper itself is like 200 pages of very closely reasoned mathematics and that paper itself relies on thousands of pages of work of earlier mathematicians. So it's hard to describe in detail what he did, but he basically showed that this elliptic curve does not exist, right? And he did this by verifying a conjecture of Shimura Tanyama and relying on earlier work of Ken Ribbit, a mathematician at Berkeley, who showed that if the Shimura Tanyama conjecture is true, which is some conjecture about elliptic curves, then the kind of curve which Fry had produced from a hypothetical counter-example to Fermat's last theorem, right, would lead to a contradiction. So as a logical consequence of Viles' work showed that work of Ribbit and Seher, when combined with Viles' proof of the Fermat of the Shimura Tanyama conjecture, the Fermat elliptic curve had contradictory properties and it would certainly not exist, right? So some of that was the Graw-Lienes of the demonstration. It kind of synthesized Viles' work is a marvel of ingenuity. It's actually a testament to human perseverance and his kind of tenacity. He worked for seven years on this problem and he synthesized a lot of work of several mathematicians and came up with a very ingenious proof and that work has lasting impact. I have used it in my work on the proof of Seher's conjecture and many people have actively currently studying how to use it for proving that there's this famous program in mathematics called the Langlands program in which a lot of algebraic numbers here is focused on and Andrew Viles' methods have led to great breakthroughs in that. Okay, so the logic of Fermat's last theorem then is the proof of Viles is that you start with a counter-example to Fermat, A to the P plus B to the P equal to C to the P and this gives rise to some elliptic curve and then a series of arguments show that that elliptic curve cannot exist, right? And so that's kind of the basic logic of the proof and so this is a picture of Andrew Viles born 1953, roughly around the same time as Gert Falkings, okay? So, but Viles of course, the proof is his where he relies upon the work of many, many people across the globe. So his work kind of truly illustrates the international kind of character of mathematics or many of the sciences that it's used work of Gauss, Kumar, Arten and all these people are from Germany and Arten was Armenian descent, Andreeve, Shimura, Taniyama, Seir, Grozendijk, then there's very important French mathematician Jean-Marc Fontaine who passed away a few years ago, Jean Tait, my advisor, Harouzo Hida, right? Some of his Hida's work was an inspiration for Viles and it's a synthesis of the work of many people, of course, combined with a brilliant argument of Andrew Viles himself. So in fact, the sort of story is it was, of course, some of you I think may have seen the BBC documentary on Viles' work, which if you haven't seen, I would encourage you to see. It's quite inspiring because Viles, it was his childhood dream of proving Fermat's last theorem because as a boy of 10 years old, he had gone to the public library in Oxford and borrowed a book in which he had seen the statement of this theorem and he could, as a 10-year-old, he could understand the proof and he could understand the statement and he was very motivated to try and prove the theorem. But of course he tried as a boy and he did not succeed at that time. Then he went on to do graduate work and again, he did not work directly on Fermat's last theorem because it was kind of a suicidal problem maybe because it seemed like some isolated problem and you could work for years on it, get nowhere and you'd be stuck, you'd probably be unemployed at the end of it. But on the other hand, in the 1980s, when people showed that Fermat's last theorem would follow from some much more systematic conjectures in Nambath theory, at that time he was motivated to actually think about the Fermat's last theorem full-time because instead, because he knew that it would follow if he proved the Shimura Tanyaayama way conjecture and somehow he had this intuition that though the Shimura Tanyaayama way conjecture itself seemed impossible, that at least if he worked on this conjecture, because it was such a central conjecture to the subject that even if he did not succeed, the mathematics that he would develop would still be interesting. So it would not be sort of just a waste of time. So somehow one of the features of one of the qualities of a mathematician which in fact, I've heard Andrew Wiles himself saying is that one should know it, one should have maybe some intuition of when a problem is ready to be solved. Sometimes you might be trying to solve a problem for which the mathematical technology I mean is not there. So the solution might lie hundreds of years in the future. So Wiles maybe had this kind of uncanny intuition that maybe the time had come though it was probably it was only he who had this intuition. No one thought that the Shimura Tanyaayama conjecture was accessible at the time he proved it but he had this intuition that maybe it was ready to be solved and he worked on it. So certainly intuition is a very strong part of doing mathematics. So the story is okay just to conclude in after seven years of hard work he announced the proof in 1993 it turned out to have a kind of gap but then on the other hand within a year he actually succeeded in proving it, right? And the proof made it to the front page of the New York Times and the proof is extremely complex, very original even bizarre at first sight it uses lots of sort of very strange looking arguments but now they were in retrospect after 25 years people have systematized the methods and now it has become a very coherent piece of theory which people keep using to prove more and more interesting statements. For example, my proof of SARS conjecture with my French colleague and nine few years ago on SARS conjecture certainly used in an essential way the techniques of vials and developments of it. So yeah, so maybe I think I'll skip these slides. So for my last time and one still many departments from all over the world still keep getting elementary solution. So the question is one might ask the question does the solution have to be this complicated because this question is very simple, right? Fermat's last theorem is a very simple looking question and the solution is extremely complicated and of course the solution is in some sense is far more interesting than the problem itself where the solution has led to much more sort of has led to techniques which attacks central questions in number theory while Fermat's last theorem just as by itself is a curiosity. So the question is, can they be an elementary proof? Now it's highly doubtful there is where it doesn't stop people from trying to find elementary proof and one gets even a UCLA here or departments all over the world one keeps getting solutions which at least as far as I know, always wrong. Because as I said, it's very plausible to come up with arguments which look kind of kind of promising but they always have some sort of hidden flaw in that. But I would like to conclude by saying that this work of Andrew Viles is like an inspiration because he worked on it doggedly for seven years and came up with this amazing proof and I did not mention the connection with Ramanujan but certain congruence Ramanujan observed in 1916. It was a congruence with respect to the prime 691 actually led to a series of developments across the 20th century which were then systematized by Seir and this Jean-Pierre Seir and Sunit and Dyer in the 60s and 70s and some of these observations of Ramanujan have fed into the proof of Fermat's last theorem. So maybe I'll stop here and answer any questions if you might have them. Thank you. Yeah, thank you Professor Khare. There was one question and it was followed by a comment by one of the members of the audience but I'll still read the question for you. So what would happen if we take three different powers in the FLT equation? I mean an equation of the form A power X plus B power Y equal to C power Z. Does it have integer solutions? Again, I would not really... I don't know. Yeah. I would support that. There cannot be a uniform answer. I think there might be some values for which it is true. But yeah. But again, an illustration of the factor number theory you can ask lots of interesting questions and as I'm illustrating personally it's very difficult to answer that. Okay, so I'll ask... I'll encourage participants to type in the questions. We still have time. About seven minutes. And in the meantime, maybe I can ask you a question which is not mathematical but I would like to know how you became interested in mathematics. When did you decide to become a mathematician? Maybe around sometime in high school I think I was always interested in physics, math, maybe also English literature. But so I, yeah. Sometime in high school I like this property of mathematics that you can somehow carry it in your head. In some ways it's a very organic subject where you don't need anything to start doing it. Even if it's a very difficult problem like Fermat's last theorem, like Andrew Weisen as a boy of 10. Even as a boy of 10 you can somehow at least start to think of something. Maybe you won't make any progress but it's something you can carry in your head and I like this property of mathematics something you can do by yourself and you need this pen and paper. And on the other hand it has this, it is quite difficult of course. So you get stuck, you get frustrated but it's a bit, it's also addictive, right? You can, if you start to think of a question especially if you fail you can either give up, right? Or you can just keep thinking about it. And at the point is that if you keep thinking about it it rewards this act of thinking about it. It's kind of a, I think it's a very sort of forgiving subject unlike, so for example being an athlete then at least you need some athletic prowesses and so on. In mathematics I think anyone can start thinking about mathematics and you can, as long as you're persistent as long as you're interested you will probably make progress and I like this quality of mathematics. I was never, I was actually never that good at mathematics, high school. I mean I was reasonably good but I was reasonably good but I was never that good. I'm sure all of you, all of the Olympiad students are much smarter at this stage than I ever was but I was interested and I kept thinking about it and persisted. Right. And maybe a follow-up question. I mean growing up, did you ever know of Math Olympiad in India? Math Olympiad. Yeah, I think in fact at that time it was in a very nascent stage. Maybe I think I even participated. I don't think I made it to the, I did do well in the Maharashtra kind of, I forget what it was but yeah, I think it existed in some form but maybe not in the highly developed form it is now in India and of course I would like to also congratulate all the people who work with the Olympiad students. I mean it's now a very sort of successful effort in India and I did participate in some form in this Olympiad. I don't, not with great success though. Right, so we have another question in the chat box along the same lines as the previous one which appeared a while ago and so someone continues to ask the question and the effect equation, what if we take objects other than numbers like integer matrices? Yeah, I think that that has been thought about. Again, I don't really know what the answer is but it is a question which people have thought about the version for matrices for Fermat. So for example, I'm illustrating the fact that my personal interest in mathematics is not these kind of difantine equations. In fact, as a graduate student, I was motivated by Ken Ribit, Ken Ribit this very distinguished number here who said Berkeley. He had made this observation which had reduced Fermat's last theorem to proving this much more structural conjecture called the Shimura-Tanema Conjecture. So I spent many years, two couple of years as a graduate student studying Ribit's work because Ribit's work is a rather sophisticated piece of mathematics. Again, I could not understand it at all when I started reading it but I kept trying for maybe six months and gradually I kind of understood some parts of it and so again, I really feel that in mathematics if you engage with a set of ideas and keep it in your mind, then kind of, even if you don't understand it at the beginning, persistent space, right? So with that work of Ribit, for example, I studied it as a grad student and I still keep using it, right? So my interest in Fermat's equation was important even on my development as a mathematician because it was hot at the time I was a grad student. Ken Ribit had just proven this fantastic theorem and Ken Ribit's techniques were very complicated using the latest mathematics. It had nothing to do with Fermat kind of methods and it was not directly related to Fermat's equation as such but it had a consequence which implied which had, it had a consequence for Fermat. So, yeah, so that, that's, so Fermat would have an impact on my development as a mathematician, yeah. Thank you. And may I ask you another question? I mean, what's your current area of work and what are the problems you're working on? Maybe briefly if you can explain to the students. Yeah, yeah. Non-technical. Many of us doing algebraic number theory our work has been impacted by Wiles's work. So I've kept on trying to develop Wiles's techniques and also one of the central goals kind of the central goals of algebraic number theory is something called the Langlands program which is a series of deep conjectures relating algebra, geometry, and analysis, right? Roughly speaking. So these are different branches of mathematics perhaps when I start as an undergrad but they actually bridged by these kind of vast conjectures which all have to do with number theory they go by what is called reciprocity laws. The oldest such reciprocity laws due to Gauss from the 18th century. So the quadratic reciprocity law. So Langlands conjectures are like a super generalization of quadratic reciprocity laws. I mean, I'm interested in proving cases. The number theory progress is very hard ones and there's very grand conjectures. And so one thinks, one knows what should be true but one doesn't know how to prove even a fragment of them, right? So there's these grand conjectures but then one proves them by very, very sort of kind of piecemeal kind of, you do it one at a time. So Andrew Wilder says his proof just proves a very small fragment of the Langlands program. But on the other end, a very important fragment and he has made a dent and one can then keep on trying to work more in that direction. Yes. Okay. So I don't see any more questions in the chat box and it was really a pleasure listening to you. It was a wonderful talk and thank you very much for your time and thank you for spending the time with us on the National Mathematics Day in India and on this occasion of the emphasis hours. So with that, I should conclude this session. Thank you very much once again, Prof. Sakhar. It's lovely listening to you. Thanks, bye-bye. Thank you, thank you all. We will now have a break of about 10 minutes and we'll start the award function at 12 o'clock. We'll be back. The room will.