 Well, good afternoon. Welcome to everybody. Thank you for joining us for this special event. So as you know, CISA and ICTP have been sister institutions for so many years, for 40 years and CISA actually is celebrating the 40th anniversary this year. And so we would like to work together as much as possible. And then this is a nice opportunity to strengthen our collaboration by the creation of this Institute, which is an E-CAP Institute for Geometry and Physics. There will be a joint institution between CISA and ICTP with the support of INFN. So we have here Stefan Rufo, who is the director of CISA, Alberto Lerdo, representing INFN, who is the head of INFN for theoretical physics, and Cumbrum Van Fajo will be the lecturer for the inaugural lectures for this event. So probably we can improvise the order. So at some point I have to give some, to say some words, which I already did. And then I think Stefan wanted to say some words, and then probably we sign after that. And then Alberto would like to say some words also, you know what I mean? And then we all leave the floor to Cumbrum. So let's welcome Stefan. So I'm very happy to be here on this occasion. It took us three years before we arrived at something, really a project that we can finalize. And I'm very happy that we signed this agreement. I would like to say a couple of things. First of all, the administrative structure that we decided to give to this institute took us some work. And I must thank the administrative offices of CISA, I see some people sitting there, for the very tough work they did in finding in the statue of CISA really the way to make the institute possible in the context of an agreement. So the institute will really have an autonomy also from the financial point of view. And we have an investment plan for the next four years. So it really became a concrete and based on ground activity. And another thing I would like to say is that the institute started really from the activity of researchers at both CISA and ICTP that are already working together in this area of research. And so in some sense the institute gives a home to this activity which already exists and will for sure boost the activity that is already present in Trieste together between CISA and ICTP. And the last thing I would like to say that this will allow CISA to reopen the building that we have here which is not yet ready to host the activity but we decided to profit of the presence of Professor Vafa here in Trieste to make the official opening. The building will be ready by the end of the summer and the new institute will occupy the first floor and also the common rooms for the joint activities. So this is all what I wanted to say and I'm really happy for this step in collaboration between CISA and ICTP. Thank you very much Stefano. Alberto, do you want to say some words? I'm very glad to be here today on the occasion of this inaugural ceremony for the new institute for geometry and physics and to bring the greetings of INFN and of its National Committee for Theoretical Physics that I have the honour to chair. When a couple of months ago I heard about this idea of having a joint institute between ICTP and CISA dedicated to theoretical physics, actually to the more formal part of theoretical physics and to geometry, I immediately gave my strong support and congratulations to the proponents because I really think that this kind of enterprises are important and useful. As you may know, INFN and ICTP have a long-standing tradition of collaborations over the years which include, for instance, several agreements that allow to financially support some activities like workshops like the one that we have now or to sponsor short-term visits of INFN associates to ICTP. On the other hand, CISA with its many research groups plays a very important role inside INFN as far as theoretical physics is concerned. So on these basis, I certainly believe that the new institute for geometry and physics will help to strengthen even more the collaborations and the ties, the strong ties that ICTP and CISA on the one hand have with INFN. And I personally hope that this new institute will provide new opportunities, a new synergy with the National Centre for Advanced Studies that INFN runs in Florence, which is called the Galileo Galilei Institute for Theoretical Physics or simply GGI. The GGI organizes several workshops over the year and also some schools for PhD students. And I hope that this new institute will establish a profitable collaboration with the GGI by exchanging information and even visitors if the circumstances will allow. So on behalf of INFN and its National Committee for Theoretical Physics, I would like to wish a lot of success to this new institute for geometry and physics. And I personally looking forward for new collaborations and joint activities in the years to come. Thank you very much. Thank you very much Alberto. So now we will proceed to sign the agreement within the two institutions. So we have the two documents here and then we will immediately give the word to Coombrun to give the inaugural lecture. So it's a real pleasure to start this new initiative with CISA in general. Thank you very much Alberto for the support for INFN. So CISA is always very thankful for all these collaborations over many years and this is a one step forward to move on, in particular this combination between geometry and physics which is very important for the whole community, for us and for the two institutions. And hopefully this will be the beginning of a greater activity in this field. One of the reasons we will hope to have this great activity is because we have Coombrun here. Coombrun has agreed to come and visit ICTP and at least in general for regularly over the years. So we thought that it was the perfect way to start this institute with asking Coombrun to give this inaugural lecture. I don't have to introduce Coombrun to you. It's a well known figure but let me say some words about him. He's one of the greatest theoretical physicists alive, I would say, without exaggeration. And he has this special talent to use very, very heavy mathematics to concrete questions in theoretical physics. And he has been developing many, many, many important results. To name a few, you're just introducing the concept of F-fury. It's a whole community working on that, which he had not only introduced but he has developed to look for phenological applications to it. And probably the one that he's more famous for is for the outstanding calculation for the relationship between entropy and area for black holes in spring theory where you can count the number of states and that coincides with the prediction that the entropy has to be the area divided by 4 which is one of the most famous equations in physics and theoretical physics, I would say. And so without further words, I would like to ask you to join me to introduce Coombrun and welcome him. So, I guess it's worked. So, first of all, I would like to thank you for having me give a talk here on this wonderful occasion for the inaugural of this institute. This is for geometry and physics. Especially here, it's very dear to my heart because I have been coming to ICTP for... Oh, it's hard to count now, 34 years now. 1984 is when I first came here as a student with the summer school and I've been trying to come here every few years or so and it's a pleasure to be associated with ICTP. So, in that sense, I really feel quite a lot of pleasure in giving this talk. In fact, even more, some of my best collaborators are from this both institution in particular Narayan from old days when I was working with him for related to topics in string theory and ongoing work still with Sergio Cicati from CISA. So, to me, to have a joint institute and me being giving a talk is a dream come true. So, it's a great pleasure. I was asked to give a public talk and I see this is more or less like a private talk in the sense that we have some of the world experts on black holes here and I'm going to be talking about elementary aspects of black holes. So, apologies to those group of experts. So, we can perhaps view it as a review of what we know for those experts. A simple review perhaps. So, without much more discussion, let me just turn to my talk. So, this is related to math and physics links provided through string theory. So, in fact, I would like to dedicate this talk to the memory of Stephen Hawking who passed away just a few months ago. So, math and physics have had a beautiful connection over many years in fact millennia and this connection has been growing perhaps in a very accelerated sense since the advent of string theory but it's not new in the sense that a lot of interesting physics has come from joint relation with math and I would say vice versa. Many new ideas in math have come because of connection with physics so it's kind of in some sense it's just one field really sometimes separated artificially to two parts. But I hope to illustrate some of the deep connections first by reviewing some transient features from past, what kind of connections there were between these two topics and then exemplified in the context of work I'm more familiar with in the context of black holes. So, let's go back all the way to the Greek mathematicians and philosophers. They believed that math and geometry should explain nature in some sense and they were very good geometers and understanding aspects of symmetry was really exciting to them. For example, they studied and understood the symmetries of solids in particular, classify these symmetries in terms of platonic solids and they thought that this has something to do with explaining nature and associating objects to them, air, earth, fire, water and the whole universe. They had the vision that there is something to do with reality, with real nature so this vision of relating math to physics is by no means new. This tradition continued over the centuries and I would just like to highlight some example for example by the Islamic scientist Bironi began to use more simple ideas of geometry and trying to calculate radius of the earth for example by going on top of a mountain and looking at the horizon and finding how much that angle differs from 90 degrees and so forth from this kind of measurements of the height and so on he measured radius to an unprecedented accuracy during his time. And not only that, the height of the atmosphere was measured believe it or not a thousand years ago and the way they did it was very simple. They noticed it's hard to believe that the idea that the atmosphere is finite and they measured its height is not a new concept. They didn't have to go outside like astronauts to figure this out. It was already figured out but it's very simple observation that when you have a sunset it takes a while before the sky becomes dark. So the theory they had was that there must be some upper limit to what this height is so that the sun that we don't see still hits those parts of the sky and we still see it but if it goes further down we don't see it because there's nothing further up. So from seeing how long does it take for the sky to get totally dark after the sunset they could come up with the notion of the height of the atmosphere and that's basically this simple geometry will give you some estimate for the height of the atmosphere and they got it within a factor of 2 or so correctly. So it was quite remarkable that you can actually do simple geometry to try to get some facts about nature. And of course we know Newton. During Newton's time this connection became really amazingly strong and clear that math and in particular calculus, the development of calculus became the bread and butter of physics. That is it developed and applied to understanding the laws of mechanics and the universal law of gravitation. This is also well known and so well documented that we all know and it's one of the amazing beginnings of the modern physics the way we think about the relation between math and physics. And the connections we view in Newton as both a mathematician and physicist in a sense, Gauss, which we usually view just as a pure mathematician, amazing work in math and his work in number theory and so on, is well known. Actually to a great extent he was also a physicist even though it's not as much acknowledged perhaps among our community he actually had a belief that perhaps the space around us is curved well ahead of his time and he actually did experiment by going to the top of three mountains trying to actually measure whether these angles add up to 180 by doing the final experiment looking from each top of each of the mountains the other top and by looking at it and assuming that's a straight line that is the light is a straight line by looking at these directions you could measure an angle from each of these mountain tops so you get three angles and he was curious to know if these three angles added up to 180 or not. It is quite remarkable that he understood that in order to measure it you need to go bigger and bigger distances he had clearly known he really knew about curvature and geometry so he thought the real world also might be curved and this was well ahead of his time clearly from the physics and it took about a century or so before we actually got to that part of the understanding. Maxwell who's a physicist now he was he basically learned to basically summarize and add make a consistent theory of electromagnetism in terms of differential calculus and so the power of differential calculus came to being in this context of these beautiful equations that he summarized all these diverse electromagnetic phenomena and discovered in particular that there must be electromagnetic waves so this is an interesting application of math back to physics and then again we have Riemann who was clearly a mathematician talk about Riemann in geometry and higher dimensional manifolds and so forth but again he was extremely deeply interested in physics and he has actually a lot of work on physics in terms of electromagnetism he's probably not very well known among physicists that he actually did write papers on physics related topics and he had this dream about connecting his work in mathematics to physics in some form but of course that again had to wait half a century before it actually happened and that was due to Einstein's work where Riemannian geometry and in three plus one dimensions ended up encoding physics, the gravitational forces by the curvature of space and time so this was Einstein's trying to basically bring out geometry into a central role, playing a central role in the whole physics and further on, Kaluzan Klein starting with Einstein's theory noticed that if you have had one extra dimension to Einstein's theory an extra circle, you would find that you can actually explain electromagnetism in the same frame where you can unify electromagnetism with Einstein's theory so these ideas about geometry, adding dimensions and all that these kind of things are going back and forth between physics and math and it's getting accelerated during the past century and this brings us to the earlier part of the 20th century where the story that I want to tell you starts so this was kind of a setup leading to the story I want to say and the story starts with this fellow, Sirinivasa Ramanujan amazing brilliant mathematician who was addressing with Hardy the following question so he was during his short lifetime, he did amazingly many things but the particular one I want to focus on is the following he was interested in the following question you take an integer for example number 5 and ask how many ways you can break up 5 to smaller integers as a sum of smaller integers so for example 5 can be written as addition of 5 ones or it can be written as addition of 3 ones and a 2, a 1 and 2 twos 2 ones and a 3, a 2 and a 3, a 1 and a 4 and a 5 so the total number of ways you can partition the 5 to integers is 7 if you count how many ways there are hopefully I didn't miss any of it so anyhow you just count how many ways there are to do and as you can see this is not that difficult when the number is small but if I take the number n here instead of 5, let's say 100 or 1000 it's going to become very difficult to actually enumerate all possible ways and they were interested in actually coming up with a formula, exact formula for this number so they in fact came up with an ingenious way to compute this number in fact exactly but here I want to just basically use the results they found only to leading order that means we look at this n instead of 5 being an n where n is very large and they found that the number of ways you can partition this integer grows for leading order if n is very large exponentially in terms of this square root of n with some parameters that they figured out so they came up with such a formula and with extra corrections that they could write all the corrections in such a way they get the exact answer actually so the way they did it was to notice that if you take the partitions of n and you make a function out of it by just taking these numbers over each n and multiplying it by a q to the n and some overall integers you get a series which can be written as 1 over the product of 1 minus q to the n product over all n now this is the answer to the question you want to find pn but this actually ended up to be not a nice mathematical object what ended up to be a nice object is that if you multiply this whole thing by an extra factor q to the 1 24th then it became a nice object now it was a denominator would you have an extra q to the 1 over 24th then this function turned out to be nice properties this is called the Dedekind Aida function and it had a nice property and in particular the property that they had is the following you define this formal parameter that they had here this q if you write it as exponential of some parameter 2 pi i times some complex number tau they notice that the Dedekind Aida function has a property that if you take and replace tau by minus 1 over tau so if you replace this tau over here by minus 1 over tau it turns out it goes back to itself up to some factor the bizarre symmetry it's a totally bizarre symmetry mathematically but it can be proven to have this property and so what they use it for was the following you take if you take tau to be very small almost zero it's just a tiny bit above zero then you find that q goes to 1 so when q goes to 1 if q goes to 1 this basically gets some of all the p of n some over all n and of course the bigger n's contribute more and more and more so as q approaches 1 this gets dominated by larger and larger values of n so by knowing how this diverges how it gets bigger as q approaches 1 they were hoping to get an asymptotic expression of how this diverges what is the contribution of p of n for large n but on the other hand because of this symmetry this is the same thing as saying instead of tau going to 0 tau goes to infinity and in that case when tau gets bigger this gets dominated by that pre-factor that we added which is q to the minus 1,24 so in other words this artificial looking term that was added by hand by Hardy Ramonujan to convert this thing to a nice mathematical object ends up having the key to the property of this the answer of how this behaves when tau goes to 0 sounds like a formal mathematical trick to compute something sounds a little artificial if you ask me from the mathematical viewpoint sounds artificial you want the different functions you change it a little bit to make it something you can say about and use that property so it sounds a little artificial but nevertheless they use it beautifully the answer they wanted and from that they deduce that for large n the partitions of integer grows in this way their method of proof was reinterpreted 60 years after their work in physical terms so physicists working on string theory reinterpreted what these guys had done in a different language in the language of physics and the language was relatively simple to explain so I will remind the non-experts about how that went so in the context of string theory we look at closed loops of strings so like a circle and we're interested in waves like harmonics on these strings and the energy of a string is more the more the harmonic is on it so the larger number of the more harmonics you have the higher the energy and the energy is proportional to n now if you are studying a particular string so if you imagine having a string in a thermal situation all of these harmonics get excited and you have some superposition of all of them so in particular if you ask the Boltzmann if you have a given temperature what is the distribution of which harmonics you get there is a probability for each harmonic which is proportional to e to the minus beta times the energy and the beta is where the Boltzmann constant times the temperature this is the famous Boltzmann distribution and so we can study this now if you have a string of a given harmonics n1 and n2 etc you can have superpositions of them you don't have to have just one harmonics you can take superposition of these and the energy for that situation will be the sum of the corresponding energies and therefore the partition function will look like e to the minus beta e this energy so it's e to the minus beta if you fix e how many states are there for this amount of energy well that corresponds to dividing e to the sum of lower integers in other words the partitions of integers that we talked about becomes the number of times a given e appears so you can rewrite this you can rewrite this partition function in this form p of e times e to the minus beta e because you fix the e and for a given e there are this many different ways to get come up with that e by this partitioning and so finding that syntotic behavior is like taking the beta to zero you want to find what is the behavior of this as beta approaches zero and this is so in quantum mechanics we now shift gears and think about this in terms of quantum theory so in quantum theory when we evolve a state we have an exponential minus i times the energy times time and if we change variables from i times times to beta this becomes like a Boltzmann distribution so in physics sometimes we call this the Euclidean rotation weak rotation going from Minkowski to Euclidean space but this Boltzmann distribution gets related to the unitary evolution and so we think of having a string like the boundary of this disk here the string and you want to evolve it in time beta in the Euclidean sense and so we can think instead of doing the Boltzmann distribution as if you are moving the string up in time so it's kind of like a string rolling in time so you have now instead of having a one-dimensional string which is a circle you have a string which is going like a cylinder going up in fact if you want to compute the partition function take a trace over all these states which means closing up when you propagate upwards gluing it back to itself and your cylinder becomes a donut or a tourist so you have this description of this circular geometry going up and closing it back to itself and this will compute physically the thing that will give you this partition function of this thermal gas of these strings now you can think of it if you think about a tourist you can open it up to a rectangle where you identify opposite sides pair-wise so if you glue this end to that end it becomes a cylinder and this end to that end closes the tourist so if you think about this this becomes a circle when you identify this side with that side so this line is just the original circle we had and this moves in time upwards and you close it back by bringing it back here as a string moving up in time and we are interested but in the limit of small beta as we said we want to take beta to be tiny so in other words we are interested in having a very very tiny beta but instead of thinking of it that way we can think about this as a string going the other way from the right to left instead of from down to up because a rectangle you can slice it either horizontally or vertically it shouldn't matter this fact is a key obvious fact from physics perspective this is a trivial statement this is an obvious symmetry and this turns out to be this amazing symmetry that Hardin or Amunujan used from the perspective of a string picture it's a triviality but turns out to be the amazing key insight since the partition function is scale invariant we can rescale the parameters so previously we had this previously we had this was beta and 1 it turns out if you can rescale both sides by a factor of 1 over beta the physics doesn't change so you can think of it as the same size as we had before 1 but this height is now 1 over beta so in other words the partition function has a symmetry so that is basically the simple idea from physics that gives you the bare bones of what we need as a symmetry that can be predicted just by translating the math problem to a physics problem so in other words instead of string going one way around time you can identify a different time the string is cut the other way and goes this way around the axis of time so you're just splitting what you mean by your space and your time and that is the symmetry that manifests in the physics language so we are interested in the limit of small beta but by this symmetry the computation becomes exponential minus the energy over beta as beta goes to small values but if this goes to small values this gets contribution only from the ground state the lowest e because as beta goes to 0 this becomes e to the minus infinity so you get the smallest value of e and the smallest value of e actually from the physics side due to an effect called chasmere effect which was studied by chasmere a long time ago gets a non-trivial contribution which is precisely that mysterious term that Hardian Ramanujan just put in by hand so that energy that comes out there is needed for the physics to make sense and that mathematically makes this object a good object so physics kind of anticipates that you need to correct it to get the correct object because of quantum effects so the quantum effect knows about this funny thing it's not funny from physics side it's quite natural to get the symmetry and from this you can recover the result that they got in other words the result of Hardian Ramanujan is a physics result written in a math language in a sense that's the most natural interpretation of it the formula I wrote so this was the first story I wanted to share the story of Ramanujan and in this case Hardi trying to compute the partitions of an integer the second story which is what I want to tell you is the story of black holes and this is of course gravity in the way we know it now starts with the work Newton did a while back and followed by the geometrization of Newton's work by Einstein and the context of properties of black hole quantum aspects of black hole starting with the work of Hawking so black holes this is a very simple picture for how we think about black holes intuitively that if you are on a surface of a planet let's say earth and you throw an object up it comes down if you make this object with the same radius a bit more condensed then it's going to be if you try to throw it up you have to have higher velocity if you want to get it out of the hole if you want to throw it away so it doesn't come back you have to make it higher because of the gravity force so in order to reach to go off to infinity you need what's called escape velocity it gets harder and harder to push it up and in fact if the concentration of the matter is enough the escape velocity could be bigger it could be bigger than the speed of light in which case means that the light doesn't escape either and that means that the object is black so to compute it from Newton's perspective you get the formula that the energy that you throw out should be compensating the gravitational one so that at infinity you have no energy and so the escape velocity will be this now if you plug in the escape velocity being speed of light you find the particular radius for which this happens which gives you from Newtonian perspective what's the limiting if you believe that speed of light is the highest speed you get an explanation you get a radius which gives you the radius of a black hole that the radius in which for a given mass not even light can escape surprisingly this turns out to be exactly the correct answer in Einstein's theory even though the explanation is not this trivial necessarily but it's the correct answer so I just keep it like this so in other words Einstein's theory predicts the existence of black holes that with these properties that if you have this much mass with this radius you get a black hole the picture we just took last night I'm just kidding so so it should be something like this that you have some black dots somewhere or sphere something and the only sign of how we see it is basically by objects falling into it at least that was what it was up to recently black holes are so enigmatic that I mean they are bizarre because when you try to describe them they have bizarre properties for example the nature of time changes when you go inside the black hole and this and that Einstein didn't believe they really existed even though his equations lead to it so he thought that there must be some other process preventing the creation of black holes so he tried very hard to try to identify what physical process prevents their creation so it was that mysterious and but we know that if you if you fall into them you hit a singularity in finite time and therefore the black holes as we know it is not understood by because there's no infinity there's no singularity in physics we believe and therefore we don't have a good explanation and I'm sorry to say that despite all these many years we still don't know the answer to that even with all the advances in strength theory so it's still understanding what happens in the interior of the black hole is a big mystery for us there are various types and sizes of black holes that they are classified by three numbers mass their mass, their spin and their charge the typical ones we find in nature have big mass and big spin but typically they don't have any charge or very little charge and then there are two basic classes of them one is stellar size that means their mass is roughly of the order of mass of sun and then there are these super massive ones which are like millions of times the mass of sun so there are these different types and we think that these super massive ones these big ones are at the center of most galaxies if not all and in particular one is at the center of our own Milky Way interestingly enough we are at a very interesting time where people are actually taking pictures of the center of our galaxy black holes so we hopefully will have an image of this due to this event horizon telescope which is this collaboration between various different telescopes trying to get an image of the center of our galaxy and they are currently analyzing the data so hopefully we will have a real picture of a black hole soon and so these are given by the objects which are swimming around it and so on that we know of the properties of this massive I guess 3 million times the mass of the sun inside the center of our black hole and there are probably many of them we don't see because there is nothing which is going around it there is no binary or there is nothing falling into them just solo objects and there will be we won't have much to say about and find them anyhow but one of these ones 3500 light-year waves they just sent a message that Hawking had during his service for him a few months ago sent towards this black hole as a poetic justice here in fact most recently that means a few years ago the black holes were discovered yet another way which I think is most remarkable that is you couldn't see it but you have their gravitational properties if you have two black holes merging since black holes respond create curvature around them as they are going around they create curvatures which are changing in the space around them and this is going over time so they create this ripple in space time which propagates with the speed of light and this ripple was measured due to this amazing experiment LIGO experiment which detected these two signals of this gravity waves by the shrinking and stretching of space around it an amazingly remarkably accurate measurement when the two different sites that they had in Hanford and Livingston give you exactly this or more or less the same signals and this is one of the remarkable things which gave you insight about yes indeed there are black holes and moreover you can actually connect it with space and time fabrics and so on so it was remarkable results of confirming not only the existence of black holes in yet another way even though you cannot see it with light but you can see the effect of them by gravitational one so now you have another window to understand black holes without actually seeing them because they certainly respond to the fabric of space time response to it so black holes have an interesting geometry next to them so there is this area near them called the event horizon so if you are somewhere outside the black hole it's like this point B and if you send the light out you can basically escape to the infinity but it turns out and this is what's called the geometry of this light cone so there are ways if you are sufficiently close to the black hole but not at the surface of the black hole so to speak you can escape if you go sufficiently fast namely with the speed close to the speed of light but if you get inside the black hole as we just mentioned even if you go with the speed of light you cannot escape and that's because in the language of Einstein's theory the geometry of light cone changes in such a way that you will be fully inside the black hole and you cannot get out so this means basically there is this limiting distance close to the black hole which is called the event horizon beyond which if you get closer you will be doomed and you cannot get out so that's called the event horizon of the black hole and the question is suppose you have a black hole which is formed and you can ask what is the fate of the black hole what happens to all the matter that it eats up and what's going to be the future for this matter and there is a hint that was understood already before studying quantum aspects of black hole and this was just studying Einstein's theory so what they found was that if you look at this black hole has an event horizon this area around it if you bring two black holes together you form a new black hole and so they found that just by solving Einstein's equation studying properties of Einstein's equation they found that the area of the new black hole the combined one is bigger than the sum of the original ones so the sum of the A1 plus A2 is smaller than the new area and this was bizarre from physics usually something going up something increasing over time was not seen anywhere in physics at least not obviously except in the context of thermodynamics the entropy so this smells a lot like entropy in the sense that entropy always goes up so they said okay this sounds like there's entropy there somehow related to the area in such a way it will explain this statement so classical Einstein's theory suggested that there's something to do some area has something to do with an entropy but it was bizarre because if you solve Einstein's equation with a given mass or spin or a charge the answer is unique there's only one black hole this is what's called the no hair theorem for the black hole there's only one such solution therefore the entropy is zero there's no entropy there's a unique solution and therefore it's bizarre to say that there is an entropy associated with black hole so somehow there must be some information there in some sense it's very intuitive too because whatever thing you have outside like a dice or something and you're throwing inside the black hole there's some information you had and goes inside you don't want to say that you lost the total information somehow you would think that there's somehow encoded in some aspects of the black hole and as you throw something in the area of the black hole goes up because the mass goes up so the statement is that that fact that you have something going inside the black hole increasing the area must also be somehow related to information that that object had so it sounded like maybe there's something to do with the entropy or the information around to study that area and so this was the kind of thought that Bickenstein was having but they didn't quite know how to actually get it precisely to work and what is this entropy, how do you compute it and what is the proportionality constant and that was the work that Stephen Hawking did and he found the answer by studying semi-classical effects in the context of black hole that is studying quantum behavior what happens when you take into account quantum properties of black hole and his discovery that black holes are not quite black and they in fact radiate was one of the most remarkable discoveries the beginnings of a quantum theory of gravity that sounded to be very exciting and basically making more clear sense of Einstein's theory and his idea was brilliantly simple the idea was that if you go near the black hole in particular near the event horizon we know that in quantum theory you can have particles and anti-particles virtually popping up created from vacuum and going back into the vacuum by the process of this production going back up and down is what is a property of the vacuum of any quantum field theory and what Hawking surmised was that what happens if you look at these pairs that you are produced near the surface or near the event horizon of the black hole well in such a case one of these pairs might be inside and the other one might be slightly outside and since one of them is inside the one inside cannot get out because we said it's inside the black hole nothing gets out so the other one can get out therefore it could look like a radiation so he came up with an explanation of how this could lead this quantum effect of pair production virtual pair production of these particles in the vacuum can actually lead to an emission from the surface of the black hole and this is what is called the Hawking radiation so in fact he found by studying this more carefully that even though that Einstein's theory predicts exactly unique one this gives you the different picture and suggest that the number of such states is e to the area over 4 in Planck units namely if you compute the area of the event horizon in Planck units for every Planck unit there's a contribution of one quarter to the entropy but he could not identify what these states are but this many states but where are they Einstein's theory said there's only one so where are these encoded so he found the entropy is area over 4 and by the relation between the area of the event horizon and the mass of the black hole we talked about and using the simple relation between the energy and temperature and entropy using the fact that the energy is the mass he came up with a temperature so the black hole has a temperature it's a thermal object it emits radiation like any thermal object would it goes like 1 over m for astrophysical mass objects this is extremely tiny so it's very hard to measure and so we will not be able to directly measure this temperature at least that's not imagined possible today but nevertheless theoretically we believe the black holes evaporate by this hocking radiation until they completely disappear now the next part of the story I want to get to is what happens if you bring gravity and quantum mechanics together for which hocking began talking about the semi classical aspects what is a complete theory how do you describe these things merged together and this is the third part of the story I want to mention which is the emergence of string theory which started actually almost exactly 50 years ago by when he wrote his formula and later interpreted and so 50 years ago the story started the basic idea now we think is that the basic elementary objects like quarks should be viewed as strings and this turns out this resolution of thinking about the particles as extended objects like strings or more generally higher dimensional objects like membranes gets rid of the inconsistency between gravity and quantum theory and strings interact simply as you go around two strings can come and join to form a new string or going the other way a string can come and split to two and this process turns out to underline all the forces between strings so joining out here or just noting the joining of two strings as you can see it's a beautiful link between geometry and physics because you see this beautiful surface here is encoding for you simply by the topology a physical interaction so it's a very nice smooth way of introducing interaction in physics unlike the case of point particles where you create or destroy particles here it looks much smoother so string theory is really already starts the whole life by geometry in some sense so it geometrizes also the interaction very neatly and it also has extended objects we call sometimes d-brains or other kinds of brains which and sometimes strings can end on them cut strings open but there's a sad part to the story at least that's what we thought that the dimension of space time is not 3 plus 1 as Einstein may have thought but actually 9 plus 1 and this was forced on us by consistency reasons so what do we do with these extra 6 dimensions and haven't we seen them around us so this was a very sore point for string theorists giving lectures like this people thought you know haven't seen the 3 spatial dimension and we say there's 6 more they're too tiny you can't see them that's the way to resolve the problem you take the space to be so small these 6 ones like little tiny spheres that they're so tiny sorry you cannot see them it sounds a little hard to believe right I mean this is this is incredible and we cannot do experiments yet because they're too tiny okay now physicists would be a bit skeptical about this kind of picture until there's some evidence for them experimentally we don't have experimental evidence for string theory so there's still some people who are skeptical however the question is is there any theoretical use of these extra dimensions other than to say that it's needed for consistency of the theory is there some physical effect in our universe in 3 plus 1 dimension that you can see at least theoretically because of these extra dimensions that's the question one could ask so in other words we want to know what are these extra dimensions good for we have learned how to deal with these extra dimensions very effectively using string theory what kind of spaces are good we have studied these amazingly mathematically rich objects called kalabiyaw manifolds or even more exotic exotic stuff we have put so we know how to study these extra dimensional objects 6 extra dimensions there are a huge number of dimensions 6 we're barely comfortable with 3 dimensions and what you can do with that let alone 6 of them so it's quite a curious thing you can do a lot of things with it but what are they good for physically so we think about macroscopic dimensions like this black surface here and at each point on this macroscopic dimension at each point in your space imagine having a 6 dimensional object like a 6 dimensional sphere or something like that and now string theory has these other objects like strings membranes like 2 dimensional objects or 3 dimensional objects and there's an interplay there could be an interplay between these extra dimensional objects and space namely you can have for example a membrane a 2 dimensional membrane wrapping around one of these spheres at a particular point so at this point in space you imagine there's a sphere and imagine wrapping the membrane around that sphere from the viewpoint of our perspective in 3 dimensional space it just looks like at one point in space there's a particle we don't necessarily see the extra fine fine structure of the extra dimensions because they're too small but we see from the perspective of the extra 3 dimensions that are observable macroscopic spatial dimension has a massive object somewhere put there because of this extra membrane wrapped around this sphere so we think about these 3 as large dimensions and 6 as small and so you can think about an object like a membrane or something could have various dimensions could be wrapping around this extra some cycle inside this extra dimension and from the perspective of the macroscopic dimension it will look like it's happening maybe at the point or maybe in some other depending on what extra dimensions there are there could be other objects this is another picture of the same kind of phenomena so the geometry of the objects and how they are wrapping these extra dimensions could play a role in describing objects we see so for example if you have a membrane which is 2 dimensional and if you take wrapping it around the circle if the circle is sufficiently small it looks like a 1 dimensional object at the end in other words a membrane wrapped around the circle looks like a string or a 3 dimensional version of this membrane wrapped around the 2 dimensional sphere would look again like a string so depending on how you put the objects on what they wrap around you get different dimensional objects left over so there is an interplay between physical objects, membranes and higher dimensional ones and the geometry of the space that are compact part of the space time so now I'm done with the third story so we have three stories I've told you the first story was the hardy Ramonujan they were trying to count the partitions of an integer and they found the formula like exponential of a constant time square root of n as for large as the number of partitions growing within and later physicists interpreted it in terms of counting string harmonics and second story was Hawking black holes and studying quantum aspects of black holes he found that he needs a number of them who grows a lot and for macroscopic black hole is huge is exponential of the area over 4 and so that's a huge number of states that he did not know where these states are hiding what are the microscopic states accounting for black holes and the third story was a string theory emerged with a random luck as a quantum theory of gravity and therefore it should have something to do with the explanation of this second story but it had the bizarre feature that has extra dimensions and we have no idea what are these good for physically what do we do with these extra dimensions and so the basic idea and this started with the work I did with Andy a while back you could use these wrapped membranes of them in string theory to construct black holes so you can have models for these black holes by taking these objects taking a particular point in your space and wrapping around these membranes around these internal dimensions of string theory and then count how many ways there are so from the viewpoint of a 3-dimensional 3 plus 1-dimensional physicist those extra dimensions are invisible so you wouldn't see them as extra degrees of freedom but if you zoom in and see the extra dimensions there might be different ways you can wrap around the object, around these extra dimensions and by different choices of these wrappings you should presumably recover this exponential area over 4 so the extra dimensions are good for potentially explaining where the black hole degrees of freedom are hiding in 3 dimensions so they are not in the macroscopic dimensions they are in the internal dimensions of string theory so that would be a very nice story in the sense that somehow the extra dimensions of string theory come to the rescue to make the 3-dimensional theory consistent so you can think of it this way so you can think of the extra dimensions here I'm drawing like a donut or a torus and so you should think about this donut or a torus at every point in space but I'm now viewing the one particular one I'm drawing here is the one on top of the let's say the origin in space x, y and z are 0 and at that point I can wrap around string or membrane or something around one of these cycles of the string of the sorry of the torus now what we studied with Andy a while back was actually some version of what we studied was the following the objects that we looked at were 3-dimensional membrane the objects not just like a string but 3-dimensional objects we took the 3-dimensional object wrapping around the 2-dimensional surface and when we did that if the 3-dimensional 2-dimensional surface was small you ended up getting 1-dimensional left over dimension like the pictures I drew for you for the membrane wrapped around the circle so the 3-dimensional object for us became a string effectively even though it was not the string we started with so the 3-dimensional object we took a surface we wrapped it around it and the left over dimension was a string it was kind of like this where you just see a string left over because I'm not drawing some of the other dimensions that we had and we can put more and more of these strings and they would get basically create a black hole in other words if you start with this if you think about this as a wrapped membrane the more and more wrappings that you did in the higher dimension would actually create the geometry of a black hole with a more and more massive object there so the question then was can you actually find the area of the horizon and relate it to the degrees of freedom of this string that was the left over question we had to deal with but the hint was already there namely if you take this 3-dimensional object wrapped around the surface you actually get a string because 3 is 2 plus 1 so this 1-dimensional object is a string exactly of the same strings that we had studied in the context of creating the Hardier-Ramunujan kind of counting and we could study these strings having these oscillations on them and count how many oscillations and how they contribute to the structure of the black hole so these same oscillations is given by this partitions of an integer and that we studied before so the same kind of objects that we already studied that Hardier-Ramunujan already studied comes to count the properties of this black hole in particular as you recall the Hardier-Ramunujan formula was the exponential of some constant times root n and if you studied by solving Einstein's equation for a given particular black hole of a given charge you find that the area of the horizon is given by 4 times that same constant times the square root of n and you see that this formula is agrees with the exponential of area over 4 and so you can actually understand the relation that Hawking would have anticipated by connecting it to the work of Hardier-Ramunujan and that's the basic link so a simple link between the math over a span of a century to physics of quantum gravity in the context of string theory back to geometry and relating it back together it was very satisfying to see this link across to the completely different looking areas so the answer to where the black hole degrees of freedom are ended up being hiding in the extra dimensions of string theory so we are not as embarrassed anymore to say string theory have extra dimensions that's where things can be hiding ok so that is the main part I want to say but now since I talked about the dedicated to Hawking this is the last time I saw Stephen Hawking this is about 2 years ago in my office I was about to go for a travel so I had given my office to him to use but I'm explaining here to him that I have a traditional cleaner clean my office for him and as you can see he is smiling so and then we had a party for him where we had the black hole cake and we are showing here a dice falling into the black hole and that represents the information paradox I was telling you about the entropy of the black hole going up when the dice falls in this is all edible by the way including the book underneath and it was gluten free because he had he couldn't yet have gluten free and he told us a story about his travel to Iran and with his healthy body in 1962 it was it was the last travel he did in his healthy body it was just a few months before he discovered that he had just before he started his graduate work in Cambridge so he was telling us about that story it was quite lovely but let's go back to string theory again so string theory is full of connections between math and physics and I've just touched about one little really tiny corner of that for example just give you a few more examples mirror symmetry and normative geometry is another example it's very rich in terms of linking between physics and math this is now more getting very well developed has been much more developed than the connection with black holes and another example is the relation between gauge theory that we know and love in physics and singularities and more precisely the A.D. type singularities in geometry singularities of geometry is that you take discrete subgroups of SU2 and you take a quotient of space by that and you get singular spaces and it turns out those symmetries are exactly related for the interesting one especially to these things that the platonic solids that the Greek had already thought was interesting anyhow and indeed surprisingly the first one of them E6 associated with the group E6 is the thing that physicists today believe is closely related perhaps to grand unification of all forces so via the connection of geometry and physics today we are looking differently now perhaps on what Greeks were telling us they may not have studied exactly the language we would understand now but this intuition that they had that good math or elegant math has something to do with reality is becoming possibly more real with the understandings we have learned in the context of string theory today there are more such things that are happening conformal theories are classified by geometric singularities these are the bread and butter of all quantum systems that we know and this amazing duality or ADS-CFT duality seems to be another yet rich area of connection between physics and geometry many aspects of this are being studied by physicists but not as much yet by mathematicians this would be an excellent area to further study in the context of its mathematical meanings so anyhow it's clear that there are a lot to do and it's perhaps the right time to focus on deepening our understanding of the connection between physics and mathematics and with that I think this newly inaugurated institute for geometry and physics is an excellent vehicle to realize this dream thank you thank you very much any questions, comments further connections between math and physics let me start myself while people think what is the connection between E6 and FHIR E6 and what? FHIR FHIR you have to ask the Greek about that yes yes I didn't realize it was FHIR go back and look at it it's good that they had the courage to connect them I think other questions more seriously? yes, Joao do you think we will be able to test the area formula for some experiments well some aspects of it we have tested namely one aspect at least the fact that the area increases over time has been tested namely in this black hole merger that people that LIGO observed the addition of the area law was higher the new black hole form had bigger area than the sum of the other ones that's not precisely what you're asking of course but to actually say that this is actually counting entropy degrees of freedom you have to know about thermal properties of black holes so measuring the temperature is extremely difficult that's such a low temperature so I think with present techniques it's impossible but we have learned not to say anything is impossible in physics so who knows maybe future thoughts of how you measure this might be possible honestly I never thought you can measure gravity waves or anything like that we have done that now so who knows identifying micro-test of black holes is not enough for addressing the information paradox right yes that's correct would you make some comments on the next step so the question here is that the fact that black hole has entropy is certainly needed as a thermodynamic aspects of black hole but whether or not the evaporation process of black hole is unitary or not whether the information that can be goes in can be retrieved from the entropy that's not enough information in fact Hawking who discovered the exact formula between entropy and area believed the information is lost for example so today we think that the information is not lost we have examples for example in the context of ADS CFD where the Hamiltonian evolves unitary we know that because we know enough about the quantum system and yet it has black hole objects so therefore we believe this must mean there must be a unitary process and therefore information cannot be lost however the method of how it is not lost is not really understood clearly so I think after all these years the belief is that it's not lost the evidence is that it's not lost but mechanism of how it's retrieved is not clear either so people have some ideas for example exponentially small effects somehow giving the information out is one of the methods people suggest as how the information can be retrieved but I think that these are these are still subject to the research and I think it's not settled yet for the questions well if not let me just tell you that there are some refreshments outside and well let's all thank Kunrum for such a great person