 for mathematicians, so and I see there are many physicists in the audience, so the first part may be to elementary for them, because I am going to start with quantum mechanics and things like that, so let me just, what I want to convey to you is why is this, first of all this looks like a rather interesting and amazing connection, you might think you know quantum black holes and number theory and both are kind of sexy words, but what do they have to do with each other, so that is the purpose of my, I do not want to give you all the details, so I just need to tell you a few things about string theory, so for the mathematicians the audience basically the main principles of quantum mechanics is that physical systems correspond to state which physicists usually denote as a ket, but you can think of it as a vector, so I just going to give you a translation, a state is a vector in a Hilbert space, then there are observables, so what is a physical system, for example an electron, electron with some momentum, an electron, a particle, elementary particle going in that direction, that is a state, it is in some state, it has some charge, it has some momentum, but I am just going to give an abstract definition of what is a state and what is an electron, so observables are operators, let us call them like P or A, so these are operators, self-adjoint operators usually, then we have symmetries, one other important thing in physics is symmetry, which is just some group G and you have unitary irreducible representations of that symmetry, acting on this Hilbert space and then physicists often talk about quantum numbers like mass of the particle or the angular momentum of the particle or the charge of the particle and what you should understand is these are just Eigen values of these operators, so you have some charge operator and the charge operator can Eigen value plus 1, minus 1, plus 2, minus 2 and that is what we mean by charge, by the quantum number of that particle, so for example electron is a state with mass m, which is some number which you do not need to know and it can have spin J, spin J, J 3 in the plus minus half and in some units its charge is 1, minus 1, so that is what we mean by a state and in particular one of the important symmetric groups G is, so first of all we live in a 4D space time, so R 3 comma 1, so it is like R 4, but with the metric is not Euclidean, so it is a Lorentz metric, the transformation which leave this metric invariant are so called the Poincare group, which you can think of as ISO 1 comma 3, is it inhomogeneous group of rotations in an indefinite metric Hilbert's vector space and you can have G to be some, so G, so therefore G is going to be ISO 1 comma 3 for sure and then it can have some additional factors like U 1 or SU 2 and so on. So, let us forget about the SU 2 part, we will just focus on these two things. So, now I can give an abstract definition of what is a particle, a particle for a mathematician is just a unitary irreducible representation of this group G, in our case just ISO 1 comma 3 cross U 1, is that clear? So, please feel free to ask questions, it is supposed to be a basics notion seminar and so if you have even a trivial question do not hesitate to ask. So, electron U 1 is minus 1 and this has a chasmere, so it has the generators of this are the momentum generators P and rotations. So, translations and rotations are the generators of this and if you just look at the representation theory of this group you will find that it has a two, one of the chasmere is P square, it is a chasmere, it is a quadratic chasmere. So, if P mu is the translation generator, if the group of translations is generated by P mu, then P square has to be is a chasmere invariant of this group. So, the representations are labeled by this, so that is M, so M is some number and then there is this Wigner's theory of little group representation. So, you have to in addition have representations of little group of this. So, for example, if you momentum can be let us call it energy you can go to the rest frame of the electron and then what leaves this invariant are group of. So, this is your time coordinate this if you like and this are the space coordinates. So, you can go to the rest frame of the electron this just has energy which is the equal to the mass and then the little group which leaves this invariant is group of rotations SO 3 whose representation theory you know very well and that is the J which in the case of the electron can only be plus or minus half. So, it is just representation theory. So, it is completely explicit. So, you might not be familiar with the representation theory of Poincaré group, but you are certainly familiar with representation theory SO 3 and U 1 and momentum is some additional thing because it is a non-compact group. So, that you just take it from me. So, any other question? Okay. So, the next thing is general relativity. So, that summarizes quantum mechanics. So, in general relativity one of the very deep insights of ours in the last century was this principle of equivalence and the basic idea was to just turn this manifold which is R 3 comma 1 into a curved manifold and that is supposed to account for gravity. So, gravity was somehow not to be regarded as some force, but it just came from curvature of space-time. So, now you have a n. So, a 4 dimensional it is pseudo because it has a this kind of us, but think of it as a Riemannian manifold. Okay, I will also add a Geyshe theory and then the metric eta goes to a curved general metric on this manifold and ISO 1 comma 3 goes over to the difumorphism of this manifold. So, that is the symmetry and I do not know. So, presumably you are familiar with, I mean I am using sort of a mathematicians terminology. So, gravity is given by a connection, there is a connection on this manifold and curvature form and the U 1 goes to the great electromagnetism Geyshe theory which for mathematicians is just a U 1 bundle over M. Is that translation good enough for mathematicians in the audience? I mean so do you understand right. So, you imagine some arbitrary connection, arbitrary metric and arbitrary curvature general, totally general a priori. Yeah, given a connection you can compute it yeah. So, omega is some d omega plus whatever it is no omega, veg, omega or something. Yeah, so the given thing is just metric and connection. So, given thing is manifold metric and connection and the U 1 bundle also has a U 1 connection. So, basically that is the summary of Maxwell theory. This is Maxwell Einstein theory, this is the summary of Maxwell Einstein theory. But the in physics we have actions you know. So, we need how is the dynamics of this to be determined, I mean the connection is determined by some least action principle and it is the least action principle. So, you take the Riemann scalar, sorry the Ricci scalar square root of G minus F star F. Where F is equal to and I suppose everybody knows what is a Riemann curvature. It is an integral over d 4 x ok, here it is a bit awkward to write it in this language, but ok let me write it like this. So, the metric defines a measure on the manifold and you integrate using that measure. A scalar and another scalar constricted from the Riemann and sorry from the gauge bundle, the gauge curvature from. And delta i equal to 0 is a variational principle you just vary this. Gives you second order non-linear differential equations and that determine. So, roughly speaking the problem in physics is to classical physics was to solve these equations. For example, one of the equation is d star F equal to 0, Bianc identity is d F equal to 0, so for the connection. These are some of the equation that follow and you have to solve for A for such equations. And the amazing fact of 19th century physics was that all of light electricity magnetism just follows from these two equations. Because light are just some wave this is it gives you a wave equation and light is just some wave like solutions of these equations of arbitrary frequency and that is the light that gives you light. So, even though they look very simple they contain a enormous amount of physics. Einstein's equations are actually much harder to solve because they are non-linear and they have much more interesting physics which I am going to tell you. Is this clear so far? Human. So, I am going to so far did I say the human bundle. So, you have a manifold with a metric a connection and a human bundle that is our setup is that clear. So, for example, here one of the solutions would be A is equal to e to the i k dot x we will solve this that is a light wave going at a momentum with some frequency. So, one of the solutions e to the minus i omega t plus e to the i k dot x. So, this represents light plus its complex conjugate this represents a light wave of frequency omega and wave moment this is called a wave wavelength 1 upon k. Yeah. So, think of this as an action of g w a and then you vary it. It turns out okay I am simplifying a bit actually it turns out that the okay let us proceed for the moment I can answer this. Is this clear? So, this is actually so quantum mechanics has been hugely successful in describing everything that we know see from laser pointers and all kinds of things and atoms and everything and this has been hugely successful for describing all kinds of phenomena. But the big puzzle was that when you try to put these two things together when you try to marry them there is a big problem. So, that is the motivation for what I am going to tell you today. So, that is the setup. So, is this clear what the problem is? So, these are really the two pillars of 20th century physics general relativity and quantum mechanics and gauge theories and this between these we understand really everything pretty much that we see from expansion of the universe galaxy solar systems everything comes from this variation principle and light and from quantum mechanics we can understand atoms and everything quarks. Of course, I have to really doing the quantum mechanics is not a trivial thing but I am just saying that these principles are sufficient for us to understand phenomena going all the way from very small to very large. But there is a big problem that these two theories are very fundamentally inconsistent with each other and the roughly to tell you it is like so in particular Einstein's theory of gravity in particular Einstein's equation gives you also Newton's law of gravity and if you the roughly the problem is that if you compute for example, the earth's gravitational attraction you know that the force between them do you remember your Newton's laws mass of the sun times mass of the earth divided by the distance between the two that is the force between the now we expect this is classical this is this is classical. Now, you expect that if you include quantum effects you will get some small correction but if you actually try to compute the quantum correction what you find the answer is infinite. So, this is clearly wrong because if you take general relativity and try to compute the simplest thing like the force between two particles then you get an infinite answer and this is called the famous problem of non-renormalizability of gravity this is a technical word for it but the main thing is that if you compute a quantum correction to even the simplest processes which you think because it is a small correction quantum correction to some small correction to some classical effect it should be small but instead of that you get it to be infinite and how to make sense of this how to get rid of this contradiction is the big problem in physics. So, unlike in mathematics in mathematics when you see that if there is a contradiction you would say that there is something you just throw this thing away in physics what it says that that is actually an opportunity because that means that each formalism is incomplete and you are trying to find you need to find a bigger framework in which they are not contradictory anymore and this has been a very time honored and very successful strategy in the past like there was contradiction between Newton's laws and Maxwell's equations and that is what led to special relativity and so on. Okay so is that clear now I am going to pass on to the transparencies so one of the enduring challenges so I hope I have convinced you that one of the enduring challenges of theoretical physics is to find a consistent framework for quantum gravity that unifies general relativity with quantum mechanics and string theory has been one of the most promising roots towards such a quantum theory of gravity and it has many things going for it that it is what we call this problem that I told you this infinity goes away if you do in string theory computation you will get a finite answer as you would expect so it is perturbatively finite then there were all kinds of amazing dualities which were discovered which tell you how to relate so that you don't need to know then there is something called holography so many there is structurally many serious physicists were kind of convinced that there is something really deep going on in the structure of string theory however there is a big however that we don't have the string theory somehow this big this all these effects become important at very high energy scale and we don't have a super LHC we are already run out of money with LHC to probe the theory directly at the Planck scale in fact we don't even know which phase of the theory so phase meaning for example for water water can exist either as vapor or as liquid and depending on whether it's liquid or vapor these properties are different and so before you can make predictions you need to know which which is the phase of the matter that we're considering and and string theory lives in 10 dimensions so and we don't know which phase corresponds to the real world so given this fact how can we be sure that string theory is the right approach to quantum gravity in the absence of direct experiments I mean unless you are religious fanatics so how do we proceed and one kind of a strategy is to focus on universal features that must hold in all phases of the theory so if you can think of some quantity like in the case of water which holds true even in whether you're considering water as liquid or water as vapor then you have some hope of getting some universal and entropy of a black hole is one such quantity which gives very precise and quantitative thermodynamic information and the main principle is that in quantum gravity it should be possible to interpret the black hole as an ensemble of states in the Hilbert space of the theory so I have told you in quantum mechanics each state is represented by each physical system is represented by state but as we will see a black hole is not a single state but it's a collection of states it's an ensemble of states and how to describe that it's so therefore there is a Hilbert subspace corresponding to each black hole so I have to tell you what is a black hole and what is an ensemble and so on and what is entropy so that is the purpose of my next 20 minutes so but I just want to give you some historical analogy because I don't think I can explain it something that you might have seen in high school physics is that in the absence of microscope one can often learn a lot about the microstructure from the thermodynamic property so you probably vaguely remember something called thermodynamics entropy second law of thermodynamics so for example some of the far-seeing physicists of the 19th century you know as earlier say 1860 had foreseen many essential features of quantum physics well before the final formulation of quantum theory which came some 80 or 90 years later things like identical particles have to satisfy some particular statistic they have to be treated how do you count them there's some n factorial it's called Gibbs paradox to really understand it in quantum theory requires spin statistics theorem and all kinds of things which require really relativistic quantum field theory which came many years later but just from thermodynamic considerations you could at least see that there was a problem even you couldn't even if you couldn't solve it and our sort of we will try to follow in the footsteps of these masters like Maxwell and Gibbs and Boltzmann to use the quantum properties of black hole in an analogous fashion and in this search you actually are led to number theory as I'm going to explain to you and it will become clear to you why so I'm just going to give a simple historical analogy see what is known as kinetic theory of gaseous is something that you learn in undergraduate physics but this was really a triumph of 19th century physics and it formed the basis for the atomic hypothesis and later for quantum theory and it started you know with attempts to explain microscopic properties of ideal gaseous in terms of microscopic atoms even though there were no microscopes at the time that could actually see or establish the reality of atoms directly in fact Boltzmann is associated with TSD I think and he committed the suicide because nobody took him seriously the atomic hypothesis and one of the fundamental concepts that makes this possible so what is kinetic theory of gaseous is that you know our room is full of some gas that we all believe is full of atoms of nitrogen I mean all of you believe but now why do you believe in that you've never seen one right how do we know that in this room is really full of atoms and if you don't have a microscope like the LHC how can you even make such a ridiculous assertion that is really composed of little atoms but that's what Maxwell and Boltzmann did and what led them there was actually very practical considerations trying to design efficient steam engines because that was the time when the industrial revolution was happening so one of the things that we know that heat flows from a hot body to a cold body but not the other way around so how can you quantify this irreversibility and that led to the notion of entropy that you define a quantity called entropy if you add some heat delta Q to a body at temperature T then you say that this entropy changes in this manner delta change in the entropy is equal to the heat that you add to a divided by the temperature right so you take a bucket of water and you heat it and you know you just keep track of how much heat you are giving to the bucket of water and if you keep the temperature fixed then that's the definition and you can do it infinitesimal this way and then the second law of thermodynamics which was formulated by Karno who was incidentally an engineer he was not a physicist he said that so but this is a this led to a very important concept in physics that in an irreversible process the net entropy always increases so delta S has to be greater than 0 and entropy is an intrinsic property of a given system and is a function of the energy and the volume of the system so S is a function of volume and the total energy so like in this room this room is full of gas nitrogen let us say and the volume of this room we know and the temperature and the total energy in this room we know and that tells you what is the entropy of this nitrogen gas so that is the assertion and then if you now do anything to this system regarding this is a closed system that you put a heater here or whatever you do in an irreversible process the net entropy has to always increase and how does that explain this our initial question that heat flows from a hot body to net body suppose you have now a hot body which you bring in contact with a cold body then the delta Q flows from a body at temperature T1 to a body at temperature T2 then the net change in entropy is the the total delta Q divided by 1 upon T2 because this lower temperature guy gained entropy and the higher temperature body lost entropy so the net change in entropy is delta Q absolute value 1 upon T2 minus 1 upon T1 and second law of thermodynamics requires that delta S has to be greater than 0 so therefore heat can only flow if T1 is bigger than T2 so is this clear this is the second law of thermodynamics then comes in Boltzmann and he made this really one of the very one of the more is this is actually one of the more subtle and deep concepts in physics it is a little harder to understand because it is not an eigenvalue of an operator the entropy like charge or momentum as I described he said that this entropy that you define in this microscopic way is proportional to with some constant the logarithm of the total number of microstates in which this whole system can exist so for example this whole system there some of the atoms can be in one corner or some of the atoms can be here atoms can be moving all over the place so there are many ways I can organize all these atoms in this room and that gives me that is equal to the entropy of this and as I said it is really one of the more subtle and deep concepts because it gives you information not about an eigenvalue of some one operator but about the dimension of a Hilbert subspace we all of us now we are used to using Hilbert subspace but this was in 1860 when the notion of a Hilbert space and quantization was not even in the horizon okay perhaps I can skip this disorder so for our purposes the significance of entropy stems from the fact that one can draw important conclusions about the microscopic structure of an object and the relevant degrees of freedom from its gross thermodynamic microscope properties so entropy therefore is a window into the microstructure of the system so even though you cannot see the atoms in this room you can actually deduce the properties of atoms by measuring the entropy or how the entropy changes when you heat the room and it is a very simple calculation again it is a you can do it on a an envelope I will do it in front of you in front of your eyes so imagine this room is full of nitrogen gas and imagine there are n molecules in this room and the typical size of the molecule let us say is lambda so therefore this room the total volume of this room has to divide up you cannot see this okay but it is divided up can you see there is this blue box which is the room and it is divided up into so many little boxes so this is the room and it has n molecules and you just divide up of each of length lambda and n of them and so now this is a combinatorics problem right given n atoms so number of boxes is therefore it is a totally simple combinatorics problem the total number of ways you can put so many molecules n arrange this molecules in the room the number of ways you can arrange n molecules in this room of size v is the number of boxes available to you to the power n because if it dilute you can put them you can put it here put it here to the power n and if they are identical particles you have to divide by one upon n factorial it is like a statistics quantum there is just a regular combinatorial statistics so it is a completely trivial calculation right you can anybody can do this but this is actually one of the very very important computations of the 19th century physics because if you put the right value of lambda which is called the thermal deborah thermal wavelength of this sorry it should not be deborah it is the thermal wavelength which depends on temperature then you get the right answer for the entropy namely this very trivial looking calculation if you take the logarithm of that and if you put lambda a correct lambda then that gives you the correct answer for the entropy experimentally meaning if you now hit this room and then what you will find is that the change in entropy will be given because lambda changes when you hit the room and the temperature goes up and precisely in this manner so the microscopic counting that we did here in a very simple way explains a macroscopic entropy so this is a very basic thing that I want you to understand for the mathematician in the audience is this clear why entropy is so fundamental in physics huh questions I mean I am sure all of you have seen this in undergraduate physics but perhaps not in a manner that emphasize the significance of it but it I should say that it is really one can give you know three lectures on entropy and it is really one of the so now is this clear why entropy is important now I am going to tell you about black holes a black hole is a space time is one of the solutions of Einstein's equation I have probably written it here somewhere it is a solution of Einstein's equation a black hole is a solution of Einstein equations and in the simplest case it just it is a Ricci flat manifold the Ricci curvature of the manifold is 0 it is a pseudo Riemannian manifold and it has a Ricci curvature so in the simplest case which is called the Schwarzschild black hole it is just a Ricci flat manifold in four dimensions with a pseudo with a metric of this indefinite signature so that is a black hole okay it is just a solution of a second order nonlinear differential equation is this clear and it is specified also much like an electron by mass charge and spin okay even though it is a solution of this very complicated equation it turns out in the end it is specified just by mass charge and spin and there is a singularity but it is clothed by an event horizon so this is artist's impression of a black hole and the fundamental fact about a black hole is that it has it is surrounded by an event horizon and what is an event horizon so this is the kind of thing that mathematicians don't encounter because if you're doing Riemannian manifolds you never will encounter such a thing it's really crucial most of the lot of the interesting stuff that happens here in physics has to do with the fact that time behaves differently from space for example a wave equation I mean see this is the solution of the wave equation the wave equation I mean elliptic operator and a hyperbolic operator are very different so this is a hyperbolic operator whereas normally in mathematics you are in a much safer territory when you deal with elliptic operators like just with a plus sign and that makes all the difference because such a thing is not a solution of except for k equal to 0 so even a photon wave would not be possible if you were just dealing with elliptic operators an event horizon is possible precisely because you have a hyperbolic because of the minus sign and basically event horizon is a one-way surface from behind which you cannot come out but will surely meet a singularity where tidal forces are infinite okay that's the definition of a black hole meaning it's like this room you know you can only come inside but once you come to my lecture you cannot go out you're forbidden to go out so that's it's like a one-way surface you can only come in but you cannot go out now a black hole is at once the most simple and yet the most complex object and understanding the simplicity is in the realm of classical gravity and understand the complexity is in the realm of quantum gravity so what do I mean by that so actually I last year I had the opportunity to go to this Kerr conference you know Kerr was a famous one of he discovered one of the most famous solutions of Einstein's equations and it was in his honor 50 years of the Kerr solution and there was a very beautiful quote of Chandrasekhar one of the famous astrophysicist in from India who said that one of his most shattering experiences in life was to realize that the Kerr solution which was the solution of this equation describes every possible black hole that we see exactly it's it's an exact solution because of what is called the no-hair theorem that means a black hole is completely specified by mass spin and charge very much like an elementary particle and this is a tremendous simplicity you know for example if I look at the table and try to look at the gravitational field of a table or of the sun it's very very complicated because the sun can have bulges it can have mountains and valleys and the gravitational field is not specified just by the mass of the child you have to know how the mass is distributed so it depends on the what I call the higher moments dipole moment quadruple moments and so on not for a black hole for a black hole you just specify the mass charge and spin everything is specified you just have one solution you just write it down and that's the description of a black hole unlike for the sun where you have to do you really need a lot more information about various properties of the sun and the event horizon as I told you is this one-way surface so the simplicity is very easy that a black hole for some reason a very complicated solution of these non-linear differential equations is specified by just three parameters very much like an elementary particle so for a while people thought maybe a black hole is an elementary particle but it turns out that it's actually not a elementary particle but it's more like a collection of it's an ensemble of states of elementary elementary particle like states but now I'm going to so how much time I have 20 minutes and am I allowed to go a bit over time or it's depending on that okay so let me maybe I'll go five to ten minutes over time so I'm just going to run you through some paradoxes based on what I have told you so far and then that leads us to number theory how how that happens so as I told you now a classical black hole has this one-way surface now Beckenstein who was then a graduate student in Princeton of Wheeler he asked this very simple-minded question that what happens if you throw a bucket of hot water into a black hole because I've just told you it can only come inside and cannot go out but then bucket of hot water as I told you has entropy so therefore the entropy as far as the outside world is concerned which can never look inside the black hole has decreased but I just told you the second law of thermodynamics is that the entropy has to always increase so Beckenstein said that the second law of thermodynamics which is so fundamental and I told you entropy is such a big thing seems to be violated if there are black holes present because you can just dump in as much entropy as you like and you will decrease the entropy around the world so then he said well but maybe the black hole also has entropy and then what you should really keep track of is the entropy of the black hole and I mean after all if you drop a hot water in in the in the sun you don't worry about the second law of thermodynamics because the entropy of the sun goes up and you keep track of everything you find that entropy actually decreases increases and doesn't decrease so he said that okay the black hole must have entropy such that second law is seemed and he gave some very specific prediction for what the entropy should be but that leads to another paradox because if black hole has entropy and mass then there is a first law of thermodynamics which relates the change in energy is temperature times the change in energy of a system is temperature times change in entropy so in the case of black hole is the mass so the black hole has mass and it has entropy then it must have temperature but you know that everything anything that is hot is always radiating something therefore something should come out from the black hole but that's a paradox because as I told you nothing can come outside of a black hole because the event horizon means that things can fall in there cannot come out so this led to a second paradox so I'm just trying to tell you that just following the way 19th century physicists reasoned about the atomic world without knowing what were the atomic constituents of matter they could actually go very far in some similar way physicists in 20th century have been able to reason about properties of space time and quantum gravity even without knowing what are the precise rules we still don't know we don't have a complete theory so this is the famous discovery Hawking in 1970s that actually the black hole even though classically it has this event horizon there is a little quantum fluctuation happening at the horizon and therefore sometimes and there is always this pair creation as you might have heard about and one of the pairs can escape and one pair falls inside so it can look like the black hole radiates in fact Hawking therefore found that the black hole has temperature precisely in a way that the first law of thermodynamics is satisfied and the second law of thermodynamics is saved in the manner that Beckenstein had proposed it so this is this led to the picture of what you call semi classical black hole in that it's not fully classical you have to take some quantum effects into account to understand that it has temperature but once you do that then Hawking gave this very beautiful this is a precise formula for the entropy of the black hole that you have a horizon of the black hole so black hole is as I told you is just a surface it is an s2 in r3 time is flowing and it has some radius we can calculate this area is 4 pi times standard formula from Euclidean geometry and that area times c is the speed of light h bar is the Planck's constant g is Newton's constant and there is 4 so entropy is always area divided by 4 it's a very beautiful formula take any black hole that you like of any mass any charge this is true in any dimension you do five-dimensional black hole four doesn't matter there is something really deep and universal about this and now this means that the black hole has huge entropy but that leads to the third paradox so L is just I can therefore just by dimensional analysis s has no dimensions area is like length square so it must be this is called the Planck length basically h bar g divided by c cube I can write as L square so that the whole quantity is dimensionless because to compare with Boltzmann we have to compare it with log of some number so the paradox three is that if a black hole has entropy then in quantum theory it must be an ensemble of microstates according to Boltzmann but then a black hole by definition so it's called black hole because it's literally a hole in space time and it's black because another way to think about the event horizon is that as you know there is a escape velocity for a rocket you have to if you throw something up it always falls down but if you go at the speed of rocket then you can escape the gravitational field of earth but if your escape velocity is the speed of light then you know that nothing can travel faster than the speed of light and therefore nothing can escape the black hole so that's another way to think about the event horizon event horizon is that the escape velocity becomes equal to the speed of light so so therefore it's literally a hole in space from which nothing can come out and how can you associate some states I mean in the case of gas in this room we imagine that there were little point what you know little atoms floating around and we put them in some box and that's how we counted the number of states and to resolve this paradox we really need a full fledged quantum theory of gravity with some well-defined quantum Hilbert space and that's where string theory comes into picture because basically that is our only candidate at the moment which has a properly defined Hilbert space where you can actually say that uh-huh a black hole can be a state like that so now I come to the first part of the title of my talk that quantum black holes black hole is simple not because it is like an elementary particle but rather because it is like a thermodynamic ensemble and this explains why it is both simple and complex and I will explain this in a moment so basically what do I mean by a thermodynamic ensemble so I told you a state is specified by some quantum numbers like m and j but for example I can have many states which have the same quantum numbers for example if I say that my total mass like in this room the total energy is some fixed but I can construct have so much energy because one particle is going that way and another particle is going that way or two particles are going in that way they will all have the same amount of energy or for example if I say that I have 10 times the mass of the my total mass is equal to the 10 times the mass of the electron then I can have those many one electron sitting here another electron sitting there all these states will have the same mass namely 10 so therefore you can also have an ensemble of states in a Hilbert space is simply a subspace with fixed eigenvalues of some operator some observables which I said are operators so I can say that for example q is equal to I want q is equal to some number minus 100 let us say and j is equal to some number 256 right and I can say m is equal to in some units some number and then all that I need to do so log omega that I introduced in Boltzmann relation as a function of this mass charge and j is equal to the dimension of this let me call this some subspace h sorry omega it is just the dimension of this is this clear omega which was the total number of states as a so in Boltzmann relation so this is a this is an important thing so if you understand this then you will understand you will go away with some something so you remember Boltzmann relation was log omega what is this omega omega is the total number of states but now I can be more precise since you know quantum mechanics that by states I mean the total number of the dimension of the Hilbert space with some specified values for the operators under consideration so if I take the total Hilbert space and look at the subspace where mass has some value q has some value j has some value these are usually finite dimensional and you just count the dimension of the operator of that Hilbert space and that is omega and the fundamental and a very beautiful relation of Boltzmann tells you that that microscopic counting is related to this microscopic right here you are talking about atoms or you know and here you are talking about like gas and so on right properties of steam engines right and roughly speaking general relativity is on this side and quantum mechanics is on this side and the black hole is on this side which is the analog of a steam engine and the question is what are the atoms that make up the black hole that is the question that string theory has been able to answer in a rather in surprising detail for a simple set of relatively simple systems but as I said because this requirement is so fundamental and is universal if for example we had failed to do it even for the simplest black hole then we could just forget about string theory we would know that string theory is wrong so even without doing an experiment you would be able to just so it is extremely non trivial that actually string theory is able to account for black hole as an ensemble of states so which ensemble of states and can you make some computations and then area upon 4 should be given by log of this omega of this ensemble is that clear so this is the this is the basic setup because of which now you can see very quickly where number theory comes in I should say number theory is here because in this case we did some very counting problem right I this was a this was a counting problem right this was combinatorics so basically counting the micro states of black holes leads to a much more sophisticated and hard counting problem and it is a combinatorics so so now I will switch to blackboard a little bit so combinatorics and number theory so as I told you in string theory now we understand a black hole is related to some ensemble of states and in the simplest example it is an ensemble of some some very simple states okay I will give you a simple example it is an ensemble of state of a string oscillating in 24 dimensions okay so string like object oscillating transverse 24 dimensions okay this is related to the fact that we live in string theory lives into 8 dimensions and there is another group called E8 cross E8 so there is some story there but basically you have this is the counting problem that you have a string like object oscillating in 24 dimensions how many ways can it oscillate it can oscillate this way it cannot oscillate that way and you have to think of some there is a counting problem that you have to do and you have to quantize this system but it basically reduces to partitions of an integers and for a given total energy the total energy is fixed so you cannot given fixed total energy you have to count this reduces to a problem of partitions of an integer using 24 colors so for example let me give you and this is actually a well known object in number in combinatorial number theory I don't know going back to Ramanujan and many people like that so this is denoted by p24n how do you calculate it so let me compute for you p of n just p1 of or a p3 of so how many ways I can I write 5 okay let me do a simpler one p3 of 3 so I can write 3 3 equal to I can write it as 1 plus 1 plus 1 with a blue chalk or 1 plus 1 plus 1 with a yellow chalk or 1 plus 1 plus 1 with a white chalk but I can also write with a 2 in white and 1 in yellow right I can have all these combinations 2 in white you can count them right this is trivial I mean not trivial it's actually a non-trivial competition and that number so instead of using now three colors you use 24 colors and for any given integer n and that's related to entropy of a black hole of a certain it turns out that this quantity is related is equal to the omega for some set of lacrosse okay but this actually gives one of the most beautiful instead of computing these p24s of n