So, good morning everybody. Today we continue with Enrico Valdinoci. He will today give the third lecture of non-local equations and the different perspectives. So, thank you Enrico. We will try to finish the... Can you hear me? Is it on? Today we will try to finish our discussion on the eight fundamental differences in laplacian and the fractional laplacian. Actually, let me mention that if you want to go into more advanced subjects, the topic that Ciavi introduced yesterday, the one of non-local minimal surfaces is probably the richest in presenting very important structural difficulties and differences with the classical case. So, since Juan Luis doesn't like eight, I just listed seven fundamental differences between classical minimal surfaces and the non-local minimal surfaces. If we have time, we will look into these differences more. But just to let you taste the flavor of the differences. So, I wrote seven. Some of them were mentioned by Ciavi yesterday. Uniform perimeter bounds, which are valid for the non-local case and not valid and sometimes not known for the classical case, sticking as properties at the boundary, the fact that non-local minimal graphs are discontinues at the boundary, the fact that the non-local catenoid grows linearly instead of logarithmic. The fact that there are unstable non-local minimal cones in one dimension less with respect to the classical case. The fact that the non-local mean curvature flows develops neck pinch types in uratities also in the plane. And the fact that non-local phase transitions are more rigid than the classical ones. This is actually probably the topic of Alestius lectures at the end of the period. So, this said, the last difference we were keeping discussing was the decay at infinity. So, we discussed the decay at infinity of the heat flow, which is like 1 over x to the n plus 2s of the fractional heat flow of the fractional Allen-Kahn equation, which is like in 1d 1 over x to the 2s. And now what I would like to mention briefly is the decay of ground state solutions for the Schrodinger equation, which is again just polynomial instead of exponential, and the fundamental solution of the fractional Schrodinger equation, this, and as I was mentioning, for large x, we have decays like 1 over x to the n plus 2s. And now some of you were asking me, okay, but why this happens? Well, the first answer is I don't know. But let me try to give you just, likely to be wrong, but convincing argument in a sense. So, what I think of this decays is that in all of these equations, the right hand side has to compete and balance with the left hand side. So, the question from my psychological point of view would like to decay as much as possible at infinity, but then looks at the left hand side and changes his mind. So, for instance, here, outside the origin, this is like zero, okay? So, in a sense, gamma will try to be as much as decaying as prescribed by the fractional laplation. So, if you start thinking that gamma is compactly supported or very fast decay, no matter what, the fractional laplation will decay like 1 over x to the n plus 2s. And so, it forces the decay to be this one instead of an exponential one, okay? So, if you are convinced that this is a good answer, this justifies the decay for shredding. Well, the decay for the heat flow, it's not convincing. So, let's see. Let's see if at least we get with this argument the right numbers. I'm not trying to sell this as a proof, but it should give somehow an idea because for the heat flow, the situation is like the following. So, you are solving the equation of this type, and now u times zero delta. So, you may think that at times zero you are very concentrated at the origin, and then after infinitesimal time, you develop a little bump. Now, you may hope that this little bump converges to zero very fast in x, but then you look at the equation and you see that the increment in time of u at infinity has to grow like 1 over x to the m plus 2 s, because even if your initial distribution, like you think you start from a delta, which is just something like this, then you may hope that after an infinitesimal time you are very much concentrated. Fine, but if this would happen, then what you would have is that the time derivative would grow like 1 over x to the m plus 2 s. So, after another infinitesimal time you would reach the level 1 over x to the m plus 2 s anyway. So, the equation pushes up u at infinity exactly for the amount prescribed by the fashion Laplacian. Now, if you are skeptical about my arguments, now you have a very strong point in your favor, because this is 1 over x to the 2 s, not 1 over x to the 1 plus 2 s, which would be the right decay at infinity. So, this tells you that my argument is not really rigorous, but it also tells you that if you fight a bit, you may make it a little bit more convincing, because what is the Lankan equation? The Lankan equation is in 1D, so in R, fashion Laplace equal to u minus u cube. So, I call v the derivative of u, and I take one derivative of the equation. This gives what? This gives v minus 3 u square v, which is like 1 minus 3 u square v, which at infinity, we know that u converges to 1 or minus 1, so this looks like more or less minus 2 v. Now you apply the argument of balancing with the right hand side 2 v, say for large x, if you apply this argument to v, you may expect that v now decays as much as is prescribed by the fashion Laplacian, so it decays like 1 over x to the 1 plus 2 s, which means that when you integrate back, you have that 1 minus u decays like 1 over x to the 2 s. OK, so this may not be a real proof, so let us now do a real proof of one of the things that I was saying was probably, me the most surprising difference with respect to the classical case that is the fact that all functions are locally s-harmonic up to a small error, so that whenever you have, you can prescribe locally any function and approximate it with functions with vanishing fashion Laplacian. So the result for any epsilon bigger than zero and for any nice u, there exists some u epsilon, such that u epsilon is, say, c2 or ck close and the fashion Laplacian of u vanishes in b1. So somehow you can extend your function u in a suitable way so that the integral oscillations perfectly coincide and the error that you make with your original function is as small as you like. Let me mention that this is a very general result, so at the beginning I thought, okay, it's already strange enough that it works for the fractional Laplacian and since I come from an elliptic perspective, I thought, well, this has something to do with ellipticity, but actually no. So the same result holds true basically whenever you put here a linear operator which has some non-local coordinates. So you can take your favorite linear operator that has two sets of coordinates, a set that I call the local coordinates in which the operator is just a differential operator of any order you like and with different orders in different coordinates and the non-local operator which is just the sum of fractional Laplacians in different coordinates, possibly of different order. And you can prove the result for this general monster operator. Proving the result for the monster operator in any dimension, it's a little bit tedious not because it's hard in itself, but because of course you have many variables and many indices and things are tricky. So just to let you see what's going on after all, I will make a simple proof in the case I take the operator just to be more general, not the fractional Laplacian, but an operator L which corresponds, for instance, exactly, exactly. So the first lecture we did this picture, maybe I repeated it. So let's see with colored shocks, it looks better.  Parabola doesn't look harmonic for whatever reason, but then you put an epsilon. Of course I draw an early heating arm but you can do it in smooth norms as well. And then what happens is that you put a u epsilon here which is S-harmonic exactly because outside this function has some oscillations then it goes to zero after a while but the oscillations that make it S-harmonic. Yes, I think that this is kind of a very nice subject of research and so concerning your type of questions and the proof I will make now I'm sure you are not gonna like it because it's too soft in a sense. It's a proof by contradiction. So the proof doesn't tell me that this is the right picture. Nevertheless there are some more advanced proofs using Hormander methods by Mikasalo, for instance, in which they can have a better control of the radius in which you have to go to zero or things like this. But somehow that is not a proof that you can do it in five minutes, I think. It's more advanced. This proof doesn't give you any information on the real picture though as I was saying the last time this is almost undistinguishable from the real picture I drew these things like I wish there is no reason for which this thing is a sermonic but then I can repeat the theorem on a larger scale so I can say okay the orange function may be not a sermonic in B2 but then I repeat the theorem in B2 so I can complete it upside to a sermonic function that looks like this in B2. So somehow it says that really any picture is good on the local case and the important part always comes from infinity which is, I mean to me sounds a bit not natural also for large s this doesn't seem to be right but it is right and it is what it is. So just a rough idea so the picture that I drew maybe not the correct picture so I have here my U and I drew here just something that looks reasonable to me but may not look reasonable to you but what I am saying is that I can repeat the theorem on a larger scale so suppose that this is now B2 I can now repeat the theorem with the orange function in B2 and say that well the orange function in B2 maybe is not the right thing that makes us sermonic in B1 but I can repeat the theorem in B2 and say okay there is a function which is almost undistinguishable from the orange function which makes the orange function S-harmonic in B2 and also in B1 so the picture I drew is essentially correct because I can change the data closer and closer to infinity and make any picture correct so there is no way to distinguish just from looking at the picture whether or not the function is S-harmonic because this is an information that dramatically comes from the data at infinity so my suggestion is to prove this result not for the easiest case of the fashion Laplacia but not for the hardest case of a monster operator so something in between my suggestion is to look at what happens for the heat kernel the fashion heat kernel good so if we do that so in a sense I am going to prove that all functions are locally S-caloric up to a small error S-caloric in the sense that it is a kernel of the fashion heat equation and so my function so first of all some reductions are the following first I can assume that U is a polynomial and the reason is somehow the storm bias trust theorem while there is a small refinement of storm bias trust usually in Italy they taught us storm bias has seen one variable and doing approximation in L-infinity but if one looks at the approximation with Gaussians basically the storm bias that holds in any dimension and for any ck norm so what I am saying is that any c2 function U can be approximated by a polynomial so I first approximate U with a polynomial and then a polynomial with U epsilon and so I can reduce to the case of the polynomial then in fact as the problem is linear I can reduce to the case in which U is a monomial say you know what a monomial is so just let me write it so U over xt say U of big x big x is comprising both the variable x and t is of the form x to m with the multi index notation and the reason for this is that the problem is linear so if I approximate any monomial then I sum them up and I approximate the polynomial then the third observation is that what it is important to do what is important to do is to produce the right derivatives at zero so the goal would be to find a function say v such that the derivatives at zero are prescribed say up to order m so the alpha v of zero are prescribed well this the bad notation because m here is a multi index so you have to put the the norm of the multi index here when I have a multi index the modulus for me is just the sum of the entries so let me call this j just not to carry the multi index notation so why is this so well because then you can just blow up your picture because if this is true you can take vr of x to be to the j v of x over r now if the derivative for instance suppose that in this way v is of the form p of x is of the form xm plus order of xj plus 1 so here you are fixing all the derivatives of v in the origin to be the ones I like to be the derivatives of u the derivatives of u are zero zero zero zero up to the derivative which is whatever is prescribed here so I say well also v as the same derivative is near zero up to the correct order this means that when I do this rescaling here I get exactly x to dm because the j is exactly the norm of m plus order r so basically when I do the difference between vr and u I get something small and so I prove the theorem so the real issue is just to understand what happens near the origin now this seems a great simplification but if you think in the classical case it's exactly what you cannot do because for instance if your Laplacian is zero it means that the second derivative the second derivative of your function have to lie in a co-dimension one space because for instance in dimension two the second derivative in the first direction have to balance the second derivative in the second direction so it's not that you can produce any number you want the sum of the two have to be zero so this is somehow the core of the proof and it's the part in which the non-local structure is richer than the local one and so I now prove three and as I was mentioning the proof is by contradiction so I look at a space v which contains all the functions h say from r to r such that l of h is equal to zero in a neighborhood of the origin well this is a non-empty space because the zero function is there and then I can call v zero in vizio I do this I take all the functions here and I compute their derivatives up to the right order so this collects an array so for any function here I compute all the derivatives up to the right order and I list the derivatives in this array so this is a finite dimensional array and say this is contained in a euclidean space and I call n plus one the dimension well n plus one is just the number of alphas for which alpha less than j holds and now my claim is that v zero is not only contained in this space but it's actually equal to this space that is really maximal span and you see if I prove this then I am done because I have found a function which is escaloric near the origin sorry I forgot to write it there exist a v such that l of v zero solution of the problem so if this is true in particular I take the array of the derivatives of u and I get this array through functions that are escaloric how I prove this well the proof is by contradiction well if not it means that this v zero which is a linear space because the sum of escaloric function v zero is actually a strictly linear subspace of r to the n plus one so a picture would be like this the blackboard is r to the n plus one and v zero has to be here or to be a linear subspace of this plane for instance so I call mu this direction and I say that of the fact that v zero spans everything is that v zero is contained in the in the space I don't know how to call it vb y in rn plus one such that new dot y is equal to zero now what do I do the idea is that I choose one h and I find a contradiction by taking the solution of the somehow of the the couple the couple diagram value problem so what I do is I look for a function phi which is the first diagram function of the fractional application in minus one one I didn't say but I'm just doing the one dimensional case so one dimensional space variable and one dimensional time variable and to make things much simpler let me actually do the case of the one half fractional application the computation would be the same with s but the exponents will be more annoying so let me take s equal to one half otherwise one has to adjust exponents and things are less transparent but if you take for instance one half ok with Dirichlet with Dirichlet datum that is I'm solving equal to lambda in minus one one and phi equal to zero now the important thing is that we know more or less how this function looks like this is something that we discussed in the first lecture we know that the boundary behavior of this function is not leap sheets up to the boundary is only held like a circumference and we computed if you remember what happens for the square laplation of one minus x square under the square root and that's exactly the same behavior so basically if I take a small delta what happens is that the phi of minus one plus delta goes like delta to the power one half basically phi looks like this near minus one and if you want in a distributional sense you can take the derivative of this expression and you can actually prove that delta positively so on this side it goes like delta to the one half and if I take k derivative of this guy it goes like one half minus k ok then this is the case I think I take h of x and t to be e to the minus tau t phi of minus one plus over lambda x I apologize with the Greeks I know that my lambda is wrong my lambda should be this one I think you understand my writing so I have to check that h belongs to v because later on I want to use h to find the contradiction so let's check that h belongs to v where for this I have to apply l to h so this guy is if I take the time derivative it's easy because I have the explanation so it's just minus tau and then I have e to the minus tau tau tau is fixed for me fixed tau I take derivatives in t I have minus tau and then h this is the time derivative and then I have to do the Laplacian to the one half of the phi well if I do that I get what I get plus e to the minus tau t and then by scaling the Laplacian to the one half is an operator of order one because the Laplacian is operator of order two and the one half is two times one half makes it one so I have to take the fractional Laplacian of all this function which is the fractional Laplacian of this function evaluated at this point but I have to take into account the scale factor and this is the only reason for which I took one half here if you do the proof with the different s you have more annoying exponent and that's why I did that ok now I remember who phi is phi is an function so instead of square root of the Laplacian of phi I can just write lambda phi so if you agree I erase the fractional Laplacian and I put the lambda but now I and this is zero because it's tau and then whatever else is remaining is h if I'm not mistaken with the signs ok fine now once you have that you can now use your use your h in your contradictory assumption written here so since h belongs to h it means that when I compute the derivatives in zero they have to lie in this hyperplane so k is that since h is in v the derivative are in v0 and v0 is in that hyperplane which means that if I compute all the derivatives h and evaluate them at zero I get zero so here I'm using the notation for which nu nu is a unit vector so it's in Sn and I write nu as nu alpha so somehow I'm writing the coordinates of nu with the same convention and I compute what is the alpha derivative now unfortunately I have to introduce a new convention I have to distinguish between time and space derivative so my multi index alpha will be thought as an alpha x alpha t so these are the x derivatives and these are the t derivatives so nu alpha is some nu alpha t alpha x alpha t and if you prefer the norm of alpha is just alpha x plus alpha t so these I just wrote these in coordinates then I have to take derivatives if I take alpha t derivatives of the exponential I get minus 2 to the power alpha t then I have to write the exponential but I will have to take t equal 0 after so the exponential of minus 2 t t equal 0 will get 1 then I have to take alpha x derivatives of phi and evaluate them as 0 when I take alpha x derivative I get tau over lambda to the power alpha x then I have alpha x derivative of phi made a mistake so let me introduce also an additional parameter delta the same computation would have worked before before it was just the translation here doesn't play any role but now I want to leave this delta because I want to have here 1 minus 1 plus delta if I don't put the delta here I go into a little trouble because phi is not differentiable at 0 so I want that 0 is well inside my domain so that's why I put the delta here that's the only reason and then I get this and now I use what I know on the derivatives of phi so I can write this guy as the sum of alpha x plus alpha t less than j nu of alpha x alpha t then I have minus 1 to the alpha t tau to the alpha t plus alpha x lambda to the alpha x and then this is delta to the 1 half minus alpha x 1 plus little o of 1 when little o of 1 is something that goes to 0 when delta goes to 0 so here I'm just using the decay of the function phi and so I know that this thing is equal to 0 now what I do now is that I take alpha x as large as possible so this nu alpha well can be 0 for some coordinates but it's not possible that all the coordinates are 0 because nu is on the sphere so sooner or later maybe the bigger coordinates are 0 but when I reduce I find the coordinate which is not 0 so just for simplicity I take alpha bar x to be the first coordinate in x for which there exists a t coordinate for which this is not 0 otherwise it means that they are all 0 so I take the largest alpha x for which there exists an alpha t in the right range with alpha t plus alpha x less than j and nu of alpha x alpha t less than 0 so if I write this this sum here actually spans with the condition that alpha x is less than alpha bar x because whatever happens above the alpha bar gives that nu is 0 so you are just summing 0 contribution ok so this reduces maybe the sum or maybe it doesn't reduce it alpha x is already giving a non-zero nu in that case you don't reduce it but in case the bigger alpha x is giving a zero nu then you reduce it to alpha bar ok so you have that 0 is equal to this guy now what I do is I multiply this expression by delta to the alpha bar this is a positive number so I just have alpha bar x here now I send delta to zero and what I get here well I get here oh sorry I multiply by delta to the alpha bar x minus one half the minus one half is useful because I remove this one half here now I send delta to zero so when I do this I have delta going to zero here so whenever alpha x is less than alpha bar this exponent will be positive and it will give zero ok so if I send delta to zero the only exponent which is remaining is the case in which alpha x is equal to alpha bar of x so let me write it like this here in principle I have alpha of x less than alpha bar of x but if I am strictly less I get zero so by sending delta to zero here I get alpha x equal to alpha bar of x ok so this is somehow alpha bar this is alpha bar this is alpha bar now alpha bar minus alpha bar gives zero so this gives one and when I send delta to zero one plus little o is equal to one now I have that this expression is equal to zero then let me multiply under to the alpha alpha bar x and divide by tau to the alpha bar x well if I do this these and these are zero but this means that you have reduced your problem by the alpha bar because now you have a polynomial in the variable tau which is equal to zero so in the identity principle of the polynomial the coefficient has to be zero but the coefficient is not zero for the way you choose alpha bar so this is a contradiction tells you that you cannot be contained in a linear space of rn plus one and so somehow it proves the theorem and it gives you that all functions are locally in the general the general structure for the general operator is somehow the same just one has to be a little bit more careful with this choice of of indices in order to reduce step by step by one what happens when I send delta to zero in the multidimensional case but somehow the structure would be more or less the same so in the next five minutes let just me recall that in the next lectures I would like to address a bit the regularity theory of non-local minimal surfaces so this is a topic that already shavi has introduced and mentioned so I will be very happy to take advantage of the material that is already introduced and so one result that I would like to present in some detail is the analog of Bernstein theorem for non-local minimal surfaces so Bernstein theorem in the classical case says that if you have minimal surface which is a graph so say a minimal graph and the dimension of your ambient space is eight or less then your non-local minimal graph has to be a plane so the classical Bernstein theorem says that if you is a solution of the minimal surface equation so it creates a graph which has zero mean curvature and n is less than always make it wrong seven, right, because I am in dimension eight in the ambient space then u is an affine function and in the classical case the dimension seven is optimal because if n is equal to eight Bombieri the Georgian justy constructed an example so this would be r8 and this would be a Simon's cone in r8 so think that what I drew here is that r8 has two coordinates x and y in r4 times r4 and what I drew here is the norm of x equal to the norm of y whatever it means in r8 and so basically they constructed saddle shape function in r9 so you have one coordinate more which is a positive here negative here and which goes logically to plus and minus infinity so this is a zero level set for the thing and this is growing here something like this and going to minus infinity on the other side so we will discuss somehow the analog of this problem in the fractional setting and let me say that the problem is still very open there are some results in this direction but we do not know whether the results will obtain are optimal and we do not have any example like this so the fractional case the complete picture is still very open but I will try at least to introduce some material that can be useful to see what's going on in this field and maybe my time is over


Here today to talk to us about what her life has taught her and the lessons that she has learned. Please welcome to the stage, Ms. Anu Agha. Good morning, everyone. Good morning. I'd like to thank the Leadership Energies Summit, Asia, for not only just inviting me, but for their extremely warm and gracious hospitality. Thank you very much. I'm going to speak today about the challenges in my life and what it has taught me. Before I talk about the challenges, let me give you a brief background about my childhood. I grew up in Mumbai in an upper middle class family and I have two older brothers. I was good at studies. In fact, I did better academically than my brothers. But not once was I told that I could join the family business whereas my brothers were constantly reminded that they would have to join the family business. And the message which I continuously got was, I have to marry and produce. Since I had good people's skill, it was suggested that I do my post-graduation in social work and being an adaptive person. I agreed. My late husband, Roenton, before our marriage had joined my father as a senior executor. And after my father passed away, he became the chairperson. He was a very charismatic leader and could attract talent. In those days, we couldn't pay the best of salaries but the atmosphere we created, giving freedom, let people come with an idea, run with it and make it successful. It was a unique work culture and in spite of our small size, we enjoyed a very good public image. Life will always throw challenges and it is up to us to convert them into opportunities or feel trapped. Whenever I feel helpless, I imagine that I'm stuck in a corner of a room and that I've forgotten that the entire room is available for me to explore. So I think that's an important lesson. I would like to share three major challenges in my life and how I dealt with them. The first was, when my husband was in his 40s, he had a massive heart attack. And in those days, they didn't do a bypass surgery in India and it was suggested that we go to the UK. And on the second day, he had a stroke. And this brilliant man, forgot who I was, didn't know how to read or write. And this experience taught me three lessons. Doctors had said that my husband would never fully recover. But with determination and grit, Roenton wrote, that's my husband, A-B-C-D, one, two, three, four, for months. And he had this much grip in his right hand. And he told the physiotherapist, I want to tie the shoelace. And she said that will take a month. But tie a big bow on your thigh and after a month you'll be able to tie the shoelace. He tied the bow throughout the night and the next morning he was able to tie the shoelace. So this lesson taught me never to take an expert's opinion as final, but have faith in yourself. As parents, we all give great importance to education because we feel our wealth can disappear but nobody can take away our knowledge. My husband's stroke was a humbling experience. And I realized that one blood clot can take away all your knowledge. So the second lesson was, nothing in life is permanent. The third important lesson was to take care of our health and not to take it for granted.  And I used to take extra effort to cook things which were good for him. And with his keen sense of humor he said, it's delicious, do cook it when I'm out of town. So one thing is we need to look after our health from a very young age. When we returned from UK after about four months, well-wishers suggested that I joined the family business. At that time it was a private limited company. Though by training I'm a social worker, destiny took me to the corporate world. I worked for five years in the Human Resource Division under a wonderful boss and when he decided to leave, I became the head. Our two children, Mehre and Kurush, studied engineering in the UK and both decided to join the family business as trainee engineers. Mehre married a man called Feroz and we sent them to UK to turn around a small company which we had acquired and which was not doing well. And my promise to my daughter was that when she had her first baby I will be with her for six months. Four years later when Mehre was expecting her baby I went to the UK. In those days HR used to look after software issues, interpersonal relationship and I did not study the hardcore business. And I said to myself, when I come back from the UK I will start learning about hardcore business. After my grandson was born and when I was returning to India after six months my husband was very happy to have me back. And he drove from Pune to Mumbai to receive me but before he could come to the airport he had a massive heart attack and he died. So I was told when I landed instead of seeing him I was given the news that he had died. Two days after his death the board met and decided that I should be the chairperson of the company, executive chairperson. By then Thermax had become a public limited company a year before my husband had died. It was a very difficult period and I was missing my husband terribly and felt very inadequate to assume this responsibility. I kept devaluing myself and kept telling myself the only reason the board has selected me is because the family owns 62% of the shares. I was fortunate that I did not encounter any resistance from within the company or from the outside firm for being selected as the chairperson. People within the company knew me when I was heading the HR and welcomed and supported me. What I had to grapple with were my own self-doubts and putting myself down. Do you think this happens with women more often, besieged with self-doubts? The second challenge I had to face at this stage was that I had a choice. Either to feel helpless and wallow in self-pity, allowing my sense of inadequacy to grow or to hold myself together and take charge of my life. Soon after I took over as the chairperson, the Indian economy went through a downturn and Thermax's performance started deteriorating. Just to give you an idea, our 400 rupees share was quoted at 35 rupees. The terrible results were a wake-up call and we had to do something drastic. I convinced the board that we have to employ a consulting company to help us turn around. There was great resistance from the male executives. They said there is no need. Their consulting company will charge a hefty fee which we will not be able to pay and please have faith when the economy turns we will do well. But I didn't buy that. And I think males find it difficult to ask for help. It comes in the way of the macho image. Am I right? Thermax had to exit from many non-core business which added to our top line but eroded our bottom line. We brought operational efficiency which meant closing down businesses and asking people to leave. My husband and I had defined businesses somewhere we give employment. And in India where there is no safety net, it was very difficult to ask people to leave. We brought in robust performance culture and rekindled the innovators spirit. When we were doing well, we were known for our innovation. But with innovation for everything that works many fail. And when we were not doing well, our appetite to take risk was low. But when we again turned around, we became innovative. We reconstituted the entire board. Perhaps this was done for the first time in India and I had to ask every board member to step down and re-appointed a new board. My daughter and my husband were used to running business divisions but they had to make a choice. Either they remain executives or they become board members. This was a very difficult choice and they in a way resented me for asking them to make this choice. But finally they decided to be non-executive and joined the board. Those were tough and difficult decisions. But thanks to all the people within Thermax and within my family that supported me, Thermax turned around in 2002. I also think that I cannot take the credit for the turnaround for me and my people but also something like called destiny was also in our fever. When Thermax became profitable, the board decided to contribute 2% of our profit for corporate social responsibility. In India, since last year, it has become compulsory that you have to spend 2% on corporate social responsibility. But we started this years before it was made compulsory. I read a striking paragraph with which I agree wholeheartedly. The business of business is to generate growth and profits or else it will die. However, if that is the sole purpose of a business then also it should die for it no longer has a reason for its existence. This is more applicable in a country like India where we are surrounded by poverty and the gap between the rich and the poor is widening. It is imperative that corporates with its financial muscle and managerial resources reach out to the society. And we have to remember that business cannot survive in a society that fails. In 2004, I decided to retire and the board appointed my daughter Meher as the chairperson. She is a chemical engineer from Imperial College, has a good grasp of finance and has imbibed the family values and is far more suited than me to assume this responsibility. I am often asked why did I retire when the company was doing so well and I was in the media glare. I am from the HR background and I talked about and preached about planned succession. I had to walk my talk so that was important and I believe it's better to have a smooth well-planned succession. The second reason was I wanted to change gears while I had the energy and get back to the social work. And the third reason was an ego reason. I wanted to quit when I was very much wanted. I didn't want to stay on till I was 75 and 80 and all around said, when will this own woman get down? If you ask me what was the third and the most difficult challenge in my life, my response would be to learn to accept death and to come to terms with it. Like most people, I imagined that death is something that happens to other people. I was paranoid about death and whenever anything unpleasant would be said, I would find wood and touch it. Years ago when there was no expressway between Mumbai and Pune and my husband and I travelled by car and if the word death was mentioned, I would make him stop the car, find a tree and go and touch it. But today I know that if I carry the entire forest with me, I cannot change my destiny. After the passing away of my husband, I read many books on the subject of death and to some extent it did help. But intellectual inputs relate to the head and have their limitations. What brings about release, healing and transformation is the heart. For years I had heard of a Buddhist meditation program called Vipassana. Has anyone been for it?  Two people? Okay, good. It's a 10-day residential program and you are not allowed to speak, to read or write. And my husband used to tell me, I'd love to see you quiet for 10 days. My husband's death had created a deep void in myself and I decided to attend the meditation program. Since you cannot distract yourself by speaking or reading, you have to go within and discover your personal power. And what do I mean by personal power? Let me give you an example. Many years ago I had a small dog called Bimbo and it used to love biting people. My son-in-law called him a piranha. And one day a little boy of about six called Rehan came to our house and I forgot to chain him. And I said, oh my God, Bimbo went charging and I said, Rehan has had it. But in the middle of the lawn Rehan stood and with great determination said, Bimbo, stop! Bimbo froze. This dog had bitten several adults and here he was frozen because the Rehan showed his personal power. In the corporate world, many executives moan and groan that organizations do not give them positional power and hence they cannot be effective but they forget that they have their personal power and if ever you forget that, remember the Bimbo story. So far I have attended four repassion art programs and daily meditated for an hour. The end purpose of meditation is to make your whole life meditative. I'm very far from that and I'm still very reactive, judgmental and impulsive and yet I have found a lot of peace in my life.   I realized that it is my choice to focus on my limitations and keep comparing myself with my husband and end up feeling small and inadequate or build up on my own unique strengths. I realized that if I go to an apple tree and say, please give me oranges, it can never give me oranges and I will not be able to enjoy the apples also. So learn what your unique strengths are and develop them.  It is inevitable. Let me repeat, we don't know if we will have the lunch and hear the afternoon speakers but one thing is sure, all of us at some time are going to die. It's inevitable. What we do not know is when, where and how we will die. It is unpredictable. Death is a social leveler and pays no attention to age, status, wealth, opposition. Beauty, royalty, glitter and glamour could not save Princess Diana. I in my own world realize that with all my wealth and all that I had, I couldn't prolong the breadth of my dear ones even for a minute where death is concerned, I learned that God is not partial to anyone. I also realize the sun rises and sets each day and we don't become sad and say, oh God, why has it gone away? Why has it gone away? We have accepted this process that the sun will rise and set. Similarly, all of us are born and have to leave this earth but since we do not accept this inevitable process, we refer to death as a tragedy. To me, tragedy in life is not the death of a dear one, but the failure to get along with people who mattered to us and the failure to invest in ourselves. Those are tragedy. Once you accept that no living person can escape death, you value the time you have on this earth and life becomes full of meaning. Dealing with death releases a lot of energy and you have the confidence that if you can come to terms with the most difficult phenomena, you are able to face any challenge in life. In 2002, I was a chairperson of Confederation of Indian Industry for Western Region and I spoke up against the killing of over 1,000 Muslims in the state of Gujarat. I was one of the few corporates who spoke up but death had given me this courage. What worse can happen to me? An untested myth is that life is fair and that if we lead a good life, we and our dear ones will be rewarded by being granted a happy and long life and the evil ones will suffer. If life were to be fair, why is it that all of us have eyes, limbs and ears when others are born without? If life were to be fair, why is it that I don't have to struggle for my next meal? In fact, we have obesity and overeat while a large section of humanity has to constantly worry about where the next meal will come. If life were to be fair, why would we be shard with opportunities while others face a blank wall? Once I surrender to the fact that life is not fair, it keeps me from feeling sorry for myself and for others. I have empathy for others but I do not pity them. Fourteen months after my husband passed away, within two weeks I lost three more people who used to stay with me. My mother-in-law, who was 96 and waiting for God to take her away, died. A few days later, my pet dog Bimbo died and my son, who was 25 years old, with his full life ahead of him, died in a car accident. The pain I felt when my husband had died proceeded into insignificance compared to how I felt when my son passed away. As a woman, sorry, by then I had learned that after the death of a dear one, pain is inevitable. You can't escape pain. Your body will ache, your mind will go through turmoil, you can't sleep, you can't eat, there is pain. But suffering is optional. Suffering happens when you keep asking, why did it happen to me? You keep feeling sorry for yourself. Why has no answer and leads to more and more misery? As a woman, I had never lived by myself. Before marriage, I stayed with my family and after marriage, I joined my husband and the in-laws. After the departure of my husband and son, my house was empty and I found it very difficult to be by myself. But today, I value my solitude and love being on my own. But this change was a traumatic process. Most of you in the audience, in fact everybody here is younger than me and you may wonder that we've never had an experience of death and why are you harping on this? Well, from a young age, it's helpful to internalize that loss, separation and failure are built into our lives and at some stage we have to face them. In fact, everything that happens in our lives, every so-called misfortune, every rejection or loss, teaches us very profound learnings provided we allow our lives to be our best teacher. As human beings, we have the ability to convert the struggle to grace or wallow in self-pity. The choice is entirely ours. My son, having lived abroad for seven years, was very concerned about poverty in India and wanted a part of our personal income to be spent for social causes. After his demise, I was looking out for a credible NGO and everyone said go to Mumbai and meet a remarkable young woman called Shaheen who was running an NGO called Akanksha. So I went to Mumbai and I got involved with Akanksha which runs municipal schools between Mumbai and Pune and we have 15 schools on a public-private partnership and my company finances eight of them. About eight years ago, Shaheen decided to start something called Teach for India which is model after Teach for America and I partnered her from day one. In a short span of seven years, Teach for India has spread to seven cities and attracts the best of talent and we work with the underprivileged students in municipal and low-income private schools to bridge the inequity gap in education. I believe you have Teach for Malaysia and Rajiv tells me they are getting an award today and I hope some of you will know more about it and get involved with it. I would like to now share some of the lessons that as a leader I learned. Thermax was an engineering company and without having a degree in engineering and having no knowledge of finance in fact hating finance, how was I able to be the executive chairperson and turn around? Coming from the HR background, I knew my people well and I could empower them. With humility, I often said I don't know and our seat helped without me feeling small or ashamed. I did not feel threatened by senior employees who were experts in their field. In retrospect, I can see that a CEO becomes a leader when she is able to lead her own life with a sense of well-being and before she attempts to lead others. While life will keep on presenting hurdles and challenges, it is important attribute of a leader to face the world with equanimity. As a leader, I have found that I need the courage to listen even though what I hear is not what I like to. I have to peel my layers of prejudice and really listen to voices of dissent. I believe that a leader can always hire skills but the key attribute of a leader is wisdom. What do I mean by wisdom? Your unique and personal learnings from life as it unfolds and accepting its rhythm.  A leader has to define success not by mere growth or profit but by the legacy left behind for the future generation. With this definition of success, we need to be sensitive to our environment and to social causes in our community. I believe that we are the custodians of our wealth we create and from our corporate and personal resources, we need to reach out not only by writing checks but by getting involved, not only for ourselves but involving our employees. My family and I avoid getting caught up in the endless cycle of consumerism and extravagance and one-upmanship. We give 30% of our yearly earnings to social causes from our own funds. I would like to share now some of the learnings about life. I think we all need to aim high and challenge ourselves but let not success at work demand a disproportionate price by making us a failure in life because work, though very important, is a subset of our lives. My definition of success is to live an abundant life. Abundance can be defined as rich in experience, full, meaningful, significant, adventurous, lived with a sense of well-being. Well-being comes from being at peace with oneself and in harmony with others through meaningful relationship. What saps our energy is not physical work but negativity, turmoil and tension. From a distance, superfluous success seems to have all the glitter and glamour for which most people crave for.  Hence, I would like to give you a thought that before you unidimensionally focus on professional success find a meaning and purpose in life. If you are not authentic and keep dancing to the tune of others you may gain fame and acclaim but as my favourite poem, The Man in the Glass says and I'm quoting, you may fool the whole world down the pathway of life and get pats on your back as you pass but your final reward would be heartache and tears if you cheated the man in the glass. We need to learn to express rather than impress. You know, expressing means saying from your heart. Impressing is very concerned. What do others think of what I'm saying? We're so afraid and caught up in what will others say about me. I have realised and have great freedom because I've stopped worrying about what other people think. When I go for my walk, I do my mouth exercises. I look at, do my eye exercise. I do my shoulder exercise and I say to myself, what I'm doing is it harming others? And if the answer is no, I continue doing because it's helping me. What helps in life is cultivating a big sense of humour, having a big dose of humour, not to be able to laugh at jokes but to laugh at myself or not to take myself or my achievements seriously. We need to internalise a profound saying that our stay on this earth is short, our roles dispensable and our impact in consequential. I believe that each of us has two sides within us, the extraordinary and the ordinary. Tell me, how many of you here are extraordinary? Put up your hands. Sad that so many of you are just ordinary. Actually, it's much easier to define ourselves as ordinary because then I can indulge myself and say, I'm just an ordinary person. Don't expect much from me and learn to live with my mediocrity. But if you accept that you are extraordinary, then you have to push yourself and give out your best. And the best example I can think of is Gandhi ji. He was ordinary and in South Africa, he discovered his extraordinaryness and we know what he did for our country. When I'm taking life easy, I need to remind myself that I'm extraordinary.  So a balance between these two is very important. I would like to end with my favorite story from Vipassana called Swimology. There was a young professor who went on a long voyage and being a professor, he took with him stacks and stacks of books. And there was an old sailor who was illiterate and he said to the professor, on this long voyage, will you please teach me something? So professor said, of course I'd love to. So the first day the sailor comes and professor asks him, do you know what geology is? Never heard that word.  So the poor sailor feels dejected and goes to his room. The second day he comes for his lessons and the professor says, do you know what oceanography is? He says oceanography, never heard of it.  Third day, do you know what meteorology? Never heard of you? 75% of your life is over. Fourth day the sailor comes running and says professor, professor, do you know swimology? He says no, what's that? He says swimming, swimming. The ship is sinking and 100% of your life is over. So before we clutter our heads with all the technologies and lodges, learn the art of living which is swimology. Thank you very much.