I'll just give a brief introduction. Apurvakhar attended the IMO TC in 1996 and 1997 before his BSTAT at ISI Kolkata and PhD in Mathematics at the University of Chicago. After teaching mathematics, including at Yale and Stanford, he returned to join ISBangalore, where he's an assistant professor in mathematics and a Ramanujan fellow. His research interests include matrix analysis and positivity, representation theory, combinatorics, and a bit of probability. And he's going to talk on groups with norms from world games to a polymath project. So what do you have, Apurvakhar? I guess I hope everyone can hear me. There's a mic working. So first of all, as Antur also said, it's a great honor to be back at the Olympiad camp, where more than 20 years ago I was on that side. Unlike Antur, I have no memory of the final day or anything. But it was definitely one of the most, I don't know, landmark or foundation of it. I don't even think of the right adjective, times in my life, because that's really as, and that's what actually goes back to what Prof. Raghunathan was saying, is it brought me in contact with all these incredibly sharp minds, and you learn what is possible. It is you learn if it's possible for you to think faster, to think better. And that for me definitely was the biggest takeaway from the Olympiad camp is I improved my own way or my own level of thinking as a mathematics problem solver. And in a certain sense, the project that I will describe to you brings me back full circle to exactly that philosophy, because it brought me in contact with some of the sharpest minds I've known, especially the last one is very famously very sharp, and the first one actually is very sharp. And again, once you meet these or interact with these people, you really, again, see maybe the next level after the Olympiad camp, the next level of how one can think even faster, even sharper. And that definitely for me was a big learning experience. So I'm very happy that it actually happened just a few months before this Olympiad camp and brought me back to the same kind of school, if you will, where I learned some more as well. Not just about the math itself, of course, there are some very fun and clever ideas here, but also about thinking about math. Anyway, so whatever this, I won't even assume what a group is. I'll try to explain something from the board. Norms everybody here has seen, so I will tell you what they are. And the point is, this is a question that doesn't require any kind of background, any thing. It just requires clever ideas. And the solution is entirely using clever ideas. And I thought it's actually, in some sense, nice enough to present it in the Olympiad camp, where as I said, the technology, the tools, the definitions you need are not so many. The ideas you will see are hopefully interesting. And they are interesting equally to people like Terence Tao, as to people here. So this was a completely, so again in contrast to the previous talk, which was very classical, which goes back to Gauss and Laplace and so on, this is on a problem that can be, as I said, asked without any maps in some sense, almost, and yet nothing was known. Not a single example, not a single counter example. And it got solved in a space of five days. But again, it's also a very modern solution in the sense that the fact that it got solved in five days makes use of the internet. It makes use of blogs. It makes use of, and it crucially made use of a computer, where human intuition sort of seemed to stop working. A computer was the next thing that provided the big. So it's a very 21st century project, and therefore 21st century talk. Without further ado, then so my, yeah, so this is actually based on, this is a full-fledged research paper, and I'm gonna try to give you about 70% of the proof, so of 70% of the paper, if you will. And as I said, it requires pretty much nothing. So joint with Tobias Spitz, he's in Germany. Siddharth Gargill, who is my colleague, and actually was my senior in college in Bangalore. Pace Nielsen in Utah, in USA. Lior Silberman in Vancouver. And Terence Tavern in Los Angeles. So here is the basic problem at hand. So I need to define for you the players in the game, and then, so there's a collection of strings, and there's a function acting on them. So here is the collection of strings. So there's, consider the collection of strings in four letters, so let's just call them alpha, beta, and alpha bar and beta bar, which are essentially one over alpha and one over beta.  Alpha, beta, alpha, alpha, beta bar, alpha bar. Things like that. There's a collection of strings, a finite line, just that's all. And only two rules, there's one rule, the two rules here are that alpha and alpha bar cancel each other. If I had alpha and that much, then I could say this is the same as alpha, beta, alpha, alpha, beta bar. Yes sir, these just go away. And similarly for beta and beta bar, and it doesn't matter which order they're from, they cancel each other. And if I have, so I'm allowed to also consider the empty string, which has no alphas or betas or these things. So you can call it the empty set, or you can call it one. Okay. And we don't, in particular, we don't assume things like alpha, beta is beta, alpha. So you can't take things past each other. The string is the string. You can't change. A word is a word. You don't say that, you know, you don't say cat is active. So you can't take A and C past each other. So similarly, you can't take alpha. So these are the, the one extra operation permitted is, I can multiply two strings. So if I have one string and another string, I can just put them together. That's all. And so in particular, alpha, beta, alpha, if I have these two strings, that's three letters, two letters, I make it a five-letter string. And then I can, the one rule I'm allowed to use is I can cancel pairs, like a particle and an antiparticle, they cancel, right? So you get alpha, beta, alpha. Okay, so this is the one thing I've permitted. So in math, we call this whole thing the free group on two letters, but forget what we call it. It's just a collection of strings in two letters and they're inverses and they're reciprocal. That's all. Fine. Now here is an example of this string with some additional relations. So suppose I could take the alpha, suppose I could say alpha, beta is beta, alpha. Then what is an example of that? Well, here are, so alpha and beta are going to act as some actions and here are two actions that can be interchanged and which have inverses. Okay, so alpha is one step to the east and beta is one step to the north. And then if I take one step to the east and then one step to the north, I get here. If I took one step to the north and then one step to the east in the reverse order, I would still get there. So alpha, beta equals beta, alpha. I don't assume this in general, but here is an example where I do assume it and then what is alpha bar? What undoes one step to the east? One step to the west. What is beta bar? That's alpha, beta, alpha, beta bar, which the additional relation. Okay, fine. So these are one step, one step, so alpha, beta denote that and then you can consider all strings of this kind and then what this string will denote is the net displacement. You start at the origin, let's say at zero, zero, where do you end up? That is the string, yeah. And of course, if you have south and north or east and west, you can just cancel them because you don't do anything. You come back to the point where you started, so the net displacement doesn't change. Okay, that's an example of what this is and the point about strings is, so here you are allowed to have more tricks, more rules to use to play around with. I'm saying even without those rules, I'm going to ask you a question and we'll see you. So for asking you that question, the problem, here is the next player, the second and last player. Suppose there's a function and I'd call it L because it's supposed to stand for length, so length of that. So I said that this stands for some displacement. That's going to be the length of the displacement. How much distance, like as the crow flies from the origin is there? So I'm sticking to that collection of strings, that free group, but you could ask the same question to model all these additional rules that you can use. Suppose the length satisfies just two properties. One is that it is a sub additive function. So if you have a string, if you have, let's see. So if you have the length of this thing, so the length of this thing, if I can write this as a product of two things, then the length of alpha, beta, alpha, alpha, beta, bar is less than the length of the first part plus the length of the second part. That's one rule. And the second rule is if G and H are equal, so this should be true for every possible pair of strings. And if G and H are equal, then now think of this example, right? So, well, okay, I'll get to the example in a minute, but so if G and H are equal, then L is actually additive. So you get double the length. And that again, so that should be very, here is the problem. Let me just first tell you the problem, fine. So suppose you have some function on the collection of strings, it takes a string and it gives you some number. It can be some real number, negative, positive, zero. It satisfies exactly these two properties. Length of G H is sub-arative and if G equals H, it's additive. Is it true that the length of alpha, beta, alpha bar, beta bar is less equals zero? And I'm gonna draw all these things in this case because I can draw these things in a second. The interesting thing was, so for those of you who know what free groups are and what norms are or metrics are, this is, okay, so if I further assume that this length was non-negative, this should essentially be a metric. And the question is, is it true that there is a metric on the free group which is additive for equal argument? And this was not, no examples are known. And is it that no examples exist? That was not, basically nothing was known about this problem. And it's very simple to ask. You can ask it after, so literally you can understand what this problem really is saying. I'm saying really is saying, not just in these pictures, after say one year of college, map. And, but at the same time, nothing was known. And so the interesting thing then was, as I said, it's a very 21st century talk because it starts on the blog, use the computer and ends in five days. And presumably in older times, it would require collaborations which would take time to form and then slowly exchange letters. And you know, it would probably take more than five days. This happened all over the world. Some people was in, somebody was in Germany, he solved something, then when he slept, somebody was in India, they solved something. When they slept, somebody was in USA and Canada, they solved something and kept going on the clock. As you can see from the, if you look at the blog, there are comments and what time the comments are made, you can see. And you can see that the comments keep happening around the end. So back to the example, I want to write down those two rules for the example and ask this question. So this was the setting, alpha is an east step, beta is a north step. Now this word simply denotes the displacement, as I said. And how do you calculate the distance? How do you calculate the length of the displacement? You use Pythagoras theorem, x coordinate, y coordinate. So more generally, let's say instead of alpha and beta like this, which were unit length, let's say my alpha was some distance a and beta was some distance b. It's still the same things happen. It's still, you can talk about the displacement. It's just that the displacement now becomes like one alpha and one beta goes to the point a comma b, not one comma one, that's all. Nothing has changed. Now given a string s, a string s meaning something like this. This is a string. Simply count the number of times alpha, of course. Call it n of alpha and then count beta and alpha bar and beta bar. Just count the number of times vector and then what is the length? Well, so the length is, what is the coordinate? It is a times the number of east steps minus the number of west steps. If you take steps backwards, then you would have to backtrack. So that's all and then plus b square times this. So and then so and the y coordinate is b times this and then by the same distance Pythagoras theorem you get this formula. Anyway, so why does this satisfy the two properties? So whether or not you had this extra rule, I claim that those two properties are satisfied. Well, l is substitutive because if you have any, so what is a string? A string takes some, start at the origin, you do something and you get somewhere else. That's string one, let's say. A string two starts from the origin and starts from this point and does something else. And the total string is this. And of course, you know that the length of one side of a triangle is less than the sum of the other two lengths. That's called the triangle inequality. That's why this is satisfied. So those two properties are satisfied for this model just because one is called a triangle inequality and the other is saying that if you start at some point and do some action, some string g and then you again do the same action, well then you're just going along the same direction for twice the length. So length is twice. So it's really saying nothing. There's nothing there. In fact, more generally if you have g to the n, meaning you take that same string and do it n times, you just go n times along the same direction for the same thing. So your length becomes n times. And now what was that question? The question was what happens to the length of this element? The question was what happens to alpha, beta, alpha bar, beta bar? Well, but what is alpha, beta, alpha, beta? You could say alpha, beta, backtrack and backtrack. And you come back to zero. And so of course, the length is zero. In particular, it is non positive. I asked was it less equal to zero? In fact, it is equal to zero. So yes, so in this case, clearly the problem is fine. Now, so here is the problem you can state for any group. This is the same question, but instead of saying, yeah, so this is how one says it formally. This is the only time I will use the word group. So you may as well see it once in your life. Well, before, if you see it later, it's very good that you will. You may see it once in the camp, or maybe you already have. Suppose g is any group. The group just means like a set of strings. So this is some set with a way to compose or multiply. For strings, we said take a string, take a string, put them together. We have some way of putting things together, a zero element like the empty string, and an inverse or reciprocal like alpha bar. So for example, oh, by the way, I should ask you. So I'll ask you later. So suppose L is a, so now here is the true problem, which were condition one was not asked earlier. I didn't mention it earlier at all. But now let's say I mention it. Suppose this length is always non negative. And the length, when is the displacement zero? When is the displacement exactly zero? It's zero exactly when you have done nothing. You are at the origin. So the origin is what you call the identity, the thing that's, yeah. So equality if and only if g is that special element, zero element, length of g and length of g inverse are the same. G inverse means you're doing the steps in the opposite direction. So you have this way, you would be on the other side, but they should have the same length. For that, I need that alpha and alpha bar have the same length here, and beta and beta bar here. For instance, and length is sub additive, which we have said, that's the triangle inequality. And there's the extra hypothesis that I had said. And that extra hypothesis is what makes all the difference. When if g is h, then L is positive. Here is the main theorem of the paper that I mentioned. G has to be abelian. G has to be commutative. Meaning, the point is this group. If I, so if I take alpha beta and alpha bar, then I get alpha beta alpha bar. This is not the same as composing in the reverse order, alpha bar first and alpha beta next. Here I just get this. This string and this string are not the same. So this group is not commutative, whatever that means. It means that you can't just take elements and multiply them in any order and expect to get the same answer. But that is forced by that extra hypothesis. So basically the statement says, if you have a group, if you know what these words mean, if you have a group with a norm, meaning this last condition, so you have a group with a metric which is a norm, that group must be abelian and that group must be something called torsion-free. And remarkable, not so hard is the statement that if you have an abelian and torsion-free group, you can have a norm on it. That's whatever that means. Forget these mumbo-jumbos, but yeah. That's the statement. This is actually an if and only if. Anyway, so, but here let me tell you the proof of this statement from the thing. So L satisfies these two properties that we started with, which maybe I should write here anyways because I'll keep referring to them. L of, what is it, G H is less equals L of G plus L of H and L of G squared is equal to twice L of G. So L satisfies these two properties. So if you believe, so if you believe from this, so I might remember my original question. The question was, is it true that L of alpha beta, alpha bar beta bar is less equals zero? If you believe that that's true, meaning G H, G inverse H inverse, right? Alpha bar is just the inverse of alpha and beta bar is the inverse of beta. If you believe this is true from just these two properties, then fine, then that's what I'm saying here. That's all I've said. If the original problem in that free group was true, was had the correct, has a positive answer, this is true, and that's all I'm saying here. But the hypothesis once said that L function is always non-negative. So how can you have this to be non-positive and non-negative, has to be zero? When can it be zero? It can only be zero if this string is the identity. So what do I get? I just got, this is E. From now, what do I do? I multiply on the right by H. So these two cancel. That's what reciprocals do. E times H is just H. That's what identity is true. And now I get G at G inverse is H. Well, multiply by G. Those two cancel and I get G H equals H. So just knowing, if you believe me that this implication is true, if this word to be true, then the group has to be committed. That's sort of very simple. Why is this true? Is this true? And it's so high. And that involves what I mentioned in the title of the talk, word games. So now I wanna try to tell you the word. So back to the original problem at hand. So in some sense, to solve this problem, you can first of all, you can try to work with as many rules on G and H as you want. You can say that G and H commute, in which case we know that this is just zero in some way. You can work with other, you can work with extra rules. Like G square is one, maybe, or X square. You can work with what we call relations in groups. This is the case. The free group is the case where there are no relations. You have no extra rules to play with. There are no extra tricks you can use. Only this much. These are the only two things you have. You can concatenate. You can put two strings together. You can take inverses. Nothing else.  If it is true here, you should be able to hope that it's true everywhere else. But is it true here? And why? Very easy to state. And as I said, it still took five days with incredibly sharp people working on it. So first of all, let's see. I claim that this is a group. So that's in bold. Why is it a group? What is the zero element? It's the empty string. Take empty string. Put anything together on left or on right to get what you put together. The empty string exactly is that. What is the inverse of alpha or beta bar that we know? The inverse of alpha was what we called alpha bar. The inverse of beta bar is beta because A is A. Reciprocals are reciprocals. For lack of a better way of saying it. What is the inverse of alpha beta? Is it that? It is not. That's correct. It is the other way. You have to do it in the reverse way. You have to write down it's beta bar and alpha bar. So if you put beta bar here, it first cancels. Then you get alpha and alpha bar. Then they cancel. So you can't just say this. In general. So for example, if you have the inverse of this nice element we were looking at, then that would be first take the inverse of the rightmost. That is beta bar inverse is beta, then alpha, then beta bar alpha. So just usual tricks. Fine. And what we want to understand is if G is, so now if G is this group, and it satisfies these two conditions for all strings G and H, then what can you say about this? Or maybe I'll even, so this was true for all G and H. This was true for all G. The question is maybe I'll make it even simpler. Not for some general G and H. Let's look at the specific string. Alpha beta, alpha bar, beta bar. These G and H are words in these letters. Let me just look at these four letters. Not some G, H, longer. Just the shortest possible system. What is this? Why is this true? And now here are where the word games start. So now I'm going to, so how does one get around to trying to prove this? Well, so here we go. So the first claim is that the length of, as I said, length of G times H is less than length of G plus length of L. Now from that I claim that you can prove this. Yes, how? One word, induction, right? Induction, great. Oh, sorry, I wrote it down, didn't I? Fine. I thought this was supposed to be a quiz, anyway. One last throwback, like, how quickly can you, anyway. The second one is that, so first of all, this one I didn't write down, so good.  So clearly you just get this is less equals, length of G plus length of G plus length of G, that's k times length of G. I claim it's equal, always it's equal, why? So some of them are easy to check. For example, length of G square, if K is two, you have two things, this is actually given to us. But from that I claim that length of G fourth is four times length of G, yes. Because you take G square, square, and then, and so by this you can therefore, save an induction that length of, for which K does it work immediately? Powers of two, all powers of two you get the same. How about length of G to the sixth? Six is not a power of two, so again, yes. N implies N minus one, X, oh wow, okay. That is very sharp, yes. So I was gonna do it, yeah, well, that's actually, yeah, just, okay. Saying that is enough, let me actually spell it out, what he said, so take any power of two bigger than that, and I'll do N implies N minus two in this case, but that's twice the application of what he said, right? So you take G to the eighth, that's eight times length of G by power of two, right? G to the eighth, as eight, there are eight of them, so take six of them, what we want, plus one, plus one, that's less than length of G to the sixth, that's length G to the seventh, because I'm using the fact that there are, this is a product of three terms, but I can use this by induction, right? And then now this is, but now length of G to the sixth by the same argument is less than six, right? And now I get back eight, so I start with eight, I get back eight, therefore every inequality in between must be an equality, this is an equality, remove the two, you get the G to the sixth, right? So you can do the same for every power, great. So this is nice, so we have one nice consequence, a specific nice consequence is for equal arguments for any power, integer power, they're equal. Now the third one is interesting, well the title, if you know what conjugation is, that's the third one is saying conjugation preserves length. So let's take any string and then take two elements, left multiply and right multiply that string with one letter or one string each, but those strings should be inverses of each other, okay? So you get some string. Can we prove that the length of S is length of G? That turns out to be slightly harder. So here is how one does, it's called string, the strings are called conjugates of each other. And conjugates, the nice thing is they behave very nicely with powers.  So you get alpha T square, alpha bar. And then you can do by N, if you take one more alpha T, alpha bar, those will cancel, then those will cancel. So in general you just get that S to the N is alpha T to the alpha bar. This is true not just for positive integers, it's true for N equals zero, because you get alpha, alpha bar, which is zero, which is just one. And it's true for negative integers, it's true everywhere. Lemma, L is actually conjugating variable. Take any two strings, G and H, just from these two properties. And the two that we deduced on the previous slide, you can get that the length is Y. Here's the proof, I don't have time, so I'll show you the proof. I'm sure you guys can come up with it, given enough time. So the proof is very simple. Start with the length of G H G inverse multiplied by some number N, positive number N. What is this? This is the length of the Nth power. That's what we did on the previous slide. Length of any string to the N is N times its length. So what is the length of this? It is the length of this to the N. As we just saw what that means, that means G H to the N G inverse. And now I use my property one. So it's less than length of G, so length of G inverse. And then H to the N, as we again know, is less equals N times length of H. That's the same thing applied inductively. This is true for all N. And now we use what the previous speaker, Anthur, told us. The method of divide by N, divide both sides, and the powerful trick is to take N to infinity. It sounds like a very small idea, but very simple thing to do. But here is what it really means. I mean, very simply what it really means. Suppose I know that A, I have some real number A, and that's less than one over N for all N. These are all positive numbers. But from this fact, for all N, I know that this immediately implies, and of course, it can only if A is less equals zero. So that's that idea that you can go from any, this is what calls the Archimedean property of the real numbers. For every real number, there's a rational or one over N smaller than it. But this idea is a very, very powerful one in some sense. And that's what we are going to use. So dividing by N, you get the length, this N goes away, the length of what you want here, is less equals length of H plus this stuff, which doesn't depend on N divided by N. And now when you take N to infinity, that goes away. So you get the less equals here, but I want equality. What do I do now? Apply the same trick, right? If G at G, if this is a conjugate of H, then H is a conjugate of this. I pre-multiply by G inverse and post-multiply by G, which are inverses of each other. So if I call this whole thing H prime, the length of H prime is bigger than the length of G inverse H prime G. So I exactly get the reversal problem, right? So I get equality. So you can start playing these games, and that's why I call them word games. You can start playing these games and deduce more and more facts. So this trick is a powerful method in mathematics and considering this thing, which is called a commutator, whatever it is, is a useful trick for proving that these things commute. Because if you can show that this string is one, is the identity, meaning it has length zero, if you will, then this equal to one, and then we saw how to post-multiply by G and then G inverse of first beta and then alpha and get that, right? So back to ask you. So we have proved all these nice properties. So we have proved, what have we seen? We have seen that length of G to the N is N times length of G, and length of GHG inverse is length of G. We have just proved these extra two properties. And the one by induction, I don't mention, that's it. Ah, thank you very much. Yes, so these are the two we have proved so far. So now, so more generally, can we find what one calls a non-Abelian group? Things don't commute. If they do commute, of course, here is an example of a group where we can prove this very easily. This distance was zero. North, east, west, south, or whatever, was zero. So find this thing or show that no such group exists. And as I said, so this somehow, so this showed up sort of naturally out of some work I was doing in probability theory, but then I actually emailed various people. Somebody at Harvard, somebody at Northwestern, Northeastern, somebody at Maryland, Seattle, Urbana, some people in Bombay, unfortunately nothing was known. And then last December, I was, I had joined ISC by then, I was visiting UCLA, where I met Professor Theron Stahl. And for those of you who don't know who he is, presumably very few, he has a very short biography. So he's the youngest participant in the IMO in history, at the age of 10. That year, he got a bronze. So he's the youngest bronze medalist. Presumably, that's the consequence of the previous one. You can figure it out. The very next year, he got a silver that makes him the youngest silver medalist in IMO history. You can guess what the next slide is. And then, you know, the only other thing he did of importance after that, award he got of importance, IMO and field medalist thing. That's where you all should go. IMO and field medalist. Anyway, so he got, but yeah, he's extremely decorated. He's one of the sharpest minds around, one of the most prolific mathematicians. He's achieved lots of powerful results and gotten most other awards. Anyway, so, yeah, he's incredibly sharp. Let's just put it at that. So, work games are terrible stuff. So this is what we discussed. So suppose, recall, these are the two properties which I wrote down here. Suppose we want to show this thing. So, okay, so, first of all, I can rescale and say that make all of these, whatever these lengths were, right? The length of each of these is some number. If it is negative, fine. If it is positive, divide by some large positive numbers so all of them become less than one. That dividing by positive number doesn't change what we want. That's the point. Zero divided by anything stays here. And positive means the inequality stays there. Now, the strategy then that came out of our discussion was to try to prove that this length is less than C for a smaller and smaller value of the I raised it. But the moment is less than, say, or less than one over n for all n. Then you get that it's less than zero. Okay, so now we come to the work games. So, this is the last time you'll be asked to play games in this camp. Or that can be solved on the board. So from this, from these alone, can you prove this is less equals, oh, I'm sorry, the third property that I mentioned, length of alpha, length of beta, all of those less equals one. From these, can you prove it's less equals four? Yes, that's trivial. That's just the triangle inequality. Okay. Can you prove it's less equals two? Yes, why so? Right, so you break it here. That's right. You do this. Now, the length of this by here is length of beta which is less equals one. And beta bar which is less equals one. And you get two. Perfect. I might have broken it differently, but exactly. So, how about four over three? Let me tell you four over three, because that's, you'll get there. You'll get there, I know because I got there. I'm sure you will. But this is what you'll do. Four over three should remind you of something. This three in the denominator should tell you what to do, cube. So, if you look at the length of alpha, beta, cube. I'm going to call this alpha, beta for convenience. Look at the length of this cube. How many letters does this have? It has four letters. So, how many does the cube have? 12. So, first I write this down. Forget the length. Alpha, beta, alpha bar, beta bar. But I write them three at a time, because I want four. Alpha, beta, alpha bar, beta bar. And then alpha, then sorry, beta, alpha bar, beta bar. There are 12 letters. I write them into groups of three. What is the length of this? This is less than the length of this. The length of this, the this, the this. Each of them is a conjugate. So, I just get the length of beta, length of alpha, length of this, length of this. Each of them is one. At most one, so I get at most four. So, size, the length of this is, at most the length of is less equals four. But this is exactly three times the length that we want. And I get four over three. Word games. The question was, can we keep improving on these word games and come down to here? And that's what he posted on the blog. So, this is where the blog starts. So, here is a curious question posed to me that I don't know the answer to. So, and then he let it have to be the free group on two generators. He described this thing. What is not clear to me is if one can keep, so that's where we ended. And then I left to fly back to India and he posted the blog. If what is not clear is if one can keep arguing like this to continually improve on these bounds and come down all the way to zero, blah, blah, blah, blah. Anyway, this feels like a problem that might be somewhat receptive to a more crowdsourced attack. I'm sure all of you know crowdsourcing. I definitely didn't at my age. It wasn't probably around. So, I'm posting it here in case any readers wish to try to make progress on. And then it started. So, conjugation invariance. This that I erase. L of G H G inverse is elevated. That came within three hours. From that first guy, that's Tobias Fritz from Germany. So, he's this very young and incredibly sharp guy. And then people spent two days trying to find examples. Out of the five days that I said it took to build the world, which you will. The first two days were spent trying to find examples that did work. Maybe, can you introduce additional rules without the rule alpha beta equals beta alpha? That is just always good to work. Without that, can you introduce it? Additional rules and then make it work. Nothing worked out. So, if for those of you who know, I'll say they tried to find solvable groups, nilpotent groups, some kind of perfect groups, maybe nothing worked. So, then they said, okay, fine. Then, then people said, okay, maybe it doesn't work. Let's try to prove it doesn't work. Let's try to bring that bound down to zero. So, four over three was we had already put on the blog. Can you beat four over three? The next bound was five over four. And I'm going to tell you how. More, see, the point of this talk is just to tell you the word game. So, here is the word game. Again, if you want, you can follow it, but this is the word game. Start with a different string. Now, you had alpha beta, alpha bar, beta bar, put an alpha before it, okay? So, anyway, you don't need to follow this. The real idea is on the next slide. This is just some example of how much more involved you have to do. If you write this down, and twice this length, then it means you have five letters, you write 10 letters, so you do something. Alpha, so this is alpha square beta, alpha bar beta bar, alpha square beta, alpha bar beta bar, and then I multiply by alpha bar square, alpha square. So, there were 10 letters already. These are the 10 letters. There are four more letters which are doing nothing. Alpha bar square, alpha square. But then I can do something clever. So, now by sub-additive, I can break off this alpha square and get two. And here I see alpha square beta and beta bar, alpha bar square. If you carefully check, these are inverses of each other because they are in the reverse order also. So, the length of a, so this is a conjugate. The length of this conjugate, you can remove the G and T inverse, and I just get these two. But this is a conjugate, this is a conjugate itself. So, these are conjugates within conjugates. I call them nested conjugates. And so, I get one, two, and two, and so, this is a conjugate, this is a conjugate, these are two, I get four. So, twice length of this is less than four, so this length is less than two. And now I can do the same trick as on that board. So, I take, I want five over four, so I take four times the commutator, I get 16 letters, and I do it cleverly so that what remains here is that extra five letter string. And then you do the math, but then you can just check that this is again a nested, this is a conjugate of four of three conjugates. So, one conjugate gives one, another gives one, another gives one, after breaking this off, which gives two, and you get five. So, you get five over four. Isn't it? You can do various kinds. So, you have a, so the point is the following. One thing is notice the repeated conjugations on the right-hand side, conjugate inside conjugate. The second is that, notice that in each step, you get a finite improvement. I'll say that in the slide right now. So, the blog post one, so now I'm telling you all these times in the Indian standard time of the minions, so 17 December and he posted it on the 16th night in UCLA. Here it was morning. In three hours as I said, literally at the dot of three hours, broke of three hours it came. Then you, there was five over four, so four over three was fairly straightforward. Five over four was not, but it was still doable. You can see why it works on five lines. 19 over 16 became harder and 22 over 23. First of all, this is an important breakthrough because the bound went below one. That was somehow, maybe a psychological barrier for people, will it ever go below one? If it does, then it will go to zero. It did go below one, but you can imagine if it took from here, the next bound was, this is 20 over 16. The next improvement was the 19 over 16. How much harder it would have been to get to here? So it was extremely clever tricks, one after the other, but the point is one has to keep reducing to get to zero and at each stage you are going to a finite number. You have to do this, any finite number of steps with getting to a positive number. It will not get you to zero. In principle, this is an infinite process and there are therefore infinitely many more and more clever tricks and that is going to be hard to do. So what we wanted ideally was something that would recurse, that would sort of apply repeatedly to itself, feed on itself and this is where sort of, maybe let's say intuition broke down or I don't know what to say, but there was a stopping block here for almost 24 hours out of the five days, two were gone and then there was a third day which was almost entirely gone with nothing and then Siddharth Gargill, who was my colleague, he programmed, he had an idea which he asked, but he was verifying their idea manually was sort of very hard. He got a computer to do it for him. So again, this is a very 21st century project, it starts on the blog, it has a computer. The idea was the following. If indeed this was to hold, then if you take powers of this and multiply by alpha, by that sub additive business it should be close to length of alpha or it should be less than one. And he did this to a computer assistant search. I will show you on the next slide how he wrote it up and the crazy thing was, so this was the, so 21st, 20th December, 22 or 23 and it took almost 22 hours and then this, so he wrote down this thing. He said here is a computer generated proof where he got to 0.816, starting literally from saying the norm of the length of alpha bar is less than one, therefore the length of this conjugate is less than one. So if you try to write down each step in a proof where you don't jump any steps, it takes you 126 lines to get to the point where you get to 0.816. And of course, I can't imagine doing it manually, maybe somebody can in this audience, I certainly can. He got a computer to generate this and he found, the computer found the path which you can, so this is what computers can actually do when programmed correctly. And so the previous, so this line says that if you take the 17th power of a commutator, these are 68 ST characters and the bars on either side. Then that is less than 13.8596 divided by 17 because of this to the power k thing and you get to 0.81 whatever, 0.814. So he did this and somebody else, Pace Nielsen said this was beautiful, my intuition before something something something. And then he literally said the first 43 lines established something, the next 30 lines of the code he went through it established something, the next 40 lines established something, the last seven lines established. See, actually took it apart and decoded the computer generated proof and found out what it was doing. And then that was the next idea that came. And that was isolated into the following result. So that's the last trick I'm gonna tell you. And I'm almost at the end, yeah, two slides after this. So this is called the internal repetition trick for lack of a better name. So suppose x, y, z, so suppose I have four strings. Just in some four strings. But x is conjugate, conjugate remember is g and g inverse. X is conjugate to something like w times y and x also conjugate to z times w inverse. Yes, so the same w shows up in both of the strings. Then the length of x is at most half the length of y and z. And the remarkable thing here is that it doesn't depend on w. And the proof is strange but I found it very nice to show you. So what you do is you write x, you write two n powers of x. Earlier at some point for conjugation, we wrote n powers of x. Now you write two n powers of x. The first n of them you write as this one. So suppose x is s, w, y, s inverse. Then what is the nth power? It is s, w, y to the n, s inverse. That's what conjugates to do. And the same way if it's conjugate to this one, it is t times this times t inverse. What is the nth power? It is t, z, w inverse to the n times t inverse. So the length of the two nth power is the length of this string. And now don't look at inequality. Look at the picture. This string is written here. s, w, y, w, y, w, y, w, y, s bar. s bar is a single t, z, w, r, z, w, r, z, w, r, t bar. Just write these things down. And now start taking one element at a time out by the subaritivity. So my subaritivity is the length of s plus the rest plus the length of t bar. So I get a s and a t bar come out. Now what remains, I don't know if you can see. If I remove the s and the t bar, I get a conjugate because there's a w here and a w bar here, which is the first one. I remove those because conjugates I can remove. Then what I get is a y and a z. I take those out. So I get s, t bar, y, z, individual term. So s, s, t bar, one, y and one. Then again I get w and a w bar. I remove those. I again get a y and a z and keep going. So ultimately I get n of the y's. All of the n y's I recover. All of the n z's from here I recover. None of the w and w bar, they all cancel, conjugates. And in the between I have s bar and 2. So I get s, s bar, t, t bar, one from here, one from here. And that's it. Divide by what now? 2 n. You get length of x less equals n over 2 n is half. This doesn't depend on n. So by that it goes. And I'll end then with a sample. I still have 10 minutes, maybe 5 minutes, I don't know. Then I'll end with a source. Here is the sample application. Here are some random elements. So x actually is not too random. x is, as I said, the computer was programmed to find, alpha, beta square times alpha. That means alpha, beta, alpha bar, beta bar, alpha, beta, alpha bar, beta bar, alpha. And then choose y, w and z in some clever fashion. Don't try to do this right now. But you can then check that x is constant. So we found somehow cleverly this element and these three. And by that lemma, the length of x is at most, what is x? x was this number. This is a string of nine letters, which is just keep writing them and stop at the ninth point, 1, 2, 3, 2, 3. This. This is my x. The length of x is less than length of y plus length of z. But y is a conjugate of two nested conjugates. That's just 2. z similarly is 2. I get 2. I just did something. I haven't done anything yet. What I'm going to show is the previous bound that was there was 0.816. Using that lemma, I'm going to beat that bound on this board by 8 over 11, which is 0.72 bar. Just using this pattern. Why? 8 over 11, so the trick obviously is to multiply 11 of these copies. If you write down 44 elements, don't write it down. Look at it here. And separate it into four copies, the 11 of them. So now what is each of them? So if you look at the first 11 letters, you get alpha, alpha bar here. You can check. And what is in between is nine successive letters of that kind. But nine successive letters I know how to do. That's two. Whether I take alpha, beta, square, alpha, or beta, alpha bar, square, beta, or alpha bar, beta bar, square, alpha bar, the game is the same. So each of these four is a conjugate of nine left string, which is exactly of the kind you want.  I get 8. And so just from that one simple internal repetition trick that I called it, we beat the existing bound on one board. I have not used any of the previous cases. I just used conjugate invariance and the repetition trick. But again, the problem is still, this works very good. It beats the best bound. But still, it's a finite improvement. I need to get down to 0. Getting down to a finite number at each stage doesn't quite help me. So how do I get down to 0? And finally, I'll conclude. It turns out that to show this thing, that trick and the conjugate invariance are enough. It turns out that it's a very, finally, the last bit is a bit of combinatorics involving, again, binomial coefficients and so on. If you want, you can compactly express it. You can explicitly write it down and take calculations. Or you can compactly express it as the previous talk did, don't worry. You can actually use binomial variables or Bernoulli variables to write it much more sharply, much more in small space. And the key idea there is that the sum of n such Bernoulli variables has average spread, not of the order of magnitude of n, but of order of magnitude square root n. The variances of order n, the standard deviations of order square root n, which is what Amser also pointed out. And that is supposed to bound. So you basically would get that n times length of alphabeta, the whole combinator of n times, or four n many letters in the string, is at most something like square root n times the constant. One can check that. That's where you need this bound of the variance of binomial. Now divide by n, take n to infinity. You get the length of this thing. It's less than 1 over n square root of n here. And that's what you need. The order of magnitude is smaller than n. You get it. You get to 0. And that's the proof. That's roughly the proof. This was finally observed by Tau at 3, 5, and in his afternoon, shall we say.  And then that was the proof in roughly 5 days. So to conclude, then what started out is a search for more and more clever tricks and games to reduce this bound on C. Ended up as a 21st century collaboration at breakneck speed. You can see the. I came back from the US, and I was severely jet lagged for the next few days. And I had a Google alert for Gmail alert for every time there's a blog. And I basically couldn't sleep those 5 days because that kept happening very frequently. Lots of people contributing. And the computer that crucially contributed. That's the word. So I'll end with two points. There are more details if you want the full proof with what is a norm and a group, and what are the motivations, and so on. That will be what I talk about at IIT, Bombay, in the colloquium this afternoon. But the second is actually what, again, I will end with something that Professor Agunathan was saying as well, is this is what I do. This is exactly what I do for a living now. And I came to the Olympiad Camp 20-something years ago, not 25, less than that. And I knew I wanted to do math all my life. And even now, you can ask as basic questions as you can write down without any mathematical background. And the sharpest minds in the world are still interested in these questions. And they actually turn out to be. So this actually has some consequences in geometry within what we call geometric group theory and quasi-norms and so on. I won't obviously get there here. But people still love to solve questions with just, can we find cleverer tricks that work? And that turns out to be of interest to people. And I went there to UCLA because I could use a grant to go there. And it turned out to be a very fruitful collaboration. And I was going there for a previous collaboration with Tao. But this came up. And I still, basically, you still get to talk to some of the sharpest minds, work on some of the simplest to state problems, try to apply some of the cleverest tricks, come across some of the most beautiful mathematics. And I really hope, again, as you did, that some of you or maybe many of you will at least pursue and keep up this interest in mathematics. Hopefully, some of you would actually pursue mathematics as a career or maybe computer science. Theoretical computer science is all math, certainly very, very interesting math. When I was in your side, the person who gave a talk at the end, one of them was Suha Chajitpur, who has now recently won the Nevenlina Prize, I believe. He was two years my senior. He was IIT number one. And he went into theoretical computer science and is a professor, not in industry. But again, there are, of course, lots of math jobs for people in industry also. Wall Street, for sure. And data analytics jobs in the Bay Area on the other side of the west coast of the USA. And there are lots of good opportunities, now more than ever, for people doing mathematics. And I sincerely hope that some of you or many of you will at least keep the interest up. You like solving problems. Yes, what problems would like you? Thank you. Thank you, Apurva, for a very exciting talk. Any question? How old are you? This question is, as I said, I came across this question from some probability research about two years ago. But until December, it was not solved. And then in five days, it was solved. This was your question? Yeah. And so that's when he said on the blog, also, he has a question asked to me by me. Asked to him by me. And this lemma is due to whom? That lemma you stated. The internal repetition trick. Right, right. That came up. So after Siddharth put up his proof on the computer and paste Nilsen, decorated it, then I think that's what Tau's bread and butter is. He takes complicated things and makes them as simple as possible. He cleans it up. I think eventually he wrote the lemma down. Terita. That's his specialty. He takes things that look complicated. And in fact, apparently, I think that's what the principal citations for him sort of said, that somehow once he's written the proof, you think, oh, I didn't think of this before. That's his job. I think the lemma, finally, exactly this form was his. But it was isolated by paste Nilsen in some sense. After Siddharth got the idea. Any other question? If there's no more questions, let's thank the speaker again. I would request Professor Praneshacha to present the bouquet to Apurva. We'll take a photograph. Thank you. This one interesting thing came up during the camp via some app message sent by Uday Prasapati to Professor Grover. There was one, Professor A.R. Rao, who was our colleague. He was born in 1909, let me write with you. He passed away in 2011, maybe 12, 1908, sorry. You didn't erase that picture? It was 102 plus. And there was an Iber of Professor A.R. Rao, who was inspired by A.R. Rao, and he was the director of the film, 102, not out. Have you heard of the movie? 102, not out. It had several major actors. The interesting thing we had learned from Professor Grover. Thank you. OK, sorry. I said he was our colleague at the camp. Yeah, he came to the camp when he was 84 and was here until 89. He stopped coming because the lift was not working here at that time. Now the lift is working. Thank you. OK, so that's all that we had for the function and the two talks. Lunch outside for all of us. Please join us for lunch.


Fantizia once lived in a large mansion, located in Charlotte, North Carolina, which she ultimately let go of in 2013. This 6,600 square foot pondfront home was set in the Glenmore Lakes development in the Piper Glen community, and this was the same mansion featured on MTV Cribs back in the day. Remember that show? This neighborhood was an exclusive development with custom estate homes, beautiful lake, and golf course views. Walking in, there was an epic double height entryway with winding stairs as well as five beds and five baths throughout. Property records show Burino acquired the home in 2007 for $1.3 million and attempted to sell it for $800K.


Hi and welcome to the session. Let us discuss the following question. Question says evaluate the following integrals. Given integral is dx upon x square plus 2x plus 5. Here lower limit of the integral is minus 1 and upper limit is 1. Let us now start with the solution. Now we have to find the integral from minus 1 to 1 dx upon x square plus 2x plus 5. First of all let us consider function x square plus 2x plus 5. Clearly we can see this is the denominator of the given integrant. Now this can be written as x square plus 2x plus 1 plus 4. Now these three terms form whole square of x plus 1 and 4 can be written as 2 square. We know a square plus 2ab plus b square is equal to a plus b whole square. So we can write x square plus 2x plus 1 square as whole square of x plus 1 and 4 can be written as square of 2.  Now this integral is equal to integral dx upon whole square of x plus 1 plus square of 2. Now we will solve this integral by substitution method. So we will put x plus 1 is equal to t. Now differentiating both the sides with respect to x we get 1 is equal to dt upon dx. This further implies dx is equal to dt.  Now this integral is equal to 1 upon 2 tan inverse t upon 2. Here we have used the formula integral of dx upon x square plus a square is equal to 1 upon a tan inverse x upon a plus c. Now in this integral x has been replaced by t and value of a is equal to 2.   Now this is further equal to 1 upon 2 tan inverse 2 upon 2 minus 1 upon 2 tan inverse 0 upon 2. This is the value of this function at upper limit 1 and this is the value of this function at lower limit minus 1. Now 2 and 2 will get cancelled and we get 1 upon 2 tan inverse of 1 minus 1 upon 2 we know 0 upon 2 is 0 so here we can write tan inverse 0. Now tan inverse 1 is equal to pi upon 4 so we can write pi upon 4 for tan inverse 1 and tan inverse 0 is equal to 0 only so here we can write 1 upon 2 multiplied by 0. Now simplifying further we get pi upon 8 minus 0 which is further equal to pi upon 8. So we get definite integral from minus 1 to 1 dx upon x square plus 2x plus 5 is equal to pi upon 8. This is our required answer this completes the session hope you understood the solution take care and have a nice day.


Well thanks for being here today. This is not going to be a big technical talk. In fact this is the only technical thing that's going to happen is I'm going to advance the slides. But I'm going to share some stories from a personal life. I've been doing this sharing my story about mental health and that's what this is about really. This is my journey with health and happiness and that I call the the iceberg. And so I've been sharing this for about two years the story. It's it's it's highly emotional so it's not just because I'm nervous I'm nervous but it's also highly emotionally charged. I'm gonna share some things that I don't really tell people when I meet for the first time. A lot of you guys I don't know. There's a lot of other people that know me. And so anyway so I think back to third grade and Lone Grove Oklahoma. So I'm sitting in the classroom and my mom and my stepdad have just gotten a divorce. We left we were living in Arkansas came back to Lone Grove where I'm from where I held from. And I'm sitting in the classroom and I just get this overwhelming sadness and lonely feeling and that's where I met it the darkness. So when I was thinking about this talk sharing these things with you look back and in that classroom is where this all started. And I think if we're really honest with ourselves this is our life and so I call it the iceberg. So my wife gave me this idea we were talking about these things we share openly about mental health subjects and she goes it's kind of like an iceberg.  So here's what normally here's what we share. The top of the iceberg is that the Lego movie theme song everything is awesome. You know like we see you guys today and you're like how you doing Cori and I'm like everything is awesome. You know it's sunshine and success. This is in the top of the iceberg are the things that are going awesome. It's the PR version of your life right. So that's what when you when we talk and we do chitchat. How are your kids Cori? How is it? They're awesome. They're great. Sunshine and success. How's how's work? Have these conversations yesterday at an event we we went to across the street and you know just house business. Hey it's great man. You know it's that it's that top level that you want everybody to see and I have a background in public relations journalism things like that so I'm really good and I've been practicing since I was a young man to a young boy to say to make my iceberg the top of my iceberg look really really good okay but in reality if we're honest with ourselves that's what's going on that's the complete picture okay so I realized this as I struggled with the stuff that happened underneath the service I'm gonna share those stories how I came to five years ago started on this journey how I came to be able to actually share stuff very intensely personal like I've been through a divorce I've been through depression because I was really good at showing you the top and the bottom was the stuff that was sinking me down and I think the challenge today for you from my personal experiences is we got to start talking about the stuff on the bottom there's a lot of people today you might have got here something's going on in your personal life in your family relationship you might be struggling with health your business might be sinking you might be thinking how am I gonna pay my bills I got the word camp us but how am I gonna pay my bills tomorrow and this is the iceberg this is the authentic human experience I believe is that the top and the bottom we got to talk about the top we should share the top with each other all of our celebrations all of our five percent happiness stuff we should say that talking to Brad Williams I want to ask him about his son Lucas that's the top stuff the other thing is as an aside I say are you sleeping are you still have your apparent cuz sometimes I don't always rejoice in having two little kids that wake me up in the middle of night and have fevers that's being honest right okay so the bottom stuff here's this list of things just trying to think this is for my own life but I bet you there's things here that resonate with you today anger frustration every time I've shared every single time I've shared and I did not for a long time share publicly many of our long-term customers did not know I went through a divorce and and remarried they think my wife today was the wife they might have met eight years ago because I'm really good at sharing just the top what you what I want you to see is right there but you look at that list it gives me heartburn thinking about a little bit insecurity every single day for me waking up no matter what I've done no matter what I've achieved no matter how many passing the back I get I wake up that criticism hate criticism felt failure fear you only hear the good stuff in the bio you only heard the good stuff right that our friend just shared about me you didn't hear the bad stuff right now I'm telling you like I don't need to walk out but money problems burn out that's the bottom of the iceberg and that's probably just a glimpse of the iceberg now at the bottom of the iceberg I want to point out is depression that's the scary stuff that's the stuff that I realized having dealt with depression having learned a little bit more about depression having talked to people who have struggled with it their entire lives that's the scary stuff I don't mess around with is when you get to the very bottom of the iceberg at the depths is where you find depression and even worse things and I don't mess around with that by the way so you probably think so Cory why are you sharing all this stuff today and it's because hey I want you to know something so you read my bio hear these things and you go maybe Cory's got his stuff figured out the reality is I don't okay so I share that because I've had some level of success success my definition definition of success maybe not yours but I've had some level of success that's all you mostly see because that's all I want you to see that's all people want to talk about but I share these things because about five years ago I got into a business group in Oklahoma City guys that had been there and done that built big companies amazing men some of the best friends of my life and we've met every month for three hours for the last five years sharing the depth this is an iceberg group we go down to the depths and every single time we have met for the last five years I repeat to myself over and over and over I'm not alone and that's a pretty good warm and fuzzy feeling to go man these guys who I think have everything put together perfectly have been there and done it are just like me they have the same problems with different names attached to it so the message today I want to tell you if I don't pour water on the laptop is I didn't tell you this would be a polished speech by the way but you're not alone so here's my story so I shared this two years ago and I said this is the first time publicly people have heard some of these things but my iceberg story I want to share with you today is from about five years ago where all this kind of started so five years ago top the iceberg look like this my lifelong dream was to publish a commercial book and in that year my dear friend Lisa said Wilson said would you like to co-author a WordPress for dummies book and I go I'm a dummy I love dummy books if I wanted to learn about investing I go buy investing for dummies and she asked me along with about four or five other people to co-author but so I got to check that off like oh my gosh I'm gonna go to a bookstore and see my name on an actual book and for dummies but hit a million in sales I don't tell you that to brag at all I tell you that it was a video game number for me oh my god we I got the statement at the end of the year whatever and I go we did it we did it we hit the high score you know I don't take all that money in my pocket but it just felt like it's just a video game it's just like you know I found this geeky way to find you know that my kind of game as business and we hit that and it was fun to see that on the statement I go we might have just like two dollars over but we did it and then back up buddy so that's our hit product that we've done and that launched that year that changed our business my life lots of people's lives have been helped by it huge thing that came out that year the second is we did some geeky things as a team we went on a RV road trip so we we got eight sweaty geeks and bless her heart Lisa Saban Wilson in an RV we drove from she flew to Oklahoma City and then we drove from Oklahoma City to Boulder Colorado for word camp Boulder and that's in an RV and bless her heart she still talks to me and likes me and our team but it was really cool we had a lot of fun and driving an RV in Denver is kind of interesting because every little inch that you move the RV and maybe if she was standing in the back you made her a little sick when he did that but so don't get that joke okay the last is started speaking at word camps and it was fun stuff okay that's the stuff you would have saw if you were in my life around my life watching in 2010 2011 that's what I would have catered and shared with you right here's what was really happened in 2010 so wife of seven years and father divorce the week after we came back from word camp Denver my team didn't know most of my what I would say my dearest friends didn't know for six months before that happened they didn't know what was going on in my life no one that truly cared about me knew what I was going through because I put myself into self-seclusion and pride ego whatever you want to call it embarrassment shame I didn't tell anybody what was really going on in my life and so when people go all of a sudden you're falling for divorce what's going on right okay my team didn't know for a while and then finally they did know and they got drugged along by it I I slept in our office we have IKEA furniture you know I was like I got no place to go because I'm too prideful to ask anybody to let me have their couch but we got couches at the office and that thing creaks and man I couldn't sleep right about that night it was it was interesting so I slept a couple of nights in the office here's this person that's run a business and had some success professionally I was on the height of my life professionally and I was at the lowest of my life personally absolutely miserable I love my job at I themes because I've had a different job for the last since I was 16 every two years I've had a different job until I got to I themes and I've been doing it for nine years and they haven't fired me so far but best job I'd ever had in that year I said I'd rather go anywhere else but here and it was the first time I didn't want to go to work and then first time in my life someone highly trained licensed person said you are you were suffering from he called it low-grade depression and offered to put me on on medicine and that was a that was a cold shower so there's my iceberg so yesterday one of the business colleagues and friends asked me score you know on your blog you write a lot about up and downs and stuff like that so how's everything going now and I was like up and down up and down like the iceberg doesn't change I might not be dealing with the same things I was in 2010 and 2011 my iceberg today that if we go back to that screen that talks about insecurity fear anger all that stuff it's still there I've added a new not nuance to my iceberg and that's my two lovely children they provide some of the highest highs and then I've never felt more like a failure than as a parent the parents just laughed because you know how it is never in my life have I thought I'm gonna mess these kids up forever there I'm gonna invest in money and so they just so they can get a lot of counseling forget school this resonating thank you for that so that's my iceberg so I started thinking what what held me back then and by the way what still holds me back there's still stuff in my eyes down below I don't share with people maybe one person maybe two person maybe the professionally licensed counselor that I pay that doesn't see me at Thanksgiving and I don't have to say could you pass the cranberries he may know my wife may know some of those things but a lot of people don't know there's still things down my iceberg I'm still working on man I am under construction forever I started thinking I was like okay it's ego well healthy ego is good you need to have healthy ego you need to have a sense of self-worth and respect for yourself right it goes good bag it you go when it gets into pride pride's good though right I should be proud of my team I should be proud of my children when they do things right but there's that negative side when you go when it becomes a barrier to seeking help and that's what back in 2010 I started thinking about I go it's self-defense so I somebody asked me the question why'd you do that I go you know it's really it's self-defense I I share my story part of I'm a blatantly honest with you I share my story because you won't have to dig it up to find it if I tell you I'm on the offense I don't I don't tell you things because I'm defensive about it I don't want you to know some things it makes me sweat makes me uncomfortable I've got shame pride and embarrassment guilt whatever those things these are the things that hold us back from actually living the true human experience at the max level the things that get in a way is my own self-defense mechanism that I put in front of me it was it was four months of the worst time in my life and no one knew until I go I told somebody the day when I was recounting the story I said I had to go back to being a kid and I just wanted to tell mom and dad I'm hurting and them to say I love you but it took four months of being an isolation crab because I was so prideful I didn't want people to know that this marriage was going to to the brink it was evaporating I didn't want to feel any kind of shame or any kind of the guilt I was feeling I didn't want I didn't want people to bask in that guilt that was the self-protective thing I don't want people and I don't want to feel embarrassed because other people see it and see me naked and just and just raw and everything and it's a self-defense mechanism I think there's a part of that that needs to stay there for me self-defense there's other part that saying you are not doing good and healthy things by locking this stuff up in the key met my one of my dear friends and my CEO or CEO he we're talking he recounts this quote I'm probably gonna butcher it he goes my coping I how do I deal with stuff is through a lot of unhealthy coping mechanisms I butchered that quote but anyway it's like we we find a lot of very bad unhealthy ways to cope and we lock ourselves up and we suffer in silence part of the reason why I share this message with you is because I know I suffered in silence and that many other do there's a stigma about mental health and getting help and counseling sharing the fact that I deal with depression I am menopressive I am whatever the DSM 5 or whatever it is manual you go down the list and and it's time that we embrace the human experience the iceberg and allow people to truly live life free part of that is loosening up the stuff that happens underneath the iceberg so I'm gonna share a couple things with you real quick might sound some random but as I've reflected on what has helped me and what I've cling to knowing I'll continue my my my DNA my habit my genetics tell me I know I'm gonna I'm gonna hide stuff for the rest of my life I'm not gonna share the iceberg stuff that's gonna be my first default thing is to know how to bury it show you the good face give my pocket swear all good and tight and give you a good smile that's my first but these things I'm gonna share with you are the things that I have to cling to I don't do them perfectly but I go back and go this are the things that saved my life continue to save my life the first is what I call my life support team now there's this codependency side you got to be careful of but the life support team was the people in my life when I shared what was going on in my marriage in my life embraced me instead of pushing me back and away it's the people in your life that rush in when others rush out when they see a big fire they're not standing out going look at that that's funny I love to see his misery it's the people that go I will pick up a bucket and douse water on this flame I will run into a burning building to help somebody those are the people that matter so before I tell you who they are these this this is their job description for me this is what they genuinely offer is oh they're open genuine loving their wizzy wig hey it's not gonna take thing for you they're what you see is what you get they're not trying to do a facade I'm an entrepreneur in Oklahoma City about six months ago one of the most talented designers in Oklahoma and one I've ever met probably in my life and I go he moved back from another state and I said how's things going he goes you know what this guy has known me for 15 minutes he goes I went without a paycheck a couple times this year and I just step back and I go dude thank you thank you for being wizzy wig thank for being the full spectrum of iceberg you could have said dude better than ever just score this big client over here no I went without a paycheck right so it's wizzy wig the question I asked myself is everything got turned upside down who would I need my business goes down the drain if something terrific and tragic happens in my life who am I gonna need to come in rush in embrace me and say it's gonna be okay you're gonna put one foot in front of the other and we're gonna see some light I have a dear friend who's going through this right now my experience share with him was left foot right foot left foot right foot baby steps do you remember that Bill Murray story baby steps to the bus you gotta Google that on here YouTube that because you left but a step in front of the other I said you're gonna have to take it in mourn wallow in it grieve in it be sick by it and then you're gonna have to vomit it out and take the next step the next day so those kind of people so if you're married you're dating significant other or spouse that's my first partner Lindsey everybody that knows me or my story has been saying where's Lindsey and I go you guys love her more than me and that's okay cuz I like her too most times my first lady I'm her first laddie but my first lady is the one that knows me and knows my BS and says are you okay maybe you should call Kyle that's that's our counselor that we we share knows there's little it stings cuz it's prideful and all that kind of stuff but it's it's true my first lady she's my first partner in life she knows me she loves me cares about me and me to her and she's my first support team my relationship with my wife is my number one priority in this world but besides my own health and happiness she's my number one relationship my second and if you're in in business I realized the importance of the sidekick I talked to you about Matt Daner he's my sidekick he tells me every day just by showing up I'm not alone we're gonna get through some stuff together this week this past couple weeks I had to let somebody go a family member that was a part of our team and that kind of sucks and he was standing next to me the whole time gotta have a sidekick iceberg friendships that's the whizzy wig stuff that's the let's get past the surface level stuff how you doing oh it's great oh yeah five five percent quarter over quarter all that stuff iceberg friendships the one that says I'm not gonna just gonna give you this I'm gonna give you the full picture we're gonna walk together that group in Oklahoma City some of the dearest friends in my life we are iceberg friends they call me I call them I drop it I go help them iceberg friendships nothing held back here it is full gamut also many of the dearest friendships I've held in my life have been in this thing we called the word press community they're sitting here most of them today those friendships that don't just give me top stuff they give me everything they love me and they care for me genuinely and I love and care for them and I want to give my time and and and not my treasure they don't need my money but my time and my love and affection to them sharing life together we got this hashtag family by choice next one is my counselor so I put on my calendar I realized I said I think the metric for me is every quarter I need to put on my calendar alert to say call call not even if you're dealing with some icebergs well the surf of stuff but you got a call call and you got to work on your life we have physical checkups in this in this country in the world we have there's I hit 40 last year and I didn't know this but now I'm gonna have to have all these new checks and everything and I got glasses for the first time I'm really cranky about it but we we we have physical checkups we go to the doctor and get blood pressure and all this stuff but we don't check up with here and here heart and soul mind what goes on underneath the surface so four times a year his name is Kyle and I we talked to them in between as needed yeah we're WordPress people right so we know how to probably push publish a little bit or open a WordPress admin but journaling is the private version I've been doing that for the last two years and every time I'm dealing with something real deep first I'll do is go to the journal and just vomit it out get it out then I can see it and have perspective on it and then I can go share it with one of my iceberg friendships one of my support team people journaling has been one of the most amazing way to get that's what's going on inside of you outside of you and just put it all on paper just get that poison out of you for a second look at it and go no no no no and now I understand it I'm gonna go share it with somebody else's journey there's three books that I would recommend to every single person on the face of earth access paper pixel six pillars of self-esteem do you remember that show on this on Saturday night live Stuart Smalley he looked in the mirror yes I'm good enough I'm smart enough gosh darn it people like me that's this book in the back no it's an incredible book but there's a section one of the best parts of it is this oh man there's probably 10 pages of just kind of affirmations what's so funny is you read it and you go that's this really trivial it feels like that Saturday night Stuart Smalley thing but what is so incredible is it's one of the first lines is I am worthy when the first time I I read that I said it in my mind I go I am worthy and it was so hard to read it let alone believe it the affirmations in the back of that book are worth the entire book I go back to them when I need reassurance to say I'm a value in this world I won't let others allow to project their value of me on me but starts from the middle that back section is worth the book feeling good is the next one if you battle with the depression I'm not it I'm not a trained lessons counselor at all but I will tell you this book has made a lot of difference for me I am not here recommending suggesting I'm not less I'm not a precision I'm simply a broken person sharing his experiences but feeling good is incredible there's a whole section about cognitive bias and you know when you we have a bad sales day or I get you know I'm looking at my email going servers down because we're not getting payment stuff and everything I'm like oh my god the sky's falling the sky's falling sky's falling and that's part of that cognitive bias and then I get to work and that's like Corey it's okay we're gonna make sales today you know the last is boundaries by Henry Cloud that was something I had to implement in my life here's your homework here's your homework because I was vulnerable with you today I'm gonna ask you to do something I want you to take pixel or paper I want you to write three things you're grateful for part of the top of the iceberg stuff is that we brag shared all the good stuff but we neglect something and that's gratefulness so I want you to say what are three things pixel or paper three things you're grateful for then I want you to be real about it and I want you to say what is happening underneath that surface that is affecting my life the people I love around me it could it may not have to be depression or cancer it could be I've got to make a change in a relationship I'm worried about money whatever that thing is your thing underneath the iceberg be honest and put that somewhere right and last is I want you to go now trusted people people that are iceberg people that are genuine they're whizzy wig they love me just for me they know some of my baggage anyway and they don't judge me by who are those people that are gonna rush in in your life now and maybe today you came here and need to make a phone call you need to step out of session get somewhere private and say I'm dealing with something underneath the surface and I need to talk and I need your help I would be willing to bet there's one person that needs to do that today last is brothers and sisters you're not alone it's human experience I hope I've shared that with you at least from my life opened it up I'm gonna go crash get into a ball and try to recharge after this but you're not alone thanks so we we got Q&A time if you want Q&A or if you just want to share something you don't have to ask a question they can come up to the microphone right here regardless if you guys don't come up I'm gonna be here till through Sunday probably a little bit worn out but I love hearing your stories every time I share this there's always been somebody that private message me and I go that is my mandate to continue sharing the story so sorry I don't have a question hey Corey hey ginger I just wanted to thank you I just wanted to thank you for your bravery and sharing your story with all of us here today and since you've been starting you know sharing online with us for the last over the last year or so it's a very brave thing to do and I'm sure it's touching a lot of people so thank you thanks you're hey Corey thank you hey two things one thing that my family does at Thanksgiving is we go around the table and what we're thankful for and family cannot be one of them it's just something else it's just that's this that's the cop out for everyone so we do something else that way second thing your iceberg support team that you mentioned I'm actually a friend of mine is going through divorce yep and he's actually staying at my house now friend of mine that I've known for 30 years so I'm his support team and when I was going through a breakup a few years ago he was my support team so I think it needs to go both ways absolutely so but thank you for sharing your story as well so in the book Happiness Advantage by Sean Aker he says there's this thing we did at our house need to revive it he said they did the study or test or whatever and I said this group of executives say at the end of the day around the kid the table my wife's very good about saying we need to sit at the table and be a family connected and his thing was to say go around the table and say three things you're thankful for and the uptick it's one of his Sean the happiness advantage is incredible but it would should have been in my book list but they're making me think about your Thanksgiving stuff right and what what we've actually did this year's up my thankful for is I was able to help out my friend yeah you know by him moving into our house and he since lost his job so he's you know at the house all the time I work from home which is kind of good and bad but with him there but but you know for him he's got a place to stay and actually my wife's son moved in as well so we were thankful that we were able to help people out because we have the space and the ability to open up part of why I do things like that is to remember so this talking there got people come up they go man a lot of people need to hear that they go you and I need to hear that I need a reminder this part me reminding I get my pride on everything's going good hey man next thing happened something and I'm gonna be back you know and it's a great reminder I appreciate you sharing thank you one of the things can say in this and I think I should have said this earlier and in my notes and everything but is that in this industry to tie this specifically to WordPress and web stuff is that there's a lot of remote workers now and then your introverts and you might be dealing with them and I feel like I hear a lot of stories of loneliness because there's no social interaction because you're on it behind a TV and you got to force yourself to go out and find this by the way people sitting next to you now today going through stuff don't forget to introduce yourself break into a circle and say hi I'm Corey from Oklahoma City who are you can I you know that kind of stuff but I think this is especially challenging I've heard these stories privately for many people is that because you're behind a computer desk in your home office or wherever there's in this self-seclusion talked a lot of people about that I think we were talking sharing somebody at Publish was saying yesterday how it's a challenge within our industry of loneliness I see it in our community we have a slack channel and all that and I there's a webinar going on and then the slack section they're talking about baseball and this and everything I go I get it it's water cooler it's social time and but still there's this heat thing that you have this warmth when you meet another fellow human being so I'm sorry I'm Randy or not ranting whatever hi my name is Amanda thank you Corey for your talk I thought it's just so moving and I love that you both offered this talk and I love that that it was accepted for this conference it's just not the kind of thing that you really hear at a tech conference and I love that our community is about so much more than that I wanted to just add a note on the gratitude fund is that they've actually done studies now with I think it was heart attack victims or stroke where they if they had the the folks do a gratitude practice every day that they had significantly better health returns for that so it not only just does you feel good it literally helps your body so anyways thank you absolutely it's the hardest thing to do though you know I want to wallow in my misery but when you start going I woke up this morning I'm thankful my heart beats I get to see people that love and all that stuff so good Paul I wanted to join everybody else and saying thank you but I also wanted to remind everybody that this next month is really bad for people who are alone and going through things yeah I know six years ago my wife passed away and that first Christmas was just unbearable being alone so if you know somebody that's alone or if you see somebody that's being alone especially this next month so important to reach out and I still have trouble with that but I'm better at saying what to do in reaching out to people than letting people reach out to me so anyway I just wanted to point out that this next month is really important excellent reminder thank you Paul hey Corey I want to ask you about the difference between confidence in the moment versus confident confidence over time confidence yeah confidence okay so I have been working with WordPress for about 10 years and it wasn't till about like three years ago that I actually made contact with anyone like in the community yeah over the past few years I joined some online groups I spoke at a word camp for the first time this last year I feel a lot better about I feel confident helping people that I have an answer for you technically you know with WordPress problems I can speak to you I can give a talk I can help you online but I have a really hard time putting things on record like I don't have a website myself I managed to somehow get hired without a portfolio because like I don't feel it's worth putting out there permanently yeah and I just wanted to hear if you had any advice for translating that feeling that you know what you're doing in the moment versus that it's worth worth it over time it's called that moment right there if you look around there's I'm gonna estimate to probably 150 people that's the moment you go back to okay then you take a next step and you remember that this day this moment that you stood in front of 150 people your peers your colleagues and you shared those things then you take a next step and you remember that and you rehearsing your mind and you don't rehearse the bad stuff your horse to get to use your first the fact that whatever day this Friday summer 2nd or whatever it is I stood I have a hard time as an introvert and you shared some story with me yesterday I took a moment and that's your moment so it begins today you just started it so you keep building on that I think about confidence the more I've been human and vulnerable the more human I get back not to everybody but for most people and then I know that's a test right but I think those two things is being willing to be vulnerable in that when you're ready in your situation whatever it is and then for me I'm and this is me I'm directly saying this is what I do and the second is moments like this thank you you bet hey David I think it's important that you are vulnerable sometimes and that you share what you're going through with other people do you think there's ever a time that you could do that that it might be negative for you do that like maybe it's not the best time to share yes yeah I mean I have partners in the room that just that have they've actually heard this before so but there was partners strategic partners in the room business colleagues I had to really weigh the cost benefit of saying I'm gonna share some baggage with them in front of that and I think that's a case-by-case personal decision I can't tell you what that is but I think there are things I should not say I just should not say it's not gonna be therapeutic it's not gonna be helpful or constructive right but I told people when I started sharing the talk they would go I'm so sorry you went through that and I go hold on it's been five years since that event I've I've grieved I've done my therapy I'm not doing this for therapy I'm doing this because I feel like I want to take a chance to be vulnerable that my message will help resonate with others so it wasn't for just give you know oh poor guy you went through a lot of stuff you had to sleep on the couch in the office was it that's pretty bad right no I didn't do that I've dealt with that personally right with in counseling with friendships just my heart and soul right but I you got to look use discernment I think and just say am I gonna hinder is it is it positive and helpful to share I'm saying and maybe more in this context you know but I mean with another person and I just kind of go with them if I'm hurting the people that love me most want to know I'm hurting thank you yeah I'm Michael from Vermont I wanted to just ask and kind of comment about like what are the objective circumstances behind this I think one is economic I mean I think like we have this really competitive kind of system we live under and and we're kind of atomized when we're like social animals and yet we're all kind of individuals and it's hyper competitive and I think also why is there such on that level why is there such burnout in the kind of tech community where people kind of live these lives where they you know they they work 40 hours and then they do everything else that kind of because somehow the tech community seems to have kind of demand that and so like what's what's behind that and how do we how do we get past that what's what you're sorry what do you think it is I mean I think that's I am I'm a socialist I'll be straight up I think that part of it is that we have we don't kind of share and support each other as much I think there's very much of I mean you see it right now in this country there's very politicized atmosphere and and frankly everybody goes their own individual house they have to worry about their own bills the family like we don't have as much support I think coming to a conference like this is the reason people do it is because we want that but it's not our day-to-day existence and that's why there's like the imposter factor everybody feels like they don't measure up because everybody's you know showing the top of the iceberg kind of thing and you know everybody kind of wonders like how do we how do we get beyond where we're all tweeting about how great we are and we actually have enjoyable lives yeah you know something you said I wanted to say and I forgot it's like I want to say so many times I told a couple people want to say hey 2016 you're drunk go home it's all I got yeah I think we got like one more minute if someone has a brief question otherwise Corey thank you very much for being vulnerable I think you're helping a lot of people by what you're doing not just the people here but other people that would hear the talk and I know I certainly will be recommending to a lot of my friends to listen to what you said I'm curious you know you talked about how friends have helped you was there any kind of a faith or a spiritual type experience in this as well or was it strictly just the friendships you have with people in the here and now that's a whole talk or with lots of alcohol answer suffice to say yes strong spiritual background with me specifically in the Christian faith a certain in three churches before I started I think longer conversation will have over drinks however your answer to the question is there was a faith based component to this for sure lots of people I mean somebody walked past me I'm gonna go I just want you know I'm praying for you and I go thank you God I like that who wouldn't want good happy thoughts whatever you know even if you don't have my particular God or whatever religious faith but yes there was a faith background for sure thank you


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