It's time now for... Johnny Dollar. Hi, Johnny. Service. Oh, holy smoke, Art. What are you doing up at this hour of the morning? I'm on the job. So am I. Listen, I got a number. Well, did you give it to him? He's some insurance matter, so, yes, I did. Well, why'd he call you? Yeah. Did he give you any details? No. Sure, sir. Promise? Okay, I'd promise. Starting adventures of the man with the action-pact expense account. America's fabulous freelance insurance investigator. Yours truly, Johnny Dollar. Johnny Dollar. Johnny Dollar. Special investigator Johnny Dollar. The State Unity Life Insurance Company, Home Office, Hartford, Connecticut. Following is an account of expenses incurred during my investigation of the ghost to ghost matter. I'd no sooner rolled over in the hope of getting back to sleep when the phone beside my bed started jangling again. Oh, nuts. Johnny Dollar. This is Oscar H. Trimley, Mr. Dollar. Trimley? I represent State Unity Live here in Lake City, New Jersey. Oh, look, are you the man who called our price over an international press service? Yes, yes, I did. To get your phone number out of being such a famous investigator and all that. Yeah, well, you could also have got it from Universal Adjustment Bureau, your insurance directory, the law and distance operator. Oh, dear, I... Well, it depends. What's this all about? Ian McAndrews. Who's Ian McAndrews? Oh, don't you know? He's the man who founded Lake City. So what's happened to him? He'll help. Oh, won't believe what? He'll help me these next couple of days. Goodbye, Mr. Trimley. I can hardly wait to grab a train. But I didn't want to ask Mr. Trimley to know that. Because I had a strong suspicion that if you can catch a ghost off guard, you'll be one up on him. Expense account item one, the promised phone call to Art Price and international press. Are you kidding, Johnny? No, I'm deadly serious, sir. But a ghost in the Little New Church? Yeah, yep, I'll keep in touch. Then I remembered Nancy. Nancy Turner, an old flame, or rather a young old flame. She'd said something one time about taking up investigation of the supernatural. So, expense account item two another time for another call. And as, uh, look, Nancy, did you ever go ahead with your study of psyching investigation? Psyching? Investigate a haunted town. A haunted town? Except that such a thing is impossible. Oh, it is? Sure. But I'd kind of thought that maybe you were still... Well, I guess we'd better forget it. Oh, no, no, wait, I... Nancy, I'll pick you up. Taxi and train for two to New York. Item four, fifty bucks, deposit on a rental car when we got into Grand Central Station. We crossed over into Jersey and hit Route 22 for Summerville and Point West. And every mile of the way, Nancy chatted away like a magpie. She kept quoting some of the stuff she had read on the subject. A lot of authorities who decided that some of the reports on haunted towns and houses and people, things like that, had decided there was something really supernatural about them. And you know, after a while, I began to wonder. Yep, I began to wonder. Sergeant Donald Bellweather, like this to clean your hunting rifle, I've seen you at the Jamesons at seven. I'm ready, honey. All I have to do is wash my hands. I'll finish this tomorrow. Be with you in the jiffy. Donald, come back here. Oh, no, what's the matter, Reba? Are you going to leave this hunting rifle right out here in plain sight? Nobody's going to bother it. The kids are asleep and we'll be gone. I don't think our babysitter will bother anything. She's got her homework to do. Those are famous last words if I ever heard them. Donald, don't you realize that accidental death by firearms ranked second only to deaths from fall, fire, and poison? But the kids are in bed. Well, sure they are. But what if our six-year-old Marshal Villain should wander out to the kitchen for a drink of water?  Yeah, I suppose so. Well, please put it away under lock and key. Yes, dear, yes, right. And for heaven's sakes, take those shells, too. You know, many accidents happen when bullets are thrown into a fire or hit with a hammer. Well, things are dangerous. You ought to know that. I know, honey. I guess I just wasn't thinking. The unloaded gun and ammunition lying around the house is even more dangerous than an itchy-fingered hunter's. Statistics prove that. OK, there. Everything is safe under lock and key.  Oh, Donald, I don't want to sound like a nagging wife, but... My dear, you have every right to reprimand me. Me, a non-commissioned officer. I should know better. But that's where it goes. We forget. That's why there are accidents in the field out hunting and in the home. We all know better. We just forget or get careless with firearms. Well, come on along. Let's get over to the jazz. OK, dear. Now, this won't happen again, believe me. You know, I wouldn't have started to clean that rifle if you hadn't taken so long to get dressed. But you sure did a good job. Oh, you look beautiful. That's my Donald. That's my Donald. Truly Johnny Dollar and the ghost to ghost matter. Nestle in among the soft, roaming North Jersey hills. But it kind of has been Tom. I saw the reason for that in the abandoned mill, the old McAndrews cotton mill at one side of the lake. The same old story, I guess, when a town's main industry closes down, it kind of goes to pot. Nonetheless, it was a charming little place. Population, all maybe four or five hundred. When we finally located Oscar Trimley's insurance office, we found a bit of a gathering there. And Mr. Dollar, that is Miss Turner and Mr. Dollar, this is Charlie Reid. I'm very happy to know you. Tony Gray. We're sort of local businessmen's club, Johnny, you know. Okay, then let's get to the point. Sit down. Thank you. Johnny, I thought over the phone that you were turning us down. Well, I changed my mind. And when I thought of Nancy and her knowledge of the supernatural. Good. That's what we need. Well, I am interested in the subject. We're all a little worried about it. Tony isn't kidding. I had a bunch, but, well, this thing has a scared. That's putting it mildly. If it really is his ghost that's plaguing us. Oh, now you don't seriously believe in ghosts. I'll tell you this. I never did before, but now, well, wait and you'll see. Gee, Johnny. Well, suppose you tell me what's going on. Well, no, no, no, no, Johnny. You'll have to see for yourself. And here. Yes, Charlie, and here. And that means waiting until midnight. Midnight? Sure, Mr. Dollar. Why, Bill? You'll see. Yes, and meantime, you can look around. Say, aren't you quite a fisherman, Johnny? I'd rather fish than eat, but now look at you. Oh, I love to fish, too.  Charlie, you can fix them up with a boat canker. Sure can. Yes, but if I'm going to investigate... Tony, you'll arrange for a place for Miss Turner to stay overnight. Be glad, too. Well, now... And Johnny, you'll stay at my home, all right? Fine, but now at least tell me... Oh, no, I want you to see for yourself at midnight. Now, meantime, good fishing. Yeah, before Mrs. Bixley starts screaming about her high-five. See you later. Yeah, and I'll come back to the office and arrange somewhere for Miss Turner to stay. See you later. See you, Tony. Now, if you folks will come over to the print shop with me, we'll pick up the keys to my boat and some tackle, and you can be on your way.  Oh, no, not a thing until midnight. Oh, and we'll all have dinner together at the hotel. Now, Mr. Trimbley, are you ready, Mr. Turner? Johnny? You fellas are the boys, I guess. Fishing, Nancy? I'll even give you some of the fast-strike hooks I use. Okay, then let's go. There was something slightly screwy about the whole thing, and I don't mean just the talk of a ghost. But when I go fishing and at company expense, well, who's to complain? So Nancy and I spent the rest of the day on the lake. Matter of fact, she caught the big one. By dinner time, we were starved, and the little hotel served us not only excellent cocktails, but a regular banquet, completely champagne. You enjoying it, Miss Turner? Mmm, I love it.  Sure, why not? Charlie, I'll tell your wife. Tony, you stay out of this. Mr. Trimbley, about this ghost is... This is a champagne, you know, comes from the old Leland-Stanford Vineyard. Oh, yes, and it's fine. Finest, I know. But it's time we talk about your ghost. Say, you land any big ones out on the lake? Yeah, Bill. Nancy got a four-pounder. But now listen, would you... There's another thing about this fine California wine. Yes, fellas, go over near the old car.  But then, finally, after a lot more food and wine and cheddar, we drove off in Tony's car. Now, I'm stopping here in the middle of town, Charlie, because it's the best place to be when things start popping. Like what? Hey, when are you fellas going to stop this runaround and start making sense? You'll see, you'll see. I'm all excited. Look, Johnny. Yeah, Charlie? You see the old tower clock? Almost midnight. So what about that? Old McAndrew's passed away at the stroke of midnight, Johnny. Personally, I think that has something to do with this. You still haven't told me with what? Wait. Listen. There goes the tower clock. Midnight. Count them, Johnny. That was four. Five. And Johnny, see how all the lights are flickering along the streets? That happens every night. And no reason for it. Look, that! A million for so... Yeah, I see them. I don't... What under the sun is that? That's the ghost. Wailing. Oh, no. Wait a minute. That scream fills the air. It comes from everywhere. That's a horrible sound. Johnny. Easy, honey. No, listen. Didn't you hear? That clock struck 13. Yes, Johnny. I did. You asked me the devils in Old McAndrew's ghost. That's why it comes out of his house every night. Out of his house? Right, Johnny. And will you see what's there? Only Johnny Dollar in a moment. You know who said, they that can give up essential liberty to obtain a little temporary safety deserve neither liberty nor safety. Those words were written by Benjamin Franklin in the year 1759. Even before the United States of America had come into being, Franklin realized that the path toward freedom was a difficult one. He knew that there would be many people who were easier paths to the difficult one that led toward freedom. These people, he felt, did not deserve to enjoy freedom. Just as there were in Ben Franklin's time, there are people today who would choose what looks like the safe and easy way to protect American freedom. They would be willing to sacrifice some of their liberties in order to be comfortable. But the surrender of a little freedom soon leads to the surrender of all freedom. Freedom can be protected only by constant vigilance and the willingness to fight to protect it if necessary. Remember the words of Benjamin Franklin. They are part of your American heritage. Temporary safety must never be purchased at the price of liberty. And now act three of yours truly, Johnny Dollar and the Ghost to Ghost Matter. Antora, this old house always left wide open. Yeah, Johnny. You sure you want to go in? Sure. Come along, Nancy. I... I'm coming. I think it was more light around... Oh! Well, that's nice. Slammed right in our faces. I knew it. Unlocked, though. Let me have that flashlight, Tony. Here. Here. Hmm. No sign of wires or strings on it. Come on. Oh, okay. Come on, fellas. Now, Johnny. Yeah, Bill? As you can see, there's just one big room downstairs here. Johnny. Easy, Nancy. Easy. Don't you see that? Look, I admit, this is all pretty strange. What a ghost. But what else? We've been over this house with a fine-tooth comb. Hey, listen. You hear that? Somebody. Somebody is walking on the ceiling. I'm looking. Oh. Yeah, that's the kind of stuff you hear. Holder guys. No easy spirits. You can hear them, but I see them. Don't you see, Johnny, it can't be anything normal or natural. Is that the end? No, sir. The old ghost has a regular... See? You hear those shutters banging? There's no wind out there. Why? Why didn't you tell me it'd be like this, Johnny? Let's... Light's moving around somewhere outside. You can see the reflection in the trees. But they're green. Ghost lights. In front of the window. Where old McAndrews used to sit on the town before he died. Give me the flashlight. Here. Strings or wires on this, either. It's the end of it. The same crazy routine every night. It's the ghost of Ian McAndrews. That's all there is to it. Well, from what you fellas have shown me tonight, it kind of looks that way. It is, Johnny. But tomorrow I want to investigate these things in broad daylight. I investigated all right, the old house, the clock tower, everything I could find. And thanks to the help of the boys, nothing. Meantime I noticed that the town, the sleepy little town of Lake City, was being mobbed. People from all over, streets jammed with cars. And as we sat down at lunch in the hotel... Miss, waitress, will you please bring me another cup of... Oh, dear, she didn't hear me. Wait, they all come from... Yeah, business is certainly picked up around here. Well, yes, I'd notice that. Any idea why, Bill? Not the least. No wonder Tony couldn't be with us for lunch. Oh, Johnny's at his print shop, getting out an extra. Hey, excuse me, excuse me. Aren't you Johnny Dollar, the investigator? Yeah, that's right. Man, can I use you? Just let me get a picture. Hold it, hold it, hold it. Okay, Johnny, thanks, thanks a lot. Hey, that photographer's from one of the New York papers. No kidding. Really? Listen, if our price at international press gave out the word about this ghost story... Oh, not Johnny. Then Oscar, Bill, we're going to leave the check with you and pull out. You mean leave town? Yep, going back to Hartford. I've investigated, I've come up with nothing. So there's no point in staying around any longer. Thanks a lot, fellas. And Oscar, I'll send you my expense account. Come on, Nancy. Sure, Nancy and I hit the highway, but for only a few miles. Then, shortly after dark, we drove back. And for a couple of hours, for three or four hours, at night, Nancy and I walk quietly up on the porch of Oscar Trimley's insurance office. You mean you didn't leave a message at my office to be here tonight? I certainly didn't. I found a message from Charlie. Are you kidding? Somebody left a message at my print shop to be here. And I got one at my radio shop. Well, I'll be done. Of course, they couldn't know your handwriting. Wait a minute. There's somebody outside. Good evening, gentlemen. I thought you'd left town. That's right. I wanted you to think so. Well, listen. The old tower clock has started to strike midnight. Oh, yeah, that's... Five? Six? Boys, I suddenly realized that in all my investigation this morning, I was being handicapped by what I thought was help. Handicapped? Yeah, what's that? I had too much help. Two or three of you were with me every second. We wanted to be sure you wouldn't overlook anything. That's right. You wanted to be sure I would overlook a few things. Wait. That's 11, 12? Well, hey. It only struck 12. No ghostly whale tonight, is there? I know. No, fellas, no. Because the ghost is no more. You found the ghost, Johnny?  Alone without your careful guidance, I finally located that sub-seller in the old McAndrew's house. You did? I did. I located electrical stuff that was making the weird sound effects, the rocking chair, the banging shutters, and so on. Very clever. Your handiwork, Bill? Sure. But Johnny... Oh, fellas, it was a wonderful publicity stunt, especially after international press was notified. But you did that? Not only for your radio and electronic shop, Bill, for your real estate business, Tony, your print shop and newspaper, Charlie on the map again which is to say the motive wasn't entirely selfish. No of course it was okay okay man and because of that and then alone I won't give you away. Provided the ghost of Ian McGandro's never walks the streets of Lake City again. I suppose I ought to really hit you over the head with this expense account but after all a cause was a kind of wording one so I'll be honest with it for a change and it was fun to have Nancy Turner along. Expense account total including mileage on the rental car less deposit thirty one fifty yours truly Johnny dollar. In our cast we're Virginia Greg, Forrest Lewis, Joseph Kearns, Russell Thorson, Sam Edwards and Bob Bruce. Be sure to join us next week same time and station for another exciting story of yours truly Johnny dollar. This is Roy Rowan speaking. United States Armed Forces Radio and Television Service.


We have to pick a MVP for all of baseball, based on what you've seen from these three this year. Who you taking?  No, you can't, but you gotta pick one. I'm gonna say Mike Trout. OK. No doubt. The fact that he signed as big of a contract as he did and he's living up to that number, I mean, come on, man. $430 million. Nobody thought that that was even attainable. But this dude right here is making it look easy. And we still, in some way, think he's underpaid. Well, look at Mike Trout. Lee's most dominant player. He's going home early. He's going fishing in October. He'll be watching the Phillies game, because Anaheim is not going to the playoffs. I can't give him the league MVP. I gotta go with Cody Belliger. He's a league's best team. His defense has been phenomenal. And he's got 42 homeruns, just like Trout. And he's continued to dominate this season. But it is most valuable player, not most valuable with team. Most valuable player of the league this year. I'm going to give it to the team that's winning the most Paul game. OK. I'm going with the Dodgers. Well, I tell you what, there has to be Cody Belliger. There should be something to be said for playing multiple positions, right? Cody Belliger being able to play center field, being able to play right field, coming to play first base. He can DH for you. He can do a lot of different things. I'm not taking anything away from Trout. Obviously, center field is a huge position. But there has to be something to be said for guys that are showing that super utility ability, especially putting up MVP numbers, like Belliger. You don't hold anything against anyone, but bottom line is they're not getting to the playoffs. You have to pick. Yeah. And the Dodgers have had a phenomenal league. You can't take it right away from them. And Belliger has been this dominant throughout the season. He started weak one. And he's continuing now. I'm going Cody Belliger. All right, so let's crowd the picture a little more. Six weeks to go in the season. Give me a dark horse wild, a card kind of guy who can get in as an MVP in the National League if it's not Yelich or Belliger. I want to go with Peter Lanza. Yeah, I was going to say, you better go with Polar Bear. I'm going Polar Bear just because. Rookie of the Year at all. Six weeks ago, the men's led that were done. They acquired Strowman in another piece. And what happened? This team is taking off with that pitching staff. And Peter Lanza is at 40 home runs out of rookie. Yes, unbelievable. I mean, that is unbelievable. If he somehow hits, what, 48 home runs, driving 120 home runs. He can do that. Can he get the 50 homers? He's an MVP race. I'm going to go with the Cunha Junior. I just think with his electricity, what he brings to the ballpark, exactly what has happened to him over the last couple of days with the Braves lineup and Snicker, being able to say like, hey, man, we're going to call you out on things you do wrong. I think that's going to jumpstart him, man. And they're going to need him down the stretch to help pull away in that division. Maybe they'll put a little heat on Yelich and Belliger for that MVP in the National League. Don't forget, Chris, we saw Yelich win it or the last six weeks of the season last hook off. Look at Frank from way downtown, baby, you know what I'm saying? It could be decided. That was nice. And the American League wildcard picture also tight with the Indians in the mix. So who's the bigger threat if they get into the playoffs of those wildcard teams, Frank? Oh. Why are you coming in for that one? He's only known this all day tonight. That's a tough one. That's a tough one to think about all day. I know you don't have good pitching. All right, I'm going to go over the mess because of Tito Frank Kona, the added balance of the right-hand hitters. You mean the Indians? Yeah, the Indians. I'm going to go with Tito because of the Indians and his experience in the playoffs. The pitching staff is still pretty solid. The offense is upgraded with, I see, up a week. And then Frame Orias. So I like this game. They're fighting for the purpose. You know, they came together with Cookie, went down with leukemia, that team really came in, rallied together and came together. I like the Indians. We were off in the first half. Right now they're playing great baseball. Yeah, I mean, with as awesome as the Indians are, with a healthy Corey Kluber, I think I would have to go with the Indians. But if not, if we're talking about a five game set, I think I'm going to go with the Mets and this is why. That starting five rotation is pretty nasty. If you think about going in there on a five game set, you got to face DeGrom, you got to face Thor, right? Wheeler, he's feeling good right now. Strowman Mets, you can mix and match those guys in the bullpen. I understand Diaz is having a tough time at the back end of that bullpen. But I'm telling you, no one thought the Mets would be in the position that they are right now. And I think, obviously, Marcus Strowman coming over there ended up, could be, one of the better moves that have happened all season long. And you're right. That's why I said all day. This has been the toughest question of the day because of that starting pitching. I know if I'm managing a team and you got to face DeMess in the first round. You're right. You know, I don't feel that great about it. Tell me about it. Because they're starting pitching. And bullpen, obviously, the Indian second best bullpen in the American League, the Mets bullpen could be an adventure. Exactly, no doubt about that. That's a concern. And Frank Kona has handled bullpens well in his history.


to more mathematical introduction to his work and the topic of today's colloquium. So, Luigi belongs to this great Italian school of analysis. Of what the Georgie I'm told has led this school for many years. And he has been a full professor of mathematical analysis at School of Normale Superiore of PISA. And he has been nominated as a director of the School of Normale Superiore. ICTP has a long-standing relationship with School of Normale Superiore, which culminated in signing an agreement for a number of joint research collaborations in mathematics and physics. Also, Professor Ambrosio has given various lectures on optimal transport and monge ampere equations. We are sure that this fruitful collaboration will continue and will be enhanced in the future while he is the director. So he has received many accolades. I will just list a few. So he has been a visiting scientist at many major institutions like MIT at Max Planck Institute, ETH, Ecole Normale. He received the Riemann Prize of the Riemann International Mathematics School in 2022, the Balzahn Prize for the Achievements in the Theory of Partial Differential Equations in 2019 and the Blaise Pascal Medal of the European Academy of Sciences in 2019. And the Gold Medal for Achievements in Mathematics awarded by the National Academy of Sciences of Italy. And, of course, he has been invited as the plenary speaker at the International Congress of Mathematics, which, as all of you know, is an honor. So he will be talking about a very interesting topic, but maybe I will leave this to Claudio to elaborate upon, but before that I will invite Professor Andrea Romani not to say a few words. Thank you, Atish. It is just to say that it's a great pleasure to welcome Professor Ambrosio, who's a well-known scientist and director. I had the privilege to meet the director most often, so I'm really looking forward to listen to the scientists today. I'm glad to see a significant part of the Trieste Mathematics community and not only the Mathematics one, which has several different strong ties with the PISA one. And so thanks to ISTP for gathering us together here and hosting this event. Just in joining me to welcome Professor Ambrosio, I just always make a point that, I mean, important recognitions come after a good theorem. So in this case, my role is to try at least with few slogans. I mean, Professor Ambrosio, of course, has been the recipients of hugely important recognitions in the mathematical community, but just in three slogans, I mean, why? I mean, so since I was a student almost, I mean, I came across his work, and I mean, he, with his collaborators and his students, he has brought mathematical analysis to an even higher level than was before, and especially in connecting it with the, now I would say almost all branches of mathematics in a much, much deeper way. So the keywords of his work are, I mean, he wrote probably the book actually together with Nicola and Nicola Gidia, Cisa and Savare about gradient flows in metric space. I think if there was probably, maybe I don't know if he would agree, but I mean, a unifying team is to take actually very classical problems and bring them at a level of an extremely refinement of regularity to also to drop from the classical picture to the modern one, regularity issues, which is not just a game per se. So this certainly, I have to mention gradient flows, for sure, which is a topic that actually became huge. So he brought together, as I said, with many collaborators, but I mean, the study of gradient flows in metric spaces, probability measures, and geometric measure theory to an even deeper level, especially study, I mean, for me, there is one piece of work, which is still certainly contains some treasures that geometers will still have to dig out properly, which is the study of currents in Banach spaces. I mean, currents, very classical object in mathematics, but typically you study them on a manifold or a smooth manifold and so on. Ambrosio's work brought this thing to a completely new level, I mean, completely inimaginable before these works. And as I said, I mean, thinking of differential geometry, which is my subject, I'm sure, there is still a lot of understanding to be done on that side. Also, even other classical problems, like the famous isopereometric inequalities, probably one of the most famous mathematical theorems in the history of mathematics. But again, thanks to his work and the work of his collaborators, he has opened up completely new lines of research in terms of, again, asking these questions in Banach spaces, in probability measure spaces, in general metric spaces, and so on. So for all this reason, I think he has been, I mean, I could go on. I mean, there is a general theme or another general theme is the connection with optimal transport and connecting with various PDE type of equations, PDE's equations, which are famous by themselves, but now there is a kind of, again, a unifying theme thanks to his work. So these are, as I said, in 30 seconds, some of the reasons why he is recognized as one of the leading mathematicians in the world. And so I think today I was actually asked to say a few words about today, the truth. I mean, let me confess it openly. I don't know. I'm eager to learn. I suspect it's going to be, and I mean, I gave a quick look to the slides before. So I suspect it's going to be another opening up to another extremely exciting side of the story of connecting analysis with now themes coming from machine learning and also, again, classical problems about putting sparse and random points in a place where essentially it's all about minimizing an energy which is a theme, of course, very dear also to our physics colleagues, but now we will see it appearing in a very different and probably less usual approach. So thank you very much for accepting our invitation and it's all yours. It's all right, okay. So first of all, thank you very much for the invitation. Of course, I am honored to be here for this joint CISA-ICTP colloquium, also on the occasion also the Ramanujan Prize awarded to Professor Foll. And I'm going to mention in my talk maybe I move here some problems in calculus of variation which have a geometric character which are motivated as we will see by also by some, let's say, more practical problems related to the theory, which is, of course, a theory still in the infancy from many points of view on machine learning. In fact, I came to this problem thanks to a conversation with Michael Luncer who is professor at APFL in Los Angles which is a specialist in this topic. So let me start to give you a plan on my talk. We start with some general motivation and then we enter a little bit into the mathematical side of the theory. Of course, I had to introduce some functional spaces which are maybe familiar to some of you where we try to study this problem in an infinite dimensional setting. Then I will mention kind of variational problem which is the kind of problem which is analyzed in the machine learning community. And then I will come really to the main part, to more geometric part of my lecture. I will tell you about extreme exposed points and these analysis that we have been doing of the so-called class of CPWL functions. This is an acronym for continuous piecewise linear functions. Okay, so, well, this comes from two papers. I wrote with different collaborators last year and more recently this year. One of the co-authors is also Camilla Brenna which, by the way, is a former student also at CIS, at Intriest and in CIS. And Shayan Aziz Nejad is a PhD student at PFL. More on the mathematical side, well, they read this paper by Michael Luncer, more on the applied side of the theory. If you want to go really more on the mathematical side, if you are interested in more on the purely mathematical side, there are these papers that I will mention by Bredius and Caryoni, which appeared on Calculus of Variation and PD. And also this paper by Boyer, Shambhal, Castro Duval and Gournet, which appeared on the Sium Journal of Optimization. Again, maybe if you are interested in this topic of using this kind of energy, I will mention the Ascian-Shat and Total Variation as a regularization term. You may have a look also to this paper. Okay, so let me start with a very informal introduction. So what we mean by sparsity? In the classical theory, which is based on Fourier series, sparsity means that you may have problems where, let's say, only very few coefficients in a broad sense are relevant, right? And when this happens, when you do the Fourier expansion and only very few coefficients are relevant, you mean that there is this notion of sparsity. Of course, whenever you have a sparsity, this is very important for compression because this means that you can use only those coefficients to reconstruct the signal. More recently, a kind of analogy with this problem, which is typically of the theory of signals, appeared also in calculus of variation. In particular, in the two papers I mentioned to you, the authors consider this kind of energy, where F, as I will explain later on, is the usual discrepancy term. For instance, in the applications to machine learning, and Fee is a regularization term. And what is important is that they were able to prove by basically by convex analysis tools that you can find the minimizer. So minimizer can be represented with a special representation. Fee, for instance, after normalization that Fee of Ubar, where Ubar is a minimizer, is equal to one. So this is not restrictive up to normalization. So you may think that this is a convex combination, right? And what are the Ui? The Ui are, here I'm repeating this formula, right? What are the Ui? The coefficients in this kind of convex combination are extreme points for the unit ball of this energy. Right, so it comes here, the geometric problem of trying to study the extreme points of the energy Fee. Okay, sorry. What is the connection with machine learning? The connection with machine learning can be explained in these terms. Well, first of all, the typical form of a machine learning problem when you try to train the network is this.  So you specify a class of functions which you want to try to reproduce. And in this class of functions, you will optimize, right? This function depends on a free parameter theta. And then you minimize, with respect to this free parameter, this is a kind of discrepancy terms with respect to the data you have. And lambda, lambda is a kind of tuning parameter between the discrepancy term and the regularization term Fee. So what happens is that, of course, these problems are being well studied in the classical theory. But in the classical theory, typically the dependence of U on theta is linear. So this is the so-called the theory based on kernel methods. And for instance, H could be a kind of Gaussian kernel. But in terms of performance, at least empirically now we know that this approach has been outpowered by the nonlinear version. For instance, the deep neural networks. So essentially, the most simple version of a deep neural network consists on a number of layers, L. And then the parameters in this problem in which you want to minimize corresponds basically to L affine functions. Right, where BL, say W1, WL are matrices and B1, BL are potentially drift vectors. So you are basically describing L affine functions. And you do the composition of these affine functions. Modulo, then of course in this case you will simply get a linear function, but what you're doing really in this case is to compose modulo, the so-called activating function. So there is a nonlinearity here. And the typical form of the nonlinearity using the application is the positive part. So you apply the positive part to all components of this vector. And then you iterate all these operations. Right, and so what can we say? Well, in this class, in this kind of parametric class of functions, sorry, maybe I go a little behind, in this parametric class of functions you have to single out what is the most natural class of functions to deal with. And it is clear that for instance a natural class is the class of continuous piecewise linear functions because this class is invariant under all these operations. Truncation, so multiplication by the activation function, translation, multiplication by a matrix and so on. And actually this is true, it is true in a strong sense because there is a relatively recent paper in 2016 which shows that actually any piecewise linear functions can be generated in this way. Provided you have a sufficiently large number of layers. Of course, the more your piecewise constant, piecewise a fine function is complex, the more layers you need that is intuitive. Also, here comes another analogy. In the classical Fourier theory it is known that sparsity is produced adding an L1 regularization. And so Unser and the collaborator proposed as regularization. Of course, you should take into account the regularization which is compatible with the class of piecewise fine functions and the regularization that they suggested that I will explain in the next slide is the space of functions with bounded dash and the total variation of the second derivative as regularization. What is in this kind of measure theoretic framework, the analogous of sparsity is that the derivative is concentrated. And in fact, if you think to a piecewise fine function the second derivative in the sense of distribution is precisely concentrated on the edges which appear, of course, in the simplest scenario of the one-dimensional case where you have the so-called splines. Of course, the second derivative will be concentrated at a single point. Second derivative, of course, in the sense of distributions. Okay, so now let me enter a little bit into the mathematical part of my talk. So first of all, we need the definition of the sobole space, W1P, which is the space of all functions in LP, which are whose peak power is integrable with the property that the derivative in the sense of distribution is still representable by a function in LP. These are by now classical spaces which are called sobole spaces. In formulas you ask these integration by parts where, of course, u over d psi is not in general the classical derivative but it is the function for which the integration by parts formula works. And if there is such a function, it is unique. And of course it is consistent with the classical, the smooth case. Of course, you can try to iterate this procedure because, like in the theory of distributions, you can differentiate infinitely many times. And so you can iterate this and define the space of functions whose derivatives up to, in the sense of distributions, up to the order k are in LP. But in our case, it is sufficient for us to stop at the second derivatives. But our second derivatives, just look at this example, will not be functions but measures, okay? And so this, we come to the next definition. What are, which are the functions whose distributional derivative is a measure is the space BV, a space very much studied by the Italian school since the work of George Caccioppoli and so on. And so again, you ask an integration by parts formula here where in the right hand side, you don't have a function but you have a measure. And also here, I gave the definition for a vector valued function. So this is true also for the components uj where j, let's say, is between 1 and m. So we have m components and for each of the components, we have the, for each of the partial derivatives of the components we have that measure. Okay. So this is the frame. So let me just say a few words for those who are not experts about what we know about the distributional derivative. Well, in general, we can split this derivative into three parts which are called absolutely continuous jump and counter part. So the absolutely continuous part is the part absolutely continuous with respect to the back measure, so the one for which you have a density. The density is typically denoted by nabla u because indeed one can show that it is the differential of u in a suitable sense. Then there is the jump part. The jump part is concentrated on the discontinuities of u and I think also to the higher dimensional example, let's say that you have, let's say you have at the top of a roof. Then of course in this case the distributional derivative of this function is concentrated on this line and the derivative corresponds is proportional to the difference between the values of the jumps from one side to the other, right? Cove? No, no, for the moment I'm speaking about, sorry, yeah, this is already for the second order. So let me just do the first order version. Thank you. So let's say that you have a function which is jumping along a set. You have a discontinuity. Let's say you have limit on both sides which are denoted by u plus and u minus. You have the normal, which are the objects which appear over there and these objects give you the so-called jump part of the derivative which as you see is a tensor product. So the first factor is a vector in Rm and the second factor is a vector in Rm. And then there is a more mysterious part which we call a contour part of derivative. I say it is mysterious because in some sense it can't be related to the point-wise behavior of the far less to do with a differential in a suitable sense while the jump part you can recognize it because it has to do with the jumps of the function. The contour part is more mysterious. Sorry. It's more mysterious. And why it is called the contour because there is the famous example of a contour vitally function which is also called the Davis staircase. It is the example of a function whose derivative is zero almost everywhere. So in such as this graph is flat almost everywhere but the function is continuous. And so you have neither this part of the derivative because the derivative in the point-wise sense is zero. You don't have this part of the derivative because the function is continuous, no jump but still you have a derivative because you go from zero to one. And so all the derivative in this case is contour part of the derivative. And of course when we are going to consider later on the class of continuous piecewise linear functions only the jump part will be relevant, right? Because in this case only the gradient jumps along hypersurfaces. Nevertheless it is important in some sense to place our problem for geometric reasons as we see in the full setting and also to consider potentially all the other parts of the derivative. This is necessary in some sense to single out the piecewise linear functions as special ones. Okay, now it comes to my space BH, the space which was proposed, the space by the way was already well known in the theory of elasticity in calculus of variation in the theory of elasticity in some sense has been rediscovered in the machine learning community which is the space of functions with bounded addition. So you might consider the space of functions which have a derivative in L1 whose gradient, let's say whose all components of the gradient du over d psi, all these functions are bb. And then there are a few results about integrability. In particular this space has been studied in the context also of elasticity by French researchers. Dementia, for instance, he proved that in dimension 2 this space embeds into the space of continuous function but maybe these are more technical things which I leave only for those who are more interested to the mathematical part. So now let me start, go again instead to the choice of the energy, the regularizing term in the variational problem. So what you can do is typically whenever you have a vector-valued measure, so here, so I'm denoting by d2u, so this is a kind of a symmetric matrix of measures which is the derivative along the j direction of du over dxj. By the symmetry of distribution of derivatives this is symmetric. So this is a vector-valued measure and what it typically you can do is to write this vector-valued measure as h u times a scalar measure which is called the total variation and h u is a symmetric, is a symmetric, let's say with norm equal to 1. So h u gives you the orientation in a suitable sense, in the vector-valued sense of this second derivative and there is a deep theorem proved by Giovanni Alberti which applies also, it is deep because it applies also to the counter part of derivative showing that h u is a rank 1 matrix. So in some sense it is more like, more alike to the jump part than to the absolutely continuous part. And this, of course, will be an important information for us. And so combining rank 1 with symmetry eventually you can prove that for the singular part of the derivative h ij, h is a tensile product of a vector by itself. And now you can decide what is your favorite semi-norm matrices or if you like only on symmetric matrices and define your energy in this way. You weight according to your favorite semi-norm h u and then you integrate. We will see that it was already observed by in the applied community that the most relevant choice of the energy of the norm is the so-called Schatten norm so where you just sum the modulus of the eigenvalues. Of course there is also the more traditional Euclidean norm. And so let me denote in the rest of my talk by phi s and phi e the two energies that you can build putting here either the Schatten norm or the Euclidean norm which will correspond to taking the square root of the sum of squares. Okay, so this is also a small remark that will play a role later on these energies behave well with respect to many operations in particular we will see also by convolution but in particular they behave well also under symmetrization. So if you take a function u and you symmetrize it meaning that you take the mean value of u maybe the most legible formula is this one. You take the mean value of u on the sphere with radius modulus of x then getting essentially a function of one variable a radial function then this energy any of these energies decreases. And then a typical variational problem which is studied in the applied community is this you minimize the discrepancy term as I said tuned by with respect to phi this parameter lambda plus my phi energy and then there come analytic questions which are related to the embedding of these spaces inside continuous functions so one can prove that for n larger than 3 actually this problem is not well posed is not well posed in general on the other hand in the case n equal 1 the problem is well posed if you consider functions just of one variable and indeed in this case what Unser and collaborators did was to compute minimizers explicitly and prove that exactly minimizers are splines are one dimensional course like those I wrote over there then there is the critical case which we studied in the second paper with Conti and Brenna in this case this is really the critical dimension because there are kind of concentration effects of the action one has to analyze them very carefully anyhow maybe this part I think will be more relevant for those of you who work already in calculus variation so we will go about this more quickly actually what we can prove also because the assumption n equal 2 I will say is not particularly realistic for applications to machine learning it's just a mathematical more mathematical question anyhow in dimension 2 one can prove that this problem is a solution but the norm for lambda sufficiently small actually if q equal 1 of course everything depends also depends on the power q that you put here if you put to here q equal 1 actually this is true you minimizer for any value of lambda as I said in this problem everything is related to the analysis concentration effects of the action along minimizing sequences and so one has to study this capacitary problem and in the analysis of this capacitary problem so you try to minimize the energy among all functions which are equal to 1 at the origin and which have compact support and the constant 4 pi emerges really from the analysis of this problem I think this is more not so surprising so let me go instead to the geometric part F are the data F are the data in your problem so you are given sorry it should be u sorry F is u is a type obviously it's u is lambda u of x i minus y i let me check if there was also before the same area ok thank you ok so instead let me go instead to the more geometric part of my talk which is I think the one more interesting so let me remind to you what is the definition of extreme point and the exposed point the definition of extreme point in convex analysis is a purely geometric definition so let's say that you are in a vector space and you have a subset S of your vector space and you say that a point a vector v is extreme point if you can't write it as a convex combination of distinct points in S for instance of course in this case for a circle of course all points are extreme points for a square the corners are the extreme points and so on there is also another interesting notion which is the notion of exposed point exposed point which uses a little bit of topology so if you have a topology you may say that a point is exposed if it is the unique minimizer for some continuous linear function what are the basic thoughts about these two notions well known since let's say I would say since the 1950 first of all in a housed topological vector space a compact convex set can be generated by the extreme points generated means that if you take the convex combinations and you take the closure you recover all the set of course if you are in finite dimensions you don't need to take the closure this is Caratiadori theorem and also there is clear implication between exposed and extreme exposed points which are the unique minimizers are clearly extreme but in general the two concepts do not coincide think for instance to this picture a kind of stadium so for instance this point is a point which is clearly extreme but it is not exposed there is no way to make this point a unique minimizer for some linear function moreover however still there are plenty of exposed points even if in principle they are less than the extreme points they are dense in the class of extreme points and so still you have the crime mill theorem you can write the crime mill theorem using exposed points ok so as I said at the beginning of my lecture the main point is to try to understand what are what are the extreme points of the unit bowl for my energy this is a second order problem so let me for the sake of illustration tell you what we know about the first order problem the first order problem is basically well known let's say it's part of the folklore on this subject so let me say let us look first at this problem we consider all bv functions so just one derivative we take the bv functions modular constants right in this way the term u becomes not only a semi-normal but a norm and you try to understand what are the extreme points of this kind of bowl computed with the total variation well in this case the answer is pretty well known one can prove that u is extremal if and only if well first of all the total variation has to be equal to 1 this is pretty clear you are not in the interior of the bowl right not only but u has also to be a constant multiple of a characteristic function so c times the characteristic functions on set e and moreover the set e is connected not really in the topological sense but in a measure theoretic sense that I'm going to tell you so how the proof works so the proof arises trying to make reasonable the compositions of u well first of all it is clear as I said that the total variation of u has to be equal to 1 so this is a necessary condition then you try to do the most natural splitting of your function you truncate at some level l and you write u2 which is the positive part of u minus l in this way we have a natural decomposition of u then one can compute this constant which are the perimeters of the level from 0 to l and the perimeters of the sub level set from l to infinity and the quarry formula the quarry formula tells you the quarry formula tells you that alpha plus beta is the total variation so it is equal to 1 and so we are basically building in this way a convex combination u equal alpha with 1 plus beta v2 so if u is extremal this means that this should be a trivial convex combination and so it should be that v1 should be equal to v2 but v1 and v2 are parallel to us gl and u minus l plus and so this means that these two functions whatever level l you choose are proportional and since the level is arbitrary it is not difficult to prove out of this information that u is basically only two levels so let's say the empty level and the level e so it is a characteristic function and then it comes a deep result by Herbert Federer in geometry measure theory which says that this is analogous to the theorems that you have in topology let's say any open set can be written as a disjoint union or connected components in this measure theoretic setting one can prove that any set with an perimeter can be written as a contable unit of in the composable components the composable means that you can write e as a disjoint union of sets in such a way that the perimeters add right of course if e were the composable if you were able to write e in this way then again you will be able to provide a non trivial decomposition of u and so e has to be under composable so this problem is pretty well understood for one derivative what I'm going to tell you is what we have been able to prove for the second order problem so again we take a connected open set and we consider all functions which are in BH but they will not be BH star again modulo affine functions because affine functions are not seen by my energy by my second order energy phi and in this way modulo affine functions again phi becomes and we would like to understand what are the extreme points of S as I said that the one dimensional case has been studied is not particularly difficult and one can prove it's not particularly hard that all the extreme points are piecewise affine piecewise linear are the so called splines and what are the natural candidates in higher dimension of course are the piecewise linear functions what do we expect about this problem well we expect this problem to be more rigid compared to the first order problem right as usual because the second derivative is more rigid than the first one for instance it is symmetric just to give you one extra information and also as we will see the choice of the norm will be particularly relevant and so what are the questions that we try to address can we characterize CPWL functions among those which are extreme well this can be done basically if you take a piecewise affine functions you may find the conditions in some sense of the support of the derivative which tell you kind of really a connectedness conditions which tell you that this piecewise affine function is extreme actually this is not yet clear for the more restrictive notion of exposed but the more delicate question which was really raised by Ulzer and collaborators can we say that all the extreme functions like in dimension one are piecewise linear or not of course a positive answer will be very good because we will tell you again in terms of this convex representation that any function can be represented as a convex representation of piecewise linear function actually we found that the answer is no it is not true and for instance if you take this radial function let's say a kind of truncated cone you can rotate maybe this picture along this vertical axis all these functions are extreme the proof is not obvious it has to use all the structures I told you the distributional derivative because in principle you have to consider candidates which are not piecewise linear so you need all the information on the second derivative on any of any BH function and so the answer is negative actually it will be interesting to know we still don't know whether the truncated cone is not only extreme but also exposed that's an interesting question anyhow we have a negative result but we may try to recover a little bit from this negative result trying to prove at least that the CPWL functions are dense in the class of extreme functions in this way we can still recover see again for people 2 otherwise you take the sum of the p powers the power 1 over p they are extreme for all the p while we are going to see actually that the question that we are trying to recover so the question at least you would like to know to have the density for this it is really essential to have p equal 1 and so this motivates in some sense even more the reason why in the applied literature they really are convinced about the use of the shutter norm the one with p equal 1 I mean ok so my next question is well we know that there are functions that are extremely and are not piecewise linear but can we approximate them in an efficient way and this is my next question I still have some time so the question is are these functions dense in energy with respect to f s density in energy of course remember here we are considering really my geometric problem so I don't need just density in LP topology which will be trivial I need the so called density in energy meaning that I want to approximate let's say in L1 but in such a way that the energies are convergent f s of u h converges to f s of u and then up to a normalization we could say if f s of u is equal to 1 so you are on the sphere generating the CPWL function to be on the sphere as well right is an approximation in energy in fact yeah by scaling you can assume these and well this is more for the expert all these energies are lower semi-continuous so what is crucial here is to get the lean-soup energy the lean-soup inequality and actually it is important to notice that this is not the question not let's say no smooth functions or bb functions because you can immediately reduce your question to smooth functions if you are able to do this for a smooth target function u you are able to do this for any function u because by convolution any of these free energy decreases by convolution why because these free energies are convex convolution is a mean value or translates so it decreases under convolution so basically we could rephrase the question is that given u not in bh but in infinity can we find an approximation in such a way that the energy converges and of course this is maybe again more for the experts if we knew that for the Euclidean norm the answer is positive then it will be true for any free energy but actually as I said before the choice of the energy for this question is relevant and the reason maybe I go about this more quickly but maybe the expert can ask to me the slides the reason is that first of all the two energy coincide the Schatten and the Euclidean one let's say on CpWL functions because on these functions you have only the jump part of the energy and then they coincide on the other end they do not coincide let's say when for instance you are just your target function is just a quadratic function so in this case you are the strict inequality and so combining district inequality with an argument by contradiction you can prove that if you have a positive answer for phi then you have a positive answer for all the other ones and so this is not possible so for phi the answer is no, it is not possible to do the approximation ok so actually we got our results in different steps maybe first I tell you what is the most elementary and classical steps so this is not particularly difficult to be proved that you can get the Lin-Sup inequality by a more or less standard approximation but you get the Lin-Sup inequality up to a multiplicative factor which depends on M and so this result of course is not optimal well this not optimal result can be obtained by any reasonable interpolation you do a Lagrange interpolation whatever reasonable interpolation you do using essentially the Poincare inequality in the single simple access of the composition gives you this inequality but of course and maybe this is clear to the experts in gamma convergence in the audience if you want to get the sharp approximation you need in some sense to adapt your frame to the local behavior of the function assume also as I said that your target function is completely smooth but of course the action matrix of you will be diagonal in frames which change from place to place and so you have to adapt your frame to the local behavior of the function and this is not so obvious because if you just change your frame in a trivial way typically the geometry of the simple access degenerate so this is very well known in numerical analysis that when you have triangulations with very small angles the constants that you get in the embedding degenerate become out of control so you have to adapt your triangulation but keeping in some sense a control on the geometry of the simple access and eventually we were able to we did this first in our paper in dimension 2 with the and maybe I just to close my lecture I just give you some idea about the construction in dimension 2 and then I will tell you about the higher dimensional case so what is the idea of the construction well as soon these are the very first steps as soon as you have your function and first of all you have made a division in many cubes in your domain has been divided in many squares dimension 2 and you know that in each of these squares here I have singled only one of these squares the action of you is almost constant and it is diagonal in this tilted in this tilted direction right so the tilted direction is the one in which your action is diagonal and it is basically almost diagonal in any point inside this square of course in a different square you will have a completely different frame you have to tilt in a completely quite different direction then if you know that this is the good direction so to speak you can do the most of your swing you divide the square into triangles which gives you let's say optimal constants the point is how to refine in some sense these triangulations near to the boundary right because when you when you are going to consider the triangulations which will be done in a close square you don't want to create jumps in the derivative and in dimension 2 we have been lucky because one can use a kind of self-similar construction so as soon as you have done already the trivial decomposition here you continue in this region and again here in this rectangle you do the trivial decomposition and then you continue and here there is some kind of self-similarity which eventually tells you that all things you are doing are under control and that there is a degeneration only near to the boundary which can be eventually controlled so this proof is using kind of self-similar construction because you see for instance the triangle which appears at this level is similar to the one which appears here and also things can be kept under control more recently with a much more complex construction with the help of Sergio Conti in Bonn we did the extension not using self-similarity but a kind of really optimization there is a kind of optimization technique for the frames we were able to get the result in the same result in any dimension as I said why this result is relevant at least from the theoretical point of view because it tells you that all the extreme points of the unit ball are at least the limits of CPWL extreme functions and so even though we know that not all extreme points are CPWL it is still true that we can recover in some sense the unit ball the unit ball of my energy the unit ball of this energy by taking not only the convex hull but this is typical also of infinite dimension by the closer the convex hull of extreme CPWL function so in some sense we recover in some weaker sense we recover the heuristics of Unser and the collaborators and I think I can stop here and thank you for the attention and of course I will be made available the slides if you are interested thank you so you mentioned before that if it were true this density in energy with respect to the Euclidean then by this argument the combination would be true for any other norm by Reshetnyak I wonder whether now that you have this result about this density in energy just for the shutter norm does it imply anything for some other special type of norm maybe I don't know made with finite number you know norm such that it's a convex hull or finite number of Euclidean it's assumably so yeah I will say in principle keeping in mind also the proof of Reshetnyak theorem I will say something which is less so a norm with respect to which the shutter norm sounds uniformly convex exactly so it's kind of a relative uniform convexity a relatively strict convexity yeah I think this is reasonable thanks you stated the density theorem for a cube omega is a cube can it be clearly extended to polyadol sets or is there some other difficulty well I think the only difficulty we had in fact in the second paper we can't we consider all the slightly more general question is let's say as soon as you are able to extend your BH function from omega to a slightly larger domain without creating a second derivative on the boundary then you can make it but it is not so clear to me whether there is a general result in this direction right for the cube you do a reflection and you make it but I will say for a general domain not so clear anyhow it's a kind of extension problem otherwise if you don't want to do the extension I think to refine a little bit of our argument while going to the boundary but since our proof was already quite complicated we didn't want to introduce even this extra level of refinement but I will say that I will guess that for a C2 domain with more complication in the proof even if you are not able to extend it should be true the comment or question I have to confess I have a curiosity I hope it's not too embarrassing I am a geometer but in passing from first order to second order in the first order problem you were studying also there was a data which were the XI, YI and in fact you had the non-degeneracy condition to be imposed which I first of all my first question was there a connection between the fact that this non-degeneracy was actually applied on an affine space with the fact that you are actually studying non-piecewise linear or piecewise linear I mean is the linearity of the functions connected to the linearity of the non-degeneracy condition the non-degeneracy condition was only meant to exclude the trivial solution because since the since the in all the variation of problems I mentioned was a second order and so it's as if you already know that your data stay on an affine plane you have the trivial solution for which it equals zero and you fit completely a u by an affine function so in that statement I was saying assuming that this is not true then the problem is imposed and this is what I was going to say dimension higher than 3 but then when you pass to second order what is the I mean this data like the x i y i I mean then the problem seemed to become different at least I didn't see about this extremes points and but I mean is there an analog problem like the one of first order that you no but the variation problem was already second order so the phi was already meant to be second order the variation problem was already for this with two derivatives not with one I presented the first order version of the geometric problem of the extreme of the characterization of the extreme points of the unique ball for the sake of illustration right because if you consider the variation problem with the first order that the non degeneracy is that the data are not constant because only one derivative right any other question if not I have to invite everybody with a phd to leave this room and also those who don't have a phd only for age reasons because it's our tradition to leave the speaker for a few minutes with students to be able to discuss freely about life and mathematics and science


Hey, welcome back to VMworld 2013. I'm John Furrier. I'm with Dave Vellante, my co-host. We are here live in San Francisco, covering VMworld for our fourth year. We started theCUBE in 2010. Dave and I was on a journey. We didn't know it would lead us. Who knows we'd be the fourth year in the headlining VMworld again for four years, but we have a special guest appearance here on theCUBE. We're excited. One of our first early guests ever on theCUBE at EMC World 2010, Steve Herrod. At the time was the CTO of a small company called VMware, and now it's exploding 23,000 people here at VMworld. Welcome back. No, thanks, this is great. You guys have grown quite a bit since I was last on theCUBE as well. So you're no longer a CTO of VMware. You transitioned out as a big time VC for General Catalyst. Tell us a little about your new role and what you're getting excited about. Yeah, it's a really fun role. So I obviously love VMware and I got a lot out of working with engineers who are trying to create new things, a lot of what you're seeing here. And I thought it'd be fun to even try and spread out more and look at all kinds of companies doing it left and right. I also spent a lot of time, we bought several companies at VMware and I worked with a lot of venture firms and I never saw myself there and I never saw someone who really dug deep into engineering groups and to technology and so I thought, why not? Might as well give that a try. So I'm trying to take a very technical look and an engineering organization look at companies and help them build really big successful things. So you were a really big impact at the VMware success as well as the whole team. Carl was talking about the success of the team but you were also blogging early. You were out in the community getting dirty with the tech geeks in the field, customers as well as engineers. Do you miss it? And do you miss the keynotes that Carl took over? I think Carl did a great job today. I think I've been replaced. So very good job on his part. Do you miss VMware? I miss it a lot, obviously. I miss a lot of the people that are here and obviously this type of event is unbelievable. I think it's the largest IT event there is now. So I can still come hang out. They haven't kicked me out yet. So compare and contrast your role here at VMware this year versus last year. Last year you would have been keynoting. How busy would have you been? And what did you do this year? You wake up late, you roll in. You know, I just, I kept coming by wondering if you'd let me show up on your show because I have nothing else to do so I appreciate that. Being humble as usual. No, it's very fun. It's really now it's about meeting interesting companies and trying to help them do something good. So I do a lot less scheduled time and a lot more meeting people. So General Catalyst is a really well-known firm, big firm. I don't have many experiences with them but from what I know about the reputation, very solid, done a lot of great deals. Tell us about the firm that you're at and what attracted you to General Catalyst? Yes, it's not super well-known in the West Coast. Very big firm based in Cambridge, Massachusetts and generally hadn't done a lot of work in enterprise infrastructure. There's a couple of companies that we're now involved with. Virtual Instruments and Data Gravity, both are here that we're trying to help out. But I saw the opportunity to help them grow on in Palo Alto, down where you are, chance to sort of expand onto the West Coast and then really to dive deeply into the area that I like which is the plumbing and the things that we do here. So I'll be driving a lot of the enterprise infrastructure deal-making that we do and hopefully making them very well-known on the West Coast as they are on the East Coast. So how do you approach investments? So you mentioned before you're trying to bring a little technical, a lot of times you know just sit down with VCs that go right to the numbers, try to figure out the IRR, whatever. How do you approach it? Yeah, it's interesting, because especially in this space that you all know very well, this is not the flash in the pan type of idea that becomes viral and takes off very quickly. Most of the companies that I'm very interested in take a long time to build and you have to get the product definition right and there's hard testing and there's really a longer life cycle for doing it. And so in that sense, I do think that two hardest things you have to do when you're starting something like this is really think about how the enterprise is going to accept products and how it's going to work there and then secondly, building a great team. And those are two things I feel I've done a fair amount of and so I think I can add that type of value in addition to the important things around how do you do financing and channel build out and all the other pieces. What's the most exciting thing that's getting you jazzed up these days? In the VCs, you see a lot of things, you see all the early stage creativity, you see some technology ideas and sometimes prototypes. What's getting you most excited? Segment, technologies, anything you can share? Yeah, I mean obviously I was working on a lot of what's going on and being announced here today and obviously I think a lot of the hybrid cloud world is very interesting, network virtualization. I'm personally very focused on what happens once you have these things in place. So trying to look a little further out, let's assume hybrid cloud is in place, let's assume network virtualization has taken hold. Now what? So I spent a lot of time thinking about what are the higher level networking services over time? How are you going to be doing interesting things around management and security in this new world? So do you think Flash is a bubble and we get a bubble with Flash and or are there blind spots that people aren't seeing with Flash if they think differently? To use the old Steve Jobs line, think differently and if you take Flash as a memory, not storage, that might open up some possibilities on the software side.  I wouldn't say it's definitely not a bubble in the sense that there's still a huge amount of things unrealized. Now it's been hyped as big as anything has, as you know. But I think we're just now seeing people that have taken a storage approach to it and now you're seeing with great companies like Pernix and even some of the stuff that VMware announced, how do you actually leverage it on the server side and sort of merge things together? So I think at a macro level what's exciting and again what you're seeing a lot of these days is how do you converge storage networking and compute into some sort of package? Can we go back to what the world looks like once the SDN and software defined storage takes place? What does that world look like? How do applications change and how is development different? How is value created differently? Well it's really interesting, just at a top level if you think about it, we behave based on how the infrastructure allows us to behave. We put things in the right place of a data center because that's the subnet that we were allocated. We file trouble tickets and we wait certain ways. So you kind of have to assume all this weird behavior that we've gotten into has to go away. And if you do that then you really wonder why am I placing firewalls in certain places? It shouldn't be that there's perimeters of the network in the modern world because I have mobile devices accessing it and things that are going on within the perimeter are just important. So you really start to just think about things that we take for granted as being done the way they currently are and just start thinking really creatively on that front. So obviously that opens up a lot of new opportunities, creates all kinds of disruption. Yeah it does, but I also think it's really core and the core value to anything that we're trying to do overall is how does it help one of the big problems that are out there? So everyone's been talking about it myself for several years too. It's at the end of the day you're trying to make things more efficient and faster moving. And you just have to really think about once that hurdles down, what's next? And that's really what most of the thought exercise is right now. Well we're psyched that you stopped by. We have Craig Nunez coming on next who's another CUBE alumni. He's been probably most of the CUBEs we've had coming on next. So thanks for coming on. I want to ask you one final question. What's the biggest surprise of the VC world that you didn't expect? You had a lot of experienced work with companies. Tell us, Sheriff, that's something that was unexpected. Yeah, the unexpected and probably the funnest thing about it is how many enthusiastic people with great ideas there are. My first week in the job, I wanted to fund every single thing I saw because it was really exciting and neat. But after you go for a while, you really hopefully develop a nose for what's not just great, but what's really, really great. And I didn't realize how many different ideas are out there that are quite exciting right now. Okay, Steve Harry with General Catalyst. Now VC, former CTO of VMware on the CUBE, stopping by sharing his insight and certainly we'll see more of you. Anytime you want to come on the CUBE, we'll want to hear the updates. Seeing what the future will look like when this stuff plays out, that's what he's working on. We'll be right back with Craig Nunes of HP right after this short break. Stay with us. Thanks.


I've got a few trees come down there. I'm out in New Yorkshire and it's raining, it's gale force winds and I'm stuck behind this turniped. Yeah, get out of the road. Stuck behind this plough jockey. There he goes, see you later. I've got my wetsuit in the bag and I'm going to go free diving for crayfish.  My bag's well heavy so I've got all my weights because I know the river's going to be quite shallow so you need a lot of weight to counter the buoyancy of the wetsuit so the bag is this heavy mate. Let's go. It looks brown. I don't know if my camera's going to survive this. It's not waterproof. Am I going to survive it? Look at it, it's a bit brown, isn't it? It's a bit bobby brown.  That might work. Or will it just catch the wind a bit too much? Look at that, look at that. Can you still see there? That thing over your head. Working or not? I don't know. Best I've got mate. In these conditions it's best I've got. I'm a qualified free diver and I'm a strong swimmer so I don't get you dried off and I've got me a 10 meter swimming badge from school so it's not too bad for me but don't do it like if you if you see this and you think oh I'll have a go at that. Please err on the side of caution. Be safe. Safety first. Look at this man. Look at this setup. I've got you on. Hey. Trying to keep you protected from rain mate so you probably can't hear me. Anyway. Yeah the river looks like it's going so quick. It's such an undercurrent that I'm probably not going to be able to stay stationary or at least you know it's going to just push me down and I won't be able to find anything. And if I'm not going to get out what's that's not going to make a very good video is it but if you don't go you don't know as I say so I'm just going to get in anyway. Stop waffling on about it and get in. I'm doing it. I'm getting in. I don't care now.  The torch because it's looks pretty dark in there. A plate for me optimistic catch. I've got a net as well my lobster net which saves me getting out. I'll just put them in there again. A bit optimistic mate but that's life. Gloves yes and then me wetsuit, socks, duds, panties. Just a little bit of conditioner into the suit mix it with a bit of water and then it just it slides on a lot easier so you just you nip off the bottoms fill this full of water and conditioner and you can use this and then rub it on you on your on yourself. If I slip and fall in fade to black in it and then I'll use that same water in here as well or I'll pour it all over my foot. Either Iver may either work, pour it all over your foot. Okay all right then. What I'm looking about now is real real as it comes. Any dog walker's gonna think I've lost me. Lost me marbles like I've slipped off the cracker. This is me nice. Now it's mainly just like a thin pointed one. It's got a bit of a serrated edge there in case you get caught in any ropes or anything but what it's mainly for or what I have it for is just to kill the fish you know brain them if you get them but I wear it because it's just safety first in it especially when you're on your you get caught in some ropes or weeds it's there it's on my it's on my ankle so I can just reach down and grab it. If you don't see you anymore and again and someone finds this can you upload it to where there's outdoors YouTube channel because I'd just like to say goodbye to everyone and thanks very much for the support over the years it's a it's really meant a lot to me. Right let's switch to GoPro. I'm just going to check the check the current check the visibility and have a few dives down see if I can see any side of any crayfish and if I do see some I'll come back and get my net and torch right here we go let's get in. That current is strong.  Got to love it there. This isn't working. This isn't working so I'm going to try swim down there and get into the side into an eddy see if the water slacks off a little bit. That's all I've seen I've been going for about 20 minutes to see one so I've got my net let's see if I can get a couple. It's quite deep here so I'm going to dive down and see if there's anything in the deeper water. Double whammy! These are the American signal crayfish as you can tell by the bright orange underneath the claws they're a lot bigger than our native crayfish. Starting to get dark now look. Result. Absolute result. It was far too quick in the middle. I'm cold mate. It was far too quick in the middle and then when I got too deep it was too dark so I just stayed in the edges in the little eddies and stuff where it wasn't so quick and once I got my eye in I managed to get a decent amount there. I'm freezing. There you go. I got a few and then I just sacked the filming off and just focused on just foraging and managed to get a decent amount like. You need a few because there's not a great deal of meat on them so it's not a bad it's not a bad load but you can't see on this camera but it is getting dark now and it's starting to rain again so better be quick. Yes mate that's me awake it that's not the normal that's the awake it but it's it's warning W column from awake it. All my brain's gone all cold brain-ed so I can't really think straight. I'm happy with that mate. Well happy with it. Fingers are a bit cold, face is a bit cold but well I'm a bit cold.  I'll do one of these ones. I'll see you there. Right woodland it's absolutely bog myrtle everywhere. I've got to chuck my wellies on but look even just the ground you can see it's just absolutely dripping wet. I've found a I've taken taking refuge under this pine tree and there's a few pine trees in this area it's creating a natural sort of the canopy is keeping me dry. The crayfish are in here they've just been prepped I've done this on multiple videos. It is quite simple you just put the knife I won't do it now but well I will but I won't show you. Once you've humanely dispatched them you see the the tail here you can grab the middle bit of the tail and just twist it to one side it'll click to the side you'll hear it click again and then you can pull it. I don't see that there and that's the track that's all it's here. All the insides just attached to that little bit of tail and that's him fully prepped and ready for the pot. Bring it to a boil and let's four or five minutes don't take long at all and then they're good to go. You can obviously pimp up the water a little bit you know some Cajun spices some lemon wedges orange wedges I think traditionally they use orange wedges and chorizo and things like this but I'm just I'm just in wood sir we're just going bareback that's all I want it eat them off see how many I got it's quite a good haul isn't it you just twist the back end off there's not there's nothing much in Ed you can suck out the brains and stuff it's quite a good to lovely flavor actually you just get these little nuggets nuggets of meat oh yeah that was hard for as well worked hard for this food mmm that is nice I'll get a claw on the go yeah the claws pretty much exactly the same same as a lobster in the way that you crack it and eat it and just you know where the meat is oh yeah man and that the claws are a lot more earthy but more like a almost nutty I find absolutely delicious mate delicious oh I'm gonna get tucked into these you don't want to watch me what do you want to watch I'll tell you what I'll show you and I'll show you one of my little landscaping projects from last week because I know some of you like to see see my work every now and again so yeah it was them so my mates building their built this it's like a granny flat I guess like a big extension at the back of the garden and we're gonna we've started but we're gonna finish it with burning timber Japanese style and cladding it with burnt timber but the whole garden was on a slope covered in brash and bits and pieces and concrete and all sorts of rubbish and so my job was to find the right level and do something with it to make it a usable space so I retained it using sleepers and I think it was a quite a nice little job so I'll let you have a quick look and there we go that's the finished patio I've just used big 900s gray Indian stone sort of brick bonded with the pattern we've run I've run you see that put pipes underneath so we can run wires through here and here conjure it all the way through for for lighting the deckings in used sleeper rails as a retainer and a step up let this loose because we're gonna run I'm gonna run a sunny flow out of here along here and up tapping to the main drain and so Shugi ban whatever they say so as you give an I think it is it's the Japanese burnt wood is started here and that's gonna clad all this building yeah some tricky levels to play with but nothing I can't handle there we go nice usable space another happy customer I smell like Shamu's ass I better get going before I totally lose me light I'll pack this up and then I'll get out of here thoroughly enjoyable for me don't know if you've enjoyed it or not if you have click the thumbs up button because apparently that helps so and all it is is that in it you don't have to do a lot just that and subscribe and hit the the bell notification below because I'm cranking out a few vids at the moment you don't want to miss them dear you don't miss quality content like this and just on a just to finish off I didn't talk much about crayfish because I feel like I've done a few videos on crayfish and I don't keep repeating myself but just for any new subscribers there don't just go randomly picking crayfish out of the rivers because there are two different sources the signal crayfish which is the American invasive species and then there's our native white-clawed crayfish which you don't want to be in them and also you sign up on the you can't go online and get a license and you've got to put in your data and say where you go in because you might you might get a load of crayfish out of one bit of river not really thinking go home you might wash them they might have eggs the eggs go into the into the ecosystem again they're so vivacious and they're so brutal they're taking over our waterways and they are a real pest so please just be mindful of it and when you're dealing with them and they are a delicious little pest but they are an invasive species so we need to be just be mindful of it anyway right look after yourselves take care namaste