Hello everyone, welcome to theCUBE's presentation of the AWS Startup Showcase. Martek is the focus and this is all about the emerging cloud scale customer experience. This is season two, episode three of the ongoing series covering the exciting fast growing startups from the cloud AWS ecosystem to talk about the future and what's available now, where are the actions. I'm your host, John Furrier. Today we're joined by CUBE alumni, Jerry Chen, partner at Greylock Ventures. Jerry, great to see you. Thanks for coming on. John, thanks for having me back. I appreciate you. Welcome back for season two as a guest star. You know, hey, you know, season two, it's not a one and done. It's continued coverage. We've got the episodic CUBE flicks model going here. You know, congratulations. The coverage on this ecosystem on AWS has been impressive, right? I think you and I have talked a long time about AWS and the ecosystem they're building. It just continues to grow. And so the coverage you did last season all the events of this season is pretty amazing from the data security to now marketing. So it's just great to watch. And 12 years now, the CUBE and running, I remember 2013 when we first met you on the CUBE, we just left the Mware, you're just getting into the venture business and we were just riffing the next 80, no one really kind of knew how big it would be, but we were kind of riffing on it. We kind of had a sense. Now it's happening. So now you start to see every vertical kind of explode with the digital transformation and disruption where you see new incumbents, I mean, new brands get replaced, the incumbent old guard. And now in Martek, it's ripe for disruption because web two is going on to web 2.5, 3, 4, 5. Cookies are going away. You've got more governance and privacy challenges. There's a slew of kind of ad tech baggage, but yet lots of new data opportunities, Jerry. This is a huge thing. What's your take on this whole Martek cloud scale market? Yeah, I think to your point, John, first, the trends are correct and the good old days are bad old days. Martek was really about your webpage and then email. Right? The email's the only channel and the webpage was only real estate and technology to care about. Fast forward to 10 years, you have web pages, mobile apps, VR experiences, car experiences, your Alexa home experiences, less not even get to web 3, web 18, whatever it is. Plus you got text messages, what's app, messenger, email, still great, et cetera. So I think what we've seen is both explosion and data, explosion of channels. So sources of data have increases and the fruits of the data where you can reach your customers from text, email, phone calls, et cetera, have exploded too. So the previous generation created big homes like responses, Eloqua, that exact target that got acquired by Oracle or Salesforce. And then the companies like MailChimp that got acquired as well, but into it you're seeing new generation companies for this new stack. So I think it's exciting. Yeah, and you mentioned all those things about the different channels and stuff, but the key point is now the generation shifts going on, not just technical generation and platform and tools, it's the people, they're younger, they don't do email. They have proton mail accounts, zillion Gmail accounts just to get the freebie. They'll do subscriptions, but not a lot. So the generational piece on the human side is huge. And then you got the standards bodies throwing away things like cookies. So all of this makes it for a complicated, messy situation. So out of this has to come a billion dollar startup in my mind. I think multiple billion dollars, but I think you're right in the sense that how we want to engage with the company branch, either consumer brands or business brands, no one wants to pick a phone anymore, right? Everybody wants to either chat or DM people on Twitter. So number one, the way we engage is different, both where, both how, like chat or phone, but where like mobile device, but also when it's the moment when we need to talk to a company or brand be it at the store, when I'm shopping in real life or in my car or at the airport, like we want to reach the brands, we want to reach us at the point of decision, the point of support, the point of contact. And then you layer upon that, the playing field, John of privacy, security, right? All these data silos in the cloud, that the game has changed and become even more complicated with the startups. So the startups are going to win, we'll do the collect all the data, make it secure and private, but then reach your customers when and where they want and how they want it. So I got to ask you because you had a great podcast just this week published and Snowflake had their event going on the data cloud. There's a new kind of SaaS platform vibe going on. You're starting to see it play out. And one of the things I noticed on your podcast with the president of HashiCorp, who was on, people should listen to that podcast, it's on Gray Matter, which is the GrayLock podcast plug for you guys. He mentioned, he mentions the open source dynamic, right? And I like what he thinks he said, software business has changed forever. That's my words now. He said infrastructure, but I'm seeing software in general more broader. Infrastructure and software as a category is all open source. One game over, no debate, right? You agree? I think he said infrastructure specifically starts to open source, but I would say all open source is one more or less because open source is in every bit of software, right? And so from your operating system to your car, to your mobile phone, open source, not necessarily as a business model or whatever, we can talk about that, but open source as a way to build software, distribute software, consume software has one, right? It's everywhere. So regardless of how you make money on it, how you build software, an open source community has one. Okay, so let's just agree, that's cool. I agree with that. Let's take it to the next level. I'm a company, starting a company to sell to big companies who pay. I got to have a proprietary advantage. There's got to be a way, and there is, I know you've talked about it, but I have my opinion, there is needs to be a way to be proprietary in a way that allows for that growth, whether it's integration, it's not going to be on software license or maybe support or new open source model, but how do startups in the Mar-Tech, this area in general, when they disrupt or change the category, they got to get value of creation going. What's your take on building on that? You can still build proprietary software on top of open source, right? So there's many companies out there, you know, a company called Rockset. They have a open source technology like Rockset DB under the hood, but they're running a cloud database that's proprietary. Snowflake, you talked about them today, you know, it's not open source technology company, but they use open source software, I'm sure, in the hood, but then there's open source companies data break. So let's not confuse the two. You can still build proprietary software. There's just components of open source wherever you go. So number one is you still build proprietary IP. Number two, you can get proprietary data sources, right? So I think increasingly you're seeing companies fight, I call this systems intelligence, right? By getting proprietary data to train your algorithms, to train your recommendations, to train your applications, you can still collect data that other competitors don't have and then they can use the data differently, right? The system of intelligence, and then when you apply the system intelligence to the end user, you can create value, right? And ultimately, especially in marketing tech, the highest level is what we call the system of engagement, right? It's the chat bot, the mobile UI, the phone, the voice app, et cetera. If you own the system of engagement, be it Slack or be it the operating system for your phone, you can also win.  Just got to own system record, system intelligence, system engagement, easy, right? Yeah, oh, so easy. Well, the good news is the cloud scale and the CapEx funded there. I mean, look at Amazon, they've got a ton of open source, you mentioned Snowflake, but they're getting a proprietary value project. So I need to ask you, Mark Tech in particular, that means it's a data business, which you pointed out and we agree, Mark Tech will be about the data, the workflows. How do you get those workflows? What's changing and how these companies are going to be building? What's your take on it? Because it's going to be one of those things where it might be the innovation on a source of data or how you handle two parties handling encrypted data sets. I don't know, maybe it's a special encryption tool. So we don't know what it is. What's your outlook on this area? I think that last point you just said is super interesting, super genius. It's integration or multiple data sources. So I think either one, if it's a data business, do you have proprietary data? One, number two, with the data you do have proprietary not, how do you enrich the data? And do you enrich the data with a public data set or a third party data set? So this could be cookies, it could be Dunham Brand Street or Zoom info information, how do you enrich the data? Number three, do you have machine learning models or some other IP that once you collected the data and enrich the data, what do you do with the data? And then number four is once you have that model of the data of the customer or the business, what do you do with it? Do you email? Do you do a tax? Do you do a campaign? Do you upsell? Do you change the price dynamically in our customers? Do you serve a new content on your website? So I think that workflow to your point is you can start from the same place, what you do with the data in between out of the side of this pipeline is where you as a market company can have that. So like I said before, it was a website to an email. Go to a website, have a cookie, fill out a form, I send you an email later. I think now you can't just do a website to an email. It's a website plus mobile apps plus in real world interaction to a text message, chat, phone call, Twitter, ad, whatever. It's like they're playing checkers in web two and you're talking 3D chess. There's a huge gap between what's coming and this is kind of interesting because now you mentioned machine learning and data and AI is going to factor into all this, you mentioned Rockset, one of your portfolios has under the hood, open source and then use proprietary data in the cloud. Okay, that's a configuration. It's an architecture, right? So architecture will be important in terms of how companies posture in this market because Martek is ripe for innovation because it's based on these old technologies but there's tons of workflows but you got to have the data, right? And so if I have a best journey map from a client that goes to a website but then they go and they do something in the organic or somewhere else, if I don't have that, what good is it? It's like a blind spot. Correct. So I think you're seeing folks with the data be it Snowflake or Databricks or an Amazon that S3 say, hey, come to my data cloud, right? Which, you know, Snowflake's advertising. Amazon would say the data cloud is S3 because all your data exists there anyway.  Databricks will say S3 is great but only use Amazon tools, use Databricks, right? And then build on top of that. But then he had our SaaS companies like Oracle, Salesforce, whoever and say, you know, use our Eloqua, Marketo exact target, you know, application as a system record. And so I think you're going to have a battle between do I just work my data in S3 or where my data exists or do I work my data as some other application like a Marketo Eloqua exact target or, you know, it could be a Twilio segment, right? That was a combination. So you'll have this battle between these giants in the cloud, easy to castles, right? Versus the contenders or the challengers, as we call them. Well, great, always chat with you on that. We always talk about castles in the cloud, which is your work that you guys put out. Just an update on, so check out graylock.com. They have castles on the cloud, which is a great thesis on, and a map by the way of the ecosystem. So you guys do a really good job. Props to you, Jerry and the team over there, graylock. Okay, now I got to ask kind of like the VC private equity market question, you know, evaluations, first of all, I think it's a great time to do a startup. So it's a good time to be in the VC business. I think the next two years, you're going to find some nice gems, but also you got to have that cleansing period. You got a lot of overvaluation. So what happened with the markets? So there's going to be a lot of M&A. So the question is, what are some of the things that you see as challenges for product teams in particular that might have that killer answer in Mar-Tech or might not have the runway if there's no cash? How do people partner in this modern era? Because scale's a big deal, right? You can measure everything. So you get the combination of a new kind of M&A market coming, a potential growth market for the right solution. Again, value's got to be there. What's your take on this market? I think you're right. Either you need runway, so cash to make it through this next two, three years, whatever you think the market term model is, or two, you need scale, right? So if you're at a company of scale and you have enough data, you can probably succeed on your own. If not, if you're kind of in between or early, to your point, either one, focus a narrower wedge, John. Just like we say, just reduce the surface area in the next two years. Focus on solving one problem very, very well. Or number two, in this Mar-Tech space, especially, there's a lot of partnership and integration opportunities to create a complete solution together to compete against kind of the incumbents, right? So I think there are folks with the data, there are folks doing data privacy security, we're folks in the workflow or market workflows. You're going to see either one, some M&A, but I definitely can see a lot of cooperation and partnership. And so in the past, maybe you would say, I'm just raising another $100 million and do what you're doing. Today you might say, look, instead of raising more money, let's partner together or merge or find a solution. So I think people are going to get creative. Like I said, scarcity often is good. I think it forces a lot more focus and a lot more creativity. Yeah, that's a great point. I'm glad you brought that up because I didn't think you were going to go there. I was going to ask that. BizDev activity is going to be really fundamental because runway combined with the fact that, hey, you know, if, you know, get real or you're going to go under is a real issue. So now people become friends. They're like, okay, if we partner, it's clearly a good way to go if you can get there. So what advice would you give companies? Even most experienced founders and operators, this is a different market, right? It's a different kind of velocity. Obviously architecturally data, you mentioned some of those key things. What's the posture to partner? What's your advice? What's the combat manual to kind of compete in this new BizDev world where some, it's a make or break time. Either get the funding, get the customers, which is how you get funding or you get a BizDev deal where you combine forces and go to market together or not. What's your advice? I think that the combat manual is either, you're partnering for one of two things, either one technology or two customers or sometimes both. So it would say, which partnerships are gene for technology, e.g. solution completers? Like you have, you know, this puzzle piece, I have this puzzle piece, data and privacy, let's work together. Or number two, it's like, who can help you with customers? And that's either A, they can be a channel for you or vice versa, or it can share customers and you can actually go to market together and find customers jointly. So ideally you're partner for one, if not the other, sometimes both and just figure out where in your life cycle do you need friends? Yeah, great. My final question, Jerry.  As usual, always awesome. Final question for the folks watching that are going to be partnering and buying product and services from these startups. There's a select few great ones here and obviously every other episode as well and you've got a bunch you're investing in. It's actually a good market for the companies that are lean and mean have value and the cloud scale does provide that. So a lot of companies are getting it right, they're going to break through. So they're clearly going to be getting customers. The buyer side, how should they be looking through the lens right now and looking at companies? What should they look for? And they like to take chances with seeing that. So it's not so much they got to be vetted, but how do they know the winners from the pretenders? You know, I think the customers are always smart. I think in the past, in Marktech especially, they often had a budget to experiment with. I think you're looking now the customers, the buyer technologies are looking for a hard ROI, like a return on investment. And before I think they might experiment more, but now they're saying, hey, are you helping me save money or increase revenue or some hard core metric that they care about? So I think the startups that actually have a strong ROI, like save money or increase revenue and you can like point empirically how they do that, will rise to the top of the Marktech landscape and customers will see that. Customers are smart, right? They're savvy buyers, they can smell good from bad and they're going to see the strong ROI. Yeah, and the other thing too I'd like to point out and love to get your reaction real quick is a lot of the companies have DNA in the open source or they have some community track record where community is now part of the vetting.  Yeah, I think open source, like you said in the community in general, like especially all these communities that move on Slack or Discord or something else, right? I think for sure, just going through all those forums, Slack communities or Discord communities, you can see what's a good product versus next, versus bad, don't go to like the other sites, these communities would tell you who's working. Well, we got a Discord channel, theCUBE now, had 14,000 members, now it's down to six, losing people left and right.  We could use some help there, love Discord. Jerry, great to see you, thanks for coming on. What's new at Greylock? What's some of the things happening? Give a quick plug for the firm when you guys working on. I know there's been some cool things happening, new investments, people moving on. Yeah, look, we're Greylock partners, seed series A firm. I focus on enterprise software. I have a team with me that also does consumer investing as well as crypto investing, like all firms. So, but we're seed series A, occasion of the later stage growth. So if you're interested, find me at Jerry Chen on Twitter or jerrygraylock.com. Thank you, John. Great stuff, Jerry, thanks for coming on. This is theCUBE's presentation of the AWS startup showcase. Martek is the series this time, emerging cloud scale customer experience where the integration and the data matters. This is season two, episode three of the ongoing series, covering the hottest cloud startups from the AWS ecosystem. John Furrier, thanks for watching.


Tom here from Orange Systems. We're going to talk about IT certifications, college degree, and well, let's get this right off the rip and out of the way because this question comes up quite a bit. I want to have this video as my go-to reply for what are your thoughts on these things. First, my thoughts are I don't have them. I do not have a college degree. I do not have IT certifications. So if that's enough to make you stop watching this video because you shouldn't listen to some guy who doesn't have certifications to talk about certifications, that's perfectly fine. That's why I like to disclose things up front and maybe you've stopped watching there. If you still are watching a video, let's talk a little bit about this because as a business owner, one, you don't need, at least here in the US, any type of certification in the business. So that is just something that doesn't come up, even though I've had a tech company and I've worked in technology since 1995, it is not required that I have certifications to do what I do. There are specific exceptions where there are licenses that you may need or certifications that have to be vetted by state agencies to be very specific in certain areas and cabling and wiring is one of those areas, but Michigan where we are located is actually an exception to that, but some other states do require certifications that you have to get for that, but that's a little different than the IT certification. So at least I'll mention because we do cabling and wiring and people ask, well, what search do you need to get started? In Michigan, the answer is not. Back to the actual topic. One of the problems is a lot of people are they see these headlines and they drive me nuts, you know, and I could sit here and be all high and mighty about it going, I'm successful in tech and I'm all about the acquisition of knowledge, not the validation of institution, but that doesn't really help anyone in their IT career. That's just a click-baity headline to say that someone dropped out of college and built some big company and it's some article you'll read and there'll be this implementation that you can do it to just because they did it. The reality is you're probably going to need some certifications to get into IT. What certifications is going to vary greatly based on where you're going and the cybersecurity world and especially some of the application pen testing world is some of the crazy side where there's certainly a lot of people without certifications to get in there, but there are skill levels or where they prove it. And one of the ways, especially when a cybersecurity side, especially if you go into application pen testing and just collecting bug bounties, well, there's no certifications. You just have to be good enough to do it. That's the bar of entry. Can you do it? Can you dive into that and go in that market? But if you're looking for a regular, I want to work at a company with a job, it's important to have certifications. They are what make you marketable. They are what make the prospecting employer, when they're going around, going through places to go, oh, person's got a few certifications. And then like the college degrees as well. Those are things that are looked at as qualifiers to kind of filter out and it's how HR departments will make some of those certifications or terminations. Now, there is another video I'll link to kind of talking about getting started in cybersecurity where a couple of people who are experts in cybersecurity that I had on my channel discuss that. I'll leave a link below and it's Xavier and Chris Roberts. And they talk about hacking, getting into there, maybe even how to get around HR and things like that. Soft skills are where I have always worked at and that's what's helped my career more than anything else. I've had the skills by reading books. I've done lots of hands-on labs. I've been working in tech for 25 years. The other side of that is I know a lot of people in tech and it's because, well, I've been social in that realm. I take an interest in what they do. I've been someone who's worked on a lot of this myself and having a lot of friends in the industry is what gets me a lot of the connections and you're probably thinking, well, I don't know a lot of people. There's a time in my life I didn't either. I don't come from anyone in my family who does any type of tech, but I will tell you those type of soft skills and going to conferences or online forums is actually where I meet a lot of people too. Over the years, I'm talking like from the earliest days, I was, I met people in the BBS dial-up days where we dial in a Baltimore systems and have discussions with people about networking and it's been ongoing. Those conversations then it moved to IRC. Now it's moved to me hanging out and everything from Discord to Slack channels randomly that dive deep into those technology topics, but having those circles of friends and technology is another part of your career path. It's kind of a conglomeration of everything together. Pick your passion, decide what you may need to get certified in, whether you're going to the network engineering side, whether or not you're going into some of the Microsoft certifications which of course are extremely popular and Microsoft and Cisco collectively represent a massive part of our industry. Those are still going to be popular, relevant certifications. Now I'm not going to really send you down a path of which one because I don't know what you want to do, but I will say that they're relevant. Now the final thing I'll answer because this is part of the question that comes at me is do you have certifications? What do you certify it in or what's your college degree, which like I said was none, but do I require that of my staff and wouldn't that be kind of hypocritical and not really? I don't require it right now as a hiring mandate, but I've also only hired mostly my friends and people I knew.  I've had a few that have come in that I didn't really know, but yeah, I did some Q&A and being that I am technical, I'm able to see through people who don't really understand what they're doing, but put it on the resume that they did. That's not as easy for your average HR person to do, but the vetting process I go through and it's not something I require. Now it is something that I think is important in any of my staff that want to go and pursue certifications relevant to their career in here. I cover that. I do provide training. IT Pro TV is one of the things I have an affiliate link down below, but we actually use it internally. They'll take the tests. I've bought books for them. These are all things that are not a problem. Now, the final thoughts are certifications. Do I think they're any good? This is a real fuzzy thing because I will admit they were really bad in the beginning and they really have changed a lot now. So I will say that my opinion, if you were to find some old comments that I had left somewhere where you don't need certifications and they're not really relevant and all they do is regressions information versus how I look at it today, I think they're a pretty important part of the industry. Our industry probably needs more of that because we have a lot of things that are not done properly and making sure that these certifications and people that are qualified to do the jobs are the ones doing the jobs. It's going to be an important part of the maturation and how we get IT more organized and a little bit better and more secure is it's going to be through some organization like that and putting together proper certifications and proper understandings of the parameters needed in order to make sure that people know what they're doing when we sit them down in front of the keyboard or whether they know what they're doing while you're up a cabling, whether or not they put things together in a secure manner when they're building an application. So that's my thoughts on it for here in 2020 and I'll leave a link below to the Chris Roberts video with Xavier and diving in a cybersecurity topic because that's an interesting one all together that they are experts in. Thanks. And thank you for making it to the end of the video. If you like this video please give it a thumbs up. If you'd like to see more content from the channel hit the subscribe button and hit the bell icon if you like YouTube to notify you when new videos come out. If you'd like to hire us head over to laurancesystems.com fill out our contact page and let us know what we can help you with and what projects you'd like us to work together on. If you want to carry on the discussion head over to our forums.laurancesystems.com where we can carry on the discussion about this video, other videos or other tech topics in general. Even suggestions for new videos they're accepted right there on our forums which are free. Also if you'd like to help the channel in other ways head over to our affiliate page we have a lot of great tech offers for you and once again thanks for watching and see you next time.


They're all close to the fire. They're staying warm by the fire. That's good. Nice toasty fire in the morning. It's hard to beat, isn't it, Amy? Good morning, campers. I think we're on day three. Look at the sunrise and the background. Fuel for the soul, soak it in. About to get that first cup of coffee and the fresh mountain aerial. And today we're gonna be doing some fishing. I have always been able to catch some trout up here in the little streams and also in this lake that we can't buy. But I've already tried a little bit and it's been difficult. And we really need to catch some because we're planning on one of our meals being trout. The camp is counting on me to get some fish today. So I'm gonna put in some work. I've fished for like an hour here, an hour there, but I've had four bites since I've been here and not a full connect. Hopefully we can get some rainbows, some coquinis, maybe even a brown trout out of the lake this morning. So let's get that coffee, let's get warmed up, get fueled up and give it a day. That's the sweet nectar that makes me and Moby in a good mood in the morning. Okay, our trout adventure begins. Like I said, it's been hotter than usual the other times that I've been up here. This is my fourth year in this general forest area. I'm wearing shorts. I can't remember wearing shorts like every day like this. I've taken two spinning reels. So I got the new 1000 size Fugue Squad Gold Series and matching the rods, but these are the, sorry, my breath, altitude. These are the micro series. So trout and panfish rods should be really good for throwing these light rooster tails. We've got a little bit of a hike. I'm going to fish a spot that I haven't been to yet and it takes a little bit of walking to get to. So we've got about three quarters of a mile walk and then hopefully it'll be worth it. Okay guys, running head cam here. Very calm conditions. I'm going to start off with this little, I think it's an eighth ounce. Yeah, eighth ounce, a little rooster tail. And just see the only thing I've gotten bit on, well, I should say that I got bit on the rooster tail in a little stream, but in the lake I've only been getting bites on a spoon and it's just like little slaps. I'll be darned if there's not any fish out here though. There's definitely fish in here. Oh, there was a trout right there. Definitely saw one surface right there. Going with this spoon, just a nickel spoon. And I've seen a couple of surface in this area, so I'm just going to experiment with different depths. Maybe they're, I don't know, I don't know where they're at. I don't see a whole lot of bug activity. Holy patooties. Sorry I didn't get that on camera for you guys. We had a hard lull, a hard lull. I didn't think I was going to get a fish. This is a kokanee right here. Oh my gosh, this guy's going nuts. So I tried the old dough bait. I wouldn't get bit on that.  I tried throwing spoons for a while. I tried the float and fly, the pistol peats. The pistol peats, it was just not going down.  There we go, okay. So I did not even want to attempt to click record on GoPro because I was desperate to get a fish in. So we finally got a kokanee. And the second cast, this lure right here, this is one that one of you sent me. You made them. It's an inline spinner. I'm going to try to get this box out because I got the whole box. So whoever sent me this box years ago, I've hung on to it because it had, you know, I don't trout fish a ton, but it looked like some decent trout fishing lures in there, some salmon fishing lures. And what do you know? Tied one of those on there. I don't even know what weight it is. It just looked good. And there he is right there. So let's give her an angle. Come on now. Give me another. I need two for me and Steph's dinner tonight. Another one. Come on baby. Yeah. Yeah. Now we're cooking with Crisco. It's a decent one. Racious. Look at that rod gyrate. Okay. That's about those fish for about five minutes apart. That's good. Get your wiggles out. Fish number two, Kokani salmon. There we go. Two Kokanis. That one's pretty shallow. This lure is getting it done right now. If I get one more of those, wife will be happy. Just a steady retrieve is all I'm doing here. Every time I've caught fish here, it's you've got to go so slow. But it's very similar to crappie fishing since I've gotten pretty decent at crappie fishing in the last few years. It's, you know, it's a lot slower presentation usually.  Set up is handling the fish good too. I'm yet to lose the fish. These, uh, these trout and these salmon, they just shake their heads so violently. I've lost so many. You really got to have a light rod. It's almost 11 o'clock, so we're going to start getting some winds here too. Midday mountain winds. That might help. Might hurt. I'm not sure. Grasshoppers are coming out and walk down here where I haven't fished yet. Try this lure. Go to these couple points. Little points. It seems to be where all the cockney salmon live. Right off the end of points. Okay, we got the lure they want. We got a little wind. And we just need to run into a couple of them. There's two subtle little points right in front of me. I'm going to try casting over. Oh, oh, dang it. It's a rock. What a background. Just need to fish to gyrate my tip, please. Give me the tip gyrations. Mule deer. Mule deer coming to get a drink. Oh yeah, look at this little point. Woo. This is juicy. There we go. Come on, baby. Be a cockney. Cockney in. They certainly know how to make their way back to the water. All right there, bud. He was getting back in the water. Oh my God, he got away. I cannot believe it. I cannot believe it. I had him pinned. I had him pinned to the ground. Holy smokes. All right. We're using to catch another one. There we go.  I'm going to try to catch another one. Death grip on those things. I really thought he was done. That was wrong. He's getting back in the water. Oh my God, he got away. Oh God, I got smoked. Smoked. Missed him. Get up in here. Got you in the trees. Take this guy up here to about 10,000 feet so he doesn't get back in the water. Holy cow. That thing is when you when you drop one of that, they like just roll right into the lake. I'm going to walk him back to my stringer. I think we're good on Coconies. We got our meal for the day. I just hiked. We saw you walking. Well, we actually went down to the overlook and we saw you out on there fishing. I went along ways. We were heading back as you were heading back too and I saw you were holding something. I am. And yes. For your dining outdoor pleasures, I have acquired three Coconies salmon. Oh, that was my bet. She thought it was a cutthroat. Yep. Coconies salmon. All right. I had four. I literally had another one at my feet. I thought I had them.  But I think that'll be enough for us tonight. I'm not sure if we'll see you at a bony. Sammy might give her a taste. Yeah, we'll give her a taste. She ate it last time when she was two. But I still need to clean them. Well, what do you think? Well, it's pretty. That's a Coconies salmon. It sure is a pretty background behind you. Yep. You can pull that in. That's going to be our dinden. Yeah. Hold them up. Say, I got a fish. I got a fish. You got three fish. You got three fish. You got dinner. Oh, I'll never forget that first time you said that. I got a fish down here.  I got a fish. You got a fish. I got a fish. I got a fish. Yay, Amy. And that's our dinner. I'm smelling it. Just like daddy. Smell it. Smells like dinner. You see their little noses? How sharp their little noses are? All right, Mommy, do you want to hang on the Ziploc? All right. All right, Amy. I wonder what these fish have been eating. What do you think? I don't know. I think they've been eating small fish. All right, just so everybody at home how to clean a trout or, in this case, it's a Kokini salmon, which I actually prefer to eat over the trout. You want to feel it? It feels pretty soft and slippery, huh? It's kind of gooey. So you don't need any kind of special filet knife for this. You can literally use a pocket knife. I'm just using my little camp knife here.  Now, a lot of the really good meat is right behind the head. So you just want to cut right, I'll show you guys. So you want to get like right there. It's better to put the fish down where you got like a steady surface. But you'll feel it kind of slip in between that little, uh, little spine there. Flip them over.  Go up through the belly there. And then, maybe we're going to poke our little thumb in here. I'm going to see what, what this fish has been eating. What do you think? Oh, I don't see much. It's like a male.  So once you get the, the inner side cleaned out, you can just take where you cut on top. And pull down with your thumb. That whole section just comes right off. And then we feed it to the fish there. We're getting all dirty, but watch, we're just going to push with our thumb right here. Push all those little nasty parts out. Give her rinse like this. And it's done. It doesn't look so nasty. Look at that beautiful orange meat inside of there. That's what mommy likes right there. All right, mommy. Zip lock. That's going in. That's good. Tell me if I'm in frame. Does this look good? Okay. He needs my little camera girl right now. Then we go up. You got it, Emmy? Everything looking good? All right. We'll pull down. You're going to pull down. Drunk the head. Looks like we got another male. All right. I'm done filming. You're done? All right. You want to hit the red button on top for me? It's just red. And scene. Let's get a fire going. Got a beautiful fire going. We're going to cook these salmon.  So I'll be able to control the fire a little better that way. I've never done it, but we'll still put them in tinfoil. We'll do the butter, the lemon, and all that stuff. But I'm not going to do them caveman style where I just throw them into the coals. I'm going to do a more controlled method. We'll see how it turns out. And hopefully the rain is going to hold up long enough for us to get these things cooked. So the rain knocked down our fire a little bit, but this is what we're going to be cooking with. It's just a Dutch oven. Cast iron Dutch oven. I've got a tripod set up, and I've got a little pot hanger that I'm going to be sitting over some paracord. This worked out really well with cooking some pulled pork last night. So I thought we'd give it a shot. I found a stick that has a little knob like that. And then I just carved a notch right in there. And I might carve another notch right here just to give me a little extra adjustability on this thing. Seven inch ice cream. Yeah. Thank you so much. I'm going to put it in your hand. Yeah, okay. Give me a little scoop there. Thank you. Pretty good. Pretty good.  I'm an experienced ice cream eater. I was a little worried bringing a four-year-old up here into the deep wilderness with no attractions or anything like that, but she's having the time of her life up here. It is awesome. These are memories she's going to be having forever. She has no problem just taking simple pieces of bark and making plates with them and serving vanilla ice cream with sawdust. It's good, man. It makes me feel good as a dad that my daughter can chicken hang in the woods. You know what I mean? Got the pot heating up now. So we're going to go over here and prep these salmon. Got your mountain tire on. I'm cold.  You're good. So you're making grits. Making some grits, yeah. Cheesy grits with salmon. Fantastic. And a little salad. Gotta have a salad. I wish she salad. Okay, so in order to make grits, how do you make grits? Well, I started out with a little bit of butter. Probably don't need that much butter, but a little bit of butter. Then you put in... You have to actually get grits. You can't just use regular cornmeal.  All right, guys. It's pretty windy, but I'm going to try walking through this. OSG's doing the grits. She's got a stir while it's getting hot with the milk. Yeah, I got to stay on this station. So it's good that we have two people on deck. Okay, put you guys right here. Camp, kitchen prep. All right, so butter. Got to have butter. Keep it basic. And I'll take... I have no idea. About a finger's width. A pat. That was it? A pat. A pat. A pat of butter. It's really like a teaspoon. Oh, I'm sorry. A tablespoon. Lessons being learned. So about like that. And I'm just going to chop it up into about four little... Little things. Four little squares. Four little deals. So we've got our coconut sitting in the tin foil like this. And then I'm going to put two of those little... Little butter buddies. Butter buddies. Little butter buddies right in the cavity. Then I'm going to sprinkle hard Cosmo SPG up in there. Get all up around the neck and shoulders. Then we'll take our other couple of little butters. Oh, I got a look. Okay. Sorry about the chaos. All right, we're going to fold this over. We're just basically going to make a burrito. Fold it up. Crease those ends. So we keep the butter in there. And then we've got a salmon burrito basically. I completely forgot our fresh lemon. Dad gum it. You were even talking about it. Yeah, I know. Well, I'm under duress here with the conditions. We've had just absolutely perfect conditions up until this point. So I'm just going to slice some lemon. Slice some lemon up.  Our little burrito sacks. I almost came back and said hello to me there. Don't want any knife injuries in the wilderness. Just relax. I know, I'm trying to hurry. It's going to be okay. I always do things worse when I'm filming. There's no question about that. It's always better when you're just focusing on what you're doing. But I've decided to record my outdoor adventures. And I like it. You better like the like button. Don't lose that. Because we need that. Put that thing up. Put that thing up. Dear goodness. These are ready, honey. Where are you at? Where are you at with those grits? You want me to start putting them on? Yeah, you can put them on. All right, we're going to put these on. I'm going to use my pot hanger as a lid getter. Put these trout in here. You can already hear them sizzling. Never tried this method before. So we're going to see. That's good right there. As the coals burn down, I'll move it another notch. Remove this string down. And that's trapping all the heat inside. And hopefully it's going to be cooking on the top as well. So never tried this method, but sure, it's going to be fine. It doesn't work. We'll throw them straight in the coals caveman style. Cheesy. You gotta have that cheese in them grits, baby. Just about there. I'm going to turn it off actually. Add a little bit more milk. In my opinion, there's no better compliment to a fish meal. Then some cheesy grits. Some cheesy grits. We have become big fans of cheesy grits. Cheesy grits. If you have a little bit of jalapeno in there, Texas style it, it is. Or green chilies. Green chilies. If you New Mexico style it. It's super delish. And we got to check in on our fish, see how we're doing. I've been kind of just adjusting this stick and string just depending on the fire here. Don't hear any sizzles in there yet. But I am going to set a timer. I'm going to set a timer for 10 minutes and then I'll probably flip them, rearrange them and then put them back on there. All right campers, it's time to check our fish. My duties are done. Your duties are done? Well, you can hold the camera then. Oh, I was waiting patiently. There you go. I shouldn't have said it either.  See what it looks like here. It's pretty good. It sounds like it's done. It smells great. You know what? I'm going to go on a lunch and say we'll leave it in the Dutch. Just let it cool over there by the camper for a minute. And then we'll check it over there. But I don't think it needs any more cooking. The Dutch oven might be the way to go. The way to go. I'm liking it. All right, it's starting to get dark. And inside the camper, it's pretty dark. So I'm going to film out here so you guys can see the fish. The fish.  But I think the Dutch oven did a great job. So how you know if fish is done is if it flakes off the bone. I'm so hungry. I might just eat the skins and everything. These coconut salmon. That skin is peeling off really nice. Man, that. Oh man. Okay. So the heat was just trapped inside of that Dutch oven. I've never done that method before. But it looks like it did an incredible job. Oh yeah. Baby. I want to take a bite of that skin too. Some healthy fats. That's better than trout. Holy cow. Yeah, it's just flaking off. It's flaking. Look at the flake. Grab that piece right there. Put it in the nom nom. Oh man. Let me get that skin. Mmm, that's a pin. Pair this with OSG's cheesy grits and we've got an incredible outdoor wilderness meal. Y'all smash that like button. I'm going to go eat this with the family inside the camper where it's not windy. We've had a lot of bad weather this afternoon. It's kind of unpredictable in the mountains, but it's part of it. Thanks for being here, y'all, and all my outdoor adventures. Subscribe to the channel so you don't miss any more. And I'll see you guys on the next one later. Steph, it's ready. Fish came out amazing.  Take a bite of this. Some own.


So Bitcoin, right now, one of you guys will win right now for a dollar. Whoever gets closest to the price of Bitcoin right now, without going over, wins a dollar. All right, so we're gonna start over here. A dollar, okay. For like one Bitcoin? Yes. And the five grand? 5,000, 10,000. Okay, 10,000. Right now, I'm gonna tell you guys the price of Bitcoin, the current price of Bitcoin right now is $19,517. So you won another dollar. Congratulations.


Let's figure out if we can get to the bottom of this. So I received a great question and it's from someone who's in a relationship and they're wondering why can my boyfriend not show himself emotionally to me, why is he so reluctant to be vulnerable. So we'll read out through this question together and hopefully it can help you as well if you're going through something like this, could be your boyfriend, girlfriend, husband, wife who is reluctant to show themselves emotionally or even if an emotional issue comes up in the relationship they run away they can't handle it they avoid it. So here's the question it says is my boyfriend a sociopath. So I've been in a relationship with my boyfriend for about a year and a half and when we get on really well for the most part there is a major problem. He's so reluctant to show how much he cares for me he pushes me away all the time. I just get the feeling that if I was gone from his life he wouldn't be affected at all. When I talk to him about this I get nothing back he closes up and gets very frustrated and uncomfortable. It drives me crazy that he won't just talk to me and tell me what he really feels. So I'm beginning to wonder if he is even capable of feeling emotion at all. He's nice to me and respectful but never truly sensitive and vulnerable. His work comes before me I think it takes all his attention and I feel that he only makes room for me on his schedule. We are all we are so different I sometimes wonder if there is something wrong with him. Is he a sociopath? Is he capable of feeling emotions? And while I want to make a life with him I don't know if I can continue to be in a relationship where there is this much distance between us. I feel like I desperately need to be loved on a deeper level and without this love I don't know if we can be happy together. Okay so what I'm going to talk about here is it's great that you're asking these questions. You may be asking the wrong question though and what I mean by that is you're wondering is my boyfriend a sociopath. Now the chances are that he's a sociopath to be honest are quite low.  There are psychopaths who are born that way. What's far more likely is and it's going to be helpful for you in this situation is something called attachment styles. So if you can start to understand what attachment styles are it will make your life 10 times easier. Okay attachment styles what do I mean? You can have there are three different types of attachment style and everybody has one of them. Okay now your boyfriend sounds like he may have something called an avoiding attachment style. So that is somebody who in their early childhood they were in a relationship with their parents and as a child obviously you're very vulnerable you can't fend for yourself at all but you reach out to your parents to have your needs met and the parent is not attuned to your needs at all. Okay they and those could be you know practical needs physical needs fine it could be enough food to eat it could be a shelter warmth it could be medical needs it could be anything like that but also emotional needs which is more common. So you have emotional problems or emotional fears or things that go on in childhood you reach out to your caregiver or your parent and the parent does not respond does not meet your needs okay. Worse again they might even ridicule you for having those needs so not attuned at all. So what that person learns then in childhood is relationships don't work okay they have a depth to them emotionally these people with avoiding attachment styles but they've learned that to reach out to another person is pointless it doesn't work and I need to rely on myself. So things you could look for for a person you know how can you tell if someone has an avoiding attachment style how can you tell if you have an avoiding attachment style someone who devalues relationships they're not a priority in life they're very very independent they keep their partners at distance they can let them get a little bit close but then push them away very strong boundaries sometimes they can be mistrustful of people especially people who are emotionally you know show themselves emotionally they are very close to people who are close to them very loyal to people who are close to them so if someone actually does manage to form a relationship with them they're very very protective of that relationship. So under difficulty you know with emotion expressing emotion and being intimate so it sounds maybe possibly that that is what you're experiencing here what you're partner it's not the door psychopath or sociopath or anything like that more than likely if they were a sociopath for instance you would probably know it by now because I think you said you've been together for a year and a half and the chances are if the person was a sociopath they would have shown themselves they would have shown a level of cruelty and the cruelty will be followed by a complete lack of remorse or empathy okay and that would have shown itself by now so he's very respectful he's very you know considerate to you but there is that lack of emotional depth that you're experiencing that speaks to perhaps the avoidant style now there are two other types of avoidant style or of attachment style everybody wants the secure attachment style the secure attachment style is everything's okay in childhood you had your needs but when you reached out to have those needs met or understood they were met or understood with compassion from the primary caregiver or parent okay so you learn that yeah relationships work I can rely on other people I can trust people and then that goes into subsequent relationships in in that person's life if they're securely attached the final one then is perhaps and this could be something that you might want to look at maybe is something called an insecure attachment style so insecure attachment style is kind of between avoidant and secure so in childhood your needs were met some of the time but some of the time they weren't so what happens is you develop a need and a desire for intimacy and relationships but you feel like you can't rely on them okay they're not consistently there for you so what happens with that well what is the the signs of someone who might have an anxious attachment style that would be things like if you're in a relationship you find it very very difficult to let go of a relationship you crave intimacy in relationships sometimes you suppress your own needs you're obsessed with a relationship the partner is always in your head sometimes you play games in a relationship or you might get jealous a lot at a time okay you play games and so like to test the other person to see if they really do love me so again you can see it's all about I need the relationship I want the relationship but I feel like I can't trust the relationship so the point of this video is an answer in this question is to make your life easier in this relationship all you need to do is understand the avoidant attachment style because let's say you have an anxious attachment style and you're with someone who's avoidant you will perceive their avoiding the attachment style as rejection of you and really it is nothing to do with you it's more to do with that person feeling like it's not safe or what is the point of me being intimate I've tried that in the past so it develops a greater sense of compassion for your partner and also compassion for yourself the work for you on a personal level here might be to figure out what your own attachment style is and your work in the relationship is to try and understand what his relation his attachment style is and how that could influence you so in a conversation with him it's probably not going to work if you're asking for intimacy or a level of emotional vulnerability okay that's probably going to be very very slow in coming with someone with an avoidant attachment style what would be probably better would be to discuss with him what his childhood was like what his past was like and help him understand where this is coming from and how it makes you feel there's nothing at all wrong with what you're doing in terms of wanting the intimacy with him because on the deepest level he actually still wants the intimacy there is he's not a sociopath it's very unlikely he's a sociopath far more likely is that he has a depth to him emotionally it's just that he feels like I tried that in the past and his defense mechanism is independence okay I learned to be independent because there is no one that's going to help me once that happens in childhood it's difficult to to change that style okay because it's so deeply ingrained in us but if you're in relationship with a person like this for a long long time and that's what you intend to do you're going to need to understand this because otherwise you'll feel like you're beating your head against a brick wall and it will drive both of you crazy okay so think about that okay what is his attachment style it's never a rejection of me or my love it's more about his coping mechanism his style of coping and how he learned to cope with life and over time with compassion you can start to break down those barriers a little bit but it is important for you to have expectations about what that's going to be like it will be a slow process he's not going to open up automatically overnight it's going to take time and if you can can be can be compassionate with him and yourself in that process you will find that you grow emotionally steadily over time okay so let me know what you think of that hopefully it helps other people too who may be going through this if you would like to work with me you can just send me a message you the link is below and also maybe subscribe to this channel like the video and all those things it helps me reach more people so thank you for watching this video and hope it helps and I'll see you next time


So this is the basic notion seminar, and so what I will do today is to tell you some basic notions of projective geometry, algebraic geometry. So I will not assume that you know anything about algebraic geometry, but I would like to motivate these basic notions by connecting it to an applied problem of the geometry of tensors. So let me start by defining the tensors I'm interested in. So I will in general fix k complex vector spaces. And I consider the tensor product of this vector spaces. So this is again a complex vector space w. And so this vector space is generated by vectors that we call in the composable. So I don't think this is, OK, I can just show. These are what we call the indecomposable vectors. So these are just tensor products of vectors in the VI's. And in general, an element of this tensor product, this vector space, is just a sum of indecomposable tensors. So the minimal number of summons in such a decomposition is called the rank of the tensor. So we have this vector space, and every vector has a rank, which is the minimal number of elements that you, and the composable elements that you need to write it as a sum of, say, primitive or indecomposable tensors. So this is our easy definitions. And so again, let me keep them here. So there are, so the people that study the composition of tensors, they're interested in several questions. So let me mention just some problems that are important in applications. So given, if you're given a tensor, how can you compute the rank of this tensor? Minimum, meaning the minimum number of indecomposable tensors so that you can write it in this form. And in general, what is the generic value of the rank? Meaning if you pick a random element in this vector space, what will be its rank? And, well, other important questions is, when is this the composition unique, and how to actually find, if you give me a vector, if you give me a vector, how can I compute a minimum of the composition of this sort? So these are problems that people study, and they are very important for applications. So I would like to mention some applications of the composition of tensors. So they are very important for computability and complexity of algorithms, algebraic statistics, phylogenetics, signal processing, and many other applications. So I just listed a few. And what I want to do today is to motivate the definition on some very classical objects in algebraic geometry using the problem of tensor decomposition. So I will concentrate in this problem what is the generic rank of a vector of a tensor. So we are going to consider these questions for some types of certain vector spaces like this. So we want to know the generic rank. And I will explain how this can be important for computability and complexity of algorithms. So this is the goal. So I will start with the application, the specific problem of complexity of algorithms. And then we are going to see how ranks come by. And then finally, we will introduce the projective varieties that are associated to this problem. OK, so this is a very simple problem, which is matrix multiplication. So if you have two matrix, A, B, C, D, and E, F, G, H, and you want to compute the product of this matrix. So we learned this in high school. So this is the usual algorithm that we use to compute the product of these two matrices. So if you look at this, you will notice that we have to compute the product of two general matrices. We have to perform eight multiplications. So these are listed here. And of course, when you increase the size of the matrix, then you will increase the number of multiplications needed. And it turns out that this is very caustic computationally to multiply matrices in this form. And then you can ask, well, maybe is there another algorithm that I can use to multiply, say, two by two matrices using less multiplications, even if I have to use more additions? So additions are sort of cheap computationally, but multiplications are expensive. And as it turns out, there are other algorithms to multiply matrices. And there is this called the Strassen's algorithm, which I have put here. And this is why I could not improvise this on the blackboard. So you have to trust me that there are some rules that you do. So we want to compute this matrix here, which is the multiplication of these two. And as it turns out, there is Strassen's algorithms tell you how to do this just performing. So here you're performing seven multiplications. And then by performing the seven multiplications, then you can just perform some additions and subtractions to obtain the product of the matrix. So what I want you to keep in mind is that there is a way to multiply two by two matrices using only seven scalar multiplications. And what I want to explain to you today is, so actually Strassen's got to this algorithm by trying to prove that you need eight multiplications to multiply these two matrices. And then when he was trying to prove it, he ended up running into this algorithm. And what I want to explain to you today is why maybe you should already guess that there could be an algorithm only with seven multiplications. So for that, now I will introduce a tensor, and then we're going to see how to interpret this problem from the point of view of tensor decomposition. OK, so I will consider my space of two by two matrices, so with complex entries. So this is a complex vector space of dimension four. And I want to understand, so my goal is to find out an algorithm, say, to compute multiplications. So I consider this linear map from the tensor product V, tensor V to V, that associates to a pair of matrices, sorry, AB, its multiplication A multiplied with B. So we can view such an operation as a tensor in this tensor product. So this is one way to interpret the multiplication of two matrices as a tensor. And let me make this very explicit. So let me tell you how if you give me an element in the composable element in this vector space, so this means that you give me, say, a linear functional alpha and linear functional beta and a matrix C, then I can compute, I can define the following linear map. So you apply it to AB, and you just compute alpha in A, beta in B, and then this is, so these are numbers, these are complex numbers, you multiply them, and then you multiply by the matrix C. So this is very explicitly how from a tensor here, this tensor here defines in the composable tensor, defines an operation in this way. And now what we want to do is to understand, so if you give me now the, if you are considering now the multiplication tensor, it can be written for some R, for some rank, it can be written as a sum of in the composable tensors of this form. And the question is, what is the rank of T? And roughly speaking, the rank of the multiplication tensor T is exactly the number of scalar multiplications that one needs to execute it computationally, the minimal number that one needs to compute it using R, using scalar multiplications. So now we are interested in understanding this, and now what I want to explain is, so what is the generic value of the rank of a tensor in this particular vector space? And using algebraic geometry, what we will see is that this generic value of the rank of a tensor in this vector space is precisely 7. So this could tell you a priori that while maybe then there could be an algorithm that requires only 7 multiplications to multiply 2 by 2 matrices. So now, so now we, so we passed from the problem of multiplying matrices to the problem of tensor decomposition. And now I want to introduce algebraic varieties in context with this problem. So let me start with a very general definition of the projective space. So we, I will define the projective space of dimension N. So this is going to be the one dimensional linear subspaces of C to the N plus 1. So we consider the vector space of dimension N plus 1 minus the origin and we identify all the vectors that lie on the same line through the origin. So this is how we define projective space and we, so a point, any point in C N plus 1 different than 0 gives a point in the projective space and we denote this point in this way. So this is what we call the projective coordinates of the point and notice that this is not uniquely defined. So whenever you have a point that is given by this coordinates, I could multiply everything by lambda and it still corresponds to the same point. Okay, so these are the projective space and inside the projective space we consider the projective varieties. Also these are algebraic varieties. So these are going to be sub varieties of the projective space, subsets of the projective space that are defined by zeros of polynomials. But now there's only one subtlety which is, well, if you are in C N, you can define an algebraic variety as the zero locus of polynomials. But now we are in the projective space so instead of considering zero locus of polynomials doesn't make sense because, you know, if maybe you have a polynomial that vanishes on these points, but if you multiply everything by lambda, it should be the same point but it doesn't vanish anymore. So this is in general, it's possible. But what we do is instead of taking just polynomials, we take homogeneous polynomials, sorry, this should be just a polynomial ring. So we take homogeneous polynomials and the good thing about homogeneous polynomials is that if it vanishes on some vector and you multiply this vector by some lambda, it still vanishes. So this now becomes well-defined. So if I take, if I fix this homogeneous polynomial and consider all the points in the projective space for which this polynomial vanishes at this point, then this gives me what we call a projective variety, an algebraic variety. Okay, and in general, so I define this, you know, as just the quotient of C N plus 1 minus the origin by this equivalence relation, but in general I can do it more abstractly for any vector space. So if I have any vector space W, I can consider the projective space of W. This is going to be just the space of lines through the origin or dimension one linear subspaces. So this is just W minus the origin, a model of the equivalence relation that identifies vectors that are multiple of one another. Okay, so these are projective spaces. These are projective varieties.  So let me remind you then, okay, we had this tensor, the multiplication of two matrix could be interpreted as a certain tensor in this tensor product, in this vector space. In this case, V and V dual are both isomorphic to C fourth. So this tensor product has dimension 64, four times four times four. And so the projective space is going to be a projective space of dimension 63, always one less. So we have now a point, though this tensor corresponding to matrix multiplication corresponds to a point in a projective space. And now what I'm going to do is, so we want to compute the rank of this tensor. So what first thing I'm going to do is to define an algebraic variety, a projective variety inside this projective space corresponding to tensors of rank one, meaning in the composable tensor. So this is very easy to do. So I consider here the product of this three projective space. So this is P three times P three times P three goes to P 63 taken. So you take points corresponding to this vectors and then you just associate the tensor product. So the important thing that I'm not explaining here is that it turns out that if you look at the image of this map, so this is called the sagri embedding. So the image of the sagri embedding turns out to be an algebraic variety, a projective variety. And actually one knows exactly what are the equations, the homogeneous polynomials that define this projective variety inside P 63. So we have a projective variety inside this big projective space which corresponds to the tensors that have rank one. So this should be thought of as the space of all tensors. And now inside this space we have a smaller variety of dimension three plus three plus three of dimension nine corresponding to the composable tensors. Now what about the ones, the tensors of rank two? How can we define them from this point of view? Well, if you have a tensor of rank two, that means that it can be written as the sum of two in the composable tensors. So this is from the point of view of tensor decomposition. Now if you view it from the point of view of algebraic geometry, what this is saying is that T belongs to the line in the projective space that joins these two points. And then one can go further if you look then of a tensor of rank r, this is going to be in the r minus one plane spanned by r points of x. Okay, so given this let me then define something in general. So if you have any, well here there is this note, not the general, it just means that it's not contained in any linear subspace, but it does not so important. So we have a projective variety of say of dimension n. Then we can define this d second variety of x in the following way. So I take for any d points in x, I consider this, well maybe I should take here for d general points in x. I can take the linear subspace spanned by these d points, so this is a p d minus one if these points are general. So I take the union of all these p d minus ones and then I take the closure. So this is what we call the d second variety of x. And so what happens, so we have the second, so for d equals two this is just the union of all the closure of the union of all the lines that are secant to x and so on. And now we can, we will want to understand and it will be clear why we want to understand what is the dimension of this variety. So we have a very natural candidate which is called the expected dimension that is very easy to compute. So what are we doing here? We are taking d points of x. So x has dimension n, so for each point of the, so for each choice of a point that means that we get n dimensional family of choices. So we have d of them, so this is d times n and for each choice of these d points we are taking a d, a p d minus one. So this is dn plus d minus one. So this is the expected dimension of this variety. So it's expected dimension but in general it may not be exactly, we are going to see an example when this is not coincide with the correct dimension of the variety. But at least no, no, this is sort of easy to compute. And well, of course, this is, this is, if this is less than n, so in fact the right, the right formula is that the expected dimension is the minimum between this number and the dimension of the ambient space. Okay, so now let's go back to our problem. Again, we had this tensor t that corresponded to the multiplication of two, two by two matrices. And we considered the sagra embedding, so this is the, this is embedding this product of projective spaces as a certain variety corresponding to rank one tensors in this big vector space. So this is our x and the dimension of x is 3 plus 3 plus 3, 9. In this case, and then we are going to consider the second varieties of this x. So the expected dimension of the second variety of x is given by this minimum. And in this specific case, it is known that the dimension of the expected, the expected dimension of the second variety is in fact the dimension of the second variety. So if one computes the dimensions using this formula while you get the dimension of the second varieties is 19, the third secant 29 and so on, until you fill out the whole space taking the seventh second variety. So this is what I said before. So this, what this is saying is if you pick a generic point inside this p63, meaning if you pick a generic tensor in this, in the tensor product of these spaces, its rank will be precisely seven, because this is what fills out the whole space. So this is why one maybe should have expected that there could be an algorithm for multiplying two-by-true matrices, making use only of seven multiplications. So this is it, this is how I wanted to motivate these notions. And I would like now to, yeah, so this is what I said, the conclusions that the rank, the generic rank is seven. Now I would like to maybe discuss some other very interesting example. We are going to consider the following space. So we are, we start with the vector space C3. And inside this vector space, I will consider the, well, I take the tensor product. So in this case, if you just take a tensor product of the same space, you can view this as three-by-three matrices. So this is V tensor, V can be interpreted as the C vector space of three-by-three matrices. But I will not be interested in considering all possible tensors in V tensor V. For some reason, well, depending on the problem that I'm working in, I am interested in considering only the symmetric matrices. So this is what we call the second symmetric product of V. So these are just the symmetric tensors here. So in terms of matrices, these are only the three-by-three symmetric matrices. So this is the space that I'm interested in. And again, I want to compute what is the generic rank of a symmetric tensor. And again, we are going to interpret this from the point of view of algebraic geometry. So again, I have a projective variety parameterizing rank one tensor. So this is very, so what I'm doing here is I'm viewing a vector. So I'm taking a vector V in C3. So I view this as a column vector. And if I consider V dot V transpose, this is a symmetric three-by-three matrix of rank one. So in this case, the rank of the matrix is precisely the rank as a tensor. So in this case, we see that so this is a symmetric matrix of rank one. And I can consider then this map that takes a vector to this symmetric matrix. Again, the image of this map. So this is called the Veronaise map. It's also a very classical object in projective geometry. So if I consider the image of this map, again, this is a projective variety in P5. So the symmetric three-by-three matrix is form of vector space of dimension six. When I consider the projective space, so this becomes a P5. And inside this P5, so this is just a P2, right? So V has dimension three. So this is a P2. So the Veronaise map is something that goes from P2 to P5, take the point corresponding to a vector V to the point corresponding to the matrix V times V transpose. OK, so again, this is a projective variety. One can tell, well, it's easy to write down actually the equations, the homogeneous polynomials that define this equation in P5. And it has dimension two, because this is just an embedding of P2 inside P5. OK, now let's consider the secant varieties to x. So if you compute the expected dimension of the second and second variety of x, this is already five. So one would expect that the secant variety to this is called the Veronaise surface is already the projective space P5. But it turns out that this is not true. So let's see. So if you have an element in the second secant variety of x, so what does that mean? That should mean that it can be written as a sum of two rank one tensors. So it should be written as a sum of two symmetric matrices of rank one. But see, if you add two symmetric matrices of rank one, you get something of rank at most two. So being in the second variety of x is equivalent of having rank at most two. So having rank at most two is exactly saying that the determinant of the matrix vanishes. So the determinant of the matrix M is just a polynomial, a homogeneous polynomial in the coordinates of the point corresponding to the matrix. So this is what we call a hyper surface in the projective space is cut out by only one equation. And when you define a projective variety just by using one equation, it's the mention, sorry, it's the mention. So this is what I'm saying. So this is the projective variety defined by the determinant. And that means that the dimension is going to be one less than the projective space. When you cut out by one equation, the dimension drops by one. So you see that the actual dimension is smaller than the expected dimension. So that does happen sometimes that the expected dimension is not the actual dimension. So it is actually a very difficult problem in general to compute the actual dimension of these secant varieties. OK, but see, OK, this is a basic notion seminar. So also I would like to not only, so I use them to define, sort of motivate the definition of projective varieties. But I also want to explain to you why we got our computations wrong in this case. So why did we compute that the expected dimension is 5 and why is it not 5 but only 4? So let me, for that I will have to tell you a little bit more about this veronizing padding that I defined here. So we have this P2, let me again, this is our map. So we have our P2 that by this map V is sent to a certain variety inside P5, our variety X. I think there's only one geometric aspect that you must understand in order to understand why we got the dimension count wrong in this case. So if you take a line in P2, what is the image by the veronize map? So here the definition is here. So if I take a line, so that means that I am taking a vector and I'm varying its coefficient linearly. When I compute the corresponding point in P5, I am going to get, so I'm taking V dot V. So this is going to be not, so a line here is going to be sent to something of the grid 2. So a line is always sent to a conic. So this veronizing map takes lines to conics. And this is enough to understand why we got the dimension count wrong. So let's try to understand this. So let's suppose that I pick the point, I point P in the second second variety of X. So let me erase this for a moment. So what does this mean? This means that it lies, if I take a general point there, it lies on a secant line to X. So this is the definition of the secant variety. Now, so we have these two points here that determine the line passing through P. So in P2, then they will correspond to certain two points here. And I can consider the line joining these two points. Now, if I take the image of this line, this is what we've just seen. They're mapped to a certain conic through these two points. But now let's look at this picture. So a conic determines a plane. So this conic here is contained in determines a certain plane. That contains, of course, this line passing through P. So everything now is happening at this plane P. But now notice, if I take any other line through this point containing in the plane, it will again meet the conic in two points. So a line in a plane, so I'm working over the complex. So a line on a P2, on a plane, always meet a conic in two points. So see what we got the wrong dimension, because when we make that dimension estimate, we counted this point, this point should be counted infinitely many times. So there is a one-parameter family of second lines that pass through this point. And so this is what is causing this dimension here, sorry, this dimension here to drop by one with respect to the expected dimension. So what I'm trying to illustrate here is that the dimension, the expected dimension being bigger than the actual dimension. So this occurs for some geometric reasons. So in this case, the geometric reason is very easy to describe. And there's a lot of questions in general on when is the expected dimension not equal to the right dimension? And in this case, what is the geometry that is behind this? OK, so let me just state now a very general problem, generalizing those that I have described. So let me, in general, I will start with, so this is going to be, this one is going to be a mix of the two examples that I explain. So I will fix some complex vector spaces, v1 up to vk, each one of dimension and i. For simplicity, let me just order them in increasing dimension. And for each one of those, I consider the space of symmetric tensors of order d in vi. So this is the subspace of this tensor product corresponding to symmetric tensors. And now I take the symmetric product of all these things. So this is a mix of the two types of tensors that I described. So there is another important type of tensor, which is the anti-symmetric tensor. And again, the anti-symmetric tensor also corresponds to a very important projective variety. So these are what we call the grassmanian varieties, or you may have heard of grassmanians. They appear exactly as the projective variety corresponding to in the composable anti-symmetric tensors. But for this talk, I just want to concentrate on, so this sort of mixed product of symmetric. So this is symmetric in different pieces. So now I consider this problem. And again, I have a projective variety so I can take. So in this projective space, I can consider all the in the composable, or in other words, the rank 1 tensors. So these are exactly parametrized by this product. So if I take points in this vector space, in this projective space is corresponding to these vectors, I take their symmetric power vi to the di, and they take this tensor product. So this is a rank 1 tensor in this space. And again, this is a projective variety, so this is a mixed situation. So this variety is called the segre veronaise variety. So this is a segre veronaise embedding. And so again, this is a projective variety. And the problem, so this is an open problem in general, is when is the expected dimension equal to the actual dimension? So this is the problem that I have been investigating. And in this generality, it's very much open. So very little is known. So I will tell you now what is known for this problem. So again, let me just, so I'm just repeating the notation here. So I fix some vector space of dimension n i. I order them by the dimension. I consider this space of tensors that are symmetric in each piece of this. So this is this tensor product. I consider the segre veronaise variety in this projective space, which is the variety corresponding to rank 1 tensors. And I want to understand when is the expected dimension equal to the actual dimension of the segre veronaise. So this problem, basically, if I can solve this problem, basically I'm telling you what is the generic tensor for every space, the generic rank or the rank of the generic vector in any tensor space like this. So the problem of computing the rank of a generic tensor of a certain type is rephrased by when computing the dimension of a certain projective variety. OK, so for this problem, let me tell you what is known. So the first case when k equals 1, so k equals 1 means that I'm just considering just one factor. So I'm looking at the symmetric tensors of some order, of vector space. So in this case, I already showed you an example here when the expected dimension is not equal to the actual dimension. And in this case, this problem is completely solved. So this is a very important paper. Alexander and Hyshevitz show in 1995, they have a list. So in the case of when the degrees 2 was in this case, this is always what we call defective. So we say that the variety is defective when the secant does not have the expected dimension. So in the grid 2, this is always defective. And then, except for the grid 2 cases, there are four cases that they list, and that's it. All the other cases, the expected dimension equals the actual dimension. So this is completely solved. The problem is completely solved when you have one factor. Now, if you have more factor, then the known results are some very special cases with few factors. So for instance, only two factors of dimension, one of them having dimension 2 or some very few cases are known. And the first general result, not just when you don't really bound the number of factors when you just allow anything, the first general result was given in 2003. And so Catalizano, Jeremy and Jimigliano show that the expected dimension is equal to the actual dimension whenever for this type of tensor products, so this type of tensors, whenever s is less or equal than n1. n1 is the smaller dimension here. So this was the only general result for any number of factors that was known until recently. And then let me just state a theorem that I proved recently with Alex Masarenti and Ricky Riste. So we improved this bound by showing that the expected dimension is equal to the actual dimension of the second variety. So here, this was a bound that was linear on n1. And here, and this did not take into account the degrees of the symmetric powers here. So here we took this degrees into account by adding this exponent here. And so far, this is the best general result for this problem taken in general, of course. If you look at fewer cases of certain types, then you may have better results. But for general results, I think this is, as far as I know, this is the best one so far. And so I don't want to explain to you the proof, but just by connecting to this, let me just say that the proof is very geometric. And we studied the osculating spaces. So here we were looking, and in this case, things were sort of very simple. But for our proof, what we do is we look at osculating spaces to this project of varieties and compute them explicitly and use them to understand the geometry of the second varieties. So this is all I had to say. Thank you. It's mysterious, right? Yes. Let me say how I can, OK. So this is, I don't expect that this is sharp or anything, but this was the way our proof went. So the proof had, so what we did is the following. Let me try just to give you an idea of what was the best method known before we introduced this approach by looking at osculating space. We had to look, so we had this variety, the variety X whose sequence you want to understand. And then there was a method for computing this dimensions, which was to take, you take general points in X and look at their tangent spaces and look at the linear space spanned by all these tangent spaces and project from it. And then try to understand the fibers of the projection. So this was the known, the way that people usually approach this problem. So what we do is we instead of considering, so the problem of this is that you need to consider many tangent spaces. And things easily get out of control if you have too many tangent spaces, you cannot control what's going on. So what we do is instead of looking at the tangent space, we look at the bigger osculating space. And in such a way that we can, so this osculating space has this property that if you degenerate these points to a single point, the image of the span of the tangent space is still going to be contained in this osculating space. And then we sort of apply the same ideas. So this log 2 appears there because this is, in a sense, it contains information of how many tangent spaces we can deform to an osculating space at each time. And then so if we can say, let's assume that we can degenerate three tangent spaces, let me see if we can, let's assume that we, in fact, we can degenerate exactly n1 tangent spaces to be contained still in a suitable osculating space. And then again, we take n1 points in general position, take their osculating space, and then again deform. And so we do it always by taking n1 of those. And then that's why that log 2 appears there because this is sort of telling us how many times we can actually do this degeneration. Maybe it was not. It appears technically in the degenerating process. So this is a good question. And I don't know. I haven't looked at them from this point of view, but maybe they could tell you something. I really don't know.


Welcome back to the channel Last we left off we had created our post type of villages and that's what we'll be storing information in our custom post type and right now we still have a couple of issues coming up when we try to add a new village we see that it still says new post here and we have taxonomies that will be coming up there not yet here so we're going to build on that today so what we're going to learn in our lesson today is how to change these particular elements here to change them to be called villages also just like our custom post type we're going to add a number of labels and then we shall advance to add on some taxonomies today so let's go into we see if the labels that are growing will be having more information coming in our array here so what I'm going to do is I'm going to spread out this particular piece of code and I'm going to use the basics of PHP that is I'm going to use what we call variables and that is I'll have a variable that starts with the dollar side now give it a name and call it label and this will store our labels array right here so I'll just cut this from here test it here of course terminate that and then I'll allow the label to be referenced here which will allow us as we add more label fields into our element here everything will be a lot cleaner so just run this here just to make it simple and easy and then what I'm also going to do here is I'm going to get this array of items right here and I'm going to call it arguments for our sexual also run it into a variable and then terminate that here and then I'll be able to reference it here as arguments so we're just going to do a little cleanup here and then this and leave it on one line if I save this I'm actually go back to our browser refresh everything is still fine nothing is broken the next thing that I'm going to do right now is that you'll find that I'm repeating most of these texts so I'll also just save them in a variable so I'll have a plural name which I'll call villages and put a semicolon at the very end and then I'll just duplicate this to have a singular name so when we save that I'm just going to get the plural name to go wherever we have villages here and just save that and then I'll just get also the singular name to go where we have the village here so save this always test your code see that nothing is broken and yeah no major changes in here so quickly just place those in and then I'll explain what they do actually mean so I'm just going to get those from code snippet that I did here once I save this I'll just explain this so in our labels we've added our name which was already there we have a singular name for the post type we have a menu name what will appear in the menus will be for the parent the parent name and then we have all the items will be called by the plural name which is villages whenever we save view or view item we'll also add that to that whenever we add a new item we'll see this coming up we'll see the edit whenever we click edit or edit item we'll see that so I'm just going to reload here when we reload we see that we'll add a new village right here when we go to the villagers section we're able to see village add a new village this would be very bad grammar not village found but we can actually just correct that by coming here and saying no village found so we'll just come here and reload here so we have no village found we have this if we try to search we'll probably get the no village sign so I think this is a very good step for us when we come here we can add a new village in our menu and if we go into our menus we're going to be able to find the villagers I'll just go to the screen options to see that we also have the villagers tagged in there so once I come here we see that we can add villagers to our menu let's say if we had content and this is made available by what we add in our arguments here we say we shall add it in the navigation menus we shall add it in the menu we shall add it in the administration bar and then we shall we shall allow the hierarchy for formatting like there will be a level of hierarchy for this particular post type then we shall also allow it to be exportable by the users if they want to export the information we'll also have it archived we shall allow it to be searched we shall give it a false query in terms of search but we'll also allow the public to query it publicly anyone can query for this information or we can write in our theme then we shall give it a menu position of 18 which is about just below the media section and above the pages so while we are talking about this particular menu position I am thinking you might be wondering how do I get this number it's not something that I guest up when you go to the docs and you look at her the add menu page functionality in WordPress it also requires a particular position here as a default argument so quickly if you scroll down to the end if you scroll down you'll find that there is a menu structure of the numbers that have already taken up so you cannot apply it to because that's a dashboard location post takes up number 5 10 is for media links is at 15 and then we have pages at number 20 so me having a number between 15 and 20 is actually very ok if I want position it that way and that's why I want to put my villagers if I wanted to put it under pages or comments I'll put it between 20 and 25 and the same goes for all the others so if you don't add a menu position in your code like right here if you don't add a menu position then automatically your post type is going to be pushed at the end just after settings then the next thing we shall do is that we are going to give it a very nice icon which is a palm tree icon coming from dash icons that is a default font that is in a classic press or in wordpress so we have our palm tree coming up here remember it was before we had a pin just like the posts and by default we get that pin however we've changed it to the palm tree so to have all that information there so I would say most of the information we need for our post type is actually done the only thing that I would like to talk to you about is the supports when you add an argument of supports you remember whenever we come to our if we go to I'll just add a new village whenever we go to our posts we have a couple of things so this is our title, we have the editor here we have custom fields showing up here we have discussions and author and a number of things that we can turn off in our screen options if we wanted to explicitly say you know what we're not going to support maybe the title and author and custom fields so I'll just take out the comments custom fields and author for a start to show you what I mean so if I save I'm explicitly saying this particular post type of villages is not going to have this information however it will have the title it will have the editor and it will have the thumbnail which are already here so if we go to our screen options we'll find that we only have just the featured image showing up only so we can decide to say you know what we also don't need the editor for now so maybe we'll just take it out and maybe at an eventual date we'll probably want the editor so if I save this and come here and reload we're going to see that our editor is gone so just have the village title and we have these items here so I think we are good to go now we shall be looking at the taxonomies and that's basically what we have in post like categories or tags you can register your own taxonomies and we're going to actually register something that is new we shall register our districts we shall register our sub parishes and all the other information and we'll make it available all this information that we've been putting here can actually be found when you go to the codex of wordpress that is codex.wordpress.org and you look for the register post type functionality so you'll be able to see what post types are reserved automatically by wordpress and then you can see this particular labels that we actually using here and that you can set up there are more that we haven't added like uploaded to this item ABC and you can actually look this up and find out more information about what we've been discussing in this particular tutorial and this particular chapter of our development so if you like this video please give it a thumbs up right there in the video or if you have any questions please feel free to drop them in the comment section and subscribe to the channel I encourage you to subscribe because you don't want to miss out on updates as we go in our series or any other series that we shall be starting up in the future thank you for watching have a good time