What's up guys? How's everybody doing? Marco's all over the place today. Pay some nice gains off the spy puts. Nice, nice. What's up? What spy puts did you get? What's up Jawa? Just messing with the mic right now. How's the, how's the volume? You guys hear me okay? All right, I got the mic set up. 435, 1011. Yo, risky, dicky. That's today. That's today. Super risky, dicky, but it looked like it paid off. We hit 434 at the end of the day. Huge move to the downside. What's up, Kip? Spendo, how are you doing? Let's take a look at the market today. It's kind of weird. FCX moved up today. It's probably a shortening opportunity here because if the market is going down, the dollar is going up. There's no reason for commodities to be up. So that should be going down. When group is seeing some recovery, oil as well. It's kind of odd in that sector too. A lot of mix man, a lot of mix. So a lot of things that are down, a lot of things that are up. Southwest airline got wrecked today with like their workers being off and stuff. They had to cancel a lot of flights. FT going downside. T-Mobile continued to die and Gap as well getting wrecked. Wow, AT&T got smashed today. Amazon too. So we saw a lot of good setups on Friday. A lot of bullish setup to the upside, but then they got wrecked man. They're going downhill. A lot of the financial sectors seem like a reversal to the downside as well. Let's check out the earnings side to see if we have any earnings for today. What's up? Good afternoon, Mario. Horace, how you doing? Depressing? Just buy puts then. Buy puts. Short the market. Short the market, Spindle. Don't let depression get to you. Beat the depression. Oil is still dead well. Interesting, interesting. Let's check out the earnings. No earnings today. It's Columbus Day. Happy Columbus Day. Hope you guys, if you guys have kids, probably have some good time to spend with your kids today. Beat your kids around too much because the market is pooping and stuff. Got some big earnings tomorrow. Got Fastano, which is like in the construction industry. And then we have smart global holding. It's like a chip sector stuff. Yo, what's up, fiat world? And then on Wednesday we got the banks starting out. So we got JP Morgan, Delta, BlackRock, first Republic Bank. So you got some of the banks heading. Got Bank of America on Thursday. Wells Fargo, Citibank, Domino's looking for recovering that. Morgan Stanley and TSM. So this week, I will say I'm probably most bullish on TSM and Domino's for some recovery there. Close out a lot of stuff in profit. Congratulations. Good job. Yeah, I was a little worried. I mean, I start some long positions on a late Thursday and Friday morning. So I don't know if you guys are here on Friday. You know, I was like having a little bit FOMO. I was like rushing to open put positions because I've shared this chart a few times on like Friday. So I was like, I was having some FOMO on Friday opening put positions because I was scared because we had this rejection, man. Look at this trend line, right? Bam, bam, smack, right? And then some more smack here. And again, again, with the other smack, you know, if spy was a girl, I would say she liked getting smack. That's true. TSM could potentially turn around the chip market. So, you know, we've seen that reversal, like a lot of rejection here on the side. Like smack it like an old money. So, you know, we've got a lot of rejection. We're finding some support in this range. So like, if I just extend this line over, right, you can kind of see a finding support there. So this line is like the top from October 3rd and October 5th. Those are, so you can see us finding some small support there.  So we'll see how that holds. But if it doesn't, like, if it continues to break under there, then we could potentially see 426, you know, on spy. That's an ideal target. I don't, I mean, it's an ideal target. I don't really want it, but I'm okay with it. You know, I'm okay with, like, getting it. So I hitched a little bit. I was having like massive FOMO. So if you guys saw me on Friday stream, took me like 10 minutes to start the stream because I was having FOMO. I was like, literally trying to buy the put contracts for spy on Friday. And that was to add a little bit today. But it was a spread. So it was like really hard to fill since it was a spread. So a spread is two contracts. And it was tricky, man. Oh, shoot. Yeah, well, you went to the restoration hardware restaurant. Which one? The New York one? That was talking about. In Manhattan, said food was meh. Oh, in Dallas. What'd you get? What'd you get? Lobster roll and truffle fries. Dang, I thought you would get the steak, bro. Lobster roll. But yeah, I feel you. It's like, it's probably not like, it's probably not the best place for food because the prices aren't even that high for the food, honestly. But the environment, you know, like, it's nice. It's all right. At least you can check it off the list. Be like, I've done it. The ribeye was like 35, right? Oh, you don't do ribeyes? How come? It's too fatty? I feel you. I feel you. Sometimes like, a lot of people love ribeyes, but I'm not, sometimes I'm not a big fan of ribeyes, too. I'm like, I, I, typically, if I had a chance, I usually like, get New York strip. A5 beef, bro, if you're ever in it. That's true, bro. If you really want fat, get the, get the A5, get the wagyu. Yo, you guys ever hear like, people eating so much wagyu that they have stomach issues? You guys ever heard of that? Because, because one time we went to this wagyu place in New York, right? And oh man, I don't know how much we blew. I think we blew like, I think we blew like $600, $600, $700, right? On a wagyu estate.  Like butter discusses me, you know, I guess it's like some of a mentality sometimes, but butter discusses me sometimes. So, but there's a few exceptions. Like I would, I wouldn't mind butter on the lobster roll, but just butter on the side would discuss me. So, but anyways, we've got like so much wagyu, man.  And I'm just telling that in like a few pieces, I couldn't stomach it because it was just so fatty, man.  Wagyu is just like, it's like melting the piece of butter in your mouth. Oh, it's like, it's just like fat, man.  Fancy name, like wagyu, melted butter is fire. So I guess to eat their own man, to each their own, right? So after eating like three, four pieces, yo, I could not, I could not do it. I couldn't do it. Like, I couldn't, I just couldn't, man. I was like, yo, just give me some fries.   Like they're like, they're like, yo, I think we got food poisoning. I don't know how we can get food poisoning from like a $700 steakhouse, but we got, I think we got food poisoning. Everybody is telling me like they're messed up. Like they have stomach issues and stuff. They're like, yo, they're like, yo, are you okay? I'm like, I'm good, man. I don't know what you guys are talking about. We went to a fancy restaurant, everything looked legit. I don't know why you guys hurting and everybody is like, everybody's like dying, right? They're like, they're like lying in bed. They think they got food poisoning and they got like a food virus because they couldn't move their stomach hurts and all that stuff. And then apparently, we learned that later on, like apparently we learned that later on that if you eat too much fat, your bodies can't, your bodies can't handle it. So I think one of you guys said here before, right? Must be American thing, wagyu stomach aches. Enzyme, it's Lou for your colon. What's this? What is it called? Yeah, you got to try, you got to try some wagyu, man. Just shove it. It is, right? Hungry. It does get, it does get signalling. So you got to take this ahead of time. Is that what it is? But like, why do they serve you so much wagyu if they know people can get sick from it? Like, don't you like, I don't get it. Like, like they know they're serving like 99% fat. Why don't they just at least like warn people, let people know. That's so weird, man. Oh, that's true. That's true. Interesting. If only we talked to you guys before we went there, it would have been a whole different experience.  What's up, Ravi? What's going on? Check out HYLN. Sure. Let me bust out the list. All right, so setting up the chart list. This is the wake of October 11. So if you guys have a stock that you guys want us to check out, let us know. Put on here. We'll take a look at it. And then I'll share the spreadsheet with you guys so you guys can check it out later on as well. Shall we start with spy today? I feel like the spy chart kind of belongs on this. Let me put the spy chart. What's up, Ian? What's going on? So we got requests for Nike, requests for what's that? IONQ? Yeah, a lot of blood today. A lot of blood. Of course, Sully, you can edit the spreadsheet. The spreadsheet is just for you to copy and paste the link. It's not for you to edit. What if somebody's come here and they just go like this, boom, and they delete everything? Be so silly, man. Let's check out the first one. Where did my Bazingo go? So you got CCO, we got chewy. Is the music just changing by itself? All right, let's check out HYLN. Check out this Highland Company, man. We're, oh my gosh. Oh my gosh, look at HYLN. Oh my god, Kip says that it's not even a bag anymore. It's a tired dumpster. Damn, Ravi said he got smoked in this. Who hasn't kind of smoked in this, bro? I feel you, man. I think everybody got smoked in this. All right, so we got some support over here. It just broke too.  I can't even hold the support, man. Because like, I got the entire stat. I don't know how I show this, bro. Oh my gosh. Why did it drop so hard today? Insider selling three electric vehicle stocks under $10 that Wall Street Bits Predict will rally by more than 55%. They got it downgraded to $5 from some firm. Goddamn. Let's check it out. Oh, it's not bad. I thought it was like $5 already, boys, but it's still like $690. Yo, I thought it was like $5, man. I thought it was like dead. I see it. So this morning, HYLN shares the trading lower after UBS downgraded the stock from neutral to sell, and lowers price targets from $14 to $5. Oh, why? Why?  I still have like 200 shares from like $10 cost average a while back. And it's the name of the game, man. A lot of these special acquisition stocks, they've been going on these downhill moves, and we lost that big support. So this was the support that we held from April. We held it again in May, and got blessed back to $13, and then we came back and we held that support again in mid-August. But now that we're finally breaking under that support, it's going to find the next leg down. So kind of like over here, like last time when it found that support multiple times in that range there, and then it finally broke down. It's, you know, it next leg down to the next level. So I will say like, I will say honestly, bro, Ravi, your best case, like depending on how big your bag is and how heavy it is, and how much you believe in the company. If you believe in the company, you like the company, and you like, you believe in their vision. Your best case here is wait for it to find the bottom, wait for it to, you know, find some support. And by finding support, you want to see like, you want to see it drop to the point where it doesn't drop as much anymore, and it sets up like a little bit reversal pattern like it did over here. And then you just dollar cost average, man, dollar cost average, and add a little bit of bigger position. And, you know, as long as you believe in the company, you think it's going to do well and come back up. That's all you could do, man. But in terms of technical wise, I think it could possibly do the same thing that it did in March. So be a little careful. Be ready for that. Alternatively, you can also sell cover calls against it too.  That's all I can say, man. That's all I can say for it. Looks like I could do the same thing it did in March 4. Good luck, man. Oh, we both doesn't allow cover calls.  I got you, my brother. What else we got? BV. Oh, BV, that big dumpster. Yo, wasn't BV the socks that started like this whole downfall? Like, I swear, I'm pretty sure it was the BV IPO that started the whole, like, downfall of like the China stock market. Like, it started the whole downfall of all the China stocks, if I remember correctly. Kips it because they took a big short position properly, realizing they couldn't make money. I don't think it'd be long. Yeah, next earnings, they got to show what's up. Check out Nike next.  Yeah, that's true. That was the other time when all the China stock fell. Party City stock holder. Welcome back. Indeed, indeed. Hopefully Uber could use the help too, because if DD turns it around, Uber would go up too, because Uber has a, has a decent sized stake in the company that they exchange for a while back. So Nike, um, ever since the news like of the, um, of the fact three and the ER situation came out, they've been on this downturn downward slope. So a lot of rejection on this trend line here. So we've got three rejections already. We had a nice bounce off the support here. So I think that that was like one of the gap areas too, or prior, like the after gap area, if I remember correctly, something like that. Oh, this line, this line is drawn from the, not from the gap. This is, this is a huge gap right here. It hasn't been filled yet. I'm still waiting for this gap to fill. But this line is actually drawn from January, February, February and March. So it's drawn from the tops like this quad. What's five? What's the, what's the, what's the word for like five X? What's after quad? I don't even know, but this like five topper, this five X top, you know, over here. So here, here, here, here, right? So didn't even draw this here. Quinn, this, uh, Quinn top. You know, thank you for that. Penta, which one is it, man? Now Penta is like eight, right? Isn't it? All right, more. Oh, Penta is five. You're right. Quinn duplicate. It would be Penta. Yes. So, so then anyways, you know, we got, we had that top there. So that was like pretty good resistance a while back. And then we gap above that resistance. So it kind of makes sense a little bit that when the stock came back down, it found support at this, you know, at this hardcore resistance that was here for a while back, you know, so the next level I'm looking for it to break is that if it gets the rejection from here, if it hits here and then just kind of rejects, I don't think that support is going to hold anymore. I think that support is going to break through. And next level I'm looking for is for it to come back down and fill this gap all the way at like 135s. Makes me almost kind of want to get like, like maybe like $140 puts at this range for like January, because I feel like it's bound to fill, like that's three months, you know? Like it's bound to fill in like three months, I think. That's, I feel like there's enough time, man. Let me see. Let me see, man. How could it not? What is my target? Target is 135. You open a position at 147 at long? Whoa. So it says if I bought a put, it would cost like $1,100. 82% return on the risk. If I do a bear put spread, I could get 128% return on the risk. But it's hella expensive. That means I would probably want to short it. Why he says it would short it for 160? I don't want to do that. Short it from 150 to 135. Yeah. It would cost like 4.95. It just goes to shows.  Like you still need a human perspective. I remember you asking me about Nike last week. So that was a good long from this range. You long from this line too, right? Because I remember you asking me about it. So that's a good area to long. This is a good area to take profit. A 153 area particularly is a good area to take profit.   You know, RSI was on the bottom. I think I mentioned a little bit about that too. So that's why sometimes you don't want to short like a dead cat.  It's what you call the dead cat bounce. So 153 to 154 areas, a good area to take profit. Could it go further? Yeah, probably. But you know, the rest of the war isn't there anymore. So that's the chart for that. I hope that helps. Let's take a look at IONQ. What is this? A lot up down. Did they just IPO or something? No, they didn't just IPO. This is a special acquisition stock that just finished the merger with a special acquisition. So Ian, you know how before we just looked at this other stock, Hyland, Qualcomm, I got you here. Add it to this. So despite the good stuff like quantum computing, track records of innovation, deployment, a lot of good things here, man. This ideally would be the investment of the future. This is like based on the wording, based on the things they're saying. I don't see how this wouldn't succeed in the future. This sounds like something that is like, this sounds like one of those Motley fool videos. A thousand dollars. If you had just invest a thousand dollars in IONQ, in 10 years, you would have a hundred thousand dollars. That's pretty much what I think of this company. It's cool and stuff, but the only thing is, the other thing is, I do know that a lot of these special acquisition stocks, there's a lot of BS behind it. There's a lot of like big projection, you know, that like they tend to over project. So a lot of these stocks that comes through the SPAC mergers, the SPAC mergers, they usually have a valuation of five years ahead already. So let's say they just finished this merger and acquisition in 2021 of October, right? They have already priced in at $10 price point, how the company was going to do by 2025. So investing at that $10 mark or whatever mark it was prior to the merger, that's already like a very high price point. I think I, from what I've seen, from what I've seen, a lot of these companies typically go back down to around $5. So I wouldn't be surprised if a lot of sure sellers are hitting it because it's a very successful strategy. If you look at it, almost every stock that finished the SPAC merger, almost every single stock that finished the SPAC merger, it's done this exact pattern. I kid you not, it's done this exact pattern. Spike up before the merger to $11 to $12, in this case $13. Spike up and then came back down. It's almost so consistent that it's like, why isn't anybody shorting it? So like another example would be like Money Lion, they just finished a merger recently. This is a great example to that theory. So they chopped around $10. Prior to the merger, they spiked to $10.60 to $11 and then it came back to the $5 range. So you're smart. You obviously noticed it. It's a common theme. So that's why I would always advise a lot of people to be careful of the SPACs. A lot of these companies are interesting and they could be potentially very good companies. But just watch out. If you like them, try to pick them up around that $5 level. Otherwise, just be a little bit careful. If you want to risk playing the short squeeze, just remember, it's a very risky thing. So you can draw the trend line from the top side here. If you want to look for breakouts or areas to take profit because, you know, consider this top line as the resistance line. So if it makes a lower low and they come back here, that's probably where you can take profit. So I'll put that on the list here. I hope that helps, man. CCL, let's take a CCL next. Because at the end of the day, my goal is to help you guys preserve capital, you know, so that at some point or another, you guys will eventually be very successful and make money. So one of the things, you know, I'm trying to do is just educate people on SPACs and like, you know, I don't want to see people get wrecked. I don't want to see people lose like money pointlessly because first for some of you guys, money is not easy to come by, you know. Right, so we got to preserve this capital, continue to build the wealth. All right, spend out, take care. Indeed, kid. So Coronable has been having issues, you know, like personally from the valuation perspective, I think they belong somewhere between the 10 to like $15 range. That's why I think they're worth from like fundamental and valuation approach. I think that's where they're at. Obviously, the stock is much higher right now. There's a lot of news going on. There's a lot of, you know, different factors. There's the whole situation with the whole COVID vaccine and everything. And that's been driving it up and down. So we had a, we saw a huge squeeze in a lot of these these reopening stocks and things like that. But even with airlines right now, we're seeing that not, they're not as bullish that people think they are, you know, they're not only getting squeezed by the COVID situation, they're getting squeezed by the labor shortage, you know, so it's just so many factors, man. Like they just can't catch a break, man. Can't catch a break. But let's forget the technique. I mean, let's forget the fundamental side. Let's talk about technicals, right? The technical perspective, we have a rising, you know, rising trend line here. If you're bullish, this would be the area to buy. If you think this is gonna, you know, if you think we're gonna see some recovery in the market, or we're gonna see some money rotate into the travel sector. I mean, winter is coming. So I don't know.  And we could see some recovery here. So watch for that trend line. But if we break under the trend line here, so this trend line is at 2340s. So if we fall under 23, chances are we're gonna climb back down again. And I think naturally at some point or another, I believe we're gonna climb back down to that $13 level. I don't know when, but I believe that the market itself will eventually find the correct valuation for this. So that's like carnival 23.4 area to see some support near the 23.4 area. If that doesn't hold, good luck with that trade. I hope that helps. Check out chewy. I think Spindle is looking long on this position. She's praying for some recovery. She's looking for that candle to hit the VWOP. But it's breaking. It's like it's breaking a little bit on the support here. So looking a little bit iffy. Chewy has been dying for, for God knows how long for like a whole month. Chewy's been dying since September. After that last earning hasn't done that great, couldn't find support at all. So it's just been running down and down. I think the only that sets really could save Chewy is, uh, next earnings, man, um, in December. That's probably gonna be the next catalyst. As we break down the support, I think it could potentially keep going. Next level is, I don't know. Next level, got small level of 59. Maybe a 51s. Shoot. I hope not. Spindle is gonna get wrecked if this happens. But this could, there's a possibility. DCA, FFD1. That's all I can tell you, dollar cost average FFD1. Next FedEx. Check that out. I thought FedEx was finding some recovery last time, but no recovery. Um, very hard company to draw a support line at because there isn't really much support between this range and this range. Since it just like, just, just, it just like started running like a dollar, $2 every day. So it's very hard to find support when it does things like that. Um, one of the next level we're looking for is maybe like 207 and 188. There's really no reason why FedEx was selling off so much because they had a pretty decent quarter. Um, they said, you know, orders are coming in like crazy and things are still very, very hot. The only problem is they don't have the workers to fulfill their needs. So that's why it kind of dropped a little bit. Let me see. Has a, okay, RSI is already high. So chances are it's probably gonna next leg down at this point because RSI is already getting topped out. If RSI was like already beat up, there could be another possibility for a bounce, but it's probably gonna set up for next leg down. Are you guys bearish or bullish on it? Um, what was that CP? OBLG. What is an OBLG company? Multi-stream collaboration technology and managed service for video collab and network stuff? For Cisco? Are they looking at stocks? It looks like they're looking at stocks, yo. What is this? Ethereum? Kind of weird. It's an interesting company. Um, not much short right now. Only less than 1% shorted. Price of sale is 2.5. Tangible book value is at 39 cents. Um, it's a tech company. Valuation could get crazy if it wants. So it got a little bit trendline right now. I think you could see some buying opportunity at $1.48 if it continued to value the trendline and keep bouncing off the bottom of it. Uh, but if it breaks, it can break back down to like 90 cents area. Interesting company. Next support, $1.48. Resistance to $18. Hope that helps. Alright, let's check out DD. What was it that asked about this DD? Oh, I mean, I mean, look at it from this perspective, right? At least DD is finding support at the $8 level. Oh, not $8. This is what? $7? $7.20. At least DD is having support at $7.20. So it's finding some support there. Um, as long as that holds, you're pretty good. Um, so I got this little range here for you. I keep buying the dip. So Dave, it was Dave in the ad phrase. Like keep buying the dip on DD, but hoping it's done the dip. I feel you, brother. I feel you, man. That's all you can do if you believe in the company and you like the company. As a long term investor, I would advise you to keep averaging down, you know. Um, but yo, definitely, please, please, man, definitely take some profit if it comes back to this $10 area. So I decided not to trade Chinese companies and also blacklist the MU. Dang, blacklist the MU, yeah. I feel you. Sometimes that's what you gotta do, man. I mean, there's so many stocks to trade. Why, why, like, why bother, right? Why bother trading the stuff that you don't like? There's so much opportunities. Like the selections are endless, but DD looks like he's been doing this move pretty well. It's finding support on the bottom of the line here, which is the $7. And it's been having a little bit trouble breaking above the $10 area. So if you got, you know, if you cross average between sevens, eight dollars, and it comes back to the $10, whatever shares that you bought down here, sell that up here, you know. And, and then if it comes back to like $18 and you have shares from like $11, you good. You can hold on to those. That's how I would play it. Don't, you don't have to be out with everything, you know, at like 12, just like sell the ones that you got at the lows at this level here. Because it looks like this is some of the level that has been struggling at the last two times. So chances are, it's a plus. Second 20. Oh, you have every call options that you can cross everything down. And the S, which is thought on SOLXL, semiconductor ETF. So we've got two triggers. We're going to look up now, look at Qualcomm and XLSL. So anyway, that's the thought on DD. I hope that helps, man. Good luck. Oh, snap up to 40. Shoot. What's up, turbo laxative. I got you. Last one, man. Last one. So you made it. That was going to be last picker for the day. I'm going to look at SOLXL and Qualcomm. I mean, in a way, I feel like we didn't drop enough on the semiconductor ETF. Kind of like the Q's. I feel like semiconductor ETF has been like, hasn't really had that big of a drop while the whole market was dropping really hard last week. So we could potentially see some move further down. Over here, we did lose that yellow line support. We tried to break back above it. It didn't happen. Coming back down a little bit. So if market remains bearish, it would most likely come back down a little bit further in terms of semiconductor ETF. But we do have catalysts this week. And this is going to be the big one because this is the mother of all semiconductors, right? TSMC. They're like one of the biggest chip makers in Asia, if not like the whole world. So they're going to have earnings Thursday morning before market open. And curious to see what they say, what their guidance is, what the shortage is looking like, what's the backlog? And what's the progress on their factories and how they're going to deal with this whole situation? So if what they say is good or improving, if they say, yeah, I understand we have shortage, but we're dealing with a situation where we have been able to make our output more efficient. We're doing a great job with it. And we're getting Apple and AMD's chip order out as fast as we can. And everything's on schedule. Then always well, always well, right? If they say things like that, then always well. But if they say like, oh snap, we're screwed. We have like double the increase of back order and things are so backed up now. And we got labor shortage and China is about to destroy our factory. We don't know what to do. It's over.  It really depends on them. It really depends on what they're gonna say. Yeah, that's true. And there's also the whole, you know, whole shortage over there too, right? So the whole like energy crisis. So it really depends on them. I think it's not a bad idea to hedge because chances are it's not looking good. Like chances are it's not looking good. I probably lean a little bit more to the bearish side. Even though I did long some TSM, I was looking for earnings run up. But things look a little bit scary, man. You know, so there's that. So this is not exactly SOXL. This is SMH. I picked SMH because SOXL is a three time ETF. And sometimes three time ETF has natural decay. So the chart is a little bit harder to chart out. This is the regular one. So it doesn't have that decay. So I hope that helps. Let me look at Qualcomm as well. But I would probably say the answer is very similar, you know. Qualcomm is also another semiconductor company. You know, we're at a good level. We're at a good level. So it's like, I guess you can even consider this whole area of the manzone, right? The manzone is where people could potentially buy the stock, right? Where people look for opportunity to buy. I would say this whole area is the manzone right here between this 121 to 125 area.  You go long here. If you're bearish, you go short here, expecting a breakdown. So it really depends on how you feel, whether you have, you know, maybe you guys have the insider information that you know what's going to happen with the semiconductor company. Or sometimes you just got to go with your guts, yo. Just got to go with your guts. If you feel good about it, go for it, you know. Impulse decision. But I mean, I would also say that if you're going to go long, please, please buy enough time, please go like one month out, right? If you're going to go long here, please at least get like November, you know, calls and stuff. So you don't get burned as badly. I'm looking at the stock and 1119, 140 calls. We're in reasonable bounce because if it recovers, we're probably going to come back to 138. So it looks good. You look good, man. It looks good. And like I said, multiple times before, man, fortune favors the bull.  It's going to pay out good.  I think we were looking at SoFi last week. Last week, October 6th, we looked at this, right? It was at a good area. It was sitting at the support, you know, on Friday, it came back down at the support of the trend line somewhere around here. Let me see, what did it hit on Friday? So Friday, you know, came back, hit the support around here, and this trend line different. So, you know, we came back, hit that support, and then today, beast, you know, like just big, big deal, don't move over, right? So it made that beast move up. I was too scared to do it because market was kind of, kind of bearish.   It was a good buy at the support area. And, and obviously it bounced off the support area because it was a good area to buy, but I just didn't feel right with the market sentiment. So I didn't get in, but we did talk about the play. And if you guys got in, they paid off well. Now I'm looking for a firm to come back down. Maybe we can come back again. If a firm can come back again, I definitely get in 110, 108 area. Come to Bobby. Check out the last one, Val. What is Val? Iron company, right? Iron or? I remember correctly. So I had the trendline drawn already. This is the support I drawn all the way from January 2020. So, so previously it was a resistance, right? So if the stock price is above the pre the line, then the line is support. If the stock price is below the line, it's a resistance. So as of right now, the stock price is above the line. So that means this is a support line. Now this line up here was a previous support line here, but now that the stock price is under, this is now the resistance line. So as you can see, it's played out pretty well. It bounced off this line that we drove from last year. If you're bullish, look for the move back up here. You know, the test at $16 range, but a little bit iffy right now with the whole dollar move and things like that. So if you think the dollar is going down, then the whole mineral sector is going to go up. Everything from copper, iron, steel, gold, silver, corn, wheat, sugar, lumber, natural gas, crude oil, everything, right? But the dollar remains strong, you know, causing a deflation, everything's going to go down. So, I mean, and obviously the other factor is the commodity issue, like shortage of supplies. And a lot of these mining companies are having a problem with that, especially with COVID. A lot of these mining companies are getting wrecked because I've been investing in mining companies. I'm trying to catch the move back up, but a lot of them are getting wrecked. I've been following the earnings reports and a lot of these mining companies aren't making as much money. They aren't finding as much minerals because COVID is mandating them not to work. It's not allowing them to mine. So it's a little bit iffy, man. A little bit iffy there. Yeah, naturally, the dollar should be dropping naturally. So a good area here, $13.57. Where did you get in at? Gold, silver, et cetera? Damn, so you're investing a lot of commodities then. Yeah, I'm a little bit worried. Let me show you something. I think gold and silver is a little bit okay right now. But I think a lot of these other mining companies like copper and iron, I think they might be in for like a little surprise. So I think we might be looking at something like this. So you see like over here, go hit that top a while back at August 2020. It's been on this downtrend move, right? Kind of in this downward pattern because the expected move has already happened.  Naturally, these should be going up a lot higher, but they've been keeping this down, pushing down. So we're seeing a lot of moves down, a lot of the commodities lately. So that was gold. We're looking at silver right now, copper. Copper is still like pretty strong. So that's why fcx is still pretty strong. But if you look at some of the other commodities like soybeans, they've been going down slope. Same thing with corn coming on this down slope, sending up almost a head to shoulder pattern.  So it's about to get a smack down here. So I think, I think a lot of other, the commodity can potentially follow the situation where it's gone up so much, it could have a little correction coming back down. So a little bit scared on that, but ideally they shouldn't be. Ideally they shouldn't be. Ideally they should be going up because dollar is losing value. If the dollar is losing value, commodity should be more valuable. That's pretty much it. I'm going to move in some water. I'm going to end it here for today. We'll be back here tomorrow to check out some of the other earnings reports like smart global holdings and fasteno. And then once we've got some interesting stuff, man. So thank you guys for stopping by and hanging out. And have a good night. Take care.


Now we're going to look at the maximum height. The dolphin will reach its maximum height at t on f divided by 2, halfway through the total time it spends in the air. This is because it'll take the dolphin just as long to go up as it will to come back down. So now we can use the equation for the vertical position to find the position at the point in time halfway through the jump. And this vertical position will be the maximum height of the jump, substituting in our values and cancelling our units. We find that the maximum height is equal to 0.957 meters to three significant figures.


Well, first of all, I'd like to express how honored I am to be part of this celebration of Marcel Berger's life and work. So he had a tremendous impact on my own work, the support he provided, support and interest, etc. It meant a lot to me. And like Tom said, we met in the early 70s. I met Berger in the early 70s and a whole group of his students. I don't know if all of them were students, but that was my sense. Since then I found out that I looked up students on the math sign-at and also on the... That's right. And so it didn't look like all the ones I knew. So I thought I knew many more. And it turns out that I think it's incomplete. But independent of that, I feel that... So he always had a large, nice group of students. And from day one I felt like, oh, these are so nice to be around and it's nice to be around Berger. And their work and the whole attitude and support really meant a lot to me. In a certain sense I feel like an orphan in Orhus because I grew up with topologists. At the time they were very, very algebraic topologists. So I broke from that and I wanted to do geometry. And so Marcel Berger was really my idol and then clinging back to the big main figures in Europe and the world actually at that point. So I thought I would talk about... I took also the title of the conference, literary. So I wanted to give like a survey, talk about past, present and future. On one of the topics where Marcel made a tremendous impact, namely manifold to positive curvature. So let me tell you immediately. So I want to focus on... I write this as M plus. So manifolds in M plus are simply manifolds with sexual curvature positive. But here comes the beyond part. So the beyond part is I cannot help also to talk about manifolds of non-negative sexual curvature. And I may even go a little bit further because they're very natural classes related to one another. Namely those that are called manifold of almost non-negative curvature. For now I'll just write it like this and I'll explain what this means in a little bit. So this of course is really where Berger has tremendous impact. And much of my own work has been concerned with this area. But I think I'll focus as much as I can on generalities in this talk. So I will focus general results as much as I can. And then things stop quickly actually, but still. So let me just remind you geometrically what this means. So one way to think about this is simply the following. If you take two vectors at a point P and you look at the corresponding geodesics. As long as they are minimal, the curvature positive means that the distance between the endpoints is smaller than the distance between the endpoints in the tangent space. And greater equal just means that they are less than or equal to. So this is the Hinz version of Topologov's theorem. There are many equivalent formulations of that theorem. And all of this started with these comparisons theorems, comparisons of Jacobi fields. The Rauch 1 theorem, it starts out comparison of Jacobi fields that start out by being zero at a point. And then there's Rauch 2, I don't know why it's called Rauch 2 because it's due to Berger. So this is about Jacobi fields that start off by having some value, but covariant derivative zero at the point. And those two are really basic results that led to the proof of Topologov's theorem. So I think I will use this board. The reason I'm doing this is I thought I had figured it all out to avoid these shadows. But it didn't quite work. At any rate, let me start with the most general results in this, if I take the area as a whole. So there are two, I don't think anyone gets offended. There are two milestone results of general type. One of them is due to Chigur and Grommel. It's a soul theorem. Originally, Mofka and Meier did some previous work with Grommel also. So it simply says the following. If M is a manifold with non-negative curvature, I assume they're complete all of them because otherwise we know for Grommel there's no theorems. So M is always complete but non-compact. This implies that M is actually the total space of a compact manifold that sits in isometrically. So M, so there exists a compact sub-manifold. And this is totally convex, meaning that any geodesic starting and ending at S is in S. So in particular it's really metrically embedded in here, totally geodesic. Such that M is diffeomorphic to the normal bundle, so the total space of the normal bundle. And this equivalence is up to diffeomorphism. All right, so this is one theorem. I like to think of it as a structure theorem, a reduction theorem if you like. It's a structure theorem because it tells you what such manifolds look like, what's the structure of these manifolds. So they're all vector bundles over something compact from the same category. So it follows here that of course S is also in the same class. And I should add that this is what's called the soul of course. If M happens to be in this class, if it happens to have positive curvature, then the soul is a point. And in other words, then M is just diffeomorphic to Rn. Which means that I won't talk about it anymore. So we have a complete classification of positively curved manifolds that are non-compact in the sense of the diffeomorphism type. So in that sense it's also a reduction theorem because now we're reduced to think about compact manifolds. So for the rest of the talk, I'll talk about compact manifolds in these classes. But before I move on to this, there's an obvious set of problems related to this, namely the converse problem. So you can imagine, given, let's see how this is working. Maybe it's not working at all. Okay, working. Only if I put it all the way up. Oh, this is not going to work. I thought I had these brilliant ideas. You see, they teach the things that I never learned at any other place but Notre Dame. If there are two blackboards, they teach you from day one, I think, that you should always start with the one that's behind because of the shadow issue. I can't even see it now. So maybe I think I'll just give up and you will have to live with that shadow. Okay, so I'll pull it down again because this becomes much more comfortable for me. But now I started with the back one. All right, I'll just say it in words. So I'll just move on to something else. So what's the converse? The converse problem is you give yourself a compact manifold of non-negative curvature and a vector bundle over it. The question is, can you put a complete metric of non-negative curvature in the total space? Now, this is known to be false in general. There are lots of examples. The first were due to Waldschap and Oseiden. And many more since then due to Bellegrade and Kapovitz. All of those examples are examples with infinite fundamental group. No counter example is known for finite, in particular, simply connected manifolds. Now, and the original question by Chig and Grimald was, what about bundles over spheres? Even that is not known, only in dimensions up to the base being five-dimensional. However, Regas, a long time ago in his thesis, proved that over spheres, stably it's true. So you can add a large enough trivial bundle and then it's true. You can put a metric of non-negative curvature on it. And recently it's been a lot of work by David Gonzalez about all other kinds of manifolds where you stable it's true. In particular, I think by now it's known, take any known positively curved manifold, then it's true all vector bundles over those if you add a trivial bundle. So that's a reasonable general question. All right, so this was the converse of that. So now let me pull this board down again and write down the other sort of general result. I can't do that now, right? Because I'm not tall enough. Yeah. This is taking more time than I planned. When I came I saw all these boards, I said, oh, I can finish this talk in no time. Okay. Yeah, I really messed up with this here. All right. So what's the other one? You probably guessed it. So this is the bending number theorem due to Gromov. And it simply says, so like I said, from now on we focus on compact manifolds. So one version of it is the following. If mn, so it exists a constant depending on the dimension n such that if m is in this class of non-negative curvature, then the sum or the dimension of the homology of m, any field of coefficients is bounded by this constant n. Now there's a more general version of this theorem that doesn't care about what the lower curvature bound is. And I should also add that the same thing is true for the fundamental group. Pi 1 of m is generated by less than some constant also. This one is a very simple, beautiful consequence of Tobodnogov's theorem. This one is really hard. It's a deep theorem. And it's sort of the starting point is actually a rough version of this theorem that says that the topology, so it was proved only after the sole theorem. But if you look at a complete manifold of non-negative curvature, then it's a compact type. There was Gromov's observation, meaning that if you go far enough away, there are no critical points for the distance function to the sole. So that means that it has the type of a compact manifold with boundary. That really is one of the key starting points in his proof of this thing for compact manifolds. But I won't say anything more about it. So now the more general result is for any curvature bound and then a bound also on the diameter. Of course, in non-negative curvature, you don't see that because you can always scale it. You can even go as far as you could say, whatever we're going to discuss from now on, we scale the diameter to be some number, okay? Maybe pi. We like pi for the unit sphere, maybe, or else. And then it's just a question about what is the lower curvature, but the minimum of the lower bound for the curvature, right? That's like one function of this class of manifolds. And this theorem says that if that function is great equals zero, then this is true. But there's a value for all of them, okay? There's a value that depends on this number, okay? However, that value is really large. And so in particular, if we go back to this general situation, it does not distinguish these classes, okay? You can say, if you look at this thing, it's kind of the same number. It's no, at this point, okay? So one of the huge problems I would say that is that we don't even know that these classes are different. Well, we do at the level of fundamental groups, okay? So at the level of fundamental groups, these are really distinct classes. Here, for example, we have Born-Emeyer's theorem, the classical theorem. So now this is positive curvature. That's a general theorem of positive curvature. So M in M plus implies pi1 of M is finite. This is the classical Born-Emeyer's theorem. It's true for richer curvature, even. And of course, this is not true for the other classes, okay? But actually, you know what the structure is. You know a lot about the structure manifolds in these other classes. So for example, in the class M0, you know that any compact manifold has a finite cover, which is at least diffeomorphic to a simply connected one with non-negative terms across the torus, okay? And what is this last class? Maybe I now should tell you what this means. So what does it mean that M is in this class? Well, it means a couple of ways to say it. One is to say that there exists a family of metrics. Can you read this, by the way, up there? Well, that's impressive, because my handwriting is bad too. But I'll try to say it out loud. So there's a family of metrics on M, so that the section curvature of M relative to this metric is greater than or equal to minus one. And the diameter of M with this metric approaches zero. So there's one way of saying it. So this means collapse with lower curvature bound, okay? This is equivalent by rescaling. You can see that there exists GI. Here's another way of saying it. So the sexual curvature of MI, MGI, is greater than or equal to let's say minus one divided by I, where I goes to infinity. So this approach is zero, getting close and close to zero. And let's say the diameter of MGI is a fixed number, maybe pi, okay? So this is why we call them almost non-negative incorrect, of course, right? And manifolds in this class have not been studied that much. But I think they really are important for understanding the whole picture and for understanding, in general, manifolds on the lower curvature bound, okay? Just like in the case of manifolds with bounded curvature, where almost flat manifolds play a central role in the work of Chica, Foucaille, and Gromov, okay? So that's expected, but it hasn't been really developed. All right, by the way, in this class, any manifold in this class is also finally covered by a compact manifold that fibers over a nil manifold with fiber being simply connected, okay? But in general, the class of fundamental group is larger. But if we move on to assume that M is simply connected, then there's not a single example known that of even going all the way to almost non-negative curvature that couldn't possibly have positive curvature. No obstructions are known, okay? Let me add, so this is, so what are the other general theorems about positive curvature? They also go back a long time, they go back to sinc, or sinc. So it's also just about the fundamental group. If M is even dimensional, this implies that pi1 is either trivial or Z mod 2. This is when M is orientable, and this is of course when it's not, okay? And if it is odd dimensional, then M is orientable. And these are results going back to sinc in the 30s, okay? And all of them are sort of like baby Morse theory in a sense. You look at the minimum length of general minimal shortest closed stewardessic or whatever in the homotopic class, and you show that it has to be trivial, okay? So this is just a simpler variation formula. So in this example here, suppose M is not simply connected, you take a shortest closed stewardessic that's non-trivial. You find, from the assumptions, a parallel field, and you just look at the variation in that direction, then you find shorter curves, okay? There are several theorems like this, and I actually all put them in the same block. I kind of sing type theorems, okay? There's a theorem due to Frankel that says in positive curvature, if you have two totally geodesic submanifolds, they intersect if their dimensions are too big. It's exactly the same proof, kind of. Going back to Weinstein, if you have an isometry and such a manifold, you find fixed points in even dimensions if it preserves orientation. And that, by the way, is very similar to Berger's theorem about zeros of killing fields, okay? So he proved that any killing field on an even dimensional manifold has a zero, and in odd dimensions it's true that if you have two killing fields, they're dependent at one point, okay? So these are the general results that I know of for manifolds of positive curvature without any further assumptions, okay? So that's end of story in that sense, okay? There's much more work that has been done, but all of it with additional assumptions. So for example, what Marcel did, of course, was to prove the quarter-pinsing theorem, and he proved other things, very interesting things for positive curvature, but these are not like completely general results, okay? So I'm trying to stick with completely general results at this point and see what else one can do. All right, so now I want to move on to some basic conjectures. This is really not working well. Can you see it? Okay, I know the theorem, so I don't need to see it. Okay, I can't see it. All right, so what are some main conjectures? Maybe I'll give up and just put this one here, and then do it reverse afterwards. Okay, so we can't avoid Hopf, and there was mentioned by Bruce earlier today. So Hopf has two basic conjectures attributed to Heinz-Hopf. One of them is that the Euler characteristic of M is greater than or equal to zero when M is non-negatively curved, and actually it should be positive when M has positive sectional curvature. And the other one, the one that Bruce mentioned is the question whether S2 cross S2 is in or not. I guess if it's a conjecture, the conjecture was if it's not, but maybe Hopf just asked the question. So, okay. Now this has two very natural, more general related questions I would say. So one is the following. If you take two manifolds, N and V, and take their product, there is not a single such example known to have positive curvature. I'm not saying conduct or anything. I'm just saying there's not such an example known, okay, yet. So, of course, we're talking about S2 cross S2. Well, I mean, this is not, if I take them to be simply connected, okay? Simply connected. So, Rp2 cross Rp2 doesn't have positive curvature. So, the zoom N and V is simply connected. And the other direction is how about you could think about symmetric spaces. And I'll use, I guess this goes back to best this notation, maybe. I think so, right? So, in fact, rank one symmetric space, okay? So, S2 cross S2 has ranked two, okay? And you could ask this question for any symmetric space of rank at least two. Is there a nut, right? Plus, I'm saying it shouldn't have positive curvature. Oh, yes. Oh, then I can't attribute this. I just wanted to do that. Symmetric space, darn. Symmetric space, thank you, of rank greater than or equal to two. Yes, thank you. Of course, that's what I meant. Yes, the other ones do. All right, so these are basic questions. And here's a basic conjecture due to bot. That says that the, if we look at the, if M is also in this class, then the sequence of bedding numbers of the loop space of M grows at most polynomially, okay? There's a, this is, I'm not sure this is the formulation bot had because it's not in any, it's not written down, but bot wrote a paper with the Berger where they looked at pinched manifold. So, in this class with two bounds of the curvature and they provided estimates for the bedding numbers, but they're way from this.  They are estimates in terms of the pinching, but they have nothing to do with this conclusion that you would like, okay? Now, if you focus, so you could also, another question is, are you thinking of any field like in Gromov's theorem or what not? Okay, but if you, if you, so here we could say F and this could be any field maybe. That's a reasonable conjecture. But it becomes really obviously interesting in particular. There's a lot of consequences if you take F to be the rational numbers because then this condition is actually equivalent to the following that the dimension of the homotropic groups, the rational homotropic groups is finite. Okay, so this follows via the minimal model theory of Solomon and so on and so forth. This is old also, right? This is also way past, okay? Now, this by itself, I'll just write down, it has lots of consequences that have been proved, topologically. One of them is that the Euler characteristic is then greater than or equal to zero. This follows from this, okay? So, the butt conjecture would imply one part of the Huff conjecture at least. The other part is that it's, so, which suggests what to try to do if you want to prove the Huff conjecture, if you believe butts. So, because being positive for the Euler characteristic is then equivalent to all odd rational better numbers to be zero, okay? So, in a sense you might as well just prove that to prove the Huff conjecture, all right? All right, another consequence is related to Gromov's theorem and it is that the sum of better numbers, rational better numbers here, is at most true to the end where n is the dimension. See, that's a good bound, right? And in general to distinguish these classes, one might want to prove just bounds that would allow you to distinguish them. That's what maybe one possibility, but that's really all I wanted to mention here, okay? Now, so, given this bleak situation about general results, I'd like to provide some other bleak news about how many examples do we know of this, okay? So, examples and constructions. So, what are, how do we construct examples? Well, the, so let me talk about some constructions first. So, if you start with a manifold m in m zero, and we look at, I'll call these projections. So, you look at like a Croson map, m to n pi, where this is a Riemannian submersion. Then this implies that this guy is also in this class, and you might be clever or lucky or whatever, and show that sometimes it is in this class, okay? That's a very important construction. Now, if you look at this class, Tom and I went out together last time. I lost my wise two now. If you look at this class, then you can take products, right? Products in this class work, of course. And then there's the bundles, and I would say in this class. What do I mean by bundles? It's the, that's the converse of this one in a certain sense. This is a very hard direction to go in. So, basically, you start with something, you start with, with something, a base in a class, and you have a bundle over it, maybe a principal bundle, maybe a principal G bundle. I'll just stick to that right now. Then it follows that when B is in this class, that P is also in this class. So, this is a way to construct many examples of almost non-negative curve manifolds that doesn't work in either of these categories in general, right? That's why this inverse, the converse of the sole theorem is very hard, right? So, to go up to bundles is very difficult, right? To construct metrics. So, these are some of the constructions. There's one other construction, and this I'll just call them special gluings, and Jeff was the first to make such interesting constructions, namely, for example, and now I can use my word cross, okay? So, if you take m and n, okay, compact rank one symmetric spaces, then you can prove that the connected sum has a metric, not a negative curvature, okay? Basically, using the, the, if you cut out a point from one of these space, you have a bundle, and you change it so that it becomes product near the, near the boundary, so that you can glue them together isometrically with a, by means of unit sphere. And there are other constructions about gluings that have to, related to, to manifolds of curvature, these one that Wolfgang Tiller and I developed, but let me just move on. So, these are the examples, and they have led to the, to, to the follow, these are constructions that have led to the following examples. So, to, to get started, you need, you need something, okay, I'll not write, the main source for examples to this day, just leagrups, compact leagrups, or leagrups with binary metrics, because they, if you take a binary metric, they belong to this class, okay? So, therefore, you can, you can, you can do all these constructions with them, okay? So, you can take quotients. So, all homogeneous spaces have metrics of non-negative curvature. The ones with positive curvature have been classified by Berat-Bergerie, and, and actually, so, the examples constructed started with Marcel's work on the so-called normal homogeneous spaces. There are three of those, two in Dimension 7, one in Dimension 13, and then Wallach constructed, classified the, the even-dimensional examples in Dimension 6, 7, and 24. I have no idea if that 24 has to do with Brentel's novel, I don't think so. But, and then Aleph and Wallach constructed an infinite family in Dimension 7, okay? And that's it, all right? So, so, so, if you look at homogeneous examples from this class, there are, of course, the crosses that were always there, and then they're the ones you have to work for to construct, and they exist in Dimension's, I'll just write down the Dimension's, 6, 7, 12, 13, and 24. 16 is one of the crosses, the Cayley plane, kind of also unusual, but there is one example here, it's a flag manifold, one exam, another flag manifold, one exam, another flag manifold. This is a Berzei's example in Dimension 13, and then there are infinite many here, okay? Taking quotients, you can also take what's called biquotients, okay? So, so these are denoted, I really would like not to move on to another board. So, so a group 8 can act on both left and right. So, you have a subgroup of d cross g acting on itself, maybe freely, and then these also have all have metric of non-negative curvature, of course, and you get many more examples now with positive curvature, okay? So, you get more examples in Dimension 7 and Dimension 13, these are due to Esenberg, you have infinitely many, and here, due to Berzeichen, you have infinitely many examples, and that was where the situation stood up until recently. So, so these are, these are the constructions and examples that you have at your disposal. Of course, in non-negative curvature, you can do a lot more, right? You can take products, you can take quotients and this and that, so that's why you get so many more examples. Now, so let me move on to see what, what you might do since, since if you're really not able to prove, let's say, Bruce's suggestion that you can find positive curvature metrics on this 2 cross 2 arbitrary cross to the product metric, or anything else, approve that hop, the answer is no, you can't find such a metric, or anything else, you've got to do something, right? And various approach, you could look at the pinching problem, pinched curvature, etc. It doesn't really need lead to new ideas, I would say. You could even look at the pinching of the size in terms of the diameter, does, does also not lead to really new insights and new examples or anything. So what you can do is, you can study all these examples and see if you get new ideas about manifold to positive curvature or what not. Or you could try to find more, okay? One way to try to find more is to say, let's try to start over again and say, let's admit it, we can't do this in general yet, let's just see what we can do in the presence of symmetries. So like Bruce explained in dimension 4, if you have a circle of isometries, you can actually prove what the manifold is. The manifold, if it has non-negative curvature, it's either S4, Cp2, S2 cross S2, connected some Cp2 plus minus Cp2. And you even know the action up to a very different morphism, complete classification, okay? And so you might, you might wonder what you can do in general. So let me just write that down here. Let me just start it. The classes M plus and zero and M zero minus in the presence of large or special isometric group actions. And it's not completely unreasonable to do it. Certainly brought insight in dimension 4, but all the known examples have a fair amount of symmetry, all of them, right? So you could ask, is that how much does it take to classify? And it actually might give you new ideas to construct new examples as well, okay? And this indeed has happened. So let me just give you two examples of large. So what could it mean that if a group G acts on M, that this group G is large? Well, one knows this obviously that the dimension of G is large relative to the dimension of M. This is called the degree of symmetry. Also studied by Wu Yixiang. It's studied topologically by many people. You could think about the rank of G being large. You could take a maximal torus or just think of torus actions that play a role also in collapse manifold-rebounded curvature. So the torus groups are certainly important. Another way of thinking of large is if the so-called co-homogeneity, which is the dimension of the manifold module of the group, so the orbit space, that this one is small, okay? I will tell you about some results. Let's see what should I do. I think I should bring back since we all... Oh, I want that one down. Maybe now I'm really not so fast. All right. So I will focus on just the rank condition. So this one, suppose you have a torus that acts on a manifold M, isometrically. And here, in this situation, I'll assume M has positive curvature. And what you can show, for example, is in this case, is that the dimension of K cannot be too big. It can be at most roughly half of the dimension. And if it is, you can tell what it is, the manifold. It's actually like a cross, okay, up to diffeomorphism. Now, Wilking showed that you can do better. You can go down to about... K is about one fourth of the dimension of M. And you can still say what the manifold is. It's again like a cross, up to 10-densil-homotopy equivalence. So this equivalence is up to 10-densil-homotopy equivalence. But there are also ways to address these questions due to Hopf in this setting. And Lee Kennert has a number of results related to this, namely answering, basically addressing the Hopf's questions. Forget there's two crosses, two. But think about the other ones for positive curvature. So with K growing logarithmically as a function of N, there are many results, he has several results showing that then you cannot have a product, M not a product, and M, the Euler characteristic must be positive. So what does this mean? So of course in this case, if here K grows linearly, it's like one fourth of the dimension, grows linearly, right? And then you can say a lot, you can say what the Betty numbers are and everything, right? So in particular this is true and they are also not products, right? That they are just like crosses, okay? But if you have much slower growth, so this is interesting at high dimensions, of course in particular, then you could still conclude that you can't have manifolds with this amount of symmetry, and M being a product of two manifolds, or examples like this. And also you can conclude that the Euler characteristic is positive, okay? Now here's the most spectacular work in this direction that just happened very recently, at least in my mind. So this is joint work due to Kenneth and Wilking. So it simply says the following. If M is in this class, simply connect it, and T5 acts on M isometrically, then the Euler characteristic of M is at least two. So this is an amazing result in the sense that this doesn't matter what the dimension of M is, right? So it has some symmetry, but no matter how big the dimensional manifold of positive courage you're looking at, it's one of those with positive Euler characteristics, and they can do more apparently than this, okay? This has proved studying the fixed point sets of this, and the topology of the fixed point sets, and so this is really something, okay? What am I supposed to stop? I think I'm about out, okay. So I will skip one part, but let me just say, let me just add then that, so in the other direction, and with the other notion of large, they say the co-homogeneity, let me, okay, let me just state the result in words. So again, Wilking has a fundamental result that says if you fix the co-homogeneity to K, no matter what it is, it could be 10 or whatever you want, then if the dimension is large enough, and it's not like one of these Gromov numbers, then the manifold is of the type of a cross, okay? If it has that co-homogeneity, okay? So for example, look at the homogenous that are gone somewhere. How many, they dried up a dimension 24, okay? And beyond that, they're only the crosses, okay? So in co-homogeneity one, Zillow and Wilking proved a sort of an exhaustive class of possibilities that is a classification in all dimensions but seven, okay? In all dimensions but seven, the result is the following. They are either a cross, a normal homogenous base, or there are some infinite family of Berserkin examples, and we know exactly what they are, okay? In dimension seven, there are the two normal homogenous bases. The normal homogenous ones. And then there's an infinite family of the Eschenberg examples that also are co-homogeneity one. And then there are two infinite families of candidates for positive curvature. That was what I was sort of driving at, right? And so in co-homogeneity one, there are like pk, qk candidates of new manifolds of positive curvature. I should say immediately that p1 is the seventh sphere, so this doesn't count, and q1 is one of the first aleph wallach example, I call it 1-1. The normal homogenous one, actually. But p2 is known to be, now, is in this class. This was proved by Derrick Hut and independently by myself Verdiani and Zillow. By the way, Verdiani did the even-dimensional classification before we did the odd-dimensional classification of these. Very recently, Derrick Hut has claimed that he can also do p3, q2, and q3, okay? So these are new examples of manifolds of positive curvature, so it sort of validates, at least, that this is not a bad approach to try to find new examples, okay? And not even to speak of co-homogeneity two or whatnot, and I had meant to talk about co-homogeneity two, and the only reason was that there you can prove that if you are co-homogeneity two, and in this last class, then you satisfy the bot conjecture, okay? Then you're racially elliptic, okay? I also would like to point out, and we'll finish by that. So if we go back to this list of examples, co-homogeneity zero means homogeneous. We already saw that all the ones, they're all in M0, okay? I don't need to think about the almost non-negative curve. Co-homogeneity one, these are three distinct classes, okay? We have this very strong restriction of positive curvature. There are lots of examples that have non-zero curvature, but there are lots of examples also that do not have non-negative curvature, okay? But they all have almost non-negative curvature, right? In co-homogeneity two, most do not have almost non-negative curvature, period. But the ones that do, you can analyze, and in particular they do, they are racially elliptic, okay? So let me stop here. Questions from now at least one question. What else is in the future? Yeah, well, I didn't have time. Well, I had a couple of things I want. So first of all, there's this question about why are only these crosses in high dimensions? What's going on? Maybe there's something to it, okay? Or is it just because we are too ignorant to construct all examples, don't know how to construct examples other than using league groups, okay? Now from that point of view, I think, so there's a lot of constructions now. We know much more now about a very interesting generalization of group axons, namely so-called singular Riemannian foliations. So these are locally everywhere equidistant, I mean leaves, but they don't have to have the same dimension. But they are locally everywhere equidistant. These are called singular Riemannian foliations. And there's an abundance, no, Marco Radesche has done a lot of really nice work on unit spheres, for example, not at all homogeneous. And so there's got to be a lot of examples of these on other manifolds, right? And there's recent work of Martin Karin, Christian Schanker and Goethe that showed that all exotic seven spheres, now we know that all exotic seven spheres have metrics of non-negative curvature. And this uses what Wolfgang and I did, but they took quotions and one way of looking at the quotions that they looked at these constructions was that you can think of them as the co-homony to one picture with like a singular Riemannian foliation where all the principal leaves are by quotions instead of being homogeneous spaces. So this is sort of like a hybrid between really general singular Riemannian foliations and homogeneous ones, right? But it shows that there are definitely interesting ways to think about these items in this context of singular Riemannian foliations. So I think that maybe one could construct examples that way. Another one was there's a big old problem from the beginning of Alexandrov geometry that says that the boundary of an Alexandrov space should be an Alexandrov space with the same lower curvature bound that's completely open. But recently, Peterson and I and a student of mine, Adam Moreno, showed that it's true for leaf spaces of singular Riemannian foliations. So maybe another source is to look at all these general objects, take the boundary of such a thing as non-negative curvature, maybe positive curvature. In some cases they are manifold, maybe you can do deformations and so on and so forth. So those are some of the things I wanted to talk about. Well, if there are no further questions, thank you very much.


Welcome back to Kids Fun Science. My name is Ken and today's experiment it's just a phase. The eight phases of the moon in order. As always adult supervision is required. What you need for this experiment is a shoe box, a box cutter, a ruler. Black spray paint is optional and a white ball. So I started with my shoe box and I selected the optional black spray paint. So I'm just spraying the inside of the black box to have a better viewing of the eight phases of the moon. Then we need to cut eight holes on the side of the shoe box and I got the measurements right here on the side. It's two by two squares, a half inch up quarter and a half inch in between. So you go ahead and you can freeze that and get those measurements. Every shoe box is going to be a little bit different. Now take your box cutter and be very careful as you know they're extremely sharp. You're only going to cut three sides of each square, two by two square. So once you have those marked you're going to do three or basically just making an open up windows on there. So we can view the moon through these different windows. So we're going to have that option by having the ball inside the box. And that hence having the optional black spray paint on the inside is going to make us help us look and see the moon a lot better. So go ahead and push those out. And so it looks like that. And then I'm just going to go ahead and do the other ones without showing you. Because I don't think I need to show you all eight. So here's three of them and I'll show you what it looks like after I get the other eight out. There is the measurements again. So you can always freeze that and look but all shoe boxes are going to be different. And so your sizes are going to vary. You just want to be able to make sure that the ball isn't too high and it fits into those holes. So you got one on each end and you've got three on each side and make sure that they're all facing the same direction. It kind of helps when you're going around the circle to view it. So there's the top view. I ended up painting the outside of the box too. I don't know why I did that. So now for the experiment I have a cue ball. If you play billiards and I put a little cap water bottle cap to hold it up and you put it directly in the middle of the box and you close it. And then we're able to do our experiment. According to NASA the moon travels a distance of 382,400 kilometers as it orbits around the Earth during a 29.53 day lunar cycle. Throughout this travel the moon waxes and wades and even becomes invisible for us for a short period. The eight separate phases are recognized during the lunar cycle and each one of them can be enjoyed in a comfort of your own porch. I start off with a small flashlight facing the cue ball that's in the middle and then I go to the other side and that's the new moon. The new moon's between the Earth and the Sun and on the side that's being lit is facing the Sun. Seeing the dark side which means we're unable to see the moon in the night sky. This is called the new moon cycle and it's considered the beginning of the moon phase. Phase two, the waxing curset moon. The moon travels eastward in the sky and a few days after the new moon we can see a slight edge or crescent in the moon and lit by the Sun. The common turn is a phase that's a waxing curset. Waxing means the growing larger or fuller. The third phase, the first quarter moon. The first quarter moon is the term used to describe the waxing moon that is exactly half lit. The moon is now one fourth of the way through the lunar cycle. The fourth phase is the waxing gibus moon. When the larger portion of the moon becomes illuminated it is called the gibus moon. Fifth phase is the full moon. At the full moon the moon's face is completely bright and we'll see an entire circle in the sky. During this part of the cycle the full moon rises at almost at the same time as the Sun is setting. The sixth phase, the waning gibus moon. The light on the moon begins to wane or grow thinner and once again we see the gibus shaped moon. In this case it's referred to the waning gibus. Seventh phase is the third quarter moon. The third quarter moon phase is when the moon is once again half lit. However this time the left side of the moon is lit instead of the right side. As the first quarter the moon is now three quarters of the way through its cycle. The eighth phase is the waning curset. The last phase is the lunar cycle. The moon appears to be a sliver in the sky before it becomes dark again. I hope you enjoyed this video and if you did enjoy this video please click the thumbs up below or share this video. Thanks for watching!