What up fellas, it's your boy Johnny and today I'm gonna be tackling a concept that we all go through with pretty much anything, phases. And particularly with this channel, the whole entire compass of men's fashion. Like with anything, we all start somewhere and then we end up in a completely different place later down the road. So here are my phases of men's fashion that I think most of us are going through or will go through. Prepare to cringe because I'm cringing already. So the first phase that we all start with and with any new style is straight up ignorance. Not even thinking at all about clothes, you just grab and go without a care in the world. And honestly, this is a pretty cool phase. You don't even have to think about anything, but it does have its drawbacks. Because you can end up with some pretty wacky outfits and that can hurt you professionally or socially. And that is why I don't really delete many of my Instagram posts because I just want you guys to see how I progress as a person as well as my sense of fashion as well. And man, look at this outfit.  I'm wearing a shirt from Anchor Blue. I don't even know if you guys know what that story is. Some ran a pair of basketball shorts and most likely I was wearing some skate sneaker from DVS.  Yeah, those were my daily wear. So that is the first phase, literally just wearing anything. Hand me down to your best friend. If it functioned, you wore it. Now the next phase for me was when YouTube was still very young and this is where you're gonna come across the huge channels of today, the menswear industry. Now personally, this is just what I came across first. Maybe you came across streetwear, but at the time for me, menswear was everywhere and nobody was really talking about streetwear. And back then I didn't really care about watching any of these types of videos. But there is one main takeaway from watching any of these videos and that was fitted clothes. Fitted clothes, fitted clothes and fitted clothes because the ladies liked it. And then you also hear you gotta dress your age and more mature. You guys know what I'm talking about. And honestly, there is nothing wrong with that. I get why Alpha M says that and all these other fashion channels say as well. But when I came across those videos, I did not care about any of it. I didn't take any of the advice except for wear fitted clothes.  You know, cop the V-necks, cop the Henleys. But even for me back then, graphic tees had to be fitted. And then this mindset of fitted clothes stuck with me until the next phase. And so the next phase of fashion for me was streetwear. But not the right kind of streetwear. Any streetwear item I coped, it would have to be fitted on me. It had to show off my gains and trust me bro, this ain't the move. Now I've always been into streetwear. I just didn't know the term back then. You know, back then I'd be skating my DVS shoes and obey shirts, skinny jeans. You know, this is back when Obey was hot. Kanye just interrupted Taylor Swift and was in his music prime. And the world was gonna end in 2012. But fast forward years later after watching Rich Lee since the Fung Bros. days, I got really into the sneaker and streetwear culture.  But you know when someone first gets into fitness and that's really all they can talk about? Yeah, that was me with streetwear. And so in this phase, I went through a phase of like arrogance, I would say, cockiness. I thought I could put together any outfit.  I dressed better than everyone. Phew. And then I decided to start a channel. And boy, were my first videos horrible. Because I was kind of a mix of menswear and streetwear. I just could not let go of fitted clothes because it was engraved into me since day one. And honestly, I just ended up giving me out the wrong information and just generic advice. The culture still grew on me a ton. And then I got sucked into this hype beast mentality of copying anything that is hot.  And so this next phase is honestly the most cringe phase. And I really only got out of it earlier this year. I would straight up copy any hype look that was trending from 2016 to the top of 2019. Control C, Control V, copy pasted what everyone would wear and what was hot. Chelsea boots, flannels and distressed zipper pants, I was on that. Pants tucked into your socks, I was on that and could not let that go. I would also straight up rip the standard menswear look of a leather jacket, black denim and Chelsea boots. You know, the badass look. Because during this time, all I cared about was impressing girls and guys. And honestly, a lot of guys are like this. They're insecure, they need validation from guys and girls. And buying hyped brands or copying the badass look was allegedly going to help you with that. But really, you're just projecting your insecurities. So I kind of built a foundation on menswear and streetwear. But it was based off other people's outfits or the opinions of others. And let me just clarify, there's nothing wrong with getting inspiration from another person's outfit. It's the straight up copying and following these rules that doesn't really sit well with me. So this phase of fashion for me was around last year. And it's just straight up copying other people's outfits or dressing for validation. Now the next phase of men's fashion is the one I'm currently in. And it's basically like point number one. I just don't care. It's pretty much like I'm ignorant again. And that's probably why I can't dress. I'm just doing me. And that's really the beauty of the first phase. You don't care about dressing for other people.  And that's really where I think most people are going to find their style. And it's at this point where I finally learned to let go of fitted clothes. See, I wear garments that I have a taste for. Instead of trying to hit these trends or dressed and pressed the ladies. Like I said earlier, all that does is just show how insecure you are. But yeah, fitted shirts, I'm cool with that. I can still wear it. Oversized shirts and crew necks. Yeah, I'm with it. Skinny denim, that's cool. Baggy pants, cool with me. Because I already have the foundation of menswear and streetwear from years of keeping up with it. And this is going to happen with you. You're going to learn to form your own opinions about the culture, streetwear, menswear, how an outfit should look. This is really how you cultivate your own style. Instead of copying others, which is how it should be. So that's why I encourage wearing what suits your personality, what suits your lifestyle. And what suits your interest to curate a taste of your own. Do you work the corporate life? Well, then maybe business casual and dapper should be for you. Breaking Bad, your favorite TV show? Maybe you should pick up a Los Polos or Mono shirt. Maybe your photographer and camera bags and messenger bags are for you. Tailor it towards you and not for the opinions of others. And who knows how long I'll be in this phase. I know there's going to be many more phases down the road. The overall message of this video is this is how you really find your own personal style. Wear what you want, not what social media wants. Because whether you're in menswear or streetwear, fashion is subjective. I hope this video helps you out. This is really kind of how my style has evolved. And I made a lot of mistakes on this YouTube channel. Giving out boring advice, generic advice. But we all learn from our mistakes. So if you guys found this video helpful, leave this video a like for the YouTube algorithm. Please, please. Leave a comment down below if you guys can relate to this. Tell me what phase you're in. Subscribe to this channel and follow me on Instagram at BlaBlaJohnnyTai. And until then, stay fly. Stay animosity free. I'm out, bruh. Peace.


Do you feel stuck in your spiritual life? Maybe you finished high school, you finished college and you recognized that. You don't have a sense of purpose or you're trying to discover God's will for your life or you're trying to find direction or maybe you lost a fire in your spiritual life. Well, I have a perfect opportunity for you. We're excited to present our hungry generation internship in the summer of 2016. I believe at this internship your spiritual life will be stretched. I believe that your walk with God is going to be enhanced and your concept of the spiritual world, the supernatural world, is going to go to another level. You will also receive tools of how to be successful in your spiritual life.  It's useless if it's not stretched. The moment you stretch a rubber band, it becomes useful. It can hold things and it can be of purpose.  You will also be getting hands-on experience in ministries. You'll be participating in ministries like home groups, evangelism, Sunday schools, tease ministry and also be equipped with the tools to use video and social media to advance the kingdom of God. During the course of our internship, you will receive personal ministry, you'll receive freedom and deliverance by participating in our monthly prayer lines. As well, you will be mentored and discipled by our leaders in our church. Alongside the supernatural, our interns will be given the opportunity to step out and practical ministry. They'll participate in various activities to serve the community, feed the homeless, help the elderly and learn how to preach and minister in a practical way. The internship is from June 20th till September 16th. That is three months and your tuition will be $1,500 which includes housing, breakfast and school supplies. The deadline is May 6th, so please let us know before the deadline if you are interested. The documents can be found on our website and for more information, go to www.hungrygeneration.net or contact us at our email internship at hungrygeneration.net. We'll see you there.


Hi, I'm Joe Eichle, the Extension Weed Specialist with North Dakota State University. I decided for the Castleton Field Day we're standing in a field in Cass County that this was a field that did not have a crop in it last year. The weeds went to seed and again we did not get a crop planted this year. So not too dissimilar from what happened in some areas of the southeast part of the state last year and again this year. So what you can see behind me, we do have some weeds that are dying and then some that are still green and so the weeds that we have in this field, a good mix of weeds but all the green weeds we still have common ragweed and water hemp that are glyphosate resistant in this field. This is kind of a worst case situation if we were to manage a prevent plant situation in the field here. Did nothing to it last year except for spray once, weeds went to seed, have sprayed once this year and we're controlling most of the weeds except for the glyphosate resistant weeds which based on the green you can see behind me we still have a pretty hefty population of those weeds that we would have to deal with. So that's one of the things we've talked about this year is dealing with weeds and prevent plant. As you can see if we have resistance spraying one herbicide alone does not going to do the weed control job that we need to do. This is traditionally a conventionally tilled field so if it ever dries up we would come out here and disc this field under and control the weeds before they go to seed. Otherwise we have a little bit more limited herbicide options some of these weeds such as lambs quarters and some of the water hemp are going to seed so we definitely want to get this managed as soon as possible through mowing some cultivation or really aggressive herbicide program that would be effective on those weeds. Some of the other topics that we've dealt with in 2020 to date I already mentioned water hemp. Water hemp is emerged very early this year compared to historically what we've seen for water hemp emergence across the valley. So on this site we've actually found water hemp that emerged on May 1st and really it was probably up a day or two before then. So typically we talk about May 15 as the beginning of the water hemp emergence window in this part of North Dakota it was May 1st this year and we did have a very cool spring, cool and wet spring as well. So what we think happened is a lot of our fields across the valley were not tilled last fall. We had a lot of water hemp seed production in 2019 because of the wet conditions which is conducive for water hemp growth and seed production and so what happened is we had a lot of water hemp seed on the soil surface. We did get a couple of days in the 70 degree temperature range at the end of April and that sun beating down the black dirt probably got that part, that germination zone for water hemp seed on the surface warm enough to trigger germination and the moisture was there. Another possibility is we just have a lot more people actively looking for water hemp so possibly just more scouts in the field looking for water hemp but you know this is a weed we typically think mid-May the battle begins, it began early May this year and we've been fighting this weed so far to this point in the summer multiple flushes and now we're still two months to go of water hemp that can germinate and emerge and become a problem throughout the year. If we have drowned out areas last night we had a lot of heavy rains in some areas and so we had that happen last year and some of those areas did have water hemp go to seed and we could see water hemp germination once the flood waters recede again in the coming days or weeks. The water hemp is one that is growing in its range of where we're finding it we'll have to fight it into the future. Common ragweed is also becoming more prevalent so here again we have glyphosate resistant common ragweed in this field getting more questions about ragweed as well because of resistance to glyphosate and our group 2 herbicides. One good thing with ragweed our group 4 herbicides like 2,4-D and dicamba are very effective stinger as well if you're in sugar beets for instance so group 4 herbicides very effective on common ragweed but when we rely on glyphosate or group 2 chemistry that's when we're starting to run into a lot of issues with ragweed control. The other thing I'll just kind of mention is we are on July 1st now so we're past our cutoff date for spraying dicamba and extend soybeans for this year. What that means moving forward the federal registrations are up at the end of this year we do think these products will get re-registered moving forward but it's really going to be until fall will we really know what that looks like as far as use patterns whether it's still used as we've used it in the past couple years pre-emergence only at this point we really don't know been getting questions on that but we'll have to see what the re-registration process looks like moving forward but I'd expect that we probably will have those products again in some way shape or form in 2021 and moving forward so that's kind of a recap of the season to date some of the issues we've been fighting as always you can call my office send me an email for those you have my cell phone give that a call as well for questions moving forward and hopefully when we're back on field days in 2021 we'll be able to see everybody in person again.


Let's go to our man John and Philly John. What's going on brother and thank happy thanksgiving to you too, man Absolutely say a real quick question as you close up, please on first majestic common stock But the stock has gone sideways now since Labor Day very specific question Tom If this rally is back up to 1150 or so What is it that you need to see if it does that that tells you you've got a See the D rally going on up towards 15. I'd appreciate it. Yeah, you get on the daily You know, you're gonna be looking for basically Five to eight million shares trying to break that level up there And I think what we have John is that what happens folks is that this is a this is a good stock to trade but it's a very tough stock because it Moves very fast folks up or down and I suspect that the way this like this week We try to get to a higher high and you can see the volumes dying on the vine So it's like this consolidation still here. So I suspect it's gonna probably try to get you know Whether it's 30 40 cents off of this level meaning like maybe the $10 low before it makes another surge You know, but I like the setup, but we know that this is a tough one, man And I don't like buying this one closer to the highs I like buying it at the bottom of the consolidation You know what I'm saying so I do great. Thanks so much. Okay, man. You have a great safe one


Hey, this is Brett, and this 2020 GMC Yukon XL SLT is stock number 13762Z. I am here at Summit Automotive in Fond du Lac, Wisconsin, your new and used SUV headquarters. This 2020 GMC Yukon XL has the 5.3 liter V8 gas engine, and it's paired up with the six speed automatic transmission. This Yukon has been fully-safetyed and inspected by our service shop per the state of Wisconsin inspection process. It has a fresh oil and filter change, all the fluids have been checked and topped off, and it is 100% ready to go. I'm going to go all the way around in this video. Inside, start it up, take a look under the hood, go over all the options, give you the most accurate representation of the vehicle that I can. Summit White is the color, and I show all my videos in 4K, so if you have HD capabilities turn them on now, because it's your best way to check out the quality, condition, options and cleanliness of the vehicle before sitting in person. If you like the video, you can subscribe to my YouTube channel, which is youtube.com slash summit auto. Click the bell notifications, you'll get updates on the videos I do each and every day, as well as having access to one of the largest catalogs of vehicle walkarounds on YouTube. So, that's pretty awesome. This one comes with the 20 inch polished aluminum wheels, they're in pretty nice shape, and it has Pirelli Scorpion STR tires. These are 275, 55R20s, and I would say they have right around half the tread left on them. Maybe just a hair under, but still pretty decent tread on there. Comes with the projector lamp headlamps, LED running lights, and the factory fog lights. We'll turn all those lights on at the end of the video so you can see just how nice and bright they are. You get the front bumper parking sensors, and the grille is in fantastic condition as is that front bumper, I didn't see any dents. Dents, scuffs, or scrapes. The hood is in very nice shape as well. Didn't see any dents or dings on there, and the passenger side front fender is in fantastic condition too. Passenger side wheel, pretty nice shape, only thing I saw is one little scuff right there. One right there, nothing too bad though. As you go down this side of this 2020, GMC Yukon XL, take note of how clean that body is, how reflective and mirror like that paint is. I take these HD videos so if you are far away or even if you're close by and you just cannot make the trip down, but you're still interested in purchasing the vehicle, you can see the vehicle, hear the vehicle, and have confidence in what you're looking at before you even get here. So when you do get here, there's absolutely no surprises, and you can make a smart and informed buying decision from wherever you're at. If this video helps you make that buying decision, let your salesman know that you saw the video, it was helpful, and that Brett sent you. Back wheel is in excellent shape, and the back tires look like they have probably about 60 to 70% of the tread left on them. Frame and underbody looks pretty good. And coming around to the back of the vehicle, rear bumpers in great shape, those have the back up parking sensors, full towing package, which includes a receiver hitch and seven pin wiring. The rear gate is in excellent condition, I didn't see any dents or dings on there. That is a power lift gate, you also get the LED tail lamps, and that power lift gate works nicely. We do have the cover for that receiver hitch, we keep those inside so they don't go missing, and you get a nice all-wether floor map back here, there is just a little bit of storage that you can do underneath there. Those seats do manually fold down flat, and to get this gate down, press this button right here. This one also has the flipper glass which flips up, so you can still get to your stuff, even if you have a trailer on the back or a boat. Coming around driver's side, just as clean as that passenger's side, no dents or dings in the rear quarter, this back wheel is very nice and clean as well. And it does have the capitalist feel-fill, which is a really nice feature, never get gas on your hands again. Looks good. And down the rest of the side, doors once again are in fantastic condition as well. This one does have the power mirrors with the blind spot mirror.  There are no rips or tears on the seats, smells very clean inside this vehicle.  You can turn those parking sensors off and on as well. Power windows, locks and mirrors, memory driver's seat, rear door locks, and Bose premium sound system. Back seats, you get the quad seating. The bucket seats back here, no rips or tears on them. They're in really nice shape. They do fold down and up to get to that third row seat. That's an excellent shape, no rips or tears back here. And you get that all-weather floor mat throughout the second row. Seats go up like so. Rear air controls and then you get 120 volt, I'm sorry, 110 volt, 150 watt plug-in and a 12 volt power point back here. Inside of the doors look pretty good. We'll hop inside, check out the miles, the radio, and everything that this Yukon XL has to offer on the interior. All right, you can see that this one has 54,831 miles. You do get a digital speedometer and compass display. Get the leather-wrapped steering wheel, information center controls and Bluetooth controls on the right, cruise controls on the left, and then your forward collision warning control is right there. You also get audio controls in the back of the steering wheel. There is your shifter for that six-speed automatic transmission. This one does come with the GMC and TELA link system. It has the factory navigation system on it. There is Highway 41 that is working nicely. You get AM, FM, and Sirius XM radio capabilities. You also get a storage area behind there with a USB jack. And then this one also has projection manager. You can project your cell phone to the screen via Android AutoRap or CarPlay. So if you have a nav system, you like better than the factory one. Project it right up there, such as Waze or Google Maps. Pretty cool system.  You do get two USBs and a 12-volt power point right there. And the passenger side format and seat are in fantastic condition. Once again, it smells very clean inside this vehicle. I don't think it's ever been smoked in, and the headliner is in great shape. This one does have the power sunroof. You get map lights, home link buttons for your garage door security systems and lane systems, and on-star and SOS buttons in that rear-view mirror. We'll start it up and take a look under the hood. Starts right up. No check engine lights or anything like that. I would personally like to thank you for checking out the video today. And hopefully from this HD video, you've been able to verify the quality, condition options, and of course, cleanliness of this vehicle all the way around inside and out. All those lights are working nicely under the hood. We have the 5.3-liter V8 gas engine. Engine bay is very clean, runs very smooth. Once again, this vehicle has been fully-safetyed and inspected by our service shop as a fresh oil and filter change. All the fluids have been checked and topped off. There is the emission sticker. And this one is 100%. Ready to go. I would highly recommend this Yukon XL from a quality and condition standpoint. And to see more pictures of this vehicle in one of our other 550 new new Skars trucks, SUVs, minivans, ringlers, you name it. We've got to go to that website right there, summitauto.com. Full pictures and descriptions of every single vehicle all at summitauto.com. And if you'd like to check out more HD videos, you can go to youtube.com slash summitauto.com. Click the bell notifications. You'll get updates on the videos I do every day here at Summit Auto as well as having access to one of the largest catalogs of vehicle walk-arounds on YouTube. In fact, in a second, you will see a link to subscribe to my YouTube channel. The upper left, a link to more Yukon XL videos that have done in the upper right. A link to this vehicle on our website in the lower left. And a link to one of our latest YouTube videos in the lower right. Click those, check us out. We're super excited to help you with this Ultra Clean 2020 GMC Yukon XL SLT in Summit. White clear coat. Thanks again for checking out the video. Remember to like, subscribe, and share on the YouTube channel. I really appreciate it. Thanks again.


Greetings and welcome to my astronomy and physical science classes. In this video, I am going to go over the basic instructions for the Solar Observation Project that you will be doing for this class. This is a semester long project, you do need to start on it at the beginning of class. You cannot just wait until the end of the semester and expect to do well on it. And essentially what we are going to do is watch the sun over the course of the semester and measure its motions. And we can do that by measuring the shadows. So there is some introductory material here that you will want to look at. And that I am not going to go through and read again. But we can also, what we want to look at is what you really need to do. And what we are going to be doing is making these altitude measurements. So we are going to start here and look at the altitude measurements. And this is what you need to do. And so what you are needing is to look at a specific time. And that is local noon. Which for this area is generally around 12.15 Eastern Standard Time. Or 1.15 Eastern Daylight Time. So that would mean you would want to make these observations at 12.15 in the winter. And at 1.15 in the summer months. And I will let you know during the semester when these change. Essentially when we set the clocks back for daylight savings time. That adjusts our clock but it does not change what the sun is doing. What we need is a specific object. A solid object is shown in the picture here. So you have some kind of pole going up. And what you are going to measure is how tall that pole is, the height. And you are going to measure the length of the shadow that it casts at that time. And record those two numbers. And really that is what you need to do right now. Put an object out in the sun. What kind of object should it be? A box is a very good object. Because it casts a very, if we use the sun up here. It casts a very nice clear shadow straight back behind it that you can then measure. You can also use things like cans.  But you can still use those. They will cast and you do want to measure from the very back portion here only to the tip of the shadow. You don't want to measure any of the shadow along the edge of the can. You don't want to do that. That will cause you to overestimate the shadow length. So you could use those types of objects. And all you need to record for right now is the height of the object and the length of the shadow. And I will show you that in the data table in just a minute. I show some of the calculations here. Don't worry about this for right now. We will come back to this later in the semester. And I will give you far more details for that. If you are really up into the math and you can go through and work on these if you choose. But I will give you far more instructions and tell you exactly what to do later on. What I really wanted to show you here is again this is all the information on the calculations. Then there are some sample questions here. You will want to come back to these at the end of the semester.  I give them to you now so you have time to think about this. I will talk about some of these over the semester and other ones you may have to look up yourself. I give you some general comments and expectations. Read through those for what I am looking for. The big thing to look at is my write up information here. And that is because this gives you all the details. Every section that I am looking for in your final write up. Again I will give you that now. I will give it to you again at the end of the semester so you can figure out what you need to do for this. But it gives you some basics and what you really want to look at right now is the requirements for the data. The rest of it you can look at later. But I asked that there be two days, two full days between each observation. So if an observation is made on February 1st, skip two full days and then make one on February 4th. So you would have to have two full days between those observations. I only count up to four observations each month. So if you wait until the end of the semester and make ten observations in the last month of the semester, you will only get credit for four of them. So you will not get any additional credit for those. You are always welcome to make more. But that is the requirement that you need to do. They need to extend over at least ten weeks. So, again, you need to start on this early to be able to fit that into the term. The idea is to get you started early. There are some limitations for that. If you are taking this in a twelve week class or a six week class or even a four week class, there are some different expectations for how long they have to last and for how many observations you have to make. I expect ten for a full semester class. That would be a little bit lower for the shorter classes, certainly. So I give you some of those numbers just to look at so you kind of look at what you have to get. And I give you the numbers here that a twelve week class should get eight, a six week class at least five, and a four week class get at least three observations. So you know how many observations you are expected to get. And again, you can look at the rest, but the rest of it does not matter until the end of the semester. And then finally, I have a data table here that you can use. You can make your own on Excel. That's fine. You will be submitting this three times during the course of the semester. And all you need to do at this point is fill out the first five columns right here. That's all you need to record. Your date, so if you're making this on April the 17th, you'd put that date there. That would be after daylight saving time, so you might be making it around 1.12 p.m., say, record that as accurately as possible. What were the sky conditions? Maybe clear. If it was completely cloudy and you couldn't see a shadow, don't record an observation. It's not going to help you. How tall was your object? Maybe 20 centimeters exactly. And your shadow length at that point might have been, let's say 20. No, let's not. Let's go back and say your shadow length at that point would probably be something maybe like 15 centimeters. But say 15.1 centimeters. And that would be what you'd record and you're done. You do not need to worry about these for right now. We will do those later in the semester. Then you just want to record those over the course of the semester. What units you use does not matter, as long as you measure both of these in the same unit. So if you have an inch ruler and that's all you've got, that's perfectly fine, an inch measuring tape. Just measure them in inches. Don't do any conversions. You just want to measure them in the same set of units. So that's really all you're doing. All you need to do right now is work on this data table and get that started as early as possible. The expectations are generally for about one observation every week to 10 days. So that's usually enough time so that you're not trying to observe this every day because of course people have busy work schedules and school schedules. So you're not trying to do this every single day just as you can over the course of the semester. Set yourself some kind of reminder so that you can try to keep on track with this project. And again, at three points during the semester you will turn in a copy of this data sheet and I will take a look at that to make sure you're on the right track, try to keep you going with the project and to make sure you're on the right track with it. So that concludes this video discussing the solar observing project, the semester long project that you will be doing in this class. And there will be plenty of other videos explaining other projects. So until next time, have a great day everyone and I will see you in class.


to everyone at ICTP, especially including you, the diploma students, so, okay, and the speaker today, I'm very happy to introduce our very own Pavel Putrov, who's going to speak on some classical invariance of links and their higher dimensional analogs. So, floor is yours. Thank you for the introduction. So, I will talk about links and their high dimensional also, but before I start some systematic presentation, let me mention some motivation and slash applications and why people are interested in links and their high dimensional analogs. So, first, well, in principle, so, there is a big interest in the subject in mathematics, which is maybe called low dimensional apology. There are many places where these links arise there. So, they're interested on their own, but they're also related, very closely related to the topology of three and four manifolds via so-called surgery procedure. Then these kind of links and the invariance, they appear in a certain notion which arose in physics, but it also found many applications in mathematics, and in particular, this is a notion of topological quantum field theory. In particular, there are some and recent interest in this notion related to some particular application to some condensed matter system and also kind of some more fashionable topics such as what is called topological quantum computing, many other things. So, this is what I'm not going to talk about. So, I'm just going to consider these links on their own. So, probably many of you have encountered kind of a notion of the usual notion of link, but let me define it anyway. So, they're kind of different variations for me, so what I will use is the following. So, an L component, so L is some positive integer number, L component link is a, so I will assume that it's smooth to avoid some technical difficulties. Smooth embedding of a, so they're not by link usually, by letter L, smooth embedding of a disjoint union of L copies of a circle into R3, the three-dimensional Euclidean space. And so usually, so we have this following embedding and the image of this map I will usually denote, so the components of the image I will denote by L with subindex I. So, Li is the image of a Ease copy of S1 here. So, let me give you some simple example. So, for example, this is a two-component link. It will be a two-component link, which will look something like this. It will be a component, the image of component one. This is some circle embedded in R3. And there will be a second component, which looks something like this. So, there are two circles embedded in R3, which of course I visualize them on two-dimensional place, but this should be naturally understood as a three-dimensional configuration. Okay. But of course, since we want to consider some, I mean, of course, this embedding, there are too many of them. We want to consider some robust information. So, what we actually want to consider, we want to consider links up to what is called ambient isotope, which I will define in a moment. But morally, you should understand that you want to consider these configurations up to all continuous kind of distortions or deformations where I allow to move things around, but I don't allow the strands of the links to cross each other. More formally, although we're not going to use this definition directly, let me write. So, links, L prime, what is called ambient isotopic and only if by definition. And so, this will define an equivalence relation on all possible embeddings like this. And usually, and what we want to consider links up to kind of links, which I mean as a topic to be equivalent links. If and only if there exists continuous map, let me denote it, g from r3 times an interval to r3 such that for any t in this interval, if I take the restriction of this map to a particular element here, so this would be a map from r3 to r3 is a homeomorphism. So, to remind you, the homeomorphism is a continuous map for which the inverse exists and it's also continuous. And more like such as the following properties. So, g at restricted to zero, so if I take a zero here is just identity map on r3 and g restricted to the end of the interval satisfies the following conditional z. So, L, if I understand links here as the maps from the bunch of circles to r3, the composition of L prime of L, this g restricted to one is L prime. So, more really, again, this just means that we kind of we allow a certain distortion of our ambient space so that we can bring the original link L to a link L prime. And of course, so given some sort of presentation of a link, it's natural to ask are they equivalent or not. And this is the kind of one of the basic questions in the not all links here. For example, one can see the following simple link, two component link. Well, let me use a second component, different color. And so this will be a link L and consider another link L prime, which is So, also, in general, we want to keep orientation. So, on the link components, so there'll be circles with a particular orientation with a particular direction chosen. And we also can see the other link, which is just a disjoint union. It's just kind of two are not separated. One can ask a question, is this so it's kind of, you would say, okay, it's kind of obvious that those link are not equivalent. They're not ambient as logic. So, this is kind of connected. So, you cannot separate. There is no, you cannot shuffle the surround and separate them. But how, I mean, how would they actually prove this? And it's actually not obvious because, I mean, if you just want to use a definition, how would you prove this? Because you have to show that it doesn't exist any map. And there are kind of, a lot of maps like this, so that you cannot bring this configuration to this.  But if you want to show that something doesn't exist, it's a bit hard. It's usually what you want to do is you want to sign some obstruction of this existence. So, this is where the invariance of links enter, one of the questions. Okay, you can also ask a question, you can also say, okay, this is a bit, I mean, here it's an obvious example. But what happens if you have something more complicated? And you cannot, it's actually, it wouldn't be obvious if you cannot disconnect. So, let me try to draw you some other example. Let's see, something like this. Okay, this is a, so you can ask the question, is this link equivalent to this link or not? So, this is, I mean, you need to spend some time to kind of convince yourself, can you separate it or not? But is there some, and it also cannot be, I mean, it's hard to be sure, like what if you are not kind of smart enough, you cannot, so you just cannot imagine a way to disconnect this component. But is there some, so one can ask, is there some systematic way to answer this kind of questions? So, method to show that, for example, two links are not equivalent. So, find an invariant, so an invariant, so invariant means it's, it will be invariant under unbiased entropy. So, for each, for each link, so let me write i, for each link, we are some, some numeric, some value, suppose a numerical value, such that the value of this invariant on a certain link is the same as the value for invariant of L tilde if whenever L is unbiased at topic to L tilde. And, so the first step, and the second step, so if you, if you manage to show, if you manage to find some invariant show, and show that these invariants are different for this particular power, then it follows, of course it follows, and it's not, they are not unbiased. In my talk, I present some simple examples of such invariants, which have, to define them, you don't need a lot of prerequisites, and so they, they can be very just in geometric terms, and they, like if time allows, I will also consider generalization of such invariants for kind of high dimensional generalizations of such links. Any questions? When this, in this case, there is, there is actually not, not really a problem because, yeah, so you can argue, well in this particular case of R3, so we can see links on R3, because any, yeah, any homomorphism of, well, if it's oriented homomorphism, well you have to, yeah, you have to specify that G is preserved orientation. If you specify Z, then you, any homomorphism of R3 into R3 can be homotoped to, to identity. So this is like, yeah, you can, you can require something weaker, weaker condition actually. So before I proceed to, to definition of the invariants, I want to define the certain, the certain important notion of the Eiffel surface. The Eiffel surface of a link component Li, so it's a particular link component, so this is a circle embedded in R3, is an oriented, also connected surface, sigma i, surface embedded in R3, such that the boundary of the surface is, is my link. So for example, if, if my link component just looks like a a not circle, so I can choose the, the Eiffel surface to be just disk, just disk such that its boundary is this circle. So let me, so of course there are, there are, there are questions about existence and uniqueness. So let me give you kind of a brief answer to these questions without going into kind of proofs, detailed proofs of this, of these statements. So the following two remarks, so it's always exist. And so there are different ways to show this. So one way to show that it is constructed by explicit, it can be constructed by an algorithm. I'm not going to go into, into very detail, so just give me, just, just, just give you, illustrate this algorithm on a some example. For example, I can see there's a following oriented link component. I'm sorry, it can be noted in some way. And then what I want to do, I want to, I want to kind of consider this not, not diagrams of the projection on a plane where the intersection, the crossing points I result. So one, one strand goes a bit above the other or under the other. And then using this diagram, I can go, this is, there is something wrong with the orientation. So if I go around, so I start at some point, I go around and I also follow arrow. And on the, on the right of my arrow, I will spun, I will, I will spun some disc. And then I do, then I do the same at other places. So if I pick other point, which is not already a boundary of some disc and do the same here. But of course I don't, this is not really discs. I have to do some sort of resolution here at this point. At this point, there is not, there is not sure way to connect these discs by, or small strips.  So if you have a, so here it kind of looks like a just a flat, just a part of the plane. But if I want to transition from here, this part to this part, I kind of, if I take some line here, I rotate this line by 180 degrees. And then I, as I do a transition to this plane. So in any case, there is a certain algorithm which allows you to explicitly construct a zephyr surface by given this diagram representation of an object. And here, as I mentioned, I don't want to go into details here. And so the second question is about uniqueness. And there is about uniqueness, there is a following statement. Any two zephyr surfaces, sigma i and sigma i prime of lambda i can be related by a sequence, by a finite sequence of the following operations. So the first operation is ambient isotope, which preserves Li itself. Okay, let me write it like this. So let me, so again, it's just, so if I have something, let's take me this on a simple case of a node. So I can start with just a kind of flat disk here. And then I can do again, certain distortion where disk kind of continuously deform this disk, but the boundary is still the same. And I do this in a continuous way. Second operation is what is called handle addition removal. And this looks like full. So if I have, I can, some vicinity of some point, my zephyr surface looks like this. I can change it. I can add a handle here and it will look like this. Essentially what I do, I take a tube, a circle times an interval, and I cut out two holes here and attach this tube like this. And the third is what is called infinity passing move. And again, kind of schematically on the example of a, of a knot. So suppose I start with just the ordinary disk and then I take, I take a big sphere. So in general, I take a big sphere which surrounds everything. And I, so I have this and I, and then I attach this handle, sorry, I cut out some circle here and cut out some circle on my sphere and attach a handle, attach a cylinder, connecting these two cut out circles. So it would look something like this. And then there's a hole here. This is essentially artifact of what we are, because we are doing things in R3. I'm not on S3 questions. Yeah, I'm not here. I'm not kind of very precise, but on the geometric level, I hope it's clear what's kind of possible. And one kind of, well, the proof of this statement is quite hard, but kind of, on an intuitive level, it's kind of, every kind of change, you can kind of argue that this is any change you can imagine. It's just these three basic operations. At the same time, the simplest possible link invariant can be defined in terms of this, the effort surface as follows. So many of you are probably familiar with this. So linking number of two component link in all our links are oriented here is defined as follows. So it's denoted by Lk, L1, LT. So my link has two components, L1 and L2, and it's defined as what is called intersection number of the effort surface associated to the first link component, and the second link component. I have to assume that for generic choice of C1. So they assume that they intersect transversely. So what exactly this means? The intersection number. So exactly this means the fall. So I have to do a sum over all points. So assuming this intersection is just a finite number of points, I do a sum of all points here, and for each point I add plus or minus one, depending on a certain orientation. So let me elaborate on this choice. So if locally the situation looks as follows, so I have a sigma one. So of course the choice, as I mentioned, so this is a oriented surface. So the orientation should be in agreement with the orientation at the boundary. So in particular, if I rent, so if there's only this example, if I rent the boundary like this, so the orientation of the surface which can be determined by choice of normal vector is will be upwards. So this is the kind of exterior, the upper part is the exterior part of my surface. So suppose the surface here is oriented like this, and the vicinity of my point A looks as follows. So it kind of pierces the L2, pierces the surface like this, and then this should be counted with sine plus one. And if it's the other way around, it goes like this, this point should be counted with sine minus one. So again this definition, it, so first of all it's not, so one has to do some checks. So first of all we kind of used a choice of the surface here, and so one can, so for this to be well defined one has to check that this is independent of the choice of the surface. And the second question is that actually invariant one has to check that this is, it did invariant on the ambient isotopia that we can use to distinguish those examples. Again I'm just going to give some kind of generic geometric idea of why this is the case. So the first remark about this definition is it does not depend the choice of sine one. So we know, well you have to believe me, but those are at least, I mean I will assume those facts to be true that how, what is the ambiguity, we know what is the possible ambiguities of the choice of the surface. And so we have to check that this is invariant on those basic relations. So in particular, essentially the most non-trivial part is, well it's also trivial, but so for the first part we want to check that this invariant on the ambient isotopia which kind of deforms my zephyr surface. So suppose, so you need, I want to, so essentially what can happen here is that we have to be afraid that there are kind of, once we do some deformations there are some points here which can disappear or appear again. But essentially what the only thing which happens is that they can disappear always in pairs or appear. So we can have an example situation like this. So we start kind of moving the zephyr surface, so this is L2, it's kept the same, but we kind of deform the zephyr surface. So this is no longer intersect L2. And so we can lose one point, but we always will lose it together with another one. And they come and as you can see, so if this is orientation like this, this comes, they came, this point came into the sum with minus one and this came to the sum with plus one. So the total sum is still, is still zero. So this is unchanged. And similarly, you can also argue that the adding handles in this infinity passing move are also can be performed such that it doesn't change because you can essentially always this tube, you can make it essentially very thin, or similar this very thin and you can always move it around so it just doesn't intersect. This three tubes that should be at doesn't intersect with the second component at all. So this is some basic idea of why this is an invariant under the choice of sigma one. And another property and this is also invariant under ambient isotope of link itself. So essentially the argument will be similar. So instead of moving kind of for the surface, we can move one of the link components. And the argument essentially will be very similar. So I'm not going to repeat it. And another nice property that although this is defined in some sort of, so it's defined for two component link, but it's defined in a certain asymmetric way. So it has some preference between first and second link component. But what is actually true is that this number is symmetric. And again, let me give you idea why this is the case. So you have to check that the difference is zero. So the difference is intersection number of sigma one with l two minus intersection number of sigma two minus this l one. But since l two is the boundary of sigma two and l one is the boundary of sigma one, this is a condensed suit. So all points here together with these points here, including taking account of proper signs, are points which are contained in the boundary of sigma one intersected with sigma two. But this is just an intersection of pair of surfaces. So this contains either circles, so this joint union either circles which don't have any boundary or it can contain intervals whose boundary a pair of points, but this pair of points is always contribute with opposite signs because they will come with opposite orientation. And so they will always cancel in this sum of the points taking account of the notation. So this is zero. Questions? One of the components is what? I mean, one of the components is not going zero. I mean, this intersection can be empty. Is there many definitions of this? Yeah, this can be given by integral, by Gauss integral, explicit. So they can be given by some explicit formula. So you kind of integrate over pairs of points like this. So you put a point here and point y here on the second component and you can write some sort of integral. So this kind of, I understand this point as a vectors in Euclidean space. So you write some, I'm probably going to mess some of this. This is probably wrong, but something like this. Or is it, maybe this is even, I think this is, sorry, what, yeah, I'm not, I'm going to write something wrong here. Yes. And the Gauss definition actually appears to something in physics. Okay. So suppose we convinced ourselves that this is a well-defined environment. Okay. So can we use to figure out the simple examples? So here, let me choose this to be a, let me choose the surface for the second guy here. So here there is no intersection. So the linking number is just zero, obviously. So here, like this. So the orientation is such that the normal vector points at us. And so this comes so that way. So the linking is plus one. So, okay, this is a difference. So they're necessarily not equivalent. And what about this guy? I choose again the second component to pick the surface. And after a bit of, well, this is, if you move this a bit around, so you can only, yeah, so, yeah, you can imagine, so you have intersection here and here. And everything else, yeah. So this goes up and this goes below. This goes below and this up. Everything else is up. So this is a, this is plus two. So they're all, so we, I mean, suppose we have this definition, then they proved that they are, they're easy check that they're indeed non-accurate. Yes, that's right. So you can, the other, yeah, but yeah, kind of, I wanted to do this more geometrically, but you can use just a plane projection. And essentially if you, so suppose you go, so you go around one of the link components and you look at the crossings. And you look at the crossings and suppose I pick my, I go around L1 and I only look, actually it's enough to only look at under crossings. So if my under crossings is like this, if the second, if the second component passes below in this direction, this is, gives me minus one. And if I have a, if the second component under crosses and goes to the left, this gives me plus one. And I just do the sum of, sum of all these crossings when I go around L1 here. Yes, you can check. Yeah, I mean, yeah, here it's obvious that the, the Lincoln is zero, but here, well, you have to show that, you have to prove that you cannot deform it in any way because the diagram is not unique. You can always change your diagram. There can be different diagrams which represent, well, for once, so this is specifically defined for a pair, for a link component with two components. You can, you can define what is called like linking, self-linking number, but for this you need to use some additional, additional structure on your, on your, on your link component, that's called framing. Then you can define linking number with itself, but you need to use kind of sort of the framing.  Then you can define, then you can set the definition of the link into a self-linking number. But, but I don't, I'm not using this here. So now, so this was something which, and the basic invariant which is defined, yeah, for kind of, for a two component link and is a basic abstraction, simple, the simplest abstraction which tells you that these cannot be, the components cannot be separated. Let me consider, well, suppose you have something like this. This is what is called Boromian rings. So, so Boromian is actually was a, some wealthy family living in the north of Italy. They had, they had this, was a part of the coat of arms. And so this is a three component link. I don't remember if I mentioned, but this is also has, this link, this particular link has a nice name. It's called Hopflink. So here, so if I, so let's me, component one, component two, and component three. It's easy to see that if I just take a pair, each pair, they can be, they can be easily separated. And indeed, there's also, which is also an agreement with the fact, if you click the link in number for each pair, this would be zero. So either, either some simple, either a kind of similar simple environment which gives, which is something, gives something on trivial here and in a zero, when they are all, when, in a zero for the situation, when they are all separated so that we can, we can, we can, we can find some sort of quantity which tells us how the triples of circles can be, can be linked together. And so the answer to this question is, can be given by the quote Milner triple linking number. If some historical reasons it's usually denoted by mu bar, they define for three component links. And so, but actually to define it, we want to assume that all pairwise linking numbers are zero. And so this is the case in this example. But this can be defined for any, for any, for any three component link which satisfies this condition. So the pairwise, they can be, they can be separated. Well, the pairwise linking, not necessarily, but the pairwise linking number is zero. So this will be, let me write in this kind of first, kind of mysterious way, but I will elaborate. So the first part is, will be a number, again counted with signs, particular signs, a number of intersection points of the triple of the effort services. So this is okay because these, the two surfaces in general, they transversely intersect over, over a line, as we've seen. But if you take the, a line, generally intersects with the third surface over a point. And so the choice of signs is determined by, the choice of signs is determined by, again, by orientations. So let me try to write some picture, draw some picture. So this is how vicinity of a intersection point can look like. So this is, will be a surface surface sigma one. And the blue will be a surface surface sigma two. Violet will be a surface surface sigma three. This is my section point. And suppose the orientation is like this. So this is a orientation for sigma three. This is orientation, this normal vector to sigma two. And this will be a normal vector sigma one. So this is a first, this is a second, and this is a third. So this, in this particular case, this will come, could reach plus one. But if I have a kind of, if this is a standard ordering of the basis vectors in R three, but if this, if I have something like this instead, then this will come with minus one. So the other terms point as follows. So we'll sum over pairs of points. So the first point will be in the intersection of l one with sigma two. The second point will be in the intersection of l one with sigma three. And I only over sum over pairs of points which satisfy a certain ordering condition, which I will explain in a second. And each pair contributes with plus and minus one. And this plus and minus one is determined by, by the signs associated to this pair of points in the same way, the pair of points in the intersection of the link components with the surface, in the same way as we had it before. And the ordering is defined as follows. So I take, I consider my points in, so this is, so I take my link component, so link component l one, and I pick a particular base point. And by picking this point, I can do, I can order points here. So starting from here to here. And so the correct ordering is, for example, the correct, I do, I get a contribution from the, from the pair of points if B is, goes later than A. So I have a configuration, for example, a pair of points can contribute if I have configuration of this type. So this is, I have something l one, this l one goes, pierces through sigma two here and sigma three here. So for example, if, if orientation on l one is like this and rotations on sigma two and sigma three are in agreement, then this pair of points will contribute plus one. And similarly, I will define contributions t two, three one and t three, t three, one, two just by cyclically permuting indices one, two, and three here. And so what I can argue by doing, so there is an, by doing this kind of similar argument, so I don't have time for this, but I can argue that this is indeed invariant. So it's invariant both on the, so it's independent of the choice of that surfaces and also invariant on the, under, under ambient as a topic of the link itself. And the way to, to check this is essentially, so there can be a certain different, you can consider certain basic deformations. And the, you can show that the, the, the contributions to different pieces here can, can be traded off when you, once you do the deformation. But the whole, the total sum is invariant. And yeah, unfortunately, I don't have time to, to, yeah, I was planning to do this to, I mean, to argue why this is invariant, but I don't have time. But let me just give an example. So I have to finish, just give me, let me give you example of the calculation of this invariant for this Boromian rings. So there is a way to represent the Boromian rings as follows. And so I'll just do this and I stop. So instead of writing durian like this, I can grow, and grow it like this. I can, so that essentially in each, each component is a serve, is a oval line in a, in a particular x, y, or z standard planes. So then the, this will be the surface just interior of the, of the, of the, of the oval line in this kind of horizontal z plane. And this will be one and they intersect of course. So they, they intersect along this line. And this, the last guy interior of this oval.  And so of course, the, the violet oval and the green oval, they will intersect along this. So there will be a single point in the intersection, in the triple intersection of remands, of the surface. And it's easy to see there is no, so if you go, for example, along this link component, we only have intersection of this link component with points with sigma two, but not with sigma one. So there will be no actually pairs of intersection of points, both in sigma two and in sigma three. So there will be all other contributions will be mu is one. And of course, for this, in this case, the mu is obviously, mu bar is obviously zero. Okay. Of course, I didn't have time to go into high dimensional analogs. So there can be analogs of surface links kind of for the, the, you can see some limon surface embedded into R four, four dimensional space. And so they're also quite natural object. And you can construct new variants in a similar way. And but yeah, I'm sorry, yeah, I'm out of time. So I stop here.


As political appointees and orders began to resign from their current status to contest the 2023 January elections, all eyes are on how those officers will be drawn in a short term. A legal practitioner, liberal social worker, and a political analyst, Olufemi Lawson, are advocating that those in charge of checks and balances adopt structures to ensure there are no gaps. There have been contentious positions as regards the constitutionality of Section 84-12 of the Electoral Act 2022. In fact, President Mohamed Bwari protested the constitutionality of the controversial provision of the newly assented electoral law. Others have argued that the section of the constitution is able to disenfranchise people from vying for electoral offices. A legal practitioner, liberal social worker countered this. It does not disenfranchise anybody because the provision of the constitution actually limits participation for civil servants. That if you are in the employment of the government of the Federation or of the state that you cannot contest for an election except you resign one month before the election. The Court of Appeal had heard that political appointees are not employed in the government of the Federation or of the state for the purpose of that section and so what that means is that if you are a political appointee you can remain in office till even contest for election wise in that other office. A political analyst Olufemi Lawson shared the same view. He insisted that the section of the law would limit the excesses of the political class. I would want to say that morally it is I don't think we even need the law to this extent to tell you that come you have an ambition to pursue in Jigawa state then why are you sitting in Abuja as a minister if truly you are a very serious candidate. One of those vying for the highest seat in government is the vice president Professor Yemil Shimbajil. There are questions about how his latest ambition could cause a divided attention in carrying out his current duties as the nation's second in command. The VP's attention certainly would be divided just the same way a sitting president who is vying for a real elections attention would be divided so and the question is what we should do is to ensure that there is proper monitoring not just a divided attention to ensure that the job and the functions of that office is properly carried out. The truth is that pursuing a political ambition is a natural distraction. Politicians will continue to aspire for higher positions. Politicians will continue to even demand for what they do not deserve. But it is the duty of you and I to say no you don't deserve it. So those who have had competent ministers as deputies should hand over to them, brief them, those who have not had should use the next few days to get them carried along so that the duties of state will not suffer. Ainec has given political parties between April 4 and June 3, 2022 to conduct primaries. This means that public office holders seeking to be elected must resign before June 3 when Ainec expects that all issues on party primaries must have been resolved. Meanwhile, the Secretary to the Government of the Federation, Boss Mustafa, ought to have issued a secular stating a deadline on when they should resign. But he has not, as President Buhari has not given directives to do so.