Welcome everybody back to the Independent Investor Channel. As Hylian investors, we need to continue to stay focused on the progress of the company. I will earmark 10 and 2 bonus items that I feel like Hylian must do to succeed and they will. All of these items, I am aiming to add a little bit more highlight around as we continue to monitor the progress with the company itself and remain steadfast and focused on better times ahead. I know sometimes we get mixed up in what we think we can do in the short term to help maybe infuse a little bit of confidence in the stock price here. Those efforts are futile and they are not helping in the face of what has been an exacerbated sell-off in the grander stock market I think has a lot to do with the contributing factor behind this wall of headwind that Hylian is up against and many other stocks. Nobody is immune to the roll-off and I think Hylian is just caught up in this tornado storm of what has been a fairly horrible 2022 in stock market investing and greater equities the bond market and the stock market alike have just been atrocious. Real estate is next, real estate should roll off and I think in this firestorm of volatile markets, I think we need to understand that the more pile on that happens with this company, the more opportunity it gives them to get this stuff right. So I thought it would be beneficial for the grander shareholder base of the company for full disclosure and disclaimer. I own a significant share position in this company I have since the beginning. I took a small liquidation for a tax loss back in 2021 of about 4,000 shares. I've since supplemented those shares and much more as the stock prices digressed on us here. I do still consider 2022 and 2023 as being bridging years. I think if you're walking across the bridge, I think 2023 is going to be the latter half of the bridge. In other words, you can actually see the other side as opposed to embarking on this journey across this bridge. I think 2022 has the potential of being the worst year in the evolution of the stock. There has been really no good news with regard to the stock sentiment as we've watched and monitored this stock price continue to fall back on us. However, the company has continued to show marked progress and in the face of all of the global geopolitical conflict that's going on right now as well as the supply chain issues that I understand is starting to loosen up a little bit as well as just general volatility in the market. I think it's quite amazing and perhaps maybe in reflection. I know it's difficult to do it now. Look at the current sentiment around the stock and expect that anything other than the fact that the company is going out of business is in order. I think that's very short-sighted and I think it's all too often the disease that is applied amongst the retail community. When looking at a specific stock price and somehow justifying your position based on what has happened in the short term as far as the trajectories for the stock and unfortunately that bleeds over into your stock decisions to just suggest that it's going to continue on. Hylian is in a phenomenal position with regard to its financial backing. Now whether or not this transition across this bridging phase that I've identified to make it a little bit more conceptually acceptable for shareholders is going to happen within the next couple of years is yet to be seen. But here are the 12 items and I'm going to march through the 10 as quickly as I can for some of you people who have complained that you're not happy with an hour and a half video. You're not happy with a 45 minute video. Perhaps maybe I can crank this out in a half hour to try to appease everybody because social media is one of those things to where I come on and no matter what I say it's impossible to appease everybody. The irony is you can pull me up on your iPhone 13 and enjoy what I feel like is the best idea that I've seen in the last 20 years of investing and that's including Google which was a phenomenal idea and including Facebook on the onset which I consider to be a phenomenal idea. I think Hylian is right up there with those phenomenal ideas different approach right in the industrial technology space. I for a living work around and inside and around vessels and when you're out to see you need to understand that certain power generating units are put on those vessels specifically for those intended power draws in certain applications and Hylian has basically done just that and I think their idea is phenomenal. I think their idea is good enough to really stand the test of time and once penetrated into the greater class 8 space I think the domino effect is really going to take hold and people are going to be forced to take notice on this opportunity which right now they are really just being piled upon with the rest of the market and when better times are to be had and they will they always do. I've been through this many many times in my young investing career I'm only 45 I've went through a multitude of these and it's my steady hand during these times of volatility where it can put a retail investor with less time and confidence in the investing decisions that they've made at ease and that's what I aim to do. For you guys that follow me on Twitter I would just like to say that my comments to Hylian with regard to their lack of transparency is specifically isolated to what they've aimed to address with their investor relations. It is not a specific knock on Hylian as a holding as a whole rather an isolated specific criticism of the company in way of what I feel like is a lack of transparency and I'm right. You guys can disagree with me all you want you can disagree with my tactics I am one that believes in free speech I don't believe in you know curbing my dialogue and having some sort of like obligation as a professional to somehow not point out the fact that I believe that they are absolutely failing in the category of being forthcoming. I mean if you compare the amount of news releases from Nicola Motors who just had their founder and previous CEO convicted of fraud on three fourths of the counts they still keep firing away and I find it ironic in the face of a company that I actually believe they have all kinds of things to say which is a lot of the reason why I push this product out that they should be doing more and they are choosing not to. Regurgitating that they finalized the Carnot Generator and pushing out as of September 13th an investor relations video and awareness which I appreciated wholly guys we are a month removed to that news and the Carnot Generator if you scrub that from the dock it was actually an old piece of news that they waited and conveniently released around this time perhaps maybe thinking that the stock was actually going to be at 20 or $25 at around this time they were wrong it generated zero hype and since the news was released the stock has done nothing but continue to retrace why because sentiment is blasted it's gone sentiment amongst the few retail communities that hold this share float it's fine but I'm talking about general sentiment in the market which couldn't be any worse than what it is right now actually it could and I foresee and have no reason to believe that it couldn't get worse and that's just the brutal reality of the situation that we have with regard to the stock price but for you guys that were offended by my Twitter tweets I guess I do apologize to you few folks I think there's a few folks out there that don't say one way or the other but actually enjoy my defense of what I am merely trying to point out as being extremely misleading on the onset now here we are holding the bag asking for answers and being provided none to date here's here's the list and as I work through this list I want you to kind of think about these things in the context of whether or not you think that Hylian will achieve these these milestones or if they are pursuing achieving these milestones or they won't and this will help you in your deliberation of understanding where I think the goods are in evaluating this company as far as where they could potentially go into the future number one achieve certification now I have a statement at the end of this and I have gone through the deliberation myself and I've answered these questions and I will give you the answer to what I think that they will or will not achieve as I go through this but achieving certification with the assistance of Cummins and other players in the network I believe that they will I will I believe that they will absolutely achieve the carb and NITSA certification which I think should be kind of a cherry on top of all this research and development that has gone into taking the hyper truck from what it was two years ago to something not even close to what it was two years ago I think they have used this public market as a as a proving ground and shareholders and share owners alike have have bared the brunt of that proving ground will it pay off into the future well that's what we're betting on okay so is it going to be something that you're going to blame the company on into the future if they actually end up turning out a product that achieves carb NITSA certification and that fleets can actually embrace as being a finalized proof of concept that can be put into the rigor of class eight trucking and actually work for those fleets out there in what they're pursuing and that is to move away from a diesel dominated future and I believe that they will I believe they absolutely will and I think we have that coming in Q1 probably of or even later Q4 as we move into 2023 I think that's going to be a fantastic milestone I'm anticipating at this point guys incurring good news and having the stock continue to just remain here at these recessed all-time low levels because evidently we're working in a stock market to where if you're charismatic and you lie and you manipulate and scheme your way to the top you can actually do really really well if the optics are well right now the optics are horrible for highly on but the fundamentals are fantastic now there are some gaps in the fundamentals the financials and some of the things that I'll discuss that I believe they are looking to achieve of course they are I'm a I'm a realistic individual okay I don't think they've put on this this Sunday dress to not go to the party okay if you just look at highly on the business the build out of the team the acquisition of the technology the leveraging of their current patent portfolio to make this thing and see this thing through to the end is there any indication that they're not going to achieve these milestones I think if from a practical perspective you take a sit back and you look at this and you say God are they really doing all this only to inevitably fail and in the eyes of the market there's a lot of people out there that are just suggesting that this company is just going to go away I find it interesting the people who really have a an in-depth knowledge of this company find that a very very far cry and at this point I think it's probably the most applicable to make those criticisms of the company and I think that it is the most convenient and easy to do right with a stock price below $2.50 it's easy to sit there and say that now I think it's going to be more difficult to say that as the improvements are made within the company and all of those naysayers those voices will drown out and then it'll flip and everybody I'll be talking about FOMO buying the stock to the moon it's just usually how it happens but in either case I'm continuing to hold true on my stock and waiting for them to achieve that certification because I think that's going to be a critical milestone in the evolution of the company and I believe that they will number two is Garner new orders now I look at their existing order book is being proof that they will continue to garner orders there's been orders that have been added to the order book from companies that I didn't even know existed okay there's been orders added by some of the companies that they've done business with and they feel comfortable entering into those orders this early on in the game I think for some of those larger companies that have not submitted orders I think perhaps maybe they're waiting for some of these milestones to be reached and when they do I believe those orders will follow suit so to garner new orders I think to expect that they will not continue to build out this order book maybe is more of an escape to thinking this is just my best interpretation of the current situation what's going on I think the recessed stock price absolutely hurts confidence but right now it's not like we're in a flourishing stock market to where Hylian has looked upon and said whoa whoa whoa this is a big red flag here something's going on here it's the it's the ladder where that the market is dragging everything down the sentiment is adjusting down so to have or expect that the company sentiment is going to increase in the face of what has been volatile markets I think is really kind of short-sighted and it's it's it's it's ill-advised at this point it doesn't fit if you wanted to criticize Hylian in a bull market and everything was flourishing and there was no supply chain and there was no global you know geopolitical tension as well as an energy crisis right now that's putting ripples through the energy market then I would say that those criticisms are are warranted but I I think to suggest that going forward even over the next coming bridging years closing out what has been a dismal 2022 and entering into 2023 that they're not going to continue to fortify that order book is probably an escape to thinking in my mind and I would suggest that you reevaluate your current posture on this idea that Hylian is going to fade into oblivion and not meet new orders I think we're going to laugh at this time when we reflect back on it when Hylian is turning out multiple hundreds and maybe even dare I say thousands of orders per quarter as some of these things kind of are met and checked off of the ledger in their evolution of the company but Garner new orders I believe that they will number three is is build out their current team roster this is something that I just sit back sometimes and I'm amazed at how well they have done on this I've always contended that Hylian gets an A plus in this category finding talent in the capacity that they are looking for talent is very very difficult the engineers that they're looking for they just hired on I believe the senior engineer for the Carnot generator project the Carnot generator project I guess you could sit back and disagree that that's not bullish I'm one to completely disagree with this I'm very excited about the Carnot generator project I released a couple of videos at the time of acquisition of that technology and I think it's going to lend itself valuable when I talk about leveraging into additional income skews as the company grows and builds out but the key to this is going to be their ability to build out a current team roster okay and they get an A plus on this they're continuing to build out their team hireings continue on a month over month basis I track this internally with the assistance of one of my good friends in the community who's who's walking this bridge with me and at some point it may just be me and him to be honest with you since people judge me on my Twitter feed as somehow the end all be all of who I am as a person and a social media content creator when at the end of the day I'm really just trying to put pressure on them to provide more transparency that's that's it at the end of the day but to continue to build out the roster to suggest that they won't is probably an escape from thinking my evaluation is that they will continue to build out that roster as we move forward number four to continue to endure supply chain issues I think is huge I saw some data just as early as this week that would suggest that the supply chain issues are relaxing a little bit we will look for more color on the next the next call that's coming up with Thomas Healy to suggest that perhaps maybe some of the parts that they are requisitioning are becoming a little bit more available if not then they're going to have to continue to endure this the greater question is is highly I'm going to survive because of the supply chain issues that's the deeper question and I believe that they will have they done a good job of enduring it thus far yes the color that's been provided by the CEO suggests that the team has done a good job of finding alternative strategies to find these components that they need and build a backlog of said components to make sure that they can beat meet their their order their order or responsibility to the orders that have already been placed but I think they'll continue to endure it and I think that this in short order in the next couple of years if not even shorter I think going into 2023 I think if the ability to dissolve away the supply chain issue that we've that we've been enduring over the last year year and a half or so will be a major catalyst for the company will the stock move probably not but will they do this I believe that they will so when you take that and you check that off of the ledger on things that needs to happen not necessarily in the control of highly on it's going to be a positive catalyst to the to the to the bottom line if we can get that out of the way number five is to continue to solidify a process with the OEMs this is one of the major criticisms I have for for highly on we've had major pieces of milestones release and the highly on has fallen short of those milestones by a long shot okay the highly on bowls they're going to agree with me on this probably in silence but not necessarily publicly me I'm a lot different I come out and I say it how I see it and they have fallen short on a lot of milestones that they've pushed forward therefore it calls into question some of the credibility and understanding that were some of those things that were talked about with the OEMs in way of OEMs hubs I remember many many quarters ago I think it was later 2020 when one of the analysts on the Q&A on one of the quarter on one of the calls actually pressed Thomas Ely on expressing an idea of how that OEM hub would be made up how would that how would that assist in providing the OEMs with that technical supplement for those customers that want the highly on solution installed off the OEM line and I thought I thought Thomas Ely stumbled through the question couldn't answer it at all and it makes me just suggest that maybe some more color is in order on making sure that once we get across the bridge that we don't have another two to three year bridge to follow while highly on builds their actual OEM manufacturing facility themselves because their relationship with Peter Bilt wasn't what it was suggested and I'm not suggesting that it's not what I'm suggesting is that we need more color around this which is critical to the company if you remove Peter Bilt from the equation we do not have a company let's just be real highly on is a symbiotic company they have to have the buy-in from another company to make it they have to we had to have Meritor and Meritor was absorbed by Cummins now we have highly on on their own still sourcing their product from Cummins yes but without that sourcing highly on would not exist okay the solution comes together as a byproduct of a lot of singular products that are put together to integrate as one solution and I tell you what without the major builder of that truck being the OEM and the Peter Bilt and the five series that's turned off in the line which is supposed to be in the initial iteration of the hyper truck ERX guys we are in trouble and it is not a knock on the company I'm not being rude by demanding this transparency okay some of you think that I am and that's fine I've suggested that if you don't like my approach and you don't like my edgy type of approach to this I challenge the world a lot and being a lot more fluff fluff than we need to be just speak normally and my twitter feed comes across as if I'm screaming at highly on I'm not I'm merely suggesting that they need to actually do this and if they're not going to do it just return shareholders money throw in the towel don't go silent on the line it makes people come up with presumptions and again the bulls are going to be like they're going to make it there's no guarantee that highly on makes it and I hate to just be a bearer of the truth in the stock market but I think there's a lot more followership and a lot more blind devotion to this company rather than just being acute and astute shareholders in the company and that doesn't mean just accepting everything that they do at face value and that everything they do and every decision that they make is golden I don't take that approach if you want to take that approach that's totally fine and again you're totally welcome to sign off good luck finding other highly on content creators out there as down to earth is me that speaks what I consider to be the truth on both sides of the coin now my edge and my emphasis is on a bullish conviction my share ownership speaks to that but I don't speak about that except for the on the onset to provide the proper disclaimer to investors to understand that that is absolutely where I come from up from a share owner position but when we're looking at a new company and to be somehow immune to scrutiny that's just where me and you differ okay that's just where me and you differ and I don't think it makes a damn bit of difference to be honest with you guys anyway somebody would suggest that somebody's reviewing all my tweets forget it they don't they don't read anything they don't respond to anything I've never been responded to silent alert is about as above board as you can possibly get and he's received one one response from highly on in response to a very very prudent question and he does that almost on a daily basis and receives what I consider to be zero and I mean zero reciprocation from a shareholder perspective okay but to solidify the process with the OEMs do I think they will I do I think there's been enough provided to suggest that Peter built is in bed enough with highly on that they will be there on that initial onset to realize and help highly on realize these orders not in 2026 but in 2024 but we'll see my tweet this week was suggesting that how many build slots or if at all they're going to need if they have an OEM hub they won't need any build slots they'll basically just merge in with the existing build slots for those companies that want to add on the highly on solution at the end of the line the beginning of the line the wherever in the line we don't know when it goes to the OEM hubs at the time where it's the easiest to do so and the chassis is the most bear to start to add these components on there when and how are they going to be added when are you asking to interrupt the Peter built defined assembly line and and and segue this highly on solution for highly on's benefit and what does Peter built get out of it we don't know we don't know and these are some of the things that I run against barriers as a diplomatic applicator and looking at this thing from face value guys and you wonder where some of the frustration comes from you guys are crazy I I I think it's wonderful that you can live life walking on a cloud me I'm a lot more realistic and I'm very aware that I have both of my feet squarely on the ground and when I see something that needs a little bit more color I don't I don't mind at all to push that stuff out there it's as if there's some sort of social media set of rules and regulations that I need to abide by to exercise my right to write free speech and you guys that are really really savvy on those rules of etiquette if you want to hold others to that no problem if you're if you're if you think that I'm breaking those rules again I would invite you to just unsubscribe and unsubscribe from every piece of media content that I put out because it just doesn't mean that much to me what means more to me is having open and abated access to a company that I follow and I think we deserve more that's it that's it and 20 years ago would have never been possible now it is sorry that the monster of twitter was created to actually pose those questions to the powers that be and demand those answers 25 years ago retail investors would have been stuck having to sit on their haunches on the farm and and wait for the sunday newspaper for this information to get delivered now we have the opportunity to release said information multiple times a day and they're choosing not to understand where my deeper frustration comes from now good if you don't unfollow me please I'm begging you I don't want you to follow me I've had to block a couple of people because some of the direct communications to me came right to my wife and that's digging a little bit deep you guys have no idea how much scrutiny I endure through social media to put this product out you have no idea zero none all you get to do is watch for my sunday video and you get to watch me in my best light come on and talk about a company that I love you have no idea about the below the surface stuff but when it starts to cut that deep I cut it off off and that's exactly what I did so you get to sit back and you get to enjoy the public version of me and that's it and if you don't like that then I'll block you from the independent investor channel because I have no qualms about doing that either because here's the thing my friends I share a vision with the retail investing audience that seemingly misses 99 percent of the time the fact that you have to continue to walk across the bridge until you get to the other side most people want to jump off the bridge 25 percent 2 percent 67 percent 51 percent across the bridge and very very few will make it to the other side very very few okay for people who watch me can actually feel a little bit motivated and feel like no they don't have to succumb to habits of the past why retail investors suck in investing okay they can tune it to me and understanding what mindset it takes to actually succeed in this racas okay so I'm sorry if my tough love offends you if it does to the point where you lose sleep over it or you feel like you want to reach out to me via direct comms do that better be above board with it because I'll cut you off because it doesn't mean shit to me okay all right number six they have to leverage their network partners I talked about this and this is going to help them achieve their certification commons is their collaborative partner now on this front to what extent I don't know all I'm going off of is what I've been told um through news releases with the company I think this is exciting I think it could lend itself to those certifications that we seek down the line okay so do I believe that they'll make this yes I do and they will okay number seven is to validate and substantiate their TCO this is something I've spoke about for the last two years with highly on imagine fleets out there that have ownership of this product okay and they're introducing it into the rigor and they're introducing the specs on a level that we know of now okay we know about them now thousand miles of range bringing the onboard generator reducing downtime driver experience hoteling etc etc okay we know about it they're in the hands of the fleets and the fleets are able to identify this total cost of ownership benefit to the fleets year over year and dare I suggest multiple years over years and dare I suggest even to take it through a lifecycle of a truck from year one to year seven to ten okay and they're able to really articulate and and and grind up that information and turn out a bottom line figure to suggest that the highly on solution hyper truck yaw racks compared to diesel provided us this much net benefit to the bottom line okay that is going to be powerful what I foresee is that these fleets are going to take ownership of this company and the domino effect is going to happen the communication the actual introduction into rigor and these products are actually going to perform in a manner that it can actually really get these fleets attention to suggest that look we can drive these routes on alternative fuels and we can do so in a way that's comparable to diesel or or superior to we can enjoy better torque and horsepower in these specific categories we can drive down our fueling cost by enjoying renewable natural gas leveraging existing infrastructure through compressed natural gas and we can really start to benefit on multiple fronts and all the while go green go green which we're going to be forced to do anyway okay we're going to be forced to transition anyway and I think when that churn of industry happens and that domino effect and communication and it really starts to speak to the bottom line on the books that's really going to be the unlocking catalyst for the value of highly on how is it actually affecting these companies that they serve that's going to be key and we are not there my friends we will be we will be and when we are that's going to be the real churn and that's going to be the real unlocking of the value the value proposition right now let me give you an analogy it's like a compressed spring and a spring can inevitably compress all the way until it can't press anymore and you can still apply mechanical pressure to that spring at its break even critical mass okay now if you think that the that the press is going to press the spring and so much as to crush the spring i.e highly on goes out of business okay you're entitled to your opinion but I think the compression is what's going on right now and it can compress very very tightly okay 250 150 350 I don't know it doesn't matter we're in a compression stage what does that matter it help you get some context and and shed some light around what I feel like what this stock price means right now it's compressed it's at all time lows it's compressed what happens when the pressure lifts and the pressure lifts in all fairness in the stock market sometimes late sometimes early but it does inevitably happen if highly on goes tits up and goes insolvent then we'll have our answer right if those of us that believe that there's a better chance that that doesn't happen and that that severe pressure on the top of the spring right now is happening and that pressure is relieved right we're going to have a compressed an expansion and we're going to have an expansion the spring effect is going to jar at the beginning of that when we start to actually get into this validation stage which unless you're a follower of this company you're not going to truly understand the potential on the top end for that expansion all you're seeing right now is the compression that's all you're seeing oh wow you know highly on's really under the press I have no idea why I do but I have no idea why therefore I'm not going to invest in it okay it's a no touch right now I'm going to go invest in Tesla okay I saw an interesting comment come through by one of the highly on discord group members that suggested all of these people that are piling into Tesla right now which I think is propping up the stock price may have a 1.5 to 1.7 times multiple suggested that potentially highly on has much larger I'm not going to disclose those numbers but much larger expansion multiple than that and I think that's smart I think that's smart to suggest that the compression with Tesla happened a long long time ago now it's been an expansion mode for the last multiple years right so when we're looking at a stock price it's easy to look at Tesla and say well that expansion is just going to continue to ultimately and inevitably and infinitely continue could be could be you're entitled to your your opinion might I suggest this for you guys and this will really help you the money is made on the compression the money is made putting your bets in before expansion okay there is going to be people who pile into this company at 10 15 2025 pile in okay there will be very very very few and the rhetoric on the landscape speaks to this there will be very very few that have the conviction necessary to actually purchase this same company it'll be the same company at 25 as it was at 250 same company okay it'll just that we have some of this compression fear actually taken out of the equation and it'll be identified that some of this news is starting to contribute to the expansion phase okay all right good stuff they need to realize integration that's key there was some discussion on the highly on discord group this week on going by different semis and looking for the highly on emblem I do the same and we're not seeing it it's a diamond in the rough right now we need massive numbers turned out and there are thousands and thousands and thousands of trucks out there on the road to penetrate it speaks to the addressable market and we need to start to realize some of that integration into the class eight fleet okay number nine is to leverage the tax incentives this is going to be huge what's it going to mean for highly on when these companies like anhyzer bush like green path logistics are coming out and they're talking about enjoying that $40,000 tax credit on the units that they've purchased this is going to be huge if companies don't know and I would be hard pressed to suggest that they don't know but if they didn't know to look at those tax incentives and say hold on a second this highly on hyper truck ERX does qualify let's look more into the solution and start to churn out some of that interest in the in the fleets that Thomas is working really hard right now mechanically to try to generate some of that interest I would suggest perhaps maybe that those efforts are somewhat futile now because those need to be more organic they need to be more of a churn within the industry to suggest that once they bring on one of their fleet partners like a return it that was featured this week on a tweet up in Canada that other companies are being brought into the fold by nature of referential and not necessarily direct contact from highly on highly on it's not going to be able to pick up the phone and sell thousands of hyper trucks every single every single month or every single quarter you guys think that that's the expectation no the expectation is that people are coming to highly on and saying hey I'd like to solidify my block of slots through the OEM how do I go about doing that and their job is to process the order not necessarily make sales okay that's how it's going to happen it's going to happen more organically and I think right now they're in more of a mechanical phase where they're trying to generate churn and that's very very difficult to do it's easier to suggest none I'll wait on the early onset of a product I'll wait I'll wait a year I'll wait two years I'll wait until other fleets put it into rigor then I'll choose my opportunity based on what I evaluate from the other company because they took on all the risk now I get the the benefit of the information granted that it's not proprietary information and that we can understand that that company either benefited from the product or they didn't and I can make my decisions that cost me nothing right and I think there's a lot of that gamesmanship going on right now in the class eight space in other words I don't think the dam has even become close to being broken on this opportunity and highly on has to be there when it does break they have to make sure that they can meet those specifications so right now over this bridging phase it becomes much more important to prove that proof of concept for that inevitable turnout of the product that's the prove it moment where these things get integrated into the fleet start to enjoy these tax incentives number 10 on the list here and I'll close it out and then I'll mention my two bonus strategies that I think they will my answers on the previous ten were that I think all of these are going to come to fruition there's been some progress marked on some very little on others but I think monitoring the company I think if you can keep your focus off of the stock number one keep your focus on the company itself I think you are going to be well positioned to understand that these are things that I think the company is going to absolutely realize going forward and I think the 10th thing is to pursue different income streams I think those different skews of incomes just came as of the last couple of months with the Carnot generator being able to provide or create cheaper energy based on what's being able to be created and pushed to to remote sites I think that's exciting I think the onboard technology is something that I think has always been overlooked with Hylian and I think that it is going to at some point be kind of a dominator in the conversation I think the equipment under the hood is second to none but I think the monitoring systems over the Hylian solution are actually really the key here in tying everything together for the potential for preventative maintenance for the opportunity for Hylian not just to sell companies a product rather to collaborate with them and become a lifelong partner going forward so when we look at these anemic orders of 10 or so just being a as a building or a block to selling said product and making said profit on that said sale perhaps maybe the value is below the surface and understanding that that relationship of those initial 10 is far more valuable to establish that customer as a lifelong customer and really let that customer understand that Hylian will be there for the long term okay and I think that's really the key in understanding that right now in this compression phase a lot of everybody misses that aspect of the company in what they're trying to do in building a foundation for this company and evolve to something to where that organic growth can start that spiraling and snowball effect going forward all right the two bonus things that I'll leave you with and then we'll sign off for today and the Sunday Sunday video and this will be the last Hylian video that I do for a couple of weeks I'm actually on a professional obligation over the next couple of weeks and I will be out of pocket so please understand if you you know want to go back and review some of the old content you can all that stuff still valid but this will be the last for the viewing audience that is devote and stays with me to the end because this is when I drop the nuggets I do that for a reason I'm not interested with pea brains who have an attention span of five minutes or less because this is a life changing opportunity look at it how you will there will be people who study and perhaps even scrutinize this body of work that I've put in on Hylian over the last couple of years it will be studied it will be scrutinized will absolutely be interpreted in a lot of different ways my intent here is very very simple simple leverage the power of social media to provide an education and an awareness on a company that I feel like has an idea of none of which I've ever seen before the idea is fantastic but here are the two bonus items that I want to push your guys this way they need to improve sentiment and that's part of the problem that I have about going silent on the line when the stock has done nothing but slip into oblivion that does not suggest that they need to come out and hype the stock it doesn't what I think they continue to do is continue to provide some level of assurance to shareholders that they're still there talk about the company I don't care you don't need to talk about the stock price has nothing to do with that it has everything to do with saying nothing which is a terrible terrible approach is that their approach I don't know it's part of the frustration okay if they can come out tell me a little bit more about Thomas Healy tell me a little bit more about where he comes from tell me a little bit more about it there's all kinds of creative cool stuff you can do we know everything there is to know about Elon Musk okay and that stuff is invaluable when shareholders really want to embrace the company and get to know them as a true investor true investor there will be investors out there that invest for the money yada yada yada me I want to know the company and when they basically turn a blind eye to me as a shareholder this is me off and I know I'm not alone in this okay I know I'm not alone in this just to put out some level of PR some level of press release some level of some sort of update or progress or not at all not at all say hey we're still enduring the lingering effects of the supply chain issue we've made marked progress on this this and this we're happy to announce this whatever but they have to generate a lot more turn than zero zero and that will in turn improve sentiment improve sentiment I believe that they will but this is the biggest delta that I have because it seems like at this point Eileen is unwilling to help themselves and this is the impression that I get a lot of people will disagree with me on this but doing nothing does not suggest to me that they're doing everything that they can possibly do I'm sorry you want to meet me in the middle somewhere there fine but accepting nothing I think again I called everybody who accepted nothing mushrooms and I do believe that have a soul have a heart okay defend your position all right demands more of the company that we're looking to be so intimate with and support going forward on this road to path to profitability the path toward mass scale up and actually seeing this integration in the fleet that we are so in agreement will happen inevitably into the future and the last thing I'll mention to monitor any is any progress globally that they make expand globally if they realize all of these catalysts how can we not look to say either yes it will improve the stock price or no it won't if all of the aforementioned list starts to become realized we are in for a ride of our lives guys we are in for the ride of our lives but we'll be during that time when I may segue the message because doing this highly on opportunity when the stock is going down is a lot less scrutinizing than if the stock goes up see I've been putting out a ton of content hasn't made a hell of a bit of difference at all with the stock of course not nor do I even believe that that's even a contributing factor I totally believe that my motive is very very sincere my motive is very very straightforward and that is to provide awareness on a company that a lot of people didn't even know existed five years ago company's been around for seven going on eight years it's been in public markets for two all right the splash that this company is going to make when they start to check off if not all of these boxes that I just talked about that are must do's to succeed for highly on how can you not suggest that they are going to see a lot more success into the future I'm a diplomatic applicator this is just what I see these are my opinions do not do not make your investment decisions based on what you hear here that's not the intent here is to drive your investment decisions now that's not the intent the intent is to share a story that I feel like is extremely exciting and I feel like is progressing toward a very very exciting decision point for the company when some of these things start to organically create churn and we enter into that expansion phase that I talked about rather than the the severe compression phase that we find ourselves in now guys I appreciate you continuing to support the message I truly do we are off next Sunday and probably the Sunday after that so two weeks of break I will be out of pocket this will be the last highly on video that I released for at least a half a month look to re-engage in the process and maybe look for better times perhaps maybe we'll identify in the compression phase is maybe moving away from us once that we get off of these all-time lows here in the company but I do appreciate you tuning in with me for the totality of video leader comments the bottom of the video if you do enjoy the video hell I would invite you guys to subscribe I'm a very straightforward applicator I do I'm not a professional on social media nor do I profess to be I think a lot of people that profess to be professionals are not even close to being professionals so who's to say it's all fair and love and war but I think you appreciate my straightforward down-to-earth talk when we're talking about a company that I think has extreme potential going forward we'll continue to monitor the company's progress help you enjoyed this video in the 12 things the highly on must do to succeed we'll continue to monitor those things going forward guys thank you so much for tuning in to the message and good luck in your investment future


On Monday, Sony owned Bungie, laid off around 100 people. CEO Pete Parsons said Destiny 2 was not retaining players as expected and pre-orders for the final shape were lower than expected as well. So what you've got is a one game company currently, that's the one game they have out there and that fan base is furious because it not only affects the company they like, it not only affects them in all of those ways but it affects them in they're not gonna get the next expansion when they were supposed to. And so everything feels off and delayed and weird. Names like Bungie, like Blizzard, like others are kind of at a dip right now. Like that's, it's just different. There's different people who are in the lead. And so you can come out and say, hey Baldur's Gate 3 sold so many copies that they're hiring 500. Nobody's talking about those successes. And not everyone published Baldur's Gate 3. Bungie did not publish Baldur's Gate 3. No, they didn't, it turns out. They wish they had, but they did.


Thank you very much for reconvening. So as I had said, we will have, we have four panelists to present two or three slides on regional freshwater budget under major use scenarios. They'll give about a five minute to seven minute presentation. We can hold questions till the end or if there is a very burning question, it can be asked right away. But either way, we have time after the presentations for a Q&A session, which will be followed by lunch. So without any delay, the first panelist to speak is Upman Udall of Columbia University. Okay, thanks. Okay, great, thanks. Thanks very much Venkat. I decided to not follow the script and talk about budgets because I found a water balance discussions interminable and not conclusive. So what I decided to do is to try to talk about something that I thought might bring things into focus from when people do all their detailed analyses. And so I've titled it a decision analytic perspective, but the idea here is that if I want to make decisions under uncertainty, which is basically going to be the case here, one of the things I have to define is after I do all these wonderful water balance calculations, what's the probability that my supply exceeds demand and over what time period and when into the future or when in the past. So that's sort of the way I'm going to pitch this and there are three points I want to make. There are significant uncertainties with any of these analyses that we do that came out in the two talks that we already heard. But what people don't often talk about are significant biases that go into all of these models and estimates that we make. And some of these become apparent as you start actually looking at the situation, but many of these are latent. So I'll talk a bit about a couple of those. The climate community has made an out of this. They bias correct all their models, which you should not get me started about because that's a big disaster. The second thing I want to talk about with regard to groundwater is the issue is not really how much water you have. The issue is, is it accessible and who is able to access it? And that is a key issue that I think this group needs to think about if they want to get somewhere with that. And I'll give you an illustration of shallow versus ground just as a macro story on that. And finally, the point I want to make is that we focus on hydrology, but it's probably non hydrologic information that may be critical in terms of defining that risk and that goes towards the access and the demand side. And there may be remote ways of harvesting that through natural language processing and so forth that we should be thinking about. So here's the first comment that I had. I want to think about this in the context of water security. And water security can be a dynamic or adaptive context if you allow that it is changing over time and you want to be able to do something with it. You can have an early warning or predictive context. You can have a retrospective context or you can have an asymptotic context where you say, well, we just want to know what is the risk in this particular place. And that could be time dependent or not. So these are things that are well-established in the water resources management literature, but not necessarily in the hydrologic literature which is highly fragmented and I never have understood that. So let me talk a bit about biases and I'm going to jump right into the issue of the biggest water use which is irrigation because I'm mad to establish that so I don't have to dwell on that aspect. Typical way we estimate that is from biophysical equations and which have been calibrated in some places and with perhaps some remote sensing data on which crops are being grown where and what state they are in. Let me narrate to you from five years of working in Northern India and Punjab what I learned about that. These equations have no connection whatsoever to actual irrigation practice. The actual irrigation practice is dictated by an agricultural extension service which says every three days for rice you need to apply two inches of water. It doesn't say don't do it if it doesn't rain or anything like that. The farmer extrapolates that and says I need to water every three days. Electricity is not reliable so I'm going to leave my pump on, okay? So the amount of water that's applied on average that we estimated from field surveys was in excess of 2.2 meters per season for rice whereas from the biophysical equations the amount of irrigation requirement I would have had would have been about 0.6 meters. Okay, so that's the bias of a factor of three. Now think about that in the context of the analysis that people present to you on the global basis and I will leave it at that. Okay, the second thing is the urban use and so forth. So here we use very crude numbers without regard to actually who's getting access and not and the big issue here is the social equity or injustice issue because the populations that are disadvantaged typically will lack access and they are the ones likely to migrate or cause problems. So those are two things that I think I wanted to bring out here. Let me skip through this and then the other point I wanted to make here was with groundwater. This is the thesis of this particular meeting but I would argue that you cannot isolate groundwater from surface water sources. And more importantly, even within groundwater the access dynamics for shallow and deep are very different and the way I would like to illustrate that is by going to this graphic that we did for the US. So the upper part is the trend of depletion in shallow groundwater defined here as aquifers which are being pumped with wells that are less than 30 meters deep. The lower one is deep. You can see deep is where the action is. Most modeling that is being done at the larger scale focuses on the shallow. It's not relevant, sorry. The other aspect that I wanted to show is that there is dramatic sensitivity to climate variability, okay? So that people understand. It rains more, there's more recharge but remember I told you it is a deep that matters. So the real issue here is that we talk about conjunctive use in academia but in reality what happens is that when there's a long persistent drought people pump the deep groundwater because that's what's accessible. There's not the shallow. So you get dramatically higher sensitivity of response to climate from the deep groundwater than for the shallow groundwater because of human action and not because of the natural system behavior. Unfortunately, most of our work in hydrology is focused on natural system behavior without mapping in the human element into it. So that's the question that I wanted to highlight there. The other thing that I wanted to quickly say I know I'm out of time here but I'll just make one more quick point on this and that is for example in the Punjab area in India when I was arguing with the government people and the groundwater hydrologists in India that this is something that is solvable and it's important that these people don't pump this much groundwater. Their reaction was your estimates are all wrong because the excess groundwater that they pump is recharging back into the system. So we should look at the net change. Well, you pump from 60 meters you recharge the upper strata which is only five to 10 meters thick. This is a delayed evaporation system leading to selenization of the soil. This is a much more serious issue than what we are talking about. So I think getting into the details of some of these things in the interaction between the human and the natural system is what I wanted to bring up. Thank you.


Industrialization has brought many benefits to society, but it has also had negative effects on the environment. One of the main sources of environmental damage is the dye industry, which produces large amounts of wastewater containing dangerous chemicals. However, recent advances in technology are helping to reduce the environmental impact of this industry by developing more efficient methods of treating and managing the wastewater. These techniques include advanced oxidation processes, membrane filtration, microbial technologies, bio-electrochemical degradation, photocatalytic degradation, and other innovative approaches. Microbial degradation appears to be the most promising option for recovering resources from the wastewater while minimizing environmental damage. Challenges still remain, however, and further research is needed to ensure that the industry remains sustainable. This article was offered by Toril Shindal, Parita Rakhaleya, Sunita Vajjani, and others.


This morning is by Sylvia Serfati in search of the abricus of lattice Thank you very much. So first I would like to of course thank the scientific committee for inviting me to give a lecture here. It's a great honor and I will start also by apologizing because I am not going to talk about something that Poincaré worked on However, you know if you think of something that Could have been in the spirit of Poincaré you have to find You think that maybe it has to have some connection some strong for connection with the physical phenomenon and some interesting mathematical content, so hopefully this is this is fulfilling this Fitting the bill And so it starts from a physical model Which is related to superconductivity? So in fact superconductivity was discovered one year before Poincaré's death in 1911 by somehow by accident by Camely owns and So what what turns out is that if you take these particular materials or alloys and cool them Below a certain critical temperature. They lose all resistivity And it explained it's it was explained later by the presence of superconducting electrons core called Cooper pairs Which form superconducting currents? and One of the most striking features of superconductors is this Meissner effect that you see here on this picture Which says that the superconductor? Repairs and applied magnetic field And by repelling a magnetic field it can levitate above a magnet and so here you see a nice picture of a superconductor in levitation The other thing that happens is that there are some phase transitions depending on the external Applied magnetic field so there is a what's called a first critical field hc1 At which in fact the magnetic field does start to penetrate in the material And it penetrates via what are? vortices So these vortices are a bit like the vortices that fluid mechanics Talks about and that that point I had been interested in Except that they are quantized so they have an integer charge and they correspond to if you want like a Place where there is a normal phase So the superconductivity is lost in the center of the vortex and they are surrounded by like superconducting currents little loops of superconducting currents and What you see is that the higher the applied field the more vortices Will be present in the sample and in fact they arrange themselves in these things that are called abricos of lattices So this is the term from the from physics It's named after abricos of physicists who got the Nobel Prize for this in 2003 and who had predicted that such patterns should be observed and later after Several years after his prediction they were indeed observed and you see that they they for there are perfect triangular lattices and they are understood they can be understood in a very used realistic manner by the following reason so you have a You have these vortices and they repel each other At the same time they are sort of confined to stay in the material by the external field So there is a sort of competition between confinement and repulsion and in order to Arrange themselves in the most favorable way. They they find these triangular lattices and this is in fact something quite difficult to Explain and to understand mathematically as I'm going to try to to show you so the abricos of prediction was based on the model by Ginsburg and Landau that was introduced in the 50s on purely phenomenological basis, so later it was explained by a quantum theory But but at the time it was it was just phenological and it it proved to be very very successful in describing in Repredicting the phenomena that happened in superconductors and so again these Earned them noble prizes. I mean Landau got the Nobel Prize for many other things But Ginsburg certainly got it for this model and so here here is the The model it says basically that the states of the sample are the things that minimize This energy functional, so just a few things here For simplicity I will restrict to a two-dimensional domain and the vortices if you want to understand them from the from the model they correspond to the zeros of this complex valued function psi that's here and The they have they come with a charge or degree Which is the winding number of the zeros in the complex plane and here? There are two parameters the intensity of the external field here and epsilon which is a material parameter and Corresponds to the size the characteristic size of the vortex core and it will be taken to go to zero and A here is the gauge of the magnetic field, so I'm not going to give too many mathematical descriptions of this but This is this is a model that With with my collaborator it and Sunday we have been working on for for many years and one of the sort of eventual conclusions of our work was that if you want to understand the minimization of G epsilon in the end you find that Vortices essentially interact indeed according to a Coulomb type of interactions as if they were electric charges In in the plane interacting with the with simple electrostatic Behavior and so that that that justifies this explanation that I was giving you before It's as if you had charges that were repelling each other and that were confined together by an external potential and in fact it turns out that there is a much simpler model that Contains these ingredients, which is a discrete model, which is also Also a sort of classic standard model, which is the model of a Coulomb gas So let's focus on this one for a moment And in this model you just have n particles or n points sitting in the plane Or on the real line. That's a variant. That's also interesting So here they sit in RD with D equals one or two and you see that we We have in this model a sum of logarithmic repulsions between the points so this thing will become very very large if the points get close to each other and we have a sum of We have a the effect of a confining potential capital V So think of capital V as something that grows at infinity say for example that grows quadratically at infinity and The the parameters here are cooked up in such a way that the two effects balance each other the effect of the repulsion and the effect of the confinement So this is Again, what's the Hamiltonian of a Coulomb gas and if it's in one dimension if the points are Constrained to live on the real line It's called a log gas Because then the the logarithm is no longer the Coulomb kernel in in one dimension So a motivation for studying This model for studying minimizers of this There are several motivations actually but one of them is that minimizers of this also happen to be maximizers of this type of pairwise interaction where you have here the product of the distance Times the product of the exponential. So to see this just take the exponential of minus W and These things arise in a completely different context in the context of interpolation and are called weighted fakety sets. I Will come back to that a bit later So this is this is no longer about physics. This is about Trying to find best interpolants for numerical computations, for example And you can also know that you can reduce yourself to products of such distances on Manifolds of for example on the sphere by choosing a potential V Appropriately and using stereographic projection. So really a motivation for this can be to study the maximization of such products and On any general manifold two-dimensional surface Okay, so the question in in this in this Setting is to understand now the limit as and the number of points Tends to infinity Okay, how do the points, you know, how do the points arrange themselves in order to minimize This total interaction. So here is a numerical simulation for quadratic potential for 29 Points and you see they tend to arrange themselves relatively uniformly In something that looks like a disc But and here is not yet very very large. So we'll see what happens for a larger end This thing is in fact more interesting even if you add a Temperature to the system and you start to do statistical mechanics So the the Coulomb gas is really a statistical mechanics model where you're looking at a probability measure on the space of configurations of endpoints X1 Xn again that live either in the plane or on the real line and Here you see you put in the standard Gibbs factor So you put it exponential minus beta times the Hamiltonian where beta represents the inverse of a temperature and Z is the normalization factor that makes this whole thing a probability And W n is as before so this thing is is Interesting in physics, but it turns out that it also has some close connection With something that arises from probability theory, which is a which is random matrix models So If you take these particular very particular values of beta Which are one and two and if you take the points to live in dimension one or two Then it turns out that this probability Here is nothing else then the law of Eigen values for some random matrix models that are like the most most classical random matrix models So what I mean by that is the following you take an end-by-end matrix and You draw the entries at random according to a Gaussian law and you draw them in an independent manner, so it's iid random variables Okay, so Gaussian entries and then you compute the law of the Eigen values and What you find is exactly this You fight exactly this law of the Eigen values So then it depends on the symmetry that you impose on the matrix So if you don't impose any symmetry The Eigen values are going to live on the complex plane And it corresponds to this case what's called the Gini Bronson ball For the case of v quadratic and beta equals 2 If you impose the matrix to be real symmetric or complex her mission Then you're going to have Eigen values that live naturally on the real line By symmetry and again the law of the Eigen values is going to follow some formula of this type with beta equals either one or two and v quadratic and these are called The Gaussian unitary ensemble and the Gaussian orthogonal ensemble So there is a very very large literature on random matrices It takes off this form and most of it exploits the fact that There is a matrix model behind which means there is some sort of algebra That allows you to make computations and to understand What the Eigen values can do But at least what What you can see from this Coulomb analogy from the analogy between Eigen values of the random matrices and the Coulomb gas model Which was first noticed by Wigner by the way and exploited by Dyson What you find is that the Eigen values of these random matrices tend to repel each other In a Coulomb manner. In fact the Eigen values interact exactly like Coulomb particles in a confined potential and Here is a plot here is a numerical simulation for the Eigen values of matrices without symmetry so Matrices with just iid Gaussian entries and what you see is indeed that the The Eigen values there are many many of them They tend to distribute it to distribute themselves sort of uniformly and you will observe that they distribute themselves in a disk and They repel each other At least you can see that they tend to be not too close to each other on this picture So one of the things that we would like to understand is something more precise about this distribution and In particular is it true that if you decrease the temperature Which in physics corresponds to? Decreasing the disorder Will you see those points? Crystallize and organize themselves according to abricots of lattices All right, so you see here There is a certain Still a certain amount of disorder, but this we know corresponds to the value of temperature beta equals 2 So what happens as beta gets larger? We can expect that it's somehow related to what happens in vortices and superconductors So this is this is really the punchline of the story already So another model just to give you another example that's of interest to The physics community is a model of Di-blog co-polymers, which is called the otaku wa-zaki model and just to show you very briefly It's a model where you have a it's a phase field model You have a function u which takes values plus and minus one and its average is prescribed You have a greens kernel or greens function here g and you see that these This this u thing can be seen as a set of positive and negative charges that interact With itself in this coulombic manner except that here you add another term that essentially penalizes the perimeter of the interface Between the plus one and the minus one phase. So this is essentially what's written here. You have two phases interacting Via a screen who long colonel and this is the perimeter of the face and the there is this Competition because the perimeter term of course wants everything to be relatively round and in one piece and the coolant Kernel interaction here Prefers some rapid oscillations between plus and minus one And so again, you have a competition between two effects. So it turns out that in this model There are many many possible regimes, but people expect that Minimizers of such a thing are periodic Exhibit periodic patterns and numerically you see periodic patterns however mathematically it seems Almost impossible to prove very few things have been proven So this is a numerical plot of what happens in this model You see you have these two phases plus and minus one represented on the picture by black and white And so essentially the the whites the white repels each other and the black repels itself and The perimeter of the interface is penalized and you see this is the effect of trying to optimize these two competing of constraints these two contradictory constraints and We'll be interested in the situation where the white phase becomes very very small compared to the black phase You can prescribe this The volume of each phase and so in that case you will see you see immediately that essentially the white things are going to become points and we are back to the same setting where you have points interacting in the plane and In fact, we will have similar results on this model as well Okay, so in in three models. I have mentioned against Bolando Coulomb gas or fakete sets and otaka was a key we have these Same ingredients in the end. We should have a 2d Coulomb interaction with large large number of points And we expect to see some sort of pattern formation in in minimizers some periodicity and even To see this abricus of lattice showing up Okay, so let's see what is known What is known and I will do it only on the second the simplest model the one that's discreet the Coulomb gas where I have endpoints x1 xn and Assume they minimize w n Then you can form the probability Distribution which is the sum of Dirac masses at the points normalized by the number of points Okay, so this is a probability Distribution and it's known that it will converge to a certain probability mu zero Which is the unit minimizer of this Interaction energy now, which is simply the continuum version of the previous discrete interaction So the this thing has a unique minimum among probability measures Because it's convex It's strictly convict and in fact all this has been understood since the 50s This is the basis of what's called potential theory Which goes back to actually gauze And which consists in understanding things that minimize such energies All right, so there is a there is going to be this mu zero which is the equilibrium measure And for example if you take v quadratic, then you compute you find that mu zero is essentially the characteristic function of the unit ball Properly normalized to be a probability measure And so what this corresponds to is exactly this Picture because you see here that the probability of finding a point is converging to the constant density on the unit ball The constant density Supported in a disc okay, so The point after that is to try to say something more precise so beyond This mean field limits if you sort of expand around this behavior of n mu zero when you you look at The error that you make compared to that You are able to find next-order terms Which should tell you something about? The microscopic distribution of the point so here is the sort of rough picture You have many many points There are going to be distributed in a set Which is the support of the equilibrium measure? we know what's the Global density of the points what's the sort of macroscopic density but as you zoom at scale root n so Because of course you're in dimension two So if you have n points in dimension two in a box of fixed size the typical distance should be one over root n So if you blow up at this scale around a point you will see a configuration of points now Which are well separated from each other and which should fill the whole plane as n goes to infinity Okay, so if you're on the real line is the same situation you have a point living on a certain segment with a certain density and If you zoom You're gonna have points that now live on the line And so you would like to understand the microscopic Arrangement of the points by understanding The sort of energy that governs The distribution of this these points on the real line or these points Sorry these ones in the plane or these points on the real line, which are the blown up configurations Okay, so there is some computations involved in doing this. This is a hard part of the work But what we find in the end is that these points After blow up they interact according to Again a logarithmic interaction, but you can see them as screened by a fixed background charge So it's what physics physics the physics literature calls sometimes a gelium So it's not it's not very important But it means that there is something that makes them hold together even though they repel each other logarithmically It's as if there was a background charge. That's Exactly uniform and negative So it's opposite and it holds them together because it has the opposite sign to the positive charges So mathematically We find ourselves with with this equation where you see here you have H Which is like the electrostatic potential Generated by a sum of positive charges Dirac's at the point and this negative background charge That I was mentioning and so long down now at the limit when n goes to infinity lambda is just this infinite discrete set of points filling up the whole plane So the point is we are able to define a total interaction energy of this set of points So I will give you formulas here, but you don't have to really follow them So essentially the the energy that we define call it w is defined based on H and It's defined essentially like this so it's essentially consistency taking the limit over larger and larger boxes of the Square norm of the gradient of H except that it has to be computed in In our renormalized way because this thing that these things don't converge So here is the proper way to do it if you're interested. Okay, so we have a definition Which works for arbitrary configuration of points in the plane and this is the point So you have an infinite number of points They fill up the whole plane instead of summing their pairwise Coulomb interactions with some sort of infinite sum you replace this Difficulty by by computing something which is local and which is more like amenable So there was this case of the line. So here I described the situation in the plane So in the line, it's in fact, it's the same. So what you find if you blow up is you find a sum of Dirac masses and What you can do is you can embed the real line into the plane and solve For this equation which means you can you can view the positive charges on the real line as living in the plane and you can compute the Potential H that they generate on the whole plane Provided that instead of having this constant one as a background charge now you put a constant Density on the real line as the background charge So you you make a sort of singular charge on the real axis which is negative And it's going to sort of balance these positive charges positive discrete charges on the real line and if you compute You can then compute the energy of this configuration in the same way as if it were in 2d And so this gives an energy and the point is that we can derive it as a The limit of the original problem. That's the main point Okay, so this is the way to compute So you have this configuration of points in the plane, which is infinite you take balls of radius r and you average of a larger and larger balls this energy and If you're in the real line you make you embed this real axis into the plane and You compute over a larger and larger strips the electrostatic energy generated by these guys, so The point is that this thing can be derived rigorously from the previous models, okay, so what we show is that after blow-up Around almost any blow-up center in the support of the equilibrium measure You converge to minimizers of this limit energy w So we did it for ginsbolando. We did it for Coulomb gases and for the otaka wasaki model That I mentioned before so so this is this is really the picture We have this picture. We know the sort of average distribution and when we blow up these guys have to minimize a certain interaction energy and Hopefully this is going to start to explain why they arrange themselves Triangularly, okay, so that's the next question. What are the minima of? This w so first thing is you might want to have a more Tractable formula for this w. I gave you a definition which is valid for arbitrary configurations in the plane Here is a computation when you assume something more which is when you assume that the Distribution of points lambda the discrete set of points has some periodicity so if you assume that it's a Pattern of points that gets like repeated Definitly the same pattern Then you can compute the w in a much more explicit fashion and you find a gain What you would expect which means you find again a sum of pair-wise interactions So here g can be computed explicitly It's the green's function of the underlying torus on which these points live This thing is a constant, which is also explicit So you find yourself with trying to minimize this sum of g of a j minus a k Over a set of say n points, which are again repeated periodically Okay, so there is a formula so even though we gave a complicated definition in At least in some simpler settings It amounts again to something you would expect and to something much more explicit a Discrete energy a discrete sort of coolant interaction So you see the effect of the background is that instead of having a log here You have rather the green's function of a torus, which is something slightly different So it's a solution to this by the way Okay, so this g if you want you can express it Even more explicitly you can solve for the green's function of a torus and you find some Eisenstein series So so g can be expressed as an Eisenstein series And it turns out that I found that such quantities also arise You can even write them on general Riemann surfaces. They arise in in Arakulov theory. They are called heights so the question here is to identify the configuration of points a1 to an which minimize such a thing such a height and Well, maybe if there are a number of theorists in the room that have ideas we would be interested I'm sorry advertising for the problem Okay, so there is one thing we can do Which is that we can minimize this formula Among configurations which are already Themselves a perfect lattice Okay, so a simple lattice if you want, right, so you take configuration of points Which are just du plus dv with u and v two vectors Such that the unit cell has volume one, so you normalize the volume Okay, so then you only have two parameters of minimization. It's actually the angle between the two vectors and the size of one of them and Then using the explicit formulas that I showed you before which means using the formulas in terms of Eisenstein series you find that This guy this function w is minimized among the lattices by the triangular lattice So this is the place where you're like because We derived this limiting interaction energy and at least we can see That it makes the difference between different kinds of lattices, so it sees a microscopy a Microscopic behavior and it can tell you that the triangular lattice is the best among all lattices So triangular lattice is better than say the square lattice So this is the one thing we can prove the triangular lattice is better than all the other lattices However, we don't know to prove that it's better than anyone else Which wouldn't be a lattice Okay, but at least it's consistent with the observations that I showed you at the beginning on superconductors where you see these triangular lattice this abricus of lattice All right, so the proof of this in fact boils down to using theorems from the fifties that were already known on Number theoretic Quantities So here you state the zeta function So you found the zeta function of a lattice so here now lambda is a lattice, right? So you some one over p to the power s Over points in the lattice So when s is equal to 2 this is critical this some would not converge And it turns out that when s is strictly bigger than 2 this can be viewed as one way of regularizing the sum In the Eisenstein series thing that we have to compute here is a sort of another way of regularizing the same sum some of one over p squared and what we can prove is that they're essentially equivalent so you can you can Transform one minimization problem into the other and then fortunately the other minimization problem had been solved So the question of minimizing this thing among lattices Had been solved and it was known that the best is the triangular lattice Okay, so here Modular functions and number theory are the things that give you the answer a little Digression which is there is another proof of this fact about the zeta function due to Montgomery So what he observes is that the zeta function is the Mellon transform of the theta function and now the theta function is this thing you sum exponentially decaying terms over the lattice And you ask yourself Again the same thing. What is the lattice that minimizes this quantity? And so Montgomery proves that again among lattices. It's the triangular Which is the best and then by using the fact that the zeta function is the Mellon transform you obtain the result for the zeta function and The reason why I'm mentioning this is that it turns out that in superconductivity again in the Guinness-Bolando model But in a different regime of external field than the one I was studying So there's a sort of two different regimes of external field Then you find this problem so you you also have an abricus of lattice and What you find is that if you try to explain why this abricus of lattice you find to a minimization problem You find a minimization problem, which once restricted to lattices gives you the question of minimizing this theta function So both the theta function and the sort of zeta function Which are Mellon transforms of each other arise in the same superconductivity model and why? Why that is so far. I can't say Okay, so this is a remark and so we are led to this question, which is just what I said before We can conjecture now That this triangular lattice is not only the best among all pure lattices, but it's the best Among all possible configurations or at least that it achieves the global minimum of This function w that we define okay, and Maybe a reason supporting the conjecture is simply the experiment, right? Since you see that in superconductors there are triangular lattices since we derived The fact that rigorously from the Guinness-Bolando equation from the Guinness-Bolando model We should see a minimum of this w While nature gives it to you nature's nature tells you well The minimum should be the triangular lattice now go prove it We have no idea how to prove it at least what I can say is that w Can be expected to be a sort of quantitative measure of disorder of a configuration of points in the plane if you think that The the most ordered configurations are the triangular lattice The most order is the triangular lattice and things that have a higher w then Would be more disordered if you want So now there is the one-dimensional case, which is particular So I told you there was a one-dimensional version of w and then you can again compute everything explicitly And then you can completely solve the question so in one dimension It's not too too difficult to show that the regular configuration, which is z putting the points Putting the points equally spaced Is the minimum over all possible? Configurations achieves the minimum of course 1d is much easier than 2d right you have much much less geometry in one all right, so What do we do once we have proven this we have derived this w as A sort of rigorous limit from these problems with large number of particles So just a few things that we do we expand The partition function We get a finer asymptotic expansion of the partition function, which was not known before We get a large deviations type result, which tells you that Essentially if you have finite temperature there is going to be a threshold Below which all the configurations live except with very small probability So w has to be smaller than a certain threshold As n goes to infinity and this threshold decreases as the temperature is decreased The threshold converges to the minimum of w which means that as you let the temperature decrease You should indeed crystallize To the minimizing configurations for this w and if you believe that the minimum is the triangular lattice this Conjecture that we don't know how to prove then it gives you a proof of Crystallization in the Coulomb gas Okay, and in 1d of course you have a complete proof because in 1d the minimum is identified and Two things I want to point out which are two things that sort of follow from our analysis Is that if you want to solve the conjectures there are two different things that would suffice but that both seem very difficult So the first thing is it would suffice To compute the minimum over periodic configurations But periodic with respect to a larger and larger box Okay, so that means if I give you again this formula All right, which is the formula when I have like just n points in a torus repeated periodically if you want if you can identify the minimum of this and let the size of the box go to infinity or Let n go to infinity then It's it would suffice to to identify the minimum Right, so it suffices to identify the minimum over periodic configurations as long as you can take the period arbitrary large But still this we don't know how to do Okay, second option is compute this integral This integral is the partition function of the Coulomb gas Right, it's an explicit integral, but with a number of variables that goes to infinity So if you can compute this integral and take the limit beta goes to infinity In the order n term you would find the minimum of W The problem is that as far as I know nobody knows how to compute this integral Except if beta equals 2 So if beta equals 2 You can recognize your square of a van der non determinants and using the fact that the square of a determinant You can expand and there's some algebraic manipulations that you can do And eventually you completely compute exactly zn But when beta is not equal to 2 it seems you cannot do the first trick and You're sort of blocked here Except in dimension 1 if you replace R2 with R1 Then these integrals are computed. They are called Selberg integrals and this thing is completely not Okay, so again two open problems So now we'd like to Maybe extend the picture a little bit So what we what we find ourselves with is in fact a crystallization problem and There are many such things in In nature and it's a more general family of problems You know you give yourself some interaction potential V And you look at the sum of pairwise interactions V of X i minus X j Over a family of points in the plane and maybe some kind of boundary conditions. I don't know and you ask ourself we ask ourselves When can we say that the minimum of this total interaction is? To put the points on a lattice To put the points on a perfect say triangular lattice so really the importance of this question I think cannot be understated because This is really the reason why matter Organizes itself in crystals right you minimize some of total interactions between Between atoms say in three dimensions typically and you want to understand Why things organize themselves periodically and what I would like to tell you is that there's very very few instances Where we can explain this rigorously Not another area where this arises there's this thing called the cone and q mark conjecture where They take a potential like this so they take this sort of minimization problem and They say if it's of the form f of x squared with f completely monotonic So what completely monotonic means is this so the each derivative each case derivative has the sign of minus 1 to the k Okay, then they conjecture that the minimum of this is achieved By the triangular lattice always So that's in dimension two and they also have a conjecture in dimension eight There is something that's the ea mood lattice that would do it in dimension 24 the leach lattice so that means that the abricus of lattice is expected to have some sort of Universal minimizing property is going to minimize a large class a large class of starch Such problems and and so you can ask the problem differently What is the class of potentials V for which the minimum is the triangular lattice? Again We don't know and so as I said there are very if you even enlarge the problem and you ask yourself Given a minimization problem minimize an energy and interaction blah When do we know that the minimizer is periodic? When can we prove that the minimizer has some periodicity? Either then in dimension one which as I showed you I showed you one example either and other than in dimension one there is Very few instances. I mean, I don't know many so it's something that's sort of Very intriguing and at the same time we don't really have any Tool to attack such a problem. So let me give you a few positive answers Few positive answers. The first positive answer is the sphere packing question so in two dimensions you take You take disks of fixed size and you want to arrange themselves In such a way that maximizes the number of disks that you can put in a box Or if you want you minimize the interaction with the first hard sphere potential Then it was proven By writing in the 80s that the triangular lattice is the minimizer, which is of course what you would intuitively expect but You know to go and prove it is something else. So This is the proof of this Now this was slightly extended by Florian tile in 2006 where he looks at the Again this sort of interaction where V is a slight perturbation of The hard spheres potential So he takes a potential of this this form which is sort of caricature of a lino drone potential Or you see there is a fixed distance Which is slightly which is very highly favored So it's very highly favored to put the potential the points at distance say one here from their nearest neighbor and For such for such a potential he manages to give a rigorous proof of Crystallization and this is almost the only proof These two are almost the only proofs that are known I would say on This on crystallization from so now if you compare to our problem We were very far off because in our problem. We have a very long-range interaction the interaction is logarithmic Whereas this thing is very short range. So We really not in good shape to apply any of their Of their results Now there is something a little Surprising even which is that there is another problem, which is soft and which doesn't look so far off from ours so I Would like to rephrase our question in the following way so you can sort of heuristically wave your hands and see that the minima the mini equation of minimization of w is the same essentially as the question of Minimizing a sort of h1 star norm of Sum of Dirac minus one so h1 star denotes the dual of h1 whatever that means some functional space Right, so what you're trying to do is you're trying to minimize a certain way of measuring The difference between the sum of Dirac's and the constant one Okay And so now there is this problem which is to measure this distance in the dual of the lip sheets functions and This thing is it is in fact the same as measuring the Vasselstein distance in optimal transportation Between the sum of the Dirac masses and the constant charge one So Concretely what this means is the following problem You're given you give yourself points in the plane you imagine it fills up the whole plane and You want to attribute to each point a certain domain of the plane of volume one and All the mass that's in this domain you attribute it to the point and you compute the integral of the on the cell See I of simply the distance between each running point and the point Xi the black point Right, so you take all the mass on the cell and you transport it According to the March counter of each transport problem on to the point Right and the cost that you pay Is essentially each unit of mass that you transport into the point you pay the cost equal to the distance to the point Okay, so this is this is the and then you integrate right so this is the total cost for each unit of mass You pay X minus Xi and you integrate over the cell and so the question is what is the best? Partition of cells and what is the best distribution of the points in order to minimize this whole thing? right, so What is the way that if you want if you want you you you can transport the mass to these points in the most efficient way Well, it is the triangular lattice and this is a There is a proof in preparation by born policy and tile which hopefully will appear soon So they managed to really prove it the best distribution is the triangular lattice and you see This is for a different way of measuring this distance, so it's very strange that for this way in fact The proof is not very very difficult. You can do it. I mean it's smart, but it doesn't seem Extremely extremely hard and when you change it into this we have no idea Okay, so here these are a few a few positive results and now Another positive results by the way is this honeycomb conjecture by solved by hails in 2001 Which tell you that if you want to minimize the perimeter Among tilings by region of equal area The honeycomb lattice does the best So now just one minute. I will take one minute to say to come back to these fakete points So you remember there was this question on minimizing fakete distribution of fakete points on a manifold so the product of Xi minus xj or more generally Sting with risk risk kernels the sum of one of our Xi minus xj to the power s so in fact this thing comes up a lot in this question of interpolation because the Optimal points give you the best way of interpolating a function and of computing numerically integrals There is a whole very interesting article by saff and Kuhler's In in mathematical intelligence sir and they explain why these things are interesting One of the reasons they give is the understanding of the carbon molecules that live on sphere You know these soccer ball types molecules, but also interpolation, etc and so here is a picture that you find here for Numerical computation of fakete points on the torus and you see You do like to believe that if you zoom around the point you see a triangular lattice picture seems very very telling in that case and so What what was conjectured in fact is that if you take this energy and you expand in powers of n You have an expansion all the way to all of one And people ask, you know to prove this thing so what we should be able to prove now Is that the order n term this constant here? Is the minimum of w so we should be able to identify this constant and relate it to this other minimization problem and maybe to prove to you that this thing is of interest to people just want to finish by saying that this was asked as Something like this was asked by smell as a seventh problem for the next century. So what happened is at the beginning of the 21st century. So this one this century Mathematicians congregated and they try to reproduce what Hilbert had done in 1900, which is to phrase problems for the next century And so this thing was recorded as smells sevens problem. So it's again is to find it's to find the minimization problem this minimization problem on the sphere, but to provide an algorithm That will give you a minimizer up to an error of log n Okay, so this thing is considered very very difficult. Of course, you want to do it in polynomial time And the reason why it's difficult is because if you look at the local minima of this interaction energy There the number of local minima is expected to grow exponentially in n and so Algorithmically, you know if you compute you have a very large chance of being stuck in a local minima not knowing how to Extract yourself from there. And so whether this is solvable in polynomial time or not Is not Okay, so I hope I've tried to Convince you that this abricus of lattice is in superconductivity, but is also sort of everywhere and That there are some interesting math problems Related to that. Thank you Thank you very much Like I reattached a great importance to the least action principle. So I'm sure he would have liked your lecture other questions again Did you make some kind of numerical simulation or computation on the quasi periodic crystals like Penrose dialing or something? We you know, some of them have high degree of quasi symmetries. So maybe some of them are just better than the equilateral lattice Did you maybe maybe you can make some computation? Yeah, I mean There are some first of all is difficult to do computations with this because again you get stuck in look You know, you cannot prove that you have a global minimum, right? It's very difficult to show that you're not in a local Sure sure sure sure, but so so far no one has found anything better So all the numerical computation seem to go in the direction of the triangular lattice Not me not myself, but some people have done yeah, and they seem to as far as you can see You don't see any quasi crystals. That's all I can say who knows Remarkless Sorry, you mentioned you you mentioned beta equals one and beta equals two and This corresponding to real and complex case But beta equals four wouldn't it correspond to the quarter and some kind of quaternionic case and can't it be for instance? simulated with or Understood more efficiently analytically So yes, be type was for it the quad quaternionic case That that's just more difficult to to analyze because the computations are more complicated from the algebraic point of view the you know probabilists work on this But in fact the point is not that the point is that you only have one to four for which you can say anything Using these and then in dimension one there are these trigonal Matrices which can work for any beta etc. But yeah, it does it Be the temperature of one half seems higher than temperature of one four Yes, would be one more So what you can do is you can see as you increase beta in these models whether you indeed see more order In the local statistics and you do you do so we did some computations which compute W for these Processes that are obtained as locals, you know local limits of matrix models and you observe that W decreases as beta increases, but You cannot make a whole curve, but it seems to fit like everything seems to fit quite well Thank you Let's talk it's up to 30