This video is going to be about prospecting lead generation and referrals. All right guys look we're going to talk about lead generation. Lead generation is something that most sales people don't know how to do. And look repetition is the mother of skill. Okay in the beginning what I'm going to teach you right now to be a top sales associate is how to generate your own leads. So we're going to start out number one with lead generation. Now let's talk about some things that don't cost any money. It's just sweat equity which means you got to show up and do the work. I'm going to start out with number one. We're going to talk about Facebook. Facebook is probably one of the most powerful things out there that you can leverage and use for lead generation. Why? Because everybody's on Facebook. So there's some things that I know. So most people between 7 a.m. and 10 a.m. in the morning when they're at work what we call scrolling. They're scrolling through Facebook. You know what they're doing? They're not working. So if you were to post on like let's say Facebook marketplace at that time you're going to have a lot of browsers. Okay now I'm not talking about car salesmen not working although that hopefully isn't you. I'm talking about sales associates that are working to post their cars on the market to catch our customers scrolling by vehicles that we have to create interest. Maybe they have an itch. They want to look for a certain vehicle. Maybe it's not today. Maybe it's in the future and you're going to scratch that itch. How you going to do it? You're going to do it by posting an ad. So I would recommend number one we want to start out between 7 and 10 a.m. in the morning post one car on Facebook marketplace. The marketplace is a place that has hot and cold in market buyers right now. Okay cold means this maybe they're four to six weeks out but if you post a car today there's a good chance that they could reach out to you and be in the market today. Alright scratch the itch. Then you have hot which means in the next day or two they're looking for a car to buy. So they're on there looking for and by the way you can post cars all day every day but I want to tell you how to start your day. So prospecting I believe can be done midday but needs to be done in the morning. Get in on the early shift. Go outside.  You say Andy what do I post? Well post something that's hot. Okay something that's unique. Okay when I say the word unique let me explain what I mean. You're taking a 2011 model and it's got 40,000 miles. Well it's an older car with lower miles that's great. You're taking a 2019 but it's got 90,000 miles. Well that means somebody can get a 19 model and it's going to be priced less than the market usually it has it priced because of the miles. Okay and that right there could generate a lot of leads but also you can have somebody that could find an attractive car like a 2011 with really low miles that even though it's pre-owned it's new to them and their family. Maybe it could be a lifted truck. Maybe it's a diesel.  I got that but there's a specific car today that I bet if I had posted somebody would definitely be interested in it. Use your gut feeling for knowing your business and find that car and then go and post it. So what I'd recommend is that you could use the dealership pictures okay but I like to take the vehicle. I like to go photo it myself. Take four to six pictures pretty simple.  Not on your Facebook page but Facebook marketplace. Go through and post it. Now your ad copy is really important okay. I want you to think about how cruise line ads work. They show a picture of somebody taking a cruise. There's beautiful people there's sunshine there's palm trees there's water you know people are laughing they're having a good time they're relaxed you know make it look like it's a once in a lifetime find. So let's talk about Facebook on your main page right? Facebook post.  Well if you have 1200 friends you have access to 5000 friends on your Facebook page. Go and start messaging or sending friend requests to everybody in your area if you're in San Antonio and get some more friends. So what you can do is that also when you put that car on Facebook marketplace you could also take and put that car on your Facebook post every day. Now that means the people that are your friends will see that post it'll pull up in their feed so when they're scrolling they'll see what you have for sale. Also you have what's called Facebook story. So you have a little story box okay and it looks like this it allows you to add a 26 second video. As a sales associate I would say you take 26 seconds to talk about the special that you have.  I can't believe we just traded for this car. Look guys if I had a hundred of these I could sell a hundred a day nobody knows that we have this car. If you're looking for a new vehicle guys I've got this once I put it on the internet it's going to be gone this is an early bird special reach out to me 999999999 guess what I have this car it hasn't even hit the website yet let me tell you about it. Boom post that 26 second video on your look look let me make sure you understand on your story okay that's your story you got your Facebook which is called your post and then you got your Facebook which is called your marketplace which means this is actually where cars go for sale and people look for cars that are for sale so that is going to be a form of lead generation I would tell you now secondly you could also post on Craigslist Craigslist is hot it does cost five dollars per post a post on there this one's free but if you have a really unique hot deal rare find or something put it on Craigslist it's great okay guys so listen let's talk about as a sales associate what it means to know how to prospect now look as a sales associate if you're on the showroom floor your service department is always humming always humming you can always walk over to service and meet somebody that's what prospecting means okay lead generation and lead gen means to actually go create an ad maybe make some ad copy take pictures and try to create someone to reach into you these customers most of the time are already in your store so you could always go over into the service drive and obviously I don't care where you're at as a sales pro as a sales associate your dealership needs more inventory it's just the truth there's a shortage of inventory around the world right now so more cars equals more sales so you could always walk over into service and say hey how you doing my name's Andy great to meet you number one I just want to say hi I'm glad you're here today um what are you in service waiting on and they can tell you and then say hey I just wanted to say this our dealership is giving more money right now than what cars are worth it's crazy we're selling cars faster than ever but they love pre-on inventory would you mind if I ask you a question if if I could give you more money than what your car was worth more money than what it was worth would you entertain selling it to us and they say well what's it worth and you say hey great question look let me get to that secondly if you did get more than what it was worth what would you replace it with if you did sell it to us would it be something less miles bigger smaller maybe newer brand new again what would it be if that number was right that right there is called prospecting guys I want to tell you to work your service drive and to work your service station you have to understand it's one of the hottest areas in a dealership the trades are the best the customer is already there in the dealership they haven't done a lot of research but who doesn't want to upgrade for a great deal so as a sales associate learn how to prospect it is very important and you'll never lack it in your service department okay guys so let's talk about number three which is going to be referrals referrals or everything guys listen men and women sales associates all the top professionals in the world in the automotive industry understand you spend two to three hours to deliver a vehicle to sell it why wouldn't you want that customer to send their friends and their family and to come back and buy all the vehicles they're going to buy for the rest of their life and make that first two or three hours you spend with them count for the rest of their life you you've got to look farther than the hand in front of your face you've got to be able to look out here and say look the best opportunity that in your man or woman could ever ask for is in the automotive industry and it comes from referrals so scaling your income as a sales associate will happen by creating a pipeline of referrals as you continue to sell so one year two year three years four years on your fourth year you shouldn't even ever have to go out and do lead generation or prospecting because you have so many referrals coming in that it's just your day set before your day's even here okay these people that you've sold one time they want to purchase something again and it's their time back around so number one how do we handle a referral okay and how do we start it so if you don't have any referral business right now number one first day call after they buy obviously tell them thank you three day call seven they call thank you thank you any questions about your car any of the way it operates that you don't know about or understand i just want to make you out maybe i'm reaching out too much but i'd rather reach out too much than maybe you not reach out to me and tell me that there's something wrong or there's a problem i just want to be there for you and serve you at the highest level and give you world-class customer service when you do that it's just finding bias to your customers that you were as great as you were that you were a genuine person you didn't just do a thank you there's your car and then if you need something else let me know you really stayed in tune with them and told them that you were different if you act like your competition they'll treat you like your competition don't this is a chance for you to really prove that you're different after the sale which will really truly create them as a referral and make them loyal to you so my goal is is that you do that first week of follow-up and then after that you know you have your monthly follow-up every month you need to call them and say how are you doing i just want to reach out and say hello and say thank you if you're not going to make time to pick up the phone you're not making time to scale your business now whenever i talk about referral something that's really important you can get referrals from the person you just sold but also the person you just sold could send you a referral okay one of my favorite ways to do it is to leverage social media and what that looks like is this i'm a sales associate i have miss johnson she just bought a brand new bmw i would say miss johnson i would love to inspire other bmw buyers out there that are like you that this is a great place for them to come do business with would you be okay with that i'd love to make a five to ten second video i'd love for you to be in it i'll do most of talking if you'd be okay with that they say sure that'd be no problem i say hey guys what's going on it's andy i'm here with mrs johnson guys look at her brand new bmw she just purchased it today she's amazing she deserves it but miss johnson what did you think about the service you got today was it great what do you think she's gonna say she's gonna say oh my gosh it was amazing you guys are awesome say miss johnson i can't thank you enough congratulations on your brand new vehicle look if you ever need anything i got your back for life i appreciate you so much that's it and then what you do is that you'll post that on your page but you'll ask you'll say would you mind if i friend requested you right now and you accepted it and then i tagged you in the post i would love for you to share the celebration on your facebook page would that be okay what do you think they're gonna say they're gonna say sure that'd be all right so now they've done add to add to profile they've put that on their page now mrs johnson has your video you made with her in it and her saying to her 5000 facebook friends that she got a great experience from you now i want you to watch this this is very very important you can't miss this part people are going to start commenting on her page or liking it on her page with her purchase when they like that i call it turning leads in the likes which is actually another form of lead generation and prospecting and what you do is that when that person likes that video you want to take that like and click on it and see who it was because you'll get notified and then you want to basically send a video and say let's say her name was ruby she liked it i say hey ruby what's going on it's andy look i sold miss johnson her car i saw your thumbs up that we were celebrating she got it number one i just want to reach out so you have a name with the face i was the one that sold it to her she's amazing look if you ever need anything from me my cell phone number is 999999999 reach out to me let me know if there's anything i can ever do for you have a blessed day boom now all of her friends got a video from the salesman that sold her the car saying you're awesome thank you for thumbs uping and celebrating with her there's anything i could ever do for you let me know here's the deal you may have one person reach out and say i've actually do got a question do you have any of these bam lead generation referrals somebody comments below never fish in the comments remember i said this somebody comments below and they say oh my gosh judy beautiful car you need to comment and say thank you so much for commenting on judy's video she's amazing she deserves it and now that person right you don't say hey reach out to me i sold the tour shoot me a text message you never fish in the comments you dm them behind the scenes and that is where you're going to get your best referrals now listen if you did this with every car you sold and you sold i don't know 200 cars a year i mean you could only imagine how many people would learn to know who you are look who are you what do you do are you great at what you do and where are you if people don't know who you are and what you do and where you're at i mean what does it matter your goal is is it is it more important for you to know people or for people to know you it's more important for people to know you and you sell cars and you give world-class customer service they need to know who you are and what you do guys if you will follow this you will have more referrals than you know what to do but always remember this consistency is key you can't you can't do things once in a while what's what's practiced in private is seen in public these are things that aren't fancy you do them behind the scenes but all the top sales associates the one percenters they get more referrals and they know what to do with it and you never have to wait on a customer for the rest of their life instead of chasing customers their customers chase them okay guys you made it to the end of the video you are the one percenters do me a favor shoot me a text message 918-210-254 listen let me ask you a question are you making a quarter of a million dollars a year selling cars are you making a half a million dollars a year selling cars why are you accepting less than what you're worth don't be afraid don't be paralyzed with fear and indecision reach out to me guys i mean 700 grand a year selling cars i will help put together a strategy and a game plan for you specifically for you how you can dominate it in the store that you're at shoot me a text 918-210-254 i got your back for life let's crush it


Hello, and welcome back to the craft box. My name is Brenton today. We are on episode 225 Of the was 101 death walkthrough last time if you haven't checked it out already.   Sorry the frigid more I Said that because it's a copy pastor of that I said in the last episode as well, but you know, it's fine And today we are going to be starting by going to the lift Which is the passage to the mine lift Rasputin spit fiends will be blocking the way I hope you can handle them. I'll do what I can Okay, I don't know what you can do, but we'll see what can you do Cesar You don't give us that you give us for us 10 resists. That's pretty good brings us up to a 54 And that's about it, but I mean 10 resistors good enough for me. I Would take that looks like we got even more penguins who managed to escape from their cages Why they rush back into the mall? I have no idea Penguins I only see one penguin in fact this other one looks a bit like the master, don't you think? Nah, I don't see any resemblance Besides he said to let the oski through no one else. We'll have to whip these ones into shape Okay, let us destroy these guys off camera and I'll be back in just a sec. Okay I am back. Let the master deal with these troublemakers. Yes. I think he might enjoy that We should get back to other duties. Hey, right Okay, good job. Um, let's go and search for Ivan down here And we will find out what he's been up to What's down here? Oh, hello Ivan the great always gets his man Even when man is slimy and skinny now Rasputin you too will feel for survival's fist We like though you truly are an imposing figure Ivan I like that, but I think I have a better use for your strength. Let me show you Ivan is feeling sleepy Very sleepy very friendly very agreeable. Yes Ivan will do what Rasputin says Ivan is feeling a little tired Yes Yes Ivan will do what the Rasputin says Ivan hello, are you feeling all right?  Yes Ivan very good Now pull the lever and take us down. Yes master Ivan will pull lever take us down. I Knew it your Oskie friend betrayed us now They've taken claim to the bottom of the frigid more will have to go the long way. Let's go Okay, yeah, I don't think he Betrayed us on purpose you see I think it's a bit different Cesar just has different motives any just Game hates me when I'm recording. I really hate this game when I'm recording. What is this all about? I don't like this why is this doing this my network is fine Fix your game Kings. I'll please for international players. That would be great. Thank you I'll be back in just a sec. Okay. I am back. I swapped over my antenna and we'll see if it works now This is the bad sort of thing So I won't be surprised if I stop in a bit, but you know what here we go There you are. What happened? They got away. That's what happened down to the bottom of the mall I don't get it. Why did Clementine go to Oskie in the first place? She said she needed Ivan's help to steal something from Rasputin, but what does that matter now? You must go after them This passage will take us deeper into the frigid mall. It goes past the pebble trough It's a cave where we have this silly old tradition To court a woman in Ervil young men are supposed to collect a betrothal pebble from the trough and give it to their sweetheart Clementine's been expecting me to present her with one and if times were better I would but right now. I just want to make sure she's safe Don't worry. She will be fine. If anyone can save her this wizard can you go on I will wait outside the mine could lock him as a me Okay, let us get on down and we will What is my father doing down here? Probably some crazy new prediction. Come on. It's dangerous We have to tell him to leave What is he doing? Oh my gosh? I'm sorry. I had no choice, but to come I know my recent predictions have been flawed. I just couldn't ignore this one my son I know how far new or if Clementine, but I have foreseen that you are in great danger of losing You don't need psychic abilities to see that Rasputin has her. That's why we need to hurry We don't have time for this nonsense right now No, son Rasputin is not the risk. You are at risk of losing Clementine's heart She's been waiting for you to court her for years If you do not collect a betrothal pebble right now from the pebble trough I have foreseen that you will never have the chance again You will spend the rest of your life alone ill unhappy. I can't allow that. I'm not leaving until you get her a betrothal pebble Fine if that's what he takes to get rid of you so be it the wizard and I will get a betrothal pebble But for the record this is crazy Okay, let us head on in here and see what we can do this time We are so close to leveling up to 110. That'll be fantastic On our way to 125 110 Basically technically would be finished with Polaris before we actually are because Polaris the level cap was 110 But what is this? The pebble trough is a sacred place for the village. These aberrations are defiling it We must purge them immediately. Okay, I'll be back once I've defeated these stone elementals Okay, I am back once again Just a few of those guys I suspect Rasputin put those things in here the rat. He wants to destroy our traditions not today Clem and I will be proof I could not have done this without you I would be honored if you would collect clementines betrothal pebble from the pebble trough Sure Let's collect a pebble Marci buku now if my father speaks truly I will live happily ever after let's go tell him the exciting news Yeah, you're exciting news. Good job. Um, that was easy not too difficult at all. We're gonna level up here I'm hyped beyond belief for 110 like I'm excited very very excited. I don't know why but I've just You know 110 is such a good level.  I appreciate you humoring your crazy old father Citar I know Clementine will be happy to know father It is I who should be appreciating you. It was very considerate of you to come down. Thank you Yeah level up be honest. That was not my only reason for coming I wanted to speak to the wizard as well. I have a new prediction to share There is something very sacred deep inside this mine Our people have always known this Rasputin seeks to steal it and you must stop him. It's the borealis gemstones, isn't it? Don't worry. We'll stop him Come on wizard. I'll explain about the borealis gems on the way Okay, let us hit on down deeper Um Say we're level 110 Level 110. I can't believe that that's ridiculous Honestly can't believe how do we get one damage? What oh? Because of our dick That is a borealis gemstone Irvill's greatest secret as far as we know This is the only place in the spiral where you can find him There are other precious gems here, but the borealis gems are magical We never understood their powers only that they were sacred We did not touch them until Rasputin forced us to start mining them My father predicted that he would pay for it. I hope that proves true Okay, let's keep going down This is where we'll start seeing the miners pit boss minoc uses a charm Rasputin gave him to keep the miners Glamored while they work is similar to the magic Rasputin used on the oski Not quite as perfect, but the miners are almost entirely under my next control Ah, there is Lambert like Clementine. He was immune to the pit boss's charm though. He's not looking too good now Let's talk to him Okay, okay Lambert with an extremely big name. Hello Mine the borealis the master needs more gemstones no time for idle talk Please leave me to my work. I am a miner and I must mine I Cannot in good conscience leave my fellow prisoners like this We need to free them before we confront Rasputin We'll have to face down pit boss minoc and get the overseer's charm won't be easy But it's the right thing to do. Come on. It's this way Okey-dokey Let's go and do this Hello. Hello. Hello. Hi there. Hi. Let's go in Let's find out what we got to do today Which boss we facing obviously put boss, but I don't know if that he's the first one we have to fight mobs first Who knows? Let's find out It's not fair How do we even get station down here these beat things have no respect? They never let us do anything And you're complaining about this be careful what you wish for comrade. Oh You'll see I think we've got trouble.  No, thank you. That is a bit too much for me Yeah, I think they should have just stayed upstairs as well to be honest because the most they're all balance And it's great. Where is these guys death and fire? No, thank you Let's see. We'll do this we will do This We have a blade hi, I think they can probably go for a E-freed.  Well, let's see. What can he do finally found him Um He can a free. Yeah, it's already good. That's great. He can also choke as well. Well, he just choked So yep, he can choke. That's fun. Yay This is annoying This is already annoying.  Oh Let's try this again on you It's about to put it on the wrong guy. That would have been bad Fizzle good. Can you like fizzle or something as well? Okay Natural attack won't do too much correct Correct. Nothing 200. That's nothing too bad 68 okay Yo boop fantastic prism And we will get another Thing we got this That's good. Are we fizzled didn't we? I'm like didn't we put up a blade?  Stop with the smoke screens and the stuns and the Minus accuracy is just stop with it. It's not cool. I dislike it. I disapprove Let's go for one of these We'll go for this one first because why not Then we will go for the spirit blade I swear all they do is just natural attack natural attack natural attack bubble skeletal dragon Honestly, that's all they ever do.  Oh, no, we got the best. That's fine minus 25 isn't too bad Anything more than minus 25 is bad And that's for anything Like minus 25 is easy to deal with for like a shield for like weakness But anything more like minus 30 35 40 45 60 65 70 75 80 85 90 no just no We're gonna discard you Because I don't think you're necessary. Did I faint the right guy? Yes, I did Yeah, okay Just checking sometimes I fainted the wrong person and that is awkward That's awkward What's this how much is gonna do like nothing 111? What a good number. I love 111s won't probably one of my favorite numbers.  I don't know why it's also why I like number 11 It's just double ones. It's cool It's cool. Um, okay, can we I Think a couple more blades and we should be good for a Lulu. Luckily we have a shed.  I'm gonna fizzle on time. I didn't fizzle. Yes, that's good Good job. I did not fizzle even with the minus 45. Yay Do He's definitely dead and hopefully you're dead as well Nope, you're not dead. That's unfortunate so close Okay Can we hurry up? Thanks We can probably go for another one And we could do this 2000 we need a Blade if we blade up we get a share that would be great. Otherwise, we will just go for Okay, we will not just go for it just as well. I didn't go for it. Jeez. That would have been bad I didn't even notice his purpose.  That would be fine. I'm like mmm. I really hope we go shared which is good We don't need this if we don't get it. No, we didn't get it. Good job Don't say we have to just cut like our whole and just to find it. Don't be ridiculous Please game, please Please be nice Thank you Thank you What's these?  Hello. This is gonna do what like 200? Nope 133, okay So bad so so bad We can always want it next if we need to Kill which we may have to do or we could do and I whatever really comes up, you know I know we've got a wanted me for tonight. So whatever. I really feel like it depends how much health he is probably There's too much for a wanted me. I'll definitely deny Okay, oh there we go fantastic Look at that. That wasn't so bad. Perhaps we gave the gulag guards more credit than they deserve I suspect the pit fiends will not be so easy Okay, let us hit up to the Pit fiend layer and we'll see who we have to go So you beat the gulag girls good. They finally did their job making a buffer We heard the scuffle you see and we prepared. Yeah, they was just a buffer. You see. Yeah What he said? So I'm gonna do this one off-camera and I'll be back in Just a bit. Okay. I am back. Hopefully killing We have two feints and a couple blades, so it should be fine He wasn't too difficult. We also boost as well. So that's even better At least I think we boost pretty sure Do we really do both of them we boost so easy peasy Lemon squeezy that did mass. Good job. Okay Gula guards for one thing the pit fiends another but the next is pit boss. My neck. I will not lie to you He makes me nervous That's unfortunate. Um, don't need health which is good. The perks of being death. Let's go Let us see For many years my tribesmen and I wandered the tundra until Rasputin recruited us to run this mine It gave us purpose where we had none. I know who you are I remember you from the Bastille. You seem determined to undermine our efforts. I will not allow that to happen Okay Let's do this Prisms just in case Well, not just in case we will need prisms I don't think he would be he doesn't cheat. I know that sorry. That's always good Let's just slap a prism down right away probably a good idea Oh He can do a life shield so that's slightly annoying. He can also Verrulion little in plague That's not cool. He can also juju. Oh my gosh He's good He's actually good He can juju. That's ridiculous That's a minus 90 Plague is minus 40 so either way carry some cleansers. Haha. He also can death ninja pig as well, which is not cool Shift sugar glider, which also gives a minus 25% to the next thing I'm outgoing spell, which is ridiculous as well But we can faint him. So that's good But Yep, yes, the shift sugar glider. Yeah, we crit as well.  It's kind of weird. I don't know They're like no one ever uses them will not as well make the mobs use them then, you know Whatever Whatever a heal. Hello. I haven't seen this for a while. I completely forgot you could heal Honestly thought I had the wrong pet on for a sec because I hadn't seen a healing forever Pet doesn't often heal. Um, so you know something's up. Oh We won't shine. Yo, we will go for you What do we do last time we must have fainted? I'm like some of this blade, but no, we apparently did not blade Must have fainted Okay, hello Um, what's this blade good job didn't fizzle great And what are we gonna do now? Well, we can ensure you that's why it's a step. Was it a potent or a normal? That was normal, so I need to wait for potent.  I Juju's three pips anyway. Good job. Good job. Not even knowing your own school spells. Yay me Yay me Okay, we'll just cut you Not making it easy for me game. We will draw hoping to find a blade good We will blade What's this a shield? Okay, that's good You know what I might actually put up the bubble if we get it as well Even though that helps him might put it up. He's gonna Lulu probably which will be interesting We'll go for a faint He bit a Lulu He bit a Lulu, you know, it's really bad. I just realized a few jujus. I'm gonna lose all of these faints in the prism I only just realized that it's bad That's really bad. I needed cleanse now anyway Okay, we've got a shed which is good. I will hit next Once I cleanse, thank you I only just realized that That's so bad That's actually really bad if he does that. I Hope he doesn't do that next. I'm gonna be so angry That's probably like a rage quit material right there for me. Jay's Jay's How many do I have I only have three blades and a minus 25? I don't know if this is gonna kill. I Don't know But I'm gonna do it anyway because I need to get the minion out We'd really like to put up that other blade, but we're just gonna deal with it for now. Okay, that's fine. I Get worried about these guys because these guys start to use shields and they're actually start to get kind of like Not good, but they use shields which is Kind of good But like just more annoying. We crit. Okay. He's dead. He's dead if he doesn't block. He's dead if he doesn't block Don't block Good. Okay. He's dead. Yay, Gigi There we go. Fantastic. That's a mess on the minion. Did you see that? Wow, I have never seen anyone stand up to pit boss mine. I cannot get squished You are even more formidable than I imagine now. We have the overseer's charm We must simply take it back out into the mine and use it to free the miners Let's not forget lumber too. Okay, let us gone free the miners And then I think we'll end off the episode there Next time we will continue on with the Frigid more is that what it is? I think it is pretty sure that's where we are, correct Correct the frigid more but let's free the miners because we deserve to free them They deserve to be free finally. Oh dear. That was just miserable. Merci beaucoup my friend Who else where's another miner that we can free Who's next to you? Hello. Oh Dear that was just miserable. Merci beaucoup my friend. Okay, it's just repeated dialogue. That's fine. I Was exhausted, but I couldn't stop you save my life Can maybe not quite all repeated dialogue. Ah, I feel better now. Thank you. I'm getting out of here Okay Dear that was just miserable. Merci beaucoup my friend Wow, that feels better.  I hope I never see a gemstone again Does this mean I'm free? Yes monami. You are free. The town has been liberated. You should go up and look for my father Thank you, Cesar and you strange and you should know however that there was a massive borealis gem on earth on these shift the other miners were calling it the borealis majoris I believe even now he has come for the borealis majoris I fear what he intends to do with it if it is within your power. I suggest you stop him Borealis majority that's why he needed Ivan's strength. Don't worry Lambert I've seen this wizard's magic Rasputin will be stopped Okay, and that is where we will end off this This episode we're slowly loading in a whole bunch of stuff over here. Don't look at it. It's cheating Next time what we will finish off the rest of the frigid more which isn't too much, but I don't want to Run this episode way too long. I'm just because I did recently Re-look at how I'm gonna be finishing off the series and there are gonna be a lot of long episodes coming up Especially in Mirage, so yeah, we're gonna kind of keep it a bit shorter at the moment So next time what we will continue on thank you for watching We're into little I can subscribe and if you're very subscribed press the we little bells You know exactly when I post another video and remember craft outside the box


Thanks a lot. Yeah, very long title. It's there anyway. Yeah, so actually, it's a very, it's really a great pleasure to be here in Trieste. I should say be back here. Because there was a sort of version of this conference six years ago, I guess, 2013. And I was a little baby at that point. I was just starting my PhD. And I came in the audience. Like, it was probably one of my first year as conferences. So it's nice to be back and I'm very thankful to the organizers for giving me a chance to be back and to give a talk on a bit of my work. By the way, this is joint work with Andei Martínez Finkelstein from the University of Omega in Spain and also Baylor University. Parts of it is already on the archive, but we made some improvements that we might update. We're putting a new paper, who knows. But anyway, I'm going to talk more about that along the way. OK, to get started, let me just pose a question which is super, super basic. It can appear already like in the morning session, for instance, doing a leases lecture. And you take, let's say, your favorite matrix, m, random, of size n, capital N, with certain eigenvalues. And you just look at its counting measure here. One of our ns, some of delta masses at the eigenvalues. And what you're interested in is in the limit when the size of the matrix goes to infinity. You have more and more eigenvalues. You're constructing a measure that's counting those eigenvalues. So you really should think of this as essentially the measure that counts eigenvalues in a certain interval if you integrate this measure over the fixed interval. And anyway, we're interested in the large and limit of this, of this random measure. Actually, there is a more deterministic side of this question, if you want to say so, which is the following. Instead of looking at the random measure because, again, this right-hand side here is really a random measure because eigenvalues are random. You can actually average that. So you take the average of your matrix of the characteristic polynomial. So now this Pn here is a deterministic polynomial. And you can look at the counting measure for the zeros of this polynomial and ask yourself about the limit when n goes to infinity of this deterministic measure now. So what I would call the measure for the zeros here, so deterministic zeros. And here the measure for eigenvalues, random. And of course, I'm saying here that the limits are the same. But this is like a calculation that they have to do sort of model by model. But it turns out that in many models, when the eigenvalues are on the real line, this is true. So the limiting, the measure for the eigenvalues happens to be the same as the measure for the average characteristic polynomial. And that's going to be the case in our talk. So for most of the time, I'm going to be focusing on the last one, on the counting measure for the zeros. And then just at the very end, I make some comment on that. Anyway, but just to have a warm up, let's look at one model, which I can say is classical right now, which is what's called the emission matrix model, emission random matrix model. So many of you know, and many of you also saw today's lecture by a least. So I have to thank her, because she introduced many of those things. So over there, if you put, let's say, so over here, we're considering like emission matrices. Random with distribution, which takes these forms, exponential minus n trace of v of m. So v here is just an appropriate function. Let's say, let's stick to a polynomial. And so just to make a connection, if you put, let's say, v to be Gaussian, you're just exactly back to the model that Elise was talking about this morning. Well, let's say we have a general polynomial, v. Well, it's not that hard to compute, to be actually an orthogonal polynomial for the measure e of minus n v. Again, let me just trace back to what now Alexander Buffett was saying this morning. And he computed, like, in that case, if he had Gaussian, he computed that to be admit polynomials. If you have a general potential, you just get other orthogonal polynomials for other measures. Anyway, so this polynomial is known. This measure, the limiting measure for the zeros or for the random eigenvalues exists. They're the same, let's say almost surely, if we're talking about the eigenvalues. And this measure minimizes the log energy, the weighted log energy on the real line. So if you want to find this mu star, what you should do, you should look at lots of measures on the real line. So every probability measure on the real line, then you compute this double integral for this measure, plus this single integral with your potential. And then you minimize this quantity over all probability measures, right? So you give me a probability measure, compute this quantity here. This gives me a number, and I minimize this number over all probability measures on the real line. And the one minimizing it is exactly the one that's going to be mu star. That's gonna describe the limiting eigenvalue distribution, in that case. There's another characterization of this measure, which is perhaps not well used, but we're gonna need, or at least we're gonna generalize that immediately, which is the following. So if you compute, okay, suppose that you know already this limiting measure, and then you compute this Cauchy transform, or sometimes called also still this transform, right? Just a transform, a transform of the measure. So as a function, right? So it's a function of z, analytic function of z outside the support of mu star. And it turns out that this function here satisfies an algebraic equation. Where this algebraic equation here, because v is a polynomial, these those a1 and a0, they're also polynomials. So you can imagine that, you know, for hz, you solve for psi, you have two solutions. One of the solution is exactly this analytic function, which is the limiting eigenvalue distribution. And I must say that this is some sort of characterization. So if you look at the equilibrium measure here, this is very, I mean, there's a unique measure minimizing this functional. So you know that, so there is uniqueness of mu star through this characterization as a minimizer. But also in some sense, if you impose some conditions on a1 and a0, there's a unique measure on the real line that satisfies the corresponding algebraic equation in some sense. So in some sense, you can think about this algebraic equation as uniquely determining your limiting eigenvalue distribution. So just have in mind two main objects here, an energy minimization to determine your eigenvalue distribution, or also an algebraic curve, or what's called spectral curve to determine the limiting distribution. Again, if you didn't follow any of these, let's be very concrete, as I told you, if you consider, let's say, V to be a Gaussian, so a potential to be a Gaussian, then essentially we're talking about matrices whose entries are independent Gaussian, appropriately normalized. Your average characteristic polynomial is a mid polynomial, as I mentioned. The limiting eigenvalue distribution is just the semicircle, the semicircle distribution on minus root two to root two. And this algebraic equation is very explicit and it's over here, right? So a0 is just constant equals two and this a1 is z and then you can solve here very easily for psi and then you can recover those series transform and from there you can recover the measure itself if you want to, right? Okay, so you're gonna now look at that picture, those type of results, but for another matrix model, which is a kind of very, I would say nature, very simple, at least generalization, which is when we consider the very old trace of v of m, but now we just add some perturbation here, a polynomial perturbation, or sorry, a matrix perturbation of this v, which is given by minus a m, where this a here that you see is a deterministic matrix, which actually we can fix to be a diagonal because you can essentially diagonalize m and fix a to be diagonal. Anyway, and this is called the external source, this matrix a, and v as before is just a fixed real polynomial, right? And for the talk today, I will fix our results, fix on actually a with a certain form, which is when you have only two distinct eigenvalues, the symmetry that imposing that a of a minus a is not a big deal, you can always change your potential reduced to that if you don't have symmetric eigenvalues, but what we cannot reduce that we're imposing possibly different multiplicities, let's say a comes with multiplicity and one, minus a comes with multiplicity and two, they don't have to be the same, right? For some of our results, we believe, although we didn't do, we believe that they can be extended to general a with more distinct eigenvalues, I might make some comments at the end, but for now, let's just, for our results, we're gonna be restricted to this situation here. Okay, so the picture that you see there is exactly what we discussed already about the standard Hermitian matrix model. So we start with a counting measure for eigenvalues or counting measure for zeros of the average characteristic polynomial, and we understand the limit if it exists. So let's say, can we write down some sort of variational problem that characterizes this limit before we saw that for the Hermitian matrix model, you minimize a log energy with a weight on the real line and you get the measure that you want. On the other hand, you can also look at a more analytic, from a more analytic perspective, you can look at a more complex analytic perspective, you can look at, let's say, your, the Cauchy transform of your limiting measure, if it exists, and try to see if it has some properties. Let's say if it's social algebraic equation, as we saw before in the Hermitian matrix model, when a equals zero, identically zero, you just get a quadratic equation in the Cauchy transform. And from that, perhaps you want to see sort of local properties for your measure, or in other words, you want to perhaps classify university classes, whatever that means. Okay, so when they equals zero, again, we just recovered the Hermitian matrix model that I showed you in one of the first slides, and so this whole description is well known. I have to make some, of course, I have to thank several authors, many of those in the audience, for lots of things that I've learned from them. For instance, if you consider this polynomial v to be symmetric, and also when the multiplicities are the same for the perturbation for this external source a, there are several partial results in the literature. Again, a very symmetric situation. Let's say you could also fix, let's say n1 and make n2 equals zero, or sorry, I should make a, or minus a equals zero, just fix let's say one, one, this thing I can value with small multiplicity. That's kind of our small perturbation, any time symbol. This has also been studied in the literature. And also when you make, let's say your external source as general as you want, but you make, let's say, just a Gaussian potential, there are also several people in the audience who contributed to that. And so there are things that are known in various different setups. As far as I could trace back, there's not much known for the situation that I'm describing, although there are lots of natural questions with natural answers already available, although no natural proof, let's put it this way. Anyway, so okay, here's the picture and what we're gonna talk about today is the following. So first, can you go from let's say, let's say a saddle point, so let's say a variational characterization and then find this spectral curve, an algebraic equation, and the answer is yes. We were able to do that back in 2016. I'm not gonna talk much about that today. Recently, what we posted on the archives, let's say the other arrows, so we somehow can start from the algebraic equation with a certain structure and then obtain the corresponding variational problem from there we can classify all university classes. So just from the algebraic equation or from the spectral curve. And again, why we started with that algebraic equation? Well, because there were all those cases before that told us, hey, perhaps the structure of the algebraic equation should be of this form. So we start from there, from the structure that we wanted, we did lots of things and then we went back and say, wait, but we have an end to prove something concretely for the matrix model, not only starting from, let's say, the ideal situation of an algebraic equation, and we were very lucky, actually, we managed to prove in the past few weeks that actually there is a limiting measure under various mild conditions, which at least told me it's a hard condition anyway, but I'm gonna tell you later. But anyway, under some very mild conditions I'm gonna show you today, we can actually prove that there is a limit of those counting measures, at least along subsequences. And any such limit along subsequences has all the characterizations that I just told you about and that I'm gonna describe today. Yep. Okay, so this is basically the picture that tells you everything they're gonna discuss today, but let's go a bit to the results. The first result that I wanna talk about too today is some sort of finite version of this algebraic equation. And that will give us a hint of the, let's say, infinite N version of it. And this version is as follows, I just have to introduce some quantities. So remember that I told you before that in the one A equals zero, in the standard emission matrix model, the average characteristic polynomial is just a tonal polynomial. Now we don't have necessarily a tonal polynomial, but we have a slide generalization, which is what's called a multi-portal polynomial. So for a tonal polynomial, what you do is just ask the polynomial to be orthonormal with respect to one fixed weight and now possible monomials except for the degree of the polynomial itself. Now instead of asking for the polynomial to be orthonormal with respect to all monomials, we split the monomials into two groups. And then we ask for our polynomial to be orthogonal to have, let's say, whoops, where is my pointer? Oh, my pointer just disappeared or my computer just collapsed. Okay, so there was a pointer here somewhere. Ah, yeah. Okay, so instead of asking for, let's say, all polynomials for the polynomial to be orthonormal with all monomials, we split the monomials into two groups and then you ask for the polynomial to be orthonormal with respect to, let's say, some monomials for one measure and some other monomials for another measure. And yes, perhaps, it'll come back. I can talk in the meanwhile. Yeah, so you just change the orthonoid conditions split into two different groups and if you look at those conditions over there, you can just put it there. If you just look at those conditions over there, if you just count, yeah, thank you. If you just count the conditions over here, I have, let's say, let's say K1 condition in the first line, K2 condition in the second line, so I fully determine a polynomial degree K1 plus K2. If I impose this polynomial to be a degree K1 plus K2 and monic, right? So these, those conditions, determine a nique polynomial. It turns out that the average characteristic polynomial satisfies those conditions, right? So it's a characterization of this average characteristic polynomial in terms of multi-portal normality conditions. And I'm going to need that. And then what do you do? If you look at that, those multi-portal normals, they depend on both N1 and N2, the multiplicity. You can play around and then reduce one multiplicity or the other multiplicity and then you can construct a vector. So here's the average characteristic polynomial and the other two are just polynomials that you obtain when you reduce the ortonite conditions by one, one way or the other. So, and then you multiply by E of minus Nv, so this is sort of a wave function, right? Well, the first result that is wave function here satisfies a first order ODE with where this R here is a polynomial, is a matrix with polynomial coefficients. I don't even want to put this as a result, as a theorem because for people who are familiar with integrable systems, this is pretty much almost trivial. It's just a calculation that you do and you verify, right? So this is not exactly a theorem, but it's there. But what perhaps we can put as a theorem is how this actually characteristic equation of this polynomial R looks like. So again, so the setups that we have a potential external source here, and we are going to look at the limit when the multiplicity N1 over N converges to a certain parameter alpha here. So I'm fixing one rate of convergence for one of the multiplicities. And then basically you take that polynomial R in the ODE and then you compute this average is its characteristic equation, right? So you just take the determinant, the characteristic polynomial of the ODE, right? And this ODE has a very special form. It's a cubic, which just followed because R is a matrix of size three. But then there's like a V prime here, coefficient. There are some coefficients A1 and A0, which are polynomial just because R is a polynomial, no big deal so far. But perhaps what's not so simple, it took us a bit of calculations to do, which is to see that actually this degree of A1 is at most the degree of V minus two. And even more special, we can compute exactly the first two coefficients here of this R. We don't know R exactly, although we have a representation for R, which is not very explicit, but we can still do stuff with that. Anyway, and then there's A0 here. It's just like connects back to the potential, the external source, and also the multiplicity parameter in this form here. Again, so Vm is the linear coefficient of V, Vm minus one is the second linear coefficient of V, and then you have A square, alpha. Anyway, it's not very important how it looks like. Just have in mind that we're just making a connection between, let's say if you give me the data V, A, and alpha, there is a way to connect those coefficients with those. Not exactly compute the whole coefficient, the whole way is zero and A0, but at least the very first few coefficients. Whoops, and second also, this perhaps like the most important, that's why we call this some sort of fine inversion of the spectral curve. Suppose that you can actually, that you know that all the zeros of the average characteristic polynomial remain on a compact set when N goes to infinity. In other words, suppose that we are in the right scaling. And then what we can prove is that if we evaluate, let's say the Cauchy transforms for the zero counting measure of those polynomials, make a shift, then essentially this Cauchy transform for the zeros goes to zero as N goes to infinity. So in other words, this characteristic equation of the ODE is, gives in the large, the large limit if it exists of this characteristic equation gives your spectral curve, right? If it exists, and again we can prove that all those things, actually we can take the limit, these are, yeah, we can prove actually that we can take the limit over here under this condition of all the zeros remaining on a compact set. Okay, yeah, so let me just give a few words to indicate how we prove that. And again, it follows pretty much like arguments on integrable systems. So if you don't know anything about that, I'm sorry, I just wanna give a few words on that. Perhaps it means everything, but it's gonna be only one slide, sorry for that. But if you have seen some of those arguments, probably gonna recognize some of the steps. So the point is that, okay, we look at this ODE as a vector value, ODE, so there was a vector psi and then there was ODE with polynomial coefficients, but actually there is a full matrix version of the ODE where actually the first entry is the one that we just constructed before, this wave function. It turns out that there is a Hema-Huber problem associated with this T, and this Hema-Huber problem has constant jumps. And once you have constant jumps, you can easily verify that there is an associated ODE and you play a bit with the structure of this Hema-Huber problem and you essentially get those conditions that I wrote here as a star, the conditions on the coefficients. There is quite a bit of calculation, I can't say two, three pages, but one can do. It's not like super, super hard. But now, as I said, we also want to take large and limit of that and making sure that we can control all the coefficients in that ODE. And there's more to do that than you have to realize that this R here is very easy to compute if you know T, right? You just invert T and then you compute R basically, right? And it turns out that the inverse of T can be constructed in terms of the biotogonal functions for the system. There is like those multi-paternal polynomials, there is associated with some biotogonal functions. And we can compute the inverse using that. And once you do the math, you realize that this R actually is gonna be a certain non-linear expression but that can be written down just in terms of the recurrence coefficients or several recurrence coefficients although in a very complicated way. And then basically, if you construct the recurrence, or if you are able to control the recurrence coefficients for this biotogonal system, you are done controlling R itself, right? It takes a bit of work, but you can do it. And basically, that's why the boundaries of zeros helps you. So if you know that the zeros are bounded, you can play around some arguments and show that at least the coefficients, the recurrence coefficients are gonna remain bounded, hence like the matrix R, the coefficients in the matrix R itself are also bounded because those can be expressed in terms of the recurrence coefficients and then you just compute everything else or start taking limits along subsequences. Yeah, and as I just told you, under those conditions, you now take the large, you assume that the zeros of the average characteristic polynomial stay on a compact set. And then you can take limit of the, let's say find an spectral curve when you get rigorously that any subsequential limit of the zero counting measure has to satisfy an algebraic equation of the form that you see on these lights. So what you see here is a cubic plus V prime or minus V prime psi square plus, oh, there should be an A1 psi here. So there's a A1 psi here plus A naught of Z equals zero. Where psi here, the solution that you obtain is the limiting measure for the zeros or for the random eigenvalues, right? So any subsequential limit will necessarily satisfy spectral curve, although we don't have uniqueness at this point, unfortunately, right? And again, in those polynomials, as I mentioned, they have a certain, those polynomials are zero and A1, they have a certain structure, right? Those coefficients, again, it's like psi here, right? So then you start, okay, I have like an algebraic equation and there is a probability measure that will Scottish transform solve this algebraic equation. What can you do with that? And that's our, let's say, next starting point. So we can, at this moment, just forget a bit about the rest and then just look at algebraic equations of this form with certain structures. So you impose a certain structure for the coefficients which are just coming from the matrix model, although at this point you don't need to have matrix model anymore. And then you impose also that one of the solutions to this algebraic equation, to this spectral curve, is the Scottish transform for probability measure on the real line. And then you wanna say, can I say something about this measure? And the answer is yes, we can. Yay. Another thing that I should say, again, as I said, you can forget about the matrix model at this level and everything I'm gonna talk about today even works for any polynomial, V doesn't have to have even degree. So the spectral curve doesn't have to be coming from a matrix model, well-defined matrix model, could come from a ill-defined matrix model, whatever that means. As long as V here is real. Okay, and our result now is essentially completing one of the pictures that we saw in one of the first slides which is, how do we go from a spectral curve to actually a variational characterization of the measures? And the theorem is as follows. So you start with this measure, mu star, which solves your spectral curve, right? And then we are gonna construct a vector, three measures where the first of the sum of two, of the first two gives back your measure. And very importantly, this vector, three measures is constructing such a way that this vector is a saddle point of an energy. Remember, before, when we were talking about the emission matrix model, we were minimizing a certain energy. Now this third measure, mu three, which is kind of auxiliary, is gonna live on the complex plane. So we're not gonna exactly minimize, but we're gonna look at perturbations of this vector in a very specific way, but I'm not talking about what specific way that is, but just look at certain perturbations, so nice enough perturbations of the vector along a certain direction, and we see that actually for a certain energy that we're gonna write down in just a moment, we see that this vector of measures here that connects back to the spectral curve is critical, is a saddle point for that energy, right? Also, one thing that might be important for people who like Him and Hilbert is that this third measure, this auxiliary measure is connected. It lives on the plane, on an arc on the plane, but if it's non-zero, it's gonna have a connected support. And also as a consequence of our results, we can classify all possible singular behaviors that could occur for, let's say, this measure solving the spectral curve. What we see is that this measure could potentially vanish like a root in pretty much the same case of, let's say, the classical emission matrix model, so power of a root type of vanishing. It could vanish like one third in the bulk, which was already described. It could happen even for V Gaussian in that case, but it could happen also that this limiting density vanishes like five over three. Although we don't have concrete examples, our arguments tell us that this is a possibility, probably when V has odd degree, not even. I don't know. But so the fact that you have this type of vanishing is not a big surprise, but it was kind of a surprise for us that we don't have higher powers of three appearing over here, so that doesn't appear. So your density, if you put an external source, you might create some sort of new university glasses, but not many. You don't get a higher order of piercing, for instance, in that case. Yeah, and I promise to you to show the form of this energy. Let's dive into that, it's a bit technical, but I said we have three measures. The sum of the force to recover the measure that solves the spectral curve, having mind the limiting eigenvalue distribution. So the first two measures here, they have totem as one, and then the difference between the force and the auxiliary measure has totem as alpha, alpha is this parameter of multiplicity, the limiting multiplicity for the external source. And the energy here involves laws of terms. There are certain log potentials of each measure, mu one, mu two, and mu three. There is a certain plus interaction between mu one, mu two, a minus interaction between mu one, mu three. So sort of, if you were to minimize mu one and mu two, don't like to be close to each other, but mu one and mu three do like to live close to each other, although mu three is on the plane. I'm gonna show a picture in a moment. And there's certain potentials acting on the three measures, which come from V and the external source itself. Okay, ugly form of three measures, what not? Just having mind at the end that we're looking at saddle points of an energy functional for three measures. And this one that solves the spectral curve is also, can also sort of be seen as a saddle point for this energy. Okay, let me show you a picture. So I'm gonna show you a picture of the supports in a specific case. So what I'm doing here, I'm taking my potential V to be a quartic plus quadratic. And I'm making my external source to run from A between zero and one. The picture that you see over here is A close to zero, right? With A is zero, this orange part, the third measure escapes to infinity and we just have mu one. When A becomes very, very small, but positive, the third measure is supported on an arc on the complex plane. The first measure is supported in a single interval and the second measure is not present yet, right? It's zero. Then you start tuning up and then at one specific moment you see that the support of the third measure merges into the first measure and then it separates the first measure into red and blue, meaning first and second measure. Now the three measures are present. The orange is auxiliary one on the plane and blue plus red describes the corresponding eigenvalues. Then you keep moving forward. You see that this orange starts to shrink. And it shrinks, shrinks, shrinks, shrinks up to a certain moment where it completely disappears. The mu one and mu two separates. At this point, I guess it's A equals a half if I'm not mistaken. And at this point you just have, it's connected support for the eigenvalues distribution. So the eigenvalues is supporting two intervals in that case. And then you just keep moving forward and they start to separate more and more and more. And at this moment, the third measure doesn't play much of a role. Yep. Okay, let me make some comments. As I said, so first off, this result that I just described to you doesn't have directly to do with the matrix model. We start with a spectral curve motivated by matrix model, but the input to get this result simply start with algebraic curve with those coefficients like this form and a probability measure solving it. And also we have that, we can write down the variational problem. So this works for V odd as well. Doesn't have to be V even to have a well-defined matrix model. For V even, we can encode all previously known results including A equals zero and also including the results when V was symmetric. Everything we can recover from that. And the main technique which perhaps I won't spend, like I talk fast, but the main technique which I'm gonna spend the five minutes talking about is actually to try to do this not on the plane, but on a Riemann surface. Try to construct those measures by playing around the Riemann surface associated to this spectral curve. Yeah, and I must say that we were successful to apply this to other situations. For instance, constructing what's called a motor bug problem for the normal matrix model. We applied similar techniques and also for us in topics of multi-partonal polynomials which don't have necessary to come from matrix models. And I mentioned especially the second one because I'm gonna show you some more pictures, but the techniques are similar but the pictures take like a week to produce so I just produced them for that example a while ago. So I'm gonna just reuse those pictures and explain to you the ideas. And it just follows. So again, so the picture you're gonna see here is again pretty much the same as before. I'm describing how the solution to this variational problem, how the support of the solution of this variational problem changes with a certain parameter. In this case, so it's not exactly a matrix model but just some zeros of our tonal polynomials. For a small parameter, the zeros split apart into red and blue. Red is on the real line, blue is on the complex plane. I'm gonna play the same game as I did before. I'm gonna start tuning up, right? So to start tuning up, they start to move up to a point where they touch each other. When they touch, things just start to move to the other side. Yes? Ah, here V is cubic. If you wanna look at this V, as again, it's not a matrix model but I'm also not putting external source. I'm dividing V as multiplying one sector and then another sector without external source. So it's multi-portal analogy. That's why I get this geometry but it's not a matrix model. But again, the idea would be very similar. That's why I just wanted to show this one. Yeah, so at this moment, after the measures, blue and red, they merge together, a third measure is created and then you might wonder, hey, but what's happening? Why exactly when they merge or when they cross each other, why do you create this new measure? And that's the answer. That's why you have to move to the Riemann surface because if you look at these at the Riemann surface, not much is happening. If you just look at a plane, something weird is happening. You have this sort of single light that one touches the other and transferring to something else. So what you should do is you look at the Riemann surface. So let me just show you what this means. So remember, although the cores here don't correspond necessarily to the cores we had before. So we're for that. But what I want you to look at first, this interval here, right? This interval here corresponds to this interval that we started with on the real line. Yep. And if you look at this curve here, this other curve here in green, this is the other cut that we started off the real line. So we started with an interval and then another contour like this. Yep. So those counters, so this one and also this one are the ones that encode your measure. And let's say what you do, you're perhaps with one specific parameter, you are able to compute those measures and then you just look at those in the Riemann surface. What are those other curves? Those other curves are what we call trajectories of a quadratic differential. So you construct this quadratic differential on the Riemann surface, which, in other words, you construct a harmonic function which is the real part of a well-defined object on the whole surface. And then you look at all those trajectories, all those curves, all those level lines, if I'm gonna say so, some of them encode your measure, right? So let's say in this case, as I told you, this one here encodes a bit of your measure and this one here also encodes a bit of your measure. Yep.   You start tuning up your parameter, things are moving around, some trajectories move here and there, up to this point nothing changes for the measures. Something now is gonna happen. So you see, so your measure was here, right? And now so here. What's gonna happen now? So now those curves are in order, so those level lines they are in different sheets, they're not touching. What's gonna happen is that some of those level lines are gonna move to another sheet. And in that case it's gonna happen that this interval now is gonna stick out and appear over here. That's when you create the other measure. But again, measure one is here, measure two is here, and then the third measure is gonna be created in another sheet. So actually they're not intersecting, it's just that one measure moves to the other sheet and then it becomes another measure. So there's nothing really, nothing single happening at that level. And then just keep moving forward and of course you have to be able to control all those parameters. As you see there are lots of curves here and there, but at the end of the day what you get is that there is like a green arc sticking out here, there is this red arc here, and there is a blue one here, and those three are the ones that give back your measure. So all you have to do once you are able to look at everything as interjectors of a quadratic differential, all you have to do is to do the formation and to control this deformation all you have to compute, all, although it might take some work, is actually to compute certain integrals and look at the signs of them, which one is vanishing first, which one is vanishing second. So it's kind of not complicated if you are able to control those numbers basically and see which one vanishes first basically. Anyways, so the deformation argument, so as I said, what everything that I just showed to you means is that we look at everything at the Riemann surface and embed everything in terms of quadratic differential. So we start with this algebraic equation, so we start with the spectral curve, there is associated a Riemann surface, which is the three sheets that I showed to you before. So they associate the Riemann surface and it starts from the Riemann equation, you just construct the surface in your favorite way, and then what you do is that you construct a certain canonical quadratic differential on this surface. In a very easy way, you just take differences of two solutions, the square, these are quite differential. It's not very important how it looks like, what's important that basically what you're doing when you do that is that you're encoding the trajectories, those level lines, and therefore the measures that you want, you are encoding them in terms of level lines of integrals of psi j minus psi k, where psi j and psi k are two different solutions of the algebraic equation, and the whole point is that if you look at those functions here on the plane, they are not globally well-defined harmonic functions on the plane, they have single lines, so the psi functions have branch points, but if you look at the surface, those are globally well-defined, so this is one globally well-defined object, harmonic in the whole surface, so it's much easier to control the level lines. And now, so again, so because of compactness, so this is like it's gonna be harmonic on the surface, and those level lines that you just saw on the animation, they are actually, those level lines are critical trajectories, and actually if you look at all of them simultaneously, this form a graph, what's called the critical graph of the quite differential, and it has a lot of nice structure. So because this graph has so much nice structure, it's a lot easier to sort of control what possible behavior could happen if you start playing around with parameters in your model. Yeah, and as I mentioned, so the measure we want to material recover from those level lines, and in exactly this external source model, what we did is that we call, we did this idea, there's not exactly the formation going on there, but the idea is the same, we just look at the spectral curve, we look at it at the level of the Riemann surface, and it turns out that this Riemann surface, okay, we can kind of describe it a bit, but more importantly we can describe the order of the pole of this function at one of the points of the infinity, and because we know the order of the pole at one point and infinity, we can recover everything, everything. We can recover the support of the third measure, and just because we know the order of the pole at one infinity, we can recover also only the possible critical behaviors that I talked to you about. Of course, it's a bit technical, it's like 70 something pages to do that, but just to say that the ideas are simple, and as I told you, we applied to all the random matrix models as well, and I've heard almost from a few cities that if you do a nice calculation one time, you call a trick, if you do it two times, you call it a theory. This is not exactly a theory, but just like a technique that has been shown to be kind of powerful in many situations for us. Again, the details of course are hidden, but the main idea is over there. I hope at least you get some idea from that, and you can draw pictures. Okay, so I talked a lot. I just want to then wrap up what we just discussed. So we started with this matrix model with external source or polynomial potential, external source and some condition on the multiplicity of one of the sources to convert, let's say, to a certain alpha, G01, and then we obtained, exists on a spectral curve with a very specific structure provided that the zeros of the average characteristic polynomial remain bounded when n goes to infinity. The point is if the limit of this, again, we're just imposing polynomials, if for some reason you know that the limit is unique, then actually you can make sure that the limit for the eigenvalues, so the random measure of eigenvalues is also the same as the one for the average characteristic polynomials. That's why I mostly focus on the average characteristic polynomial. And actually also, again, we're imposing that the zeros remain bounded. We pretty much believe that this should be always the case because of the scaling n that we're putting in the front of the potential. And for instance, if we can show that, let's say the absolute value, the maximum absolute value of eigenvalues of the random matrices remain bounded, we say with positive probability uniformly as n goes to infinity, then we are done, which I thought, well, this should not be, we don't need any refined, any refined constants in the front, and so I thought, well, that perhaps should not be hard, although we haven't spent much time. We started thinking about this idea a few days ago, but then I asked Alice today and she said, well, you're asking me a hard question. How can you expect an easy answer? So I don't know if we can prove yet, but okay, is that conditioned at this point? So I don't know if we're gonna be able to get rid of that. Yeah, so also I must say that in principle, our calculations indicate that we can extend the existence of the spectral curve exactly in the set of the dimension to you, when you have also possibly different, more eigenvalues, distinct eigenvalues for A. So the existence of the spectral curve in the large and limit, that argument, we haven't done the calculation yet, but when we were completing the calculation for the two different eigenvalues, it seems like it can be extended for several. So we don't know if we're gonna do it right now, but again, so the first part of the talk kind of can be extended, but not the second part. So for the second part, when we start with the spectral curve and construct the variational problem, that's the corresponding spectral curve is gonna go to a degree higher than three, and that can be quite complicated. But anyway, in the case when you have two distinct eigenvalues, so the spectral curve is degree three, then we are able to obtain this variational problem, right? And as I mentioned, I guess along the way, also it was a surprise to us that we could actually classify all possible singular behaviors by describing those trajectories as well. We could rule out lots of other singular behaviors that in principle could appear. And as I mentioned as well, we apply those techniques to other models. And it should stop right here, thank you.


ve a long term solution like talking around with some of the budget and the wind and changing off the rust spots and the wind spray the whole thing and getting all the light out of it it's not broken through at all it's solid out there floor is solid but there's a lot of so much stress it's definitely something that's going on tomorrow of the year this looks like with some maintenance this could be a few more years but we can make it weather proof and there's there's a couple of gaps on one of the front doors that potentially something could be there it looks like they made something they got trapped in there's a pile of fur over one point so no I think it was a little work now the other piece of that we talked about potentially potentially not having that certain material that we have purchased you know that is something that she's done she's done that makes we can't get money to the state to help on that we get the money over the next couple of years there's also money there's money in $725 I think and then it wasn't I think actually that was something that Cindy mentioned was maybe them getting a bigger annual appropriation so that currently what I put in there now is money because we have to generate we have a contract with Yankee generators so they come in and deal with that I guess it's still just this 1200 but there's still money left because I think that I've paid the I feel like Yankee generator was about $725 so there may be money in the budget I'll double check that all because we may have money already built in the budget or if we Yankee generator that annual contract is $700 then we already have money built in here so we could be buying supplies so it may not be in the budget but yeah thank you so much I think we can get a couple of years out of it in one of the years because the prices are just out of sight yeah well and not necessarily the size that we want but yeah we're going to get something much larger than necessarily the well right also right now there is material in there but there is also I'm sitting here telling me a hundred why so we've got that it's being stored in common schools because they were concerned about putting it in there and if you put them in plastic tubes and put them in there it's going to fill up on the town management before you get through the sessions I got that that the we'd met from the 25th if they had any amendments to it would it be good to prove them as no need thanks for I didn't know Eric Derek you didn't want to be written in as Derek my legal name is Derek I'm fine with that most people know me as Greg I go back to Alan he always calls me Derek I would call you Greg and then he's looking at me like I'm talking about the same person I'm like so yeah I don't know what I have this problem to so any emergency story for the president of the second phase 2 I often spearhead investigating the problem not solving come on so if we don't if you don't approve that and you're still stuck investigating well yes I suppose technically you could do that I think it's the way it is I got 52 on the part of my partner there you go into the committee minutes and the defendant so one favor and there were some other committee minutes in your chapter was there anything else for the board there's a light bulb oh what is it about Eric that I don't know I think we got no money I think so we just picked the damn color so the board is that broke so I was concerned then I was not happy really happy I've got three years okay I'm going to go up with I'll make a fine slack and go up with me so that I can lay the entire thing out and then we'll head out up there and tell them you don't go through to exactly what needs to be done so since it's not I'm not going to go over the hard part I'm going to be great at it but I don't know what you're going to do this is the third time I've done it okay I'll get a fine I'll get a fine I'll get a fine I'll get a fine I'll get a fine I'll get a fine I'll get a fine and we'll go up to take a look at that thank you and speaking of trees we're going by this little jet I think it's going to grow I've been on your list the strong trees and I know that you've got to settle, you've got to think it's fine you've got to think it's fine you've got to be quite better at work I actually want to date it's on this side of the road oh it's not Jets it's going to wind you up to that it's leaning about 40 degrees into the road I know we've had a conversation about that so I'm going to see Adam tomorrow so I'll ask him about that because we had a conversation about that we did talk about it trees it couldn't be that he talked to somebody else if he didn't want to but it's maybe he talked to somebody he had a couple with him and he was waiting for him so I'm going to ask him Adam it's moving back now it's going to be long it's on this side I'm already talking about that do you work across the road do you work around do you work back here on trees


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Today we're going to look at fold stability. So all these structures, both for fibrous, globular and memory proteins, we've gone through the last two lectures. I'm not going to try to relate them to the fundamental concepts that you picked up from physics. We're going to be looking a little bit about their size distribution and essentially what is it that makes a protein a protein and show that this is very much due to fundamental statistical mechanics. Based on what we talked about, in particular, in the lecture on globular proteins, I'm going to come back to this concept that we had a number of different classes or whatever we call it. We had the globulins. This is an example of the myoglobin fold. We had these large, mixed alpha helix and beta sheet regions that were called alpha slash beta, that they're literally in the same part of the protein. And then we had these alpha plus beta proteins where you had the alpha helixes and beta sheets in these joint regions. Now, it's not entirely easy to classify these based purely on math, but you can probably see the difference here, right? Purely alpha, alpha beta mixed, and then alpha beta segregated. I didn't have room for a pure beta sheet protein here. And the question that we're going to concern ourselves with today is roughly where I ended the lecture on globular proteins, that we had this observation originally by Cyrus Schocia that they were in the ballpark of 1,500 folds, maybe 2,500 or 3,000 if you ask Mike Levitt today. But it's a relatively small number. And more important, it's finite. It's way smaller than the number of sequences. So most sequences that fold into proteins, at least, appear to find one of these folds. Why? Well, it's even more extreme than that. Forget about 1,500 or 2,000 folds. I would argue that the vast majority of all the sequences that we see, if they're actually proteins, they're going to fit onto a very small set number of folds. Maybe 20, 25% of those 3,000 folds we saw. That's why I spent some time to go through simple stuff size, as the globin folds, the four helix bundles, those beta sandwiches and everything. So of course, given a sequence, I don't know exactly which one of these folds is going to be. But with 80% probability, it probably will be one of the 20% most used folds. So nature appears to be pretty conservative here, and re-use things. Or there is something fundamental in these folds that makes them very suitable to harbor many sequences. And it's going to turn out that it's mostly the latter. If you compare this to something else, say RNA. RNA, we haven't talked that much about it, but RNA, too, actually folds. Remember, this single strand will fold up into some sort of three-dimensional structure. It's usually not quite as rigid as proteins, apart from, say, the tRNA. That you had a very fixed structure. Then on the other hand, you have DNA. DNA in principle just has one fold, right? That double helix. Technically, it has a few slightly different versions of it. You have an A helix and a B helix and the Z helix. But the first approximation is just a single double helix. And that is partly because DNA is even more concert. You have the phosphates that need to face water, and then you have the bases that need to be paired up. So one way or another, evolution has selected for this. So with proteins, there is some sort of feeling that we need the diversity, and we're going to be looking into the cost of distortions in this and why we end up having a relatively small number of them favored.