All right, so it's time for a second hot topic and today being technophile Tuesday We're taking a look at the impact of technology in agriculture. I have my guest or lack of humor Who is the founder of my food angels? You're welcome to the breakfast. Thank you, Marine First of all, tell me how is it possible for a farmer to look so glamorous so cute Thank you Compliment I am in the food sector I am a Mental backward integration What we do is we ensure that homemakers Be as a professional is half food not only half wood But know what to do with the food that they have so that's what my food angels is all about All right, I'm going to discuss that with you in detail at another forum not on this this Tuesday Let's talk about technology in agriculture. How much of technology do we have involved in agriculture in Nigeria today? So I'd say the the adaptation of technology in agricultural sector in Nigeria today We're seeing gradual stakeholders Performance in that sector, but the truth of the matter is we could have more we have we have ever about 80% family still doing in the In the in the normal I think how we grew up who I'm caught last way we could adopt more when it comes to our Greek We have a good sector of people who are It's getting momentum. Technology is getting momentum in that sector That is the young people are beginning to embrace more eating more healthy and also ensuring that they are They want to know where the food is coming from and how the technology that is used But the large-scale farming which is what controls majority of what we eat is still being controlled by the normal traditional way of farming Like you said To educate the farmers, but the truth of the matter is as much as education is important also funding is important So we have a lot of farmers in the space They probably have the technique in the knowledge of the technology but was the access to the funding and that's where the and that's where we need more more Private sector and even our government's like they're Coming to play in that in that space. Currently we were we're facing tomato Ebola I'm not sure if that's a widespread news right now, but tomato Ebola is one that could have been curtailed with the right information and So it's from a Parasite called to ta to ta absoluta Hmm and what it does is that it attacks your tomatoes if you check the price of tomatoes right now in in the farm It has kind of Exactly we we got a basket of tomatoes in mouth for 60,000 on on Saturday. It's on head of now If you go into research in what how we could even curb that it comes from even seed rotation It comes from from from knowing what to how to introduce some certain parasites that can attack that Qta absoluta, you know, how do we harvest Nigeria? We harvest we have us when it is ripe But in in modernized economies, they don't have us when it is ripe They have us when the when the plant itself has reached a certain maturity physiological maturity and then they have the harvest and they put in cold rooms We don't have that cold storage facility. We have it, but not in in excess Not not definitely not enough not enough that can control what the population eats So and then that's which which we have already makes the prices skyrocket You know, so we there's a death of technology. There's a gap. There's a wide gap that needs to be breached Okay, well if you just joined us You just finding out the reason for the high cost in tomato today While you're buying a bucket of that small bucket at 7,006 5 and the basket at 60,000 is because of what she's described as tomato Ebola now This lack of storage facility, which is part of isn't part of the technology. Yes, definitely It's part of it. We still suffer from it for from that. I mean growing up. We do know I do remember that one of the reasons why we can only eat mangoes for instance when mangoes in season and not eat It when it's out of seasons because we like the storage facilities and that has affected every other crop and even our yam and all of that How are we going to get out of this? Literally, it goes down to more people picking interest in this sector where we are More people the government themselves also picking more interest. We don't we appreciate what the government has done so far and we need Even more research being done For instance, like you said, you know, we other countries actually have planting all year round We don't wait for season. You don't hear tomato season or or mango season in other countries It's all it's produced all year round. How well how are we not able to do that here in Nigeria? It's because there is a gap and that gap can only be filled by those who are Currently playing in the space spreading the word and then those who are not in the space right now But have the funding to come in now a lot of people want to spend money around when it comes to a Greek a Greek sector is very Lucrative it is very profitable, but guess what? It is not the quick fix It's a long-term investment And so we need the big money money people to come in and who understand futuristic planning to say we want to do This and do this work on Nigeria the value chain in agriculture How popular is is that to people how maybe I should put it this way Do many people know about the value chains in agriculture and how they can read from them? So yes, there's education that is to be done also that space we have people who probably just want to play in the space of Planting for instance meanwhile in from that planting you're generating Waste that's another part of the of the value chain in that Greek You know you generate we start waste can be turned into money can be turned into into fools and all of that We have that space we are currently I'm playing in the space of of you know aggregation and Distribution, you know, that's another space. You have another space where they're talking the technology You know, so not everybody is keen into we see more people these days when they say they want to invest in a Greek They probably want to invest in them a Get which quick scheme which a lot of people are falling scam victims for Yeah, but we need to for anyone who wants to come into that week space. They need to literally be educated on where to start Where a Greek starts really even from the seed formation, you know So what do you want to play because there's so many there's so many It's not that you've alluded to this Scamming that's going on Tons of some radio stations you hear some people promoting whatever whatever and I imagine that's part of where this coming is coming from It's any kind of regulation and to because time is no longer on our side I haven't really talked about Government involvement because I do know that government cannot solve all of our problems Food security is a major should be a major priority of any responsible government so the two things Regulation to set people from all these kind of scammers and to the level of government involvement in bridging some of these gaps that you've revealed So yes, there is some from a regulation, but we could have better. We could have we could have more education even in this that space, too Then when it comes to government participation, yes, like you rightly said government cannot do everything The government is definitely they have their work cut out for them in this space I would I would I would advocate more of even Public sector the private sector sorry coming into the space and helping us Make sense of this our agri space All right, so would you say that technology is a future of agriculture farming in Niger Technologies the future of everything is it safe It's not complete so controlled technology because you could actually have technology that has now evolved into getting you know Human beings exposed to cancerous Casino genic as the genetic Properties, but yes controlled technology. How are we importing the technology that and to fit into our own climate? Yeah, you know, we can't we can't import technology that was made for for instance the cold temperature Regions, so this our own space. We need to come up. That's why I said research needs to be done in this our own space to know What the technologies are? Efficient for our own space also to avoid harm to people exactly Environment. Yes. Thank you so much. Oh, that would you know for your time? I wish you had more time, but I'm sure we'll have to do this again It's been my pleasure all right some other time. Yes. Thank you so much. Thank you Oh, my father my food angels has joined me to take a look at the impact of technology in agriculture Well, I'll come back. It's the homestretch couldn't give you sports this morning But um, we do have the quote of the day just you have the quote of the day Yes, I do and it is by Daniel Bell and it is on technology Technology like art is a story and exercise of the human imagination. That is really wonderful Sorry in March exercise For today, but if it is for being a part of it, my name is Justin And I am Maureen do have a splendid day. I remember to join us tomorrow morning for the breakfast


From Bahrain, it's theCUBE. Covering AWS Public Sector Bahrain. Brought to you by Amazon Web Services. Hello everyone, welcome back. This is theCUBE's coverage here for AWS Summit. We are in Bahrain in the Middle East. Cloud computing is changing the game. Telling all the top stories. Cloud computing decrees are now being offered. Amazon's reach is now operational. Our next guest is Mays Rihon, Chief Technology Officer at Aramex, big global provider of logistics and transportation services. Welcome to theCUBE. Thank you, John, thank you for having me. So you were on a panel this morning. We power tech, it's a women in tech panel. Congratulations. Thank you. But you're also the CTO of a really big logistics and transportation company going to the cloud. Yes. Tell us a story. So I work for Aramex and Aramex is a global leader in transportation logistics. We have, we're proud to have a diverse culture and we have 30% of our management are females. So it's only natural to have a female CTO. But for Aramex, the whole digital transformation journey and the cloud adoption started like all other enterprises with the whole cloud waves started and we found ourselves competing with the corpus that are not competing with the classical competitors from the logistic industry, but rather competing with innovators that are companies that are really consuming the best out of the cloud in terms of speed and agility. So we had to transform and we created digital transformation roadmap and we fired multiple programs and this is how it started. And now we're proud to have a big data lake on AWS and we moved a lot of our business processes to be derived from our machine learning logic that's there. We're modernizing our landscape and moving. So you guys are big logistics, a lot of compute powers needed, a lot of IT, now moving to AWS. Yeah. Which also on a logistics business they move a lot of packages around their business. A lot of compute, a lot of storage, a lot of IT. Why the shift to the cloud? What was the reason? Agility is the most important thing and being able to create MVPs for all the ideas that we might have to support the business, but also furthermore because we needed massive computing power because whenever you're in the services sector the number of transactions that you process and the speed of processing those transactions needs to be tremendous and that's the power of the cloud. Scaling up at the end of the month when you want to invoice with 100K items this is what we were looking for. So digital transformation is complex for you. You have IT, you have your back office, you have a huge organization. IoT must be a big part of it too because you've got to keep track of everything. How big is the IoT or the industrial IoT component of it? Well, for our industry, IoT is very, very relevant. It's going to be transformational on shipment tracking, handling, and it will shift the way we do business. One of the main drivers of building our big data on AWS was the fact that we want to feed IoT findings and sensors reading to the cloud. This is something that we're very serious about. We're actually, I wouldn't say that we haven't seen most of the innovations that are related to that. And I think we had some success stories but for certain sectors like pharmaceuticals and so on but the minute those sensors would be commercialized and feasible enough to be attached to each and every package, we will be the first to... We love talking about Cloud 2.0, a whole new generation is coming. You have compute, you got storage. It's either on-premise or it's going to be in the cloud. The network now is important because it's got 5G and other radio frequency capabilities, tracking, real-time data, do you move compute to the data? So an entire paradigm of computing is shifting. Yeah, we know shifting from the traditional way of doing things where you have your own data centers and the whole communication and networking setup is built to service this architecture of local data centers. Introducing the cloud would be a challenge. How do you connect both of them? But the way we did it, because we were very serious about the cloud move, we prepared right from the beginning. So we knew the locations that we selected for our data centers and how close and how we're going to connect them with our own data centers for the hybrid setup. When we run both, we took it very seriously and we changed the whole ecosystem that comes around. So you got your data lake now, so analytics are important. Yes, data lake, we moved our data lake and we moved our BI as well, reporting to AWS. And we created a layer of logic, data science logic, to derive business processes like your last mile delivery, geocoding, address prediction, as well as understanding market trends and everything. So how's it going with Amazon so far? Good? It's doing very well. We've had this data for so long and it's a wealth of knowledge of the end, specific for our industry. We wanted to bring the best out of it and we could do that. And you have in-house developers. Can you talk a little bit about the dynamic? Is it mostly, you had that for a while, right? Mostly developers? Yes, we do have. For years, actually we've been an in-house development company. We were building our own applications like on a very wide range with very good and strict production and DevOps processes. Now we're shifting our development towards the cloud. We're very open to start developing on AWS and to adopt even low-code platforms, whatever makes us move faster and in a more flexible way. Mayes, you were on the panel with the We Power Tech, Women in Tech, talking about diversity. How has diversity changed with the technology innovation? Actually, the diversity within the workplace is a very significant and important driver for innovation. And we at Aramics, we're very proud and we're considered differentiator and an Aramics edge to be a diverse culture. We have more than 80 nationalities, 30% as I told you, of our management team, our females, that gives you a wealth of knowledge and also allows you to bring in different age groups, different formats of thinking, different knowledge and backgrounds that really makes change management easier and you don't really feel any resistance to progress. Well, you're an inspiration. What advice would you give for women out there watching? Young girls to professional women, looking at their careers, what's your advice? Everything is doable. We understand that we come from a region where there has been cultural challenges. However, I don't see them anymore, especially when you work with big enterprises. If you create a balance at home and gender balance within your family, then you're okay. If you can do that, you can do anything. You can do anything. I have to tell my wife, she's amazing. If you have four kids, she could be the CEO. Thank you so much for coming on and sharing your insight. Great stuff. Thank you. Reporting here in Bahrain is theCUBE coverage. I'm John Furrier. Thanks for watching. We'll be back with more coverage after this short break.


Now, because it was pushed back to March, it gives you actually a couple of months less to adapt to this, can the Navy carry out its mandate under these massive cuts? Well, look, we're going to, you know, our job is to protect and defend the American people at sea, and we're going to continue to be able to do that. But it is going to be harder in the long term to meet some of those future missions if the uncertainty continues and if we can't work our way through this. The goal, as I said, is to preserve the readiness of forward deployed forces. That preservation will come at the risk of some near term training and readiness, but we're going to try to preserve it and protect it as much as possible.


my beautiful coffee that I needed so much oh my god it's so festive too oh my god so cute the star of the show I love it I've been here before but it didn't look like this I was gonna say I really love those things but you were talking about something are those new what you're sitting on no you have the couch well it's so cute it's honestly well I love the bud light thank you um I keep up margo so it's not like I wouldn't didn't know this I like the new nightstands I did see those yes you have such a spacious apartment you really do I was thinking about that on the plane yeah because I was like honestly I just didn't I don't know why I didn't say that last time I was thinking it well it was much more spacious last time because there's nothing here no I feel like it feels even more spacious with stuff in it yeah well anyways good morning everyone all right guys um hi by the way I'm in New York I don't think I really started the vlog I'm visiting Margo we're going to a concert on Monday and then I'm recording Monday Tuesday but I'm here for the weekend well what is today Saturday Saturday to Wednesday morning but here's my outfit I have this sweater um that's Princess Polly jeans are also Princess Polly boots are the Jeffrey Campbell ones that I'm obsessed with gonna wear my normal like H&M coat not Uggs not Uggs everyone um and that's my outfit for today so it's kind of cold right now we will see I have these we have our loot combs coasters when did you even get these that's funny um we have all the skims pajamas we are at the new Nordstrom that everyone's been talking about forever I have to do a little exchange to give Claudia a gift but it's beautiful here so beautiful probably at least a cafe it's so beautiful here we got the exchange we're gonna go give this to Claudia and then get food see I like those two because I just love pajamas all of them guys it's so Christmasy wow just feels so good wow really just really right especially Christmas we're at lunch we're so hungry like dying you look so pretty right now the colors are like perfect wow you got diet coke really happy about this I cannot believe that is a small as we're getting two of these they're so big they're the five-foot candles we're gonna bring one to jockey yeah let's see it yeah it's gonna be us in the streets yes you can't even see I mean I have so much work I close I meant to say like match these but oh wait we should switch them so we're like you were we'd like switch the colors so we're actually like crazy oh yeah but I'm saying these are shorts and it's freezing oh I get what you're saying now as we're back from shopping we dropped off some gifts we dropped out we were literally Santa we were actually Santa I imagine my Hillary crying don't we all mad I get that I need to wash my hair we have time I need to get some things together mainly myself yeah I'm very good that active haul that active haul you have so many this keychain reminds me of some sort of childhood like hey remember on how you slide make things in those ovens those like yes yeah this is the key chain to give you guys some background eventually I'm gonna get up and get going but we went shopping we went to lunch we walked around I walked around in heels the whole time because I am really growing up your feet are beyond us yeah but like not as bad as they should I mean yeah we did a lot I'm wearing heels tonight too so so you should rest your bones you want to take a bath I'll clean my bath yesterday yeah I really need to do that I just like you don't you have to like start over from scratch with your hair that's fine I'm doing my hair so it's just like doing it once yeah that would feel a lot better but it'll be fine we have a long time we're just gonna hang out I am Margot right now we've been back I showered I don't even last on our vlogs I'm really tired I didn't even want to drink but I need to get an espresso martini the only thing that will keep me up for the night Margot is doing her hair we're gonna go to one of Claudia's shows tonight but I feel like I'm Margot guys we're ready for the night I'm gonna show you whoa whoa whoa whoa I'm telling you everyone uses for photos it's very very bright I need to turn that time but um anyways we're gonna go to dinner soon because I keep overheating so alright guys we just had dinner we're off to the show to express the martinis I think that's gonna fix my energy levels I'm feeling good about right guys we made it Margot made the playlist this will tell you very soon it's called pre-show playlist and he hid his ball right on my spot and I have a lot of other playlists you can follow yeah what's the ball? um we walked here it was very quick and very cold yeah honestly I am yeah I get it you're fine I'm overheating it all the time we're done but it's so exciting here you are every time I take a shot that someone talks about take a shot not like take a get a get a shot oh wait I have it I check for addresses first guys we have left over McDonald's um I fixed Margot's clock we're wearing matching pajamas what is going on with this I don't know what that is McDonald's is ready to smoke in whatever you're missing now that's pretty oh you have some of this you don't have mushroom barley soup what? I have mushroom barley soup maybe not for now that was so random that was good in the fridge that's good we'll see you guys in the morning so you should get two cheeseburgers today yeah I got the double double so I have one later I thought I had oh I thought a double double it was like um um sorry you're right I'm wrong um I got two singles then smart yeah but ahead uh I can't take it seriously when I look at the viewfinder we're watching this because we're only you saw the tag on yours I have a tag I don't mean I mean I'm sure you just got it yeah oh my god I'm just obsessed with my dog like that's really it that's all you have to know she's just like I I smell the food I know I feel bad well we'll see you guys in the morning everyone happy Sunday we're still matching um well it's actually like noon we're going to watch a movie we're not doing shit today yeah we're not matching dramas we ordered um I'll say ables and then I just ordered a bagel so yeah uh it's just really such a cozy do you like my bed cold yes you know I had this bed in um LA did you yeah of course I didn't mean like the color like the bad color I meant the comfort yes it's very comfortable um yeah we're gonna watch a movie I'm just trying to get involved I know it's okay here but nevermind bye she said you have to pay me for that I'm out yeah wait mad it is 515 I'm feeding Mads wait I don't know where her other bowl is today has been like the laziest day um I'm currently feeding Magnolia right now Margot is recording the red heads right now so she just left I'm feeding Magnolia I'm gonna take her out and then we're just gonna order food in tonight and we have a workout at 6 a.m. today has been the laziest day ever but it was very very very needed like if I would have gone and done something today I would have not survived the rest of the week tomorrow and the next day I have a bunch of recordings I'm actually I'm gonna see like Brooke Danielle Margot Katie for a little bit we're all like really busy this was like a last minute trip and also it's like Thanksgiving weekend so no one's even in the city but I am here until Wednesday morning I'm actually going because we're going to the glucose concert tomorrow night that's like the whole reason I came to begin life and then we just like made it a trip um and then I scheduled some recordings and stuff and I'm staying here yeah I just am so tired like we didn't even stay up that late I just like stayed up I just haven't really been sleeping well the past like a week so we watched a Christmas movie today we laid in bed we ordered food I got a bagel I took one of Margot's cruise because I'm cold Margot has got back and our sushi just arrived I want like a candle and then we'll eat oh wow I love that for us guys it's the first side of Hanukkah Margot's also vlogging but I have to say the representation of Christmas versus Hanukkah yeah it's very exciting yeah well honestly I that's why when I walked in and I saw like actually Minora I was like oh my god it well I mean it's before the tree yeah I mean it might look like it's two trees and one tiny Minora yeah I think it's supposed to be this way oh hey no mouth how long she had alcohol I'm a shaker time it's tough it's even able how they can air shelf on a guy our hotel night oh hey no mouth how long she awesome it's the way she just watches you and you put it in the window to show everyone the miracle oh that's cool all right guys Margot are gonna watch a movie I'm gonna bet four hours six a.m work out so fucking tired oh my god I cannot believe you even did to this but for the record by the way I was about to say to you you're more than welcome to back out if you want no I really am into it um but for the record I I'd love to bring this up that when Margot when I met Margot I thought I was so unrelatable because I would go work out early in the morning and look at her now look at me now yeah well hope you guys enjoyed this vlog I'll see you guys in the morning the vlog is over well I'm doing two I'm doing a weekend in my life and like a week in my life oh yeah so we'll see you tomorrow bye he's like well I hate the song but he's like real religious and he was like but I love you and he was like I just want what's best for you and all this stuff and I was just for me with my story it's really cool because here you have this song that's you know explicit and all this stuff and it's funny and it is definitely part of my personality like


European diplomats argue over who shall ride first on a funeral train. A great popular song hit is discovered by accident. Can you imagine that? Well, friends, this is Lindsay McCarry back with you again to present with the able assistance of my cohorts of the Nimble Tongue, another set of strange facts in this session of Can You Imagine That? After just a minute and a half, we'll be back. So be with us, won't you? Try at astounding you in this broadcast of Can You Imagine That. It's seldom that we have the dubious pleasure of receiving advice from a burglar for which may I add, let us be duly grateful, but here's once it happened. On the last day of May, 1911, the chief of police of Jersey City received the following letter.  So far, I have been unable to get away with anything white-wired. In all the houses I have visited in the last three months, I have not gathered up enough to pay me for my energy. I am beginning to think that Jersey City is a collection of tight watts, signed, the slippery kit. The police buttoning in annoy me greatly. Fine thing. Well, now let's see what the next item is. To most of us today, probably the archaic purple pageantry and pomp attended upon state affairs and some of the older kingdoms and empires of Europe seem a little out of key, as it were, with modern progressive civilization. For instance, well, Theodore Roosevelt used to tell a story which aptly illustrates this. While he was traveling abroad after his second term as President of the United States, beloved King Edward VII of Great Britain passed away. President Taft cabled TR and asked him to represent his nation at the funeral ceremonies. Roosevelt at that moment was in Vienna, but promptly cabled his acceptance and soon after discovered that a special train was being made up in the Austrian capital to carry the various monarchs and other emissaries and their parties from southeastern Europe to Windsor. Among these were two personages who were soon to find themselves embroiled in a question serious to them and of great pith and moment to their countries. It's our of Bulgaria and the Archduke Ferdinand of Austria. Just as the special train was about to pull out. Pardon me. Are you a member of the party of His Excellency Archduke Ferdinand? I am his special representative. And I hold the same honoured position in the entourage of His Imperial Majesty, the sir of Bulgaria. Manuel, you're vicious. His Imperial Majesty desires to inform His Excellency that His Imperial Majesty with his party will occupy the coach immediately following the engine. But the sir of Bulgaria riding in front of a representative of the ancient house of Habsburg recently. Well, well, of course. One moment, please. Manuel, you will please inform His Majesty that His Excellency the Archduke representing the Emperor of Austria will occupy the coach immediately following the engine. You insist upon this? Positively. The order of coaches will be, first, the coach of His Excellency, the Archduke. Second, the coach of His Majesty, the sir of Bulgaria. Next, a dining car. After all, my dear, Austria is centuries old, I'll tell you. Release, my apologies. I shall communicate your desires to His Majesty. And so Archduke Ferdinand did occupy the first car immediately behind the engine, and as a consequence, the car into which sifted the largest quota of certain cinders. But this international affair wasn't over yet. Not by any means. A few hours later, as the train rolled along over the picturesque countryside, the Archduke became aware of a gnawing sensation in the pit of his stomach. He was hungry. So, he dispatched his emissary to the next car to contact the representative of the monarch of the Balkan state. Manuel, Manuel, please, you wish to speak with me? Yes, sir. You see, His Excellency the Archduke wishes to gain permission of His Majesty to pass to His Majesty's coach on His Excellency's way to the dining car. Wait here, please. His Majesty regrets that there are so many problems of international consequence involved in His Excellency's request, and there are being no precedents upon which His Majesty can possibly make a decision without the assistance of His legal and diplomatic advisors. And inasmuch as His Majesty's legal and diplomatic staffs are a great number of miles away in the capital of Bulgaria and inasmuch as this coach must be concluded to be for the time titular of Bulgarian property, it is deemed unwise to grant His Excellency the Archduke of Austria a visa to pass through His Majesty's coach. What do you mean to say that you mean that? Unfortunately, His Excellency regretfully must wait until the train stops in order to pass around His Majesty's coach to the dining car. Bah! Yes, indeed. And the Archduke of Austria did wait until the train stopped at the next town before he could go around the Bulgarian coach on the station platform and into the dining car. And then, after he had finished his meal, the train was again on its way and he had to wait in the dining car for another stop so that he could re-enter his own coach next to the engine. Can you imagine that? And we wonder whether or not in the following few years such incidents as this might not have lost their extreme significance for Ferdinand of Hapsburg, particularly on that day in 1914 when he and his wife gazed into the muzzle of a student's revolver at Sarajevo, the revolver that was to end their lives and touch off the tender of the Great World War. You know, chance remarks have often been the inspiration for some of our most successful popular songs. I'm sure that you've heard of many of them, but I always find it interesting to trace some of these stories down to their original source. Here's one which resulted in the composition of an old-time number you all know. A number of years ago, a young actor named John F. Palmer had just completed the current theatrical season in a tour of the play The Two Orphans, starring Kate Claxton. He returned to his home in the Harlem section of New York City, maintained in his absence by his sister Pauline, to await his next call to the boards. One day, Pauline and her usual morning routine went to the door of John's room and in the midst of a din created by an old-fashioned German band playing nearby, called to her brother. John! John, breakfast is ready! John! John, do you hear me? Breakfast! Oh, dear, that old band. John, breakfast! Just a moment. All right, all right, let the band play on. Subsequently, John Palmer, satisfied that he'd heard enough band music for the moment, joined his sister to breakfast. As Pauline was clearing away the breakfast dishes, he turned to her brother. John, that remark you made a while ago. Remark? What remark? When you said, let the band play on. What about it? That's a good title for a song. A title for a song? What are you talking about? Yes, the band played on. Huh, the band played on. That's not bad, sis. Not bad at all. All right, then, get busy. All right, slave driver, give me some paper and a pencil. In a short time, John Palmer had penned the lyrics of a popular song telling the exploits of one Matt Casey who exhibited the uncontrollable desire to waltz with a certain luxurious strawberry blonde during certain provocative moments while the band played on. Palmer concocted a melody to fit the verses in chorus, but like so many of us whose heads ring with the greatest song of all, he didn't possess the necessary education to place the musical notes on manuscript paper. But with the help of a trap drummer in the orchestra of the Old Third Avenue Theater, John finally was able to present his song to a music publisher. A music publisher? I should have said many music publishers for John Palmer made many futile journeys to publishers' offices trying to sell his song until at last he abandoned hope of disposing of his masterpiece and turned to writing for the stage. One day while sitting in the office of a friend, borrowing the use of a typewriter to type a play manuscript, John unconsciously hummed the chorus of his song. Come in. Yes, sir. My name is Ward, Charles B. Ward. Oh, yes, Mr. Ward, of course. I remember seeing you on the stage several times. And your name? John Palmer. Yes, Mr. Palmer, who was humming in here a moment ago? I guess I was. Oh, and what was the name of the song you were humming? Oh, it was Little Diddy I wrote some time ago. I never sold it. It was called The Band Played On. The Band Played On? Mm-hmm. And you never sold it, eh? No, sir. Now, Mr. Palmer, I have recently turned to publishing music on a rather small scale. I like your song. How would you like to sell it to me? Sell it to you? Sure. Let me buy it from you, and I'll publish it. Yeah, yes, of course, I'll sell it to you. Sure, I will. Well, good. Drop over to my office when you're finished here and we'll talk turkey. And so because of two elements of chance, John Palmer's chance remark to his sister and his chance humming of his song, he was able to dispose of it to Charles Ward. Ward made several changes in the melody, placing the melodic pattern on the accent of notes upward instead of downward as in Palmer's original melody, and thus came to be known as the writer of the song. Yes, it's a great idea to possess tenacity of purpose, courage, ambition, and all the other attributes of normal success, but every once in a while, chance does play a part in winning a goal in the game of life. Matt Casey formed a social club that beat the town for style and hired for a meeting place or hall. When payday came around each week, they greased the floor with wax and danced with noise and vigor at the ball. Each Saturday, you'd see them dressed up in Sunday clothes. Each lad would have his sweetheart by his side. When Casey led the first grand-mama when Casey led the first grand-mars, they all would fall in line behind the man who was their joy and pride. For then, man, he would waltz with the strawberry one and the man play. He'd glide across the floor with a girl he adored and the man play. His brain was so loaded, it nearly exploded. The poor girl would shake with alarm. He'd marry the girl with a strawberry curl and the man waltz with a strawberry blund and the man play. He'd glide across the floor with a girl he adored and the man play. There you have it, ladies and gentlemen, another session of Can You Imagine That? This is Lindsay McCarrie inviting you very cordially to be back with us when we visit your station again. And until then, goodbye now.


So let me first share my note in the chat. I have put the PDF and also let me send the link again. Okay, so welcome to the third lecture. And here is the plan for today. First, deformation invariance for naive counts. Second was spines, tropical curves and the tropical modular spaces. Three, sketch of proof of the connected component theorem. Four, toric tail conditions in families. Five, gluing formula and independence on the choice of torus. Six, structure constants and associativity of mirror algebra. And seven, convex finiteness. Eight, boundary torus connection and finite generation. It's a lot of contents today. If we don't finish today, we will continue in the next lecture, the last parts. So let's start with deformation invariance of naive counts. So recall from the last lecture that given any spine H from some nodal metric tree to the essential skeleton of U, H is some piecewise affine map. So given any spine H and a curve class beta, we have the naive count of skeletal curves and H beta of a spine H and a class beta. The definition was pretty straightforward once we have the skeletal curve theorem. Question, what properties do the numbers and H beta enjoy? The most wanted property is deformation invariance. We want the invariance of the count and H beta under small deformations of H. So in fact, this property determines directly the viability of the whole project of no Archimedean curve counting. So it was the first property that we had to check before embarking on the project. Here is the rough idea for deformation invariance. We consider the space of analytical curves in U analytic and we have the map taking spines, taking associated spines of the analytical curves, and we reach the space of spines in the essential skeleton, SKU. And we are given some spine H in the space of spines. Intuitively, the count NH beta is just the cardinality of the fiber of this map SP over H. Let's ignore curve classes for the moment. So intuitively, the number of analytical curves of spine H is just the cardinality of this fiber. So for the invariance of this count NH beta under small deformation of H, we want the map SP to be somewhat atow over a neighborhood of H. And more precisely, recall that the counts NH beta are defined via evaluation of an internal market point PI. Assume for simplicity that our H is an extended spine. So let's first assume that H is an extended spine. Here is an example of H. We have five infinite one valent vertices V1, V2, V3, V4, V5. They either map to the boundary of the essential skeleton or they map to the interior. For example, V1, V2, V3, V4, they all map to the boundary and the V5 maps to the interior. The lag containing V5 shoots up vertically. It's contracted. Let P, the bold P, be the top of weight vectors at infinity. So here we have P1, P2, P3, P4, P5. They are weight vectors at infinity, meaning just the derivatives. And if V5 is an internal vertex, this derivative P5 is just zero. Recall that each PJ also lives, is also an integer point in the essential skeleton. And we have an explicit description for the set of integer points in the essential skeleton, which is just zero union positive integer multiples of essential divisorial evaluations. So recall, if the derivative PJ is zero, J is called internal. And if the derivative PJ is nonzero, J is called boundary. And in this case, we write PJ as MJ times VJ. Like here, some positive multiple of some essential divisor and assume that each, so this is grid letter nu, assume that each nu J is given by a component dJ of d. Now we have a proper modular stack. Properness will be important in the moment. We consider the modular stack m bar yp beta consisting of rational stable maps f from a nodal rational curve C with marked points PJ to y of a class beta such that each boundary marked point PJ maps to dJ with order greater or equal to MJ. So here, we say order greater or equal to MJ, because we want a proper modular stack. So we also allow some components of the curve to go completely into the boundary d. Otherwise, we cannot have a proper modular stack. And inside this proper modular stack m bar, we have the sub stack m up beta more relevant to our accounts, consisting of curves whose intersection with D is exactly given by P. In other words, each boundary marked point PJ maps to the open stratum dJ with order exactly equal to MJ, and there are no other intersections with D. So this m up beta is what we are really interested. And it leaves naturally inside this compactification m bar for any internal marked point PI. Meaning that the PI maps to the interior U, we have the natural map phi i from the proper modular stack m bar to the module, the space of domain curves times y by taking first. So to here, it's the modular space of an pointed stable and pointed rational curves. The first factor of phi i is given by stabilization of domain. Because for a stable mapping m bar, the domain curve might be unstable. So we take first factor of phi i is a stabilization of domain. And the second factor of phi i is evaluation at the ice marked point in order to obtain the eight dollars of the map of phi i. As we said that for deformation in variance, we want some sort of a darkness. And in order to obtain the eight dollars of phi i, we consider two more sub stacks. First, we consider the sub stack msd consisting of stable maps with a stable domain. In other words, there are no bubbles in the domain. And the second, we consider a further sub stack m smooth inside msd consisting of stable maps such that the pullback of the logarithmic tangent bundle is trivial. And note that these two sub stacks are in fact spaces. In other words, they are not a stack because stable pointed rational curves do not have non-trivial automorphisms. So that also implies that our accounts are positive, non-negative integer numbers as opposed to some rational numbers. Because eventually we are just working with modular spaces instead of modular stacks. And using the deformation theory of curves, we have the following smoothness theorem. So finally, I give the precise statement of the smoothness theorem which we have used already several times in the previous two lectures. We have the following smoothness theorem. First, phi i is a tau over this smooth subspace m smooth over the subspace m smooth. And the second, this subspace m smooth is sufficiently big. I mean, if this subspace m smooth is empty, then it's not very useful. So we have to show that m smooth is sufficiently big in the sense that for any fixed modulus mu inside the space of stable and pointed rational curves, there exists a risky dense open v inside u such that when we restrict our modular stack mu p beta to mu and v, it's contained in the subspace m smooth. So here, yes, yes. So here, the subscript denotes three images and usually it's clear it's pre-image by which map. So here, mu means that we consider the subspace consisting of stable maps whose domain has modulus mu and v means that we consider the subspace of stable maps whose ice marked point maps to v. And this second statement says that for any fixed domain modulus as long as the ice marked point goes to a risky dense open subset v, then the stable map automatically lies in this smooth locus m smooth. So it says that m smooth is very big. It's not only dense open inside this m up beta, the good modular stack. It's actually dense open over every fiber mu. So this is the smoothness theorem that we have used many times and we will be using again today. So it can be proved by the deformation theory of curves. Now we consider the following commutative diagram. We have our map phi i from the space subspace m smooth taking domain of stable map and evaluation at the ice marked point. And we also have the tropical, at the tropical level, we have the map phi i drop from the space of spines. The space of spines in the essential skeleton with infinite directions given by bold p or top bold p to this product. Similarly, the first factor is just takes domain of a spine and the second factor is given by evaluation of the ice marked point in the spine. Then we have two vertical maps, sp and rho. The left vertical map sp just takes the associated spine of stable maps inside our modular space m smooth and the right vertical map pro is the usual tropicalization map. On the first factor it sends every analytical curve, every stable and pointed rational analytical curve to an extended tropical curve with n marked points. And the right hand side is on the second factor, it's just the, the retraction map from the analytic you to to the skeleton to close the skeleton. What is the closure of skeleton here? Oh, so here, the close here, because the skeleton of you lives in the identification of you and it also lives in the identification of why. So we take a closure inside the identification of why. Why is the natural, why is the SNC compactification of you to this closed the skeleton? And yeah, so now I stated the following theorem, let's state the following theorem, the connected component theorem, which is the one main theorem of today's talk. Let S inside the space of the spines be a transverse extended span. I will give the precise definition of transverse in a moment. Just for Maxim's question, I want to say that this bar has, it's really has no importance here. I write this bar just to be able to receive a map from you analytic because it might, because it might go out of and but finally what really matters is the interior. So let's just ignore this. Okay, so the statement of the connected component to zero. We consider S a transverse extended spine inside the space of spines. I'll give the precise definition of transverse in a moment. Then there exists an open connected neighborhood. VS of S and a Zarisky dance open are inside this, which is the first I push forward VS VS is open subset here I push forward the VS. And I pull back by row. So there exists open neighborhood. VS of S and the Zarisky dance open are in this push forward pullback of VS such that when we restrict our modular space and smooth to VS and R. Meaning that we consider the sub space. So it's just a pre image by a VS and R. And the meaning is that we consider the sub space consisting of stable maps whose spine belongs to VS and whose ice marked point maps to R. So this sub space is a union of connected components of the proper modular stack M bar restricted to R. And this is the connected component theorem. It says that this space when we restricted sufficient. When this space sufficiently restricted becomes a union of connected components of this proper modular stack restricted to R. So why it's important because an immediate consequence of the connected component theorem is that by the properties of the modular stack M bar and the smoothness theorem. We deduce that the restriction of phi i to this. This space we denote temporarily by MVS R. The restriction of phi i to this over R is finite a top and whose degree is exactly the count and I S beta. Recall that we defined in the last lecture the count and I S beta using. Skeletal curves and it's just a naive kind of skeletal curves and this we defined using degree of or length of some. Zero dimensional analytic space of skeletal curves and this fiber is just one particular fiber here. So the degree of this finite data map gives exactly this count of skeletal curves we defined in the last lecture. It's the count of skeletal curves with the spine S curve class beta and evaluating at the ice marked point. Hence this connected component theorem. Especially this finite a dollar consequence shows that the count and I S beta is constant for all. For all spine S inside this connected neighborhood vs.  First this shows the invariance of the count and S beta under small deformation for a transverse extended spine. So here we have some I but as we explained in the last lecture by the symmetry theorem. The place where we evaluate has no importance at all. So this shows the invariance of the count associated to small deformation. For any transverse extended spine S. Then we prove. For all transverse spines. By studying the Toric tail condition in families. Which I will explain. A bit later. And deformation invariance does not hold in general for non-transverse spines. Even though for the proof of the associativity of mirror algebra. And also for the proof of wall crossing homomorphism. We need a slight general generalization for non-transverse spines. Which we call almost the transverse spines. And second remark. Actually we had to prove a stronger version of the connected component theorem. We can further assume. That R intersects every fiber of the projection. From this push forward pullback of the of the open subset VS. To the modular stack of stable and point directional analytical curves. And what does it mean. It means that when restricting to R. R is a Zariski open of this space. And by further assuming that R intersects every fiber of the projection. It means that when restricting to R. We are not going to miss any modular of the domain curve. And this will be important for the proof of the gluing formula later. Where we need to consider degenerate domain modulus like this. Which contains some node. And which a priori might be missed when we restrict to a Zariski open. Because this kind of domain modulus containing nodes. They are Zariski closed subsets. They are closed conditions. So a priori if we take a Zariski open we may miss this domain modulus. And that will be bad for later proofs. We must for later proofs we must be able to degenerate domain modulus as we like. So we'd better have R big enough not to lose any interesting domain modulus. That's the second remark. Third remark. Recall that finite eta is equivalent to being proper plus eta. Finite eta is equivalent to proper plus eta. Properness is a question of compactification. While eta is a question of transversality. Compactification and transversality are two pivotal themes of enumerative geometry. Ideally we would like to treat them separately. The compactification and the transversality we want to treat them separately. However here the proofs of the two properties are intertwined for two reasons. First if we want the stronger version in remark two. Here where we do not want to miss any domain modulus. Then we must apply the smoothness theorem above with some properties conditions. And second properties prevents analytical curves in you from escaping to infinity. That is escaping to the boundary D. If we want to establish properness purely via tropicalization. We must show that tropical curves in the essential skeleton SKU do not escape to infinity. That is do not escape to the boundary of the essential skeleton. Which is just the closed essential skeleton minus the essential skeleton. So if we want to establish properness without using smoothness. Then we can try to do it purely via tropicalization. So we must have better control on tropical curves. We want to prevent it from escaping to infinity. But it's extremely complicated to consider tropical curves with components in the boundary of the essential skeleton. And especially the modular space of such tropical curves. If we think about components of tropical curves mapping to the boundary of essential skeleton. And then what is the modular space for such tropical curves how they deform. It gets very complicated at the level of combinatorics and elementary topology. And the task is greatly simplified if we use the smoothness theorem within the proof of properness. So that's the two main reasons that it's difficult to separate the two issues. The compactification and the transversality inside our proof. And they are intertwined in some way. So that is the statement of the connected component theorem. Which is the main theorem of today's talk. Now let's go to the second section. Was spines tropical curves and the tropical modular spaces. In order to better understand the behavior of non-alchimedian analytical curves. We need to study the associated tropical curves. Question, how do we take tropicalization of analytical curves inside your analytic. Recall that skeletal curves have canonical spines. Then what about non-skeletal curves. Moreover, the spine is only part of a bigger tropical curve. So how do we obtain the whole tropical curve. The idea is the following. Unlike spines of skeletal curves, tropicalizations are not canonical in general. They depend on the choice of some model. Here we will work with toric models. Recall we have our log-calabi-o containing some torus. And it's contained in some SNC compactification y. We denote by d the complement of u in y. Lemma for producing a toric model for our log-calabi-o. After replacing our pair yd by some toric blow-up. There exists a toric compactification yt dt of this torus tm. Such that the birational map pi from y to yt induces a bijection between the generic points of the strata of d. Intuitively, this bijection means that yd is simply a blow-up of some toric pair yt dt whose center does not contain any strata of dt. Here is an example. We have our toric surface yt dt associated to some fan. Then we make blow-up at two points on the boundary. And we obtain a log-calabi-o pair yd. u the complement of d in y is log-calabi-o. And the two blue curves denote the exceptional curves. Let's introduce a notation. Let e in y and e t in yt denote the complement of the isomorphism loci of the toric model map pi. In the above example, e is just the two exceptional curves. And the e t is the center of our blow-up. Oh, that's a small comment. Yeah, the whole picture shows a good direction, kind of generalized cluster varieties, sort of Lamp-Pilevsky blow-up. Yes. Devisors in a toric devices. Yes. Yeah, thank you. Thanks. Yeah, and should we talk to some kind of notion of mutation for such pictures, which you never used in your approach? Because we don't really need the mutations. The mirror algebra is built directly from the structure constants. Of course, later we can show that the structure constants are somehow invariant under mutations. Yeah, but I want to say that eventually, for counting non-archimedean curves, the assumption that it contains, the log-Halabiac contains a torus is not necessary. Although it will become much more technically sophisticated. And here, we just take advantage of this torus to make many things more transparent. And many of the geometric ideas here that we explain actually applies easily when we consider the more general situation of counting without a torus. But all things concerning tropicalization, it gets simplified in this case. Okay. Yes. So, okay, so now let's just take advantage of our torus and it's easy to tropicalize in the toric case. So recall, how do we tropicalize curves in the toric case? Given yt dt, a toric variety with a structure torus tm, m being the co-character lattice. For example, if our base field is complex numbers, then the torus tm is just m tensor with c star. We have the valuation map from the identification of the structure torus tm to mr, m tensor with r, which is isomorphic to rn. So if you take some basis, then this map is just a coordinate-wise valuation map. And this map compactifies to a map from the identification of the toric variety to a compactification of mr. So this compactification is given according to the fan of the toric variety. Now, so we are recalling how to tropicalize curves in toric varieties. And given any analytical curve c mapping into our toric variety, recall that any analytical curve is homomorphic to some infinite graph. If it's a rational curve, then it's homomorphic to some infinite tree. And we consider the composition of the map f with this tropicalization map or retraction map tau t. This composition has a natural factorization through tau c and h, where tau c from c to gamma contracts every path in c that are contracted by the composition of f with the retraction map tau t. So it's easy to think intuitively c is an infinite graph. f is an analytic map from c to the analytic space. The analytic space, it's very complicated, especially in higher dimensional case. It's difficult to imagine, but it has a nice retraction map to just Rn, some compactification of Rn. So the composition of tau t with f is a map from an infinite graph to Rn. And this factorizes, so this map may contract infinitely many, so it's a map from an infinite graph to n, and it contracts many, many things inside of this infinite graph. So we just contract them all. We factor this map through gamma, where we contract all the paths that are contracted by the composition, and it factorize like this. We make each an immersion. It's a theorem from tropical geometry that if we do this, if we contract every path like this, then gamma is just a finite metric graph, and fh is piecewise affine and balanced. So balanced meaning that the sum of weight vectors around every vertex is zero. By weight vector, we mean the derivatives. So here is a picture of a tropical curve. This green graph is gamma finite metric graph. It maps in a piecewise affine way to R2, our plane. And it's balanced, meaning that, for example, at this vertex, we take the sum of the three weight vectors, the three derivatives, they add up to zero. So this tropical curve, h, is essentially a combinatorial object, this map h, after this contraction. All the contractions, it's essentially a combinatorial object. And this is called the tropical curve associated to the analytic curve f from c to the analytic, f from c to the analytic, analytic, toric variety. Let me make a remark for this tropicalization. So if we have some marked points, pj inside c. If we have some marked points, pj inside c, let gamma s inside c denote the convex hole of all the marked points. Then we call the restriction of the composition of f with the contraction. The restriction of this composition to the convex hole gamma s, we call it the associated spine. Recall that in the last lecture, the associated spine is just f restricted to the convex hole. Because in the skeletal curve case, the spine maps automatically to the skeleton of u. Here, for a general curve, which might not be skeletal, we restrict it to the convex hole, but we need to compose with the retraction map tau t, which depends on choice of models. And in this case, when we have some marked points in the definition of tropical curve, it is natural to require that the contraction tau c, this contraction does not contract any edge of the spine. So that the spine is a subset of the tropical curve. We don't want to contract any edge of the convex hole of the marked points. So that this is a subset of gamma. In other words, we want h to be an immersion outside the spine, but on the spine we just wanted to be the spine. So this is natural to have this requirement because a priori in the definition of tropical curve, everything that are contracted by the retraction map, we just contracted them. But when we have marked points, then the convex hole of the marked points, they are the spine. The convex hole of the marked points is by definition, the domain of the spine, and we just don't want to contract anything in the spine. Otherwise, spine will no longer be part of tropical curve and it's difficult for some reasonings. So we just relax a little bit on the immersion condition for the tropical curve. But I mean, we ask in the definition of tropical curve that this should contract every pass, because otherwise, there, of course, we can contract less, but there's no canonical way of choosing what to contract. Or what not to contract. This is how to tropicalize analytical curves in the analytic torque variety. And we want to tropicalize analytical curves in our original y analytic. This is easy. We simply compose with the torque model. So we have a torque model map pi from y to y t. And we just compose with the torque model and then we tropicalize inside the torque variety. It's also possible to deal with the indeterminate locus of pi. But let's ignore this for the moment. A question, what do tropicalization of analytical curves in our modular space MUP beta look like? Recall that this modular space consists of rational stable maps in y of class beta, whose intersection numbers with D are given by the top of P. Here is an example of an analytical curve in our modular space. So it touches the boundary D in three points P1, P2, P3, and it also touches the two exceptional divisors. When we tropicalize, we get this tropical curve. The red sub tree denotes the spine and the two green rays denote the twigs. This is how tropicalization of analytical curves in our modular space look like. In observation, since our analytical curve meets D only at the marked points Pj, the twigs cannot meet the boundary of MR at arbitrary points, because most of the boundary of MR is just tropicalization of D, except the subset ET-trop inside the boundary of MR, which is of co-dimension greater or equal to one. So recall that in the lemma of Toric model, ET is the complement of the isomorphism locus of pi, like the center of the blow-up, and by the statement that the Toric model induces a bijection between the generic points of the strata of D and the generic points of the strata of DT, it implies that the tropicalization of ET is of co-dimension at least one inside the boundary of MR. So definition, let wall inside MR be the image of all possible twigs. Then, by the balancing condition, together with the above observation, we see that the subset wall in MR is a polyhedral of co-dimension greater or equal to one. And we can make it finite polyhedral by bounding the degree of twigs. Here is an example, the picture of walls for the example of blow-up of Toric surface we gave above. ET-trop, the tropicalization of the center of our blow-up is just two points at the boundary of MR. And by the balancing condition, we can see that the walls consists of two lines and then infinitely many rays in this sector. Next observation, by the balancing conditions again, a spine can only bend at wall. That is, a vertex of a spine is balanced unless it lies in the wall. Because if a vertex is not balanced, a vertex of a spine is not balanced, then there must be some twigs attached to this vertex. And by definition, the wall contains the image of all possible twigs. So a spine can only bend at wall. This gives strong constraints on the shape of spines, especially for transverse spines. Transverse meaning that transverse to wall. So since wall inside MR is of co-dimension greater or equal to one, we see that for a transverse spine H, like in this picture, this red curve, a transverse spine H, like this red curve, if we require the image h-gamma to pass through a fixed point x inside MR, then we can no longer deform the map h. In other words, it becomes rigid. We see that a priori we can translate this gamma. Not really translate, but this gamma we can deform it to something like this. It can only bend at the wall. So if we push it up, it bends here and then it bends again. It has one-dimensional deformation space. But if we fix a point x, then this spine becomes rigid. Furthermore, if we perturb a little bit this x inside MR, then the spine deforms uniquely. This is called the rigidity property of transverse spines. Let's give the precise statement. For simplicity, we just restrict to extended spines, i.e. not truncated, but rigidity holds for general spines too. It's just more complicated notations. So let's restrict to extended ones. Consider the map phi-i-trop from the space of transverse spines in MR with infinite directions given by P to the product where the first factor takes domain of the spine and the second factor takes evaluation of the ice marked point. And let S be a transverse spine inside this space. Then there exists a connected open neighborhood vS of S inside this space of transverse spines such that the restriction of phi-i-trop to vS is a homomorphism onto its image and is open. This is the precise formulation of this intuitive picture. So we just denote S bar by S bar, the image of S in the target, and vS bar, the image of vS in the target. Then the rigidity property implies that vS is a connected component of the space of spines restricted to vS bar, meaning that vS is a connected component of the sub-space where we ask the ice marked point to lie in vS bar. And this looks familiar to the connected component theorem above. For the purpose of deformation invariance, we stated the main theorem of this talk, the connected component theorem, and this is looks familiar, which the connected component theorem claimed that this restriction of M smooth to vS and R is a union of connected components of the proper modular stack M bar restricted to R. So let's recall the notations from the connected component theorem. We had the commutative diagram, the phi-i from the modular space M smooth taking domain and evaluation at PI. At the tropical level, we have phi-i-trop from the space of spines taking domain and also evaluation at the ice marked point. We have two vertical maps. The left vertical map takes the associated spine and the right vertical map is just the retraction map. And we have a transverse spine inside the space of spines and the neighborhood vS of the transverse spine. We denote by S bar the image of S and vS bar the image of vS. And R was a Zariski open subset of this push forward pullback. Assume we can extend the map SP to some SP bar from the proper modular stack to the space of the spines in the compactified skeleton MR. Then by some hypothetical continuity of this extension SP bar, the connected component statement one implies immediately that this proper modular stack restricted to vS is contained in the proper modular stack. The proper modular stack restricted to vS bar is a union of connected components. It just follows from this continuity of SP bar and this connected component statement one. And then statement two, which is the statement of the connected component theorem seem to follow from the smoothness theorem. So this seems to be a quick proof of the connected component theorem. But unfortunately, this reasoning has two major flaws. Maybe let's make a five minutes break and after the break I point out the flaws in this reasoning. If you have questions, do not hesitate to interrupt. So let's restart. This simple idea has two major flaws. Do you hear me? So this simple reasoning has two major flaws. First, the statement one was only about transverse spines. But we applied it to all spines. Non-transverse spines can be terrible.  Then the bands and the weights, they are completely out of control. Because if a spine lies in the wall, then the twigs can be attached to spines anywhere. So we don't have any control on the bands or on the weights for the spine. Consequently, there is no reasonable modular space of non-transverse spines. Why am I seeing some arrows on the screen? So nonetheless, tropical curves always satisfy the balancing condition and have a nice modular space. In fact, we can prove that the space of tropical curves in MR with infinite directions given by P, whose associated spine belongs to VS, is a union of connected components of the space of tropical curves, whose ice marked point maps to VS bar. But it still falls short for remedying the flawed proof, because we must consider not only tropical curves in MR, but also in MR bar. That is, we should also allow components of the tropical curve to go to the boundary of MR. This is not theoretically impossible, but we give up because the combinatorics involved become too complicated. And secondly, spine alone, this is a more serious flaw, spine alone cannot guarantee that a stable map in the proper modular stack lies in the good modular space. That is, we cannot just use spine to impose that our stable map meets the boundary D only at the marked points, because entire components of our curve C may lie inside the boundary. So we cannot really apply the smoothness theorem to conclude. One can try to remedy this flaw by considering tropical curves in MR bar, but it's not easy. So after all these discussions, we are finally ready to sketch the proof that works. I spent some time explaining why the heuristic proofs do not work, because with Sean we also spent a long time in order to figure out a correct proof. So let me give a sketch of a proof of the connected component theorem. First, we need the theorem of continuity of tropicalization. That is, the tropicalization map from the modular space M smooth to the space of tropical curves with infinite directions P, this tropicalization map is a continuous map. Consequently, the composite map that takes the first tropical curve and then associates the spine is continuous over the transverse locus, the locus of transverse spines. We claim continuity only over the locus of transverse spines, because there is no nice topology on the set of non-transverse spines. And the continuity statement follows from a general continuity result in a previous paper of mine, or in the paper by Regnation, which is proved using formal models in my paper and using log geometry in Regnation's paper, respectively. It is also easy to give a direct proof in this special case for this specific modular space. We do not need the neither formal models or log geometry. So now let's recall the statement of the connected component theorem. We have this commutative diagram, the natural map phi i taking domain and the evaluation of the ice marked point. At the tropical level, we have phi i drop at the level of spines and the map vertical map taking associated spine factors via the space of tropical curves. Then we are given a transverse extended spine S in the space of spines. The claim is that there exists an open connected neighborhood VS of S and a connected Zariski open R of the push forward and pull back of VS such that when we restrict M smooth to VS and R, we obtain a union of connected components of the proper modular stack restricted to R. Here is the sketch of proof. Step one, the rigidity property of transverse spine implies that there is an open connected neighborhood VS of S in the space of transverse spines such that VS is a union of connected components of the sub space of transverse spines whose ice marked point maps to VS bar. This is just a follows from the rigidity property. Then up to shrinking VS, we have an analogous statement for tropical curves, but we can remove the transverse condition. So the space of tropical curves whose spine belongs to VS is a union of connected components of the space of tropical curves whose mark ice marked point belongs to VS bar. Then by the continuity theorem, the subspace of M smooths whose spine belongs to VS is a union of connected components of the subspace whose ice marked point maps to VS bar. And since the right hand side is as I risky open inside this proper modular stack M bar, we deduce that M smooth restricted to VS is as I risky open inside the proper modular stack M bar restricted to VS bar. And the connected component theorem claims that we have a union of connected components. So it remains to show that it is the risky closed after restricting to some dense open subset are in inside this pre image of VS bar. Step two, we prove the following claim. So in order to show that it's the risky to prove the risky close closeness, we use some closed analytic disk to detect to detect the risky closure. So we consider a map of five from the closed unit from a closed unit disk to M bar to the proper modular stack M bar restricted to VS bar. In other words, we consider a family of stable maps in M bar parameterized by a closed unit disk with the condition that the ice marked point maps to VS bar. Such that the image of the punctured disk lies in in the smooth locus and also has a spine in belongs to VS. Then the claim is that the central fiber belongs to M UP beta, meaning that the central fiber has correct intersection numbers with the boundary. Sorry, you said it's closed. So when put an upper index smooth and low index VS to this five zero five zero five of the punctured the disk. Yeah. So this is a condition on this family. Yeah, yeah. But if it's closed, then we want to prove that five zero also belongs to the same set. It's we want to show that it is closed after further restricting. Ah, so it's just yeah. Without further restricting. It's not true. So that's why it's just an intermediate step. Yeah. So we consider a family of stable maps. Parameterized by a closed unit disk. The fibers outside the zero. They are very nice. They lives in the smooth locus and also their spines belongs to VS. Then we claim that the fiber, the central fiber over zero. Although it might not be as nice as the general fiber, but it will still have the correct intersections with the boundary. So here we see that the central fiber is a degeneration of the generic fiber. And we can have some new bubbles appear. But they will not be too bad. They are still, they still have the correct intersection numbers with the boundary. And let's first stabilize the domain curves, contracting all the bubbles. And then so after contracting all the bubbles, we get a family of stable pointed curves. Then we take a trivial family of small caps around every marked point. So it's always possible if we shrink the disk. And we can always take a trivial family of small caps, let's say of size epsilon around every marked point. P1, P2, P3, P4 as in the example. Then we take a preimage under the stabilization map. So here the preimage is again a family of small caps. But a priori, the preimage may contain some extra bubbles. Step 2.1, we show that there exists a compact analytic domain K in U analytic such that the boundary of each cap, this orange cap of every fiber maps to the compact analytic domain. The idea is to use the rigidity property of transverse span plus the continuity zero. So first, we control the boundary of these caps. And the second, we show that the body, this white part, the body of each fiber maps to K. In particular, the body is disjoint from the boundary D analytic. Because our goal is to show that the central fibers still have correct intersection numbers with the infinite with the boundary divisor D. So we want to prevent bubbles inside the body from moving to the boundary. And we use a compact subset to prevent this. And to show that each body must lie in the compact subset, we use the affinities of U and the maximal modulus principle. So we cannot prevent the body from touching the boundary without the affinities of U. Step 2.3, we show that up to shrinking the size epsilon of the caps, the caps of the central fiber cannot contain any bubbles. For this, we need to use the fact that the set of walls has a co-dimension greater or equal to one. And we use the continuity theorem and also the stability condition of stable maps. Now combining steps 2.1 to 2.3, we see that the central fiber meets the boundary divisor D only at the marked points. And the tangency orders are also good by the curve class beta. So then the claim in step 2 follows. The really essential part is to show that in this degeneration procedure, we cannot have bubbles moving from the body to the caps and finally reach the boundary divisor. So finally, we can conclude from steps 1 and 2 and the smoothness theorem. Recall from step 1 that the modular space M smooth restricted to VS inside the proper modular stack M bar restricted to VS bar is Zariski open. We already proved Zariski openness. It remains to show that it is Zariski closed after further restricting to some dense open subset R. And for this purpose, in step 2, we try to use some family over a closed unit disk to detect Zariski closure. So the claim in step 2 says that for any family in the proper stack M bar restricted to VS bar parameterized over by a closed unit disk such that the punctured disk lies in the good locus. Then the central fiber has correct intersection numbers with the boundary. This implies that if we take Zariski closure of this M smooth restricted to VS inside the proper modular stack M bar restricted to VS bar, the Zariski closure must lie in this modular stack consisting of stable maps with correct intersections of the boundary. It's just a reformulation. And let's denote this Zariski closure temporarily by this M bar VS. And we obtain a proper map of Phi i restricted to this Zariski closure by the properness of the modular stack M bar. Here we use the properness of M bar and the claim we can take R, this dense Zariski open R to be simply the complement of the image by Phi i of this closure minus the good part. So we just, we want to get rid of the bad part and we just the bad part is the complement of the good part. And we take image of the bad part and then we take complement. And the image is Zariski closed by the properness of this map Phi i. So claim is that this R satisfies our purpose. Proof of the claim, the pre image of R lies in the good locus by definition because we have take out all the bad locus. So the Zariski closure of this M smooth restricted to VS and R inside M bar restricted to R lies in M smooth. It's just by the fact that this is Zariski closure. So this implies immediately what we want that this is a union that M smooth restricted to VS and R is a union of connected components of M bar restricted to R. But when we take a complement of the image of the bad locus, maybe the image of the bad locus is just the whole thing. And we have to apply the smoothness theorem to show that R is actually big. It intersects every fiber of the projection. So in particular, R is not empty. This completes the proof of the connected component to zero. I'm sorry, it's a bit long to explain the idea, the sketch of the proof. But that's really the essential for me the most essential statement for the non-Archimedean curve counting without this deformation invariance, nothing will work. Okay, so this is a sketch of the connected component to zero. Why am I also seeing something on the screen? Can people not annotate on the screen if you I think maybe if you annotate on the screen then everyone will see on the screen. So let's go to the next section. Tauric tail conditions in families. Recall that for counting curves with boundaries associated to truncated spines, we must impose some extra regularity conditions on the boundary so as to obtain a finite dimensional modular space. When our log-Kalabiya variety U contains an algebraic torus, Tm, we can take advantage of this torus and impose a simple boundary regularity condition called Tauric tail condition. We have introduced the Tauric tail condition for skeletal curves in the two previous lectures. For the deformation invariance of counts associated to transvert to truncated spines, we must study Tauric conditions in families. So first, let us extend the Tauric tail condition to non-skeletal curves. Question, how to specify a tail inside a rational curve? Recall from the symmetry theorem that adding or removing internal marked points does not affect the counts. So we can specify tails by adding as many extra internal marked points as we want. For example, consider a rational analytic curve C with marked points P1 to P4, P1, P2, P3, P4. The red subtree is their convex hull. If we want to specify a tail containing some boundary marked point P1, for example E equals 2 containing this boundary marked point P2, we add an internal marked point PS here and consider the preimage by R of the path connecting PS and PE. Where R denotes the retraction map from our curve C to the convex hull of all the marked points. The red subgraph plus the pink one. If we want T to be a genuine tail containing the boundary marked point PE, we should choose PS sufficiently close to PE so that T does not contain other boundary marked points. If this is anywhere else, it may be T may contain other boundary marked points, then it's not a tail. It's many tails. And we denote by T star to be T minus the boundary marked point PE, the punctured tail. Recall that the toric tail condition asks the punctured tail to map to the torus. The example shows that in general if we want to impose tail conditions for stable maps in our modular space M smooth, we can specify a tail by picking an internal marked point PS and a boundary marked point PE and we should restrict to the subset theta inside the space of domain modular where the preimage T that is preimage by R of this path does not contain other boundary marked points. And then once we specify a tail, we can impose the toric tail condition. We have the following lemma that of equivalent formulations of toric tail condition. For any stable map in our modular space M smooth, whose suppose that its domain lies in the subset where tail makes sense, then the following are equivalent. First, the punctured tail lies in the torus. Second, the whole tail lies in the isomorphism locus of the toric model pi. Third, the tail does not intersect the exceptional locus of the toric model. And fourth, there are no tweaks of the tropical curve, F attached to the path to the path PSPE inside the spine. So this is a picture of the tropical curve associated to F. The red sub tree denotes the spine and we have some tweaks. Then the toric tail condition equivalently says that there are no green tweaks attached to the path connecting PSNP. This part is three of tweaks. And now we have the proposition for toric tail condition outside the wall. Consider let N in M smooth with a good domain modular be a subspace such that F of the marked point PS does not lie in wall for all F in N. Then the subspace satisfying the toric tail condition and tail inside the N is a union of connected components. So I recall that for deformation invariance, we need to show something is a union of connected components in another so as to obtain finite a tallness. And here we are doing it in two steps. First, for the extended spine, we have the connected component zero. And the second, we have to show that when we impose the toric tail condition, we further cut out union of connected components. Yeah, so since the toric tail condition is the second equivalent formulation here is an open condition. So the openness of the inclusion follows. For closeness, we use the equivalent formulation for and pick any sequence F lambda in the subspace and tail conversion to some F in N. More precisely, we do better pick any net instead of sequence because the underlying topological space may not be first accountable. For any F lambda consider the associated tropical curve like here by Toric tail condition for there are no tweaks associated to the path PS to PE for any F lambda. In particular, the leg LS this leg containing PS must be contracted. Then by the continuity of tropicalization for the limit F, the leg LS this leg must also be contracted. And moreover, if any tweak moves into the this path connecting PS and PE under the limit, then the tweak must attach to the vertex PS bar, which is the intersection of the leg. L E and the leg LS. We see that this is impossible because by our assumption, the image of PS does not lie in wall and the leg LS is contracted. So if it's not it's not lying wall, there can be no tweaks attached. So F must satisfy Toric tail condition for this shows closeness. And this proposition plus the connected component theorem implies the invariance of the count NS beta under small deformations of any transverse spine. Here the spine can be truncated and the transverse means in particular that the finite one-valent vertices do not map to wall. Second remark for the proof of the associativity of the mirror algebra and the proof of the wall crossing homomorphism. We will need a deformation invariance for not only transverse spines, but also certain non-transverse spines. For example, for the associativity of the mirror algebra, we need to show that the spines like the following SL, SMSR, they all have the same counts. Imagine that we deform the spine SL to SM, we move it a little bit to the right, and then to SR. In the middle, we get a non-transverse spine SM. And similarly, for the proof of wall crossing homomorphism, I will explain wall crossing in the next lecture. When we need to verify that the ring homomorphism, we want to verify that the wall crossing transformation is a ring homomorphism. If we multiply by a theta function that is parallel to a wall, we will need to show some deformation invariance as follows. So we want to show the deformation invariance associated to such a spine when we move it across the wall. This is important for showing ring homomorphism, showing that wall crossing transformation is a ring homomorphism. And this edge can just be parallel to a wall if we are multiplying by a theta function that is in the same direction as the wall. So eventually what we are interested in are just the equality between the counts associated to SL and SR. But to prove the equality between the counts associated to SL and SR, if we deform SL into SR, it's inevitable that we encounter some non-transverse spine SM. In both cases, when a finite end maps to a wall, we observe that in both cases, when a finite end maps to a wall, the derivative at the end is contained in the wall.  Question, do we have deformation invariance for almost a transverse spine? The answer is yes, but the above proposition does not apply. We must restrict to skeletal curves. So that's another application of the theory of skeletal curves. And in fact, for other applications of skeletal curves such that for the symmetry theorem, it's possible to do it without skeletal curves. We can have other proofs, but for this almost a transverse spine, I don't know how to do it without using skeletal curves. And that's really important, these almost transverse spines because that's for the proof, that's for the associativity and also for wall crossing homomorphism. Without associativity, we have nothing. So let's explain how to get a deformation invariance for almost a transverse spines. Recall, we have our map phi i from the modular space m smooth, taking domain and evaluation of the iso-marked point. And we denote by isk the pre-image by phi i of the skeletal. Tony, what does let i mean since it's notation isk? i means we are evaluating at the iso-marked point. Ah, it's a skeletal isk here. Bigot, it means inverse of skelet. But it's just a notation. Yeah, inverse image of skeletal. Yes, so it's just notation, inverse image of skeletal, it consists of skeletal curves by the previous lecture. So now here is the proposition. Toric tail for almost a transverse spine. Let n be a subspace of m smooth with good domain modulus intersect isk. That is, we only consider skeletal curves. Let n be an open subspace such that for any f in n, f-trop of p-s lies in some polyhedral cell sigma of wall. Then the linear span of sigma contains this derivative p-e. So this is the condition for almost a transverse. And once the condition of, once f is almost a transverse for all n, for all f in n, then the subspace satisfying the toric tail condition is a union of connected components. For me, that's really the next, after the connected component theorem, that's the second most essential technical result for the whole theory to work. Now, again, openness follows from the toric tail condition too, because it's an open condition. For closeness, let f in n be a point in the closure. Since n in isk is open, the restriction of phi i to this n tail is also open by the smoothness theorem. Because we have this open inclusion and by the smoothness theorem, we have a dial map, a dial map is open. So we have restriction of phi i to n tail is open. So we can find a net f lambda in n tail convergent to f such that the associated tropical curve f-trop has a transverse span for all lambda. So this is this blue, two blue rays are part of our wall, and this red tree is the spine associated to f lambda, and we have some tweak. So let i lambda denote the interval in the spine of f lambda connecting the point p s bar and the nearest branching point. Toric tail condition 4 implies that the associated tropical curve f lambda trop contracts the leg ls containing p s. And moreover, it is balanced on this i lambda union, the leg le with derivative pe. So the Toric tail condition says that there are no tweaks on this path connecting p s to pe. But since the branch nearest branching point is a bit away from p s bar, by the balancing condition at p s bar, we see that moreover this part i lambda is also good. It's balanced everywhere, also containing i lambda with the good derivative pe. So if in the limit f-trop of p s lies in some cell, sigma for example in this cell, our assumption says that the linear span of sigma must contain this direction pe. So we imagine that when we move this tropical curve to this cell sigma in the wall, we see that our assumption is satisfied. The direction of this leg lives inside the linear span of the wall. Then the transversality of the span of f lambda implies that the image of all this part, this lower part, does not meet any such cell of wall. Because it's out of wall and also it's parallel to the wall, so it cannot meet any such cell. Therefore, by continuity of tropicalization for lambda sufficiently big, the distance between p s bar and the pre-image of all walls has a positive lower bound. Since when it's out of the wall, it's just parallel to the wall, therefore when we compute the pre-image, we throw away all this sigma. And then we obtain a positive lower bound between the wall and this point. So this prevents, this will prevent that any twig, this green twig slides through this interval i lambda and reaches p s bar in the limit. So the limit f still satisfies the toric tail condition, completing the proof. So here the main idea is to have a lower bound between this pass and the twigs. We want to prevent any twig from sliding into our pass, so we try to find a lower bound of distance between them. And this is only possible if we restrict to skeletal curves. A remark, this proposition implies the deformation invariance for almost a transverse span. It is necessary for the proof of the associativity of the mirror algebra and the proof of the wall crossing homomorphism. That's all I want to say for toric tail conditions. And now in the remaining five minutes, I'll give a quick introduction to the gluing formula. Recall that we have defined the counts of curves with boundaries associated to truncated spines by imposing an extra regularity condition on the boundaries, namely the toric tail condition. In this way, the counts of open curves are translated into special counts of closed curves. So it is natural to have some skepticism at this point. Do the counts we define really reflect open curve counting? Or is it just some false advertising? We would like to relieve this skepticism by establishing the next important property of our counts, the gluing formula. The idea is the following. Roughly, the gluing formula states that we can glue two open curves along two opposite boundary components, like here, and form a bigger concatenated curve. Then the counts of the concatenated curve should be the product of the counts of the two initial curves. Here we have two open curves, and we glue them at the boundary. We obtain a concatenated curve, and the gluing formula states that the counts of the concatenated curve should be the product of the number of this sort of counts for the first one and times the counts of the second one. So this is a more convincing evidence that our counts really reflect open curve counting. And it is also an essential ingredient in the proof of the associativity of the mirror algebra. Maybe let me stop here for today, and next on Thursday, this Thursday, I will continue from here. I will give a quick proof idea of proof for the gluing formula. Although in the two dimensional case, the gluing formula was in some previous paper of mine, but here I'm giving a more conceptual proof. And then I will talk about the idea of proof for associativity. But the main topic of the last lecture will be wall crossing transformations. How do we get wall crossing transformations from our counts of skeletal curves? And how do we compare the resulting wall crossing structure with the wall crossing structure of gross-hacking keel condensates in the case of cluster algebras? So if you think that some technical details in today's proof are difficult to follow, in the next lecture, there will be some more statements and with less proofs. So thank you very much for your attention.


Hi everyone in this video we gonna talk about that how to find the minimal value in a list and how to return that value. First we'll make some list let's call it my list and let's put few elements inside it. If we want to find the minimum value of our list we'll use function min and inside that function we'll put as an argument our list my list. So this function will return the minimal value of this list. I am wondering what will happen if we have for example I don't know list of strings, high, hello, maybe good morning and so on. What we will get by calling minimum function and providing our list. Do it like that. Okay and we get as the result good morning so obviously minimum function returns the minimal or maybe we could say the first element lexically or dictionary ordered if we are watching those elements as terms in some dictionary. So good morning will be first in lexical ordering or in some dictionary. That's all for this lesson. Thanks for watching. Happy