I pervaded out half-hour ago, I would like already a kitchen, as if he's almost a brother of mine. That's definitely the way I felt as I traveled over the last, I would say, 20 or 30 years to the Muslim world and to different parts of it. And one thing pervades, and I haven't commented on Pakistan, I lived there briefly. In 1993, 1994, I taught there at Punjab University in the city of Lahore. And as a journalist, I want to say quasi-academic, I did a fellowship at Oxford University where I studied the influence of Islamic architecture and I've written a lot about music as a journalist. And so my background is really someone who's curious about different areas and the way, in a sense, people and cultures connect. Even if at the higher level, people are saying, we are disconnected. So that's always been my, in a sense, my outlook is to look for ways that people are connected, even though governments and leaders may say we're not connected and we shouldn't be connected. So I did write the book, actually I'm writing a book now, but I wrote this book in 2008 called, All America Travels to America's Terrorism is on the Roots. And it was one of the first books, and I think it's still the only book that addresses the historic interconnectedness between the Arab and the Muslim world in the United States. Now, I wrote this book because I was frustrated at not seeing these connections articulated in the popular media.  Oh, thank you. Is this better? Yeah, it is better. Okay, so I'm not going to repeat what I said earlier. Did everyone hear me well enough? Yeah. Okay, good. So I'm not going to repeat what I said earlier, but I will just say that... So part of my background is that I'm a little bit distressedful of what politicians say about the Middle East. A little bit, if not more than a little bit. Or anything else. Or anything else, right. And so as journalists, I'm trained to sort of step back from events and to ask questions about, well, why is that? And why is not that? So Islam has the... Is that better? Yes. So... Okay, thank you. So in my research for my book, I found that the Islamic world, Islam has been with America from America's founding. And that was a surprise for me to find that out, especially in the wake of this, you know, tragic event. And this, of course, is 9-11. And you can see the Twin Tower buildings on fire about the collapse. So one of the things I found in my research is that, you know, the connection between the Muslim world and America is so profound and so deep that people aren't, in a sense, people aren't even aware of these connections, partly because of the media, I think. Partly because the media doesn't talk about these connections, partly because a lot of politicians say there are no connections, or they don't know themselves about these connections. And so in a sense what I found in my book is that these connections, I won't say have been white washed from history, but in a sense it's more of a nine neglect. They've been, you know, kind of chunked aside for other things. And so 9-11, though, you get, of course, a lot of extremism.  And when I saw that, you know, in my research, I was personally offended. And of course we're offended by things like this, and of course we're offended by things like this after 9-11, some American taking it out on an immigrant Muslim family from Pakistan and Dallas. But I realized, you know, part of the reason that there's this culture of Muslim hatred is that it begins, in some sense, at the governmental level and in some sense at the educational and academic level. And I don't know how many of you know this book called The Clash of Civilizations that Remake in the World Order, but it's by Samuel P. Huntington. And he said in his book, a multicultural America isn't possible because a non-Western America is not American. Now one thing you should know about Samuel P. Huntington is that he was an advisor to the Bush administration and that they listened to him resolutely. Now I'm not saying that Samuel P. Huntington, you know, in a sense, led to this backlash against American Muslims, but we can't ignore people like Samuel P. Huntington when we look at the backlash. We can't say these are isolated incidents. We have to say these are connected in some ways to other things. And this is one other thing that I think was too often ignored after 9-11. And you know, of course I wrote this book when Barack Obama was trying to become US President. And at one point in 2008, he was, this was his political opponent, John McCain. At one point in 2008, somebody out of the McCain rally says, you know, Obama's era. And McCain says, oh no, ma'am, he's a decent family man. And some people said, oh look, he's defending. But no, actually in a sense to me, this is where in popular American culture, Arab and Muslim is a buzzword for he's, you know, fill in the blank. My mother might say he was sugar. But you know, you could fill in all sorts of names. He's a blank. So to me, no, that's, this was a bad sign of the culture for me. And of course, it extends into 2010 when you have Sarah Palin manufacturing, helping the manufactured partners around a moffs that was supposed to be built in Lower Manhattan. And then in 2000, actually no, this is right this year, we have Donald Trump. You know, somebody says we have one of his rows. We have a promised country. It's called Muslims in New York for President of Israel. And Donald Trump's reaction is right. He's mad. He doesn't say, well, no, first of all, what's your problem? You know, he doesn't say that. It's more like he's skating along with this right wing, excuse my language maybe, audience member. Then we have Dan Carson, thank you, would never support a Muslim candidate for president. So my book in a sense was to say, wait a minute. You know, let's put a stop to all this. Look at the history of Islam in America. That's what I wanted to do. And so I wanted to start by looking at the history of Islam. And I wanted to start it actually in Muslim Spain, because in a sense this is where Islam in America starts. We had, of course, you know, Muslim rule over Spain for about 700 years. And in the year 1492, after what was called, now called the Reconquista, you know, Muslim rule over lower parts of Spain ended. And so this is a depiction of the surrender of Spain in January 2nd, 1492. Now we all know in our history books that something else happened in 1492. Something very big. And that was this man, Christopher Columbus, landing in the Bahamas. Not to literal, I mean, now is the United States, but the Bahamas. And so that happened in 1492. And what the Reconquista was about, and what Columbus's journey was about, in a sense was about expanding empire. It was about expanding the Spanish empire. But it was also about getting rid of Muslims in Spain. And also, by the way, Jews, Jews in Spain, but that's another story. The Reconquista and Christopher Columbus' journey was about expanding Spain's power without Muslims. Spain, in fact, banned. They banned Muslims from going to what they called the New World. If you were a Muslim, you could not travel with Columbus. And you could not travel on any journeys to America unless you were a good-out Christian. Now, one of the ironies I found in my book is that Spain, of course, because of 700 years of Spanish rule, was essentially an Arabized country, was a Muslim country culturally. And so Columbus' title, the very title that he devoted himself to, was Al-Burante-Demar-Arash-Hoshiana. That's how he signed his name, Al-Burante. Now, Al-Burante derives from the Arabic word El-Amir, which means the commander. And it means the commander of the ocean, ocean sea. So, in a sense, Columbus himself was using Arabized Spanish, but to get to the New World, he also relied on Arabic knowledge, including decades and decades of these triangular-shaped sails that Arabs navigated for years before Spanish navigators adapted them. And without these sails, Columbus would not have made it to America as quickly. These are the older sails, the more square sails that they used. And of course, Columbus relied on Arabic knowledge in terms of astronomy. And Al-Fardani was a famous 9th century astronomer whose guidelines for how to navigate the world Columbus relied on. And here's one of the things I dug out in my research. And that is Fernando Columbus saying there are five reasons, top five reasons Columbus made it to America. Fifth was Al-Fardani, fifth with Al-Fardani. Now, Columbus, I found this in an obscure book. Columbus actually thanked, you know, there's this perception that Columbus and Spain were entirely against Muslims, which they sort of were actually. But it gets complicated. It gets complicated because here's Columbus saying in 1501 about a decade after, you know, it's good, good discovery in America.  With one of the men in the church in Aleman, Latin, Greek, Jewish, and Morse. Morse meaning Muslims. So here's Columbus flat out saying he's indebted to Muslims. Now, this is not very well known in any history book I read when I was a kid, nor afterwards. You know, frankly, until the time I found this in an obscure book. But what I found essentially was a whole history of Columbus relying on Muslim knowledge and Muslims personally to get to America. And so in a sense, America is indebted to Muslims. Is indebted to this interconnectedness that Columbus, you know, learned from in Spain. Now, this connection between America, Columbus, and Islam and Islam now is not very well known. And, you know, this is one of the things that I wanted my book to kind of, you know, elucidate to announce. But I didn't want to stop there. I did not want to stop there. I wanted to go back to Spain. And those of you who've been to Spain, you know about the Alhambra, this really incredible Spanish palace and mosque, essentially, in Granada. And architecturally, when you go to the Spain, you see this beautiful work above doorways. This is called in architectural terms, alf and alfines, A, L, F, I, Z. Now, you know, again, you find this everywhere. But I was surprised to find this Islamic architecture in the Alamo. Now, the Alamo is one of these iconic, you know, American buildings. American kids who grew up, you know, they always, they always learned the history of the Alamo because of what it represents. It represents the expansion of America, you know, over Mexican territory, right? And millions of people passed under that doorway. But what they don't realize is that, again, above that doorway is an alfine that takes its architectural dimensions from the Alhambra. And this is one of the ironic things. Spain, when they came to the New World, did not allow Muslims, but they did allow Muslim architecture. They wanted it. And in fact, the Alamo is part of a series of Spanish missions, Spanish missions that have Islamic architectural influence. And so when you go to the Alamo, no one is handing you out literature saying, welcome to the Islamic-inspired Alamo, Salam alaykum. They don't say that at all. And so I dubbed this out, again, through obscure research. I either find this out through obscure research, spending lots of late nights in libraries. But when I was in the Alamo, say, in Antonio, I went to one of the sister missions and it's called Mission San Jose. And this is it. Now, this is what it looked like when it was first built, you know, centuries ago. Now, I'm hearing some wows, and I think I know why.  In a sense, it's a trademark of Islamic architecture for that time period in Spain, for that time period in Spain. So it goes back centuries, right? But again, when you go to San Antonio, Texas, you realize, wow, Spain brought over Islamic architecture, even though people don't know that. And this is the shame to me. All the examples I'm going to show you are examples of how Islam has been disconnected from American roots. And I'm trying to reconnect. My book is trying to reconnect those roots. But this, to me, was a stunning example. Stunning, yes. Did you put Spain there on purpose or why? Oh, yeah. That said, no, that's my bad. San Antonio, Texas. But I actually know you're right. Thank you. That's sort of like San Antonio's name better. Thank you. And so, anyone want to guess where this is? Thank you. Yeah, this is New Orleans. When I went to New Orleans for research for my book, this is 2007, I was really surprised to walk around New Orleans and realize, wow, I thought the famous New Orleans, this is the famous French court, right? Where's the French in this? And at this point, I had done a lot of research on Islam. I'm thinking, am I going nuts? Because I'm seeing Islamic aesthetics here. And in fact, I was surprised to learn that you see this on the streets of New Orleans that Spain ruled over New Orleans from 1762 to 1803. I did not know that. I did not know that. And it just so happens when they ruled over New Orleans, a lot of fires damaged a lot of buildings there. So a lot of the buildings in New Orleans are actually Spanish-style buildings, but they're really Spanish-Islamic style. And again, it goes back to the Alhambra. But the other thing that was shocking to me, this is another photo from New Orleans, is this. Now, why would this be shocking to me? Well, because it's a beautiful courtyard. This is what Spain did when they ruled over New Orleans. They said, we're going to put in these beautiful courtyard buildings. But of course, this is from the Alhambra. Now, but this is where we can't stop, because to me, like history is always telling you, reminding you, it's complicated. It's complicated. So we can't just go back to Spain, of course. We have to go back, that's another view of the courtyard. We have to go back to this building. Now, and if you want to guess what this building is. This is a depiction of what the original, an original mosque, this is Mohammed Mosque in Saudi Arabia, its courtyard. Its courtyard architecture. And this is what happens in history. So, Muslim rule over Spain, they get kicked out, but the people who are kicking in that said, we love your architecture and we're going to adopt it. And when we come to America, we're going to put it into New Orleans. Now, again, when you go to New Orleans, there's nothing that said, welcome to the Islam-inspired French Quarter of New Orleans. You don't get that. You don't get that at all. You have to guess in a sense. And again, it's another example of how things get disconnected. People think of it as the French Quarter.  So, by the way, they were inspired by Islamic aesthetics. That's what's missing. That's what's all missing. And so, this is a church in New Orleans, the Immaculate Conception Church. Now, you know, people who know Islamic architecture will say, of course, those are menarets. Those are menarets. Now, yeah, this church did adopt Islamic architectural, as it's kind of motif, menarets. By the way, I do want to say this, that menarets actually were adopted from Christianity, from according to my research in Syria. This was centuries ago. So, in a sense, it would be adopting. But, for example, how culture was, again, they borrowed from each other. It's all about giving taste. And so, well, let me pass on these. This is a map of Spain ruled over, just part of the United States. And I want to say, well, these are not aberrations. You can go back to any part of the world. West Africa, for example, people know, of course, about slavery how, you know, 60%, upwards of 60% of slaves in the United States were from West Africa. Now, of course, Islam was a long, had a long presence in West Africa. And yet, you know, a lot of Americans don't know about Muslim slaves and their presence in America, including, excuse me, Ibrahim Abdul Rahman, Prince of Slaves. He's perhaps the best known Muslim slave in America. He wrote a book. He met, at the time, he was really well known because he was emancipated. And so, he met in the White House with this man, John Quincy Adams.  Here's another well-known Muslim slave, Omar Ibn Said, who wrote his memoir in Arabic. He, a lot of Muslim scholars, of course, know him. I wrote an article on him for the magazine called the Aramco World, the Saudi Aramco World. He's a pretty incredible example of Muslim steadfastness in America. He was asked for years to give up his Muslim identity he refused, even though he seemed to convert. He seemed to convert, but he did not. And I'm not going to play it for you here, but there are a lot of Muslim scholars who say that even the call to prayer, he doesn't hear an example of someone, but that even the call to prayer helped influence American blues music. How was that? Because a lot of Muslim slaves when they came to this country were still, had still learned these kind of aesthetics, you could say, usable aesthetics. They brought them with them. And that when they were in this country, they would, you know, do something called field jobs, field polars, and they would use the same kind of intonations they'd get in the call to prayer. And then at the last, they call it melisma. Melisma, and that's the sort of wobbling you hear like in different notes, kind of different notes in the same thing. That melisma you can hear in early blues music. And here's an example of an English, this is a quote, an English, he's an economist who went to West Africa and he was saying, oh yeah, I would occasionally see, they would occasionally sing a melancholy song and they would earnest, you know, learn his prayer. And he saw that he thought this was music. He thought, so people outside of Islam who don't know at that year, in those years who didn't know anything about it, would confuse in a sense the call to prayer from music. So, but again, there are scholars who say that there is a connection. And they even say, for example, look at Billy Holiday, you can hear the echoes of Islam music and her music. So another example, this is another example of this kid, that's these, of the, and after he's apologized, of course, to study these connections, he gives this great quote called, Africa and the Blues, which goes into all sorts of details. But I want to get into this connection too. This is Iran, and this is, you can't see it last name, but this is a very famous, Ralph Alder Emerson, right? Very famous philosopher and academic. He's known these days for one of his famous quotes is, a foolish consistency is to have godland of little minds. That quote was actually directed at George Bush, I'm sorry to say. He was in office. I'm not trying to make this a Republican or Democrat lecture at all. But I'm disappointed. And here is Ralph Alder Emerson. He loved Persian poets. He loves Hafez and Saddi. And in fact, he wrote often about their poems. And here's an example. That's the second one. I was as a jet concealed me in my brain read. Emerson thought that was from the Quran, but it's actually not. It's from a different book, but it shows you to the extent to which that Emerson looked to the Quran, looked to Persian poetry, and looked to it for knowledge. In fact, this to me, this is a really good example because Emerson first got interested in Persian poetry, but then the Persian poetry got him interested in Islam. So, and again, a lot of people know this about Ralph Alder Emerson. But in a sense, he was in love with Persian poetry and by default, you could say Islam per visitors in the name of Saddi. And here's an example of a book it's called Bullistan. And when it was first released in the United States, Emerson wrote the introduction to it. Now, I'm going to get into some sillier examples, but I think it's easily profound. This is the guy on this pinky barnum. And he's famous for barnum daily surface and that's some General Tom Thumb. But he said he was he was America's first billionaire. He was America's richest man in the late 1800s. And I have him in here and I wrote about him because when he built this house, this was his house, and he built it in Bridgeport, Connecticut. You can't see it. You want to guess what he named this house? Oh, Alhazel. My brother's a good guess, but no. He named it Iranistan. Iranistan. Now, P.T. Barnum was a wealthy man. He could fly anywhere he wanted. And so he built this house after seeing this, the Royal Pavilion in Brighton, England. And because P.T. Barnum could essentially get anything he wanted, he said, I want this house. I want to live in this house. And so then he built this, Iranistan. Now, when he saw this house, he thought he was building, I don't know, I don't know what he thought he was building. I think he really thought he was building an Indian house in India. And in fact, he was sort of right because this, the Royal Pavilion in Brighton, England was inspired by the Taj Mahal. That's the fact. Now, the Taj Mahal, any of you who've been there and know about it, it's essentially a Muslim building, right? There's a mosque on the grounds. But it's him doing his law, in a sense, mixed together. But the Taj Mahal actually was inspired by, guess what? This building, the Imam mosque in Esfahan, Iran. Now, that's one example. That's one architectural example. This is another. So P.T. Barnum didn't know it, but he was actually right when he called this building, Iranistan. Because ultimately, the architecture goes back to this building. And again, this is the disconnect that people don't know about. Oh, yeah, the Taj Mahal, right in Pavilion, the Royal Pavilion in Brighton. But no, actually, go farther. A little bit farther. That's when you'll find it. This is the source, right here. And if you go to Bridgeport, Connecticut, the Iranistan burned down, but you will find this. You will find that at Iranistan Avenue. But you'll find a mosque not far from Iranistan Avenue. And so even, this is one of the little ironies I found in my research that you go to where Iranistan was, you'll find there's now a small but growing Muslim community there. Small Muslim growing community. Just a few more slides, I think. My book gets into the language, the Arabic-influenced English language. Giraffe comes from, of course, the Arabic word Zarafa. And a lot of these words came through Spain, of course, Alcatraz, a very famous island, of course. When you go to Alcatraz, there's no one saying, welcome to the Arabic-inspired, you know, blah, blah, blah. No one says, As-salamu alaykum. No one says that. It comes from, and of course, there's a misspelling out of Alcatraz. It owes its origins to the Arabic word Ahathas, which, according to my research, originally meant the white-tailed sea eagle. So when Spain came across this island, they said a lot of words there. But again, you know, that's my example. Here, to me, is a really more profound example, though. The state of California, right? Now, the state of California, you know, any of you want to guess where that name came from? Yeah. You're definitely on the right track. There's a very... This is what California looked like on an early nap. They thought it was an island. Spaniards who came to this side of the country said, oh, this was a great island. And in fact, there was a very famous novel at the time of this discovery. Actually, essentially before... definitely. And it had... it was a novel about a mythical island full of bull and the island, you know, and by the way, it's in these history books. This is my son. He's now 14, but at the time he was, I think, around 7 or 8. And his book talks about, oh, by the way, you know, California comes... the word California comes from a Spanish novel that uses the name California, but they don't say that that novel bases California on the word caliph or caliph.  But there's all sorts of references to caliph and Islam. And it's incredible. So when California is named, it really comes from Islam. We live in the state of Islam, basically. Now, you can take that anyway. Those of you who are Muslim will agree with that. Of course. But California, much to my surprise, also, here's a more profound... You can argue which one of these is more profound. A relatively profound one of these examples is. But again, to me, hey, mom, guess what? You live in the state of Islam. What do you mean by that? Well, California comes from the word ultimately derived from the word caliph or caliph. I didn't know that. Well, I didn't know that either until I researched this book. Yeah. That's the stuff. Now, I did know from my years of research in journalism that in the earlier centuries, 19th century, 18th century, the prophet Muhammad was referred to as Mahomet. And if you go to Mahomet, Illinois, and you ask people there, wow, this is the only city in America that's named after the Muslim prophet Muhammad. Peace be upon you. They will say, what are you talking about? And that's what they told me. And I talked to someone who was from Mahomet, Illinois, who said how dare I refer to that city as something related to the prophet Muhammad. But this is a fact, actually. This is a fact. Now, I didn't research this well enough, but you can say to your friends and enemies, if you want, America has a city named after the prophet Muhammad. That's a fact. Now, I will say this is a fun fact. You can't say that the ice cream cone, this is a very well known story that the ice cream cone derives from the 1904 World's Fair, and it derives from Abe Dumar who was at the fair, who was from Syria, who ice cream was sold at the fair, and there's this Syrian flat, you could say, brad or whatever, called Salabiyah, and then he had the idea of kind of, you know, changing the shape and putting ice cream in there, and thus was born the ice cream cone. Now, Abe Dumar is a Christian-Syrian, right? And my book, you know, is about arrogance as well. And here's an example of how, the Dumar legacy lives on. And so, and made me think, wow, this is a silly example, but I think it's a good example, again, of this cultural interconnectedness, cultural interconnectedness. I'm sure some of you know the Shriners whose original name was each in Arabic order, the Nobles of the Mystic Shrine. They're very much an Orientalist group, and, you know, my publisher decided to put it on the cover of my book for whatever reason. That's fine. But they had such an ingrained history that this guy was a member of the Shriners. And those of you who are older, like I am, they recognize him. Yeah, J. Edgar Hoover. He was a Shriner. So he worked as on his head. He was, by the way, J. Edgar Hoover wasn't a real fan of his law. I can say that much, for a fact. Especially Malcolm X. But that's a whole other chapter. It's a whole other chapter. But as another fun fact, Warren Harding was also the president around 1921. He was also a Shriner. And in my research, a bunch of Shriners went up to the White House. It's hard to believe now. They went up to the White House around 1920, when Harding was president, and they said, Salam alaykum. They said that in maths. And Harding said, what do you think he said in return? Yeah, alaykum salam. He didn't know what he was saying. But again, Arabic, Orientalist culture, whatever. It has a way of seeping into the culture. And it's hard to imagine that 100 years ago, they were speaking Arabic from the White House in this very, you could say, sort of silly way. It's a way, nevertheless. I'm going to go past this. Now, how many of you recognize, anyone recognize these guys? Yeah, the doors. I've loved the music of the doors. I interviewed Ray Manzarek, the keyboardist at the very bottom. And why would I interview him for this book? I wrote. Why would I interview him? Well, here's why, because when I hear songs like The End, and those of you who know the music will know The End, when I hear that, I hear Arabic musical influence. Now, I didn't know that, but at the time, and I'm sitting with my son, I say, you know, I have to find out. I'm going to write this guy. He's saying, hi, Mr. Manzarek. I'm writing a book on the connections between Arabic music and American, you know, 60s music. Can you talk to me? He said, yeah, absolutely, I will. And here's what he said, much to my surprise. He said, the Arabic minor harmonic sense is an endemic part of the door music. Now, he also said rock and roll and Arabic music, you know, they're great together and we have to get together with as long. Now, he told me that and I was bowled over by that, really bowled over by that. He passed away a couple of years ago, but it's a shame and those of you who really know the door music will know the song I'm blanking on the name, but I think when the music's over and it has a lyric in it that goes, Persian Night's Babe, you know, that's one of the lines, and Ray Manzarek says that's the over reference to Islam, over reference to Islam.  It wouldn't be great if people like Ray, you know, Jim Morrison died in popular culture, but they don't, of course, and again, this is an example of how many people I have read my book know this, probably relatively few, and I think that more people knew about the connection between the doors, and it's not just the doors, it's Arabic music everywhere, and Robbie Kriger, he's a guitar player with a sarod, he loved Indian music and of course there's a connection between Indian music and Arabic music, sympathetic music, sympathetic strings and on the historical and he was very, Robbie Kriger was very into classical music, which of course is influenced by Arabic music, and of course, you know, those of you who know this pulp fiction will hear this song, I don't know how many, I can't play it for you, but the song called Mizrulu, and if you know that song by this guy, Dick Dale, so that's the song, now Dick Dale, here's a good example of how, again, disconnected, Dick Dale's real name is Richard Monsour, and he's one half coalition, one half Lebanese, right, and so Mizrulu means the Egyptian, it means the Egyptian, Masr means it's the Arabic word for Egypt, in my research, you know this guy, Elvis Presley, now this is how many of you have been to Graceland in Memphis? How many of you? Well, I mean, Elvis Presley loved Khalil Gibran as the prophet, he wanted to make a movie out of that, and he had the prophet on his night stand when he passed away, so again, a small example, but I can go well with an example of how even people in popular culture, like Elvis Presley loved Khalil Gibran on the past of this, and then, you know, of course I know a lot of you know the Eid stamp, which came out just before 9-11, and I interviewed, this is the Eid stamp, very famous, very famous, and I have to say he's a Muslim, he converted to Islam in his probably, I guess, 30s, and he the reason I bring him up right now is that he was also a tattooist when he was in his 20s, before he was Muslim, he was a tattooist, right, and so I asked him a lot of questions like, well, Muhammad, you know, how does tattooing relate to the Khalil Gibran?  I go, but Muhammad, it's against Islam, you know, you're supposed to get rid of those. And he goes, Jonathan, there's no use crying over spilled milk. That's my, I'm sorry. But the reason that comes up now is that, you know, here's an example of someone who, you know, adopted an Arabic on her right arm, and this, the Arabic she adopted was al-Azibah, and probably maybe I mispronounced it. It means strength of character, strength of will.  Yeah, until the end of the year, exactly. So, but, you find all sorts of examples in the culture like this guy, Akbar from Star Wars, Admiral Akbar, Admiral the Great, but I want to go back to something, because the first image I showed you was of twin powers, and these are the twin powers, and you can see how Minori Yamasaki, this very famous architect, created the twin towers, and at the base of the towers, you can see these spirals that he made. These are inspired by Islam. These were at the base of the World Trade Center towers that were destroyed on 9-11. Now, why would Minori Yamasaki do that? Because Minori Yamasaki was a fan of Islam's architecture, and if you go to Dabran, Saudi Arabia, this is a, you built this airport terminal in the 1950s in Saudi Arabia. You built the World Trade Center in the 1960s. And so you can see the connection between the Islamic spirals there and there. Now, and this, again, we saw this building earlier. This is the Shah's mosque, or the Imam mosque in Esfahan. This was Minori Yamasaki's favorite building. That's by the New York client. What's your favorite building in the world? This was his favorite building. You know, again, the World Trade Center destroyed on 9-11, ironically enough, was built with Islam at its base. And here's Minori Yamasaki. Now, I'm going to end this talk on personal note. When I did a fellowship at Oxford University in 2005 and 2006, and it was about Islam and architecture, and I went to this town, and if anyone guesses where this comes from, I will give you all sorts of stuff. Good honor? Good honor? Good honor, no. It's a town in France. Leon, that's a very good guess, though. It's about an hour or two from Leon, but... Go into this. Was that? No, that's a really good guess. That's another one. Okay, I don't have to give my whole life away. Thank you. I think Le Puy, on the other hand, is Le Puy. Now, why is this town important? Because in the 11th century, this is one of the towns where the Crusades dominated from. We had people in this town who said, you know, we're going to go to Jerusalem and we're going to get rid of the Muslims from the Holy Land. From this town, they said, we are going to be, you know, essentially the world of Islam. Now, why would I go to Le Puy for my research on Islam and architecture? If you go to the most famous building there, this is the Cathedral there. This is not a good image of it, but if you walk up the steps and you go to this huge doorway, you'll find Arabic writing there. Khufiq writing. And why is that writing there? Because in the 11th century, despite the Crusades, despite people at the highest level saying, we will not do business with stain and Muslims, at the more profound cultural level, of course there were business happy between Muslims and non-Muslims in Europe. Of course there was. Coins were minted, you know, they were sent back and forth. And so Arabic writing was ironically enough, or whatever, unironically enough, in the very cathedral from which the Crusades happened. This to me was a very profound experience for me because you realize, again, I said this at the very beginning, at the higher level we have people saying, we're at war with Philip de Blanc, we're at war with this, but really at the more profound one-to-one level, people are people and we mix with each other, we take back and forth, we go back and forth. And so that's why, this is not in my book, but that's why this was one of the many profound experiences of my life, that even during the Crusades, at the very human level, Islam is integrating in Crusader talent in France. And it's still there. The evidence is still there.  This is the Dome of the Rock in Jerusalem, one of the holiest buildings in Islam. It borrowed its architectural dimensions from the Church of the Holy Sepulchre. It's a time in all dimensions. We're taken from the Church of the Holy Sepulchre and then it moved on. And again, this is an example of how cultures always go back and forth. Islam borrows from other cultures. Cultures borrow from Islam. It's a two-way street. And I will end my talk this way. Nizar Cavani, a great Syrian diplomat and writer, saying that any culture should be open to all other cultures. It should affect them and be affected by them continuously. Cultures should strive to bring the world together in the Unified Street. This to me is my mantra. This is one of my mantras. This is what I live for. Because I don't believe, when you believe, when you live in a world of absolute truth, you're saying, Islam is this and then we're not. Judaism is this and that's that. When in fact, you know, we're all mixing together architecturally, linguistically, musically, food-wise, you name it.  But in a sense, at this point, I feel as much connected to Islam as it do in Judaism. And I end with other cultures. I'm not just limited to one. I'm looking to other cultures for inspiration, for stories, for other things, just as the way other cultures have done forever. And will do forever. So that's my talk. Thank you. We'll open it up for questions for Jonathan. Right now, the individual will also ask Aliyah Tai and Imam Karakumar and I'll take the broader questions. Please. What's the name of your book? It's called All America Travels to America in Arabian and Latin Roots. And what's your last name? Coriel. Yeah. I'm sure you know about it. I don't know if you include it in your book, but many of the names we use for stars in the West are primarily given by the Arabic astronomers. No, absolutely. And that is, you know, that's... In the academic world of science, of course that's yet is acknowledged more readily than it is outside of that world. In science, yeah, there's a huge indebtedness there. And it's not going away. We'll never go away. Whereas, you know, in a sense, the culture I write about is more popular culture. Even though that does get, you know, that does kind of see a little bit in the popular culture. But yes. Yes, please. Yeah. Thank you for your research. It's very well documented, very well researched. There's one aspect that I feel missing and that is the impact of the Holy Qur'an on the constitution of the United States. Did you get a chance to do any research on that? You know, as if he's asking about the influence of the Holy Qur'an on the constitution. Yeah. Not as much as I wanted to. I think a lot of people know that Thomas Jefferson studied the Qur'an. And actually, a party of my research studied Arabic and learned to write a little bit of Arabic. But no, that was not a huge connection for me to research on the Qur'an. Even though I think it's an important connection to elaborate on. Absolutely elaborate on.


It's a pleasure to be back here. So as Emmanuel Imou said, I was here for twice two months, and I enjoyed a very productive stay here at EHUES this year and last year. Now, I have to confess right from the beginning that what I'm going to talk about, in fact, has nothing to do with, I guess, signal transmission and information technology and no Shannon. So I'm a little bit, sorry? It's OK. Thank you. So in that sense, but I couldn't. I mean, I never worked on these subjects. I mean, there will be a randomness coming up, but in quite a different way than in the previous talk. So this being said, so I mean, now it's after lunch. You can relax. It's not important for this program. So I want to start by telling you what kind of, I mean, starting with a historical example, what one understands by a random medium and what one understands by effective behavior and briefly mention what more recent applications might be in that area. And then I want to address two specific issues we've worked on. And so one of them, in a certain sense, is motivated by numerical analysis. So really understanding the numerical errors that very popular ubiquitous engineering method makes. And the second topic has to do, I mean, now if I would be bold, I would say it has to do with uncertainty quantification. So understanding the error one is making or understanding the fluctuations, the amount of uncertainty characterizing the amount of uncertainty in solution with random data. OK, so here is, by the way, this is probably a pointer also at the same time. So this is, as I said, kind of a historical example. Probably that's the first time in the physics literature a problem in effective behavior of random media was posed and solved in the famous work of Maxwell on electromagnetism. And he was interested in kind of effective conductivity or effective resistivity. And so in the back of his mind, he had the following situation. He had kind of a slab of a conductor, a homogeneous conductor. So homogeneous conducting medium. And he was imposing a voltage difference between the bottom and the top surface. And as we know, since the medium is homogeneous, this voltage difference leads to a homogeneous current field, which here is indicated by the green errors. And there is a proportionality between the strength of this homogeneous current field and the voltage drop divided by the height of the slab. And that's a material constant. And that's the conductivity of the material. Or one over that constant is the resistivity. So that's kind of, in a certain sense, the nice homogeneous situation. And then he was imagining a slab not formed of a homogeneous material, but formed of a heterogeneous material. So a material where you have, which is made out of two bare materials, where, let's say, you have a background matrix, which is made of this material with, let's say, spherical inclusions of another material with another bare conductivity k1. So you have kind of a mixture of a material with conductivity k2 and with conductivity k1. And clearly, since this material is no longer homogeneous, even if the voltage is kept constant down here and kept at a different constant up here, the current which you get is no longer constant. The current density is not constant. In fact, will be some complicated vector field. But nevertheless, you imagine that, or Maxwell imagined, or Maxwell knew that if this, the height of the slab was large compared to the typical distance of these inclusions. So if there was a clear scale separation between the characteristic length scale of the heterogeneity and the length scale of the sample, then this heterogeneous behavior, in fact, behaves like an idealized homogeneous behavior in the sense that there is a clear proportionality between the voltage drop and the average current which flows through the sample. And then back then, of course, you couldn't resort to a computer to determine that effective behavior. So he made the assumption of a dilute regime. So he made the assumption that the typical radius of these droplets is much smaller than their distance. And in this dilute regime, he was able to actually derive, theoretically derive an asymptotic formula for the effective conductivity of this mixture in terms of the bare conductivities k1 and k2 and in terms of the volume fraction p. So that's the first example of such kind of, in the physics literature, of finding the effective, even identifying an asymptotically correct formula for the effective behavior of a random medium. So nowadays, people are interested, or people don't want to kind of restrict themselves to situations, to these dilute situations, where you can, in a certain sense, you can understand the behavior, at least asymptotically, by paper and pencil. But they want to treat, let's say, realistic situations like the permeability of a porous rock or the elasticity properties of a certain mixture here, it's carbon and polymer. And then, since there is no diluteness here, they have to really resort to numerical simulations to extract from kind of the specifications of the statistics to extract the effective behavior. And the engineering method, which is used is the method of the representative volume element. And that was the starting point for Antoine Gloria and myself. We wanted to understand the error one makes in that method. So it was, in fact, that was a question which I heard from Wayne and Eu, that there was still some work to do to understand that from a more mathematical or rigorous point of view, what are the errors made in this very common method. So what I want to do in this talk is I want to first tell you this story. So kind of the error analysis of this represent. First, I'm going to explain to you what this representative volume method is. And then I'm going to tell you why it would be interesting. It is interesting to have an error analysis and what the error analysis is. And then I want to go to, I mean, if I have the time, still address a second issue which has really to do with the characterizing the fluctuations of the solution, characterizing the fluctuations, in this case, of the current field, and so characterizing the variances. OK, so let's start with this representative volume element method. So from a mathematical point of view, this heterogeneity of the medium is described by a coefficient field or a tensor field. So something which expresses the possibly anisotropic, microscopic conductivity of the medium or whatever material property you're interested in. And typically in these applications, that's a uniformly elliptic, as we say in PDE theory, a uniformly elliptic coefficient field or in differential geometry, you would say a metric. And then once you have this metric, you can form a differential operator. We call it an elliptic differential operator which contains the physics. I mean, for instance, if you stick to the field of conductivity, this operator acting on a function would be equal to 0 if u is a potential that leads to a stationary current. And so that's from a mathematical point, that's the abstraction. That's what we're dealing with. And now in random media, you have in a certain sense incomplete information on how this coefficient field looks like. So in a certain sense, you just have statistical information, which anyway you think is much more appropriate than the detailed local information. So mathematically, that means you're not working with a single coefficient field, but you're working with a probability distribution on the coefficient field. So the coefficient field is a random field. So something which depends both on the position in space and on your realization. And throughout the talk, it's perfectly fine to keep the following simple example in mind and the numerical simulations which are going to show to you are based on this example. So we call it kind of Poisson type example. So think of your space, our d-dimensional space. Think of d equal 2 or 3. I mean, the numerical simulation, I'm going to show to you it's a two-dimensional, but theory doesn't care for the dimension. So the red points here are distributed according to the Poisson point process in all of space. And then around every of these points, you draw a ball, you consider the ball of radius, let's say, 1 quarter. If the density of the Poisson point process is 1, which means roughly speaking, the distance between the points is on average equal to 1. So then you look at the union of all these balls. That's the blue set. That's a random set. And then as a very simple example, you take this coefficient field to be equal to 1 times the identity on the blue region and lambda times the identity where lambda, I think, is 1 over 10 in the experiment. So some number different from 1 on the complement. So that's a very simple example of such a random coefficient field. Before I go to this slide, so here you have a simple, very easily, I mean, a random medium which is described in very simple terms. I mean, in a certain sense, you have three parameters. You have the number density of the Poisson point process, which I arbitrarily set equal to 1. You have the radius of the balls, which I choose to be 1 quarter. You have the contrast of the medium, which, let's say, is 1 tenth, so a factor of 10. And that's it. So you have three parameters. And now you want to extract this effective behavior, the effect on large scales, the effective resistivity or conductivity of this medium. And so in a certain sense, you want to go from three parameters to another finite number of parameters. So you start with something very low, the very low-dimensional description, and you want to get extracted very low-dimensional information. But the map is not explicit, so there is no simple formula which would relate these three numbers to the d-square numbers you're interested in. So therefore, what's done in engineering is this representative volume element method. So in order to get an idea of what that is, it's really good to think in terms of this physical application. So the functions, you should think of the functions as electric potentials. The negative gradient then by Maxwell is the electric field. The coefficient field has the meaning of conductivity. So if you multiply the electric field by the conductivity, you get the electric current. And the current is stationary if its divergence vanishes. So what's the purpose of the effective behavior, the effective conductivity of the medium? It should provide a linear relationship between the average potential gradient to the average current. So the microscopic conductivity is the proportionality between the local field and the local current. And the effective behavior should give you the relationship between the spatially averaged field and the spatially averaged current. That's what this is supposed to be. So how do engineers this in practice? They artificially introduce some finite size domain. So the simplest thing is to think of a torus. So they introduce some box and identify boundaries. And then in this finite size region, they sample their medium according to the same specifications as in the whole space. So they look at the Poisson point process with the same density on the torus and exactly the same things. So now you have kind of sampled what you think is kind of your right statistics, but not in all of space, but in the torus. And then they solve D, where D is the dimension. So in two dimensions, just two simple linear partial differential equations on the torus. So they seek a kind of a periodic function phi i, which solves this elliptic differential equation. So that's I can write this on the blackboard because that's important. So you take the gradient of x i plus phi i. And this is supposed to be periodic. And you want this to be equal to 0. And of course, I can write this here as e i plus gradient phi i, where e i is kind of the i-th unit vector. So this function is called the corrector because it does the following thing. You may also call it harmonic coordinates. You start from the f-line function in the i-th coordinate direction, and you want to correct it in such a way so you add to it the function phi i that the resulting function is harmonic. So this is why sometimes these coordinates are called harmonic coordinates or why this is called a corrector. So that's the problem which you solve. And then you take the current, which belongs to these harmonic coordinates, so this expression, so the field multiplied by the conductivity. And you average it over the torus. And that is what you take as approximation to your homogeneous coefficient. So let me also write it here. So a hom e i. So that's what's done in, that's the representative volume element method. And here is an numerical simulation to just give you a flavor of what you're doing. So you're picking a coordinate direction. Here we picked the first coordinate direction in our two-dimensional model. So we're looking for potential, which is a periodic perturbation of the potential that linearly grows in x1 direction so that its current is divergence free. So here you see the potential lines. Clearly, they're kind of dense in the region where you have a low conductivity and they're sparser in the region where you have a high conductivity because there it's more like a conductor where the field should be expelled. And those are the flow lines. That's the current which belongs to this. And now you take this vector field, you average it over the torus, you get a vector, you get two numbers, and that's the first row of the effective behavior. And now you do the same thing in the second coordinate direction. So in this case, kind of the level lines will essentially be horizontal. The gradient will be mostly pointing in the e2 direction. You get the same thing. And you take the average of this field and you take this as the second row of your matrix. So that's exactly what you're doing here. And so, by the way, it's not surprising that this number here is exactly equal to this number, kind of the symmetry of the resulting matrix is, in a certain sense, built in. And now the question is, how much should you trust these numbers? And intuitively, it's clear that you shouldn't trust these numbers if this representative volume element is too small. Then it certainly has nothing to do with the effect which you would capture. So these numbers should become better and better. The larger this representative volume element becomes. And that's kind of the error analysis we were interested in. So before turning to this, one should kind of also point out that the answer which you get here is a random answer because it depends on the realization. So it's not yet a deterministic quantity, but it's a random answer. It's very clear that if you sample your medium once more, you will get a different solution, and therefore you will get a different average. So this is still fluctuating a random number. But on the other hand, it's also intuitive that as you make the size of this box larger, and here I didn't draw it as a larger box, but I rescaled it back to unit length and made the ball smaller. That's mathematically equivalent. The variance should go down. So that's what we call the random error. And that's indeed what one sees in the numeric simulations. So here you have three different realizations of the random medium. Of course, you get three completely different solutions, three completely different current fields, and three quite different matrices. They differ, let's say, by 10%. So that's what I said. I mean, the answer which you get is what we would call a random variable because it depends on the realization. But if you were, so here again is what we saw before. But if you do the same simulation, which a much larger representative volume element, then you see that the fluctuations clearly go down from about 10% to about 1%. So the question is, yeah, go ahead. Do the results rely on the assumption that there is speculation of the Boolean model? They don't rely because that's not necessary here because we didn't put the conductivity equal to 0 in the complement. If we would allow this, remember that this coefficient field was defined to be equal to the identity and equal to lambda here. And I think lambda was chosen to be 1 over 10, but positive. If this lambda were allowed to be 0, then you would get in the situation of percolation. And that's, of course, also a well-studied problem. But it's not because that, of course, has additional difficulties or additional challenges. I'm not addressing that in this talk. So here, so therefore percolation, whether it's percolating or not, will play a qualitative, I mean, some quantitative role, but not a dramatic qualitative role. Does that answer the question? OK, so that's only one type of error. There is a second error, which is slightly more subtle, namely that even if you disregard the fact that this is a fluctuating quantity, a random quantity, by, for instance, taking its expectation, it still would not be the right value because of the following phenomenon. When you go to the torus, you've kind of falsified the statistics. If you unwrap your torus into the plane, I mean, you have kind of a periodic coefficient field, which means you've introduced kind of spurious long range correlations. Instead of having a Poisson process, you're actually looking at the wrong statistics by doing this. So that's a systematic error, kind of a bias. And also that effect one's seeing. So if we're just looking at the expectation of this quantity, of course, numerically, we're looking at the empirical average. But let's say we, when Alkriman did many, many sampling so that we actually access the expected value, we know that because of symmetry considerations that it's isotropic, so it's just about this number. And then even this number, so even after taking the expectation, still depends on L and only converges to its final value as the size becomes large. So there are two types of errors involved in this problem. I mean, there is kind of random error, which comes from the variance of the answer. And there is a systematic error, which has to do with using the wrong ensemble. And of course, so qualitative theory, which is 30 years old, tells us that both types of errors go to zero. And in fact, we have a nice Pythagoras rule in probability space. The expected square error is the sum of the square of the random error plus the square of the systematic error. And both of them go to zero. And now we were interested in at what rate do these two errors go to zero. So why is that kind of a question of practical importance? Because it depends, it would determine what type of numerical algorithm you're using when you want to infer the effective behavior. Because when an engineer does that, he has two knobs to turn. He can either look at a few very large samples, or he can look at many moderately sized samples. And it's clear that looking at many samples will only affect the random error. It will kind of attenuate the damaging effect of the random error. It will have no effect on the systematic error. So therefore, in order to find the optimal, I mean, if you have a certain number of unknowns to spend, because you don't want to wait three years, but just three weeks to get your result, you have an optimization here. If you want to reach a certain error, you can either do it this way or that way. And which way is the better depends on the scaling of these two errors. So therefore, this is a relevant, from a numerical point of view, a relevant question. And of course, since it's a relevant question, people have kind of looked at this for a while. But it's only since fairly recent years that kind of one got a satisfying and optimal answer. Just let me mention, because it came up in the previous talk, the mathematical tool by which we get many of these results, in fact, is the logarithmic Sobolev inequality. So we're using kind of an infinite, we're using concentration of measure logarithmic Sobolev inequality to get some of these results. But I will not mention, I won't talk much, or will not talk at all about the methods here. So the final result we're getting is the following. The systematic error is much, much smaller than the random error. The systematic error, I point out the difference here is not a square. So the systematic error is about the square of the random error. So it definitely pays to look at many, many realizations because you have to first make sure that you make this effect, the bad effect of the random error small by looking at many realizations, and only then you have to worry about the systematic error. And since we wrote this, kind of the things have kind of progressed. But by now one even understands that the fluctuations are approximately Gaussian. So even the error not only has the scaling, but it also has its own structure. And that's a little bit what I want to talk about. And the second part of my talk, how much time do I still have? About 20 minutes or so. OK, so that's much more than I will need. So the first part was about these kind of error estimates for this engineering method. And the second part is about the fluctuations of the solutions. So that's now changing a little bit perspective. So why does homogenization pay? Why does kind of this theoretical concept pay? Well, it's related to a separation of scales. And that is something which I already mentioned in the context of Maxwell's example. His separation of scales was the separation between the typical size of the domain, which was the width of the slab, and the typical size of the medium, which was the distance between the inclusions. So we have a macroscopic scale, which is either set by the size of the domain or by the characteristic size of the right-hand side of the equation, and kind of the microscopic scale, which is set by the medium itself, so the diameter of the balls, the typical distance between the balls. And let's set this one equal to 1. So we non-dimensionalize in this way. So we have this scale separation between a large geometrical external scale and this much smaller intrinsic scale. And now let's think of the simple elliptic equation with the right-hand side, f. So we put this length scale into the right-hand side, which means we're looking at the right-hand side, which varies on a very slow, on a very large scale, which we call l, and whose amplitude we scale like this so that the amplitude of the solution is of order 1. So homogenization tells you that instead of solving this very complex problem, where you would have to, at the same time, resolve the small scale of the coefficient field and the large scale of the domain of the right-hand side, you can solve a much simpler equation with the effective behavior with the homogenized coefficient there. So qualitative homogenization tells you this fluctuating solution, this random solution, here in blue, is, in fact, pretty close to this non-random, non-oscillatory solution. And of course, also there, you want to understand what is the error. And in fact, in periodic homogenization, that's very well understood. Kind of the typical fluctuations happen on scale 1. And the typical deviation from the limit is of the order of 1 over l. And in fact, the same is true in the random case. And so when you really want to quantitatively compare the solution of your heterogeneous equation to the solution of the homogeneous equation with the effective behavior, you go back to this object, to the corrector, because that helps you to understand the error. So here, again, is kind of this schematic picture, which I drew here. So these functions phi i correct the affine behavior in such a way to get a solution, to get a harmonic coordinates, to get the solution of this equation. And now, therefore, instead of comparing u directly to u bar, you should compare u to a modulated version of this corrector. So that's what I tried to explain by this picture. So down here, you have the true solution, which kind of oscillates around the homogenized solution. And in fact, you get a much better approximation of the true solution by taking the tangent to the homogenized solution and by doing this construction on top of the tangent. Then you get something where also the gradients are close, not just the solution. And in fact, the relative error is like in periodic homogenization. So here, you wouldn't see the effect of randomness in a sense. That's a result which tells you random homogenization is not worse than periodic homogenization. But in fact, there is much more structure in the random case. And that has to do with the fluctuations. And that's the story which I want to tell last. So let's suppose you were interested. So here, again, is our solution of the heterogeneous problem, which we don't really want to compute. So we seek kind of theoretical understanding of it with the right-hand side, which has the right scaling. And let's suppose we're not interested in the solution in an extremely point-wise way. We're not interested in understanding the solution in every point. But we're just interested in understanding certain macroscopic observables, like in oil recovery. What's the mean flow rate? So macroscopic observables would look like this. You would take some spatial average of your solution u. And the spatial average is on the same scale as the solution on this macroscopic scale l. And now the question is, can we get a better understanding of the fluctuations of this quantity? So that's now getting a bit in this business of uncertainty quantification. Can we not just understand the size of these fluctuations, but can we really characterize these fluctuations? So a while ago, we found out that they have the order which you would expect from central limit theorem scaling. So 1 over ld. So you see that this is very different than the previous error estimate, where in a point-wise sense, the difference is 1 over l. And here the average error is more like 1 over l to the power d over 2. And that, of course, means that the next natural step was, well, can we understand the limit of this quantity if we put it onto the right scale? Can we characterize these fluctuations? And the first thing we, and now we mean in kind of the small community that's looking at these problems, was to plug in the two scale expansion, so the way that you get a pretty good approximation to your heterogeneous random fluctuating solution by doing this construction of taking the homogeneous solution and modulating it with the corrector, that was the first guess. And then you would be drawn to understanding the covariance of the corrector. So therefore, what we first looked at was kind of getting a better understanding of this object here, the covariance structure of the corrector. And together with Jean-Christophe Mourin, who is a probabilist in Lyon, we were able to kind of show that there is a limiting covariance structure on large scales, and that it's, in a certain sense, even if we could characterize it in some way, it has the homogeneity of the Green's function. So here is the covariance function in blue, and that would be the homogeneity of the Green's function. So it's homogeneous of order minus d over 2. So I mean, he was a probabilist. And of course, if you see something where an object, a random field, where the covariance structure is the one of the Green's function, you think of the Gaussian free field. But in fact, it's not the Gaussian free field. It's not equal to the covariance structure. It's not equal to some Green's function. So that was the first interesting finding. It's more subtle. And then Jean-Christophe Mourin with Gu from Stanford even found out that what motivated us in the first place to look at this quantity, namely the fact that we could understand the variance of any solution by looking at the variance of the corrector, is not as simple as that. Because both of these limiting kind of variances exist, but they're not equal. So it was not the right idea to use the two-scale expansion in the simple way in order to reduce the variance of any solution to the variance of the corrector. So that was a little bit of a puzzle. We don't quite understand the covariance structure. And then very recently with Miltia Durinx, who is a PhD student of Antoine Gloria in Brussels, kind of we found what I think is the right way to understand these fluctuations, at least to leading order, and to characterize them. And what we introduced is something which we call the homogenization commutator, an object which is built on the objects which any way you have to compute on the correctors. And it looks at the difference between the current and the field to which you apply the effective tensor. So it's a very simple object. And we called it commutator because in a certain sense it's the difference between immediately applying with the microscopic conductivity or doing the homogenization and then applying with the effective conductivity. That's a random matrix field. And the first observation is a completely deterministic one, namely that the fluctuations of our observable can be in a path-wise, in a point-wise way, described by this kind of played back to this quantity, to this capital XI, to this homogenization commutator, by solving in a joint problem. In a joint problem, remember that G is kind of the observable we are interested in. We're solving in a joint problem on the homogenized level that defines the V-bar. And if we have the U-bar, which is the homogenized solution in the V-bar, we can to leading order with a kind of precise relative error characterize the fluctuations.  So if you look at the correlation function of this object, of this tensor valued object, and you put it on the right scale, then essentially you just see a peak at the origin and then a flat 0 square.  So it's characterized by a single 4 tensor. It's a 4 tensor because this object was a 2 tensor. So its covariance is a 4 tensor. And if you put these two results together, you get a complete asymptotic characterization of the variance in terms of this new object Q, which describes the covariance of that thing. So that's the, I think, nice and also pleasing characterization of the limiting variance. And the nice thing, from a practical point of view, the message, in a certain sense in America, I would say, the take home message is that if you're interested not just in the solution, but you're interested in the fluctuations in the variances, because you're interested in uncertainty quantification, you don't have to do any new work. Because anyway, to get the effective homogenized behavior, you had to resort to your representative volume element method. You had to solve for the harmonic coordinates. You had to find the harmonic coordinates. You had to solve these problems which I talked in the first part of the talk, because they give you the effective behavior, or at least an approximate effective behavior. But if you have these objects, then you might as well look at the capital xi at the homogenization commutator, or at least on the approximate level. And you may look at this Green-Cubo type formula, which is a proxy for their covariance structure. So without any additional numerical effort, you have access to, if you want to get a home, you automatically, without any additional effort, you have access to this 4 tensor q. And with help of this 4 tensor q, not only can you characterize the leading order behavior of your fluctuating solution, you can also characterize the leading order behavior of the variance of observance. So therefore, I would hope that this is an insight which can be used in practice. So that brings me to the summary. So what we've been working on quite extensively in the past years is making this type of homogenization of random media more quantitative. And here, I've given you two examples. I mean, understanding the error in the representative volume element method and an example of, if you want, uncertainty quantification to leading order, which turns out to not be computationally more expensive than what you have to do anyway to get the effective behavior. So that's it. Thank you. Is there any questions or comments? So first, a question. So you have discussed fluctuations at the level of the central limit theorem, essentially. Can one use the same techniques and work out the large deviation scale to see what happens to? That's a good question. I think, to some extent, that can be done. And I think that has also, in parts, has already been tackled by the probability community.  But as you say, kind of looking at the large deviation scale is another type of question. So my feeling is that this could be done and that this is, in parts, done already in the probability community. But that this requires different types of techniques. And a second question is, since you alluded to in the first slide, this limiting rare perturbation, so very few balls, can one also use this Miami expansion to get, I mean, first order, second order, and perhaps an expansion with a sort of chaos expansion on the Boolean model? So now you mean an expansion in the volume fraction. Yes, in the intensity, which is, yeah. And so I think that has been done. I don't know. I don't think it has. So two of my co-authors, so Missia Durantz and Antoine Gloria have, in a similar model, looked exactly kind of they've proven kind of, I mean, they've shown that this is an analytic function in the volume fraction and kind of developed, if you want some kind of multiple way of getting to all the series. So that can be done not with the Mali-Avain, not with these tools. I mean, in most of our approves, we use, if you want, something like Mali-Avain calculus because we're taking the derivative with respect to the noise, which, in our case, is taking the derivative with respect to the coefficients. So we try to understand how solutions, and that's at first a completely deterministic question, how sensitively a solution at this place depends on changing the coefficient at this place, which is described by the non-constant coefficient greens function. And that's really computing the Mali-Avain derivative, and that then we put into kind of concentration of measure the logarithmic Sobolev inequality machinery to get most of the, many of the estimates.


I'm Andy Warmer. I run Atlantic Edge oysters in Angle Bay in Pembrokeshire. We grow top quality oysters and we've been operating for the last two years. We've got 150,000 oysters on the farm at the moment. That ranges from 70,000 small seed through to part grown oysters up to the full size market size oysters. Growing oysters relies on that pristine marine environment. The oysters we grow here are so clean you can eat them straight off the beach. And we've got our part to play in maintaining that environment. So everything we do, we think about how we're operating, what we're doing. We try and adopt the best practice to make sure we're as sustainable as possible. That goes from where we purchase our small seed oysters, how we grow them, even to our packaging that we take to our chefs. Our customers are now really valuing local food, asking where their food comes from and how sustainable it is. We've got a really good relationship with our chefs. They come down and visit the site, tell us what they want for different plates. We can tell them when we've got new products coming through. If you eat an oyster straight from the sea, it's straight away big hit of salt. When we put them through the purification tank, we reduce the salinity. You can taste more of the oyster rather than the salt. And that came from feedback working with one of the chefs in the area. They help us, we help them. That relationship is vital. We work within a triple SI, so we need to be small scale. But by producing high quality, we can make a viable business. We're supplying country pubs, seafood bars, even top restaurants and hotels. We're pretty much on schedule to double production in the next two years. That'll give us enough to get into the big cities. We get an interest from people overseas, from Singapore, Hong Kong. I just like the idea of having our oysters being enjoyed in the other side of the world.


From VOA Learning English, this is the economics report. Ghana arrested more than 100 Chinese nationals in June. They were accused of illegally mining gold. In February, Zambia seized a Chinese operated mine because of safety concerns. Zambia has decided to seize property from three foreign oil companies, including one from China. Some observers say these and other incidents appear to be a reaction by African governments against Chinese investment on the continent. China became Africa's top trade partner in 2009, yet some Africans believe they are not getting enough for their exports. Wright Simons is with the Imani Group. He says China is no longer known only for giving loans and sending workers to build hospitals and roads. He says China is now a business partner. Ghana has taken steps against foreigners involved in shallow mining for gold. Only Ghanaians can get government approval for this kind of mining. Wright Simons says some Ghanaians rent the permits to foreigners who have the money and training to actually mine the gold. Observers say nations like Niger and Gabon want to get more for their natural resources. They want more money, more local jobs, and better living conditions. The wide demand for oil, coal, iron ore, and uranium has made their negotiating position stronger. Some nations are also concerned about abusive or illegal operations. In Gabon, the government says it is planning not to extend operating permits to three companies, including ADACS, which is part of China's SinoPEC. Many experts believe the partnership between China and Africa will continue to grow, but now perhaps on more equal terms. For VOA Learning English, I'm Laurel Bowman.


Let's finish our very short introduction to modeling data in our with a brief discussion of regression, probably one of the most common and powerful methods for analyzing data. I like to think of it as the analytical version of e pluribus unum, that is out of many one, or in the data science sense, out of many variables, one variable, or you want to put it one more way, out of many scores, one score. The idea with regression is that you use many different variables simultaneously to predict scores on one particular outcome variable. And there's so much going on here, they'd like to think that there's something for everyone. There are many versions and many adaptations of regression that really make it flexible and powerful for almost no matter what you're trying to do. We'll take a look at some of these in R. So let's try it in R and just open up this script. And let's see how you can adapt regression to a number of different tasks and use different versions of it. When we come here to our script, we're going to scroll down here a little bit and install some packages, we're going to be using several packages in this one. I'll load those ones as well as the data sets package. Because we're going to use a data set from that called us judge ratings, let's get some information on it. It is lawyers ratings of state judges in the US Superior Court. And let's take a look at the first few cases with head. I'll zoom in on that. And what we have here are six judges listed by name. And we have scores on a number of different variables like diligence and demeanor, and whether it finishes with whether they're worthy of retention, that's the RTEN retention. Let's scroll back out. And what we might want to do is use all these different judgments to predict whether lawyers think that these judges should be retained on the bench. Now, we're going to use a couple of shortcuts that can actually make working with regression situations kind of nice. First, we're going to take our data set, we're going to feed it into an object called data. So that shows up now in our environment on the top right. And then we're going to define variable groups, you don't have to do this, but it makes the code really, really easy to use. Plus, you find if you do this, then you can actually just use the same code without having to redo it every time you do an analysis. So what we're going to do is we're going to create an object called x. It's actually going to be a matrix and it can consist of all of our predictor variables simultaneously. And the way I'm going to do this is I'm going to use as matrix. And then I'm going to say read data, which is what we defined right here, and read all of the columns except number 12. That's the one called retention. That's our outcome. So the minus means don't include that, but do all the others. So I do that. And now I have an object called x. And then the second one, I say, go to data. And then this, you know, blank means use all of the rows, but only read the 12th column, that's the one that has retention, our outcome. So following standard methods, x, those are all our variables and why that's our single outcome variable. Now, the easiest version of regression is called simultaneous entry, you use all of the x variables at once, throw them in one big equation to try to predict your single outcome. And in our we use LM, which is for linear model. And what we have here is why that's our outcome variable. And then the tilde means is predicted by or as a function of x. And then x is all of our variables together being used as predictors. So this is the simplest possible version, and we'll save it into an object called reg for regression one. And now, if you want to be a little more explicit, you can give the individual variables, you can say that our 10 retention is a function of or is predicted by all of these other variables. And then I say that they come from the data set us judge ratings that way I don't have to do the data and then dollar sign before each of these. That'll give me the exact same thing. So I don't need to do that one explicitly. If you want to see the results, we just call on the object that we created from the linear model. And I'm going to zoom in on that. And what we have are the coefficients. This is the intercepts start with minus two and then for each step up on this one, at 0.1, 0.36, so on and so forth. You'll see by the way that it's changed the name of each of the variables to add the x because they're in the data set x now that's fine. We can do inferential tests on these individual coefficients by asking for a summary. We click on that. And we'll zoom in. And now you can see there's the value that we had previously but now there's a standard error. And then this is the t test. And then over here is the probability value. And the asterisks indicate values that are below the standard probability cutoff of 0.05. Now we expect the intercept to be below that. We see for instance, this one integrity has a lot to do with people's judgments of whether a person should be retained. And this one physical really, you know, are they sick? And we have some others that are kind of on their way. And this is a nice one overall.  And what it means is that these variables collectively predict very, very well, whether the lawyers felt that the judge should be retained. Let's go back now to our script. You can get some more summary data here if you want. We can get the analysis of variance table, the ANOVA table. And if we click on that, zoom in. There you can see that we have our residuals and the Y. Come back out. We do the coefficients. Here are the regression coefficients. We saw those previously. This is just a different way of getting at the same information. We can get confidence intervals. We'll zoom in on that. And now we have a 95% confidence interval. So the 2.5% on the low end, the 97.5 on the top end in terms of what each of the coefficients would be. We can get the residuals on a case by case basis. Let's do this one. And when we zoom in on that, now this is a little hard to read in and of itself, because they're just numbers. But an easier way to deal with that is to get a histogram of the residuals from the model. So to do that, we just run this command. And then I'll zoom in on this. And you can see that it's a little bit skewed, mostly around zero. We've got one person way up on the high end, but mostly these are pretty good predictions. We'll come back out. Now I want to show you something a little more complicated. We're going to do different kinds of regression. I'm going to use two additional libraries for this one is called Lars that stands for least angle regression and carrot, which stands for classification and regression training. We'll do that by loading those two. And then we're going to do a conventional stepwise regression, which a lot of people say there's problems with this, but I'm just going to show that I'm going to do it really fast. There's our stepwise regression. Then we're going to do something from Lars called stage wise, similar to stepwise, but it has better generalizability. We run that through, we can also do least angle regression. And then really one of my favorites is the lasso that's the least absolute shrinkage and selection operator. Now I'm running through just the absolute bare minimum versions of these, there's a lot more that we would want to do explore these. But what I'm going to do is compare the predictive ability of each of them. And I'm going to feed into an object called R2 comp for a comparison of the R squared values. And here I specify where it is in each of them, I have to give a little index number, then we're going to round off the values. And I'm going to give them the names say the first one stepwise and forward than Lars and lasso. And we can see the values. And what this shows us here at the bottom is that all of them were able to predict it super well. But we knew that because when we did just the standards simultaneous entry, there was amazingly high predictive ability within this data set. But you will find situations in which each of these can vary a little bit, maybe sometimes they vary a lot. But the point here is there are many different ways of doing regression and our mix those available to whatever you want to do. So explore your possibilities and see what seems to fit. In other courses, we will talk much more about what each of these mean, how they can be applied and how it can be interpreted. But for right now, I simply want you to know that these exist and they can be done at least in theory in a very simple way in our


Only the HMIS team or the M&E team, the program managers, who all is involved? If you take the purpose of this report, who needs it and why do they need it? And out of this report, were any actions taken or are any actions going to be taken that result in demonstrable changes? So the bottom line is does this report change anything? Or does it just become a nice report in a glossy folder stored in a glass cupboard? The second aspect of this data quality review is to look at the data and to fix it. And it's about checking for outliers for obvious data errors and then correcting them, checking for missing months of data and filling them in. Specifically, if you know a facility has worked, but no pieces of paper have arrived. For the WHODQ tool, there's something that says annual report. You click that open and the first option you have is data and you can select the group. It could be call, it could be immunization, it could be maternal health. For the period, you then select the year that you want to run the annual report for and how many years before that do you want as reference so that there is some background and then you select the organization unit. This could be the whole country, sub-level, sub-national or sub-national times two. The choice again is always yours. You then get a report that is completed and it will show up in the WHODQ tool. You have the option of printing the report and you will be able to save it as a PDF. So this report, the one I ran was nine pages, looks at completeness, timeliness of facility reporting, completeness of indicator data and a whole of guidance and indicators showing you what's happening with completeness with timeliness of the data. So this is what this report looks like. So the questions are, so here is this report. Is this report helpful? If it's helpful, for who is it helpful? What guidance can you gain from this report? Is this report a guarantee that the data is correct, complete, consistent and timely? So all sorts of things that are looked at with part of the development of an annual report on data quality using the DQ tool. The second section of this presentation looks at the data quality review and data cleaning The use case we are going to talk about is South Sudan. Fortunately or unfortunately it was done using DHIS 1.4, not DHIS 2. We are still looking for use cases with DHIS 2. And it was done for the years 2012, 2013, 2014 data. And here is a map of South Sudan and a flag if you don't recollect it. So the first question is, why do we bother to clean the data? And the problem is that poor quality data makes the data unusable. You can't use it for accurately estimating indicators, coverage of performance, coverage of service delivery and that means that all that effort that you went into to create and to collect and to capture and to clean the data from all the facilities, there is no return. So the data is collected but it's not usable. We also know that if there are a few very obvious data entry errors, people can take this as a sign that all the data is untrustworthy and that we know is not true. Experience has shown that on the whole generally HMIS data is generally reliable. It's not perfect, it will never be perfect, but it should be good enough so that it can be used. If we go about correcting any of the obvious errors that we see, the data then becomes usable and acceptable. And usable data and acceptable data is much better than unusable data which has no value. It just stores up space in the computer and costs money. What we do know is that in spite of everybody saying well we do validation checks and we have quarterly data reviews, data errors do creep in and they frequently remain undetected and uncorrected and stay there to haunt you for years. And as we said South Sudan was using the AirTrace 1.4 at the time of the data cleaning process. So some background to what happened in South Sudan. First of all it was a minimum indicator based data set implemented in 2011. So it was based on indicators and from the indicators we defined the data elements, we put them into data collection tools and it was two pages with one page for EPI. And there was a decision made in 2013 to provide a report for 2012, the annual HMIS reports because we now had a year of data. However on reviewing this data it quickly became apparent that the data was not good enough to be used. We could not use it for the annual report. And so the decision was made that we would go back and we would clean the data so that it would become usable. And so that this was then done at state level, sub-national, and we also then combined some states for convenience just so that in terms of time and in terms of distance and travel. The precinct at these data cleanup workshops was the state M&E officer and the county M&E officer for each county. And they were responsible for creating the data. Because it was the county they knew what was going on in their area, they could easily contact the facility, people responsible and ask questions. They did all of that work about fixing the data. It took approximately 45 days per group of states and it was a very labor intensive process with everybody working early in the morning to late in the afternoon in the heat and dust of South Zealand to get this work done. So we first made some crucial decisions. So the first one was that we were going to concentrate on the head count, knowing how many people access the health services.  So head count. Then we looked at children with diarrhea, pneumonia and malaria because we also defined them as a vulnerable population. We then looked at maternal health in terms of anti-natal visits, deliveries, life births, IPTP and the things relating to maternal health because we also see that those were the crucial things to have correct. We ran validation rules. Absolute and statistical were run and any violations triggered were noted. The same process that we use in DHIS too. Extreme outliers were identified and corrected using tools in 1.4 which include interpolation and regression analysis. Interpolation is making an educated guess based on data before and data after. And regression analysis, we're not going into a mathematical discussion this afternoon on this. The good process was we would concentrate on replacing the missing months of data. In other words, we know that a facility worked, but for some or other reason the piece of paper never arrived. And there are many explanations and reasons for that. Insecurity, unable to travel, no stuff, whatever, but we know the facility worked. We replaced missing data. And we looked at validation rules with compulsory pairing. In other words, if you have one, you must have the other. An example would be diarrhea treated. You must have diarrhea reported. We fixed those up. We looked for extreme outliers and corrected those at facility level. Specifically, if a value materially altered the indicator. It became important enough to correct the data. And in this process we looked both at raw data and at indicators. Because frequently on raw data you may not see that something is wrong. But when you look at it as an indicator that that data element is part of an indicator it becomes apparent that something is not right. So some of the actions and some of the things that we did we developed a decision tree for data outliers. If it was out of range and then there was the services, we first of all looked, did the data elements hang together? Did they make sense when you looked at them? Then we did, using regression we recalculated a value that made no sense. If the data elements did not hang together, in other words they made no sense. We would recalculate the key data elements. The most important ones and provide a comment. Again using data from previous months and months after. And if we had data there was no services that were provided by the health facility. We were just deleted. So this is some of the ways we looked at trying to fix outlier data. Trying to fix missing data. We looked at was the facility actually providing any services? And if it wasn't, then we closed it. If we realised that the facility did not work for one month, or maybe even two months due to security, due to no staffing, whatever, but the services were not provided. We wrote a zero and we gave a comment. There was no longer missing data. Facility did not function. So it's not missing data. And if we said that services were provided by the health facility and the data was missing, we corrected it using interpolation and regression analysis to fill in the gaps. So why was it possible to do this in South Sudan? Well, there were a few reasons. So first of all, it was a very, very small data set. There was only two pages. The local staff, the M&E officers were eager to be responsible and to fix their own data. They wanted good clean data. And they took great pride in having data with no violations. And there was a lot of hands-on support and supervision to guide people through this process. Well, what were some of the results? First of all, we had cleaned data for three years, resulting in an annual report for each year. What countries are doing an annual report each year for your HMIS data? Because we had data for three years, we were able to measure service delivery over time. Were counties and states getting better? Were they staying the same? And one of the most crucial aspects was the development of the capacity and the knowledge of the M&E staff regarding both HMIS and DHHIS. And as we all know, when you start out in a country, one of the biggest issues is an updated facility list. So a lot of work went into fixing a facility list. So in conclusion, has all the missing data, where you know a facility work but there's no report been added in, have you fixed all your arctlias, have all your validation rules been fixed or explained? And then the question is also who takes responsibility for this? Is it important to do this in your situation? And so we are left with two exercises. The first one is about the WHO annual report. So you either use your own country, DHHIS to instance or the demo one. Run the annual report for a selected or unit and data element or indicator group. Print it, which will allow you to download in PDF format. Then review it and discuss it. How can you use this report? We know that there are quite a few of you working in the same country and I think that this would be a very nice exercise for you as a group to have a look. Then in terms of the data reviewing corrections, either your own DHHIS to the demo instance, using the WHO DQ to find the missing data and arctlias. What needs to happen to replace these missing values or correct the obvious data arctlias? And here is a difficult question. How will you decide what the new value should be? Thank you.