in Sao Paulo, Brazil. It's great to welcome all of you available all around the world. We're having a co-streaming in two continents. We are in the American continent with our friends from the US office and also Brazil office. And in Europe, we connect to the Brussels office. RLS is a political German institution in Germany that are political institutions that are connected to international cooperation. Considering the German party event, there's a rule that the parties have the right to nominate foundation to receive funds to provide political information all around the world. RLS is a foundation connected to the D-Link Party, which is left-wing party in Germany. Rosa Luxemburg Foundation was founded some 90 years ago. And here in Brazil, we have operations for over 16 years now that are over 24 regional offices all around the world. And we provide political background to people all around the world. Today, we are pleased to have the second event in the series of three to discuss mobility. The basis to the set of three events are linked to books that will be published or have been published. Today, we are discussing anti-racist mobility, which is a book that is soon to be published in partnership with the publishing house Autonomia Literária. Autonomia Literária has started to sell the book today. You can connect to their website and purchase the book. We, from Rosa Luxemburg Office Brazil, we are going to have issues of the book as well to use in our courses and also to share for free. It's going to take some time to have the books available because of the cruel pandemic we're facing. Our office is closed, but soon the book is going to be available and we're going to make the PDF available so that people can download the book. The book is highly important and it has been organized by Danielle Sancini in partnership with Paiki Duki Santarene, one of our speakers, as well as Rafael Orbergaria, one of the speakers too. What is the book about? Anti-racist mobility means going and coming is exerting democracy. Over the period of the pandemic, some people who had never faced problems on coming and going, going to shopping centers, restaurants, and even crossing borders, those people started feeling that the world started to shrink. Due to social distancing measures, coming and going is no longer an option as well as in the past.  Also, transport system in Brazil is racist because it's unequal and the bus trip forces people and keeps people away and people who are poor, they truly demand on public transport to exert their rights to come and go. Transport is based on the capitalist order. Most people can't exert their other needs. Also, we're talking about the forces that want to turn over those barriers in Brazil and worldwide. Today, we're discussing anti-racist mobility. This struggle is connected to a series of three events and we are discussing plenty of different aspects connected to mobility. We're in partnership with the Brussels office, the New York office, and I would like to welcome my colleagues from the New York office and also the Brussels office. Catarini Costa is going to mediate our event. She's the communications coordinator for the Brazil office. We are more than happy to get started. Kazembe, could you please talk about the relationship we have with the New York office and also the work you do and later on, Manuela from the Brussels office? Kazembe, your phone is mute. Thank you. Hello, Test 1, Test 2. Hello, good afternoon. Good afternoon. Good to talk, everyone. Good to have been. My name is Kazembe Balagun. I am a project manager here in the New York office of the Wilson-Burk-Stiftung. We're really honored to be presenting this program with our colleagues in Brussels and also in Sao Paulo. A deep thanks to Danielle and Catarina for the organization of this afternoon and for this very important conversation that we're having. The New York office of the Rosa Luxemburg-Stiftung works primarily to act as something of a bridge between progressive actors in Europe and their counterparts in North America. And part of our work is to fight against racism. In the United States, the fight for black freedom started with another Rosa, and that would be Rosa Parks. When she decided to stay to sit in to desegregate buses. And since that time, the question around transportation and race have been intertwined. Today, we have had an issue with the question on race and transportation is either more severe with the advent of COVID-19. We're really happy to have this series going on.  And I will point to folks to check that out on our YouTube page, Women and Mobility series. And also our events that will be coming up on April 22. And I believe I'll pass it at that point. I'll pass it on to Manuela. And thank you so much for the wonderful invitation. And I'm looking forward to a great afternoon. Yeah, thank you very much, Kazembe. Good evening, everybody. Hello from Brussels. Good afternoon. I'm really honored to speak here very briefly on behalf of the colleagues in the Brussels office. We are really honored to organize this series with you. And of course, racism is also an issue in Europe. People of color, Sinti and Roma, are discriminated and treated unequally and also in public transportation, of course. So this is an issue as well here. And we are really happy to organize this event with you. Thank you. Good afternoon to all. My name is Rafaela Obergaria. I'm a researcher on mobility. I'm one of the book organizers, anti-racist mobility, the book that we're publishing soon, in partnership with Danielle Sanchini and Paiki Santareng. I would like to tell you that it's so good to be here with you today to discuss mobility. Torg said something which is highly important, which in this context, it's difficult to think about mobility and commuting in a city in the same way as we used to think about before the current global sanitary crisis. The current sanitary crisis shows us that the main way to fight the virus in low income populations is keeping social distancing. So instead of commuting, plenty of people have resorted to remote working remotely.  I believe that this book is core for us to discuss the current moment. Daniel, Paique, and I, we met at an international seminar and we were discussing free public transportation and fighting inequality. At that international congress, we started discussing those ideas. I started reflecting upon mobility. From 2017 on, when my cousin was killed and she was trying to go to university and her commute was longer than three hours. And when she was trying to get on the train, her body was cut in half and her body was exposed for over an hour. And from that moment on, I started reflecting upon mobility. And we know that some territories are seen and that the transport policies are treated differently. And we are aware that transport being able to exist, being able to have access to public policy, it depends on the possibilities that we have to come and go within the city. The book that is already available for sale in Autonomia Literárias' website, there are plenty of reflections for us to about how the bodies are affected by transport, how black bodies feel the transport, that are very important articles. And we know that mobility and transport are highly elitist when we consider how things are put together. Usually architects, engineers, those are the people that decide on mobility. And it's important that more people decide on mobility measures. If a black woman would be the one to decide on that, my cousin wouldn't have died. That's why it's important that we discuss each and everything on the book. We have put together a set of things. And there are even poems available on the book, scientific articles, articles by collectives. Danielle wrote one of the texts. And there is one of the texts that caters to the perspective of a homophobic attack that LGBTQI communities face. This book shows us that it's highly important for us to reflect upon the strategies and tackle inequality. It's important that we consider mobility and place it at the core when we discuss the rights to the city. Thank you so much. And it was so important to work with you during the pandemic. It was so hard. And we talked about it on the book. And I believe that we need to keep on discussing. Thank you so much, RSRLS, for providing support to us. Rosa Luxemburg Foundation has given voice to black women, trans gender women for us to discuss public policies available in Brazil. Thank you so much. I would like to give the floor to Katarini Flo, who is going to mediate such an important conversation. And we're going to reflect and connect with the other office. Thank you, Rafael. Thank you, Kazembe. Thorbi and everybody to be here today. I believe this will be a great conversation, very important, one indeed. And this web seminar, it's an international web seminar. It's a collaboration among these offices, Brussels, New York, and Sao Paulo. And our activity, there are three debates. The first one was held in February 25th. It was a conversation about women in the city. If you have one more information, you can just log into our website and check all the information from the upcoming events. Thorbi and Rafael have mentioned that we're about to launch a book, Anti-Racist Mobility. You will soon have it in your hands or on your screen. It's a collection of articles between all the offices of Rosa Luxemburg Foundation. There are two versions, two covers. One is red and the other one is yellow. And these colors have a lot of meaning. They represent diversity. This book, I had the opportunity to reach most of the articles. It's a very strong work that talks and covers our reality. And today, we will talk about the right to mobility against racism. And mobility, it's about our daily life. It's how we commute to live and do our daily tasks, go to the school, go to work, come back home, have fun. But even those activities are natural or very common. They are not accessible to everyone. The lack of public transport and the high prices of ticket fares are one of the barriers that affect mostly low income and inhabitants of peripheries and urban centers. When we talk about racism and right to mobility, we're talking about the human right, but also about spatial segregation, discriminatory policies, and discrimination in general. And before I invite all the panelists to talk to you here today, I would like to read a brief of a poetry. It's a poetry by Elisa Lucinda. She is a Brazilian artist. I'm a huge fan of her work. And you can find the entire poem in our book that is about to be launched. But let's have a preview of this poem. I've lived my entire life afraid of missing the train. I've always lived away from dreams, money, education, away from the certain kind of art, away from rest. I've planned my life to not miss the train. And I did the math. Distance plus a city departed equals to a path crossed pointlessly. And to help us to think about this issue, I would like to invite all the panelists that will join us today. We have Paique Duques Santareng, member of Movimento Passe Livre, and is one of the organizers of our book, Enteraceous Mobility. From the United States, Marianne Jones, from the Organization Transportation Alternatives, Deb Chatterjee from Community Service Society in New York, and straight from Sweden, Anna Niggat. I hope the pronunciation, I got the pronunciation right. But she's one of the authors of the book and is also one of the founders of Planca.New. And thank you, thank you so much for all of you being here today with us to discuss such an important theme. So to kick start our debate, I would invite to the floor Paique Duques Santareng, who has the master in anthropology, and a PhD student in architecture and urbanism from UNB, the University of Brazilia, a very important educational institution here in Brazil. So it's very good to have Paique and his approach here. He has militants in several areas, especially in the Black movement for the right to the city, urban mobility, popular and traditional cultural movements. And he fights and juggles for the construction of popular power. Welcome, Paique, to our board. You can open your mic, please, and say hello to everybody before I ask you the first question. Good afternoon. I would like to quickly sing a song. Birimbao has played. I should keep moving. I can't be late. I can't be late. My tribe, my people, it's so far away. I can't be late to get there. I live far. So this is a quick translation of the song Paique just sung for us. I have many issues to propose to you here today. Thank you, Paique. My first question. Paique is one of the organizers of the book, Anti-Racism Mobility. And straight in the introduction of this book, the authors and the organizers state that the transport system in Brazil is racist, unequal, segregating, and excluding. How is this expressed in our everyday life, Paique? And what are the consequences of such expression? If you could please relate, basically, that this pandemic moment we're living in, as Torgue well pointed out when opening this board. There is a lot of people staying home or trying to, of course, due to the pandemic, to try to avoid contamination. But do you believe that such constraint has changed for the black population or no answer? So I'll try to bring together these two questions, two issues for you. Paique, before you start, could you please speak louder? OK. OK. Can you hear me better? Great. To approach mobility and racism, the basis of this book, what we should do, well, at first, we must see and observe what's happening in Brazil. We have a genocide project going on since the beginning of this country. And it's a very straightforward, this structure wants to exterminate the population in this country.  But we also have other forms of extermination. So this book talks about genocide in a different approach and also about racism and how mobility, transportation, public transport and racism relate with each other. Obviously, I want to be very clear with you all here. This structure is reproduced and it increases somehow when we are living in the middle of a pandemic. And the issue, the core issue of this pandemic is we must keep social distancing and we can't commute as we did earlier. So if we stay together, if we go to the streets, that will be a problem. And it is, as we're seeing here in Brazil.  We have a lot of racism fronts happening at the hospitals, at the public transportation, at the unemployment issue. Therefore, I say that the pandemic is here to shed a lot of light on those racist structures. The book has seven chapters, also an introduction. And basically, those chapters cover how mobility is linked with different aspects of our daily life. Our understanding of mobility, at least in the first chapter we put that it's quite clearly. We say that mobility, it's a construction somehow, a social construction with a racist intent. Mobility in Brazil wasn't built on the commuting and circulation of people. No, here in Brazil, we wanted to keep people, certain kinds of people apart from each other. So there are several ways of... Paike has the mic off, unfortunately. I received a call, I'm sorry, I'm back on. So the freedom and the intervention on black bodies plus all the forms of the trafficking of people since slavery times, and now with mobility interconnected with the genocide project going on in Brazil, we can have a new look on how mobility supports surveillance and repression. By doing so, the huge population of black, young men in jail right now, this is the anti-mobility, but this is also part of the project of mobility in Brazil that follows this genocide flow. And definitely, mobility will manifest itself differently depending on your race, on your gender. And especially in Brazil, we see that racism clearly expressed nowadays. There are different nuances if you are a man or a woman or a trans. And of course, we can also talk about mobility in terms of religious, racism, like temples, Afro-Brazilian temples that are being attacked right now. So when we talk about mobility, we also talk about resistance and how we can be together and as Maroon did back in the colonies. It's also important to understand mobility beyond the physical aspect of mobility. In the fourth chapter, we understand mobility as a mode of conversation, of exchange, and of resistance. And mobility is also present in economic politics. We discuss in this chapter, especially how the racist mobility impacts the economics, but less. But not least, I wanted to try to briefly cover all the book, but I only have 10 minutes, of democracy and help our part of this struggle. All the struggles that have been reported on the book, people from different nationalities and all over the world, they engage in anti-racist language. I'm talking about the book a lot because I believe that it's important that we focus on the book for the discussion. But we also bear in mind that it's a global struggle. There are plenty of poetry, short stories, and other articles available in the book as well. Now, during the pandemic, the fact that we can't breathe, also the fact that society is forcing us to commute and we're targeting that on the book. I would also like to pay homage to Vilma Hayes. And guys, now I got a bit emotional and I forgot one of the names. So I would like to thank the people who drew the cover and I thought about the book and I got emotional and lost focus. Paiki, I agree with you. It's an amazing book. And I must tell you that while I was reading the book, there was a song by Elza Soares, a Brazilian singer. And her song wasn't exactly about mobility, but on the book, on the song, Elza Soares talks about the fact that the cheapest meat available in the market, it's black meat.  So when we were reading the book and discussing the book, it made me think about Elza Soares' book, Elza Soares' song. And before we move on to the next speaker, I would like to say to people who are following us through the channels, be it YouTube or in other channels. You can follow us in Portuguese or in English. I would like to say hello to Glaucia Peira, Rafael Calabria, and later on I'll talk about other people who collaborated on the book. If you who are listening to us would like to ask us a question, please type in the comments. We're trying to get your question and have our speakers respond. Now moving on, we are talking to Mariam Jones. Mariam Jones, she joins the NGO Transportation Alternatives, which is based in New York. Most of her work is related to providing improvements for bicycles, pedestrians, and public transport in Bronx. She's also the editor of the new Socialist Feminist Magazine looks. Mariam, thank you so much for joining us today. Feel free to delete it for your early remarks. And then I'll ask you a question. There is a small delay because we have simultaneous interpretation available now. Hi, everyone. Thanks so much to Rosa Lux for inviting me to be a part of this panel. Thanks so much to Kazembe for reaching out. So yeah, I want to talk about the ways that mobility, justice, and inequity ought to tie into a bigger, more holistic conversation about human rights and historic injustices. One question that comes up a lot in my work is what keeps us safe and who keeps us safe? Safe street advocates often stop and start at street design. Street design is essential from keeping us safe for cars. We need better street design, and we need to separate people who are traveling in one mode of transportation from people traveling in another mode. That is how we prevent crashes and fatalities, which is so important. However, the over focus on cars and street design ends up centering streets and cars. Mobility, justice is a framework that brings us back to focusing on the well-being of people who travel through those streets rather than centering the well-being of streets. Our motor transportation is only one aspect of what makes us safe or unsafe in public. As many of us know, when we pass through public space, we're displaying our race, often our class, gender, ability, age, citizenship, and all the securities and insecurities that come along with that. So in addition to the threat of traffic violence, the possibility of being detained or worse due to your appearance need to factor into street safety and what really makes us safe. Urban street safety often relies on making public space safer through police enforcement. That is asking police to do the work of keeping us safe by ticketing or arresting people. Ultimately, this just turns people into prisoners. For communities of color, our safety is not increased when we are pushed into the vicious cycle of over-incarceration, whether in prisons or in detention centers. Policing, as many in communities of color are well aware, is rooted in historically discriminatory practices and is tied to the history of keeping undesirables within red line zones or pushing those who dare enter white suburbs, keeping them out through this vigilante violence or white domestic terrorism. So mobility justice, this is really about putting forth a vision for street safety that foregrounds the relationship between policing and planning and the extent to which one can deny people of color and especially people of color with non-conforming gender identities, basic freedom of movement, and how not being able to move freely is attached to fear, trauma, which can be the result of racial discrimination, ticketing, incarceration, and deportation. And so importantly, the racialized policing of public spaces is not a thing of the past. It has never ended, and it's really important that we fully acknowledge it. Furthermore, the call for safe streets is also connected to urban renewal and displacement, which is really harmful for questions that relate to race and class. Everyone should have a right to the city, and no one should be priced out of it. Pricing out is what happens when certain voices are left out of the conversations on development and property value. This is why mobility justice calls for the creation of public processes that can distribute public resources equally. Mobility justice demands that it's historically marginalized groups and communities be heard as full partners in all planning processes, not just asked to approve or go along with decisions after those decisions have been made. When it comes to changing the status, for another point, when it comes to changing the status quo of our car centric world, there's a complicated relationship between communities of color and car ownership. Owning a car is seen as part of the American dream, not so different from owning a home. Cars make people feel powerful and safe. They allow us to return to neighborhoods where we can no longer afford to live, but where our friends, jobs, schools, and social lives remain. So the positive view of cars shapes the landscape of transportation in communities of color. So we need to think about how we talk about the reality of car travel. And the reality is that mass car travel produces toxic air and violent deaths, and that these burdens disproportionately fall onto the shoulders of communities of color. So we need to frame these truths in ways that can balance out the car's symbolic importance, while also respectfully challenging that symbolic importance. The right to self-determination is also a key part of mobility justice. Public transportation spending decisions that were made in the middle of last century coerced people into cars in the car-based world that we have today. These decisions created winners and losers, long race and class lines. The winners are typically white middle class highway communers who gain access to suburban home, space to spread out, and clean air. The losers are the workers, people of color, who remain in the city and saw their neighborhoods polluted, caught up, and divided by highways. This is why we need diverse conversations and this diverse voices in this conversation. We can't recreate the mistakes of the past where people of color were left out of transportation and infrastructure decisions. Another problem with street advocacy in the US we often see is what we call Copenhagenization. That is the calls to Copenhagenize US cities. This is the idea that we must make the US look more like Europe, underlying this idea as a colonial mindset, that Europeans are seen as more sophisticated and European models for public space can work well in all parts of the world, regardless of the context or background of the people who live and are a part of that community. This is a colonial mindset because Copenhagenization, like colonialism, is about imposing the preferences of white males who are able-bodied and rich on the bodies of the rest of the world. It champions the structures, frameworks, and narratives designed to uphold this idea that Europe is superior. Outside of Europe, there are a number of global cities that we can gain inspiration from. And of course, going back to self-determination, communities of color should ultimately be able to make these choices. So wrapping up, communities of color in the US are often told that we're unfit to design our own futures, to guide public spending, or understand the real issues at hand. Our communities are given limited access to information, and we're judged by how well we respond to engagement and interventions imposed from the outside. Then when we don't fall head over heels for the decisions made, for the decisions that we're locked out of, we're seen as regressive or opponents of progress. Liberating communities that face hardships and obstacles should be a central aim of safety and street design. It's really important that we break with the old framework of safety defined by exclusion and create new ways of thinking about safety that can allow us all to collectively grow. True safety means addressing socioeconomic, cultural, discriminatory, physical barriers to accessing public spaces and having the freedom of movement. Truly safe streets and real mobility require radical inclusion in both the planning and implementation of such projects. Thank you. Thank you so much, Maria. Maria, I would just like to remind you that there's a delay. There's simultaneous interpretation available. I would like to make a comment. It's quite a personal story. And as Paiki was talking about the book, so many things came to mind. When Maria talked about exclusion structures and a strategy to keep Black and poor people away from public spaces, and then it reminded me when I went to South Africa, I would say that I'm a bit childlike as regards history. And when we study history at school, it's different. You know, when you go to one place and you see that the things that you read on the book are realities at a place, but emotionally to see it, it's shocking. When I first went to South Africa and I went to different parts in the city, I went to very wealthy neighborhoods and I also went to impoverished neighborhoods. And I've noticed things that I had read about. And that's why I would say I have a childlike perspective of reality. And reading, seeing what you read in reality, it's shocking. The moment I went to the Upper Tide Museum and I saw a spatial segregation and all the violence targeted against Black people who were the original people in that area. And that's really shocking. I was reflecting upon that. And I was also discussing that with my husband. And my husband is Black too.  And then I connected the struggle, the story of struggle that I've read to what I was facing there. And then I decided that we would go to places where we would see more Black people. So I refused to go to certain areas where it would be, you know, white neighborhoods. I've decided that certain neighborhoods I wouldn't go. We would just go by car. Then when we came back to Brazil, I've started reflecting upon that. It's not that I'm going to compare Upper Tide to Brazil necessarily, but the choices you make, the choices you make about the places you're going to go within the city, it's impacted by exclusion structures. I'm from Rio de Janeiro and I've moved to Sao Paulo a few years ago. The moment I moved from Rio to Sao Paulo, the structure of exclusion shocked me too. Some things, they were available on TV and you know, I wouldn't realize. And here in Sao Paulo, everything seems to be far away. And I saw on TV a group of children from the outskirts. They were sent to a park. And those children, they had never seen trees because in the outskirts, in the favelas where they live, there are no trees. And some of those children, they had never left their outskirts. And those children, they don't even commute within their own cities.  And it's truly shocking when we realize that. I'm not going to keep on talking because I'll get emotional just like Paiki. I would like to invite our new speaker, Depriquia Schapergi. I truly hope that I'm pronouncing your name correctly. She is a senior political policy analyst. And she works with employment, inequality, and mobility. She holds a PhD in economics by Brown University. The organization where the bureau that works are connected to a fair transportation price for low-income communities. I would like you to talk about the unfolding of that program and what results do you expect to achieve? Thank you so much. Thank you, Catherine. And thank you so much to Kazembe, Daniel, and everyone else for setting this up. Thanks to everyone tuning in. This is really, I'm honored to be here and so excited to be part of this, talking to all you brilliant people and listening to your insights. My name is Depriquia Schapergi. I work as a senior policy analyst for Community Service Society. Our role in recent transit justice issue was securing low half price metro cards and subway, metro cards for subways and buses for New Yorkers. So I'll tell you all about the motivation behind that project, behind that campaign, how we got it going and how we achieved it. And I will also talk mostly on the economic justice as is the racial justice issue of mobility. And I'm very glad that Marianne went before me and her discussion of the streets and the cars are exactly complementary. They round out our discussion about New York's transport system, because I'm going to talk about public transport. She talked about feeling safe. And availing the streets through bikes and cars and the repercussions that they had. So New York City, as you know, it's a gigantic economic engine of a daytime population of 23 million or more and has a GDP of 1.5 trillion. And the subway system, which is one of the main organs of this economy, it has a rail daily ridership of around 5.5 million or so. And the bus has about 2.2 million. So these are astronomical figures meant to give you a sense that although it seems to be a big economy humming along, a significant share of New Yorkers do not get to partake in this economy. So it came along, one of the ways we realized that other than from the lived experience that we observe it every day, is the CSS, Community Service Society, does an annual survey of low income New Yorkers every two just to get an in-depth understanding of their opinions, their well-being, what are their struggles, what are their hardships. So we go really deep into employment and benefits and what their experiences have been. So one of the things we realized in during one of the 2015 survey was that one in three low income New Yorkers would not afford subway or bus fare. And a disproportionate effect with a disproportionate effect on low income blacks and Latinos. So the high cost of fare was actually preventing low income New Yorkers from meeting basic needs, like going to a doctor's appointment. 25% of them postpone attending a medical appointment or failing job opportunities, more than a third, said that they would not pursue job opportunities that were too far from their home because they did not know how they would pay for the metro card, the fare for the subway. And these rates when you look at among people of color were significantly higher. So among Latinx workers, 35% said that they would postpone attending a medical appointment. And 43% of Latino workers said they would not pursue a job opportunity away from home because of lack of subway fare, but the percentage was only 6% for white people and it was around 25% for black people. So going on from these numbers, what emerged is that the commuting costs were proving to be a real burden on New Yorkers. Another way to see it is that if you take in commuting costs out of the equation, you would actually lift about 150,000 New Yorkers out of poverty. So it was adding to almost two percentage points to the city poverty rates, like that much of a financial strain that it was putting. So then the fare fairs campaign was launched with an op-ed, there were rallies, we partnered with our other riders advocacy organizations like Riders Alliance. There were countless testimony about it in the city council. We got other elected officials on our side. And after about three years of relentless persuasion and advocacy, the mayor stepped up and worked out a deal in the budget with the city council speaker. So, and since then it has been a huge success, about 230,000 New Yorkers have signed up, which are quarter million. And the money that was set aside was around 106 million for the first round and then it was because of pandemic and so on, it was reduced to 40 million. But CSS continues to monitor the adequacy of funds as we are coming out of recovery and if usage of the subway is increasing and whether we would need more funds. So that is what motivated the campaign and how the campaign unfolded and what we achieved. Now, let me talk a little bit about the justice issue of this. And one thing to remember about this is that mass transit is the only option for most low income New Yorkers, especially for people of color. So it seems like an obvious fact for this panel and also for everyone else, but that is a crucial fact on which everything else hinges. If households that are better off, they can avail taxis, they can avail ridesharing apps like Uber and Lyft and they might even have their own vehicles like cars and bikes or whatever. But for low income New Yorkers, this is all that they have mass transit. So anything that happens, any policy or any kind of like, whether for better or for worse, policies affecting mass transit has huge repercussions for low income New Yorkers just by the design of this problem. So the price of the metro card, even with fair increases, it does not pose much of a burden for middle and high income households. But for low income households, affordability to a metro card can mean anything between having a college degree and not having a college degree between visiting the doctor's office or not and between showing up for a job interview or not. So you can understand how crucial a difference it can make. And these actions go on to have cumulative repercussions for not only one person's life, but also intergenerationally. Secondly, the fair design system is designed to reward bulk purchases of rides. I think that is probably also the case in the mass transit systems of other cities. You know, you're like a weekly or a monthly pass costs for a ride cheaper than a one ride when you don't buy just one ride. And it makes total sense in economic incentive sense, but it is designed to reward those who already have more money and are not typically living paycheck to paycheck. So you see the little bit of a regressivity that's already inbuilt in the way we are designing the fair system. Then another angle I want to point out is like a lot of middle and high income workers, predominantly again, white and not people of color, they get commuter tax deductions through their employers for using the mass transit. So this further subsidizes their cost of using mass transit and it is basically a tax revenue that the state and the city forego to subsidize travel of those who are already well placed. Now, when we encountered people struggling to afford transit and we talked about them through their experiences, some of them said that they were aware of the supports available from government agencies and all, but the amount of bureaucratic red tape cutting it would take to avail themselves of it was not worth it. They typically do not have the time or the luxury to go and sit in a government social service office for one whole day to avail themselves of assistance. And how does this percolate? How does this look in the long run? What are the effects of transit affordability struggles? So the repercussions of increased difficulty in commuting effects, not only the current workers, but research has shown that even children growing up in households in such households are likely to earn less than their peers as adults. Like this is based on research by Raj Chetty and Nathaniel Hendren of Farwood who show that the positive effects of commuting, of riding with others, of being exposed to like, Catherine said to the parks, the museums, those have a long lasting intergenerational effect in developing cognitive abilities and leading to much better long run achievements in the earnings and incomes and stable life outcomes and so on. So by making transit affordability difficult, we are depriving millions of children of color these opportunities to fulfill their true potential. And the fact that they had, like we saw in our survey, that people of color were giving up pursuing job opportunities because they couldn't afford the subway. So think about what that means. Like the unemployment rates for people of color are already high, even they were even high at the peak of recovery for these groups because they have been systematically and structurally market, but they cannot even avail the job opportunities that are there because of the transit difficulty. And given the nature of New York City, geographically, people of color are more concentrated in outer boroughs, far from central business district where transit fare burdens can reinforce the economic and geographic isolation of some of the most economically disadvantaged families. Like Paikik said it the best, and in the first sense that Brazil's transport system seems racist and equal segregated, and we can just replace Brazil with New York and it would continue to be more or less the same. And finally, I would touch on the fact that the transit affordability has been made into an issue of criminalization of poverty. So if someone is, like I said, there are millions that we saw were struggling to pay the fare.  And you would be desperate when you would enter illegally. There was a very strong correlation between the race of the person entering illegally and the likelihood that they would be arrested for fare evasion. And this was, you would be arrested and you would either have to pay $100 or more fine, which is a big burden for most people. And of course, for someone who doesn't have $3 to pay the subway fare, $100 isn't prohibitively expensive fine for them. And in the worst case, they would be arrested and they would have a criminal record for their background, which would show up in their background checks. Restricting further their possibility of finding gainful employment or attending college or even finding a place to live where if their landlords are doing background checks. And the thing that makes it more poignant is that these fare evasion arrests, they had nothing to do with the level of neighborhood crime or so all it had to do was the race of the person and where they were located. So in neighborhoods that were high poverty and predominantly black, that's where the fare evasion arrests were happening. They weren't happening in other neighborhoods even where the violations were probably at the same rate. So this is something we found in our research at CSDIS. And as you can understand, for any kind of arrests, it is often informed not by the nature of the crime or whether the crime actually is happening. It is informed by explicit and implicit biases based on a race, age and gender. And for undocumented immigrants, the consequences were even more serious who could then be deported or separated from their family forever. So all for the crime of not being able to pay $2.75 for a subway metro card. And the final point I would like to make about transit justice specific to New York City is that the subway system has been in majorly in deficit for many years now. It needs immense amount of like a 12 billion or so funding to be injected for capital projects. And why is this again a racial issue? Because what happens with the lack of repair and maintenance of the MTA hurts people of color way more than it hurts white people. The delays because of signal malfunction or because the doors get stuck or whatever. If someone high or middle income household white person or they arrive at their job 10 minutes late, it's fine. Nobody even takes notice of that. But for the below income people of color working at an hourly wage rate and jobs which are very precarious which they it's totally possible that they would lose their job just because showing up late. And this was happening repeatedly and in New York Times put out an article saying that the MTA is failing its citizens at the residence of New York City in big time. So you see how each of the disadvantages are compounding each other. They're living in outer borough because you're priced out from the central business district. You are having to rely entirely on mass transit and the mass transit fails you because of its quality. And finally, all of that combined with affordability difficulties lead to a problem that is as perpetuating inequities as any other system. And finally, I would just like to say that of course fair fairs was a success. We still have the program. It has been shown to really help people especially students and people trying to take classes in the evening to improve their curriculum and get better jobs. But the fight doesn't stop. The fight hasn't stopped criminalization of poverty and continues. And I'm really happy that we are having this conversation and there are many layers to be filled back. Thank you. Thank you so much, Demetria. I would like to say hello to Elica Theles from Salvador. Also, Cristiane Gomez, our co-worker. I would like to say hello to João Marcelo Zim, Marjorie Alencar, Murilo Valério Guimarães and other people that I have not been able to mention here. And I would like to remind you if you're following us on YouTube or Facebook, you can send us some comments. If you're following us via the New York, YouTube or the Brussels YouTube, send us questions. For us to continue, I would like to invite Anna Negar. She's from Sweden. She's an activist and co-founder of Planta New. It's an organization that provides free transport. And she's currently the editor to a newspaper in Iava, a neighborhood known by the diversity and the number of immigrants. Anna, thank you so much for joining us and considering the translation delay, I'm even going to ask you the question already. Anna has written the chapter, International Solidarity in the Anti-Racist Mobility book we're launching. In this article, Anna, you said that racism is a problem in society and public transport systems are miniature versions of the cities. So could you please tell us more about this statement on the book? Okay, so first of all, thank you everyone for your presentations and it's so nice to be here. And I'm really honored to take place in this seminar. And to answer the first question, I would like to explain like, why I said that the public transport system is like a mini-versino society is because you can see how we have all the systems that we have the control system that we have also in society with the police. But here we have the guards, maybe the ticket inspectors. In Sweden, the politicians are often using the public transport system as some kind of test area for the new reforms that they want to test. For example, they wanted to try to prohibit the begging in society and they first wanted to make a test and prohibit begging in the metro system. And they make up rules for how to behave within the metro system you pay to access. And then you have to prove yourself like worthy of being there. We have seen a lot of problems with racial profiling in Stockholm metro system, for example, where the police uses the ticket inspections and the borders, the barriers as like borders within the country because they can't just check people on the street and ask for ID or to see their documents. So they use the public transport systems because they have existing barriers and borders to check people, they ask for the ticket and if they don't have a ticket, they can also ask for the passports and documents. I don't know if that was an answer to your question, but I can continue with my presentation if that's okay. Yeah, as mentioned, I am a member of the Plankapuna Collective and it's a campaign advocating free public transport and we're also organizing fair dodgers. We're not making the public transport free yet, we're just promoting it and campaigning for it and also organizing fair dodgers in a solidarity fund where each member pay about one euro, I don't know how much that is in dollar, maybe one dollar or something like that, $10 a month and if someone gets caught and get a fine, we pay that fine together. And yes, I live in a suburb called Rinkeby, it's located in the Järva area in the outskirts of Stockholm. It's approximately 20 minutes from the city center and I'm gonna start by telling you something that happened a year ago last March because there was something that started to happening and it was the first sign of the COVID-19 pandemic that reached Sweden. The first cases came to Sweden through which people that have been skiing in the Alps and they came back to Sweden and when they were driven home by tax drivers from the airport, they infected the tax drivers and many of them lived in my area in Järva and also in other similar areas in Stockholm. The average income is low among people in Järva and many people live in overcrowded apartments, often in several generations together. So I started noticing more and more ambulances coming and picking up neighbors and the people that are new started to get sick and soon it was a well-known fact that the COVID-19 outbreak was massive in our area and many people died, especially the Somali diaspora was hit very, very hard. The Swedish authorities were slow to respond but it started to recommend people to wash their hands and if it was possible to work from home but most of the inhabitants in our area are working class people, the taxi drivers, bus drivers, shop assistants, restaurant workers, nurses and care assistants and for them working from home is not an option. So most people are depending on the public transport and I don't think that class society has ever been more visible than during the pandemic and the overcrowded buses that was an existing problem before it's never been more of a problem. And today, one year later, statistics have proven that low income is the highest risks of being exposed and infected and also to die from COVID-19 and immigrants are overrepresented among all cases, especially immigrants with non-European background. And I bet these patterns are more or less the same worldwide. The picture of the overcrowded bus in COVID times full of working class people, most of them with black or brown skin, it's a perfect illustration of how power structures are linked together and reinforce each other. There are more power structures that ranks people based on class, gender, ethnicity and even among means of transportation, there is a hierarchy and we call that the traffic power structure. The traffic power structure establishes a hierarchy among all means of transport. At the top, there's the automobile and then we have bikes, public transport bicycles, pedestrians and on the very bottom, all people that uses different kind of aids to move around. And the different resources allocated to all these groups reflects the traffic hierarchy. The stereotypical motorist is a white male and he has a relatively high income and the majority of the global population does not have access to the car. As a means of transport, the same thing applies to aviation as a means of transport but more extreme.  The crisis that already hits hard on the global south and forces many people into climate cost refugeship. Most suggest that this is something we're going to see more and more of in the future and many people will be forced to flee their homes due to climate related catastrophes and conflicts. Where there are borders, they will be met by violence and control systems. And at a closer look to these systems and the barriers, you can see that they are the same inside of the borders and outside. The same companies provide fences and gates for conflict zones for the control and to the public transport systems. The person who managed to escape from war and terror in their home country could face the very same gates and fences at the border of Europe in a refugee camp and later on in the public transport in the European city. Mobility within the city is primarily an economic issue. Mobility in the world is not only about your economical resources but also a political issue. Where the nationality of your passport makes a huge difference. Many European citizens choose holiday destinations in countries whose citizens are rejected as refugees at the walls of fortress Europe. This proves how strongly migration control and the denial of free movement are connected to the traffic power structure. As anti-racists, we have a role to play here. The Swedish author and scholar of human ecology, Andreas Malm, talks about how climate activists have to become more involved in the anti-fascist movement and take action against border control and migration laws. And I believe he's completely right. In our Plankappungnu activist collective, we have a cooperation with our local branch of the activist network, Noenysa Ligel. Noenysa Ligel works to provide support to people who are forced to live undocumented after being rejected, having their applications for asylum refused. Their demand is permanent right of residence for all people who have arrived in the country and who wish to remain. One of the main projects of our campaign is Pia Kassan, our solidarity fund that has been keeping the organization going for over the years. And as I said, each member pays about 10 euros to get the insured. But for commuters risking more than just a fine upon inspection, it's no use. So that's why we are also doing warning actions. We cooperate with migrants rights groups and the ticket sharing campaigns and actions. The pandemic is not over and no one knows what the world will be when it is. But in times when freedom of movement have been restricted and complicated, it is more important than ever to protect the freedom of our political movements. And as activists, we must integrate the anti-racist struggle in all practical work that we do. It's not just a statement. It's how we analyze things, what you say and how you act in everyday situations. Whether you choose to join a climate justice struggle an urban rights movement, a fair free public transport campaign or if you focus on refugee rights anti-racism must be integrated. We are one kind and we are one planet. So let's just move together instead of at each other's expense. Thank you. Thank you so much. Before we move on, I'm going to read some of the comments. Cristina Fernandes Augusto is excited about the book. She compliments Rafaela Bergaria, Paique Santareia and Daniel Santini, who are the book organizers and she also contributed to the book. I would also like to come to deliver Jo Pereira remarks. Jo Pereira also congratulates all of you for the book and what she said. What you said about anti-racist mobility is highly important. Now we're going to start our last round. I would like to invite all of you to please deliver your final remarks in this second round. I'm starting with Paique Santareia. I'm going to introduce him again. He's one of the book organizers, the anti-racist mobility book and he belongs to Movimento Pasi Livre. Paique, the floor is yours. I would first like to thank all of you and it was great to hear all of you. Paique, please speak louder. I would like to thank each and every one of the speakers and everybody who is also listening to us via streaming. I'm so thankful that this book, we thought about Brazilian reality, Brazilian context and it's so nice to see that our struggle is global. When we consider racism and mobility and the violence connected to it, racial violence against ethnic racial minorities all over the world, especially in colonizer countries, it's global forms of violence and this violence is linked to the African diaspora. The structure of violence and the structure of mobility is diasporic based. Also, we need to consider our struggle and consider anti-racist struggle as global. When we consider the Brazilian case, I would like to highlight some of the elements in the racist mobility. Somehow I believe that this is linked to our debate. I would like to say that when we consider Brazil, all of the elements connected to mobility, they have been organized from the 19th to the 20th century when slavery was abolished in Brazil. When racism is no longer a slavery system and becomes a genocide system. So the structure of racism will design society as a whole. If we also tell how can we commute in which spaces? Determining which work spaces are for and also the leisure spaces. But specifically talking about our work relation, this tells a lot of how we deal and transform and change the spaces. So our racist unconsciousness prevents us from commuting freely in the city space. And we feel and see ourselves as ugly and we feel ashamed of commuting certain spaces. This is a result of such a racist structure. And our public system, our transport system in Brazil is based on merchandise. And this has a racist background because enslaved Africans were considered merchandise. So the transportation in Brazil was built on how colonizers could commute and transport merchandise. So the individual transportation and public and collective and mass transportation, they are very different. Mass and collective transportation is a technology based on product and merchandise relations. So we see a clear background. It's the same line of thought from the slavery times. Unfortunately, black people was the first global commodity. That's sad. Struggling against racism and of course incarceration. I've spoke about this earlier, but the struggle and the ways of how mobility is placed in the country. It also has repercussions on racist, anti-racist mobility, individual transport, ways of transportation, modes of transportation and the access to public spaces. They are also linked to this racist background. And it's a very in-depth struggle. Beauty of the Afro-Brazilian population is precisely this. We were able to sustain and resist as Marus back in the 19th and 18th century and we kept producing life even though the government, the authorities wanted to exterminate them. So this extermination project, this eugenic project has been going on for a while here in Brazil and the black population has won because it's creative and creating new ways of commuting and occupying the space. Anti-racist mobility is a technology that we as a population can drink from so we can live happily. This is our invention against racism. Therefore, our book is a book about hope. Thank you very much for your attention. Good afternoon, good evening. I don't know, it depends on where you are in the globe, right? I hope our motto is a ticket gate-free life so I hope, I'm very thankful to be here today. Hafa, Daniel, thank you so much for bringing me into this project. I'm speechless, honestly, speechless. You two are, you have a great potential of creating and managing life and new ways of life. And I would also like to thank the publishing house that supported us and put this book out in the world together. And thank you for all that are here today listening to us. Oh, and Silvio made it, definitely. He's such a great person to me. Thank you Paiki, thank you so much. Yes, the book is about hope and everybody should read it. One of the things you've mentioned, it's precisely this. Transportation is something very black in Brazil. And this is highlighted in all the chapters. I would like to move on with our board and invite Marian Jones to the floor. I'm sorry for the noise, the background noise, but let's move on. Marian Jones, could you please open the mic and give your final statements? And thank you, thank you so much for being here with us today, Marian. Yeah, thanks again to Kazembe and everyone at Rosa Lux for having me today. I guess I'll just reiterate the handful of points that I made earlier. I spoke about safety and public transit a lot because in my work it involves being out on the street, often in the Bronx, talking with New Yorkers of color on what's important to them. So to all street safety advocates, again, I just wanna reiterate that collective safety and self-determination is really important and it's something that we should foreground in our work and if we wanna build a mass movement to really take back space from cars and ways in which activists can support that work is by recognizing what constitutes, recognizing that what constitutes safety is different for different people and should be defined by those who are most economically and legally vulnerable. We should all try to take the time to witness and address the vulnerabilities that they face. In addition to that, we need to shift the conversation from streets to bodies. Focusing on people allows for us to move beyond the constraints of focusing on streets. It also makes space for centering the lived experiences of marginalized people. You know, especially for white comrades, I would really encourage that you think about the ways in which you may have benefited from the over-placing of communities of color or those living in poverty and also think about the ways that you and your community has been harmed by this. We also really need to call out and fight against the institutional racism within planning processes. These processes often leave out communities of color and implicitly imply that communities of color aren't a part of their vision for the future. You know, conversation about cleaning up streets is often code for gentrification and trying to criminalize people. In addition to that, you know, the data that goes into deciding what infrastructure needs and what investment needs a community may need, that data might be based on transit usage or the amount of people on the street. It doesn't take into account the lived experience of community members. So this is a type of knowledge that should be prioritized and should be given the same level of respect as other kinds of knowledge. And like I said before, moving beyond Eurocentric solutions as the default, the movement for street safety should look to Central America, Southeast Asia and other places. And again, there's a long history of discriminatory transportation policies. So it's natural for communities to now have a complicated relationship to exploitive and polluting systems like highways. So like highways and like colonialism that is outside people coming into their communities to tell them how things should be organized. There's a lot of distrust in that regard. So yeah, I think public spaces need to be safe and accessible for all. And the way to do that is really complicated and does go beyond street design. So thank you, that's all I have. Maria, thank you so much for attending this panel. You've contributed a lot to our debate. Just a reminder, there is a small delay because we're having simultaneous interpretation. So don't worry. And I would like to share a comment by Kazembe, the project manager in New York. He said, this debate made me think about the image of essential workers struggling, trying to breathe inside train wagons. This is a very strong and representative image of our days. These paramount workers are so exposed to risk nowadays. We really should stop and think about that. What is essential in society? How we should treat people, generally speaking. So that's quite a good comment. Thank you, Kazembe. I would also like to thank George Pérez Filho comment also from the office in Sao Paulo. And to all of there are leaving comments in the chat. Unfortunately, I won't be able to share all your comments live. But now I would like to invite to the floor Deb, so she can make her final remarks. Thanks. Thank you, Catherine. And what is especially pointed to me after listening to everyone here is that we all kind of echoed each other using different words, different languages. But really, none of us were saying things that was completely new to the other. So imagine like it boggles the mind that from New York to Sao Paulo, to Brussels, people of color had to deal with the exact same problems of excessive incarceration, excessive use of force on them, being pushed to the periphery of productive economies, both literally as well as metaphorically, not feeling safe, not feeling secure, and basically not having a right to self-determination, which is basically the ability to realize your own potential. So we are robbing people of that by making transit and public spaces unsafe. I feel like the unequal transit justice, transit issue is just one part of a system of marginalization that has been imposed on black Americans and blacks in other parts of the world, along with many other dimensions of discrimination in housing, employment, and other spheres. So anti-racist mobility needs to be a multi-pronged approach, like we discussed over here. And one of which incorporates making transit affordable on which I was speaking. And the other thing that I want to highlight is that we fought for fair fairs because we saw that in our survey that about a third of low-income New Yorkers couldn't afford subway or bus fares and the rates were exceptionally higher for people of color.  So although our campaign took off and we now have the program in place, which is supposed to help 800,000 New York City adults who live at or below poverty line in their transit, and they're supposed to save people upwards of $700 per year, we are totally cognizant of the fact that the fight doesn't end. And why? Because the fight for fair fairs is an economic justice issue, but it's a racial justice issue, but it's also an environmental justice issue. And we all would benefit from better, more improved, more reliable mass transit options. And the transport system in New York, it is a microcosm of the transport philosophy that guides the transport system in rest of US. It is highly unequal and it reinforces existing racial inequities. So the regions that need most connection to jobs are leased up by mass transit. And that is, you know, kind of like a, it is mirrored even in New York City's transit system. Now, and I would like to end on a note that Kazembe based in the comment that you read, Catherine, that about essential workers. So essential workers in New York City continued majority to be people of color. And they were forced to take the mass transit through the pandemic, even when sanitization was inadequate. So this further exposed these communities to the virus with the result being that the morbidity and mortality rates from COVID-19 is disproportionately higher in communities of color. So we basically demanded an immense sacrifice of them without making any accommodation arrangements. And it risks not only these people's lives, but the lives of their entire communities. And given the composition of the essential workforce, we treated them more as expendable than essential. And we hope we should not be repeating this again. And I would like to end on that note. And thank you all for having me. Thank you so much. I would like to comment that on the anti-racist mobility book, there is an interview with one of the leaders in the movement of anti-fascist delivery people. And they talk about the relationship between the exploitation of work for delivering services. I think that this interview is very interesting. So please read it when the book is available. I would also like to say that Ana Nigar's organization, Plan Canu, there is a book by called Transit Power Structure. This book is available in our website, the Brazil Office website. Ana, I would like to invite you to deliver your final remarks. Thank you so much. Thank you. And as Deb just said, I feel also that we understand each other and we have a lot of common cross the globe. Like even though I never been to either New York or to Brazil, I can really relate to how you describe the transit systems and the situation with the structural racism. And you were talking about fair fairs and we believe that the only fair fair will be a zero fair because if you only had like, if you paid a minimum fair, you would still have to have the ticket system, the control system and the border system that in itself is very violent and also expensive. So our suggestion is to just take away all the fairs and have a zero fair financed either by taxes or another solution. I think that what I would like to like summarize everything by, I feel hope from all the things that we have in common and also I feel that when we organize ourselves as commuters or as citizens, when you go find things that you have in common like in the workplace, you organize in your union, you can organize in your local area, in your housing projects, but as commuters, we also have a strong voice. If you go together, you can organize a fire strike as we did, you can also just, I've seen examples of people getting together as public transport users and achieving both like environmental, local environmental changes by a more secure space and also starting a collective of activists and using this as a ground, this mini society where you can see the surveillance systems, the racism, the control systems, the borders, it's very visible and by targeting it within the public transport system, it can also be very efficient. And then you're like level up and we target the real borders, the donation states and the structural racism without any prior society. Thank you. Thank you so much. Thank you all speakers. It's an honor to me to hear all of you and I believe that we have quite an open perspective when it's great to notice all of the remarks from different territories. I would also like to highlight that Blanca News book that we have translated here in the Sao Paulo office. Tarifa Zero BH is the collective responsible for the translation. I would like to thank each and every one of you who joined us. I would like to invite Manuela Krop to deliver her final remarks. Manuela. Yeah, thank you very much, Katarina. I do not have a final remarks. I just have an announcement of our third event. As Anna just said, a fair is zero fair and the third event of our small series, Write to Mobility, will be on steps towards free public transport. It will take place on Thursday, 22nd April. We will send you the invitation and we will welcome your participation. We will have interpretation to English, Portuguese and German. Thanks a lot. Rodrigo. Rodrigo, Katarina, all of you are live. Thank you all. See you the next event.


Now, there is another question that comes. So, basically, what we are saying is, this is, if you like, you can call this as the anti-derivative. A function capital F, A B to R is called anti-derivative of a function f. So, one introduces, if f dash of x is equal to f of x, for every x belonging to A B. So, that relation that we are saying here, so one gives a name that capital F is the anti-derivative of small f. So, the question arises. So, fundamental theorem of calculus is applicable whenever there is an anti-derivative for a function. So, the question is, what are the functions which have anti-derivative? What is the class of functions or is there some, we can say that, if a function has this property, then it will have anti-derivative. So, we want to answer that question. So, question is, which functions have anti-derivatives? Keep in mind, anti-derivative is not unique because we are just saying f dash should be equal to f. So, if you take f plus a constant, that derivative also will be same. So, it is a class of functions. Two anti-derivatives differ by a constant. So, to answer that question, let us first observe a small result. So, to answer this, what shall I call? Proposition.  Suppose it is a continuous function, then we know it is integrable. Then the claim is, there exists a point c belonging to a, b such that f of c multiplied by b minus a is equal to integral a to b f x dx. So, let us try to understand what is this theorem saying. Because f is continuous, the right hand side integral exists, every continuous function. But what is the left hand side? This is f of c into b minus a. So, this is the area of your rectangle with base as a, b and height as f of c. So, if you interpret it geometrically, it is very nice in the sense that if this is a and this is b, this is your function and you are looking at this area below the graph of the function, that is your integral. So, this is the right hand side. So, this is the right hand side. What is the left hand side? It says there is a point c in between. There is a point somewhere here, c. So, if you look at this height, that height is f of c. So, this is a rectangle. So, look at the rectangle. So, area of that rectangle is same as the area that is the area below the graph of the function. So, there is a point c in between in the interval a, b. Say that if you look at that height and the area of the rectangle with that height, that is equal to the integral of the function. So, this is the reason why it is called, this is called mean value theorem for integrals. So, this is called mean value theorem for integrals. Why mean value? You can also interpret this. So, this the claim star, if you interpret it as f of c is equal to 1 over b minus a integral a to b f x dx. Then what does geometrically right hand side represent? Do you think the right hand side looks like an average of the function? f of x is a value at a point in a, b. Sum up the values. So, summation is integral a to b divided by the total number of values that is the length of the interval. So, right hand side can be called as the average value of the function on the interval a, b. Right hand side can be called as the average value. So, the theorem says if f is continuous, the average value is attained at some point in between. The average value is attained at a point c in between a to b. So, that is another way of saying. So, this is we can call it as the average value.  If I look at this number average value, this is less than or equal to capital M times b minus a. Because what is that? That is, you take the partition with n points a, b only. m is the value. So, supremum capital M on the interval a, b. So, m times the length. There is the upper sum with respect to the trivial partition and this is b minus a. We have taken it down. So, let us do not write it there and less than or equal to small m. Is that okay? m times b minus a is less than the integral upper sum, lower sum is less than the integral less than the upper sum. The interesting thing is, if I look at this value, this is a number which is caught between the smallest value and the largest value of the function. So, what does intermediate value property say? This must be attained at some point. That is all. So, by intermediate value property, there exist a c belonging to a, b such that f of c is equal to 1 over b minus a a to b f x dx. So, that implies whatever we want to say that f of c times b minus a is equal to integral a to b f x dx. So, there is a point inside the interval in the open interval a, b. While writing the theorem, I should have said in the open interval a to b. You can specify. That is what intermediate value property says. If there is a value alpha, there is a value beta and something in between, then that must be attained at a point in between. So, that is intermediate value, mean value theorem for integrals. We will see an application of this in fundamental theorem of calculus part 2, which says the following that let f be continuous. Let us say f is a function which is continuous. See, actually, what we are trying to do is find out an anti-derivative for the function. What could be an anti-derivative for a continuous function?  It says the relation between f and f. If I know a small f, I can define what is capital F from this. Let us write that straight away. That equation, let f of x be defined as integral a to x f t d t. x belonging to a, b. For every point x in a, b, let us, because f is given to be continuous, so it is integrable. This function capital F is defined, then f is defined and f dash of x is equal to f of x. So, this is the function. So, this theorem says every continuous function will have an anti-derivative. As soon as you are able to recognize the anti-derivative, you can get back the integral. So, our class of all continuous functions have anti-derivatives and for such functions, computation of integral is straight forward by fundamental theorem of calculus. Let us prove that. This is the case. Proof. I have to find a derivative. Let us take a point x belonging to a, b. How do I compute the derivative of capital F? What is the definition? f of x plus h minus f of x divided by h, limit h going to 0.  So, what is that quantity equal to? That will be equal to integral a to x plus h minus integral from a to x. I have not stated those properties of integral, namely that the integral is additive over the intervals u. So, integral from a to c plus integral from c to b is same as integral from a to b. So, I should have stated that first, but anyway, let us use that for the time being. Namely, this is integral from x to x plus h ft dt. This is integral a to x plus h minus the integral from x plus h and this is a. So, integral from a to this minus the integral from a to x. So, what is left is integral from x to x plus h. So, that is what we are saying. So, f of x plus h is integral a to x plus h minus f of x that is integral from a to x. So, what is left is integral from x to x plus h. So, let us continue with that. So, f of x plus h minus f of x is equal to that quantity.  So, it is continuous in the interval x to x plus h also and just now we proved the mean value theorem. So, there must be a point in between x and x plus h. Say that this integral is equal to f at that point into the length. Length is h. So, let us write before this. Let us just now. By mean value theorem, there is a point. It depends on h belonging to x to x plus h. Here I am taking h as positive. Just for illustration, h could be positive or negative. x plus h could be on the left side or on the right side. It does not matter actually. So, c h such that integral x to x plus h f t dt is equal to h times f at c h. So, put this value in this. So, we call that as 1. Call this as 2. 1 and 2 imply f of x plus h minus f of x is equal to h times f of c h. So, that implies that the ratio f of x divided by h is equal to f of c h. To compute the derivative, we should take the limit as h goes to 0. So, take the limit. So, implies limit h going to 0 of f of x plus h minus f of x divided by h is equal to limit h going to 0 f of c h. And what is that equal to? c h is a point between the interval x to x plus h. h goes to 0. f is a continuous function. So, c of h will go to x. So, f of c plus h must go to f of x by continuity of the function f. So, that is equal to f of x because of the continuity of the function f. So, that proves. So, implies f dash of x exists and f dash of x is equal to f of x. I just said this h could be positive or negative. If x is the interior point, if x is the end point, then what will happen? You only have on the right hand side. So, you only have the right hand derivative. So, taking care of that, those are minor things. So, you can take x in the open interval, then plus minus does not matter. And at the end points, you can just have one sided limits. So, you can prove for everything. So, derivative at the left end point means it is the right end derivative. And derivative at the right end point means it is only the left end derivative possible. So, every continuous function has an anti-derivative and for every function which has anti-derivative integral can be computed. So, that is what is the importance of fundamental theorem of calculus. So, let me give you one application of this fundamental theorem of calculus, which is quite useful or of importance rather. So, let me guess application. Of course, it is of great importance. This theorem allows you to compute integrals without going into the limit operations. And it has implications in historically in Fourier series and so on. But let us look at a simple application, which we will all without knowing Fourier series, you will understand. I think we start using a function called the log function from our school times. I think probably send it is 8, 9 or somewhere, a log function. What is the definition of log function? We do not know. How is log defined? For us, log is defined by the log tables. Or another example is what is a trigonometric function, sin theta. We define sin theta as the height over base. But if you want to define sin function as height over base and you want to prove that function is continuous, how do you prove it is continuous? How do you prove that function is differentiable? Because we start using these facts that sin x cos x, all are defined functions, well defined, periodic and so on. Sin is continuous. In fact, differentiable derivative of sin is cos. So, if you take the definition of those geometric definitions, you do not get those properties easily. You have to assume those properties. Also, the log function is a function which has some nice properties. What are the properties of the log function? Normally, we have log of, what is log of a, b? Log of a plus log of b and then we have log of say a raise to power n is equal to n times log a log of a by b. Whenever we define, we say it is log of a minus log of b and so on. Is there any such function with these properties? Does such a function exist at all or not? Not only that, when we come to slightly higher classes, we say log function is differentiable and log derivative is 1 over x. We start using that also without any proof. The reason is that these functions are not easy to define. For example, you will look at the polynomial functions. Constant function is a constant polynomial. Function fx equal to x, x square x cube. They are sums, linear combinations. All are polynomial functions and you sort of feel comfortable with them. You can define them rigorously. You can define a rational function p by q, where p is a polynomial, q is a polynomial. But there are functions which cannot be obtained from polynomial functions in any way by any algebra. You have to go to the concept of limits. That is the reason these cannot be defined. For example, you must say that I can define sin x where the series kind of a thing. But that is an infinite operation. It does not stop somewhere. There is a limit involved in it. We will come to series later on. Let me just try to illustrate how a fundamental theorem of calculus can be used to define what is called the log function. We will not go into all the details of this, but we will at least initiate so that you can try to prove it yourself. It is reasonably easy. The clue lies in this thing. The derivative of log is 1 over x. If I integrate 1 over x, I should be able to get back the log function by fundamental theorem of calculus. One defines ln of x to be equal to integral. What is the value of a log of 1? That is taken as 0. Let us start as from 0 and go to x, 1 over t dt for x bigger than 0. If it is less than 0, what shall I do? I can write x to 0. So, x to 0, but with the negative sign, log of 1 is 0. What is log of a negative quantity? 1 is less than 1. What do you think of log of half? It is minus log of 2. That is why this negative sign is coming, x to 0, 1 over t dt. This automatically says log of 1 is 0. I will not go into all the details, but let me just say what is obvious from this? You are right. Bigger than or equal to 1, less than or equal to 1. Of my oversight, log of 1 is 0. For negative quantity, log is not defined. It is only defined when x is bigger than. So, this is a function defined. ln is a function defined and 0 to infinity taking values in R. It is a function defined only for positive numbers. Log is defined only for positive. greater than, less than 1. x is less than 1. What correction she has said? I should have applied it here also. log of 1, you want 0. So, 1 to x integral, that is, when x is bigger than 0, when it is less than 0, you can write 1 to x, but normally you write lower limit to upper limit. So, with a negative sign. So, that is what. So, this is a function defined for all positive real numbers. What is obvious for this? 1 over t is a continuous function. 1 over t is a continuous function. So, ln of x is defined. Continuous function is integrable. So, ln x is defined. One consequence. Immediately it is defined. And what is the derivative? What is the derivative of ln x? Fundamental theorem of calculus. Again, derivative of ln x is 1 over x. That gives you the derivative also. It is differentiable and derivative is 1 over x. Is that okay? 1 over x, 1 to x, derivative is, that is fundamental. Anti-derivative. Just now we said that f of x was the integral a to x. So, it is 1 to x. So, derivative is 1 over x. If the derivative of, so it is a differentiable function, derivative is 1 over x, x is bigger than 0. The derivative is everywhere positive. 1 over x, derivative x bigger than 0. So, ln x is a function with positive derivative. Go back. I know something about the derivative. What does it say about the function? If the derivative is strictly bigger than 0, the function should be monotonically increasing. So, ln of x defined this way is a strictly monotonically increasing function in 0 to infinity. And derivative is 1 over x. So, what is the second derivative? If you like, 1 over t, derivative 1 over minus, what is the derivative? Minus 1 over x square. X is bigger than 0. Anyway, x square derivative is everywhere negative. So, it is a concave down function. See, all of your calculus, all those results that we proved, derivative helps you. So, it is a continuous function. It is a differentiable function. It is concave down. At 1, it crosses the x axis.  What happens to ln of x? As x goes to infinity, one can use this itself, 1 over t to show that as x increases, the area keeps on increasing. It goes to infinity. So, it keeps on increasing. And one can also prove those other properties that ln of 1 over x. They are very nice actually. One should try to read the proof that ln of, even all those properties, ln of a, b is equal to all these properties, can very easily be proved for this definition and give beautiful applications of calculus basically. So, one observes that the graph of this function ln of 1 is 0. So, the graph should look like this. It never meets the horizontal and never there is no asymptotes of, this is asymptote, but there is no horizontal asymptote. It keeps on increasing.  So, as a consequence of this, ln x is a bijective function from 0 to infinity to r, which is differentiable. So, it should have a inverse function. And what is that inverse function? It is the familiar exponential function e raise to power x. So, that is a precise definition of what is the exponential function. It comes from the definition of ln of x. So, exponential is the inverse of the bijective function ln of x. And one can use to show that, because this function is, ln x is a differentiable function, inverse function also is differentiable and derivative is 1 over the derivative of the original function. So, you get exponential is a function which is differentiable with derivative itself. Derivative of e raise to the power x is e raise to the power x. All these come because of this fundamental theorem of calculus. So, we call this as ln of natural base e. Then, you can define ln of other bases and so on. So, that is definition of what is called transcendental functions and one of the transcendental functions is log and the exponential function. Same way you can try to define what is trigonometric function sin. We can try to apply the same kind of trig. What is the derivative of sin? Cos, but I do not know cos. Derivative of cos is sin. So, I am stuck. But what you do is to define sin, it is good enough to define sin inverse. I can define sin inverse in an appropriate interval. So, minus pi by 2 to pi by 2. There sin is sin of minus. It goes the value is a one-one function. So, if I can define sin inverse, I can define what is sin as inverse of like here we are doing. Exponential is the inverse of log. So, sin inverse, what is the derivative of sin inverse? 1 over 1 minus x square. So, that probably we can integrate. So, one tries to integrate that to get sin inverse and show it as all the nice properties and the inverse of that is taken as the sin derivative of sin is taken as cos and all these properties are proved. So, that is the definition. The way one uses rigorously fundamental theorem of calculus to define functions which cannot be defined algebraically. So, I am just giving a glimpse. If you feel interested, you should try to read some book. If you want to know a reference, come and ask me. I will tell you a reference where you can read these things. So, basically, we have tried to define integral. So, let me just go back and look at, we looked at integration via upper sums and lower sums. There is another way of defining this integral. So, basically, what we did was, intuitively, we tried to capture the area below the graph of the function in lower and upper sums and we had to have the condition that if you want to define upper sum, you want to define lower sum, then your function must be a bounded function. Otherwise, infimum and supremum does not make sense at all.


What's hot in risk science at the moment? Here are just five areas that will be on my personal radar this coming year one big data We've got really good at generating data mind numbing the huge amounts of it So much of it in fact that we're having to invent new ways of storing and handling it But assuming we don't drown in the coming data deluge Exciting new ways of working with massive amounts of interconnected data are beginning to revolutionize how we understand predict and manage risk to see health Okay, I made that one up. It stands for cloud health think eHealth, but way cooler ubiquitous sensors smart phones cloud computing and other technologies are coming together to transform how we collect and use information in ways that can reduce risk and increase well-being three responsible innovation Instead of inventing something and then spending the following decades trying to fix the unintended consequences Why not try to avoid future risks up front as well as being good for people's health in the environment? It also makes perfect business sense, which is why the science of risk is being used increasingly in the development of everything from advanced biotechnology to 3d printing for Hedology, okay another made a word this time from author Terry Pratchett Unfortunately, all the physics and chemistry and biology of risk is worthless without the Hedology of risk Understanding why people do what they do and using it to make the world a safer place Another way of putting it risk science without social science is dumb science five complex interactions The more we solve the obvious problems in risk science that man eats arsenic and dies kind of problems The more we realize that we've just scratched the surface of what affects our health and how Some of the most intriguing areas emerging in risk science involve the complex ways our bodies and our environment interact with each other And the ways these interactions affect our health So that's the risk bites roundup of cool stuff to look out for in 2013 don't forget to let me know of all the things I missed and have a safe and happy new year


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