Hi. Hi, everyone. Thank you for being here. It's my great pleasure and privilege to introduce today's speakers. Elizabeth Alcesser. Liz Alcesser has been since 2012 an assistant professor in the Media School at Indiana University, Bloomington, as well as an affiliate faculty member in the Department of Gender and Women's Studies and the Cultural Studies Program. However, she will be starting a position in the Department of Media Studies at the University of Virginia very, very shortly. Liz works at the intersection of cultural media studies, media studies and disability studies. Her research and teaching interests include media history, access and literacy as well as social media, participatory culture, celebrity and performance of the self. She is the author of Restricted Access, Media Disability and the Politics of Participation from NYU Press last year, and co-editor with Bill Kirkpatrick of Disability Media Studies, which is forthcoming from NYU. Meryl Alper is an assistant professor of Communication Studies at Northeastern University and a faculty associate here at the Berkman Klein Center. And prior to joining the faculty of Northeastern, she earned her doctorate and master's degrees from the Annenberg School of Communication and Journalism at the University of Southern California. Meryl has worked for over a decade in children's media industry. As an undergraduate at Northwestern, she was the lab assistant manager in the NSF-funded Children's Digital Media Center, slash Digital Kids Lab. She interned with the Education and Research Department at Sesame Workshop in New York. Maybe you've heard of it. In post-graduation, she worked in Los Angeles as a research manager for Nick Jr., conducting formative research for the Emmy-nominated educational preschool television series Nihao Kai-Yin and the Fresh Beat Band. Meryl is the author of Digital Youth with Disabilities, MIT Press 2014, and Giving Voice, available behind you, Mobile Communication, Disability and Inequality, MIT Press this year. Today, I've also seen her writing in The Guardian, The Atlantic, Motherboard, and Wired. Ryan Boudish is a senior researcher at the Berkman Klein Center. Ryan joined the Berkman Klein Center in 2011 as a fellow and the project director of Herdikt. In his time here, Ryan has contributed policy and legal analysis to a number of projects and reports, and he's led several significant initiatives relating to internet censorship, corporate transparency about government surveillance, and multi-stakeholder governance mechanisms. I should also say Meryl and Liz have each published outstanding books in the past year, they're in the center of my field at least, and while Giving Voice by Meryl and Restricted Access by Liz offer rigorous analyses of sort of lives lived with disability in the 21st century, they're also offering very fundamental reconsiderations of what it means to study media and communication and technology, and both books are totally worth your time, and it's a great privilege to have you all here today. So, I'm going to hand it over to Meryl and we'll start today's event. Great. Awesome. So, Liz and I will be playing off one another a little bit in the sense that each of our books focuses particularly on a key term, mine, voice, and Liz's access, and so, as you might have read in the introduction to this event on the event site, can we talk, we think is a really evocative question, one that will pull and threads through each of our discussions, but thinking that it pulls upon ability, collective notions, and kind of actions of what it means to participate. So, my presentation is can we talk about voice? So, in my work, just to kind of also pull together some of what Dylan so graciously said, I study the social implications of communication technology with a focus on the role of digital and mobile media in the lives of young people, but particularly young people with developmental disabilities, so that's in particular autistic youth and young people with significant communication impairments, particularly related to something called childhood apraxia of speech, which is basically when the brain has difficulty coordinating the body parts that are needed to talk. So, I think about communication across different levels. So, some of these young people, instead of talking in what you might think of as the traditional sense, use something that's sort of like what Stephen Hawking who's pictured in the above picture here, what he uses, but instead, nowadays, instead of having to necessarily use a device that is bigger, more expensive, breaks and takes a long time to replace, you could potentially use what I've pictured on the bottom here is an iPad with this one app called Proloquo2Go, and I will point here to the screen. You can select text and icons and it'll fill in this top white bar and you can press the bar and speech will be output, and the language, the software is a little less sophisticated than what can be created in a bigger computer than that, but it can do a lot of work. So, for those unfamiliar, some of these technologies are called, sometimes they're called voice output communication aids, speech generating devices or augmentative and alternative communication devices, which ironically is a mouthful to say. So, I'm just going to say AAC for shorthand in this talk. So, because the users of these technologies, like I mentioned, don't talk in the traditional sense, and because they use speech generating devices to communicate, the popular press has historically referred to these types of technologies in a way in which the users of them get figured as voiceless. So, the top headline says, it's from the Los Angeles Times, it says, electronic help for the handicapped, the voiceless break their silence. That's a headline about a technology called the Canon Communicator. So, Canon, the company, you might think of as cameras, produced a device that was specifically focused on voice output or sorry, electronic voice generation. So, voiceless here, 2012, pretty similar headline. This is about the iPad giving voice to kids with autism. So, but the question I'm really interested in is, what does it mean for technology to give voice to the voiceless? And who does that phrase actually help or hurt in the process? So, to answer that question, I'm going to discuss three things. I'm going to talk about first the broader significance of this phrase, giving voice to the voiceless. It's a phrase you might have heard but not necessarily taken a critical angle towards. Why it's an important concept to critique, especially for people with disabilities. And third, how thinking differently about voice and voicelessness in this way, I think can more broadly create meaningful change around technology and ethical considerations more broadly. Speaking of ethics. So, before I go much further, I also want to make clear that I do not personally identify as having a disability. I'm also a white, cis, straight, upper middle class woman. So I'm sensitive to the power inherent in interpreting and sharing the experiences of others through my analytic lens. But I also believe that disability is at the heart of the human experience. And I think this picture here, some gets at that. So it's a picture taken by Tom Olin at an ADA March in the early 90s. So people of various racial backgrounds, people with various physical and whatnot disabilities marching under a banner of Martin Luther King Jr's quote, injustice anywhere is a threat to justice everywhere. So I think that there is something that's really brought out in this picture is that despite structures that systematically isolate and remove people with disabilities from the center of society, we have to think about the ways in which, how we define what it is to be human. And then even within that, I would say because there's the MLK quote here, about the intersections of disability with other kinds of identities and other potentialities for marginalization as well. So with that being said, what does it mean to give voice to the voiceless? What does giving voice mean? So we might locate its origins biblically. So in the new international version, many different biblical versions, Proverbs 31.8 says, speak up for those who cannot speak for themselves, for the rights of all who are destitute. So not only do you get illusions about voice and speaking, there's also a class dimension to this as well. We might locate in terms of how this is traced through different professional groups, different actors in the public sphere, journalists. So this is a screenshot of the Society of Professional Journalists, their code of ethics. And one line from this is that journalists, a key journalistic duty, is to be vigilant and courageous about holding those with power accountable. Give voice to the voiceless. Moving from just sort of actors to also thinking about other kinds of technologies, we can think about an endless list of things, whether it's civic media, Twitter, or open data as pictured here, as sort of giving voice. This is from the Open Data Institute Summit 2015. Speaker's talk is citizen empowerment giving a voice to the voiceless. All too often though, we consider this background. Disability becomes instrumental for another purpose outside of just disability focused issues. Tends to represent something broken for technology to repair. So consider this is Microsoft Super Bowl commercial from 2014. So long after Apple had its big Super Bowl commercial in the 80s, took until 2014 to Microsoft to have its entry point. And disability is front and center here. It features NFL player Steve Gleason who lost the ability to produce oral speech due to ALS. And the ad proclaims that the Microsoft Surface Pro, which is pictured here, has given voice to the voiceless. And this gets exemplified by Gleason himself providing the voice over for the commercial. So we can stay and I would encourage you, we don't have the time for me to play the commercial, but I encourage you to take a look at it in its entirety. But we can say then that giving voice to the voiceless means a couple of things. It means that voice is used as a metaphor for agency and self representation. That voicelessness is imagined as a stable and natural category. So the voiceless is a thing that we can locate and as a sort of immutable thing. And technology then gets figured as this sort of direct opportunity, this frictionless opportunity for expression. So there's a lot to critique about each of those kinds of claims. But why do I think it's particularly important to do so? Particularly at this moment in time. So that's because based on the ethnographic research that I conducted, despite these widespread claims to give voice to the voiceless, communication technologies that are intended to universally empower are still subject to disempowering structural inequalities. And especially for people with disabilities. So in my book, Giving Voice, pictured here and back there, I argue that efforts to better include disabled individuals within society through primarily technological interventions. When all we do is fetishize and focus on the technology for whatever kind of commercial or affective reasons, we miss the opportunity to take into account all the other ways in which culture, law, policy, and even the design of these technologies themselves can marginalize and exclude. So the book is based on a 16 month ethnographic study that I conducted of young people who use the iPad and that pro-loquo to go app kids about three to 13, and I spent some time observing them, getting trained at how to use the technologies at home with speech pathologists. I follow them to different user groups that young people would use to talk to one another and I went to parent conferences. I also started to interview different kinds of assistive technology administrators that were in the local Southern California area and lots of variations across kind of better, more resourced and less resourced school districts, larger and smaller ones to get a sense of what were the other kinds of systems that were shaping the adoption use or potentially non-use of these technologies. So in terms of culture, and I'm just gonna give you three examples kind of quickly, most speech generating devices are in English. The ones that are given to kids in US schools at home, that is not something that everybody uses to speak. You automatically can create a disconnect there between what a home culture is and what a school culture is. So one specialist that I talked to said there are hundreds of languages in these schools, one of the kids I work with at home, his parents speak Korean. Any kind of communication system, they wouldn't use it because they don't speak it, it's a big issue. We are stuck just doing school base which is fine, that's our job, but it's hard. It's hard to support them across the board because we can't. So we could say that your voice is given, but then it's also simultaneously muted. With respect to law, assistive technologies are also quite bluntly, born of a world in which half of the people who die at the hands of police have a disability. There's a 2016 report from the Ruderman Family Foundation, if you wanna take a greater look at that. But this is something that Danny's dad Peter tapped into when he talked about a fear of how a police officer might mistake his son reaching for his communication device as reaching for a weapon. So he said I need him to be able to gesture yes and no. If a cop's asking him questions and got a gun on him, no cop in the world's going to allow him to grab a talker. So this awareness of the limits of any given piece of technology in a particular context around justice and injustice was something that participants were keenly aware of that is not necessarily reflected in this broader discourse. So giving voice can also run the risk of being silenced and quite literally permanently. And lastly, all of this has to be understood in a larger policy backdrop. So school district policies what I found is tend to promote their financial investments, protecting those more so than promoting students continued growth. This is something that Moira's mom Vanessa relate in a story. So in Southern California kids have been throwing the iPads apparently into pools, this is what the mom was told. And because of that the school decided that they were not gonna allow the kids to take those iPads off campus. Even though they were federally mandated to provide the child a way in which to communicate with others. So we're bounding that within the school. And the ability to challenge that is completely shaped by one's access to other kinds of resources, financial aid, legal assistance, social capital. So Vanessa had said to me the school district changed their policy. Said that iPads only remained on campus which was in violation of Moira's IEP. I wrote them, she even wrote them, she said this is in violation. I'm asking that you give me a window of opportunity to purchase her a device for the home. One morning I was like, I don't wanna send this iPad to school. I of course gave it to her and it didn't come home. So we could say here also voice, yes, voice is given, but then it's taken away. So how does this kind of one particular kind of case get at some of these larger frameworks with which we understand technology and ethics? So my overall takeaway is that we should keep voices attached to people. So I'm drawing here on, there's a historian, Catherine Ott. She's at the Smithsonian, written an introduction to this book. It's a picture of the cover of that book. It's called Artificial Parts, Practical Lives, Modern History of Prosthetics. And she writes, focus on the materiality of the body. Not only or exclusively it's abstract and metaphoric meanings. Keeping prostheses attached to people limits the kinds of claims and interpretive leaps a writer can make. So I think as well, staying very close to the body. Staying very close to the material and embodied aspects of voice. The only way for us to understand the uses and abuses of voice in relation to other kinds of inequalities and injustices. I will just go through two applications of this in terms of what I use with my students to talk about politics in two ways. Politics and sort of big P politics, so like electoral politics. And little P politics, which is power. And it's sort of its various manifestations. And those two things are related to one another, but it's a simple way to kind of split it up. Trigger warning, there is a picture of Donald Trump on the next slide. I'm just letting you know. So with big P politics, we need to keep voices attached to citizens and our democracy. Despite Donald Trump's demagogic insistence, that he literally is our voice. This is New York Times July 22nd 2016 front page of NewYorkTimes.com. This is right after Trump's acceptance speech at the RNC convention. Trump pledges, the headline is a picture of Trump sort of smiling in a very large close up version of him smiling in the background projected on the screen. It says, Trump pledges order and says, I am your voice. Let's think about that though in relation to ways that people with disabilities potentially have some quibbles with that. So this is a screenshot from CNN's projection of at the DNC self-advocate, disabled self-advocate Anastasia Simoza directly responding to Trump's call. Saying, Donald Trump doesn't hear me, doesn't see me, and he definitely doesn't speak for me. So this pulling through of ways in which voice is getting used and abused in particular ways, it is not something that people with disabilities are, they are the ones we need to look to and draw upon sort of histories of resources in which to grapple with the uses of language in ways that more often exclude than include. On a technological aspect, nowadays a lot of interest in voice activated technology, so Suri, Alexa, and in some ways those can be really accessible. Those can add if you have motor limitations, other ways to access. But we have to think about what kinds of voices get picked up. This is just a headline that says, voice is the next big platform. But then here's another headline from Scientific American, why Siri won't listen to millions of people with disabilities. There are particular ways in which voices are recognized or not recognized. Let alone just the kinds of voices that can be produced by a given piece of technology. So ideas about the normal here and what it means to have a voice are more critical considerations. So to wrap up, technologies that give a voice to the voiceless can also reproduce structural inequalities. Having a voice and being heard are not necessarily the same things at all. And they're also not just about technology. But social, cultural, and economic resources. And having access to which is unevenly distributed. And as my book, it centers the iPad, but it's interesting because I am really interested in what some people might call an edge case or a sort of outside case. But I really believe that there is something to think about marginalization and participation that is really actually super central to what we're all trying to get at in terms of understanding what it means to participate. So we need to keep voices materially attached to people and how we build our technology or else the risk is tantamount to dismantling. Or if we can say that the structure of democracy has been stable to begin with, also an open question. But at stake is really not only which voices get to speak, but who's thought to have a voice to speak with in the first place. That's my talk. Let's turn it over now to Liz. Yeah. Yeah. So thank you for having me here today. I'm happy to, thank you, have a chance to talk about this work in conjunction with Merrill's work because we've been batting around some of the same ideas regarding access, voice, participation, and technology and disability. I've been framing my work as essentially cultural studies of technology, attempting to understand how technologies reflect and reproduce particular dynamics of power. And how users of technologies can push back upon those constructions and challenge these sort of received ways in which technologies are developed along certain assumptions.  So to start off here, we have some images reflecting a sort of pervasive utopianism in talking about the internet, worldwide web, and related technologies. At the top right is an image from MCI's 1997 Anthem Commercial. This young person appears speaking in American Sign Language right before text that reads there are no infirmities. The time 2006 person of the year was you with a big reflective cover. And then this bottom photo is a screenshot from a Yahoo advertisement in 2009 called It's You, prioritizing this kind of individual empowerment and excitement around new technologies. At various points, these technologies have been understood as democratizing, globalizing, something that can eradicate racial, gender, and disability difference. And something that can open economic and social opportunities. From the hype of cyberspace to the celebrations of web 2.0, we see stories of technology are often stories of endless possibility. In restricted access, I'm attempting to intervene in some of these celebrations by investigating digital media accessibility, the processes by which digital media is made usable by people with disabilities, and arguing for the necessity of conceptualizing access in a way that will be more variable and open opportunity in new ways. So after all, I argue that if digital media only open up these opportunities to people who are already relatively privileged in terms of their ability to access technology, then their progressive potential remains unrealized. If not simply transformed into a means of upholding those very inequalities. Now what is media accessibility, web accessibility? This is something that I often illustrate with this slide, which is just a screenshot of a homepage of the New York Times as run through the web accessibility and minds online accessibility checker. This is an automatic software tool that will check the HTML and associated code of a web page and flag with little red or yellow icons where there might be a problem. So in this case, the page is being flagged for not describing the image that reads New York Times, for not describing the small images, and for having some incorrect form usage. Now accessibility is a fascinating case because it is a very granular process. Essentially, web content accessibility comes out of non-governmental policy sources, such as the World Wide Web Consortium. It has also been taken up in various legal contexts. So there are laws in the United States that require accessibility in some contexts. And there are arguments that the ADA requires web accessibility in many contexts. However, these policies are written in such a way as to facilitate the use of consumer technologies with the kinds of adaptive and assistive technologies that Merrill gestured towards. Things like screen readers, alternative input devices like tongue typers, joysticks. These technologies are often key in allowing people with disabilities to use technology. And accessibility ensures that software will work with those technologies. However, accessibility generally has to be implemented by individual companies, developers, website operators, and is therefore a highly distributed phenomenon. There's no automatic way of understanding where this happens. Thus, a lot of my research has involved tracking digital media accessibility through the policy makers, people working with the World Wide Web Consortium, people working in government, in academic contexts, as well as with developers, consultants. Sometimes marketing departments are in charge of accessibility. Internal standards, a lot of major corporations have their own accessibility standards that are different from what we see in the public sphere. And so in these terms, accessibility may be understood as highly bureaucratic and technical. It creates a kind of baseline from which there's the possibility that people with disabilities may then access and use digital media. In thinking about accessibility, however, it's important to think about the terminology because accessibility like access is an often used term that is not always attached to these kinds of specialized meanings. We often see accessibility invoked to refer to new possibilities. The graphical user interface made desktop computing more accessible to a large number of people, even as it very much shut down access for people who were visually impaired. So we see access deployed in various contexts. And additionally, access to media and information technologies has been addressed in a wide range of academic literatures. From digital divides work to work on public broadcasting, community television, media literacy, and media policy work. But in all of these areas, access is dominantly figured as something that is had. Do you have access? A sort of unitary and universally desired endpoint. Do you have access? It is good to have access. And in addition to this sort of positive and linear framing, the concept of access is often deployed in such a way as to stand in for availability. You have access to the telephone lines if they connect your house, even if you don't, say, have a telephone. Affordability, this is a subsidized service in some way, therefore it is more accessible. Or consumer choice. You have access to 590 cable channels, whether you want them all or not. So access is a flexible term. But when we center disability and accessibility in their specialized senses, the gaps in some of these literatures and usages emerge. In fact, it seems that access is inherently variable, dependent upon bodies, contexts, and a host of other factors. When we say, check Facebook. We're potentially engaged in a wide range of technological and social practices that vary from person to person. As argued by Canadian disability scholar Tanya Tichkowski, quote, every single instance of life can be regarded as tied to access. To do anything is to have some form of access. Thus, rather than think of access as a binary or a linear progression, disability studies encourages us to conceive of it as a continually, relationally produced means of engaging with the world. So we don't have access, we are doing access. Now, in restricted access, I use this sort of a jumping off point for thinking about how then can we study access as an infinitely variable, complicated phenomenon, right? This is starting to sound impossible if every construction of access is different. And thus, I've been using the metaphor of sort of an access kit illustrated here with a sewing kit with a pair of scissors, some safety pins, needles, a thimble, other things you use for sewing. I'm not a sewer. However, I use this metaphor because I like the idea of a kit in that you can use it all together to do what it's intended for. You can use this to sew. Or you can take pieces and parts and use them differently. You might use the scissors to cut up something in your kitchen. You might use the safety pins to make a punk t-shirt or signal your safety in a post-Donald Trump world. You may recombine these in different ways. And thus, in sort of figuring the access kit, I thought about what are some categories of questions? Ways we can dig into access that will allow us to look, perhaps, through some different lenses at how that access is being created. And I'm not going to go into detail here, except to say that I sort of loosely grouped these into categories of regulation, use, form, content, and experience, which I can talk about later. And together, they encourage us to think about access as a relational phenomenon, drawing attention to what a cultural studies perspective might call the articulations of bodies, technologies, institutions, geographies, and social identities. So access is not one thing but many, not an end point, but also not a beginning. Nico Carpentier has referred to access as a precondition for participation. Before we can participate, we must access. But through the study of digital media accessibility for disability, it's become evident to me that the production of access is an ongoing part of participation in a digitally mediated society. Now, one of my favorite examples in the book is the case of Tumblr. As some of you probably know, Tumblr is a multimedia micro-blogging platform that is characterized by the sharing or re-blogging of posts across the network, the formation of interest groups, and a lesser emphasis on individual identity display than many social networks. It is however populated by user-generated content and thus not obviously bound by the legal and technical requirements faced in government, educational, or e-commerce spaces. Perhaps as a result, Tumblr is formally inaccessible. It is difficult to add alternate text to images even if you wanted to and knew how. It features infinite scroll, which can be a challenge for many assistive technologies, and it uses very limited markup features to indicate importance. Additionally, the content is highly variable and often animated, adding additional challenges from an accessibility perspective. So from a sort of top-down perspective, the inaccessibility of Tumblr seems like a problem. However, in my work, I've tried to couple the institutional perspective with a more on-the-ground user perspective. I did roughly 25 interviews with disabled users about how they used these technologies and why and what was frustrating. And in these interviews, I've got on the one hand people telling me that they contacted Tumblr and talked about the accessibility policies and were just totally rebuffed. Tumblr was not interested in talking to them, did not change anything. However, they also pointed to group pages such as accessibility fail and accessibility as other places where they were in fact finding community and using this platform. In some of these cases, users were adopting and adapting Tumblr, sharing experiences of microaggressions, sharing accessibility knowledge, teaching each other workarounds by which to make a site more accessible. Furthermore, this kind of grassroots accessibility revealed some different meanings of access and the values associated with it. While accessibility is often thought of as a matter of law, policy, or technology, or the provision of services in a kind of charity model, many users were much more likely to talk about it in terms of effective and cultural dimensions. Many prioritized feeling welcomed rather than merely accommodated or being included as members of a community rather than as afterthoughts or having their non-technical needs met. For instance, many disabled Tumblr users praised the site because its large social justice community meant that trigger warnings were commonly used. Trigger warnings, as we saw with Donald Trump, are a brief indication of when and how content might be upsetting for someone with a particular kind of trauma and they're well beyond the scope of technological accessibility policy. However, as one interviewee told me, trigger warnings make a site accessible to me, indicating respect for the emotional and social needs that can often accompany disability. Building out of such examples, I end restricted access by talking about cultural accessibility as a means of moving towards a more accessible and just future. This moves beyond sort of technocentric notions of accessibility or accommodation and aims to highlight the inner relationships among technological and economic access, cultural representation and production, and access to community in the public sphere. Not simply universal design, cultural accessibility prioritizes the ongoing perspectives and visibility of people with disabilities and it may best be achieved through sort of participatory collaborations between users, policymakers, industries, and others. I've illustrated this concluding point with a screenshot of actress Teal Sherer who created a web series called My Gimpy Life, which she funded through Kickstarter. So already we're seeing sort of a host of contemporary digital media technologies brought to bear. And in this case, Sherer also prioritized disability community and access on-screen and off. The web series had an on-screen credit for the person who produced the closed captioning. The Kickstarter page developed over time into more of a community space than a fundraising space. And we see a range of relationships and connections forming that potentially enabled the formation of community and the movement into a larger civic and public sphere from inclusive cultural spaces. Ultimately then, I would argue that access is not simply a prerequisite to participation. Access and participation depend upon one another. Just as access enables participation, so does increased participation by diverse people make possible the expansion of access. And I will wrap it up there so we have some time. Okay, I'm going to start with one question for the three of you and then we can open it up as quickly as possible to Q&A. So it strikes me that all of our work is constantly playing catch-up with lived experience. And Ryan, I'm thinking of your work with Herdick is in some way always trying to close that gap between lived experiences of blockages or clogs or censorship online and the point at which there's greater public awareness about those blockages. And scholarship is by design sort of laggy because of the time it takes to dwell on things and the time it takes to publish things. So I wonder how each of you think about lagginess with regards to lived experience in each of your projects. And maybe we can start with Ryan to bring you into the conversation. Yeah, sure. So I'll just first preface my response by saying, you know, as Dylan mentioned in my introduction, you know, I spend my sort of work days thinking about access to technology and who controls these sort of elements of the web and the internet and our technologies. But from in my personal life as someone who wears hearing aids, I think a lot sort of in the very specific use case of how that technology enables and limits me personally in different ways. And so I found the discussion from Liz and Merrill today really interesting and important. And so on this question of lagginess, you know, one of the things that really jumps out at me and I think picks up on something that Merrill was saying was this question of, you know, technology reproducing structural inequalities and something that I think is on that point is interesting to me is that I see a lot of convergence going on in technologies that, you know, that as Merrill's example showed, you know, that people can use iPads, which are consumer technologies, to do things that, you know, earlier might have required, you know, going through a medical specialist or getting very expensive medical technologies. And, you know, in the hearing aid market, there's a lot of movement now to allow companies to sell things that aren't quite hearing aids but do essentially everything that a hearing aid could do and there's a lot of pros and cons to that approach. You know, there's the potential that it could lower the costs that a lot of people who don't get hearing aids can suddenly get hearing aids but, you know, no longer are they having it fine-tuned by a medical professional and all of that. And so as you converge sort of mainstream technology and technology that helps people with disabilities, in some ways I think that you can turn Merrill's question or prompt around and say, you know, in what ways is all technology reinforcing societal and structural inequalities? And, you know, I think Sarah Hendrin has talked about all technology is assistive technology. You know, we're not naturally born with the ability to get our emails on our wrists and, you know, and yet technology enables us to do that. And so in what ways is technology that all of us are using in assistive ways, reproducing things that maybe we should be taking a closer look at?  And so parts of the world are simply not mapped and those places don't get access. And so there's an example of where technology, taken out of the sort of disability context, but something which you could characterize at a very basic level is accessibility technology is itself potentially going to reproduce the structural inequalities, the places like the favelas in Brazil that are very heavily populated but are not mapped will not have access to these technologies. And so anyway, I'm not quite sure that answers your question about lagness, but that's at least, you know, I think that there's just some bigger questions to me about technology in general and how that's reproducing these inequalities and, you know, and I think it does sort of raise these questions of, you know, from a lagginess perspective that, you know, we have to sort of think of these things in their broader context and not just, you know, in a disability context anymore. I'll just say something very briefly because then I want to make sure that we have time for questions but just talking about lag and delay and that not, you know, whether that's a negative or a positive thing or an inevitable thing but I immediately thought of when you brought up, you know, sort of the relational, the act of access and it is a process and not just a product thinking about with speech-generating devices that it can take a while to create a message for it to be then output for somebody to say. The fluidity with which one might be able to potentially, depending on what kind of motor impairment they might or might not have but the patience that is required for a conversation partner to even if you've got a technology that works well, it's like top of the line, it's fully charged, that's a whole other thing. I can't talk about the thing that doesn't have any juice that the patience is required of somebody else to follow a pace of conversation that might not be a one that they themselves and they themselves enact or used to having with another person. So that process, that patience and that's something that is learned that is something that somebody who doesn't have a speech disability would like have to be able to become better equipped at. So think about the kinds of social and cultural and personal equipment that is needed for participation. Like that gets sort of added to the list here. Just thinking about temporality in that way.  I'm from Colombia and as we are not, I mean, as we don't have that many resources, so we have to come up with creative solutions. So the main problem in these kind of issues is the economy of scale. As the population is not big, the market is not providing solutions for them. So for example, in the case of deaf people, we create this relay center with a sign language. So a person who is deaf could connect to an app and this remote person can translate from sign language. So he can, I mean, the deaf person can present an exam or have a consultation with a doctor or any kind of communication. So this is one solution of the economy of scale. And the other one was we buy a country license for a screen reader. So I think one license is like $1,000 per person per year. But if you buy a country license, it's less than $1 per person per year or per computer per year. Because we buy thousands and thousands of licenses, so we can install a license in every internet cafe, in every school, for example. And people are not paying, actually they are not paying because it's so cheap to charge for. But for example, schools pay a little bit and we gather all this money and buy a country license, which is tremendously more cheaper than buy one by one. Yeah, I hadn't heard about country licenses. That's really fascinating, I want to know more. But I will say in terms of scale, we may think about the sort of things that Ryan brought up with mainstreaming as being one way in which mainstream technologies are taking on assistive functions which enables a different kind of scaling. When we are talking about assistive technologies that are developed as such, they're often very expensive because there's a small market and a lot of research that goes into them. When those can be deployed in consumer devices, some of those costs go down. But as I think Ryan indicated, sometimes oversight goes down as well. You don't have a medical professional adjusting the hearing aids. I've been doing some research on emergency lately and you don't really have very good connections to 9-1-1 when you're relying on an app to dial it for you. So yeah, there are ways in which that is changing. That was a good question. I just had a question about the differences between adults and kids and particularly, I think that there is often talking about voice and voiceless and many times, kids are voiceless either because they simply aren't at the emotional or intellectual place where they can talk about what's going on or because legally their parents speak for them. I know from my personal experience when I was five or six, the last thing I wanted to be doing was wearing hearing aids and I didn't want people to ask me about them and if it was my choice, I would have just taken them out, but luckily it wasn't my choice. I was wondering if you could talk about some of the differences that you guys have seen. In particular, you quoted some parents talking about their experiences up there. I'd be interested to hear about how these issues of voice and voiceless and access are maybe different or different challenges emerge when you're dealing with adults versus kids. I'll hand this to Marilyn just a second. I've worked primarily with adults and in part that's because when we are looking at disability spaces, there's a lot of attention often to K-12 education and to particularly what can be done to help children and there's often a drop-off in terms of when those children become adults and so by looking at online spaces where people with disabilities were engaging with one another and creating disability culture, I think I get an interesting perspective on what happens after that and that's less structured space, but obviously research on kids. Yeah, I think the kid focus is partly just related to my expertise in background more than anything else and even then like I, 13, tends to become my cut-off. 14 in the US, you're supposed to at least federally have a mandate to talk about transition to adulthood and that's where I sort of stop even though you can be like 30 and really be into ELMO and in my first book I talk in digital youth with disabilities talk about age appropriateness and sort of the fluidity with which sort of radical spaces can potentially be created related to interest or related to different cultural spaces like theater performances that have sort of sensory inclusivity sort of mixed age, mixed abilities of all different sort of kinds and I think that with the book a lot of the research in terms of the kids that there's the parents that are quoted I think to not overprivilege while the quotas may that I use here in the book there's a lot of descriptors of behavior and of interactions with kids and other individuals I did not have the skill to interview some of the kids in terms of their capacity to use because some of these were not using the whole point was that they didn't have a reliable access to communication and so the challenges of then doing that work outside of like triangulating different sort of behaviors in different kinds of expressions or vocalizations or you know in kinds of spaces I'd say though for like my next book project which is focused on the experiences of autistic youth growing up in the digital age and different kinds of ways in which communication happens I'm grappling with that right now in terms of the interviews that I'm doing with directly with kids the ways that I talk with them about their media practices but again the ways in which some of that is oral and some of that is not and so part of that is sometimes the challenge of presenting field work to an audience and the legibility of that as opposed to just sort of having a video or another kind of recording so that sort of gets at like our methods and the ways in which we then like make our research our evidence visible and the ways in which certain kind of visibilities can unintentionally privilege or reflect certain ways in which the research was or was not conducted but yeah Hi, I have one comment about the giving voice to the voices I really like the point about how voices is seen as a mean for agency and self presentation and I was just thinking about if you change the headline to something different instead of giving voice to the voices to something like listen to the unlistenable it will be a totally different like focus on instead of on the person who needs to be given the voices it won't be on behalf of us to train our listening capacity so that's also I don't know whether you've thought about that Yeah, so listening and speaking and the dynamics between those things or something that I talk about more in the book and that gets a little bit to there's a phrase I really really love a media justice scholar Tanya Dreyer talks about the partial promise of voice so voice is in completion the partiality of it to fully say that we have any kind of grasp or pin downableness of it because that understanding of respect of a message being acted on and a promise being kept and that's kind of partly in larger public sphere discussions but I think that that point about listening whether one is able to be listened to or not again we can think about that in terms of like a biological individual level, a social level a political, you know what the mechanisms are for feedback but also some of that can also reinforce who's in power in the first place and in what ways can that still enforce a and us them a powerless, again an essentializing idea of having and not having of giving and not giving Hi I have a comment and a question I had the great pleasure and I will say some humility about ten years ago I was teaching at Northeastern for adults and one of my students was a 74 year old blind man who lost his sight at 32 and I learned like a day in the life of someone who is disabled and I had to rearrange my entire how I was going to structure an exam because we were in a computer classroom and he had to go in a special room and if they didn't have the jaws they would have to work with the Northeastern disability office to have someone come and have a reader read the exam to him and I learned something the Massachusetts disability I say oh just go to the bookstore and get a volume 6 or version 6 of the book for the class and the one they had for the Braille was like version 3 you know things like things that we just take for granted it's just very humbling another time I was at an event there was a company I will not play it up but it was 3 play media they had an event at the faculty club where they were talking and saying that many times when they have events here or classes they have closed captioning and they said many times foreign students to help them learn English are using it so that's like the number one reason in addition to disability so my question here is we're in an area where we have so many startups and just like until recently a secure code is an afterthought it's like yeah we'll do it later disability for many places is like yeah yeah whatever if you know it's not an education or government is there anything that can be done to teach like the CS students that are coming out of course 6 at MIT here at Harvard the people who are before they start their careers to incorporate it into design so it's not yeah yeah for those you know they can make it as part of how you learn how to create so you will not have these incredible disparities in you know accessibility one thing I would say is to read histories of people with disabilities as actors in the history of the development of computing so the idea of that it's more like you're not adding on disability or adding like the recovery of individuals with disabilities in computing history engineering history is really central I think to that idea of not developing a sort of charity model of disability pedagogy in a field like CS and I'll just add to that I've done some work on how web accessibility was explicitly an afterthought in teaching web development for many many years in the sense that it would be the last chapter of the book right once you've learned how to do everything else maybe you'll look at this but maybe you probably won't and that's something that's borne out actually in a lot of computer studies curriculum they don't have courses on accessibility and basic lessons don't incorporate it as something that you do as part of the process the International Association of Accessibility Professionals is a young organization maybe for five years old now that's explicitly attempting to address that by making some sort of best practices for CS education and offering some certifications for people who have actual training in accessibility to use once they go out into the job market and then of course there's a whole world of universal design and design for disability and design literatures focused on how to incorporate diverse users at an early stage I was just gonna say that I'm actually somewhat optimistic in this sense right now because I think that when you look at things like wearable technologies and there's so much more focus right now in the mainstream and I think this gets back to this convergence point there's so much more focus right now on human-machine interactions and artificial intelligence and a lot of the technologies that are necessary to make wearables better to make augmented reality better to make autonomous vehicles better like the improvements that have been made over the last several years in computer vision technology like all of those things I think will help on this lagginess question that I think it's as more technology and these startups are thinking more about how machines interact with the physical world they're solving some of these problems that maybe have traditionally been the sort of afterthought problems and they're not approaching it in the mindset of how do we solve problems for people with disabilities but I think the applications are getting closer and closer so that it's not such a leap to figure out we design this thing now we have to figure out how to apply it in a whole new context but it's actually like we now have something that can identify what's going on in this room because we need it for our artificial intelligence technology and that makes it super easy to design something for someone with a visual impairment so I'm optimistic on that so just a quick comment on that last bit there is an industrial thing that's called teach access if you do a web search on this that's a consortium a number of the big companies are trying to put together curricula to distribute throughout a bunch of universities for specifically integrating it into the CS curriculum there's a lot of trouble there because a lot of the industries are trying to hire people and nobody knows anything about it and so this is actually a pull from industry to try to be able to key that up a little bit so it's something to look at so I just had a question a lot of the regulatory issues and accessibility have to do with things around either livelihoods or access to government services or these things that are really very instrumental in sort of getting things done in your life right and I'm wondering if you could speak a little bit to issues around entertainment or just sociality just interacting because as much more of our lives become mediated then the access to these things become much more critical to just our lives and I don't see as much discussion of that in a lot of the disability discussion at least I think the place where you see the most discussion of that sort of thing is in captioning particularly in the past several years as Netflix captioned its content both the activism around that and then the 21st century communication and video accessibility act right took some steps towards prioritizing that kind of access but I think it's a really interesting question to think about content and what we're gaining access to and making sure that access to video games and access to pornography are still kinds of access and people with disabilities are not less entitled to things that we think are morally dubious than our other people so there's certainly some tension there because government doesn't want to get into that if they can avoid it but I'm encouraged because I see that that's also happening in informal ways major league baseball did what's called a structured negotiation where instead of a lawsuit they worked with disabled community members to make websites and streaming baseball games more accessible and so that's something where the mandate for MLB to be accessible is not really there but through some processes of introductions and collaboration you can actually get to places where that content is being addressed but it's very much not from the W3C I'll just say there's a chapter in the book that's about centering on like the question is like what is an iPad for and that there were these real tensions around whether an iPad was for that app exclusively or whether it was also for all of the other things that any other person might use it for and a lot of things that were related to issues of our own taste related to issues of ownership the idea of whether you had to pull different of those pieces of hardware to delineate and make distinctions between which of those things were for but for me the real lightning strike and that was I was doing an observation and the speech pathologist I was with had very negative things to say about YouTube even though it was clearly this thing that the kid enjoyed that motivated them to use this app in the first place to communicate but there was lots of values about well kids in their iPads and their boobs and like a shutdown and the ways in which that particularly extra marginalized families who maybe didn't have access to or the ability to mobilize resources I want to also phrase it as that way around English language, mobilize resources around community members who had other kinds of access to other kinds of resources, social capital the cultural capital to push back against that person in any way so yes the idea especially because an iPad designed to be a consumption technology not necessarily for creation and somewhat for circulation but just thinking about the people wanting to take advantage of all of these things that can be done but some of the professional pushbacks around expertise and it's a mainstream technology but it entered the home through the teachings of somebody with professionalization certain sort of things attached to that so more of that in the book okay thanks everyone again there are oops I'll just say there are books for purchase at the back of the room and thank you so much for coming out Liz and Marilyn will be here now a round of applause for her


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So breaking news, Yannick Sinner's coach has come out overnight and said that Djokovic may only play one Grand Slam this year, which means if he's unvaccinated, he won't get access to the other slams. This is what he had to say over on Twitter. So this is what Sinner's coach had to say. Ricardo Piatty had to say over on Twitter about Djokovic's chances of playing other events. Djokovic may only play one Grand Slam this year, the French Open. You must be vaccinated to play in the UK, Wimbledon, and outside Europe. So the Australian Open and the US Open. So Djokovic, he might only be able to play the French Open because Paris don't have the same rules as what apparently the UK and the USA are going to have going into tournaments. Because remember last year at the US Open, we had players playing that were unvaccinated at the US Open, but all the fans, all the media had to be double dosed. So an Australian Open, we know what's going on here, right? Everyone has to be double dosed, including players. And it seems like they're sitting a trend at the Australian Open for the next season, for the whole season. The French Open are doing their own thing it seems, but the UK look like they're going to bring in, you know, mandated vaccines for Wimbledon this year and maybe follow the trend of the Australian Open. And same with what about Indian Wells, what about Miami, those American tournaments in March? Is Djokovic going to be able to play them if he's unvaccinated?


Another Bemidji area man has died of an apparent heroin overdose. This time, Beltrami County authorities say it is a 21-year-old cast lake man who was found unresponsive early this morning at a home located off Power Dam road east of Bemidji. Emergency workers attempted to revive the man when they arrived but were unsuccessful. Others in the home say the man had used suspected heroin earlier in the night. If you enjoyed this segment of Lakeland News, please consider making a tax-deductible contribution to Lakeland PBS.


We sort of went through the film to digital transition and file-based transition and cloud and especially, you know, camera to cloud was a natural progression of that. It actually sort of brought some definition and some some rules back into a very chaotic file-based world. Everybody had a lot of drives and they were shooting a lot of different formats and they'd be overall all over the place on drives and cloud actually brought organization back to that. So I think that was really our inspiration was how do you allow for all this flexibility but maintain some professional organization back into it. We're coming out of a 10-year cycle of an industry that's transitioning from physical film and tape to file-based. So the next logical progression is cloud-based. So the idea of camera to cloud in some quarters in our industry is kind of heresy because essentially what we're saying is you take the most valuable file, the highest resolution file and put that in the cloud. And we've been proposing that for years but it's been a non-starter because the industry is used to vaults and physical management of that precious thing from which all their revenue gets generated. And here we are saying put it in the cloud. So in the last two years in particular, the idea of that is actually socializing a lot better. So I remember when we first started saying camera to cloud you would get reactions like violent, you know, people are really upset with that idea like what the hell are you talking about? But now, especially because of the growth of OpenStack, it's not only being socialized, it's actually happening in real life and that's what the demo was about. The idea of the archive as a separate entity is slowly ending because what studios are starting to realize is that the concept of a living archive, something that starts in pre-production and production and then it carries all that media and it holds that media through distribution is actually a far better monetary model because they can have all this content available to distribute over IP anytime that anyone wants to buy it. So I think the idea of a sort of dead archive that's separate is coming to an end. I believe also that the Sony hack was actually a positive turning point and that studios realized that they didn't gain security by doing it themselves. They gained security by reaching out to communities such as OpenStack and having partnerships with companies that day in, day out do cloud, do management of cloud, do security on the cloud. So I think that was a big turning point for us in actually a positive direction. That this was inevitable and they really needed to seek out more competency than they had internally. The biggest challenge there if you even look at the Library of Congress is you have an industry that's premised on 100-year archive. So even though there might be the best technology in the world, they're thinking will that company be around in 25 years or are they going to be acquired by another company who we're going to be dealing with then. So their challenges kind of transcend technology. It's more social and political and they used to have control over it in a bonded vault and they knew exactly where it was and it transcended technological change. And there again the solution is OpenStack because you're not tied to an individual vendor. You're not tied into proprietary storage. You can run it privately. You can run it publicly. So you still have control over your archive and your cloud archive independent of another company that may not exist in 20 years. So it's standard right now for all shows, feature film and television worldwide to back up on an LTO tape. So that's how they manage their liability. What we're doing is doing that but we're also putting it in the cloud. And to me I reckon it's a heroin. Once you give them the ability to store everything and then say oh I'm going to take that away from you. They can't live anymore. So that's how we're basically getting everybody addicted is we're putting all their data in one location and that's it. Once you experience that it's kind of like having a Dropbox account for all of your camera negative. So it's been 100% acceptance because now you have access to your stuff. It's like a no brainer. But we're doing a practically show after show and once you experience it that's what changes the head. I believe you know for us looking forward into the future I mean I think we're going to ride this cloud wave for quite a long time and it'll sort of grow into other areas. Largely right now it's a storage conversation. It's an access conversation but we're already starting to see movement into compute infrastructure too and render on the cloud and all these different things that you can sort of store a mezzanine file format and distribute from that mezzanine file format and get whatever you need from that through compute services. So I think that's really the next step for us. With the software we built critique the idea is to take that from Hollywood and make it available for anyone who's doing video production. So it's basically taking those best practices this crazy thing where you know productions have to coordinate with hairdressers, set designers, DP sound and somehow all that chaos comes into a very polished program. We think that that type of behavior is going to transition to just humanity in general. Kind of like the YouTube generation of people that are going to start to behave ever more like Hollywood. So it's great to be in Hollywood but to actually transition our IP, our software and our services into the greater video world. That's what we see as our future and really the last 15 years we define as file based and we think the next 15 years and beyond are going to be cloud based. So for us it's just like a very natural wave that we're still writing.


There's so many things that argue in favor of perhaps considering Mike Vrabel as a strong candidate. The experience, the link to the area, the way I think he's a player's coach in a way that Bill maybe is not, but I also think the crafts are very methodical. Even though it's been a long time since they made a coaching switch, the things said in the offseason by Robert Kraft indicating that he felt that Gerard Mayo was the heir apparent. The things that he said with the letter to season ticket holders and the press release saying that Mayo would be the successor. I think they're very methodical in the way they do things and the fact that Mike Vrabel has kind of cannonballed into the coaching pool here, I don't know if that would necessarily cause them to alter the course of what they're planning to do. So to me, you know, we reported a month ago about that a decision had been made and the Patriots would be parting ways with Bill Belichick. That remains what I believe will happen this week in the near future at least and I believe that they will still stay on the course with Gerard Mayo as the next head coach.


So, first of all, good morning, and I hope you had a good day yesterday, so you got a chance to see how our days will go, so the structure will be, I think, very similar throughout the two weeks. We have three lectures in the morning, and then we'll have these exercises, and it's really, really, for us, it's really key that you sit down and try to do whatever you can, because this is the best way to absorb, otherwise we keep telling you things, and you don't really learn if you don't try yourself. Okay, and we are here to help. One small comment, I think the tutorial session, I don't know, I think yesterday it was wrong at some point on the schedule, it's at 2 p.m. So I think it was written 1.50, I'm not sure, maybe it wasn't, but, okay, it starts at 2, and also I had some questions. So the room is booked from 2 to 4. On the schedule it's written, this is wrong. Why coffee break, ah, maybe, ah, it's confusing. Is coffee break at 3.30? I thought coffee break is at 4. Okay, I don't know, at some point we will have food. Okay, there are two things to correct in the schedule. So first of all, the room is booked from 2 to 4. In reality, we wanted to put on the schedule from 2 to 3.30, because we don't want to force everybody to sit here for two hours. So let's say from 2 to 3.30, we hope everybody will come and try after 3.30, if you are tired, feel free to leave, maybe some of us might leave too, but if you want to keep working and try solving your problems, you can stay until 4 to work. And I thought coffee break should be at 4 every day. And I think the other days it's at 4, so I don't know if today I will check. At some point it will be coffee break, I think it should be at 4. And today in the afternoon, we will have a socializing networking event, specifically for women and for gender minorities, so whoever self-identifies with women. And again, this will be also a type, it's not on the Adriatico, I already asked for this to be corrected, but probably it will be corrected only later. And it's also in the terrace, the same place where we had the socializing yesterday with the ice breakers, we will have some small groups discussion. And again, it's like a networking socializing opportunity. And we also have some questions about gender and mathematics and some discussions together. So hope to see all the women and gender minorities, people who later. Okay, so time to start with mathematics. And first I will give you the solution. I think most of you also did it, but let me first complete the theorem. We were proving yesterday. So we saw that if you have a rotation and the angle is rational, every orbit is periodic. But if the angle is irrational, then every orbit of the rotation is dense. So again, dense means for every, I pick any other point on the circle. And I pick an epsilon, and an epsilon neighborhood is like an arc with size epsilon. And dense just means that at some point my orbit will fill, will enter this arc. And here I wrote, when I write distance, again, you can think of this distance as measuring the smallest arc length between the two points, okay? Maybe divided by two pi, so that the total distance, total circle is one. Okay, and we did two steps. We showed, first of all, if the rotation number is irrational, then the orbit never close up. So you never get two same points. And then we used pigeonhole principle to say that if I chop my arc, my circle into capital N, sorry, this is one of them. If I chop my circle into capital N equal arcs, then, and I look at N plus one points, all distinct, automatically distinct, then I will find two points which fit in the same arc. And again, here I need to divide, my arc length has to be divided by two pi. So I want to say that the arc length between these two points falling in the same arc will be, and this is RL. And this distance is less than one over N. And as many of you did, what do I have to do next? So the key observation is to remark that R alpha is an isometry, which means that if the distance between R alpha of Z and R alpha of omega is the same than the distance between omega, Z and omega, this is obvious on the picture. So if I have two points and I rotate them, the distance is the same. So this implies plus the fact that it's an isometry, implies that I can apply R alpha to the minus k, if you want. Which is also, it's invertible and it's also an isometry. And what I get is the distance between Z and R alpha to the k minus L of Z is also less than one over N, okay? And I'm assuming, for example, that k is greater than L. Otherwise, you can do the other way around.  And I can choose N such that one over N is less than epsilon. So now I have my point Z. And if I move by this multiples of the rotation, if I move by the rotation of k minus L, which is a subset of the rotation that I had before, I look at these points which I get when I apply power k minus L of my rotation. These points of the orbits will be very finely spaced because I move by a small angle, are less than epsilon space, okay? So I move by a very small angle less than epsilon. So it means that every time I apply this power of the rotation, I displace myself by some quantity which is less than epsilon. And that means that if I have a target window of size two epsilon, I'm bound to fall inside. Any two epsilon arc will be visited. And this is what I need to do for density, okay? I think many of you got this yourself. So the main thing was to use the rotation isometry property. You can maybe see it differently if you want. Also explicitly writing the formula mode one, you can see it, okay? So you can actually say more about an orbit of a rotation. Not only it will be dense in 0, 1, but I want to also state a theorem which was originally proved by Weyl. And this tells you indeed that the orbit is also equidistributed. So for every alpha irrational, and again, irrational is key. For every alpha irrational and for every point of the circle, the orbit under the rotation of that is equidistributed. And now I will give you the definition, IE. And let me say I will tell you two properties and actually sketch the second. So I can pick an arc and try to understand on my circle how often pick an arc. And maybe I will write it on 0, 1 as an interval, AB. So for every, let's say, interval AB in 0, 1. Sometimes I switch between multiplicative notation on the circle and additive notation on 0, 1. This is also 0, 1 with opposite points identified. So if I start plotting my orbit and recording how many times I visit this arc, there is no reason why this arc should be visited more often than any other arc of the same length, because everything in the rotation is homogeneous. So indeed, if I look, I can count. So this is cardinality. If I can count how many times between 0 and n, how many k's between 0 and n are such that our alpha to the k of z belongs to AB and I divide by n. The claim is that this will actually tend to the length of the interval, okay? So in particular, it does not matter where my interval is placed. The number of visits that I have to this interval only depends on the length. And actually, the average number of visits, if I count how many times I visit and renormalize by the total number of visits, this will tend to give me the length of the interval. So somehow this point is kind of approximating the length as a measure. So this I think is very intuitive statement and this is a good way to think of this equity distribution. In general, you say that the sequence is equity distributed module one. If this statement is true, you have a sequence of points on the real line and take the mode one. If they satisfy this, I say that they're equity distributed mode one. And more in general, essentially you can have a stronger statement. So for every continuous function, and again let me write it on additive notation like this. For every continuous function from the circle to a, and for every, I will write x now, instead of z, I could have written x also there. For every continuous function, I can do the following. I can take one over n, the sum of my function evaluated along the orbit of x. So I have a function defined on the interval, on the circle. And I sample my function at the values given by my orbit, and then this will converge to the integral of the function. So a couple of remarks is that, so we will see in the, utors, lectures, you will see, if you haven't seen already, you will see Ergodistic, Birkoff Ergodic theorem, and you will see a similar result. But what is really crucial in this statement is that this is true for every point of the interval, and not almost every point. And another thing which is key is actually this, in general, as a remark, this, the systems for which two holes, two is basically equivalent. Substitute the rotation with another map, two is equivalent to unique ergodicity, which I will not define, but probably either Irene or Davide might define. So you keep this in mind if you already know it. And I will just sketch, first of all, you could try to, two is not really directly related to one. So if you could take as function the characteristic function of the interval AB, you would immediately, the statement of two would give you this statement. But in general, you cannot do directly this, but you can try to convince that actually by some approximation argument, two implies one. So it's not obvious, but try to get convinced, especially if you haven't seen these things before. And let me just give you a sketch of how to show two. And this is, again, this nice argument due to vile. And it also gives you a criterion known as vile criterion for unique distribution. And the idea is to actually look at some special functions. So exponentials. So now I will use these complex numbers. So let me call em, em of n, the exponential, e to the 2 pi i m. Sorry, I need a bigger chalk. e to the 2 pi i, em of x. So are you bothered by my em? Actually my English, British students, hate my em and em. So in Italy, this is em, and this is em. And in the UK, this is em, and this is em. So I guess my em always looks like an em, and my em looks like a strange wiggle with too many legs, but okay, don't bother. You can, I always try not to use em and em when I teach in the UK. I try to use k and l to avoid confusion. But here I think I have only em, so if you write em, it doesn't matter. Okay, so, and you want to look at this function, these exponentials. And I want to prove, so the claim is that two holes for functions which are these exponentials, for exponentials. And this is what I want to sketch. And let me assume also that em is not zero. And a remark, so what is the integral of this exponential? So remember this exponential is nothing else than cosine. No, sorry, there is no i. No, there is i, sorry. Okay, this is, no, there is no i.  Okay, for the absolute value.  This absolute value is one. Okay, so let me, okay, maybe say that we want to prove it for cosine and sine actually for, so think separately of real part and imaginary part. So we want to prove it for real part and imaginary part. Okay, so for cosine and sine. Okay, so the integral is zero. So what we want to understand is what happens when I compute these sums. So what happens if I compute these sums? If I compute these sums, so m of rk, I think I'm making a mess. Apologies, I think I'm making a mess. What do I want to do? So I want to compute, I want to compute what is the exponential of, I want to compute the exponential of, so I want to look at the orbit of the rotations. I'm confusing myself between additive and multiplicative notation. So let me just say, sorry, I made a mess between the two notations. So I want to compute it at, yeah, okay, r alpha to the k of x. Let's say like this, which is just x plus k alpha modulo one. And this is just e to the 2 pi i k, this is 2 pi i m x plus alpha. That is just e to the 2 pi i alpha times a m of x. Okay, sorry, did I forget something? M, did I forget something? E a m alpha, is it, did I forget this one? K, sorry, e to the, k alpha, thank you, yes, okay. So mx is here and mk alpha, yeah, okay. Thank you, I'm happy that you're not paying attention. e to the 2 pi i mk alpha times e m of x, okay, is it correct? Okay, and this I want to think of as some number elevated to, it's like some lambda to the k where lambda is this complex number. e to the 2 pi i m alpha, okay. So it's this number power k. So maybe we will go, we will go here. So if I just want to say that if I compute these sums that I still have in the statement. So if I compute one over n, the sum from k from zero to n minus one of e m of r alpha to the k of x. Plugging what I just wrote over there. This actually becomes the sum from zero to n minus one. And this is lambda to the k. It's this lambda that I wrote over there times e m of x. So you see what you get here, and actually this, this I can take out of the sum.  So I need to take absolute value of everything.  And then you want to use, so what is this sum? So this sum is just a geometric sum. So you just get one minus lambda to the n divided by one minus lambda. And the key thing is that in absolute value, everything which is in this sum is actually, because it's a geometric sum, so you know this formula for the geometric sum, this is actually bounded by some constant. So all this quantity stays bounded, but you have in front an n. So as n goes to infinity, it actually goes to zero, okay? Okay, so I may stop a little bit there, that's the sketch. And so what you get is actually for e m for these exponentials, for the real part and the imaginary part if you want to think of them as complex functions. Actually, you do have these two holes. So the conclusion is that two holes. And now for say real part, imaginary part of e m. And then what you want to say at this point is to finish, that I will not do. So what you should check, check what happens for m is equal to zero. And then once you have it for exponential, you can extend it to f continuous. And essentially you're just using that continuous functions can be approximated by trigonometric polynomials. Actually, okay, maybe extend it, okay. And maybe this I will leave to do. So this basically gives you a sketch of two. And once you have two, try to go from two to one, okay? Okay, this I think concludes the first properties of a rotation. And so we know in rational rotations have dance orbits and the orbits are equidistributed in this strong sense of vinyl. And as I said, this criterion of using exponentials is actually can be used not only for orbits of the rotation, but if you take some other sequence on the circle, something called y criterion tells you indeed that if these complex exponentials get equidistributed, then you say that the sequence is equidistributed, modular one. So for the rotation, it just comes from this small computation. In general, you have to prove it, but you only have to verify it for this special functions. Okay, so the next step I want to, as I said, I want to kind of review Gauss map and continued fractions. So we will step out of the world of rotations and we will look at a related dynamical system. And this dynamical system is actually not entropy zero. It's kind of more in the hyperbolic world of Hannah's course. But it will act for us as the zooming lens, as the renormalization machine in order to study rotation. So now and tomorrow we will have a break and jump to the Gauss map. And see how continued fractions relate to symbolic coding for the Gauss map. And then we will use the Gauss map as a renormalization machinery and relate the Gauss map back to the rotation, okay? So break for a second and I promise you that I will define the Gauss map so that she can give you an exercise on the Gauss map later. So I will first talk about the Gauss map then later continue fraction and the relation. So this Gauss map, it's a map of the unit interval. From zero one to zero one. And it's the following map, g of x, okay, is equal to zero. If x is equal to zero, that we have to consider separately. And otherwise, if x is different than zero, it's equal to the fractional part of one over x. Well, this symbol that I'm using now, so maybe I will, okay, we're, okay, let me write. Another way to write this bracket is also one over x, module one. So if this notation is commonly used, you can take real number x and write it as fractional part. We saw it already, we saw it also yesterday in Hannah's course, right? So this is the fractional part, which is the same than taking module one. And this is the integer part. So the smallest, largest integer which is less or equal to x. Okay, so let's plot the Gauss map. So you may say, okay, how do I plot? I'm sure many people have already seen it, but if you haven't seen the graph of the Gauss map, maybe it takes you a second to understand what this map is, so let's do it. So implicitly, this definition gives you a piecewise defined function. So how do I know what is fractional part of one over x? So first of all, I have to understand what is the integer part of one over x. So the integer part of one over x can be any number, can be one, can be two, it depends on how large, how small x is. The smaller x is, the larger is one over x and the larger is the integer part. So for which x is the integer part equal to n? Well, exactly when one over x has to be greater than n, but strictly less than n plus one, right? Because this is exactly when the integer part will be equal to n. But this is the same, then saying that x has to be less or equal to one over n, but greater than one over n plus one. Okay, so if I am in this range, so maybe we will call pn. We will use it also later. We will call pn this interval one over n plus one, one over n. And we will write this as x belonging to pn. And I think I still cannot do my plot. So maybe I'll finish my, so what happens? So if x belongs to pn, then Gauss of x is one over x minus integer part of one over x. Will be equal to one over x minus n. And this is true for one to the, for every n. So you have basically countably many branches. It's a piece was defined map which countably many possibilities. Again, I'm sure some people have seen this plot hundreds of time, but if you haven't, then. So I can plot one, one half, one third, one fourth, one fifth, and so on. So this is my p1 interval, this is my p2 interval, this is my p3, and so on. And on the first interval, this is just one over x minus one. You can check, you can plot it's zero at one. And it's, I know the answer, so I do the plot. So it's a piece of a hyperbola, which actually goes exactly from one, one at one half. Let's check at one half, it's two minus one, which is one, and zero at one. So if I plug one is one minus one, which is zero. And similarly, if you plot, so this is one over x minus one. And on p2, between one third and one half, I have a very similar picture. I also go from one to zero with the decreasing hyperbola. And this is one over x minus two, and so on. So this is countably many branches, each of which are pieces of hyperbola. And each of which are surjective. So if I call, say sometimes I can call GN, is the branch, G restricted to PN. And this is a surjective, it's invertible and monotone. Okay, maybe now I should have done this reality check later. Who had already, who had never seen the Gauss map? Almost nobody, so I could have, okay. I think I should have done this before, I think I spent time to draw the plot. Okay, so in any case now, Irana is sure that everybody can do an exercise. So I want to connect this map with continued fractions. So many of you told me that they've seen continued fractions. Have you seen how the symbolic coding of the Gauss map is related to continuous fraction entries? Who has seen symbolic coding of the Gauss map in relation with continued fraction? Okay, so I'm still doing something worse. Okay, and remind me again, who had never seen continued fractions? Okay, so I will just write for, just quickly write them. You can read more if you want, I can give you a reference on continued fraction, but let me recall you. So let me write what is a continued fraction. Continued fraction, and sorry, I miss my chalk, continued fraction. So most of you have seen, so if I have a number x, and let's say that x is irrational, I can actually express it uniquely in an expression of this form. I can write it as a fraction 1 over a 0 plus 1 over a 1 plus 1 over a 2. Dot, dot, dot, dot, dot, dot, dot, dot. And this is unique for irrational. And you can actually also do it for rational. So if x is rational. So there is a finite expression. So for x rational, you can do the same, but at some point your continued fraction will stop, okay? And actually, if it's rational, you can try to say in this case it's not unique. And if you want, you can try to find what are the two continue, there will be two continue fraction expansions. You can try to explicitly write them. And by convention, we write an expression, sorry, x in and as I wrote it, it's only in 0, 1. Otherwise, you can put an integer in front. Sometimes people call a 0 the integer part, and then they start from a 1. Here, actually, I will restrict the numbers in 0, 1 because those are the only ones we are working with. And then I start from a 0 because the coding is slightly nicer. Okay, so this notation I will just write like this, and this brackets, a1, a2, an, and if it's infinite, then you'll just put the dots. Okay, so the claim is that the continued fraction of x is actually related to the orbit of the Gauss map. And I think Hannah will spend more time on coding later, but I just need to tell you what they mean by coding this map and what is the itinerary of this map. So I have this nice partition. I have this partition into the intervals P1, P2, Pn. And these intervals are this joint, how I define them. And so Pi intersected Pj is empty if i is different than j. And also, their union is actually 0, 1 with 0 out. They kind of accumulate towards 0. So if I add 0, this is a partition of 0, 1. So what I want to do is to iterate my Gauss map. So I look at the orbit under the Gauss map of x, of a point x, do my dynamical systems with Gauss. And I want to record at each stage in which of these intervals my iteration lands. So I take my map. If you want to do the trick that Hannah told us yesterday, I draw the diagonal, I take a point, and then I go to my map, to the diagonal, to my map. I mean, I apply the map g. And as I move around, I look in which of these small intervals I land. Look at the record, record the Pn, the intervals, the Pn, the Pn which are visited. So maybe let me write it like this. Definition, the itinerary, the itinerary. And I'm going to call it ai, not by ai from 0 to infinity. The itinerary of this orbit, this will be a sequence, is the sequence of integers, ai. By the way, sorry, I think if you have still continuous fractions, you know this. But if you haven't seen it, I should do what I said that makes no sense if I don't tell you what are these ai. So the ai are integers. So positive is non-negative. So ai greater than 0 are integers in this natural numbers. And they are called entries of this continued fraction expansion. So otherwise, this is not true. You can write it in infinitely many ways if I don't put restrictions on ai. So yeah, this is, so again, the few that haven't seen continuous fractions, I will post some lecture notes and I spend a little bit more time on them. But I already bored people going too slowly on the Gauss map, so I will not do more on continuous fractions. OK, so the itinerary. The itinerary is the sequence of integers, ai, such that, so gi of x belongs to pai. So I look at gi and I look at which pn it belongs to and that I record. So I can meditate for a second if you haven't seen it, but this is just telling you. So this means that the iciterate length in pai. So each of this ai will be a number from 1 to n. And I'm assuming that I'm not hitting 0, which actually will be the case, but I will tell you later why. OK, let me assume that x is irrational in this definition. Otherwise, this is not well-defined. And again, it's in 0, 1. And I claim that you are not going to hit 0, so this will be well-defined. And now, OK, now I can go here. And now the proposition, or the key link, is that, so I can x is irrational, OK. x is irrational, ai, OK. x is irrational with itinerary with respect to Gauss ai. Then this itinerary is really giving you the continued fraction expansion. So this is a 0, a 1, a n. So I can recover my point as a continued fraction. It's actually not especially deep. It's just in some sense. The Gauss map is really made in order to do that. But it's a good point of view to think of a continued fraction as symbolic coding. This is what I mean when I say the continued fraction is the symbolic coding of the sequence of entries in the continued fractions are giving you the sequence which codes records where my orbit lands and which branches of the Gauss map I will be using. And I think we did it in their own order because Hannah, you will do symbolic coding for doubling map in relation to binary expansion. So the same relation that there is between doubling map and binary coding, there is between Gauss map and continued fractions. OK, maybe just let's check this. So first let's remark again that by definition of itinerary, by definition of plus the definition of pn, we have the following, that if I look at Gauss to the n, 1 over Gauss to the n of x. This is equal to an. Sorry, did I say it right? Yes. So if you want again, let me say it again. By definition of itinerary, g to the n of x belongs to p an. This is the definition of itinerary. And if you remember what is pn and what I said at the beginning, that the integer part is equal to something exactly when I belong to p of that thing. So is it this clear to everybody? Or did I skip too? Is it OK? You want me to? OK, so gn belongs to pn. Maybe if you want, you can add an equivalence here. And this is just saying that no, I don't think I can. Here I'm also using the 1 over x belongs to, sorry, x belongs to pn. This is what we did a little bit before. x belongs to pn. It just means the integer part of 1 over x is equal to n. That's what I'm also using in this. OK. And then basically, you just have to do it by induction. So again, I think I will leave you the details. But so let's just do first. Let's do it for n is equal to 1. So in particular, for n is equal to 1, we have that n is equal to 0. Equal to 0. We have that a0 is the integer part of 1 over x. And that means that g of x is equal to 1 over x minus a0. And now you can solve when you get that solved for x. So x is 1 over a0 plus g of x. You always check my calculations. I know, so I'll check with me that I did. So I bring a0 from the other side and then take 1 over. And then I think you can kind of see how it will go. And so if you want, we can do one more step. Then if you continue for n is equal to 1, if you want, you have that integer part of 1 over g of x. Again, by the remark is actually a1 is equal to a1. And that means, again, that g square of x, which is the fractional part of 1 over gx, is actually 1 over gx minus a1. And what does this give me? Again, I have to solve for g of x. g of x is, hopefully, it's 1 over a1 plus g squared. Let's check. So I bring a1 from the other side and take 1 over. And now this bit, if you plug it inside the previous bit, I'm not doing the proof by induction. I'm doing the first two steps, which I think are more educational for the first time. If I plug g of x inside this, you see what you get after two steps? You get 1 over a0 plus, and I have to plug g of x. So 1 over a1 plus g squared of x. So you see, you kind of start seeing that basically the orbit of the Gauss map iterates of the orbit and code the reminder you have to put if you truncate this continued fraction at, finally, many stages. So I think that the easy exercise for today will also be to, I mean, I know it's not difficult, but just do the induction. Finish the proof by induction, because just to work yourself through practicing this. So maybe complete, complete the proof. Just do the inductive step in general. And I'm almost out of time, but I have just a couple of minutes to maybe tell you some properties of this link between continued fractions. And so maybe I'll tell them as facts. So the facts that I'm not going to prove, but they're nice to know, is that actually, if you look at the orbit, so if you have the orbit of the Gauss map, you hit 0. So gn of x is equal to 0 for some n. If and only if, if and only if, what? Yes, if and only if, x is rational. So irrational numbers, sorry. That's why if you assume irrational, I told you you can be sure that you don't hit 0. And another fact, what are the periodic points of the Gauss map? Ah, maybe another more important fact before I tell you that. So if x has a certain continuous fraction, a0, a1, dot, dot, dot, dot, dot, then what will be the continued fraction expansion of g of x? Yes, exactly, a1, a2. So the Gauss map acts as a shift. So g acts as a shift on the entry as a shift. So it moves the entries by, it's really a shift in the sense of symbolic dynamics, actually. And then you can find out which points are periodic for the Gauss map. So which points are, how will periodic points for the Gauss map look like? Ah, what, how about, sorry, sorry, sorry. I went too fast. How about fixed points? If I have a fixed point for the Gauss map, so if, so fixed points for the Gauss map, there are countably many fixed points because I intersect the diagonal many times. And they basically have to have the same entries. So if you have a point which is a continuous fraction entries is k, k, k, k, k. This will be fixed by the Gauss map. And for example, this is, this is exactly what the golden mean look like. So I think I hope I'd say by, so for example, the golden mean has 111111 by continuous, the simplest number from the point of view of continuous fraction. It has periodic entries given by one, and it's fixed by the Gauss map, which is immediate by this remark that the Gauss map shifts the entries. And the next one is the silver mean, which I don't remember, it's probably square root of two. And this is called silver mean and so on. And finally, if you try to see what are periodic points, I went too fast from fixed points to periodic points. But periodic points are related in general to quadratic irrational. So quadratic irrationals are numbers which have essentially this form, a square root of d plus b over c, where here a, b, and c are integers, and d, a, b, c, and d are integers. So this indeed, the golden mean was an example, square root of 5 minus 1 over 2. And quadratic irrationals, so x, there are also solutions of a degree to a quadratic equation. So basically, quadratic irrational, x is quadratic irrational, if and only if it's a pre-periodic point. So pre-periodic. So pre-periodic means that g, there exists an n such that, there exists an n such that the gn of x is, what am I saying? Periodic, yeah, it's fine. It's just, this map is not invertible, right? So no, there exists an n such that gn of x is periodic, let's say like this. And I think I run already out of time. But for example, this implication, the whole, but you can actually try to prove an easier case. So if you have a periodic point, for example, of period 2, try to prove, so try for period 2 and then try to prove that if you have a periodic point, x periodic, and if you want to try period 2 first and then do the general case, then it's quadratic irrational. So more in general, this pre-periodic just means that the entries will have some beginning, which is whatever, and then they will repeat periodically after some time. OK? OK, thanks. That's done, yes.