Let us begin this learning dialogue with a reflection spot. You have read on a blog somewhere advice to e-learning designers. It says that in order to create e-learning modules, the main thing one needs to focus on is how to effectively and attractively put up information on an online platform. Do you agree? Pause this reflection spot and when you have made your choice, please resume. What might go wrong if you follow this advice? You might have included some PDFs and uploaded it or you may have done it better and you may have created animations and videos and images which have all the required information and uploaded it. Maybe you have a couple of exercises here and there. Still what's missing in this scenario? Attention to learner motivation. Opportunities to the learner to immediately apply and perform some activities related to the content that they have learnt right after they learnt the content. Immediate practice. Formative assessment and feedback right after they have done the task on whether they are on the right track, do they need to improve and so on. Explicit activities to foster pure learning between learners and which can help form a community of learners. So all this actually points to what's missing is ensuring that there is connect to the learner. The learner is forgotten, their motivation, their needs, their prior knowledge if all we do is think of how to attractively package and put up information. The other thing we have forgotten is the established theories of how people learn. We have seen various challenges associated with e-learning. So now let's try to see how to address these challenges. One way to address these challenges is to adopt a learner-centric approach. What does learner-centric mean? It's essentially a broad pedagogical approach, in fact a mindset almost which comes from constructivist philosophies and the focus in a learner-centric approach shifts from how well did we convey the information to what did the students actually get out of it. It shifts from did I teach well to how well did the students learn. The characteristics of a learner-centric approach include the following. Learners are co-creators of their own knowledge. They construct their knowledge as they interact with the content and with the other activities in the learning environment. This means that prior knowledge affects how they learn in future, what they learn and how they learn. Learners interact with the content and this means they go beyond watching videos, reading text and listening to something. Learners work on different activities sometimes collaboratively. In fact, that is one of the key aspects of a learner-centric approach and they immediately apply their learning from what they learnt in the video or by doing the activities and they get feedback on it from the platform. Learners control their pace of learning, what they do next. They feel included. Learners feel included. There is a personalized element, personalization element between the learning environment which includes the content, the activities, the discussion forums, the way the language is and the learner who is interacting with it. Let us pause at a reflection spot where you can think and in fact reflect on your past experience and write one instance where you may have experienced a learner-centric approach. When you are done writing, please resume. Some of you may have thought of a classroom scenario where there were active discussions between the teacher or the facilitator and all the learners. You may have as adult learners, you may have experienced a workshop where there was an activity you were doing collaboratively, doing problem solving or doing group discussions or perhaps the context where you experienced a learner-centric approach was an online course or other e-learning content where there were certain personalized elements or certain activities which followed the characteristics of a learner-centric approach. In face-to-face context, some of the flavors, the entire spectrum of a learner-centric approach includes active learning, collaborative learning, techniques like think-pair-share or inquiry learning in labs, peer instruction and so on. Now the question is to bring in a learner-centric approach in e-learning. Every e-learning project has its own challenges but in each of these, a key requirement for the designer is to design e-learning content and the entire learning environment to promote engagement of learners, to support them to learn perhaps difficult concepts and apply the core concepts to persevere through the e-learning module and so on. So a learner-centric approach becomes critical. So what are the requirements of a learner-centric approach in the context of e-learning? The content and the learning environment must provide mechanisms to enhance learner engagement. The designer should make sure that learners have sufficient support from the material itself since they have to learn these on their own. The e-learning should enable personalization, it should contextualize the learning to relatable scenarios and help the learner acquire a deeper understanding of the content. A learner-centric approach in e-learning does foster peer learning and being a part of a community. It attempts to cater to diversity and individual needs of learners. So the focus of this course is that in the context of e-learning given the medium, the format, the challenges, the lack of synchronicity between people how do we design the content and the activities and the entire learning environment so that learner-centric principles are followed so that learners go beyond watching and listening and executing prescribed procedures? How do we make sure that learners in fact think and figure out things and express their opinion, learn from peers? How do we foster learner engagement? How do we promote learner engagement? How do we try to make sure that learners feel included and they persevere? So in the rest of this course we will focus on pedagogical design principles which in fact originate from learning theories and help e-learning designers ensure that learner-centric principles are followed and these requirements are met. Thank you.


Hello everyone, I'm James Milan and this is Million Dollar Gift series, which as you know focuses on the efforts of volunteers and people putting the work in behind the scenes for all of our benefit and we like to highlight those issues as often as we can and of course right now in what we call pandemic times around here, that is more important than ever. Today I get to talk to Elizabeth Greenhall who is the vice president at Project Bred, she's in charge of communication and development. Elizabeth, thank you so much for joining us. Thank you, James. And we just want to talk about the work of Project Bred and how of course it has been affected as we as all of our work is by the current circumstances. I'd like to start by just asking you, you know, I think a lot of our audience would be familiar of course with Project Bred as an entity, possibly without knowing anything about what you actually do, mostly because of course of the high profile of the Walk for Hunger each year. I've taken part in it three or four times myself over the year. Thank you. It's been supportive and with good reason. But tell us a little bit first of all, what does Project Bred do? Sure, so Project Bred the Walk is just a part of of our organization, but it is where our roots are and sort of how the organization came to be. And the Walk ultimately is about, you know, a commitment to taking care of one another and to, you know, coming together as a community to lift the larger group up as a whole. And so as after the Walk, which was founded in 1969, had been through some years of doing it, it just became so clear that there was a lot more that needed to be done and a lot of expertise with people closely associated with the Walk that Project Bred began expanding beyond the fundraising component. Right, being a fundraiser, right. Right. And so today we really act to connect people with resources and have a big focus on federal nutrition programs like the Supplemental Nutrition Assistance Program, which is a staff or formerly called food stamps. And in addition to other programs, school meals, summer meal, free summer meals for kids and teens. And our approach is really about solutions that are focused on families, kids and communities. And trying to make sure that there's a sustainable way to support these folks who are struggling to afford food right now. Yeah, and you know, we have spoken to some other folks in the last couple of weeks about the fact that there are a lot of different front lines, one might say, in, you know, our collective fight against this pandemic. And you are certainly on one of them, the food, the programs that you mentioned that you're already heavily involved in, summer lunches for kids or food, you know, provide provision for kids in the summer at school, the SNAP program, etc. All of these things have been directly affected and dramatically so by their current situation. So let's talk a little bit about the food insecurity ramifications of what we are all going through. Let me just invite you to start to walk us through what it is that we, what you are already looking at and facing, but what we also are looking at as this moves forward from here. Yeah, absolutely. I mean, I think to start, it's worth just noting that this is completely unprecedented, you know, Project Red has been serving the community for, in the state of Massachusetts for over 50 years and in all of those years, we have never seen anything like this. But before the coronavirus arrived in Massachusetts, we already had a pretty significant problem with hunger that often went unnoticed. And so, you know, in February, if you were asking me what hunger looked like in Massachusetts, we would say it was impacting about 9% of people in our state. And that works out to about one in 11 households or one in nine kids. So far too many large, large number. I mean, if, you know, that's, it is a thousand people or more, I'm sure. Absolutely. And when you consider that that Massachusetts is, you know, a wealthy state in a wealthy nation, well resourced, like well educated. It's really unacceptable that we are kind of in that position. And of course, with the situation that we're in right now, we're seeing this problem that was certainly there be, you know, accelerated and exaggerated in a very short window of time. So some research that was supported by the USDA just came out around food insecurity in Massachusetts since in the month of March. And it was about 38% of people who were saying that they, you know, were food insecure. And just for your viewers, when we talk about food insecurity, what we mean is people who have some uncertainty around where their next meal will come from. They can't, you know, reliably say with confidence that they will have, you know, whatever that next meal may be. Thank you for that. You anticipated my question perfectly. And I do think it is really important that people understand what that means concretely. Food insecurity means literally that at some point or in some way, families are not sure or people are not sure where their next meal is coming from. Right. And I mean, you can imagine the implications that that has on, you know, the stress on a family, other long term like health and economic outcomes. And so I think when we think about Massachusetts as a community, it's really important to recognize that it's not just that individual experience of the person who might be struggling, but it's the whole family unit and then the whole community. And as a result, you know, our state isn't, we're not reaching our full potential, right? And we can't achieve the equity that we want until we can make sure that everyone has the food they need to thrive. You know, you were you have already mentioned the SNAP program. And one of the things that that has really struck us in some conversations that we have had recently is the idea that for instance, if food pantries are under, let's say in higher demand than they, again, an unprecedented situation like depression era. And when what we've heard that has really struck us is that there are stigmas attached, of course, to to both going to a food pantry or, you know, enrolling in the SNAP program or in in what people had traditionally called food stamps. And as between those two things, it seems from what we've heard that people are more or have a bigger problem with enrolling in the SNAP program than with going to a food pantry. Curious about, you know, whether you have any idea or explanation for why that is. And also what we can do about that. Right. Yeah, I mean, it's a it's a really interesting topic. And I think, you know, one of the things that project bread, before you know, we went into crisis mode had been looking at is really starting to unpack those barriers and understanding, you know, misconceptions or fear of perceived stigma around programs that were intended to support people. And I think on the, and this is, you know, anecdotal, but but I think what we hear on the side on food pantry is that that is like a community based program. And, and so in many cases, you know, people feel you know, a sense of support or, or like an intimacy with the organization, just to feel understood. On the flip side, you know, I've definitely talked to families where they are as a kid in high school, who doesn't want to go to the food pantry because he's afraid Somebody's going to recognize them as classmates are volunteering. Yeah, exactly. And when it comes to stigma around SNAP, I think it's hard to it's hard to quantify. But I will say that SNAP isn't well understood. And so a lot of people confuse SNAP with welfare, which are, you know, very different things. And I think that the SNAP program, you know, feels like official in some way, because it is a federal nutrition program. And so I think that people feel like they're somehow like putting down on their record, you know, that they're struggling or or it's, it's, you know, it's, it's difficult to kind of overcome that barrier. But I think what we want people to understand is that SNAP is proven as the most effective anti hunger solution that we have. It achieves a scale that is, you know, untouched by any other program. And it brings money into the Massachusetts economy because it's federally funded. It gives people the purchasing power to choose their own food, you know, if you want to buy the food that's, you know, we all want comfort food right now. But comfort food to me is different than comfort food, maybe from someone, you know, with a different heritage or background. And so we want people to be able to purchase and the food that they, that it makes no sense for them, whether that's cultural relevancy, or to accommodate any dietary restrictions in the home. And really, that's kind of the beauty of SNAP is that it does. It's not putting people in a position where they're asking for charity. It's actually giving people the financial resources to purchase their own food and in a way that's most convenient to them, whether that's, you know, at a big supermarket or at, you know, a corner store. If you work odd shifts, you know, you can do it at night or whenever you want. I think there are a lot of barriers on the emergency food side on that food pantry system, because it is so volunteer dependent that that they're struggling right now. Because we got to keep everyone safe. And so our feeling at Project Red is that there's a number of people who are visiting food pantries right now, who don't know that they're eligible for SNAP have never found themselves in a place where they're looking for assistance in the way that they are now and maybe have no familiarity with what's available. And we really want to help people understand that this is a program that exists. It's proven it helps lift people out of poverty. And, and it's sustainable, right? You don't have to keep going to a food pantry, figuring out that transportation piece. It's the majority of it's happening electronically. Right. And so who is and how does one recognize that that you who is eligible for the program?  Yeah. So I think, you know, on the eligibility question, like all things, it depends. But I would say there are like, that's reason that's one of the reasons that organizations like Project Red exist is to, you know, help people who are curious about what resources might be available to them. So you can call Project Red. And, you know, we can quickly pre screen you for SNAP. So you know, whether it's worth, you know, going through that application process, you have support through the application. I mean, anyone who's applied for unemployment, or tried to do their own taxes, government paperwork, and is can be really challenging. And, and especially so if it's not in your native language. And so Project Red really helps people find out the maximum amount of supports that they are, because, you know, we know that we need comprehensive supports for people. So we, you know, if you call us and we pre screen you and you're eligible for SNAP, we'll help you through that application process. And we'll also talk to you about free free meals for kids that are near you, we can tell you about food pantries that are open or closed during this period of time. So really, we help people navigate what resources are available to them. And, and that eligibility piece is tricky for a lot of folks. And that's one of the ways that that we help. Okay, so I mean, the answer in part, as you said, is it varies a lot. But you, Project Red, great resource for to figure out whether one is eligible. Yeah, and I would say that, you know, on one, so two things. One, I think there's a lot of opportunity right now to in to really push for making SNAP something that's eligible for that more people are eligible for. The broad as possible scope. Yeah, exactly. So I think people who can should should, you know, try to educate themselves and do whatever they can to advocate for that. But then on the eligibility piece for those who might be, you know, needing assistance right now, there are some resources out there on Project Red's website, we have, you know, some basic information that can help kind of let you know if it's a good place for you to start or not. But again, I think that when you're in crisis, there's nothing that compares to having a kind non judgmental person on the other end of the phone, which is, or the chat, which is what we do at Project Red. That's fantastic. I got to say, as a brief aside, just dealing recently, as I have been on behalf of an older relative with just starting to weed or weighed into Medicare, boy, could I use a resource like that, you know, so I can really appreciate what it is that you are offering. I'm wondering, of course, you guys would have thought about this and made provisions for it, non English speakers and immigrants who might be wanting to or non non English, non native English speakers, I should say, would be wanting to apply. I assume that you you, you can can meet those needs as well. Yeah, so great question. And I think it's a really important point that not everyone thinks about. And I think we all have a responsibility to be thinking about it. Which is that, you know, hunger before COVID-19 was here, you know, we already talked about that this is a real challenge that exists in Massachusetts. And we know that that certain communities and certain demographics are disproportionately impacted by it. So where that, you know, it's one in 11 households, that rings true when you look at the state average in certain communities, you know, it's one in three households. That's right. So it's really, it's really important to be thinking that through, especially people who have, you know, the power influence to be informing the the response. And at Project Brand, we do a couple things to help drive that sort of more equitable, or more intentionally equitable planning and process. So we are always making all our resources available in multiple languages and doing the promotion and awareness building around those resources in multiple languages, trying to connect with people as many people as possible. You know, our site is available in English and Spanish. And then we have translation services for 160 languages through our concrete. Yeah. And one of the great things about, about, you know, Greater Boston and the state of Massachusetts is how many cultures, you know, cultural and linguistic diversity. It's pretty good for this. Very good. Yes. And so we want to recognize that and give support that's really tailored to people. So, you know, when you're in crisis, you want to have as few barriers as possible. But then the other piece that that we are doing is, you know, trying to find opportunities to create platforms for people who are living with this experience right now or experiencing food insecurity to inform the policy and programmatic response. I think that for a long time, hunger, along with many other things has been something that the solutions have been very one directional and usually, you know, like it's generally older white men that have been driving this policy. Absolutely. And so I think, you know, it's really critical that we recognize that, you know, the best way to support someone is to ask what they need and then help figure out how to get it, how to get it to them. And so being really intentional about community engagement is important. And then also research. So looking at patterns of hunger and food insecurity in Massachusetts over the past 50 years. Project Bright is located in East Boston. So we are definitely, you know, well aware of some of the barriers that exist in communities where there's a lot of gentrification happening. So really thinking from a from, you know, both a tactical and on the ground level, but then also making sure that that's getting turned around and brought back into the policy conversations. Yeah, speaking of policy conversations, I'm curious about, you know, the mix of programs that you guys help people to navigate. I assume that there is a mix between those that are administered by the state and those that are administered by the federal government. All right, do you find or are you finding the current moment that it is easier to work with or either of those entities is being more flexible as seems required at this time? And I guess what I'm really asking in some ways is the federal government's role in all of this has not been very impressive so far. How has your experience been with that, I guess, and also with the state government administered program? Yeah, absolutely. So for a number of these programs are administered at the state level, but funded federally. So I think that, you know, it's it's sort of a blended answer, I guess I would say I'll say that Massachusetts just as a state has incredible leadership in this area. So really the national leaders on responding to the hunger crisis coming out of or not like we're coming out of it, but resulting from from the COVID-19 pandemic are congressional leaders. So Congresswoman Ayanna Pressley, Congressman Jim McGovern has been a national voice on food insecurity the entire time he's been in office. Much longer than this than we've been dealing with this pandemic. Absolutely. Congressman Kennedy. So I think that there's a lot to be said for, you know, using whole lifting Massachusetts up as an example of ways to make these things work. I think that pandemic, so something called pandemic EBT has has recently been approved, the governor announced it. And this is something that provides funds to that are federal funds to families who normally are eligible to receive for your reduced price meals at school. And, you know, Massachusetts was one of just a handful of states that actually applied to receive this funding.  And I think we really have to applaud the collaborative leadership here in Massachusetts. You know, I think, regardless of how you think about things or how you thought about things before or, you know, perceived like dramatic differences in policy agenda, it's really clear right now that people in the community are getting put first and and that our state is is out ahead leading on this and trying to kind of setting an example and showing in real life what the potential impact of these programs can be.  So that that really is it's wonderful to hear. Not only again that there is collaboration afoot, but also that it's going to have and more importantly that it will potentially have such concrete and direct results to to to attend to the needs of people that really deserve it. Absolutely. Yeah, it's, you know, I do this all the time and I'm I'm like continually just excited about the how much Massachusetts is pushing for. It's really nice. Yeah, it's it's we're doing OK in this state, you know, by by being residents of this state, I have to say a couple more things. One actually goes back to something you said earlier and that was when you were talking about, you know, people going to food pantries versus not versus, but yeah, pantries and the SNAP program, etc. And you were saying that one of the virtues of SNAP is that it does give people the means and allow them to make the choices of what food they want for themselves. Of course, that's a great thing and right in line with our American values, etc. It also made me think, hmm, how good are those choices going to be sometimes when you are giving those choices over to all the individuals? Are they going to choose well? Project Bread get involved at all with providing counsel or advice or guidance around healthy eating and healthy choices in the supermarket with your food stamps or otherwise, you know, program or otherwise? Yeah, I mean, I think it's a good question. For us, I think, you know, we believe that people can make that choice for themselves and like the best thing to do is to make information available. But I think that trying to limit how prescriptive we are is going to make it easier for more people to access and use the program. And I think, you know, when it comes to health, a lot of us know what's healthy and what's not healthy and still make healthy. I might be living on ice cream right now, but stress eating. And so I think like giving people the right to make those decisions for themselves is part of how we think about it. But definitely there's an education component to make sure that that information and those resources on healthy eating are available and a lot of our partners do that very well. And we try to do, we try to influence the health factor around these programs primarily through the school. So working with school districts to make sure that those three meals that are provided are quality food and you know can help meet the nutrition needs for kids during the day, especially a lot of kids count on school food for half of their daily making sure those calories count. Yeah and I just want to clarify that and asking that question, I wasn't assuming, believe me, that you guys were being prescriptive in any way. I just wondered whether again there was an element of education as you were saying. For sure and and I think that's actually, you know, it's an interesting point. It's sometimes a tension point in the anti-hunger community just from a strategic standpoint. But we're really about access and affordability and I think, you know, we don't, I think we know that there are others who are better suited probably than we are. As I just admitted, I had ice cream for dinner last night. No, you were actually, I thought that was all theoretical. I mean, yeah, it's theoretical. I know that that that Project Bread itself is not, you know, not in any substantial sense of volunteer organization and that really the volunteer aspect of things come from the walk for hunger. That's right. A lot of ways, but I'm wondering if people who are watching are interested or engaged and galvanized by the conversation, is there, you know, beyond the walk and, you know, we should not, just not to minimize that, but is there any, you know, thing that people can do or ways that they can get involved that would be helpful? Yes. So the development person in me has to say, Donnie. And I think that right now, you know, it's been really encouraging and like makes you feel good about humanity to see the outpouring of support. And I think that what's critical and particularly important from individuals who are contributing, whether it's five dollars or five hundred dollars, is that a lot of organizations, that is money that gives flexibility to be able to react, you know, some funding that comes in in other ways is, you know, sort of restricted to a specific program or a specific geographic region. And obviously, like, that's great too. We take that money, but in the crisis response mode, we really do need flexibility because things are changing so rapidly. So every, I think every dollar that comes, particularly from individuals, is really helpful. But on the sort of more action side, I think there's a number of things people can do. Social media is an easy one. You know, Project Red is constantly promoting the resources that are available to others and getting the word out is hard. It's hard to break through messaging and and it right now in particular, right, because there's so much. And so getting the available resources in as many, in front of as many people as possible and and networks as possible is really important. Project Red also has an action team. So this is a group that have raised their hands to say that they're interested in policy and, you know, receive a newsletter where we sometimes say, hey, can you call your elected officials and say that you support this and provide some education there. So that's another way. And then you touched on the Walk for Hunger and, unfortunately, obviously we had to cancel the event this year. You know, it brings thousands of people together. Thousands. Thousands. And it's a really powerful, I mean, you've said you've you've done it. Yep. Thank you for that. I was mentioning to my producing colleague before we went on the air that 20 miles seems like a long time, but or a long way. But when you do it, you know, I've done, I did the full 20 a couple of times and congrats. Didn't feel that way.  It's a special morning. A lot of the, you know, a lot of the time it really is. So anyway. Oh, we'd love to hear that. Yeah. And I mean, I think the other thing that kind of is relevant to what we've been talking about about the walk is that because the walk kind of grew out of a grassroots movement and had as much to do with awareness and raising, you know, forcing people, not forcing people, but like helping to put that food and security piece out in front of people back 50 years ago means that now that's still part of what we're doing and a lot of amazing events in Boston are strictly fundraising based and the Walker hunger is certainly that for us. Fundraising piece is important, but we've made the conscious decision, you know, we don't have a fundraising minimum. You, for those who can afford to pay a registration fee, like we make concessions, it's free people 18 and under when it happens. It's free to everybody right now to do it virtually. And so I think part of the beauty of the walk and part of that experience is really that it brings a good cross section of all of the best parts of our city and our state together. And so, you know, as an organization, I think it's been challenging to know that we're not going to have that opportunity to to be together as a community. And so when those decisions were first made in full transparency, it was hard, you know, it's like breathing. And so I think that are what we've done is kind of switch focus and try to use this as an opportunity when people feel really isolated to leverage the walk to bring people together. So we've had people doing steps challenges with their neighbor while everyone's social distancing. We've had people doing virtual bake sales and doing other kind of challenges through all of the means that we're communicating and that's been so powerful. So I think, you know, encouraging people to still use the walk. And I should say that obviously this year we've made some changes to the way that the money raised through the walk will be used. We're putting it directly to COVID-19 response work. Whereas in years past, you know, after the walk, we we branched out to a number of organizations that period of review and holding out to make sure that, you know, we have all the applications and everything has been eliminated. Right. Okay. So it's coming in, we're getting it back out. Right. And that's what generally happens with the money that comes in you're saying through the walk, which is, as we know, a big haul. And so then you guys redistribute that to local organizations. Far is the scope of the organizations that you are supporting and working with. Does it go all the way out to Western Mass, wider New England? Is it mostly in Eastern Mass, Boston based thing? You know, how far do you go? We are statewide. Okay. Through and true. I mean, I think what we are, what we look at from a funding perspective is, is where is that the most need? And, you know, this year we've launched or last year we piloted and this year launched a program where organizations can leverage the walk to fund race for themselves and they get to retain 60% of what they raise, which means that we can, you know, the money that is raised elsewhere can go to the communities in the state that don't have that kind of capacity within their own community to fund race. And so we're really focused on The great situation. That's, that's Yeah. Yeah. It's been, I mean, it's new and it's been a change, but there's been a really positive response to it and we were kind of worried about, you know, what would happen because this is its official launch and it's actually, it's been great. It's been an important part and people are really rallying around this and we're, we're excited and encouraged to see it and think it will be lasting. That's great. Hey, before we wrap up, I just did want to ask you, you had mentioned before about, you know, getting the word out around resources, etc. Of always a challenge and very important part of what you're doing. So let me give you a chance before we end just to cover anything that we haven't covered or and or talk about and let us know about any resources that have not been mentioned just yet. Sure. So I think from a bias, but from a resource perspective, part of what Project Red does and kind of part of why we exist is to be that central hub of resources for folks who need support. And so the best way to see the full suite of resources available is by going to projectred.org and backslash COVID-19. And I think that there is a list of everything that's available as well as our partners. There's a map of where you can get free meals for kids. There's information about the various federal nutrition programs and some of the other supports happening. So that's definitely a place to send people. It's also for people who want to better understand what the hunger crisis that we're up against now looks like. It's a good way to to get more information. So I think just as the general public, some of the most important work is raising the awareness. So, you know, you made a really important comment before that there is a lot of there are like a lot of different front lines. And I think there are a lot of different crises that are coming out of this. And so just helping people remember that hunger is one of those. And in particular, hunger is important from an equity standpoint. And so just keeping that in top of mind and helping to engage those around you to be thinking that way as well. And then obviously promoting resources and we certainly would love to have anyone who's interested in supporting to join the virtual walk with us and start a fundraising page. And, you know, like I said, it's free to do and every dollar counts and every dollar is going to support communities who really need the help right now. Thank you. Thank you very much. Thanks for your time today, but also for the work that you are doing and facilitating and that Project Red is doing. It is seriously important work and it is to our collective benefit always and especially now. So thanks. James, thank you so much. This has been a real a real pleasure. I love getting the chance to talk about this and it's so great to speak to someone who clearly is equally committed to the community. Thank you. We have been speaking to Elizabeth Greenhall who is Project Red's Vice President of Communications and Development and you have been watching Million Dollar Gift. I'm James Milan. Thanks for joining us.


What is happening? I'm the Sikes. The Tiger is back. And this time I brought the Italian Tigeress Fulvia. So Fulvia is going to be narrating this with me and what we did was you're not going to see me in this video at all.  She wanted to see what LA Fitness was like, what American gym was like. So I brought her. They let her come in for free, which was really cool.  And I started her off. She told me what she wanted to work. And I said, okay, try this.  She really never did these specific ways. So she just had a fun running a circuit. So we started out with the inclined dumbbell bench press. And as you can see here, she likes to add a little bit of a twist to it. Is there anything you would like to add to this Fulvia? No, not yet. Okay. She's still nervous, even though she's not on camera right now. It's just voice. She's still nervous. Oh my god. There's a cool world out there. Maybe I think the weight was too light for me. I could have been using some heavier weight. I talked to her about doing 10 pounders, but she was like, no, I want to do different stuff. So I was like, okay. So there she... Here too. I could have been using a heavier weight, but... She's doing her tricep extensions here, ladies. Now, Fulvia will be taking a full real deal. Personal trainers certification course. She wants to get in bikini competition shape and train people all over the world. So... I want to go to Dubai. For what? For training people in Dubai. You want to go to Dubai to train people.  Because it's beautiful. It's beautiful in Dubai? Okay. Isn't that... Indian country? Indian city, right? Where's Dubai at? Dubai? Okay, let's keep going. Is it Middle Eastern? Yeah, kind of.  No. When the hell did you say you wanted to go to Dubai and train? I don't remember you saying that. No. Okay, so anyway. Yeah, now she's going to pull her pullovers. Okay, it's going to work your chest also ladies. And abs. And me and... Yes, it does work abs too. Come on, add something. This is narration. I've never done this before. Sorry guys. That's what she said. I'm just looking. She's just looking. Yes. So now she wants to teach the ladies how to do squats. If you have any problems doing the squats, you can start off as pretending to sit on a bench, for example. Always keeping your back flat as much as possible. To not bend over... How do you guys say? You don't want to lean forward, right? You want to stay erect. Speaking of erect, man. I know all you perverts is looking at her doing them squats, man. Talking about ski-ski skiing, man.  So then she just ran right over. I want to do pull-ups. Let's go over and do pull-ups. So she starts doing pull-ups. I think at this point she realized that the weight... She wanted to change the weight for her assisted pull-ups. And keep in mind, y'all she's not used to any of this equipment. This is her first time on a lot of the stuff. So y'all want to harass her about her form and all that other stuff. Leave it alone because she came to LA Fitness and I just was throwing stuff at her. So I was like, oh, I want to work my chest. Okay, well let's do incline bench press with dumbbells. So she's never really done that before. So if things were off, be forgiven. Dick Watts, because I know how y'all can be.  And your pinky need to be totally curled around the dumbbells for perfect for...  No, no, no, but anyway if they have anything to say, any advice, I would take it. Like, I think it's everything useful. Oh, you hear that? You know how much advice you're going to get now? Yeah. 90% of my viewers are men. You know how many men want to come up to women in the gym and tell them how to correct their form but they never do? Oh boy. You're asking for something that you're going to wish that you never got once you get it. Oh, yeah. So, uh, yeah. The camera kept going out of focus a little bit. I kept having to refocus it. I don't know what the problem was with my camera. It's my phone camera, which I'm not used to, you know, shooting with. Let's do deeps. Okay, let's do deeps. You want to do deeps? Let's do deeps. Stop it. So, let me show you something here. You notice how the handles are twisted out? Yeah. Well, it's fine. When you're when you're doing dips more people pull them out like that to work the chest and triceps and you fold them in it's going to bring your arms in closer which is going to focus on the triceps just so you know. Yeah. Yeah, this is probably the longest workout video that I've ever done, I think. It's like almost 15 minutes long some of my bitches.  I don't say like that. Pieces of trash. Oh my gosh, this video is so funny. Look at that guy. Oh my gosh. She was crack it up when she saw this. My, okay, my form is the correct one and the guy is doing something crazy. So guys, please don't do it. Girls, please don't look at him at all. Thank you. He's moving more dynamically. Just what you do not have to do. Right, you don't have to do that at all.  I'm not a dynamic mover in the gym. I like slow power controlled movements. I like them like that. But you know, he was just doing, I mean, you can still make games like that, man. But you know, it's different.  Snap City, man. Yeah, so I actually didn't expect her to do this. I went from front lat pull downs to back lat, rear lat pull downs and then she switched it up like this. I'm like, okay. Fulvi has been watching a lot of films and trying a lot of different movements in the gym. Even here could have been using a different way. For the easy one. Yeah, we wasn't super serious, you know, when we went in. I kind of was half gung-ho. I was half cocked when I got there. Like, I kept running into a bunch of people I knew when I was talking to them and introducing her and everybody loved her. They wanted to know where she was from, what the accent was like. They were all nice to her, yeah. They welcomed her with open arms. Woo! Yes. I always control the weight. Yeah. Then she wanted to show how to do squats a different way. Yeah, here I'm not going to be deep lower as I used to because my knee is injured, so I'm doing whatever I can. Yeah, she ain't going deep. She ain't walk just the nuts hanging out, man. I'm happy to correct they believe me. Yeah, so she's trying to direct me.  I was like, don't worry, they can see you. Now, I believe Elliott Hulse showed a squat similar to this near the wall where you can test your flexibility. So yeah, we know she's not going really deep. Yeah. But we ain't making excuses, but how long was it since you been in the gym, actually?  A month because of my knee. Oh, that's right. She fell in the damn subway running from a mugger in London. Seriously though, no, no, it wasn't a mugger, but she did fall and she hurt her knee and she hasn't been working out at all. But she did want to see the gym and that's the reason why I brought her because I know she would have been way more gum-ho. Now, right now, she's on the peck deck. She's never done a peck deck like this. And I said, go ahead and try it. So she was like, shouldn't it be wider? And I'm like, yeah, you can make it wider. Yeah, so just so y'all know also Fulvia used to weigh around 200 pounds. She used to be a big girl, man. Yeah, yeah, girls. If I did it, you can do it. So, yes. I lost a lot of weight. Yeah, she's the type of person where like when people see her now, they don't know who she is. You know, her classmates don't know who she is at all. Yeah. Yeah, so she's starting to feel it. See how big her eyes got? She's starting to feel the peck deck there. Yeah, she ran to a...  See. Just want to say, never forget to breathe in and breathe out. When do you have to breathe out? You breathe out whenever you do the movement. Exactly. When you just... When you... Oh, they're gonna like this one, man. I was trying to get an angle that wasn't too raw. But she actually... I was wondering why. I was like, why do you do this movement? Why? Because this is asking for men to stare. And she was like, I do it because sometimes she has back pain. So, that totally relieves any type of back pain. Shit, yeah. You work out your legs and you don't put any... How do you say? So, I'm gonna do a leg press but it's more vertical. See that, y'all? It was funny. She didn't want to do it at first. I said, what? She was like, I want to do this? And I was like, no, because they stare at me. She's like, no, they won't stare at me. I said, look, whatever movement you want to do, just do it. And we'll discuss it during the video. And then she wanted to do some tricep extensions. These overhead tricep extensions. Although I feel you should be leaning more forward with this one.  Yeah, I should... I should just move the forearm. Yeah, I like leaning forward a little more and moving more head down. I need to improve this arm. Yeah. So, as y'all can see, we was kind of pussy-footing, you know?  You traitor, right there, rude.  We was in there having a good time. Making a film for your enjoyment. Okay? Because y'all can still learn something from this. This is another way to squat. Yeah, it's pretty easy. You girls can do it. See, you work on your arms and lower body as well. At the same time. Yes, yes, yes. I wonder if this is the last... No, this isn't the last movement. The last movement is going to be abs. So... But yeah, y'all. This is Fulvia's last day, so she's going to be leaving. She's going to be leaving us going to London. London. Where she primarily stays. And I've been planning to make overseas trips all over the place. And London has always been one of my stops. And I will be there.  Hello everybody. Hi mate. I want to say mate, but mate's my Australian voice. So of course that's her doing her abs. She likes to crunch down and crunch to the side. To the right. That's why you do it right for the obliques. She's still shy. She's still shy. I'm just observing. You're observing your own form. So this is her first time really seeing herself work out like this extensively. She made a couple of videos with her friends, but not like this. If you liked the video, let us know. We will do more next time. We are together. Thanks for watching. Love you subscribers.


All right. Hello. Hi, everyone. My name is Ted. You can find me online in most places as Drenmi. I work for Tinkerbox, which is a local Ruby on Rails agency. And I'm here to deliver the five random Ruby tips of this meetup. So for this time, I actually picked a theme, so I sort of broke my own rule there for five random Ruby tips. Recently, one of our junior developers, Daniel, started a daily Ruby Cata session, which brought me back to a website which I registered for in, I think, 2013. One thing this website does is when you sign up, you need to list your skills. And for reasons that are lost to me now, I listed my skill as hiding things. So the idea now is to help anyone who has to work with me by presenting five tips on finding things. And more specifically, finding things with ActiveRecord. Because the reality is that most of us doing Ruby full-time need to work with Rails. And so we need to work with ActiveRecord. But I like to point out that there's also a great gem called SQL. So if you have the time to work on a pet project or something where you want to try something new, I definitely recommend trying that out. So the first tip is a new method in ActiveRecord 5. And the way you know if you're using ActiveRecord 5 is if you're using Rails 5, then you're using it. So there's a new method called left joins. And if you imagine an extremely simple domain model where you have a company that has zero or more users and the user belongs to the company, and you want to query out companies based on some property of the users, you first need to join the user's table. But if you're familiar with SQL, the problem here is that once you do join, any company that doesn't have any user will be left out from the results. So say that you want to list all companies that don't have a billing contact. You will miss out on all the companies that don't have any users at all, which you might have actually wanted in your result. So you can see this is the SQL generated by using just joins. And if you use left joins, it will replace the inner join with a left outer join. And this will give you the companies without users as well in your result. The second tip is also new for ActiveRecord 5. So they actually introduced the OR method. Previously, you could only do OR by including a SQL fragment using a string with the actual SQL statement in it. And the way it works is you can chain it onto your relation, but you need to pass it another relation as the argument. You can't use an options hash like you do to where, for example. And this might look a bit clunky, but there are good reasons for this. So the reason it took so long to get this seemingly trivial feature is because once you start making more complex queries, there can be a lot of ambiguity of, like, where does this OR actually end up and how are different ORs and adds grouped together. So there were two or three pull requests made to Rails that were rejected to implement OR with the options hash until someone actually suggested that let's just do OR and you can pass it a relation. And the SQL that's generated from this looks similar to this. So it just changed the OR onto the where clause. Third tip, checking for the existence of record. I chose to include this because I see this a lot in code reviews as a chance for improvement, so I think maybe it's a method that not everyone is aware of. So frequently people need to do some guard clause or something based on the existence of some record based on query criteria. One of the common ways I see is to use where and then chain on any at the end, which works fine. Another method I see is to do a count and then check that the count is greater than zero. So the downside of these two, apart from not being super explicit, is that it will actually go and count everything. So to check if there is a single record matching your criteria, it will go and count all the records matching the criteria. So what you can do instead is to use the exists method and you can just pass it the criteria. The biggest benefit to me is that it's more explicit. I want to check if this record exists in the database. But the other nicety is that it generates a SQL query that basically short circuits itself when it finds the first record. So it only needs to go and find one of them. So the fourth tip is how you can create on-the-fly through relations using the joins method. So here I've set up another super simple domain model where you have offices having many employees and each employee has one role. So here you might not actually want a through relationship because your office might not have roles. It might not be super clear what that means. So you can actually use the joins method and you can pass it a hash like this and it will join on both these tables. So we will essentially have a on-the-fly through relationship that you can use in your query. So the generated SQL looks like this. First it will join the employees onto the offices and then it will join the roles onto the employees. And now you can query out offices based on properties of the roles of the people working there. The last and final tip is how you can merge queries with the merge method. And I think this is very powerful for the longest time. I only thought the merge method was like a poor man's concatenation thingy that you could use to merge together different scopes and still get a relation out. But that's actually not what it does at all. So here we have an even simpler domain model. We have books and the books belong to an author. And now we can do something like this. We join the author onto the book and then we select where the author is a Nobel Prize winner. And this looks fairly straightforward but you can also do it like this. All right. First it's the generated SQL. You can also do it like this. And now we might ask what is the purpose of this. This doesn't look much different from the previous version. But we must remember that this query about authors being Nobel Prize winners is very simple in this case. But in reality this query can be arbitrarily complex. So you might want to select some client based on some different complex criteria. Which means you can now contain that implementation detail inside your client's model as a scope instead of duplicating it into other places. So what merge does it actually goes and automatically merges your SQL queries together. So it merges the SQL queries not the actual results. And this is just to show that the generated SQL query by using merge is identical to the one where we hard coded the criteria inside the query. Right. So now that I told you how to find records I will give you a one non random tip how to find a job. Because we're hiring. The caveat is that we are only hiring seniors at the moment. We had tons of applications for junior developers and found some really good people. But now we really need more seniors as well. So the reason you should join is because we care a lot about code in thinker box. And in fact test coverage is one of our KPIs. So we currently have 98 percent test coverage on our client projects. We also believe in giving back which is why one of our KPIs is community outreach. So I'm actually racking up some points as well while I'm here. We also work hard and we play harder. We play a lot of football. We play board games, rock band. Anything that can be played is played. If this sounds interesting and you think that you're a senior level developer then hop on to the website tinkerbox.com.sg. Click into the careers section and then you can send us an application from there.


Okay, so I'll start my series of lectures. So this is about so-called trace functions of a finite field. And so this will cover a lot of ground, but most of it is joint work with, especially, Jean Fouffry and Philippe Michel. And there are pieces which are joint works also with Sattelgangouli and Guillaume Ricotta and ongoing joint work with Paul Nelson and Florent Jouves and Will Savin. And some stuff is in the Polymath 8 project. So a whole bunch of people.  So the mathematics we apply is the series developed essentially around here, between 50 and 30 years ago by Grottendijk, especially Deline and Katz, Lomo and a few others, culminating with the Riemann hypothesis of a finite field due to Deline and the applications. So what we are interested in is the applications to analytic number theory or to number theory more generally, but most applications that I'm going to discuss are really analytic number theory. And what I'm trying to do during these lectures will be to present this part hopefully for the largest possible audience so that you can also apply it to your own problems in an efficient way. So what I'm going to do is that we're going to look, especially at the Riemann hypothesis of finite fields, and at first begin by seeing it as a black box. And then fairly naturally we'll see that, so this black box, to apply it to use it efficiently one needs to know a little bit of what's inside. And over the course of the four lectures, the goal will be to introduce more and more of what's inside the black box so that the applications become more and more refined. And in the end, what turns out to be the case is that one reaches quickly or not so quickly, but one reaches points at where you cannot just take, let's say, especially the works of cats and apply them immediately, we reach open questions, very interesting open questions, even from the point of view of algebraic geometry where hopefully insights on aging numbers here are useful. So this will come towards the end, but today I'm going to begin by trying to present a version of the Riemann hypothesis of finite fields in an extremely general version, but as hopefully a very applicable one as a black box and we'll learn a bit of what's inside the black box to be able to apply it from this point of view. So as a first kind of reference for what I'm describing, so there's a trace function survey from PISA. So the PISA trace function survey that Philippe and Fouffry and I wrote, which is available on my web page, for instance. Okay, so the main objects will be these trace functions. So we'll be interested in prime number P. And so I say prime over finite fields, really what I'm going to describe certainly applies to any finite field, but because our applications are mostly over Q, the finite fields will be just, for the most part, just prime fields. So fpz not pz, the prime field. And we are looking at certain functions. So trace functions will be certain functions, complex valued functions defined on fp, which have very algebraic flavors. So trace functions, so algebraic structure. And the actual precise definition is quite involved. So I think that by the end, I should be able to give a full definition, but I want to try and show that it's possible to begin a little bit like when you study automorphic forms at the beginning. First time I saw at least myself the Peterson formula. Well, there was a Bessel function. There were Bessel functions. But I just started, I knew in such and such a place I found some bounds and I found some identities in Grazstein-Ritzchik. And you can build a lot just by knowing that at some point you want to know a bit better what is a Bessel function, but certainly you don't need to start by studying differential equations abstractly to study automorphic forms using the Peterson formula. So I'm going to take a little bit these special functions upwards at the beginning. I'll give examples and at some point we'll see that we need to know a bit more where these functions come from. Otherwise, we're kind of stuck when applying the Riemann hypothesis. And then I'll open the box further and further toward the end. Hopefully there'll be a full, strict, rigorous definition of trace functions. Okay, so what are the examples? The basic examples, so the first examples we'll see are additive and multiplicative characters. So first just additive characters, E p of n, so E p will be exponential of 2i pi n over p, chi is a multiplicative character of F p. So these are the more elementary ones that come in, but then the number theory very quickly you reach things where you replace the argument by a polynomial or a rational function. So in both of these I think that F is F1 over F2. F1 and F2 are, let's say, polynomials with integer coefficients. F2 is non-zero and they are co-prime. And F2 is non-zero mod p, so we can divide at least for most n, we can compute F2 of n and compute F of n modulo p and compute the inverse and so on. So these are the simplest examples and then one very quickly learns that it's useful to be able to mix them, taking multiplicative characters times an exponential. And these objects, in I think number theory, they don't arise usually just like that. We define this function and we want to say something about it. They come out in a natural way and what one is interested in, very often, maybe most often, is not so much one value or average value, so sums of these things, meaning exponential sums. So these come usually in applications in the form of sums. The first one may be that one encounters is following Gauss, something like E p of n squared and ranges over F p. This is a quadratic Gauss sum. Quite quickly also called Gauss sums are Gauss sums attached to characters. So as long as one is interested just in upper bounds for this, so it might be that you want something more refined. So for quadratic Gauss sums, the most important application doesn't just want the size, it wants to sign the argument of the Gauss sum. But in many applications, it's really the upper bounds. And as long as you just have these two and you only want upper bounds, you're in very good shape. In a few cases where the modulus of the sum is known, so here everything is nice, meaning if I call the first one S1 and second one S2, then S1 and S2 have modulus code of p and S2, this is when Ca is non-trivial. And then you might want to tweak these things. Sometimes you see that extra parameters come in, a in fp, and this would then still be true for a non-zero. But basically also one quickly realizes that these are essentially almost the only ones where you can actually compute the modulus exactly. So I guess maybe an even more trivial case is just the full complete sum of just a character over a group, in which case when the character is non-trivial it's zero. So almost any other sum does not give a simple formula either for the sum or for the modulus. When you take a sum and one of these special functions are already wrote down, so I guess maybe the other nice exception are Jacobi sums. Sallier sums even do have some formula, but it's not actually such a nice simple formula. And here the kind of basic case is the Clusamansum. So classical Clusamansum in one variable, which I will denote by kl2. So I think this was the notation implicitly used by Tau also, which for us it is convenient to normalize by a score of p immediately. And then this is something which sums over invertible elements mod p and ep of nx plus inverse of x computed mod p, often denoted x bar. So Clusamansum were introduced somewhat implicitly by Poincare when computing the full expansion of Poincare series and explicitly and studied already quite deeply by Clusamans to study representations of integers by quaternary quadratic forms, integral quadratic forms. And for these, so they have many, many, many applications in attic numbers theory, and there is no explicit simple formula to tell you what this is or what the modulus is you need just estimates. So here one can only hope and it's a generic situation. Now one of the important features of this example, which already throws you way beyond the theory of André Vêle. So the best estimates for Clusamansum, the integral Clusamansum are due to André Vêle. But now if you start thinking of this as a function of n, so an important feature is that these sums themselves as functions of n are kind of nice. And so it's important to define it also at zero which I will do by sending it to minus one or a square root of p and that's the right definition. Actually that's the definition that comes in there. Naturally I don't touch anything. So is also such a special function, modulo prime. And more generally many families of exponential sum parameterized by some element in fp will themselves be nice functions in this sense which immediately will give us a lot more applications of the Riemann hypothesis than just playing around with the first example. So from the point of view of more classical things you should think of this like KL2s are analogs of Bessel functions in a figuratively precise sense and so saying that we want to see these as nice functions is like saying that Bessel functions are just as nice as the exponential function essentially or like power functions. So in classical analysis the first interesting functions are exponential functions and power functions and when you start going to Bessel functions these are just as nice once you start learning about them and so on and this step of going from characters to Cluson and sums or such exponential sums is also of this kind. So the crucial feature that the Riemann hypothesis tells you is that if you look at these trace functions so whatever they are we already have a few examples I will give some more in a second provided you restrict to a certain subset which are defined by some kind of irreducibility or normalization condition then they behave like quasi-ortogonal vectors in a finite dimension real-world space and this quasi-ortogonality gives you code cancellation in a generality which is way, way beyond what the serial your Andrevelle can give you for just one variable character sum. Okay, so our h will be a statement so now I have to here say a little bit on where the functions come in but then this almost a definition but not quite a definition will be replaced by again a series of examples and then the statement of the Riemann hypothesis. So what if you speaking what happens is that you have certain algebraic geometric data which can be defined in different ways so it could be garrow representations of the rational function field in one variable which is not quite the same but you could also think of this as automorphic forms for this field but automorphic forms of arbitrary rank are just classical GL2 things or you can think of this so the geometric point of view is something called eladic shift of a certain type and it doesn't really matter whether you understand any of these words for most of these lectures so we'll see that it's there are questions at some point that you can only answer if you really have an idea of what one of these means the most convenient I find is the eladic shift point of view but the easiest to define would be the garrow representation so that's probably what will emerge as a definition as time passes now these are algebraic geometric data and you can think of them really as analogs of automorphic forms so you know an automorphic form over a number field is a relatively complicated object but with many nice properties our garrow representation is a nice object which is highly algebraic at least structures with many environs and we will send these to the space of functions let me call it c of fp which is the space of functions k defined on fp with values in c so we have some highly complicated objects with problem normalization these typically define countable infinite sets and we send these to this five dimensional vector space of dimension p over the complex numbers so the trace function is an invariant which again if you think of analogies with automorphic forms or even with Dirichlet characters modular large modulus q you should think of this maybe as being the data of the first values of the character or the first free coefficients of a modular form so it's not full information about the modular form if you think of the classical case of gl2 forms you just take let's say the first q to the 10 echo eigenvalues or something like this or q to the 1f echo eigenvalues but this is enough to get a good handle on the algebraic object as well so there's one very important fact that is analytically completely crucial so from the point of view that we have some complicated algebraic object and then we just take this invariant which is the trace function the trace function only a priori contains some of the information and the map is not injective so in particular there are literally infinitely many objects which have the trace function identically zero modulo p so if you just think of the trace function by itself as a function defined modulo p doesn't tell you that much because it comes from infinitely many sheaves or representations and some of these are highly complicated and some are a little bit less complicated so we need a way of measuring the complexity of a trace function so that we can work with objects with small complexity as a way of saying that we have a control on the function we get and that the function gives information so it's like saying that if you just look at the value at 2 of the Dirichlet character it's not telling you that much but if you know that the conductor of the character is extremely small then it's giving you a little piece of information so there is an invariant which is an analog quite close analog of the analytic conductor for L functions of so either of these objects or sheaf representation automorphic form and we will usually work as p varies with so for every prime p you would have a trace function kp which as p varies so they vary really with p but they come from sheaves or whatever which have complexity bounded uniformly in terms of p from objects with uniformly bounded complexity ok and so by abuse of notation we will write c of k for so maybe the smallest complexity of an object which gives rise to a trace function k so let me say that I will fix so these objects you don't necessarily need to know whatever they are at the moment let me call them representations let me call them rho and t rho will be the trace function and assume that trace function is k and I will use c of rho for the complexity of representation ok and as you will see just like you can say a lot about Bessel functions or you can use Bessel functions quite efficiently just by knowing that they occur in certain formulas and satisfy certain bounds at infinity and certain integral identities one can say quite a lot about trace functions one can apply them quite successfully just knowing many examples bounds on the complexity and the Riemann hypothesis which will come in a second so first more examples with examples of the complexities ok and so whatever I'm going to say later if it seems abstract you can always go back to these examples and usually even with just these examples and a few basic operations you already have lots of interesting features phenomena and what you need for many applications so the abstract theory is useful but is not necessary to already say quite a lot ok so first there are the examples I already gave I have to tell you something about the complexity of these objects so we take a rational function as before so now I'm going to take the polynomials in Fp of x co-prime F2 non-zero and so in the theory because we are looking at objects which are a little bit like primitive Dirichlet characters in contrast with just all characters so the primitivity means we have to be very careful with the value at all so which would be the prime numbers or integers in the classical case so we cannot just say that let's say at the point of interminacy we don't say what happens so where F2 as a pole we also have F as a pole we have to say precisely what is the right value of the function this is important for some statements to actually work so precisely so we define k of n to be Ep of F of n when F2 of n is non-zero not p so in this case F1 of n is non-zero because the two polynomials are co-prime so this thing makes sense and zero otherwise so for k of F of n if F2 of n is non-zero and F1 of n is non-zero so for multiplicative characters we also have to say what happens when we evaluate the multiplicative character at zero and there are two further cases so if k is non-zero sorry if k is not the trivial character and F1 of n is zero we take zero so the classical extension of a Dirichlet character at zero is k of zero is zero if k is non-trivial but if k is trivial it should be one so it's one if k is equal to one and F1 of n or F2 of n is zero so to a certain extent this is technical so in I think number theory if you were to have a different convention for zero you would just switch the value at this point and do trivial estimates for the switching but for the algebraic statement it's crucial to have the right conventions and normalizations so that's one so we have to say what is the complexity in both cases the complexity is bounded I mean I didn't define it but I can tell you this is going to be bounded by the maximum or the sum let's say of the degree of F1 and the degree of F2 with an absolute implied constant would be more precise when one has the correct definition of the complexity but for basic applications you just need to know so how is this useful it's going to be useful by saying you take F to be a ratio of integral coefficient polynomials which are fixed and then you take P to be a variable prime going to infinity and if F1 and F2 are fixed the conductor of whatever you have as a trace function modulo P will be bounded independently of P so that's the first example second examples are hyperclustrum and sums so let's pick an integer r at least 2 and kr of n and P is r ok so here I have to be careful so 1 over P to the r over 2 r minus 1 sorry sum over x1 to xn in Fp xr thank you such that the product is equal to n of ep of the sum xr sorry and that's when n is non-zero and when n is equal to 0 again one has to be careful and if I remember things right it's minus 1 to the r divided by P Philippe do you remember I think it's that ok anyway so it's important that it's there when n is equal to 0 so there's a way of computing these values right so it doesn't agree with r what? so r equals 2 I had written 1 over P yeah so is it r minus 1 over 2 I always forget this one so ok I'll write this for the moment so that it coincides with the case n equals r equals 2 and I'll correct that the next lecture because this is something I always forget ok so for r is equal to 3 these are the 3 variable custom ensems or 2 variable custom ensems that Tao spoke about when explaining how one handles the so-called type 3 sums in John's work all the ternary divisor function in arithmetic progressions you can write it this way so if n is 0 I mean if n is a multiple of P that means n equals 0 then you have to use the second formula anyway and if n is non-zero then the x i have to be invertible yeah yeah or everything is different over Fp all my functions will be different just over Fp ok so you could actually write Fp star yeah but I mean I always forget how you don't defend this the first and if you write it correctly then the n equals 0 case is given by the same formula I'm not sure if the way I wrote it down will actually give the right formula for n equals 0 that needs to be r minus 1 the exponent 1 to the power r minus 1 well yes and no because I I mean for i is equal to 2 I switch the sign I mean the right definition of minus 1 over the code of P times the sum for i is equal to 2 which I didn't do so I just anyway so it's not an important thing I'll clarify this the next time but for i is equal to 2 you should have yeah you're right it's probably r minus 1 anyway ok so this is a deep fact is that these are trace functions and this is due to the linear and I can show you that this is indeed a deep fact by the following first kind of again another fact about the complexity so I didn't tell you about the complexity too much but there are two properties of the complexity which are maybe I should say this after the next example but I might forget so let me say it now so it's a little bit this is kind of a semi-digression but it comes at a good point because the strengths of these remarks will be clear so two facts about the complexity are important so the first it's a little bit like a height in the sense that when you look at the set of trace functions with a suitable normalization which I will discuss later with complexity bounded by a constant this is a finite set so the whole set of trace functions is infinite just like the set of D.H.A. characters is infinite but when you bound the conductor of D.H.A. character you get a finite set and this is a similar feature so this is a rough way of saying it because we need a little bit more information before we can state this precisely but roughly speaking the set of K with complexity less than or equal to X is finite up to trivial modifications of the trace function so this is a relatively rough statement but it gives an idea already of what is the content of it and so as you maybe know is if you take the analytic conductor as defined by Ivanets and Sernak the same statement is true for automorphic forms over a number field and the second remark is explanation of why this is really deep a basic fact about the complexity so one has already a first way of showing that the complexity is interesting the way it's defined it has the probability that the L infinity norm of K is bounded by the complexity so whatever complexity you have to measure how complicated is a function one way that a function can be complicated is by taking large or very oscillating values and here we're saying that the L infinity norm is bounded by complexity and here what I didn't say which I should have said so with the complexity of KLR mod P is bounded by one with a constant depending only on R so let's apply this fact so KLR functions with bounded complexity and the fact that the L infinity norm is bounded by the complexity whatever it is so corollary immediately is that there exists constants AR non-negative such that the hyper KLR sum and P is bounded by AR for all n mod P and all primes P now because I divided by let's say for R equal to I divided by code of P this is a weak form of the veil bound for Clusum and Somes so if A2 was equal to 2 then this would be the veil bound for Clusum and Somes which comes directly from knowing that it's a trace function with this type of properties and so it's not surprising then to see that to prove this that these are trace functions in the sense I'm mentioning already the Riemann hypothesis of fine fields in a non-trivial way How complicated is the 0 function? Bounded complexity it depends the way you define it but I think with our precise definition it would be 1 Oh no, okay I thought you think it's possible to have a trace function which is identically 0? Yeah with large complexity but the smallest complexity would be 1 and actually it's an interesting question to find the smallest complexity of a non-zero object with trace function 0 and these I actually don't know so the upper bounds we know is P or lower bound so we know trace functions with complexity roughly P which have, sorry we know objects representation with complexity roughly P with trace function 0 and what is the actual threshold I don't know So already this is you see that just knowing that an object like Hypoclus and Somes have a trace function without knowing anything else so this is a type of information that maybe in 10 years there will be an analog of Grazstein-Riegic for trace functions and this is the type of thing you would find in the first two pages of this hypothetical motivic Grazstein-Riegic and you wouldn't need to know anything about veil bound or hypoclusom and Somes bound you would just say this is a trace function so it's bounded by complexity, complexity is bounded and then you apply that okay so this is somewhat similar to the fact that Bessel functions decay like one of a code of X at infinity in a very rough analogy it's a much deeper fact but okay so now let me go back to after this small digression to more examples to show you the variety of type of things that can arise so another fairly general class of trace functions are those which you obtain by counting solutions of certain polynomial equations so first example would be you fix a non-constant polynomial F mod P it could be a rational function also but I do it for a polynomial and you count for a given N how many X there are in Fp with F of X is equal to N and you just count them so this is integral value it's only bounded by the degree of F because F is not constant it could be 0 or fairly frequently but it is something so this is a trace function and its complexity is bounded in terms only of the degree of F so then so K is a trace function with C of K again up to an absolute constant bounded by the degree of F and therefore if your polynomial let's say X squared which is an integral polynomial then you get something which has complexity bounded independently of P and in this case of course it's not how to compute the number of solutions to X squared is equal to N it's 1 plus N over P and the last example which is again just a very special case of something more general is if you try and count not the solutions to this type of one variable but number of points on the family of curves and the simplest example is really interesting example is probably the Legendre family of elliptic curves so you take K of N to be P minus in this case one has to normalize in this way so the number of solutions in F P squared of Y squared equals X, X minus 1 X minus N so this is done in such a way that the formula is correct also for an equal 1 or 0 so that works for every N so is a trace function I have to divide by the code of P with my definition so it's the AP of this elliptic curve divided by the code of P so it's a trace function with complexity bounded as P varies and you can basically replace this family of elliptic curves by almost any family of algebraic variety depending on one parameter that you can think of up to technical issues what exactly are they mentioned? Well I mean you might have to extract the right at the bad points well first there's the bad points and also if you take a surface boundary instead of this would only be the right thing if there's just middle dimensional but for families of curves except for the doing the right thing at the bad points this would be the right this would almost always be a trace function ok so this already gives you some insight and so one of the things you can see is that once you have this already just with these examples any statement which is valid for any trace function which there will be such statements will immediately give you many many different types of statements which very often I think number theory would otherwise be kind of treated one by one with maybe different ad hoc methods and we'll see examples of some of the things we did with Fouffry and Michel where we have statements which are valid for every trace function so that looks a bit abstract at first but once you have a list of examples then you see we have actually lots and lots of statements ok now I can state the Riemann hypothesis right I guess I need a bit more room for such a big program so I think I'll keep the examples visible at least their definition if I do it this way so because you will see that I'm going to state the Riemann hypothesis I cannot quite state it just with using only the words I've already introduced I will need to introduce at least one extra property of trace functions which I will then interpret in a concrete way to see that the statement at least implies almost immediately the qualitative version of veils estimates for exponential something one variable so I keep this definition because I want to be able to refer to them so because the extra word I'm going to introduce will be a conditional trace function will be always satisfied for the example one which is the classical exponential something one variable case so this is essentially the lean but it uses also work of dot and d can show other people but these are the two main ones and the really essential work is that of the lean ok so p is a prime and we take k1 and k2 trace functions modulo p with complexities at most c1 and c2 respectively so I could say equal to c1 equal to c2 I want to emphasize that usually complexity you don't really know what it is because it's not such a very meaningful number what is meaningful is being big or being small so and you have to think let's say of c2 equals 10,000 c1 equals 10,000 and there's already lots and lots of such things as the examples above already show ok now we need an assumption k1 and k2 so the statement would be a quasi-autogonality statement but for that we will need some assumption so we assume k1 and k2 are what is called geometrically reducible so I will have to explain what it is and I will do it afterwards you can already take as information that in case one this is always true so in case one if you have k1 and k2 both of this type so one additive one multiplicative or both multiplicative both additive whichever way then it is always automatic then there are two cases so if rho1 and they are associated to representations let's call them whatever these underlying objects which I call rho1 and rho2 so if rho1 and rho2 are not geometrically isomorphic so it's another piece of information I will have to explain and I will do it afterwards and in case one in the case of one variable character sums it will again be quite clear what this means then we have code cancellation when we correlate k1 with k2 so we sum over fp k1 times k2 bar so it's an inner product and we get code cancellation with implied constant depending only on the complexity and one can compute that the upper bound can be 3c1 square c2 square with the definition we've taken of complexity so this is uniform code cancellation as long as you have the assumption and as long as c1 and c2 are bounded I mean it's even uniform in such a strong way that you don't need to have c1 and c2 bounded with p and still you get extremely good bound so that's the first statement let me put the second part here so I still have the examples on the blackboard Is this condition independent of the row one and row two are sort of unique? Right, but the condition to have this is that row one and row two are it's a condition about row one and row two so what this will reveal is that once we have the second version what this will reveal is that there can only be one row one with small complexity and with a given trace function so there's a gap between the complexities of the trace function that's a little bit what actually I was asking okay so second statement is what happens in the other case so if row one is geometrically isomorphic to row two okay so first thing is we don't know what that means but it's just a consequence which we might be able to falsify meaning that if we falsify this condition then of course we're in case one so then first information is that k one is proportional to k two for some alpha with modulus one it's really proportional everywhere so including possible bad points that has to do with the fact that we're using implicitly correct definitions of the values at every point it's not orthogonal but quasi normed meaning that the sum of k one over x k two over x conjugate well of course it's alpha times the sum of modulus of k one over x squared so we cannot expect cancellation and in fact what we have is more like an asymptotic formula this is alpha times p so in particular if k one is k two this is exactly p and up to an error which is again cancellation and the same bound as in the case here so I have still about ten minutes so once we understand a bit better what these conditions mean and we have tools to handle these conditions checking them and so on then this is really a an extreme general black box version of the most general form of the Riemann hypothesis of finite fields so it's not quite equivalent because we're still working with just prime fields and sums with non-viable but it's it's basically the whole thing from the at least from the anti-normacy point of view I mean there are other consequences of the lean's work which are more algebraic geometric which this is not really trying to catch okay so now yeah if you take the product of two trace functions I guess maybe it's not quite trace function because of that actually like products of two Dirichlet characters which are primitive might be primitive you might have to change at the bad points the value to get something primitive but there's a unique underlying kind of primitive object that arises from this Emmanuel is twice the character also a trace function uh yes but then it's not geometrically reducible it would be chi plus chi two different components if you multiply chi by a complex number of modules one you can say that it's still a trace function that's geometrically reducible but if you multiply by two I mean there are different ways of doing it but the allowed way is to do chi direct sum with chi and then the trace and this is not irreducible anymore because otherwise it wouldn't work how about the k1 equal to k2 how about the k1 equal to k2 you either you're in this case and then you you say the RU1 jump to chi isomorphic put RU2 then k1 rpk2 but how about k1 not equal to rpk2 if they are geometrically reducible then this is true and otherwise this is going to happen in situations where the complexity is so large it's only can happen that they are not geometrically they are not geometrically isomorphic but k1 is equal to k2 this can happen but then the c1 or c2 will be extremely large so that this bound will be trivial so you see this bound is very powerful when c1 and c2 are small enough that essentially the product is less than the code of p because what is the trivial bound maybe I should have said that so the trivial bound for this sum k1 of x, k2 conjugate of x so trivial bound is p times the L infinity norm on both and we know this is at most c1, c2 so you see that if your complexity is too large then this is meaningless give no information so there is no contradiction so what does it mean to be geometrically reducible so geometric irreducibility and isomorphism yeah in the second statement that c1 is equal to c2 yes so the invariant is an invariant of so these objects one and row two they are really meaningful up to sorenizomorphism class and in particular c1 equals c2 in that case so what does this condition mean so here we have to enter a little bit into where these functions come in so if k is a trace function of a certain object this implies there exists a homomorphism so row is really a homomorphism of a certain group pi, pi1 to a certain five-dimensional vector space so it's a five-dimensional representation you can think of v as a complex vector space this is algebraically not the right thing to do it should be an laddic vector space but it's only fine algebraically to think of it such that k of n is the trace of row of certain elements Frobenius elements depending on n acting on a certain subspace vn so vn is the invariant under inertia which I don't want to go into right now but the subspace okay I'm going to ignore this second information so there is certain object so row is really a map from a certain fixed group so this depends only on p into a five-dimensional vector space so gl of a five-dimensional vector space so if you have a representation you can ask that the representation is irreducible and this is almost what we want so precisely there exists a normal subgroup p1g of p1 and to say that row is geometrically irreducible will mean that there is no this does not exist so means first v is nonzero and there does not exist w inside v not equal to v nonzero stable under the action of the subgroup okay I'm still trying to keep all my examples but it's becoming difficult what? no there exists such a normal subgroup so okay more precisely so I can tell you what is p1 then so that there is no ambiguity so p1 is just the Galois group of a separable closure of fp of t so it's a group that depends only on p obtuseomorphism and the normal subgroup is the Galois group of the same big field so algebraic closure of this function field modulo the smaller group of rational functions with algebraic coefficients so it's not I mean in some sense these are not such complicated objects they are quite mysterious from the group's theoretic point of view and often one doesn't need to know really what they are but that's what it is okay so we have this condition and therefore in particular we suppose that the dimension of v is 1 so this happens in case 1 in the first example so meaning character values add a rational function additive or multiplicative so one has to know that this is the case but this is the case then in this definition you don't need to know anything v is of dimension 1 you're looking for proper subspace so it is automatically geometrically reducible so that's an example okay next geometrically isomorphic so rho1 into glv1 and rho2 into glv2 geometrically isomorphic if and only if a linear map linear isomorphism from v1 to v2 so they are the same dimension there's a linear map such that okay I have to write this diagram correctly so for every element in the group whatever this group is this diagram commutes it just means the action for every g in the subgroup sorry gamma I should call it pi1g so the first instance to begin with what you should think of at the moment as what does it mean to be geometrically isomorphic means trace functions are the same that's what you should think of and we'll see more refined versions of this in a second okay now it's becoming harder and harder to okay I'm starting to get a bit over time but since I'm an organizer I'm allowed that's not the right one so I want to show how this is a qualitative form of vales estimates so just by knowing this you already can apply it well enough to recover vales estimates for one variable character sum okay I guess I have to erase that for example and I'll just do the kluson and sum case for simplicity that's the most interesting one for many applications so r equal 2 so in that case k1 of x to be the kluson and sum you take ep of nx plus x inverse with the usual convention that it's 0 when x is 0 when there's no n going too fast so you have a non-zero element which is fixed let's call it a to avoid confusion and k1 of x is ep of ax plus x inverse for example 0 and is 0 for x modulo p for k2 you take the function 1 which you can see as ep of 0 if you want so it is a trace function and by example 1 and by what I just said so complexity of k1 is bounded independently of p and complexity of k2 also and they are not only geometrically reducible because of rank 1 as I said and therefore now I have to erase that start with these examples again tomorrow so we did use from rh in the form I stated that the kluson and sum with parameter a of p is of modulo is bounded by the code of p with an absolute implied constant provided these two things are not geometrically isomorphic so whatever rho1 and rho2 are not isomorphic geometrically so can it be that they are geometrically isomorphic suppose they are then what happens that means in particular by case 2 if they were geometrically isomorphic k1 would be proportional to k2 k2 is a constant so k1 would be a constant and as soon as you have one element on 0 this function cannot be constant because this one this is 0 so the constant would have to be 0 but then it doesn't work so as long as you have a field with at least two elements you are fine but k1 is not constant because p is at least 2 so we get a qualitative form of vales estimate without having to at all except a black box so I'll stop here and tomorrow I'll start to try and do more complicated examples so I'll try to start doing the sum that comes up in Jean's work and see where we get stuck and why we need a bit more information I think we're buying version of Rh so we could take this in our product of k1 and k2 and say scale that x and k2 by some n and then normalize by one of the repeat would that also be as what do you mean? so I'm trying to generalize the construction of Kluestring and sums here so there are sums of some k1 times k2 where one of them is with a scaled argument so by scaled argument do you mean I want the sum of k1 x times k2 and x right so this will be no problem so we'll see trace functions not only we have these first examples but they also have a rich formalism and in particular if you replace x by n times x for any non-zero element of the field then you get another trace function with the same complexity or you could replace it by one over x works exactly but now this sum is a function and then you want this as a function of n then this will often be a trace function there has to be conditions but for kind of standard random examples this will be a trace function and the complexity will be bounded solely in terms of complexity of k1 complexity of k2 this is highly non-trivial so this will require very recent work of cat on multiplicative convolutions but this exists, yeah in that example where f is x squared and the k of n is 1 plus this doesn't cancel an average because it's not geometrically reducible as you can see so one of the consequence already of what I said is that for small complexity the two estimates I wrote are incompatible so the fact that there's cancellation if you take k2 equal to 1 if you take k2 equal to k1 if sum of k1 squared modulus squared is essentially p this is equivalent to geometric irreducibility for small complexity so one quick way I'll say it tomorrow if you are given a function you know it's a trace function you want to know whether it's geometrically reducible or not, compute the mean square if the mean square is close to p then it is geometrically reducible and we'll see examples so finally you can check that closed sum and sum are geometrically reducible this way the mean square of closed sum and sum is easy to compute and it's essentially p just by i actually so I'll say this tomorrow in more detail there's not irreducible is there some natural way to break it up into pieces so we know there exists a decomposition into sums of geometrically reducible pieces there's a little technical tweak but essentially it's true it might not be easy to find them so in principle you find them by taking inner products it's like a little bit like an orthonormal basis of a Hilbert space I mean there are infinitely many basis vectors so we have a fine dimensional space but somehow with infinitely many basis vectors we can compute all of them to determine the decomposition so if you cannot compute it because you have extra information it can be very hard so a number of the open problems we still are fighting with in various applications is because we have something we don't know that it's geometrically reducible and we don't know exactly what are the components so it's one of the fundamental problems that can arise in applications