Welcome. Did the coronavirus just crush all hopes of a fairer world, or is this the perfect chance for a great reset? This is the Global World Economic Forum's Jobs Reset Summit. So far we've had three days of visionary leaders debating from the words of business, government, civil society and the media, reshaping growth, jobs and skills with the main topics of the last few days. Today, equity, inclusion and social justice. Our guests are McKinsey's Kevin Sneeder, Ashley Shelby Rosette from the Frequent School of Business, and Laura Thompson from the International Organization for Migration, as well as Fumzilla Mlambo Nunguka from UN Women. She'll be joining the conversation a little later in the show.  It's a tricky one. I've been spelling it all my life. It's not an easy one. I get checked at almost every second airport for explosives. But it's nothing to match the anxiety of growing up as a gay man. That was very difficult in the Australian media landscape. It changed very much when I came to Germany and it's changing every day. But for a lot of minorities, the world hasn't changed enough, especially in the workforce. And a second lockdown is now here. Corona is putting a lot of the development and progression back into reverse. It's something I'd like to ask Kevin Sneeder about. Is it within this contracting economy harder or easier to embed social justice? Well, I think the reality is if we think there's a reset and the need to hit a reset button, maybe we should also be hitting the panic button because we've reached this moment in time where so many forces have come together that a more fundamental question has to be asked, which is, are we now at a point where if we don't do something, the society cohesion that we think is so important to progress is going to be lost? And I think the answer to that question is yes, we've reached that moment. There are so many numbers, Ben, that point to that reality, whether it's in gender, social justice, and many other aspects of equality. Let me just give you a few and put them in the context of the pandemic. Because arguably there actually had been real progress. You talk about what you had seen personally in Germany and other parts of the world. And I think we could point to some numbers. In gender, we could certainly point to progress. And in race, we could point to slow but meaningful progress. Income inequality gaps beginning to close. However, the pandemic really has been a moment of reset. And the answer why? Because it has disproportionately impacted those who are most vulnerable to it. And they often are the minority groups. In fact, it's intersections, the places where different types of minority experience come together. We see that most profoundly. Let me just offer a few numbers. Black women are three times more likely than white women to report the death of a loved one as a top priority in this crisis. That's another aspect of the fact that we know the mortality rate has been higher in the black population. Black women are twice as likely than other women to say they do not have strong allies in the workplace to help them in this moment. And one in three have considered leaving the workforce completely in this moment on the back of the pandemic. Those numbers are in the United States. Frankly, it gets even worse in other countries. And that's why I think we talk about reset. But I think we need to add to reset a real sense of urgency, an emergency, a panic moment, a moment when the progress that has been made could go backwards. I could go on, but I know in the interest of time, let me just answer your question. It doesn't have to be a negative moment, though. There is still time to respond. There is a moment when we could realize that actually flexibility, the opportunity to rethink the progression in careers, the chance to use this moment when there is a lot of stimulus and support being injected into economies to target it in a way which addresses some of these injustices and inequalities. That's there too. And I haven't even begun to tackle the topic of the type of injustice that we saw come to fore on the back of the killing of George Floyd and other incidents around the world. But let me stop there. I know there's a lot to cover in this session. Well, actually, let's stay in the US and pick up on those numbers. What sort of gaps are there? There's a lot missing as far as minorities and ethnicities go and when it comes to policymaking. I mean, a bunch of white guys write the code when it comes to tech and tech could have been so unbiased, but it doesn't look like that's the way it's going. What are some of the gaps in data and findings? Yeah, I think building very much on what Kevin has described then is that when we look at the gaps, what we have to fundamentally look at is the data. So when we look at the data, we think about the way in which we have to disaggregate the data, the way in which we have to have to interpret that data and making that data availability in order to see some change in the reset that we would like to see. So one of the stats that's oftentimes quoted in the United States, probably the most popular one, is the notion of the unemployment rate, the jobs rate. People anticipate this rate every month. So back in February, that rate was around 3%. In April, it was around 16%. Recently, it's been around 8%. But if we start to disaggregate that number, we look and see what's that unemployment rate for blacks versus whites. For blacks, it's around 13%. For whites, it's around 6%. If we disaggregate it even more and add an intersectional lens that is encompassing both race, gender, and age, we see that young black women, again, building off the stat that Kevin mentioned, young black women have an unemployment rate in the United States of 25%. That fundamentally says that when a black woman is looking for a job, she's experiencing something very different than is a white man. And if we attempt to try to dismantle and address this inequality with a sweeping brush that's the same for all, it's not likely going to result in the change that we would like to see. So number one, we have to consider the aspect of disaggregating this data and looking at it at a more granular level in order to try to adjust some of the change. Number two, we want to consider that how exactly are we interpreting this data? Again, recognizing those fundamental distinctions and acknowledging that these experiences are embedded in larger systems and larger structures that we have to address. So if we talk about the research, we know that, for example, whites are more likely to be called back for a job even when they have a criminal record, more so than our blacks. We know that resumes with white-sounding names are more likely to be called in for interviews than our blacks. We know that blacks are actually encouraged to whiten their resumes in order to get jobs. Once in the organizations, we know that if you choose to advocate for the diversity, that there can be serious repercussions in terms of perceptions of competence with regards to blacks and women, but women, but very little with regards to white and men. And we also know that fundamentally, and this is speaking from my own research, that there exists this aspect of a white standard of leadership. And what does that do to the aspect of not just getting into the organization but matriculating up the organization? And the third aspect I'd probably say when we talk about data and gaps of research is the availability of the data. That is, we don't know what's happening inside the organizations because the organizations don't make that data available. We have seen, you mentioned tech, right? We have seen some advances over the last few years in terms of transparency with these various diversity reports in tech, but that's been a very, very small example. We saw recently, Price Boardhouse Pupers also chose to have a diversity report. But we're going to understand what's happening in these organizations and try to adjust, to try to influence this inequity. We have to know, one, what are the numbers? And two, what are the organizations doing and whether or not what they're doing is actually working. Like Kevin, I could talk on and on and on, but I think I'll just kind of leave it there for now and see what questions are posed. I like the fact that we're talking data, but what about tech and what about people with disabilities? They've often been pioneer users in that sector. Do they hold the key to some of these answers here? Ashley, maybe you can answer that. I'm sorry, in terms of disability and... For people with disabilities, they're often being pioneer users in tech and using programs that a lot of other people would never have needed to use or come across. Could they hold the key in some ways to finding solutions to see that we see equity put at the center of policymaking? Yeah, I think we know that whenever we attempt to try to resolve disparity in considering marginalized communities, it causes a certain level of innovation. It causes us to think in ways that we have not necessarily previously needed to think of. And so previously needed to think. And I think that can be a source of trying to incorporate and be more inclusive in our processes that we have in our organizations to help minimize these inequities. But as you mentioned, then we have to have a willingness to start to address these marginalized communities. I think we have a willingness to address things like disability because people fundamentally agree on that. When it comes to things like race and gender, those are a little bit more difficult. So I think it can be an in-road, but it's not an ending, if you will, because of the way in which the distinction in terms of how we look at disability versus race versus gender, versus religion, versus, you know, all the other disparities that we have. So a beginning, but definitely not an ending. I'd like to bring in Laura Thompson now from the International Organization for Migration. Laura, what signs of change are there that this moment of flux that we're talking about could actually be used to proactively embed a greater equity in the new economy when it comes to migration, which is the field you work in? I think we have to look a little bit to what the pandemic has brought to migrants, and I would say there are two signs of it as everything in life. One part is the negative side, and obviously there is a loss of jobs that has been extremely important for migrants, particularly those that were undocumented. Reduction in remittances is another big part with some of the places even going to half of the remittances that they were sending in the past, like in Central America. Thousands of people that were stranded in the middle between transient countries or some of them in the destination countries that were not able to return. They didn't have enough funds to cover their state. And other type of things like anti-migrant sentiment, xenophobia, and this type of reactions. On the other side, there have been a lot of very positive aspects as well, because very quickly governments realized that migrants were part of the solution to this pandemic. And these applied in two main sectors. First of all, in the health sector, where they saw that there is a huge amount of doctors and nurses that are foreigners and that are serving in developed countries particularly, and that were very present in the response to the pandemic. But the other side was the agriculture and distribution supply chain sectors where again migrants were predominant. So there were a number of, I would say, innovative measures that were taken by a number of governments that are quite positive and I think open a good opportunity. Now I see two or three opportunities that are going to be very important for the future. The first one is global health for everybody. This crisis has shown that providing access to health to migrants, including the undocumented migrants, is not a generosity, but it's a total need if everybody wants to be safe. So that is extremely important. The second element that I see as a very important element is that this crisis could have been the beginning of a change in the discourse about migrants and the negative perception about it because it has shown the positive contribution that they make, but also with regard to skills. Very often we talk about high skilled people are welcome but the lowest skilled people are not welcome, and this crisis has shown that the lowest skilled people are important too. And the third element that I will bring that I think is going to be quite innovative in the future is the possibility of thinking about the labor market and how this virtual reality that we have created the capacity to work is going to have an impact in the labor market, whether we are going to begin talking seriously about virtual labor migration and how governments will need to adapt in order to allow that in a much easier way. Joining the conversation now is Fumzilla Mlambo-Guka from UN Women. Fumzilla, tell me just how badly has this pandemic exacerbated inequalities? Can you hear me? I can. Okay, thank you. Thank you very much. Well, women are disproportionately affected by this pandemic, and there are really three ways that we have identified as particularly concerning noting that there is no crisis that is gender neutral. We've learned from previous crisis, the Ziga, the Ebola crisis also showed us the gender dimension of these crisis. And Ebola, for instance, the restriction of the mobility of women severely affected their livelihoods. And in this pandemic, we have seen that we saw that men were able to recover after the pandemic and go to pre-pandemic level of economic activity. But for many women, they lost everything without insurance and capacity to recover. And of course, women are also in insecure employment, mostly informal, and that does not give them some of the cushioning that you are able to have when you have a formal employment. Women are also overrepresented in the hard-hit sectors of the economy, such as tourism, hospitality, and the service sector. And thirdly, the crisis of unpaid care. Women's unpaid care in families and communities has been too long taken for granted, but in this crisis, it reached a new high. And one of the biggest takeaways from this crisis that we should all work for is that we should have proper policies to reduce unpaid care on women, to remunerate it and to redistribute it. Whether it is through parental care, it is still accessible and affordable childcare, we need to make sure that we do not have this sustained cushioning of the economies through free labor of women that is not recognized. So women are taking much harder to recover, because if there isn't a second way of dealing with unpaid care, they are unable to be ready for the labor markets again. We are hoping that both privates and public sector will be seized with finding a way out for the women so that we can build back better for women as well. Thank you. What is the way out? What's your advice to women who've pulled themselves out of work, out of jobs, to care for kids, to look after households? How can they get back into work if that's what they want to do? Well, it's not about women having to do this just by themselves. This has to be a partnership with the other stakeholders who have the adequate authority and resources to intervene. And one of the ways is indeed addressing the social, through social protection, so that we allow for women to be in transit to a better and a more liveable life. But also the fiscal stimulus that many governments have announced needs to target women much better and to make sure that going forward women do not go back to the pre-pandemic levels of informality that makes them non-beneficiaries of interventions by government. From what we know at this point is that less than 18% of what is available in the massive fiscal stimulus that are there is targeted and is likely to reach women. And yet women are amongst the majority of those who will be unemployed. The ILO estimates that about 500 million women will be out of jobs and many of them, not unless there is intervention, they will never be able to go back to a normal working life. Let me bring up then this need for an equi-covery, a quote from Anna Marie Tousey who was involved in these panels just yesterday. The old normal didn't work for many of us. How do we then, Kevin, how do we put equity at the centre of things? Well, here's the opportunity. We've obviously painted a picture that amplifies the urgency of the matter. But there is the potential for a more diverse next to workforce and technology does have a fairly fundamental role to play. So I recognise this does not apply globally because access to technology indeed is one of the issues. But let's hold on that thought for a moment. One of the realities of the old normal was there was a lack of geographic flexibility. The labour pool that you sought to recruit from was the one in the immediate area. And quite often that brought issues of race and other matters into a place where it was impossible to make progress. Now there is the possibility to be much more flexible about from where one recruits. And that geographic flexibility can be a counter to some of the issues that get in the way of a more diverse workforce. Secondly, there is the potential for greater flexibility in order to help on work life and other balance. Now, I recognise that flexibility isn't always working. At the moment there's a fine line between remote working and sleeping at the office and many people are sleeping at the office and that is disproportionately impacting women and getting in the way of progress. But let's recall one of the biggest obstacles to progress for women and indeed people at the intersection was the reality that there was a lack of flexibility in their lives. There is that potential now. The third is a more perhaps a softer thought, which is I think we're all rapidly realising we've got the physical virus dreadful as it has been in the damage it has wrought. But there's also the mental health challenge. And with it many employers I think are beginning to understand that one has to think holistically about health and the importance of health. And I think that also can allow for a better attitude and approach to some of the issues we've been discussing. So my hope is that a combination of flexibility, geographic mobility, a greater appreciation for the full issues associated with equality can indeed come together to make this a moment where the reset does allow progress to meet and it does indeed depart from the old ways of working that clearly have got us to a place that is not sustainable when it comes to racial justice and equity and other parts of equity. Ashley we've got some viewer questions coming in and I want to pose this one to you about privacy of data in some countries. It's even illegal to understand questions about diversity background. What can companies do to collect the right data. Yeah, so I think that notion of variability across the world, and what is or is not acceptable to be reported, I think is really something that we have to consider where as we know, on the United States you can but you choose not to you know in other countries that you know that actually by law you can't you can't do that. I think you still have to hold yourself accountability accountable for what's happening in your organization, even if you're not able to share that information out. And so the idea that we would like to see change, but we are not benchmarking from where we are want to move. It's fundamentally a problem. You can't expect to progress if you don't know from where you are progressing.  It's similar to revenue right if you decided that any company decided that they want to increase their revenue by two four and wanted to do that within a certain period of time. They would set that metric out there in some shape form or fashion, whether you have to couch it in those things that perhaps may be a confounded with it so if you're not able to report race are you able to report socio economics status those things that tend to to to correlate with them very strongly, but to say that you know we can't do it because we're not allowed to do it so therefore we will not address it, and then also want change at the same time. Those things simply don't work hand in hand. So by whatever means you can be accountable that that your country allows you to do so, then that's what you should seek to do. Another thing that you have to really understand if we're talking about metrics is that it's also about culture, the culture of the country, the culture of the organization, we say we want change but change does not feel good to us. Change is not innate and the way in which we would like to operate, we would instead like to operate in homogenous environments why because similarity attraction paradigm tells us that that's what feels good. Change that's going to cause us to perhaps give up something or first to interpret that we have to give up something as opposed to trying to recognize there can be enough for all, and it does not have to be a zero sum game so altering our culture. When we go into these organizations and we go into countries, you'll be hard pressed to find, well, mostly hard pressed to find people who do not espouse equality and equity and aspirations for for you know these overall diversity equity and inclusion and initiatives. Everyone says yes to the idea, what they say no to is the policies and the procedures and the practices and the internal lens that has to take place in order for that change to to happen. What we know is that when we have the proper motivation, we can accomplish almost anything. So the question is when you say that this notion of you know we don't we're not allowed to have these metrics we're not allowed to do things. But when you have proper motivation, we can accomplish almost anything. So to say these things sometimes can just be an excuse, if you will, and I would challenge you to get above the excuses to in order to move these metrics forward and to have a true jobs reset as we are describing it here. But again, you have to know from where you're moving. Otherwise, you don't know where the progress is actually taking place. Laura, what do we have to do, what do governments have to do and companies as well to prevent things going back into reverse. You were telling us about how fantastic migrants are now considered as far as global health goes. What happens when the pandemic's over. Well, again, I think it is, it is, it has become clear to people that this is this is the access to health is not generosity act, but but really a necessity. I am not seeing everything positive because we have also seen a lot of xenophobia and reaction against migrants saying that migrants are the ones bringing the virus into the countries. So there is always the two sides of the same story but I am talking about the opportunities that it can bring. And I think governments have realized now, for example in the health part that migrants, including those that are undocumented, have to be included in the response in the mechanisms that have been created for response for from the health perspective, everybody has taken it in a certain manner in a more easier way. Now from a socio-economical perspective, the majority of that have been provided and, you know, trying to facilitate access to jobs, maintenance of jobs and things like this have totally ignored migrants, particularly those that are in irregular conditions because they're blind to the system. And they have to understand that very often governments have understood now that they need to address those things. So we have seen here, for example, in Geneva in Switzerland, queues of people looking for items to eat, something that was shocking the whole society in Switzerland because nobody thought that there would be that amount of people within the lockdown that will not be able to get basic food needs. Nonetheless, that has come out. And I think that has created also a reaction of government saying, OK, these are the blind spots that we have. And these are things that we need to address seriously if we don't want to get into this situation again. And as we said, we are closing, getting close to the second lockdown maybe in Europe. And these eventually could be solutions that are already developed or included into the second response. So you're also talking about once this pandemic is over, what if it's not over for a long time from now? Kevin, what's your longer outlook as far as equity goes? Because this could drag on for quite some time. What if we're stuck with this crisis for another year, say, or another two years? Well, I think there's every light that we're stuck with it for quite some time. I mean, I think we cannot speculate as to exactly what length and much of it hinges, of course, on the progress of the vaccines. But there are a few things which I think it's worth remembering as we think about the long term. And I'll come at it from the point of view of business, which is a reminder that there is a business case for racial justice and equity that is powerful in the extreme. And it's in the data. If you look at the performance of the top quartile, then they are 36% more likely to have more diverse teams. There is just this, I don't think I could argue it's causation, but I can argue correlation between high performing organizations tend to have more racially diverse, more gender diverse teams. And there's lots of good reason why that is. So I think Ben, part of the answer is this is a compelling point of progress from an economic point of view, from a business point of view and from a societal point of view. If we are going to be in a reality where this is also a pandemic that has long lasting effects, it is vital we act with that reality in mind that I just described. And it's why I believe that if it's longer term, the stimulus and the other measures that come to be deployed, they should reflect the economic sense and the social sense and they should ensure that we target resources, economic support, other activity to make progress. Because if we don't, the long term consequences of continuing to allow what is happening now to evolve without intervention really would be extraordinarily damaging. And we could speculate as to how they would manifest. But Kevin, I've read those same numbers in many reports from the World Economic Forum over many years. Fumzila, why is it still so hard for a black woman to make a career? Well, I think the discrimination, racial discrimination that is as well and alive. Opportunities are much more accessible for people of different racial groups. And there isn't enough recourse for discrimination. It is not easy for someone to go and say, I didn't get this job because I'm discriminated because people tell you you are playing the racial card. So it is still a very, very hard part. We've had policies on affirmative action and on racial equality. But actually, you cannot change people's behavior, attitudes that easily. It is still hard. There's just still some stubborn attitude that we have not been able to change and to convert into equity that empowers women, who don't even need black women, who don't even need affirmative action, who just need their skills to be respected and recognized because they are not asking for a favor. They have what is required in the job. Well, what's going to change that if the numbers don't? Because they are very convincing and Kevin keeps mentioning them. But I don't see that change. What's going to really change things? You know, in companies and countries, if we're talking about representation, for instance, in parliaments and so on, where you provide enforceable measures that require certain percentage and diversity to be realized. That is when we have seen movement. It is just not possible to rely on the goodness on people's heart. Those who are shareholders and those who are decision makers in these companies, shareholders need to demand the changes. And those who are running the companies need to be held accountable for diversity that is real and seen in the workplace. We are seeing that though. I mean, we are seeing shareholders demand companies to be more responsible, more equitous. We're seeing a completely different generation grow up with completely different demands and respect for one another. Ashley, is it changing enough though? Not certainly not enough and not quick enough. And to the extent that as you say younger people maybe are more open and committed to diversity. The generation of those who are still in charge is not as committed. And therefore there has to be a way in which the expectations for diversity can be enforceable. Otherwise, even where you have policies if they don't have teeth and enforceability, they don't mean anything. Ashley, what's it like in the corporate world? I think there are a couple of things happening here that I hope we're able to capitalize on that the pandemic has shed light on because what we had happened during the pandemic are two central things. We had the social crisis, the racial unrest. We also had the economic crisis and we had the health crisis. And so what it allowed us to do is to not look at one or the other, right? But to actually look at two things simultaneously. Looking at the economic inequality and also within the context of the social injustice and say that it's not one of the other but we have to do both. I think as a result we have an opportunity to capitalize on something that hasn't happened in my lifetime. Not to say that it hasn't happened but hasn't happened in my lifetime. And that is we actually have corporations starting to want to engage in an integrative solution as opposed to a one-off solution. What we saw pre-pandemic was the notion that there is underrepresentation. So what we need to do is that we need to hire more. But that was it. There was not a lot of focus on retention or altering behavior or community presence or communication. And so this idea, this notion of a more integrative approach and trying to attack the issue from multiple domains I think is ultimately what's going to see a result in the change that we would like to see. That's really has highlighted the notion of actually focusing on changing attitudes. That's a very, very, very slow process. And it's going to take a whole lot of time and I don't know that that's going to happen. We've been trying to change attitudes for a long period of time. You have to actually change the structure of the entity. Focusing not just on behavior in terms of compliance but also internalization of these ideas. Kevin talked about the business case for diversity, but that needs to be in conjunction with the moral case of diversity. And so I think with the pandemic and seeing both the economic difference and the social justice differences in terms of just both of them at the same time is that perhaps we actually can adjust both at the same time. Because what history tells us is that when an economic crisis comes, that oftentimes the focus on diversity is the first thing that falls off. And so you choose not to put your energies and your monies there, but what this pandemic has forced us to do with the statements that all the corporations have been made that we're currently analyzing. With the commitments that the leaders have made. And then with the empowerment that the employees and the workers have demonstrated in their commitment to this is to say that this needs to be a much more integrative approach and not just trying to tackle it by focusing just on the numbers. And I think then what we're also seeing is that putting real money behind and resources behind this change. Now, is it at the level that it should be? No, but is that level that we haven't really seen condensed within such a short period of time before? Absolutely. Kevin, were you trying to say something there? I think it was actually background noise, but I will come in because I agree with that. I think there is the risk that we just focus on one element. Recognize that if we did that, we've been doing that for a while, we're obviously not going to succeed. So I completely agree that this requires a combination of actions. I do think if one wants to be optimistic and it is important, I think in the midst of all the challenges that we face, I think the level of action now and the intensity with which this is being discussed. And that's a soft metric is definitely and unequivocally far greater than I've seen in many years. And there is this moment when I think the case fraction has become urgent. The appreciation, as I said earlier, of a more holistic view of what's going on coupled with some very compelling numbers that have been around for a long time. It almost feels as if we just need another shove to really ensure that we can actually make some progress. And I hope that conversation like this, actually a very tangible outcome, which is a renewed appreciation for the urgency with which the action is required, because this does have to break with the past. Well, talking about a nudge, would tax incentives, this is a question coming from a viewer, would tax incentives created to diversify institutions and all management levels be effective to close this generation commitment gap? Ben, do you want me to say that? Yeah, sure, Kevin. Look, I think I think this does require intervention. I don't think it's just naturally occurring. So the intervention can be in the form of incentives. And I think if that means tax, just as we're not discussing the environment today, but we know that environmental change when it's captured with economic incentives and opportunities to bridge the gap between the future value and the current reality of P&Ls that are very stretched will help. So I do think there are opportunities here. I'm not going to get into tax policy per se, but I think there are real opportunities to motivate action. And that could be in the form of incentives that are economic. It could just be in the form of more transparency around progress. And I think that's one of the points I heard Ashley make, which I think is a different point from privacy issues. It's just saying a lot of corporations that are publishing numbers, they just didn't publish in the past. And that will be helpful because I think it means people are far more accountable. And I think that accountability is actually the most likely way in which we're going to get progress. Briefly, another question from Zilla for you. What is the likelihood of decisions by governments, institutions and businesses that could lead to systemic change on embedding equity and inclusion? I think you do need carrots and sticks, so to say incentivize change and have certain specified goals and milestones that need to be reached. And at the same time, have some sticks to make sure that if people do not comply, there is consequences. Having said that, in my country, South Africa, there was a time when we were trying to incentivize change exactly with carrots and sticks. Some companies would rather pay a fine rather than make the change. So, you know, you have still to couple it with the moral case, ensuring that you work towards deploying people in those spaces who believe in the change that needs to happen. The power and the role of trade unions, which I feel has underperformed when it comes into bringing about changes that highlight and enable gender equality, maybe not so bad on racial equality. So there is a bigger role for trade unions, especially now after the pandemic when the situation can be so dire if we don't act together in a positive way. Laura, one more question for you because you were talking about remittances before, and along with that dependency on remittances and work being cut off for low middle economies. How are we going to tackle the domestic violence and growing number of suicide rates during this current scenario? Well, that's a very difficult question to be honest. Remittances, I think at the moment that there is the opening of job markets again, remittances can come up again. It will require a little bit of time but we will come up again. But domestic violence is something that we have seen increasing substantially and that will have to be, I think a lot of governments are trying to address it. Kevin talked a little bit about mental health and the comprehensive approach that we need to look at now. And that is essential because Poncillo will also tell you about the amount of women that have been, you know, aggravated during this crisis. So this is a fundamental element. But I want to come back a little bit to the point of how we make this change reality because I think it is important that we all recognize that social and political pressure has forced all of us. And I'm talking private sector but also public sector to become much more transparent and much more open to diversity and have forced all of us to look at ways to facilitate that diversity within our own organizations and that responsibility. So I think it is something that is happening. It is certainly not happening at the speed that we want and with the results maybe that we want. But today we all are reporting about things that when I enter, for example, the UN system long time ago, nobody was thinking about not even taking into account. And today, these are priorities for all of us, for all of the heads of agencies. And the same thing applies for the private sector. We work with the private sector in a lot with a lot of companies about how to protect migrants, how to keep supply chain systems clean of modern slavery. So these realities are taking place and we cannot ignore those neither. I think, despite the fact that we are in a difficult year, there is a lot of progress that is happening and certainly we will need to continue pushing ourselves to make it happen quicker. It was fantastic to hear your take and everyone's take there on the various aspects involved here as far as equity inclusion and social justice goes, especially during this pandemic, post pandemic and well depending upon how long this pandemic really goes on for. From Zilla for your take there on on women and getting so many of those women back into work, the policies that governments need to adopt and the flexibility that they need to adopt according to Kevin, as far as geo mobilizing goes and also looking at the full issue here and the intersection of minorities and gender and race which Ashley also looked at and spoke about also the the moral case and the business case. I think it's so important that shareholders around the world have started to realize that there's a business case involved, which they can profit from as far as inclusion goes, but there's also that moral case that isn't being taken up around the world by everyone. Fantastic insights from all of you thanks for the viewers from around the world for your input as well and your questions. I really appreciate that. And it's been fantastic having you all here, Kevin, Ashley from Zilla and Laura to debate equity inclusion and social justice.


Hi everyone, Joe for Jaspy's casebreaks.com coming at you with a four-box random team break of 2020 Panini select baseball everyone gets a random team in the MLB here and Buy one of the first 20 spots in this break and get an early bird spot I think an all-card ship in this select by the way It's usually the standard with select because of all the different tiers. All right. Here are the early birds right here Oh, that's only 19. Here are the early birds right here first 20 All right, there you go first 20 are the early birds who can catch that extra spot the delicious delicious worm Let's roll it. Let's randomize it three and a four seven times for each list or for this list name on top gets it And then it'll be different dice roll for the team assignments one two three four five Six and seven then final time after seven Chad B Congrats to you. You are the early bird That has caught the delicious delicious worm do birds really like worms and Why do you have to be early to catch a worm or worms more active in the morning? Someone explain this to me Someone who has a deeper knowledge of of bird law and worm law. What are the rules? You dice roll for the names and teams. Let's roll it. Let's randomize it another seven times four and a three one two three four five six and Seven then final time after seven Justin Humphrey down to Dino Well, let's get the other tab right here All right, four and a three seven times for the teams and Seven then final time after seven times. We've got the white socks down the Braves after seven All right, so Justin Humphrey with the white socks Chris Miller with the blue jays Chris to heart you got my Dodgers Jeremy with the Padres Dino with the Giants Logan with the rays Jeremy with the Nats Edwin with the Rangers Dwayne with the Mets Neil last but Mojo Marlins Matt with the Reds Dino with the Rocks Jeremy Monahan with the Tigers Chad with the spot you won the tribe the Indians Chris Red Sox and Cardinals TJ You got the Kansas City Royals Oliver with the brew crew calling with the dime bags Dino with the Stroh's bill With the Mariners Chad B with the Orioles Dennis with the A's Matthew V with the Yankees Chris with the Phillies Adam with the Pirates Jeff with the Cubs Mark with the Angels Andrew with the twins and Dino with the Atlanta Braves All right, let's sort by column B by team. I'm gonna pause the video really quick Actually, no, I'm not gonna pause the video There you go Remember TJ. This is 2020 Select baseball. There's the fresh case right here Alright, so these are four box breaks. Let's if you can see back there. There's stacks right there I'm gonna make a stack of four right here Let's make a stack of four right here and let's make a stack of four back there So on the dice roll, I'll choose that one right there. We'll go We'll go one two three four five six on those stacks Good luck boxes Four one two three four Middle box right here. So these we will save for next time We'll post another one. We can run this back later tonight And there are your four right here You can see the other four and the top camera right there We can probably run that back a little bit later tonight if you want to All right, any trades? Maybe no trades about which Well, we've been kind of slammed the emails mix up, so I appreciate your patience on that we'll get to it But if you're talking about the the base cards, I don't think we're not we're not even gonna have those ready for a few weeks But I would just email us again in a few weeks and check in on that All right, it looks like no trades All right, we're gonna close the trade window before Matt and TJ make mistakes Nobody would don't trade away the Royals and Yankees in this one Jason Dominguez and Bobby Whit jr All right, let's go. Let's print TWC trade window close. Let's print and rip all right There's the final print out right here There you go, you know TJ with the way things are trending these days that wit base is probably Bowman that that that base could go for a lot of money too Does Jason Ming is any good? Well, we don't know yet, but he's selling incredibly well in the On a secondary market, I'll tell you what if this was a picker team break the Yankees and the Royals would be the most expensive Teams and the picker teams. Let's just put it that way Now with all the hype to Jason Dominguez is getting and Bobby Whit jr. If if they turn out to be halfway decent That'd be really great for the hobby if they try to be regular all-stars even better and a star among all-stars Be even crazier felt a little perilous there New stack right here. I'm pretty I'm almost certain that that some Dominguez stuff is in select There's mookie best artistic impressions for the Red Sox And that Yachty will go to the Cardinals Chris Maxwell to 199 There's Mauricio Dubon Willia Demis Cracked ice to 25 We've got Byron Bucks into 75 Jersey and autograph Minnesota Twins that'll be for Andrew Eloy Jimenez line green to 99 Bobby Bradley Jersey and Otto That will be for That'll be for the try that's gonna go to Chad three out of 99 We do not blame all of that is on our Instagram channel So it breaks live on Instagram. Let Jason know that you're in in the chat once you're once you're there That would be Jason Dominguez Matt V and a Mike trout relic Angels that goes to mark. I thought it might be a low number in there, but no no worries Blaine. Good luck Mike Soroka to 149 and Ozzy Albee's nice relic for the Braves Dino with the Braves Gavin Lux rookie right here too for the Dodgers Christa Hart. All right next box Restaurant impossible. I don't want to watch this This is kind of a sad show Go to the feud restaurant possible is usually like Restaurants failing because there's deeper family issues there And then there's a lot of arguing and crying and then there's Then there's resolution and then there's more crying and that's Too sad See Harvey makes me happy though Family feud makes me happy I like betting on the overs and unders on fast money. I like yelling at the contestants I like I like figuring out who's gonna get kicked out of the family Next box good luck There's Freddie Freeman to 149 That's right. Logan. Yeah, too much crying too much crying here at work. Anyway, I don't need it I don't need more of that Jose braille you 28 out of 35 Jersey and autograph a lot of crying a lot of crying about oh Select baseball hasn't been good to me White socks Justin Humphrey And the Aaron judge will go to Matt in the Yankees. That's to 199 We got two backwards cards here 37 out of 99 Aaron Saval For the tribe Chad B. That's his early bird spot. Is this another autograph? Patrick Sandoval atomic auto nice back-to-back autos there In this box Wait, that's three autos, right? Should we let Panini know? We should send one back Every time we're short an auto everyone's like we got to get an auto back from Panini The opposite is true too right when we get the bonus auto we got to send one back Louise Robert just so everything's even in the world Yeah, that's definitely a machine error by the way TJ is absolutely right Louise Robert for the white socks Justin Humphrey Especially if it's back-to-back like that Mike Clevender at a 199 tribe Malcolm, what's going on going well? I'm in a nice chill evening here Garrett Cole For the Yankees Mike trout silver There's our eras he's a Kino Nice relic there to 99 red legs Matthew now Dallas Keichel to 99 Justin Dunn Seattle Bill With that one Bill Bell Gary Sanchez Christian Yelich all right, so that box was pretty loaded Maybe that was a hot box that double that double auto that was definitely a mistake if your neighbor's a nudist What is something that you hope she does outdoors? And this this is ripe for some non some non family-friendly answers These guys this family's keeping it clean She tried not to say anything inappropriate Josh Bell and there's a nice Cory Seeger die cut Diamond level that's the shortest print Dodgers that'll be for Chris to heart and a boba shet out of 199 Autograph nice could be a big name Could be a big name in the hobby We hope Chris Miller in the blue Jays He definitely has the pedigree Oh have okay Eddie Rosario to 149 Andrew with the twins Got him on my fantasy team We've got Josh Rojas jersey and autograph for Colin Smith and the snakes 204 to 209 still not sure why they have 209 is the ending number, but there it is we got Randy here Ken Giles max-free We got AJ puck Big AJ puck relic for the A's Dennis Foley Hermann Marquez to 199 for the Rockies Kyle Lewis relic for Bill Bell and the Mariners It's out of 250 who's got the Rockies Dino with the rocks Luis Robert silver That's for the white socks that'll be for Justin and we got will ya Dennis to 199 for the raise That's gonna be for Logan It's a great chance that we break some noir tonight great chance So get into it By those spots if you bought one already Buy two if you bought two Buy three Buy a third if you bought three buy a fourth let's move that along gummy bears TJ is asking eat them one at a time or a handful One at a time I'm a one at a timer The same way I eat M&Ms plain M&Ms same way I eat Same way eat Skittles one at a time Sunflower seeds though just a whole handful at a time Look like a chipmunk and a lot of times when I'm eating sunflower seeds I'll have them one side of my mouth I'll transfer the husks the shells on the other side of my mouth and then spit them all out at once I don't know if you can get zebra autos Matthew Matt not to my knowledge. I think they're just parallels. There are Jason Dominguez zebras possibilities I think prospect cards Yeah, there's a lot a lot of interesting stuff in here. All right, we've got out of ten One out of ten gold. That's a nice patch right there Danny gruyon Davy gruyon that is That is for the fight and fills Chris Maxwell with that one one out of ten There's phenom Jason Dominguez obviously This stuff isn't gonna go as as sky high as Bulman stuff, but still very strong right there Jason Dominguez Matthew V with that one like I said before Yankees and Royals are gonna be consistently the most expensive teams throughout the season The Renzo Kane brew crew that'll be for Oliver Michael Brantley silver Jeff McNeill One of 49 Corey seager. This is TV PG Corey seager Dodgers autograph Christopher DeHart Nice, I need a need a nice bounce back season for him this year Genio Suarez to 199 Mike Trout silver They think it's a good answer. Good answer. Good answer They got Boba Shet again for the blue jays Chris Miller. He's of his jersey this time Too late. It's only four bucks Snoozed. I don't know. Maybe not. Maybe we got the night off. I might be early tonight You Darvish white prison to 50 everyone's saving up for tomorrow You know what? Big new release tomorrow folks 2020 Panini contenders draft picks football Well, we're chasing all the 2020 guys Joe burrow Tua Jerry Judy Henry rugs I know I'm kind of tired of seeing I can finally price new teams. Yeah Another Davey grew you on again I get to learn all the new players. That's what it's gonna and what team they are on So it's a fun time of year and a Gavin Lux rookie card at the very end This is four boxes from a fresh case ladies and gentlemen. We'll have more in the store We can run this back tonight. I'm Joe for jasky's case breaks comm and I'll see you next time. Bye. Bye


However, he was being spiritually inclined. He joined the Ramakrishna Mutt as a renunciate upon being impressed by the life and work of the Vedantic philosopher Ramakrishna Paramahams. His initial name was Brahmachari Brahmachaitanya. He was renamed as Swami Vidyanathananda after receiving the saffron robe in 2008. Swami Vidyanathananda is a monk at the orders recorders in Belur Mutt. He was professor of mathematics and dean of research at the Ramakrishna Mission Vivekananda University in 2015. He is currently a professor of mathematics at the Institute of Fundamental Research, Mumbai. He has widely published and presented his research in the area of hyperbolic manifolds and ending lamination spaces. His most notable work is the proof of existence of canal thriston maps. This led to the resolution of the conjecture that connected limit sets of finitely generated flying end groups are locally connected. He is also the author of a book titled Maps on Boundaries of Hyperbolic Metric Spaces. At that, Prasamaham also received the prestigious Infosys award last year. Prasamaham. Yeah, it's a technology thing. It's tough, so I'll start as soon as the technology gets in here. No, it equals function. OK, thanks a lot. Thanks very much. OK, so I will try to begin an answer to this question, which is the topic of my talk. And I'll assume that everybody here knows calculus. I'm going to move them out, but that's about it. And there are two problems in mathematics which have a fairly longish history. One is, I mean, sort of from the beginning of mathematics there have been sort of two directions, one involving numbers, one involving shapes. So numbers have given rise to number theory, algebra, et cetera, et cetera. The other side has given rise to geometry, ecology, some sort of analysis, and so on. So both these questions occupy some of the greatest mathematical minds for any guesses, couple of thousand years, actually. So there are two very specific problems. And they were solved by two of the greats. One was Galois, the other was Gauss. And Gauss solved the geometric problem. Galois solved the algebraic problem. And being a geometer, of course, my personal bias prejudice was geometry, so I'll answer the geometric problem. And that was this. I mean, of course, nobody knew what hyperbolic geometry was, and it looked like. But the problem was posed by Euclid. So when we start off doing geometry in school at the very inception, we are introduced to Euclidean geometry, first introduction to formal reasoning.  So there's a whole collection of that, but this is a very short summary of Euclidean axioms in two-dimensional plane Euclidean geometry. Since the previous talk was largely a life science talk, I can't resist making this statement. Apparently, the part of the human brain that is involved in asking apparently childish or meaningless mathematical questions, which very often lead to profound things, is exactly the part of our will, has a sizable overlap with the part of the brain that invites very new information as a child, that meaning, say, within the first two or three months. So asking these meaningless questions, apparently meaningless questions, is definitely something that helps us acquire a lot of knowledge. And some of our profoundest questions come from here. So I'll try to see. I mean, so Euclid was exercising presumably that part of his brain. And he posed a question. So he laid out axioms. So I'll just summarize the axioms first and then get to the question. So first of Euclid's axioms, there are four axioms, and then there's a fifth positive. So four axioms, first one says that given any two points in the plane, you can join them by a straight line. So this is what we, in today's terminology, we would call a straight line segment. The second says that if you have a straight line segment, you can extend it infinitely in both directions. So that by infinite extended object is really what we call a straight line. The first object is what we call a straight line segment. Third axiom says that given any point on the plane in the radius, you can draw a circle. Fourth axiom says that all right angles are equal to one another. What does that mean? It means that all other angles are wrong angles. But what, yeah, so what exactly does that mean? I mean, so what is the right angle? How do you define the right angle? So you take a straight line. So already you have a notion of a straight line given by the first and second axioms. How would you define it? So basically, you take that straight line, take an arbitrary point on it. So now you have a canonical angle. There's an angle coming back, coming from minus infinity to that point, and then going off to plus infinity. By set that angle, that's the right angle by definition. What the fourth axiom says is that that angle that you've got is independent of the line you chose and the point on it that you chose. So that's the formal content of this. Before I come to the fifth postulate, let's look at these four axioms a little more carefully. And what these axioms are really doing is that they are defining the players of the game. What are the objects that we are going to play with in order to build up our geometry? The first axiom defines for you a line segment. In terms of what? In terms of what are the things that are sort of going there, we have been quietly shopped under the carpet. There's a plane and there are points on it. So these are things that we don't define. But with that data, we are going to define the players of our game. Given any two points, you are able to define a line segment. That's what the first axiom does for you. So all these four axioms are in some sense defining players. Second defines a by infinite straight line. Third one defines a circle. So a circle has a center and a radius. That defines the circle completely. And the last axiom defines the notion of an angle. Now let's come to, so these are in some sense definition axioms, first four. The fifth guy is different. It's a different status. What does it say? It says that you take two straight lines, assume these are straight. And now you let them follow a particular line. And suppose that the sum of these two angles is less than 180 degrees. What that axiom says is that then these two lines, if they are extended in this direction, they're going to intersect. This was the original formulation of the fifth postulate. There's a crucial thing here. If these two is less than 180 degrees, it's going to meet on this side. By the same axiom, if it's less than 180 degrees on the other side, it's going to meet on that side. What if it's exactly equal to 180 degrees? Because then we have no idea. So the two lines don't know which side to meet, so they don't meet. That's not a proof. But essentially, you can crank it up to a proof. And that reformulation that the lines do not meet is called, I mean, that's really called the parallel postulate. So the fifth postulate is equivalent to, say, that through a point, so do not meet gets a special name that's called parallel. And that's what it says. You have your straight line. You have a point outside it. There's exactly one line through this point on the plane, which does not meet this line. Basically, from here, you draw any line. They just make an angle such that the sum is 180 degrees. And then that line will not meet. So the fifth postulate says that through a point, not on a straight line, there is one and only one straight line to the point parallel to the given line. This formulation of the fifth postulate is called the parallel postulate, maybe later. Now, what was the problem that Euclid posed around 200 dc or earlier through the fifth postulate from the remaining postulate? This was his problem. If he was, yeah, I mean, I'm not sure. I mean, I have not read the original. But I guess, in hindsight, the problem should have been posed as, can one prove the fifth postulate from the remaining postulate? The sort of inside this question, there's a kernel of perhaps the deepest theorem of mathematical logic in the 20th century lurking. So yeah, 200 dc, 20th centuries is a kernel. All right, so now that's before we get on to the maths, let's see the answer a little bit of story type. So what's the history of the problem? Zivians of people who are current. Finally, it was cracked by Gauss. Started thinking about parallel lines in 1792. In 1824, he wrote to a friend. Tarnus was a friend. He writes, the assumption that the sum of three angles of a triangle is smaller than two right angles leads to a geometry, which is quite different from our geometry. At that point of time, our geometry meant Euclidian geometry. But which is itself completely consistent. So the sum of the angles of a triangle, three angles of a triangle is equal to 2 pi. That's also equivalent to the parallel possible. So basically, you look at a triangle, draw a line through here, this angle is equal to this angle, this is equal to this. So sum of three angles, this is 180 degrees. And from that, we can go back to the parallel possible. All right, so there's a little bit of mathematical gossip. Perhaps an apocryphal Gauss did not publish this one. And there's sort of questions as to why he did not publish this one.  By 1824, Gauss was an established mathematician, and he did not want to appear wrong in public. There's some reason that is given, perhaps, perhaps not. However, Gauss is not, so basically, the discovery of this geometry where the sum of angles is less than 180 degrees is not a popular literature attributed to Gauss, which is, so who are they attributed to? People who published it first. But before, they published some other history. So you know of these hyperbolic functions, hyperbolic trigonometric functions, cosine hyperbolic e power x plus e power minus x by 2, sine hyperbolic e power x minus e power minus x by 2, et cetera, et cetera. So these functions came in trying to compute areas and other menstruation problems of the foregames. So in the 18th century, some of these things were actually computed by Lambert. And again, then after that, the discovery was made simultaneously by two people, a Hungarian mathematician by the name of Wally I, and a Russian mathematician by the name of Lavazhevsky. Lavazhevsky's book is completely unreadable, because it's got all the sort of bits. Apparently, somebody, I think, Gauss really went through that basically in order to understand Lavazhevsky's work, it's extremely computation-intensive, very technical. And in order to understand what he's trying to say in 500 pages, you have to read all 500. So there's no way you can sort of get the crucial idea. I mean, if you, I mean, people who do math research sort of, well, OK, let's get the main idea, then we'll try to work out the medium. You can't do that. You have to go through every tree individually in order to understand the thoughts. That's sort of a style that the publicists get adopted. As good or as bad as any other, I guess. Anyway, so he published his paper on the principles of geometry in 1829, 13, and a couple of years later, Absolute Science of Space was published by Bollywood, which is much more intuitive. Anyway, at this point of time, what were they trying to do? They were trying to formulate a geometry in which the fifth postulate would be violated, which means what? Which means that, OK, reformulation of the parallel postulate sum of the angle of a triangle is 180 degrees. So they were replacing it by the axiom that the sum of the angle of a triangle is strictly less than 180 degrees. That's what they were doing. And then they were trying to bend up a geometry. So less than 180 degrees has nothing to do with hyperbola. So the term hyperbolic geometry came much later when hyperbolic geometry was really put into a certain context. And again, in hindsight, it's a wonder that it was not discovered earlier because almost all geometries in a certain fairly precise sense are actually hyperbolic. This Euclidean geometry is some exceptional thing. And it is only this sort of perverse habit that we have grown into that tells us to think that this Euclidean geometry is the natural thing and that polygeometry is not. So I'll try to justify that so that it's not just a sales experiment, mathematically accurate. OK, so it was formulated by, I mean, the word was coined by Felix Klein. But the person who really brought it into mainstream mathematics and tied it up with zillions of other areas is Pankai. Just after Klein, Klein had an ongoing competition, one side, on this side only, because he was always doing it with Pankai. And anyway, so I mean, I think what happened was that he was trying to find out, discover certain groups. Pankai found them out, and he called them Phuxian groups. Klein was very annoyed. He said, Phux had nothing to do with it, which is perfectly true. And then, so then I think Pankai went up one more dimension, these were two-dimensional objects. He went up one more dimension, discovered some groups in three dimensions. OK, I'll call them Kleinian groups. So Klein had nothing to do with Kleinian groups. Phux had nothing to do with Phuxian groups. Pankai had everything to do with Phux. So anyway, so this tying up various things, I mean, things which are, even today, applicable in number three things called modular forms. All of it, I mean, has origins there. Certainly geometry, certainly topology, but also things which look far like numbers there. All right, so now let's get back to the mathematics from the gossip, and let's restate the parallel postulate in terms which are more precise. So let's, what is the parallel postulate? It says that given a straight line, L, name them now. But crucially, we define the context in a plane P, in a flat Euclidean plane P. See, if you define two straight lines as parallel if they don't meet, then in three dimensional space, you have infinitely many lines, right? So you have one line running on this flat on the floor. You take this point on the top, you take a straight line, and you just rotate it in a plane parallel to that line. You're going to get infinitely many lines, right? So you're talking about two-dimensional flat plane Euclidean geometry. So that has to be set in a plane P. And a point X on P outside L, there exists a unique L-prime line on P passing through X and not meeting L, which we call parallelism. And now the Euclidean problem becomes prove the parallel postulate from other axioms of Euclidean geometry. But the thing is, look, I mean, this is how a lot of mathematics is really done. There's a problem that is posed. And there is an unstated assumption, which is obvious to everybody. But we don't spell it out. So the problem was being posed in the context of Euclidean geometry, which means all of it is being done in a plane P. Yeah, yeah, everybody knows that. You spell it out.  So if you've not stated it out, if you've not exposed the vulnerable part of the problem, there is no point of attack. And so what the reason it took 2,000 years is that a large part of the time went in trying to understand what is the point of attack. There were several things, apparently this poet, Omar Hayam, he attempted this problem for a long time. He attempted this problem with a bunch of others. Yeah, it's not a who's who. You go through the history of these problems, I forgot on the list. And there's a very impressive list of mathematicians who attempted this problem. So basically, once you stated the assumption, the basic question that we need to ask now is what is the context of the problem? So that means the problem is deep. This unstated assumption, which is really at the background of this entire problem, is the place where you need to attack it. The question is, what is a pain? That which was undefined, some points on it, et cetera, et cetera, we don't know what it is. Some axiomatic thing is a piece of paper. That's not a definition. So what is a pain? That was the question. We don't know the context. Basically, the question was asked because there was a common shared context, but it was not formally clear. It was not something you could manipulate. You could manipulate the players, you could manipulate lines, angles, circles, et cetera, et cetera, et cetera. But you were not allowed to manipulate the underlying structure, which is a pain. So what exactly is the pain? Can we understand it in some more fundamental terms so that we can manipulate? That was the question. And that's one of the main reasons why this problem took such a long time. And the answer to this question actually comes from a very different area of mathematics. That's why it took a long time. And once we have defined that plane peak, just note that in the Euclid axioms, you have this line segment, and then the straight line is defined in terms of that, angle is defined in terms of that. So what we would like to do ideally is start off with the motion of a plane peak from that, derive the motion of a straight line. So a straight line now becomes an emergent mathematical object, not something intrinsic, not something defined, but something that emerges from the definition of a plane peak as an object which has some characteristic properties. So that is our goal. That is the goal of these three questions. And then motion of parallelism, once you have a motion of straight line, the motion of parallelism is clear. Two straight lines are parallel and they do not meet. It's just a one. So basically what we'd like is an a priori definition of a plane, and then from that, define what straight lines are. OK, good. So the answer for Euclidian geometry is R2 equipped with a certain metric. Metric means a way of measuring distances. So what is that? It's ds squared equal to dx squared plus dy squared. What is this? This is just an infinitesimal way of saying what to Pythagoras theorem is. a squared plus b squared equal to c squared. But if a is very small and b is very small and you want to sum, this is the only place where I need calculus. So basically what is this infinitesimal way of using the Pythagoras theorem? And it's packaged in this one neat formula. What's the meaning of this formula? There's something infinitesimally small. It's square equal to sum of two other infinitesimal squares apart from some vague philosophical content. How are we going to translate that into maths? So if you open that package, and that's a formula, it's a package, what does it signify? It means that if you have a smooth curve on a plane, r2, then the length of that curve can be calculated. And this is where integral calculus or differential calculus comes in. Calculus comes in. This is computed by a standard formula that we know from high school that integral of ds is, well, you parameterize your curve x of t, y of t, take the derivative x prime t, y prime t. And then you look at x prime t square plus y prime t square. So basically, at every point you're looking at the two derivatives, you're applying Pythagoras. That is giving you values to a certain length of a tangent vector at that point. And then you're multiplying it by dt. So that's the infinitesimal amount that you're traveling along the curve, then you sum over t. So basically, behind all this, there is a fairly sophisticated amount of thinking that has gone on. Calculus came 1,500 years or 1,580. So basically, it's 70 plus. So maybe some 1,700 years already of that 2,000 years has been captured. So in order to manipulate the plane, we want something, some more fundamental structure. So this structure came with Newton with my life minutes, much later, 1,400 something or the other. And this is a fundamental advance. I mean, we start doing our calculus in maybe class 11 or so. But we don't realize that today it's become part and parcel of our toolkit. But it was not there before that. So people were not able to solve this Euclidean geometry problem before that basically because there was no fundamental model. OK, so how many people are familiar with this thing, this formula A on the board? Very good, enough. So that means I can carry on with that. So yeah, so this is the length of this curve. And then what is what's going on behind this is that you use some parameterization of the curve. Now, if you are given this formula for, I mean, if you define a Euclidean plane this way, do straight lines have a certain property, have a certain special property? Yes, students, yeah? With respect to this distance? Yeah, yeah, yeah.  Yes, yes. Come up with whatever answers you know.  The ratio of dy and dx is constant. The ratio of dy and dx is constant, OK. But I have told you just this one formula, dx square where is dy, dx and du, yeah? The length of path is very good. The length of path is very good. So in terms of this, this L of sigma, the straight line is the unique line which minimizes distances between points, yeah? All right, where length is some object calculated according to this formula? Fine. So now we have, once you have this formula, yeah? Once you have a way of computing lengths, shortest lengths are exactly what defines straight lines. So now the straight line has become an emergent object, right? It is defined in terms of this metric, in terms of the plane with the metric, yeah? It's no longer an a priori thing, which you have to introduce as an axiom. Good. Let's proceed. So the straight line is the shortest distance as per the previous formula between them, right? Parallelism now is just as before. To buy infinite straight lines are parallel if they don't be. Good. So now the whole thing boils down to what is a plane P? Now if you have a plane P with that formula, yeah? Then prove that two parallel lines, I mean that there is one and only one line parallel to the first one. So if you take, so what does it mean? If you take your definition of a plane which was not stated in Euclid's axiom, to be ought to equip with that metric, then the parallel postulate does hold, right? Which means in order to address the question, can you prove the fifth postulate from the remaining four axioms? What will you have to do? This is basically trying to expose the point where you can attack the problem. It means you have to go back to that formula, ds square equal to dx square plus dy square and modify that. With that formula, the fifth postulate will have to hold, right? So if you want to have a hope of saying that what you want to do, you want to change that formula, see that the first four axioms are satisfied and the fifth axiom is viling. And if you want to prove it from those, then what you'll have to do is that for all possible formulae that you put in here, that satisfy the first four axioms, the fifth axiom is all the same. So now the point of attack is very different. So we'll have to manipulate this one simple-looking formula. So we'll have to basically, I mean, this is really okay. So hindsight is where it's 2020. I mean, the person who really brought in this perspective was somebody who in whose second thesis examination is called the Hamiltonian Schiff. Gauss was the examiner. Bernard Riemann. So that's how this, yeah. I have stories to say about that. Okay. So let's try to modify that formula. So how do we want to modify it? Take some subset of R2. And now it's kind of, so basically, okay. So the other thing to notice is that we wrote dx2 plus dy2. Again, there was something that was unstated. The coefficient of that dx2 is 1. We did not write 1. Oh, everybody understands it's 1, right? So because everybody had to understand, nobody writes it. And 1 is not treated as a constant, but 1 treated as a constant function over R2. So basically understanding what is unstated in the problem is what probing this problem is about. So what do you do? You replace, that's why you can hit the problem. So you replace 1 as a function over all of R2 by arbitrary functions now. Say f of xy plus g of xy. So now what is the general geometry? dx2 equal to f of xy dx2 plus g of xy dy2. What is happening? f and g are positive. And the length is computed according to the formula. You again parameterize x of dy of t, x prime t square plus y prime t square, type earlier it was 1, 1. Now you replace the first one by f, the second one by g. And now you again have a generalization of that previous formula. Now the problem from having absolutely nowhere to attack now become is given you a huge amount of choice. All positive smooth functions f, all positive smooth functions g. How are you going to try it out for all f and all g? So there are sort of two basic things that go on in mathematics. One is you want to take a problem and work very hard once you know that there is a and find some way point of attack. That's what this discovery of this theory of calculus gave us. Once that calculus was discovered, now there is some huge amount of choice. Now you want to find one particular metric and test it out for them. But now it's like hunting for a needle in a haystack with cardinality 2 bar c. So what is the cardinality of the collection of functions? That's some huge thing. Real line has cardinality c. And you want to choose one metric out of these. So now there is basically really a reaction to this story. There was no technique available prior to calculus. Now you have so many metrics available, you don't know which one to choose. So the motivation to choose the one, so there are two. So the second thing, the previous thing only accounted for some 1700 years or maybe something like that. The next thing, choosing the right metric has to come from again some kind of motivation. You have to have some clue, some intuition that if you look at this direction, maybe you're going to get a metric which is going to give you something, some new interesting kind of geometry. And then you have to play, exercise that part. So before we get into that, so the notion of a straight line in this metric is replaced by a notion of a geodesic. So it's a path which minimizes the distances but according to this revised formula. And all our concurrences that we studied in high school, angle to angle, side to side, SSS, side angle, side, etc. All of those are examples of what are called isometries. So what's an isometry in this context? It's something that preserves that infinitesimal form of the formula. If you preserve distances in the infinitesimal level, then you integrate out something, then the length of paths will be the same. So an isometry is something that preserves the infinitesimal form, hence preserves all lengths, hence preserves minimum length. Hence preserves the distance that we started off from in Euclid environment. So we have these two notions. So what are these? An isometry is a map which means that you take xy going to x1, y1 under this map I, infinitesimal thing of satisfied means you take f of xy dx square plus g xy dy square that is the first ds square. And after you've transformed it's x1, y1 dx1 square plus x1, y1 g of x1, y1 dy1 square. These two infinitesimal lengths must be the same. So an isometry, if you, now you just set it out into a formula which you do not have any other information except calculus. That's something that you can manipulate. So this is what an isometry is. Now, so the first part of the talk I think we have been able to say that okay, why was, why were mathematicians stuck because Imani's geometry was not there. But then we'll have to choose, we'll have to choose a particular f, particular g. Simplifying assumption, at least choose f equal to g. Mathematics is always trying to find, right? Plus simple thing if it works then why should we complicate our lives unnecessarily. So choose f equal to g, for instance okay, even then, even once you realize there's one function f, smooth positive function, there's so many other. So how are we going to do that? That choice was guided by something else in a completely different part of mathematics. This is what was that Poincare plant story was about. And that's the exposition that I'll be taking. You choose the Euclidean plane but not all of it. You choose the upper half plane, yeah? Vibrator than zero. And equip it with a certain metric where you take the Euclidean metric and scale it by 1 by y squared, yeah? So f of x, y equal to g of x, y equal to 1 by y squared. And you're taking y greater than z. Why am I made this particular choice? There's a short answer because Poincare turns out to be the problem. But the question is why did Poincare come up with this formula? That's a completely different story. It comes from a different part of mathematics which will define, I mean, it could be the topic of a completely different topic. So I won't tell you. So there is a little bit of a rabbit out of a hat here. So from this uncountable collection, we picked out one metric. So once you have, you are given, now let's try to compute what the state lines are in this geometry. And then we'll ask, we'll find out whether the first four axioms of Euclidean are satisfied in this geometry and whether the fifth postulate is also satisfied or not. And then we'll have an answer to Euclid's question. So the first thing is, now we have this one simple formula to manipulate with. First theorem says that vertical straight lines. So what is the space you're looking at? The space you're looking at is x, y. So this is y axis, the x axis. H is the half plane. That's called the hyperbolic plane for a certain reason. And now you're computing.  Which means you take any two points here. So that's x0, y1, and x0, y2. And you take any path between them. You want to show that the length of this path calculated according to this new formula is less than equal to the length of any other path, sigma. So you parameterize sigma as sigma of t is x of t, y of t. So what is l of sigma? So sigma of 0 is equal to x0, y1. So sigma of 1 is equal to x0, y2. What is l of sigma? This is equal to x prime t square. So f of xy, g of xy, both is 1 by y. So 1 by y of t square x prime t square plus y prime t square. And then this is equal to ds square. So you want to integrate from 0 to 1 to the length of this whole thing. In utility, in case you would have just taken this, but now you have to divide it all by this. Now this guy is always positive. So this is always going to be greater than equal to reduce x prime t to 0. So what does that become? This is integral 0 to 1, y prime t by y of t dt. What's that? That's log of y of t, integrated from 0 to 1. This is the log of mod y2 by y1. But what is this? x prime t with equality if and only if x prime t is identically 0. So anything where x prime t has non-zero values somewhere will happen greater. So what does x prime t identically equal to 0 mean? It means that the x coordinate is constant. That means what is the vertical line. So therefore what have we established? We have established that the vertical line from here to here is a huge unit of distance minimizing path. Not only that, it is the distance minimizing path because any other path has greater length. So there is a unique guide to any pair of points which minimizes distances. So first axiom of you could satisfy it. Or at least for pairs of points which are above each other. Now from this collection of geodesics, we want to collect compute new geodesics. We want to join any two points. The two points which don't lie on the same vertical, cannot be joined by a vertical line. So we would still like to join them by geodesics. So what is a geodesic joining two other two points? To do that, we like to get new isometries. So what are examples of isometries? One simple exercise that you can do is you look at f of xy equal to x plus a comma y. So then you look at d of this one. This is x1 is equal to x plus a. So dx1 is equal to dx. That shows you that the ds square and ds1 square are both equal. So these guys are these translations in the horizontal direction. They are isometries. So if you have proven this whole thing for a particular vertical, the y axis for instance, and you translate it around, you're going to get all vertical lines, all of them are geodesics. Good. Now we come to another here. So this was one computation. Now I'll get to the second computation in the stock. And this is actually something that came from the vertical optics, the magnetic optics. So here's a fact that you look at, again, you have your unpoly plane. And this is your y axis. This is your x axis. And you look at, say, the safety circle centered at 00, radius 1. And look at the inversion, the spherical inversion. Assume this is a mirror, and you invert in that mirror. So there's a thing. Oh, why am I getting this in my mind? So what is spherical inversion? There's a picture way of looking at it. Suppose this distance is capital R, then this is 1 by R. And this theta is the same. This is exactly what you get pictorially. If you invert in a spherical mirror of radius 1, the theta quartet remains the same. The radial quartet becomes a recycle, 1 by R. So that's the next claim. Inversions about semicircles. So basically you can do this about a circle of radius capital R. But for simplicity of computation, we do it for radius 1. So there's a way. There's a nice way, a formula for writing this in terms of complex coordinates. So z equal to x plus i times y. This is x, and this is the imaginary axis. And what happens is that you are sending this guy. So the right coordinate for this is the spherical coordinate or the circular coordinate, polar coordinate. For that the complex numbers are better suited. But this is just a computational tool, nothing more. So this inversion is an isometry. That is the claim. What was our formula? ds square equal to dx square plus dy square by y square. And then the complex conjugate is becomes x minus i y. So can you write this in complex coordinates in dds? So this is dx plus i dy, dx minus i dy. What is that? dz times dz bar. And what is y? y is z minus z bar by twice i. So x plus i y minus x plus i y, which is 2 i y, divided by 2 i dy, y square. So you've now expressed this always in dds terms. And what is this inversion in this setup? It's become very simple. So that is our f. z goes to 1 by z bar. z goes to 1 by z bar if the radius of the circle was r. But there's a certain constant, which I keep messing up. Every time I do this computation, so I'll choose chemical r equal to 1. So if z goes to 1 by z bar, what does dz go to? This goes to minus 1 by z bar square d z bar. What does z bar go to? It goes to 1 by z. So dz bar goes to minus 1 by z square d z. Now you plug it all back. So dz, dz bar by z minus z bar by 2 i whole square. This was the ds square on the left side. What does it transform to? Minus 1 by z square d z bar times minus 1 by z square d z divided by 1 by z bar minus 1 by z divided by 2 i whole square. Which is the same thing as this is z z bar. This is z minus z bar. That's what it becomes. This is a multiplication. So what happens? This is minus. This is minus. This gets cancelled. This z square z bar square cancels off with this z bar and this z. So this is equal to dz dz bar by z minus z bar by 2 i whole square. Which means the formula is preserved under this inversion. Good. That means inversions about semi-circles centered at the origin. If you put r square, this is the same computation. What's up? These guys are isometries. But we know that translations in this direction are isometries. You can compose all of this. So actually you can shift this origin. Fixing an origin is something that we attribute to the plane. The plane itself does not care which fellow is there. The plane is much more democratic than us. It has its origin could be shifted anywhere on the real life. So this translation is what tells you that all these points are interchangeable. So really it's not, this is a formula because you set up these coordinates. There's a difference between affine, euclidean geometry and coordinate geometry. Quadrants are something we put in order to get maps, to do computations. But the isometries basically is a way of looking at it physically from the point of view of the plane itself. So inversions about semicircles centered anywhere along the real life because those are things that you can get by translating by isometries are going to be isometries. Using that, now we want to get, so the last thing is that we want to get new geodesics. We know that vertical lines are geodesics. Image of a geodesic under an isometry is another geodesic. What have we got? We've got these vertical lines as geodesics. We've got these inversions about semicircles as isometries. So images of vertical lines under inversions about semicircles centered on the real line are going to give us new geodesics. That's the last thing that we want to do. There's no computation but a simple euclidean geometry problem. This is our picture. So if you...