and these workshops and think through these topics. And really excited to have him kick off this morning. Thanks very much, Beth. It's great to be here. For those of you who haven't met already, I work as a consultant in a group team called the Capital Science and Policy Practice at Willis Towers Watson, which is a global risk advisory company. But my background is I'm a volcanologist. So my knowledge of the science of floods is not very good. But if you want to know about erupting volcanoes, then I'm the guy to come and see. And but most recently, the last 15 years or so, I've been working in the developing parametric insurance solutions for developing countries on the whole. And so I'm going to talk a little bit today about that and the relevance of Earth observation data. I'm not going to talk a lot about flooding, to be honest. I do have a couple of slides which are specific on flooding. But I'm going to give you a feel for why I think the time is ripe right now for moving things forward. There was a lot of potential end users out there, to David's point from early yesterday.  And then just kind of end with a few kind of key points, which I think will be good to take into the rest of the day. So what's the global setting for natural disaster risk management and risk financing? There's certainly a hugely increased political profile of natural disaster risk. I think you're all very well aware of that. Climate risk in particular. And there's a need, therefore, for practical tools to manage the consequences through building resilience. And insurance is part of that because the insurance industry does this as its day-to-day job. It's not the whole solution, but insurance as a discipline and as an institutional form that deals with risk, analyzes it, and can put a price on it, which is extremely important. It's not necessarily the answer to all your money questions, by the way, but it does bring risk into the real economic and financial sphere, which is critical. Example is the global risk outlook from the World Economic Forum. This is the, I'll show you on the next slide, the 2018 version, where environmental risks, the three furthest into the top right corner, which are both the most impactful and the most likely. So the biggest global risks are all environmental risks. So definitely high on the OECD kind of agenda. Last week in Barilochi, there was the first ever insurance forum for the G20. Argentina are the chair of the G20 this year. And so, you know, and various other places where it's coming to the higher and higher up the agenda. And this is both in emerging, both in developed economies, but also in emerging and developing economies. And in the latter, the sustainable development goals. As David said yesterday, Sendai framework, the Paris Agreement, the first time in the UNFCCC process, actually kind of, I felt the big thing in Paris was the identification that actually, we're talking about risk management in the climate space and what are the tools, insurances obviously mentioned specifically, but there's much more to it than that. And then there's this thing called Insure Resilience, which is a G7 initiative under the German presidency, the chairmanship in 2014-15. And I'll say a little bit more about that as well. This is just the world economic forums, global risk outlook. And you see the top right is the really big global events or the green ones are environmental ones. So you can see the top right corner is the, or the top right quadrant is the area of real interest from a global perspective. And you can see all five of the environmental risks that they capture are in that quadrant. You'll all be very, very well aware of the Thai floods in 2011, that was another wake-up call, I think, more from a commercial perspective of the interruption to global supply chains. So all of this is kind of feeding into the narrative that climate risk in particular is becoming more and more real, both on the political side and on the commercial side. We've also developed over the past 30 years or so a massively increased ability to assess and understand risk. You know, we talked about this, others talked about it a lot yesterday. And from the insurance perspective, that's captured in the catastrophe risk modeling world, which starts in the late 80s coming out of academic science and has been now absolutely completely mainstreamed into the global, certainly the re-insurance market. So the global, the ultimate risk takers of catastrophe risk, it's completely mainstreamed. Some catastrophe modeling is done for every single deal of consequence on a global scale, and even at the insurance level, certainly for the big markets, it's a normal day-to-day tool that's in use. What this has done has led to much greater stability and efficient capital deployment in the global economic system, can manage big risk events much, much more effectively and efficiently than it was before. And it's also, you know, it's opened up a new investment class. So insurance-linked securities have been growing year-on-year since 2008. They started before that. The collapse of Lehman Brothers actually put an interruption to it because Lehman were involved in almost all of the cap-on deals that were in place in 2008, guaranteeing the return, actually not to do with the risk side. The primary risk side. But ever since then, it's been growing and it's still a pretty small asset class from a global perspective, but definitely growing. And that's an indication that risk as a commodity is kind of more, much more accessible, if you like. The deployment of cap models, I think, is going to expand, and it could do that fairly quickly. The TCFD, the Task Force on Climate Finance, Climate-Related Financial Disclosures, which is a financial stability board, which is a G20 body, which was set up after the global financial crisis in 2010, I think. The TCFD is very active, it's chaired by Mark Carney, and their outputs are starting to feed into regulation of the asset management and banking sectors, for example. They're starting to put tools in place that the insurance industry has been using for 20 years, also, into a much broader range of economic activities. And we see this particularly quickly in happening in Europe, but it's also happening in the US as well, and I think will grow in its import over the coming years. Another interesting thing, this is a relatively recent development, or certainly recent to me, is that I've come across it. The force-major concept in contract law is starting to be coming under a lot of questions, particularly related to climate risk, and this is all part of the overall, well, wait a minute, we can quantify this to some extent, so we need to, we can't just use it, say an act of God has gone the way, so, and I think, again, that along with the TCFD and other things is gonna really have an impact. So I think the understanding of risk and the quantifying of risk, particularly climate risk, is going to have a lot more end users in the coming years, and a lot of those are big financial actors. This is just to illustrate the role that cat models have played in insurance, in global risk pricing, let's say, and this is an index which a guy happened to do every year, it's basically a rate online. So a price index for global cat risk, if you like, you can see both the volatility and the, and most recently the pricing has been falling, and this is independent effectively of insured losses. So it's not because we're having less insured losses, we all know that we're not, and certainly uninsured losses are growing even more quickly, but so this is a result of us understanding and deploying capital more effectively. So what about in the development context? Well, I think we've, disaster risk in a pure development context has been, talked about quite a lot, 29 billion a year in the 77 poorest countries in the world. I think this was 2016 numbers, I believe, and of which international aid, so humanitarian support generally is about 8%, insurance about 3%. So from an insurance perspective, but also the tools and services that go around insurance have a very low penetration rate. And the thing that I've always tried to convey when I'm talking to ministers of finance in developing world governments is that just because we've identified a risk doesn't mean that we've created it. It's there already, it's having some impact on your ongoing economy. It's somebody is paying for that risk. And so when they compare the price of an insurance policy to what they're paying now, which they see as zero, it's always gonna look expensive. But actually, if you look at the underlying impacts on development on individuals, et cetera, then it can be incredibly cost effective to manage it more effectively. The obvious things that we see in the developed world as well or even more to the fore in the developing world, financial selling of assets in Africa, I'll talk a little bit about African risk capacity later, but one of the big driving forces for ARC is not because insurance in and of itself is a great thing, but if you can get in after a drought before people start selling their assets, then their economic, their fall down, the economic ladder is massively reduced. So there's a four to five times benefit, we think, and maybe in some situations even higher than that of acting quickly and more quickly than the standard humanitarian response. So, but we all see, I've spent quite a lot of time in Haiti over the years and the level of exposure to natural disasters there is huge, we all know that, but there's, until recently, there haven't been very much of a link made between that and the very slow economic development that's going on in Haiti, but to me, they're intimately linked. So, but then there's also, I think over the last few years and probably less than five years, this kind of thinking has come actually into the humanitarian space in a very compelling way in terms of both understanding the causes of humanitarian disasters, but also responding more quickly to them. And for those of you who are not aware and are interested in this space, I very much recommend a very accessible book called Dull Disasters by Daniel Clark, who was at the World Bank for a long time, now works in the British government, and Stefan Durkon, who was the chief economist at Difford for in the UK for many years, a professor at Oxford, very, very, very accessible, very readable. And basically it's saying we need better planning for post-disaster action, we need evidence-based decision-making, and we need financing, but also the appropriate responsibilities are assigned between the different stakeholders. So it shouldn't be governments de facto are liable for everything or donors in the humanitarian space that we have to sit down and think rationally before these events happen, how the response and who's responsible for which elements of the response. This is very much what we're trying to do with ARC. It's starting to build the kind of infrastructure, if you like, that will enable this kind of approach. It's not gonna work all the time, and it's certainly not 100% of the answer, but when you have somebody like Stefan who's worked his entire very illustrious career, academic career, and applied side as well, I think it's worth listening to. And then there's the bigger global development policy context, and David spoke a little bit about this. This slide is probably a year and a half old now. So it was, I didn't update it, but I mean, these four 2015 and early 2016 international agreements and events really have shaped probably the next 25 years of potentially of action, maybe 25 years is too long, but certainly the next 15 years. Sendai, Addis Ababa action agenda, and then that feeds into the, that's the financing part of the Sustainable Development Goals, Paris Agreement, and the World Humanitarian Summit, particularly on the humanitarian side. I'm gonna skip that. I think you're probably all aware of the Sustainable Development Goals, but almost all of them have some risk element to them. And this is just a highlight. You don't need to take in the slide, but there's something called the Ensure Resilience Global Partnership, which is a German government led initiative which came out of the G7, Ensure Resilience Initiative. And then there's something called the London Centre for Global Disaster Protection, which is a DFID funded program. Both of these have basically sprung up in the last year and both of them have significant funding and are gonna be supporting development world efforts in this particular space. So those of you who are not aware of those two initiatives, I very much point you to get to know them a little bit and I'm happy to provide introductions as well to the people who are running those two programs. The other thing I wanted to, before we kind of move on to the next section, I just wanted to kind of highlight that there is much more engagement in the developing and emerging sovereign and sub-sovereign space on these issues, and particularly on the financial aspects of them or the quantitative aspects of them than has been growing significantly over the last 10 to 15 years. I think disaster risk management, when I first got into the space 25 years ago, was all about kind of response and response actions. It started to go into hazard assessment, but it's now, I think there's a much better understanding of the full risk assessment, the full economic consequences, et cetera. I personally have talked to probably 35 Ministers of Finance or Ministers of Finance for 35 countries, I would say, over the last 12 years, 13 years maybe. And all in developing world countries. And so there is a hook into almost every developing world country which can be expanded fairly quickly to be talking about what we're here to talk about. That includes both not only insurance as a kind of a tool, but also a assessment of disaster risk, ownership of risk, and then different risk management tools. And the sharing of that risk between individuals, communities, local, regional, government, et cetera. At least the concepts of those are starting to be understood actually implementing, kind of joined up risk management plans across the whole country and including all of those different elements is far from being mainstreamed, but I think the understanding, at least those conversations are starting. I'm gonna skip over that last one, but just a couple of practical examples. ARC I've talked about a little bit already is covering a big chunk of the kind of theory of identifying risks, planning, financing, decision processes, and then delivery and beneficiaries. That kind of, this is more in the humanitarian space. And also the Red Cross, Red Crescent, has been doing some really interesting stuff on forecast-based financing. And similarly, triggering even pre-disaster action. And if you'd gone into anybody in the Lloyds market or anywhere else in the insurance space five years ago and said we want to do an insurance policy which triggers before an event happened, you would have been laughed at a town. But we are having those conversations now. So, the reason why we're having those conversations is because of something called parametric insurance. And I'm gonna kind of, I'll skip through this fairly quickly, obviously the slides will be available, I presume, if they're not. Then, so you can kind of delve back in and I'm very happy to answer questions. But I did want to spend a little bit of time on it because I was at a volcano conference a couple of weeks ago and talking about parametric insurance. And while volcano colleagues, very technical people, all of them didn't have a clue what I was talking about. So, I've assumed that you guys, most of you guys don't have much of an idea what parametric insurance is. But I think it's important to understand.  You argue about what the losses are actually generally. If you're a homeowner and you have a loss of justice you're not gonna come to the same conclusion about what your losses were, there's the loss of justice. So, it's not quite cut and dried but that's generally the way that indemnity insurance has developed. And it's been around for more than 300 years and it's certainly served the first industrial revolution extremely well. I haven't put the plot here but if you look at the plot of UK GDP growth in the late 18th, 30th, 19th century against insurance penetration, they're tracking exactly. A mill owner would not invest in a mill if they didn't have fire insurance. But in the modern, the third and fourth industrial revolutions we don't see that, it's not happening. But what we have seen is a start, the development of this new form of insurance it started in the weather risk markets in the US for energy companies mainly who wanted to be able to hedge out their additional costs for either cooling in very hot days or heating in a number of cold days more than a normal number of cold days in a row for example. So, there's a pretty active weather derivatives market in the US and increasingly in the European Union as well. Little bit in Japan where there are about 20 standardized indices these run off ground stations on the whole in big cities and they can be bought, they're traded on a daily basis. So, it's mainly through the Chicago mercantile exchange but so weather risk has been traded for probably 20 years and is pretty much mainstream but in a fairly small niche market. What we started to do in the about 13 years ago was look to see how these same principles could be applied to bigger catastrophe events, hurricanes and earthquakes were the two that we kind of started with and I worked with the World Bank in 2000, started working with them in 2004 and five to set up the Caribbean Catastrophe Risk Insurance Facility which used parametric insurance to insure against for the governments to insure against hurricanes and earthquakes. So payments for a parametric insurance policy are made on an agreed scale based on the movement of an index. The index is selected so that it's a good proxy for loss damage or impact. Basically it can be in its form most similar to regular insurance, it would be a proxy for loss as you would adjust that loss but it can be very much kind of looser impacts which makes it a really interesting tool for in the development space. So as a very simple example there, less than a certain amount of rain falls over a certain amount of time than a contract triggers. And what you need is that real time monitoring of that index and as I'll come on to say later, Earth observation is a very, very obvious source for that information, particularly in the developing world. And if the trigger is threshold is met then the payouts released. There's no second guessing it, there's no going in afterwards and seeing if the trigger actually did what it, or the event did what we thought it would do on the ground. The contract is based on the index itself and that means that the payout can go up very quickly. So that's the other huge benefit of parametric insurance is that in the Caribbean for example, the governments have received their payouts and 130 million plus of payouts so far in the 10 years, 11 years that it's been operating, all of them have flowed within 14 days. Nobody gets an insurance claim sold within three months generally after a big natural catastrophe. In the Caribbean, Sean knows this very well, there's a big hurricane in the Caribbean, the regular loss adjustment community is about 10 people across the English-speaking Caribbean. 10 people after a NACCAT in any of the islands, let alone one that's hit multiple islands, is the work of 100 plus. So you just cannot adjust claims very quickly and that goes for everywhere in the world actually. If you look at Chile after the earthquake in 2012, was it, there were more than half of the claims were outstanding a year later. I think Maria, I'm not sure of the actual numbers but I know that the Maria actual loss estimates are moving all the time because the adjustment process is still ongoing and this is, we're more than a year past Maria now. So, and then just add the indices are almost always measured by or at least verified by a third party. I say almost always because I have a case right at the end where we have managed to put together deals where the measurement is not totally external to the client but that's the norm is that to take out the moral hazard of pouring water into a rain gauge or something, you wanna have it, the measurement made by a third party and public domain sources are obviously perfect really. I mean, they're ideal because they provide fantastic transparency. So it's very difficult for either the markets, the risk takers or the clients to argue when you say, well, I'm just gonna go to this website and take that data from the USGS, from NASA, from NOAA, whoever it is. So I've run through some of these already. The parametric is very good for disaster response financing because of the speed and as I said earlier, money quickly is much, much more valuable in all disaster response, but particularly where you avoid sales of assets like a slow onset events like droughts but also in hurricanes as well. There's very good evidence that injection of liquidity and ability to both the government to spend money and also the communities to spend money maintains the markets and aids recovery hugely. The contract settlement is very clean as well. It takes a lot of the political angle out of insurance and off the record, I have many stories of where kind of insurance is used as a political tool and with parametric, you can eliminate that effectively. And so the Minister of Finance has to sign on the dotted line for a contract and then it's a contract and it's on an index which is verified or measured by a third party and there's not much wiggle room there. And as I said, there are some now being, we're not thinking about doing forecast-based financing and triggering on either a forecast or certainly the obvious case is upstream flooding, which is gonna inevitably move downstream and getting a week advance in the big river basins, for example, in Niger River Basin, those floods take two weeks to move down the basin. So that really is, there's great potential there and for no regrets investments really. The other thing about parametric insurance, I just wanna mention and I'll give you some examples of this is that it's very bespoke. So it's really pretty flexible. If you're buying the insurance for your house, you basically buy insurance for your house and it's the value of your house. Governments ensuring themselves, what value do they want to insure from? Yes, they can insure all of their assets but most of the governments I've worked with don't have any clue about where their assets are, let alone what they're worth. So whereas with parametric insurance, you can model out what the impacts are gonna be across a broad range of government activities including property loss but also loss of tax revenue, all of those sorts of things can be looked into and then you can find a proxy for those in an index and they can buy the index and they don't have to buy 100% of the index, they can buy relatively small fraction of that index but it will respond in the way that they need it to. So if there's a bigger disaster where they'll need more money, they'll get more of a payout but it may not be anywhere near all the money that they need so parametric insurance is unique in that way that we can effectively, we can effectively tune it to exactly the requirements of the client. I'm gonna skip over these slides a little bit just in the interest of time but those are the two end members, a simple parametric which I just described right at the front what we would call a binary so it meets the trigger threshold of an index and it pays and indemnity is obviously the loss adjustment side but there are actually a couple of additional forms of parametric insurance in the middle getting increasing in complexity but also reducing the effective difference between I've called it basis risk here but the effective difference between what a parametric would pay out and what an equivalent indemnity policy would pay out if you look at a kind of a normal property basis for the insurance and the modeled loss basis is effectively we use we use a catastrophe risk model it's just locked in advance and we generate the pricing from using that catastrophe risk model with historical or stochastic storms for example for cyclones and then when the real storm happens we run the track data from the NHC through that same model and what comes out as the modeled loss is what we pay against so that's quite equivalent and actually from a contractual perspective effectively what we're saying is that we're gonna agree to not go through the whole adjustment process we're gonna accept that the model is gonna give us a reasonable estimate of the loss and we agree to pay on the basis of that reasonable estimate so we can get quite sophisticated with those kind of models but we also have found that explaining a modeled loss versus a parametric index for example can be quite challenging in the development context so those are just the takeaways I've talked about those already so a few practical examples of earth observation in particular being used for parametric risk transfer in the development context and these are all projects that I've personally worked on there are others but I think I've covered all of the main forms that the parametric insurance has been used in again I'm not gonna go into detail I did have slides for all of these different schemes but the slide that was getting pretty heavy so we've used Prim for extreme rainfall as probably the outside of earthquakes and hurricanes which I haven't covered here because they're fairly mainstream I would say extreme rainfall was always the first thing that I got asked about after we launched the CRIF every single minister of finance everybody in the Caribbean said well what about rainfall? Hurricane models so you guys probably know do not capture rainfall very well and so Trim was pretty much all there was ground station density is nowhere near good enough and doesn't capture hurricane rainfall very well in a way so we used Trim in the Caribbean but this particular thing we did was a micro insurance program in Haiti for Foncose which is the biggest MFI in Haiti has women micro entrepreneurs who are the repayment rate for Foncose is absolutely incredible it's 99. something percent I mean these women are incredibly good clients of the micro finance institutions the one time that they can't handle is when they get blown away by a hurricane washed away by a flood shaken down by an earthquake so this was a program I was working on before they're actually right before the earthquake and we tested it with the earthquake as though the clients had insurance in place even though they actually didn't but we had the whole thing pretty much set up so they were given payouts or had loan forgiveness as though the insurance was in place and building on that experience we put together a program and it was in place for a couple of years and I'm not gonna go into the detail but I mean it worked as advertised the main problem was that it was because the risk is so high in Haiti it was just too expensive to add it onto the loan the lender's cost was they just couldn't bear that cost we got some money to pay for it for a few years but then when that dried up it just wasn't sustainable and this goes to well who owns that risk in Haiti the development actors have put millions and millions tens of millions, hundreds of millions of dollars into Haiti year after year after year in the development context so they're paying for it to support this would be a heck of a lot better use of that, a small chunk of that money to be honest and supporting the economic engine of Haiti to get past events that they can't control I think is a pretty good model but so that's one example we did NDVI for dairy farmers in the Dominican Republic they were worried very particularly about a loss of production for when forage levels dropped and so we got it something for them and then we've done a WRSI based drought modeling in ARC and Elka here at the back there and there's a few great posters on the flood stuff that we're doing but ARC started out as the first policies were droughts using WRSI based on satellite rainfall so those are kind of rainfall related for both high rainfall and low rainfall and using both direct satellite rainfall estimates and greenness in the seas a couple of a few that are kind of on the drawing board volcanoes, we are in the middle of doing a project with the World Bank on just exploring what's possible for volcanoes volcanoes are pretty complicated there isn't any one signal that is going to tell you that it's about to erupt there are a multitude of signals many of which can be obtained from space and so give you the kind of coverage and kind of uniformity of data that one would need these are interference rings from INSAR on a volcano in Alaska and so ground deformation has definitely increased in its ability to be routinely monitored but is not there yet and then other signals are being starting to be processed routinely on a global level so there's definitely potential there I think we're not there yet but I hope that, David we've spoken about this you know as a potential end user parametric insurance I think can drive the kind of right developments in some of these spaces wildfires is another obvious one I don't think there's been a wildfire parametric yet but it's certainly something that I think we're starting to get to the point where it's we have enough of a history of what it looks like and then the routine monitoring which is very well developed now of course so there's plenty of wildfire and indemnity insurance but I don't know that it's been done on a parametric basis yet and then another project I'm involved in is ensuring coral reefs and there's been a lot of interest growing interest in the ocean risk space it's not something that the traditional indemnity markets have really paid very much attention to there being those who owns the reef whose value is from it but cleaning up the reef very quickly after a hurricane is incredibly important so just having a relatively modest policy which triggers on a hurricane index which can generate some funding to have a clean up team it can be really valuable and then as I was looking for some stuff yesterday I just came across this headline which I think was after the Hawaii actually it was after Florence but in the context of what happened at Kilauea which I just thought was kind of amusing that volcanoes even though incredibly challenging to model is actually covered in on Hawaii insurance policies but that flood is not I'm gonna skip that, that's in the Philippines I'm seeing your two minutes I'm gonna just very quickly run through actually I'm gonna skip the CRIF one but I've said a little bit about that it's a potentially really good vehicle it now covers both the Caribbean and Central America and a really good vehicle for channeling some of the risk quantification as well as just insurance it's called an insurance facility but it's really talking to a much broader range of issues our arc I could talk for hours and hours about arc and would love to but I think the really important thing about arc is that it's really taken the kind of things that Daniel Clark and Stefan Durkan talk about in their book about the preparedness, tying the preparedness to the financing, having the early warning and the insurance product being linked together so that there's continuity between the messages that the governments are getting about risk about how it can be managed about the importance of preparedness in arc the money the government has to come up with a detailed budget on how they're gonna spend the payout that detailed budget and action plan is audited so we're holding their feet absolutely to the fire in terms of how they're using that to make sure that they're using it to the not only to avoid corruption et cetera but also that it's going to the things that where it has the most value because it's arriving very early what are the investments that we can make right now to reduce the overall impacts? Then one flood example before my I think is what's gonna be my final slide is the flood example, it's not in the, it's in the US but I just thought it might be interesting I'm not gonna go into the details but it was a post Harvey Houston thing that came up it was from a Levy Improvement District which those of you who know the US well these are tax raising entities who have who are mandated to look after the mainly the engineer aspects of Levy systems and so what this particular Levy district which is an extremely good Levy district it's a very, very good reputation and but they had flooding outside of the Levy because they couldn't pump the water high enough quickly enough during Harvey because two things one the rain rate was too high and secondly the water level was too high in the river so those two things combined were what they wanted to be covered for and so their engineers came up with a table of okay this is where we're gonna be in trouble these are the conditions of gauge heights and rainfall rate and so we put together a parametric deal for them and they ultimately they didn't buy it unfortunately but it was very interesting to go through that process and have them think about what it would take and also what they would spend that money on and they said yeah absolutely if we'd had a relatively modest amount of money a couple of million dollars to give out to the community so it wasn't insurance for the individual householders it was an ability to generate some funds that the Levy district could spend in the immediate aftermath of this to help the community and interestingly I'm a very quantitative guy when we took this deal to the international markets we went to a few markets and the pricing differential that we got even though it's US, the data's pretty decent we got rainfall, long history rainfall flood height not so long but the gauge I think was 20, almost 20 years worth of gauge data, so pretty decent data and the differential between the pricing we got was three times differential between two kind of mainstream re-insurance markets it was a fairly small deal but even so the amount of kind of wiggle room when a broker like Willis goes into the room and says okay this is the risk it's all quantified for you it's a parametric, what price can you write that at? There's still a huge, huge difference so there's a lot of stuff around the edges which is important to pay attention to as well and then just my final slide just a few kind of out there questions which I hope we can address the rest of the morning in the breakouts to some extent The one thing I always highlight to the EO community in particular is that to do insurance you need to have a probability of events happening and you basically build that up or you need to have some way of knowing of rooting that in history you can model it out, you can do simulations but when you're talking to a client if you can't point to some historical events where which would or wouldn't have happened would or wouldn't have triggered then they won't believe what comes out of a model so yes having the best real-time data is great but if you can't have some kind of representation of what that data is now capturing for historical events then it's very difficult to use that best data if you like in an insurance setting so that's a flag, I'm going to... Yeah, so and I think the GPM and David I've mentioned this to you I think and other people have as well the GPM is fantastic, you know we know that the real-time stuff is great we've done the comparisons with Trim and it's hugely better we can't use it yet for rainfall parametric because we haven't got the history so we are literally, we are ready to go with three deals I know of in the developing development space with as soon as that history comes out within a couple of weeks we'll be able to switch it over so just and I know that you know that but it's really important in this setting to have that reliability of data so and by the way putting that together is absolutely incredible and the word that there's a poster there by EAR by John and co-authors on the flood modeling that the AAR have done for ARC the footprint modeling our biggest thing on our to-do list when we were looking for somebody to do this for us was we needed to have a history we needed to be able to go to our African governments and say we have these flood footprints going back to 2000 and where is John? 2007? 98? So you know to have that I think we said 2001 to start with I think because that's where RFE2 goes back to but yeah, further the better and that's so I'll, Elka can probably answer that question better it does contribute but because we're using flood footprint not from a model basis but from an actual what you're seeing then that would, I think that would contribute to understanding how that flood footprint got there but we need, we wanna see that flood footprint and know what context that flood footprint happened in to be able to demonstrate to go back to a big flood in 2001 for example and say look, this is what the tool that will be working in real time this is what it came up with when that flood happened that's really, really important and so it's that, the second point I wanna highlight is that I really think that EO data has massive potential because of it's global it's global footprint, it's uniformity of coverage it's ability to be objectively processed or all of those things standardized delivery and NASA, I have to say NASA not just because they're sponsoring this but and NOAA as well, I mean they're really the availability of those data sets the incredible sciences behind them is absolutely fantastic and I just, without that it would have been very difficult to do what most of what we've done particularly outside of the earthquake and hurricane space so more of that and definitely you've got an end user which I think is gonna expand potentially very quickly David and potentially also beyond the developing world I think that some of the stuff we've been doing in the developing world actually has huge application in the developed world as well and then the last thing was the forecast based financing which I think has a lot of potential and I think we'll see a lot more of those kind of deals happening and initiatives in the next few years thank you very much for your attention. Thank you so much Simon, what a clear description of many use cases that we can really I think dig into as a community and think about how to move forward. We're a little over time but we can take some questions for Simon before moving to breakouts, lots of hands. So that's a good question. I think depends on the sophistication of the parametric instrument that we're trying to develop in some concepts context, reanalysis is really important. An example would be the best track hurricane data set which is a reanalysis and I think without that we would be, the parametric insurance for hurricane risk would be a lot less used and a lot less useful and so and then if you go to the model lost parametric then we're using a lot of the same tools that the CAT modelers are using anyway. So we just I think concentrating on the hazard side of it rather than on the ultimate risk side. So we wanna know what the ultimate losses are gonna be or have a good estimate of them so that we know that they proxy what we're trying to cover but really the concentration is on getting the hazard probabilities right because that's what the, so it's something I didn't really emphasize that the global risk takers love this stuff because the probabilities and therefore the risks that they're taking is really down to the hazard and so and that's an area they feel comfortable with if they don't know about it then they can find out about it. Dealing with stochastic hurricane track sets and CAT models is something that they do and they're very comfortable with but particularly at the hazard end of it. So and if you can define the risk more tightly then you're gonna get better pricing from the market. So African drought risk for a country in the Sahal if you'd taken that even to the most adventurous underwriter 10, 15 years ago they would have said, no, we're good, thanks, right? Just because they wouldn't have known how to quantify it not necessarily because it was a high risk just they didn't know how high it was. We've now we're now transferring African drought risk into the international markets. We have 24 markets on the ARC re-insurance program. It goes to 40, 45 re-insurers effectively. So we're sampling the entire market and we're getting the rate we're getting is absolutely incredible in terms of the small margin over the expected loss cost. So it's an incredibly efficient way of transferring that climate risk. But yeah, I think the re-analysis and the modeling is very important especially in the helps to build the understanding on the ultimate risk takers. So you can start to do things that are really cost effectively. More burning questions, yeah, Guy? No, I think that's a really good point and I think most people you taught to on the insurance side would think that it was a given. So I think that's a really useful point and I'm happy to act as the point person to make sure that we join our voices to you guys in that. I should quickly mention that there's something called the insurance development forum which my boss at Willis has helped to put together and that's basically an initiative between the World Bank, the UN system and the global insurance and re-insurance industry and it includes all of the industry. It's not just a few key players, it's everybody. And they have actually just appointed a secretary general anyway, they now have a secretariat after kind of getting into spin up mode over the last 18 months, two years. And so this is something that, this is kind of thing that they would definitely be interested in taking up and being an advocate for. So we're starting to develop some of the institutional forms rather than it being coming from individual companies or individuals, it can be from the industry as a whole. All right, last question and then I'll give you. Yeah, and I just let me mention that the, I mean, what I think what you guys, NASA did with the trim was just incredible. You're putting out data until we were really worried about that because it would have, if you'd switched it off, we would have had to cancel all of the policies that we had based on trim. But the fact is that you, we had the time to make the necessary arrangements. So yeah, that was a great example of something that, yeah, it's gonna come to the end of its life, but you, yeah, so, but a good point. All right, last comment or question and then we'll move to breakouts. I think six years is pretty difficult from a global perspective. I think you could probably start to apply it in particular cases where you could build a history of, you know, in a particular area that was now part of that global monitoring. You could build a picture of what history would look like. So I certainly don't want to stifle innovation, absolutely not. I was just making the point that until we can deploy, there's a barrier to deploying a new tool immediately and that barrier is that what that tool, how that would have behaved or what it's trying to capture. I had a conversation with somebody yesterday about, well, what about if, you know, new technology could, you know, could say whether a pixel was flooded, you know, a 10 meter pixel was flooded in the middle of a city now, would that be useful? And the fact is that you, you know, if you put your mind to it, we could probably put together a history of when that pixel was flooded from various sources. It wouldn't be uniform, but if we could do that, then we can use that data because we can say, well, now we can measure that down to that level. We couldn't before, but we've, you know, we've put together a history. And then, you know, the point I just made about the, you know, the price, different pricing in the markets, that's very much about the story that you tell around the data. And the easiest story to tell is, well, here's 30 years of history from this bird. And, you know, that's what we're going to use in real time. And, you know, so, and that's a very easy story to tell. But I think we've gotten better at telling stories around, you know, putting together data sets, which are a little bit more complicated and multiple, multiple sources as well. And I personally feel pretty confident that, you know, you can start to do that, but it's very much on a case-by-case basis. And some of the, you know, one of the drawbacks to that is that it does end up being kind of costly in a verticomas. And, you know, when some of these deals are fairly small, that cost is difficult to bear within the, you know, so as a kind of research challenge, yes, we could definitely do it, but a practical mainstreaming of these kind of tools into insurance, you have to be careful about those additional costs. Thank you so much. Okay, so we're running a little bit over, but may I suggest grabbing coffee and taking a quick, you know, five to 10 minute break on the way to your breakout session?


Will we have homework this week? This was okay? We don't. But this is, you know, it wasn't about some things. Right. Then we'll be very psychotic. Okay, guys. So, you have homework to do. And there won't be another homework. So I'm going to go to the next week because you'll be studying for your exam. The exam will be by popular demand, or at least not popular protest, a take-home exam. And so I'll distribute that after the problem session. We'll do a review, time for a more review, on Friday. Are we from Friday? I don't know. And then we'll do in class the next Tuesday. All right. I will discuss more the rules of the game when the time comes. Or what's allowed and what's not allowed relative to the take-home. But not flexible. So the exam covers the center of the material in lecture notes 8 through 16. The associated podcasts are there. And through the problem set that's due today. Yeah. Which one? From October 30th. Last Thursday, the notes aren't posted. Not every lecture has, in terms of the date, has a lecture because we don't cover all the material. Sometimes it's just a podcast because the lecture notes extend over multiple days. Okay. All the notes are being posted at this stage. So that includes the discussion of quantum dynamics. That is, high alert picture, short alert picture, abatic evolution, sudden evolution, all that kind of stuff. The wave-mechanical representation in quantum mechanics, position of momentum space in your function. It's running a wave packet. In solutions for bound states, in 1D scattering states, all that kind of stuff. And the studies you've been doing, we've been doing in recent weeks on the online oscillator. Okay. So that's a little bit of stuff. We'll discuss it more as the time approaches. All right. So just to sum up what we were talking about last time. We talked about this idea of the quasi-physical states in phase space. They're also known as coherent states. And we obtained those states by a displacement, a phase-based displacement on the ground state of the symbolic oscillator. Okay. And that the phase-based displacement operator is a unitary operator which takes this form or equivalently in terms of the dimensionless X and P variables that looks like that. Okay. And the thing that's special about this state is that it is an eigenstate of the annihilation operator with a complex eigenvalue of alpha. Okay. So it's a kind of the closest approximation we have to a simultaneous eigenstate of X and P. Obviously, it's not a simultaneous eigenstate of X and P. It's an eigenstate of the annihilation operator. But the annihilation operator is this linear combination of X and P. So it's an eigenstate of that non-remission operator. Okay. And it has some very nice properties. It is, its expectation values are just the X and P values that's related to the real and imaginary parts of alpha. So it has its mean value is centered at X is in the X operator and its mean value is centered at P. And moreover, it is a minimum uncertainty weight packet with equal amounts of uncertainty in X and P just like the ground state of the harmonic oscillator is. And moreover, as a function of time, if the Hamiltonian of the system.  These states are a set of states that exist in Hilbert space whether we're talking about a harmonic oscillator or not. Okay. They're just a set of states in Hilbert space and we can talk about representation, representation, coherence states regardless of whether we're talking about a harmonic oscillator in just those states. However, if the system is a harmonic oscillator, to say if we're talking about a particle in a harmonic well, then if at time t equals zero, we started in one of these states, one of these coherent states, then at a later time we stay in a coherent state. And the mean value, or as I said, the eigenvalue of the annihilation operator just evolves according to the classical trajectory. Okay. So X and T are just the classical harmonic motion of the particle. Okay. But if this is a wave packet, then, you know, we start, this is Gaussian state, the Gaussian wave packet just oscillates back and forth. Exactly like the harmonic motion does. Completely undistorted. So the wave packet is completely undistorted. It doesn't spread. It doesn't breathe. It doesn't do anything. It's just that wave packet that just goes back and forth in the harmonic well. As a classical particle would do. Okay. The other thing we discussed about these coherent states is that, of course, if they're going to have something that has a well-defined phase of oscillation, kind of know where it is in what the phase of the oscillator is, then it doesn't have a well-defined number of excitations. Right? It doesn't have a well-defined amplitude. It has to have some uncertainty in its axis because there's a number phase uncertainty rotation. And we saw that explicitly if we decompose this operator in terms of the number of states. It is a superposition of number states. It has a huge number, which is the square of the amplitude, just like you would have classical. But it also has some fluctuations in the number of excitations. And those fluctuations we calculated were given by the square root of the mean. And that's characteristic of a Poisson distribution. And we see that explicitly if we look at the probability distribution of the, which is the square of the amplitude in the number basis, then that's given by this, which is a Poisson. And in phase space, we kind of look at this and say, what this state looks like is some kind of packet that's localized in phase space around this value of x and this value of p with an uncertainty bubble around it that's completely isotropic. And classically it would just be this point in phase space. It's the pure state we know exactly what it means. Quantally, we can't define the states as having definite x and p, but we can define it as a kind of minimum certainty packet that surrounds that point in phase space. And it has this uncertainty, right? One half or one over root two in the other. And this state is a minimum uncertainty in phase. Just looking at this geometrically, the number phase relationship is a minimum uncertainty. Yep. Added to the phase in the alpha. You added time to it on the previous board. Adding and adding time to it in summation is trivial, right? In what way? Just bring it into the... Sure. It's a linear operator. So we can bring that in. We can bring that in. And that's u of t on that. And then what is that? Good question. If it's our monocosmic. e to the i h bar t or h t or h bar? Yeah. And what is... So that's e to the minus of h t over h bar on that, which is what? For this hematomy. So that becomes e to the minus i omega t n plus a half. Right? And then... Right? Now of course the overall phase I can factor out inside here. I don't really care about that. So putting all that back together. e to the minus i n. And then there's some overall phase out here that is interrupted. Right? But look at what that is. This can be combined with that. You're getting ruffa. Right? So this becomes alpha e to the minus alpha t to the power n over the square root of n factorial times... And I'll just put that in here too because I can't. It doesn't change anything, right? And so that's exactly what we said. That u of t acting on this is equal to that, which is alpha t, classically. Classically the phase just rotates clockwise. So... That's not good, right? What's the worst? What's the drama's history of the mankind? It's a lie. You didn't want to try it any time. Okay. So that's why this is called the Kwasi Classical State. The center of the weight packet moves exactly along the classical trajectory. And moreover, it's uncertainty about that classical trajectory is the minimum possible. Okay? And that is what makes these states the Kwasi Classical States. But there's a little bit more to the story than we have. I just want to conclude our discussion with one state. I want to say a little bit more about the classical limit. What does it mean? So, yeah, you know, what do we want? What do we...in what way... In what sense is classical physics different? From quantum physics. And there's a lot of ways. And so there are lots of ways in which we talk about classical only depending on which of those ways we're looking to match. One of them is we have classical dynamics. That is to say we might... One thing we might want to see in the classical limit is that we have, you know, Hamilton's... Let's just talk about dissipation-free classical physics for a moment. We have Hamilton's equations of motion, Newton's laws, right? We might want, in some way, that the... ...dynamical evolution of the system is described by Hamilton's equations of motion. That's one thing we might like to see in the classical limit. Now, we talked about this a little bit before. We know that we get this in the short wavelength in WKD. We see some recovery of this, right? We saw that in the ion states, and we also saw that if we have a wave packet that is made up of a superposition of energy eigenstates that have short wavelengths relative to the scale over which Petro is changing, then we have geometric optics, and that geometric optics is Hamilton's equations. So that's one way we see that. The harmonic oscillator, though, is kind of special in the sense that, for the harmonic oscillator, what's special about the simple harmonic oscillator is that the frequency of oscillation is independent of energy. If I take my oscillator, if I put it over here, just a little bit, having given that much energy, the period of oscillation is the same as if I started up here. Go back. The period is exactly the same. The frequency is independent of the amplitude. That's special. Only the harmonic oscillator does that. If I had any other potential, then this guy would roll down from back. But that period would depend on where I started it. And we kind of see this part of that as the fact that the Heisenberg equations of motion are the same as the classical. We have x dx dt, it's p over n, and dp dt is equal to minus the spring constant. That's the forces minus kx. That's just like classical, and what that means is that no matter what wave packet I chose, no matter what wave packet I chose, the center of the wave packet will follow the classical trajectory. If I take the expectation value of this, the expectation value of this follows the classical trajectory. Of course, if at time t equals zero, the mean value is zero, then we never see any evolution at all as in the stationary states. But if I took any wave packet, it doesn't have to be this place harmonic oscillator, I mean the place ground state, no matter what I chose, it will just follow the classical trajectory. So if I had some other funky thing, and maybe I have something that looks like this, it will follow the classical trajectory by having a ellipse around there. If I had any shape whatsoever, because every point on this curve kind of goes around at exactly the same frequency. That's what I just said. So it doesn't matter where it is, it goes around undistorted. So there's more than just the fact that the mean values follow the classical trajectories. We want more than that to say that something is classical. And that has something, but there's another aspect that distinguishes classical from quantum, and that has to do with prediction of measurement outcomes. So in classical physics, we can assign a joint probability to both x and p. There's some probability on measure x and p. Classical physics, we would have a probability distribution on x and p. In classical statistical physics, that's what you do, right? That might be the Boltzmann distribution. If it's in thermal equilibrium or if it's non-equilibrium, whatever it is, right? Okay? And the fact that we can do that means that we can calculate some, if we were to measure x and p or x or p, we know how to calculate that from this probability distribution. And this probability distribution has some properties. These are numbers that are always positive or non-negative, I should say. And we typically normalize the distribution to... In quantum physics, generally for business, the assignment, because x and p don't commute, now it's a little bit subtle why those two statements are the same, that they are. I mean, we know for one thing, if x and p commuted, then we could write down simultaneous eigenstates of x and p, right? And they would have both these. This would be a simultaneous eigenvector of x and p. And then we could define, you know, some probability distribution of x and p as that. We could try to do that. But these states don't exist. There are no states that are simultaneously eigenstates of x and p, right? If I like to find a state that has this form, I cannot do it, because x and p don't commute. Yeah, Steven. You said, would I generally, would it apply to what's being case, what they do with this kind of curve? Yeah, we'll come to that in a moment. Yeah. So, that, because this is not true, I can't do this simultaneously. I can't generally do this. Okay. So, now we do know, as you've been studying, there's this thing called the figure function, which is called a quasi probability distribution. The property that it has the correct marginals. So when we talk about, this is the, if I have a joint probability distribution on two random variables, then the marginal probability distribution is the probability distribution on one of those random variables, irrespective of the other. So we said the marginal probability to have a certain value x would be just summing over all other possible, because that's the margin of the distribution. And we demand that this equal for a pure state, the square of the weight. And the probability would be the momentum space. Why is this possible probability? Because this thing can be negative. Which means that it doesn't represent the joint probability to find x and b. Because the probability can't be negative. So, one of the things that we would demand from classical limit is that in the classical limit all of the figure functions are positive. And what distinguishes in some sense a classical state from a non-classical state is whether the bigger function is positive or negative. And this comes back to the question that Stephen was, when I wrote General Leo's thinking around the fact that there can be states where this is positive. In fact, you're doing one for homework right now. The coherent state. The coherent state is a state whose linear function is everywhere positive. Yeah?  Exactly. Okay. Right. So, let's take a... I'm going to give away the answer if you're going to derive it more. But let me just show you. Yeah! If you can give me the homework answer. I think that's a graph for the last part too. Okay. Come on, computer. Tom, let's do what we can see first. I need to turn it on a little bit. Just take a moment for the first bullet. Yeah, let's get to the bullet. So, this is a picture from the website of quantum optics, but it's the same view. So, what am I showing you here? What I'm showing you are the bigger function. This is taken from a general article. As a function of position momentum here, a position is called q and p. Okay. And we're looking at this for four different states. Guess what this state is. Ground state? Could be the ground state. Indeed. Or any coherent state, which is just a displaced ground state, displaced to some mean value of q and p. Okay. Notice, as you suggested, it's just a little hill centered at this. The bigger function is everywhere positive. Okay. So, I can just think about this as if there were a classical distribution of possible complex amplitudes. I just don't know which one it is. This is another example of a state with a positive bigger function. You know what that state is? That's a squeezed state. A squeezed state is a state which is also a minimum circuitry wave packet, but does not have equal uncertainty in x and p. Okay. And these are the marginals are being shown here. And these are the projections onto the screen there is by integrating over all p. You get the distribution of x. Integrating along x, you get the distribution in p. Right? What is this state? Like the n-excited state. Yeah. Which one is it? Second. Another guess. The number of p's. The number of nodes. Excellent. What does this have? It's got one node. Right? This is the square of the wave function. Right? It's not the wave function, it's the square of the wave function. And this is the wave function, square of the wave function in p. And this is the square of the wave function in x or q, whatever you want to call it. Notice in phase space, it's exactly what you would expect. In phase space, if I have the excited state, so here you know x and p or q and p here's the ground state. The first excited state is localized around some circle. Right? With that amplitude. So it must be that the wave function is rotationally symmetric around the origin of phase space. It must be. And that's what is being shown here. Notice that this state is negative. Right?  It's got to be negative. So this kind of state is highly non-classical. And in fact, if I had a very excited state even though the WKB projection would have peaked around the more localized near the turning points in terms of its shadows of x and p, it would be really negative. It's only when I make a superposition if I can get rid of it. What about this guy? What's this? Something you might have seen in your homework. Close. Kind of like that. What's that? Which homework did you say? Come on, you did the homework. You've read the solutions. That's not a problem. This can be a problem on a oscillator, can it? It's not an eigenstate of a harmonic oscillator. It is not. Can they do part of it? It's too... Oh, could they be two different delta-wells? Or is that what she said? Yeah, that's what she said. It was too Gaussian. Too Gaussian! That's exactly what she said. This is mischroding your cat. Now! Look at the joke. It is in a... This is a state which is a superposition of being here with this position and this momentum or being here with this position and this momentum. If I were a classical distribution I would have some probability or some probability of my amplitude. Some probability to be here and some probability to be here. But because I'm in a quantum superposition of being here and being there then I have quantum interference between those alternatives. And I have all this quantum weirdness in the middle. Notice, what is the... This particular guy is in a superposition of two positions. Right? That's okay. So it's in a quantum superposition of two positions. When I look at the shadow on the position screen well it has a probability distribution here or there. Right? That's the square of the wave function. What does it look like in momentum space? It looks like that kind of thing. You know what that is? That's the double slit interference pattern. Right? What you see if I have a double slit and the particle is either here or there in some superposition then in the far field I see the Fourier transform. The Fourier transform is the momentum space wave function. So the negativity here is an indication of the non-classicality of the state. Okay? If we got to the classical limit what we want is somehow these negative pieces have to go away so that I'm left with just either this or that rather than a this superposition with this. And there's no way we can get that dynamically out of just a closed system dynamic. To get rid of this and we'll talk about this and we need decoherence. Decoherence is what gets rid of these negative fringes and gives us these two alternatives. We mentioned decoherence at the beginning of the semester and hopefully we'll talk more about it as the semester progresses. But one of the messages that I want to take away from all of this is that one of the modern perspectives in quantum mechanics is to think about classicality versus quantum mechanics of the statistics of the probability distribution of what you can measure. And moreover how those statistics can be used as a resource for processing information. So a classical state even though this is, you know, a state of quantum mechanics for all intents and purposes the statistics I would see if I did measurements would just look like classical Gaussian fluctuations. Whereas the statistics I would see from these guys would be nothing like that. And it's that distinction between what you can see in a measurement what statistics probability distribution can generate is one of the ways that quantum mechanics is distinguished from classical mechanics. It's not just about Hamilton's laws versus the Schrodinger equation. It's about the intrinsic statistics of the measurement state. So a quantum harmonic oscillator in some sense is in some sense always classical. Because because it's linear if you start in the ground state and apply a harmonic force you're just going to move this classical bubble around to make something non-classical and need something non-linear. Very good. So that concludes for now what we're going to talk about relative to the harmonic oscillator. Let's continue on the backboard. Yeah, let's turn up. We've been talking about the quantum mechanics so far. We've been talking about one degree of freedom. We've been talking about we've talked about state. It's been one half of the time. In which case our Hilbert space was the set up was two-dimensional complex vector space. Right? Or we've talked about a particle moving one phase of the image. Hilbert space was the space of square normal magical functions on the real line. Right? That's what we've been talking about. How do we talk about more than one degree of freedom? Now let's first review how we do that classical physics. How do we deal with that classical? Well, let's look at some examples. We might have two particles moving along the line. They have positions and momentum. Or we might have a situation where we have one particle moving in a plane. They're both examples of two degrees of freedom. And the way we do it is quite simple. If the phase space is what we call the Cartesian product of the phase spaces for each degree. So if I have X1, P1 for particle 1 and X2, P2 I take the Cartesian product of those two phase spaces and I have coordinates X1, X2 and P1. That's the Cartesian product. And the total dimension of the dimension of the phase space is two times the times the number of degrees of freedom. Now, of course what coordinates we use is not necessarily. We have lots of choices. For example, in here I can talk about the position of particle 1 and particle 2. Or I might talk about the relative position between particle 1 and particle 2 and the center of mass position of particle 2. That's a perfectly good set of coordinates. And then there will be a conjugate momentum to this P braille and P center of mass. And I would just I would have the relative momentum the relative position as my description. That's perfectly good. Or in this guy, I could say for example instead of talking about the X and Y coordinates I can have the cylindrical radius scripty row and angle. That's a perfectly good set of coordinates.  It's a tensor product but those two space are a partition product. Okay? Alright, so that's classical physics. What about quantum physics? So now let's talk about quantum mechanics quantum description. Let's for example just consider for example two degrees of freedom two particles or one particle which is called that I could call so it's two degrees of freedom having X A P A and X B A and D represent the two degrees of freedom they could represent two particles or they could represent the two coordinates associated with the plane. Now of course in the quantum description these are operators moreover so X A and P A satisfy the congenital communication relation for degree of freedom B but A and B know that that has to be true if we think about for example the fact that P is the generator of translations in position so if we let's define the momentum let's just think about this for example 2D as an X component and a Y component what we expect is that if I do a translation by some amount delta X on the X operator I translate this by an amount delta X where the translation operator you recall is e to the minus i over h bar which is approximately 1 minus i over h bar P hat delta X delta X is small so if I look at that infinitesimal chain over here let's look at the I component of this the I component i over h bar P J sum over J which is X she knows X's and these things have character no these are numbers second line down on the line should it be X hat ETX Y hat ETY here yes indeed they should and this is supposed to be equal X i minus i over h bar and so this must equal this is going to work this has to equal i h bar the chronicer delta so what we see here is a general rule as we'll understand more in a moment observable variables associated with different degrees of freedom commuting so I can simultaneously talk about the position along X and the momentum along Y quantum mechanics allows that alright what about the Hilbert space in this case for this particle moving in a plane defining representation so my wave functions I can I can since I can simultaneously specify this two degrees of freedom I can specify a joint amplitude for being having X a and X b such that this square is the probability density to be near X a and X b the joint at the same time and I can talk about the joint momentum now the position momentum so these wave functions if they're going to be probability issues like normalized they have to be in the set of square normalizable functions on the plane so this is the Hilbert space we're talking about there are functions of two real variables which are square normalizable of course I can talk about the position and momentum space wave functions related to one another to the Fourier transform no reason I can't in this case because X and p for different degrees of freedom can be I can write down a joint probability distribution that's kind of mixed I'm going to know what got a half of a tilde on it maybe I'll put it on this mark I don't know what we would call it we don't have a name for it per se but I can do this there's no reason I can't that's to say this thing is I'm just for a transforming with respect to b so I can write down a joint probability distribution by squaring this joint for the position of part a and momentum of part b it's not the same thing as the Wigner function which had to do with X and p for a given degree of freedom these are for two different degrees of freedom now there is a special set of states which are product states suppose that side of X let's write this one suppose that my joint probability distribution separates into a product of two functions then of course the joint probability distribution factorizes the probability particle probability density for that where this is the square of this wave function of this wave function the product states are in some sense uncorrelated that is to say the probability of finding this has nothing to do with where b is the two probabilities factorize but not every state is of this form not several which has a special name in detail later they're called tangled states suppose I have a state of the joint system which is some amount of that product and some other amount of this product chi and eta are not equal to sine of phi then the probability distribution for this thing is a factorize but this is in the Hilbert space because I can always take if this is in the Hilbert space this is in l of 2 of r2 and this is in l2 of r2 then by definition any superposition of them now moreover every state in Hilbert space our two-dimensional Hilbert space not two-dimensional Hilbert space our Hilbert space on the two-dimensional plane can be made up of superpositions of product states how do we know that well, let's look at a basis the two-dimensional superposition of product states yes so every state this is in every way function in this Hilbert space can be expressed as some superposition of products not every state is a product state but every state is a superposition of product states that is to say there exist basis vectors that are product vectors exactly so that's what I'm about to say so this is a basis these states the simple harmonic oscillator number states they are a basis Hilbert space on the line so I claim the following history before I say that what is it going to be a basis that means that the set we know is complete it needs to be a basis and if I look at that let's look at that let's look at this as in position numbers that says that the sum over all the number of states because on the basis un of x un of x prime is delta x prime that's what it means to be a basis now I claim the following history to be a basis as I say to take a product basis of all the basis vectors for degree of freedom A and all the basis vectors for degree of freedom B how would I prove that? well what I want to be true is that when I sum over all these guys both from their infinity of this should be star of course but they're real that's why they're off but this should be a two dimensional delta function that's the completeness function let's say this is a two dimensional delta function is that true? sure it's obviously true because it just factorizes I have the delta function for A and then the other sum is the delta function for B so what this says is that I can form the joint Hilbert space as a super position of product states from the individual degrees of freedom and that has a technical name it's called the tensor product not to be confused with the Cartesian product so what this says is that the Hilbert space L2 on R2 is equal to the tensor product of the two Hilbert spaces for each degree of freedom so this is HA and this is HB and this is the joint Hilbert space of the two degrees of freedom you can find this formally well let's talk about the Dirac notation here so let's so let's consider some Hilbert space which I've formed as the tensor product of the two Hilbert spaces let's suppose I have a ket psi A which is inside Hilbert space HA and a ket phi Hilbert space B then I'm going to define a product state just like I did with the weight functions but I'm just going to put this tensor product symbol in the middle but a state in the Hilbert space is obtained by looking at the tensor product that's just like multiplying the weight functions with one another often sometimes we are sloppy and we leave out the tensor product symbol and just put them next to one another but it is understood we often always write tensor products symbol there I'm just going to leave it lying on the side yeah B.a yeah that should be right so how does this learn the rules of the game well the tensor product has a following kind of properties it's linear let's just say if I take a constant a scalar complex number and multiply that's equivalent to I can multiply this guy or I can multiply that's the same thing the inner product another part of linearity of course is that if I look at the a superposition tensor product like this that's the sum the inner product so let's suppose I have a product state the inner product between this product state and this product state that's just you take the inner product first case and you multiply that number by the inner product on the second degree you know that's exactly what you would have you had two weight functions right I mean this is very fancy notation but if I had two weight functions that were side one of x a side by two of x b so this is one weight function and I look at the inner product of this with the other weight function or I look at the inner product between those two products base how would I do it I'd start this multiply it by that and integrate over the two variables right and that would factorize this integral and that input that's what this is saying in a more abstract form okay and finally the last one I'll say about this is so suppose A is an operator that acts on the word space A map vectors in the word space A to other vectors in the word space A and I have an operator B that acts on the word space B then I can define an operator that acts on the joint of the word space of the two values for example the product operator is A tensor B how does this operator act we get it acts such that if it acts on a product speed then the first operator acts on this degree of freedom and the second operator acts on this yeah so the operator can only act on an object that's in the right space though you couldn't act that no way it's an operator acts has a domain and the domain is defined by the hover space upon which it acts now we can be a bit here's an example just finish this last thought out so let's consider particle moving x, y, y now if I write the x operator that's the operator that acts on the x degree of freedom really this is a shorthand for x tensor the identity on line because it's really a shorthand we don't always and we rarely would write it this way it would be incredibly pedantic and p, y for example really means the identity on the x coordinate and the momentum operator on the y coordinate that's really what it really means now why do these guys commute? well you see because let's look at the commutator of x with p, y why do operators have different degrees of freedom to commute? because of this structure right this really is the commutator of x with the identity on y commuting with the identity on x and the momentum on y that's what it is so let's write that out that is equal to x tensor the identity on y times identity on x momentum on y minus the identity on x tensor momentum on y times x tensor that's just writing out the commutator in absolute order detail now how do you write this? I didn't tell you that I didn't tell you that if I had A tensor B times C tensor B that is A C tensor B that follows just for you how they act on the states so now you tell me what's this times this so this is x times the identity tensor the identity this times this tensor this times that what about this? 1 times this this times 1 operators acting on different degrees of freedom compute because they will always have this structure always in fact on this degree of freedom it has the identity on all the others if it has x and this is the degree of freedom it has the identity on all the others and the identity commutes with this so it's always the case that observables associated with different degrees of freedom compute so for example the spin that we know spin x y is a no commute but the spin along x with particle 1 commutes with the spin along y particle 2 it's only when they run the same particle that's the degree of freedom we'll continue this


Welcome to Learning English, a daily 30-minute program from the Voice of America. I'm Ashley Thompson. And I'm Dan Novak. This program is designed for English learners, so we speak a little slower, and we use words and phrases, especially written for people learning English. Coming up on the program Monday is President's Day in the U.S. Ashley explains the history of the holiday and shares some interesting facts about our presidents. For the Science Report, Brian Lin has a story on the effects of a possible collapse of ocean currents. Later, Jill Robbins and Andrew Smith present the lesson of the day. But first... The third Monday of February is known as President's Day in the United States. For nearly 100 years, America honored its first president, George Washington, on February 22nd. That was his birthday. But the date was not a national holiday until 1968. That year, the U.S. Congress passed a measure known as the Uniform Monday Holiday Act. The measure meant that some public holidays would always fall on a Monday. Today, the country honors its first president on the third Monday in February, and not on Washington's real birthday. The holiday is now commonly called President's Day. Many say it also honors Abraham Lincoln's birthday. The nation's 16th president was born on February 12th. The federal government, however, still recognizes the holiday as Washington's birthday. The Uniform Monday Holiday Act gives workers a three-day weekend. It also gives shops and marketers a chance to have special President's Day sales. In January of 2021, Joe Biden made history when he took the oath of office as the 46th president of the United States. At 78 years old, he became the oldest elected president. The man he beat was the previous record holder, Donald J. Trump. He took office at the age of 70 in 2017. Biden and Trump are leading candidates for the Democratic and Republican presidential nominations for the 2024 presidential race. They will likely face each other again later this year for the presidency. As the first billionaire president, Trump replaced John F. Kennedy, the 35th president, as the richest man to serve as president. Kennedy still holds the record for the youngest person to be elected president. He was 43 when he took office. Kennedy is also the youngest president to die in office. He was assassinated in 1963 in Dallas, Texas. He was 46 years old. Another presidential assassination put the youngest person in office for the president. In 1901, Theodore Roosevelt, then the vice president, became president at the age of 42. He took office after William McKinley, the 25th president, was shot and killed in Buffalo, New York.  The country's ninth president only served 32 days, the shortest time of any president. Another Roosevelt holds the record for the longest time in office. Franklin Delano Roosevelt was the 32nd president. He took office for 4,422 days. After his death, the 22nd amendment was passed. It limited a person to two four-year terms as president. Most Americans know that the two Roosevelt's were related, but they may not know exactly how they were related. Here it goes. Franklin Roosevelt's wife, Eleanor Roosevelt, was also his fifth cousin once removed. And Theodore Roosevelt, the 26th president, was Eleanor's uncle. That makes the two presidents distant relatives. But they were not the first relatives to both serve as president. That title belongs to the Adamses. John Adams was America's second president. His son, John Quincy Adams, was elected as the nation's sixth president. The Bushes are the other father-son presidential pair. George Herbert Walker Bush was elected as the 41st president. His son, George W. Bush, was the country's 43rd president. Not everyone can run for president. The U.S. Constitution says a person must be at least 35 years old. A person must also have lived within the United States for at least 14 years. And they must be a natural-born citizen. But the meaning of natural-born citizen is not exactly clear. People read the rules in different ways. By some reasoning, the first natural-born American president was not George Washington or John Adams. It was Martin Van Buren, the eighth president. He was born in 1782, six years after the signing of the Declaration of Independence. All seven presidents before Van Buren, and William Henry Harrison, the ninth president, technically were not natural-born citizens. They were born before 1776, when the American states were still British colonies. I'm Ashley Thompson. Scientists are warning of the possible collapse of an Atlantic Ocean current system that would likely lead to worldwide climate changes. Researchers identify the ocean system as the Atlantic Meridional Overturning Circulation, or AMOC. They say the system plays an important part in moving or circulating salt and warm water around the world. This process helps control world temperatures, affects carbon dioxide levels, and supports worldwide water flows. The AMOC system brings warm water to northern areas and carries cold water south. The U.S. National Oceanic and Atmospheric Administration, NOAA, explains. The circulation also transports nutrients necessary to support many kinds of sea life. The process begins as warm and salty surface water flows northward toward Greenland in the Atlantic. Scientists created a complex climate model in an effort to simulate a possible collapse of the AMOC. The computer model was able to measure a sudden weakening of the ocean circulation. The team said in a statement. The research was led by scientists from Utrecht University in the Netherlands. A study describing their findings recently appeared in the publication Science Advances. For thousands of years, Earth's oceans have depended on the AMOC circulation system, which acts like a conveyor belt to control climate. The system is still operating but has experienced slowdowns in recent years. The driver of the conveyor belt sits off the coast of Greenland. As more ice melts here from climate change, more fresh water flows into the North Atlantic and slows the circulation, said Rene van Westen.  Van Westen told the Associated Press his research on a possible AMOC collapse does suggest we are heading towards a tipping point. He added, however, that while his team found clear evidence the system is moving closer to collapse, we're not sure how much closer. In the current system, cold, deeper and fresher water heads south past both Americas and then east past Africa. The system also pushes saltier, warmer ocean water from the Pacific and Indian oceans past the southern tip of Africa. This water flows to and around Florida before continuing up the U.S. east coast towards Greenland. The Dutch team simulated 2,200 years of AMOC flow. The researchers added to the model expectations for future human-caused climate change. The results predicted an abrupt AMOC collapse after 1,750 years. But the researchers noted it is difficult to provide a realistic date because the process could be influenced by many unknown circulation events. Making a prediction is also difficult because current observational records are too short to provide an exact estimation of collapse, the scientists said. The team said more physics-based measuring tools are necessary to create early warning systems. The team noted one difficult area to predict is around the tip of Africa where it is difficult to produce realistic estimates of circulation flows. Measurements around that area have a big effect on how much the AMOC slows, the researchers said. This value is getting more negative under climate change, Van Westen said. He added that when the AMOC reaches a certain level, the change does not happen over time but would be more cliff-like, he added. Van Westen said an AMOC collapse would mostly affect climate in Europe. The researchers say a collapse could reduce temperatures in northwestern Europe by 5 to 15 degrees Celsius. Their study predicts the change would also extend arctic ice much farther south, raise temperatures in the southern hemisphere and change worldwide rainfall. Some scientists warned the climate effects could cause worldwide food and water shortages. In addition, the team said the simulation predicted a possible 100 cm rise in European sea levels if the AMOC collapsed. Van Westen warned that once the collapse happens, the resulting climate effects are nearly impossible to change. Another member of the team, Hank Dykstra, said staying clear of such a tipping point is necessary to avoid the devastating consequences on climate, society and the environment. I'm Brian Lin. Brian Lin joins me now to talk more about his science report. Thanks for joining me, Brian. Sure, Dan. Glad to be here. This week's report dealt with a new warning about changing currents in the Atlantic Ocean that could one day collapse and cause climate changes around the world. What is the main cause of the changes currently affecting this major current system? So, the researchers involved in this study said the main way changes are happening is that fresh water from melting glaciers and the Greenland ice sheet is diluting saltier water in the area. This prevents water from sinking and this limits warmer surface water from circulating effectively through the current system. The report pointed out the researchers involved in the study think it is just a matter of time before this Atlantic current system collapses. But they also admitted the difficulties involved with trying to make exact predictions, right? Yes, that's correct. The model the team created to simulate a total collapse was covering a period of more than 2,000 years. And while the researchers said it is difficult to put realistic timing on an actual collapse, they are very confident it will eventually happen if climate conditions stay the same or worsen. I should note too that last year a similar study predicted a complete collapse of the same current system somewhere between 2025 and 2095. Got it. Thanks again for joining me, Brian, and thanks for your report. You're welcome. Thank you, Dan. Now it's time for The Lesson of the Day on the Learning English Podcast. My name is Andrew Smith and I'm joined by Jill Robbins. Hi, Jill. Hi, Andrew. Our lesson is based on our video series Let's Learn English. The series shows Anna Mateo in her work and life in Washington, D.C. Here's Anna introducing herself. Hello. My name is Anna Mateo. In today's lesson, we'll be teaching seven expressions in English. You will also hear a bit of music from Anna and her special guest. And we're going to use an audio scrapbook to travel back in time to a previous podcast. An audio scrapbook? That's right, and we'll explain that expression in just a few minutes. But first, let's begin with Lesson 23 of Level 2 of the Let's Learn English series. Anna learns that she'll be interviewing one of her favorite musicians, a woman named Emma G. An interview is a conversation where a reporter or journalist asks questions to a well-known person. Or it can be when one person asks another person a list of questions to find out if they are suitable or a good fit for a job. Ms. Weaver tells Anna she'll be interviewing Emma G. And Anna gets very excited. Hello. Anna, this is Ms. Weaver. Ms. Weaver. Tomorrow you'll be interviewing musician Emma G. Emma G, I will be talking to Emma G. Now let's hear how the interview begins. And welcome to the music today. Musician Emma G will be joining us here. She is a wonderful songwriter and an amazing singer. And I love her. Let's give a warm welcome to Emma G. Right here. Well we can't wait to hear you sing. I listen to your music a lot. So I feel like I know you. Is that strange? Anna, listening to music can be a very personal experience. So no, that's not strange. Good. Because I made something for you. A video scrapbook of our friendship. I took photos of you and added myself. I worked on it all weekend. That's strange. Is that even legal? Emma G thinks it's strange that Anna took photos of her and made a scrapbook. The scrapbook has photos of Emma G and Anna together as if they are close friends. There's that word again. Scrapbook. Let's explain. A scrapbook is a book with empty pages but on the pages you can put photos or newspaper articles or other things to help you remember special experiences. A few minutes ago I said we were going to use an audio scrapbook. That means we'll revisit a previous podcast and listen to an explanation of the future continuous tense. You've already heard the future continuous several times today. Here are some examples. In today's lesson, we'll be teaching seven expressions in English. Ms. Weaver tells Anna she'll be interviewing Emma G. Hello and welcome to the music today. Musician Emma G. We'll be joining us here. And here is our explanation from a previous podcast based on lesson 45 of level one of the series. She will have to wait but our listeners can notice right now a special way we can talk about the future in English. It's called the future continuous tense. Listen again. Marsha, we just left DC. We'll be eating lunch in about two hours. Instead of saying we're going to eat in two hours, Anna says. We'll be eating lunch in about two hours. That's because it takes some time to eat. For actions in the future that take some time to do, we can use the modal will plus the verb be plus the main verb with ing. For example, listen to this exchange. What will you be doing tomorrow afternoon? I'll be working from noon until 6 p.m. We often use the future continuous when we specify or say the amount of time we expect the activity to take. If we don't know or care about the length of time, then we can just use the expression going to, in this exchange. Do you have any plans for the weekend? I'm going to try to clean my room and maybe go to a movie. Okay, we're back. It's handy to have a scrapbook, isn't it? It is. Handy describes something that is useful or helpful that makes a job easier. Now let's get back to Anna's interview with Emma G. You can listen for the future continuous tense. Now, Emma, let's talk about your schedule. Tell me, I mean tell us exactly where you will be and when. Exactly. Sure. Well, in early June, I'll be performing at civil venues across Austin, Texas. That's my vacation week. I can meet you there. We can tour the city. We can have lunch. Yeah, maybe. Then in mid-July, I'm going to be teaching at a music camp in Chicago. I will be visiting my sister then. She only lives six hours from Chicago. I could drive over. We could hang out. Yeah, maybe. You know, maybe now is a good time for some music. That's a great idea. Now let's hear a song from Emma G. I wrote this song a couple of years ago. It's called Rarity. It seems like Anna might be obsessed with Emma G. Because Anna wants to visit her in two cities. If you are obsessed with something or someone, it means you think about them all the time or almost too much. The noun form of this word is obsession. And I think Emma G. is becoming wary of Anna. Maybe that's why she chose to sing her song called Wery. Maybe so. Wery is an adjective that means you are being very careful because of something that could cause problems or be dangerous. Wery is spelled W-A-R-Y. For example, you should be wary of a hairy bear. And you should be very wary of an angry bear. I think you should be wary of bears in general. I do too. Notice that we use the preposition of after the adjective wary. Now let's look at two more expressions we heard. Venue and the phrasal verb hang out. Listen again. Well, in early June I'll be performing at civil venues across Austin, Texas. A venue is a place for an organized gathering of people, such as a concert, sporting event, or special meetings such as a convention. Most venues are indoors, but some are outdoors, such as a farmer's market or a stadium. And the phrasal verb hang out is just an informal way of saying spend time together. It's very common in American English. But remember, it's informal. So we usually use that expression with friends and family. Yeah, like Andrew, maybe we can hang out next weekend. That's right. Maybe we can. So to review, today we looked at the expression's interview, handy, scrapbook, wary of, venue, obsessed by, obsession, and hang out. Before we go, here's the end of lesson 23 of Level 2. Anna gets to sing with Emma G. They sing a bit of a song that came out in the year 1983 by the group named The Police. The song is named Every Breath You Take. And you can hear the future continuous tense in that song. The songwriter named Sting says it may sound romantic at the beginning, but it's really about an obsession. He wrote it after the breakup of his first marriage. Here's Anna and Emma singing the song.  I'll be watching you. And we hope you'll be watching lesson 23 of Level 2 and all of the Let's Learn English series. And we'll be doing another lesson of the day soon. Remember that you can also find us on YouTube, Facebook, and Instagram. Thanks for listening. I'm Jill Robbins. And I'm Andrew Smith. And that's our program for today. Join us again tomorrow to keep learning English through stories from around the world. I'm Ashley Thompson. And I'm Dan Novak.