Ie ddim yn dweud y dweud o'r cyfrifloedd Cymru yn ystod i'r dweud. Felly, wrth gwrs, fel y dyfodol, cyfnodd cyfrifloedd yr oeddwyr rhaglen o'r cyfrifloedd ar Weill, ac mae'n fyddon ni'n gael bod yn cael ei ddweud am y penderfynol gyda'r cyfrifloedd cyfrifloedd cyfrifloedd. Mae'n bwysig yma, mae'n dwi'n gweithio'r cyfrifloedd, a'i ddim yn cyfrifloedd yma. I want to talk to you today about the current public health situation and about the new variant strains of the virus, which we are monitoring closely. I'll also update you about our vaccination programme, which continues to go from strength to strength. I'll start with some positive news. The public health situation is continuing to improve across Wales as cases of coronavirus fall. Overall, the seven-day average rate has reduced to around 50 cases per 100,000 people, and all local authority areas in Wales are continuing to report fewer than 100 cases per 100,000 people. This is the lowest rate that we've seen across Wales since mid-September. The R number remains below one, and the positivity rate has fallen to 5.4%. The number of people with confirmed coronavirus cases in our hospitals continues to fall. The latest figure shows that overall number of people in hospital with coronavirus cases continues to fall. In yesterday, there were fewer than 450 people with confirmed coronavirus in hospitals across Wales. That's the lowest number since the 17th of October. As more people are vaccinated in Wales, we may be seeing a faster-than-anticipated fall in the number of people admitted to hospital. We have also seen a reduction in the number of people with coronavirus in our critical care units. This is really positive and suggests that we're over the worst of the second wave of the virus in Wales, thanks to everything that you have done over the last couple of months. By staying at home, you have kept yourself safe and you have helped to keep Wales safe, helping to control the spread of coronavirus. But this doesn't mean that we can afford to let our guard down and forget the basic steps which have helped to keep us safe throughout the pandemic, or to allow us to relax all the restrictions in one go. So we will be taking a careful and gradual approach towards lifting the restrictions, starting with our top priority of getting our children back into school. We'll also be following the very clear advice from our experts to monitor the changes we make for their impact on infection rates. This is particularly important as we emerge from this lockdown because the even more infectious Kent strain of the virus is now the most dominant form of the virus across Wales. This is one of the new mutated strains of the virus, which has emerged in the last few months in the UK and elsewhere around the world. We first became properly aware of the Kent strain in December, but it was probably driving some of the high infection rates we saw across Wales in late November. Now, as you'll have heard before, viruses mutate all the time and it is common to see new strains. But some of these new mutated strains of coronavirus could cause the pandemic to take a new turn, causing more problems as they become a source for more infections and potentially serious illness. In the case of the Kent strain, this is a more infectious form of the virus than the original version of the disease that we had become used to. A number of other new variants have emerged from overseas. To date, there are 24 cases of the South African variant in Wales. Detailed investigations have identified the majority of cases have links to international travel. Investigations are continuing into a handful of cases. These include targeted testing of close contacts to identify the source of infection and, of course, to help prevent any further spread. We're closely monitoring these because of the risk of severe disease and because they may limit the effectiveness of our current vaccines. The UK has introduced some tighter quarantine restrictions to people coming from countries with these new variants where they are most prevalent to try and prevent these strains from gaining a foothold across the UK. New variants can emerge at any time in any part of the world, but they are more likely to occur if we have high levels of community transmission. That helped to explain why we need to keep rates of infection under control and as low as possible, even as more people are vaccinated. We also want to get rates of the virus as low as possible before we begin significant relaxations because as soon as people start mixing again, we know that the virus will start to transmit again. That's why we can't forget the basic steps which help to keep all of us safe. To keep our distance, to wash our hands often, to wear a face covering in indoor public spaces, working from home wherever possible and keeping indoor places well ventilated. I want to finish with that brief update about vaccination. For the last few weeks, we have been working with a reduced supply of vaccine. We currently have the best vaccination rate in the UK and a greater proportion of people in Wales have had both doses of the vaccine than in any other UK country. Last weekend, we passed a significant milestone of a million vaccines administered within the first three months of the programme. We're now getting very close to delivering the millionth first dose, another significant milestone for this truly remarkable programme. All of this has been thanks to sheer hard work and determination of the many hundreds of people working both behind the scenes and in clinics across the country. My thanks and my pride go to everybody involved in this truly remarkable effort. I also want to thank almost a million people across Wales who have taken up the offer of vaccination. You are doing your bit for our significant and unfinished national effort. Every single vaccination really does count. Every dose is another step towards a brighter future. The vaccines are safe and effective and I urge everyone to take up the offer when it is their turn. I certainly will be doing just that. We are expecting a further increase in vaccination supplies in the coming days, so we can accelerate the delivery of vaccines to people in groups 5-9. We remain on course to meet our target of offering vaccination to everyone in this group by the middle of next month. Thank you very much. I'll now take questions from journalists and as usual, all answers will be broadcast live on our social media channels. The first question today is Felicity Evans from BBC Wales. Minister, thank you very much. Will NHS workers in Wales get a pay rise above the projected rate of inflation in the next financial year? We're waiting to see the outcome of the pay review body. We expect to publish the evidence that we've provided to that pay review body. If not today, then in the very near future. We certainly have not done what the UK Government has done and tried to set an artificial ceiling on the pay rise that should be provided. We think that our staff do deserve a pay rise. We're asking pay review bodies for evidence for our determination and advice on what that should be. We want to make sure that the UK Government are in a position to properly fund that pay rise. I understand why so many staff across our NHS will be feeling hurt and upset and a little angry at the evidence that the Department for Health and Social Care at the UK level have provided. As I say, when our evidence is published, you'll see we have not taken that approach. The schedule date for the Senate elections is getting closer and there is increasing concern among candidates that Covid restrictions at the moment prevent them from communicating effectively with voters via leafleting and door knocking and that sort of thing. Some of those restrictions are being eased in England from Monday. What's the plan here? Well, it's one of the things we're looking to consider as we go through this next review. So I think the First Minister may be in a position to confirm something next week, but we recognise that for the election to function we want all parties or candidates to have the ability to communicate with the electorate. And we'd like to be in a position to safely move from having paid leaflet delivery to the sort of things that people can do with volunteers. We may not be in the same position as England when it comes to canvassing though. We're going to need to think about the advice we get and also the reality of how the public would feel about having someone like me or another band of followers from different parties coming around and knocking their door wanting to talk to them. So we've got to think about what the public health advice says, but also there's some public expectation here. I hope that we can make some progress. If we continue to see a continued fall in the case rate, then that should mean we have other choices to make. And that will obviously affect how the election is run as well. Thank you, Fistie. I've now got James Coyne-Smith from ITB Wales. Thank you, Health Minister. You mentioned the pay review bodies. What is Welsh Government recommending to those pay review bodies would be a fair pay rise for NHS staff? We haven't set a percentage recommendation for pay review bodies. They regularly consider the evidence that's put before them. They then go through a process. They then make recommendations to us. And you recall, of course, James, that it wasn't that long ago that trade unions side were asking us to engage openly with pay review bodies, but then they wanted to know that our starting position would be that we would accept the recommendation of that pay review body without setting an artificial ceiling for that. So that's the approach that we've taken. So it is a different approach to the UK Government. And I look forward to receiving the recommendation of the pay review, but I all be it that it's entirely possible, James, that recommendation will be provided either during our pre-election period so I couldn't make a decision on it or after the election when there could be someone different in this place speaking to you and the country as the Health Minister. Thank you. On the handful of South African variant cases, are you saying essentially you don't know exactly where they all might be and how worried are you that that variant might be going through the communities in Wales undetected at the moment? No, what we're saying about the handful of cases, we're still investigating all the links around them. So we're still doing our contact tracing around those people. It's been highly effective to date the numbers of people. We are talking about a small handful of people when we have that confirmation we'll provide it. But as you know, we've taken an approach that is enhanced contact tracing and then testing of contacts to give us the assurance we have caught up with the spread of this particular variant. And that's been effective to date in helping to manage and to contain the South African variant and that's again a different approach to the early approach England took that had wider postcode testing but I think they too are looking at a more targeted approach. So I think people can have assurance that our contact tracing service and our public health agencies are doing the right thing to help keep all of us safe from the potential for new variants to emerge here in Wales. Thank you, James, and I've got Flakey Lynch from S4C. Finan dda, Wili Thog. Now with the on S4C, now with the on S4C understands that a number of senior Welsh Government officials working closely with the First Minister have tested positive for COVID-19. Can you confirm this? I believe the BBC reported on this yesterday that's been confirmed that there have been a couple of positive cases and so a number of people are following the advice given my test rate protect to isolate and work from home including the First Minister. I'm here because I wasn't in at that time and so that's why I can come here today but we have been trying to do the right thing all throughout this pandemic and the way we conduct ourselves here in the Welsh Government with space between people, hand sanitizer, frequent hand washing but this does show that despite all those control measures it is still possible for coronavirus to sneak through. That's why the control measures matter because without those we can be confident that there'd be more cases and it also helps to reinforce why there is a cautious approach out of lockdown. With your confirmation there that positive cases have been confirmed within the Welsh Government staff have the Welsh Government's COVID safety measures failed to protect the health of its employees in this case? No, as I've said, it's the recognition that even with all those measures in place it is still possible for coronavirus to get through because as you'll be aware one in three people we understand are asymptomatic, don't have symptoms so there can still be in a position where a spread can take place so that's part of the reason why we have regular testing. When people do then get symptoms it's why we reinforce please get a test and self isolate from the moment you have symptoms you'll be protecting other people and even then it's also why we have set such store by contact tracing and that's been really effective in this case too and people should take real heart in the fact that the contact tracing service here in Wales isn't just a public service so we haven't spent lots of money on private consultants that eye watering sums you see from England don't apply here in Wales but actually our service is more effective we reach more people than contact tracing services do in England and we reach them faster. This is a reminder that even with everyone doing the right thing with rates as they are it's possible we'll see more people who are doing that thing getting coronavirus. If fewer people do the right thing we'll see further instances of the virus spreading that's why the path out of lockdown is so important for all of us why if we all want to be able to emerge out of this together that's why we all still need to do the right thing and not act ahead of the restrictions that are being eased because we could compromise the further freedoms that we all want to enjoy. My family too and I'm sure yours too, Blakey. Thanks for your questions. I've now got Adam Hale from the Press Association. Thank you Minister. You're boss the first to start some stinging criticism yesterday of the UK Government of Boris Johnson's engagement or lack thereof with the Welsh Government during the pandemic. Nothing new there but with people's minds very much on the future about how we reopen and rebuild the country post Covid. In your opinion are governmental relations between Wales and England worse off or just over a year ago better off or has nothing changed? Well we've seen a fluctuation over the year so I recall that there were regular COBRA meetings through February and March those were initially chaired by the UK Health Secretary then the Prime Minister chaired a few of those and we went into a UK-wide lockdown then the Prime Minister was unwell and then there was another COBRA meeting that he chaired after returning then that hasn't returned since then. We haven't had a regular pattern of the Prime Minister engaged directly with the other national governments of the United Kingdom and we've been in a better position if that were to happen but what has improved though from a long period of radio silence effectively from the UK Government for a number of months now there have been regular meetings that Michael Gove has chaired and those have been useful but as we all know Michael Gove isn't the Prime Minister and so whenever there's a conversation with the UK Government there's a need to then wait to see if that really is going to be the position taken because UK Government ministers need to go back to their boss to confirm what they are and aren't doing so that's unhelpful in our relationships and it would be much better for all of us if you put the politics aside of how people do or don't want to vote which party they prefer it would just be much better for all of us if the Prime Minister did meet regularly with the first ministers of Wales, Scotland and Northern Ireland and I don't think that's particularly controversial You said earlier this week in the health committee although you'd love to see music festivals, sports events with large crowds going ahead this summer the advice that you're receiving from experts is that those things are perhaps unlikely to go ahead at least in Wales can you expand on what you're being told with that why these things might not be possible even if transmission rates are tumbling, vaccination programmes going as well as it is at the moment and the data suggests a more positive picture Well we're bound to publish some more evidence I think we'll help to explain the position that we're in and the approach we're taking so our technical advisory group giving us scientific advice and public health advice that's led by the Chief Medical Officer Department and it's because the vaccinations are really helping us but a substantial amount of the fall in case rates has come from the lockdown measures themselves and the support that people in Wales have provided and again I want to say thank you to everyone who has supported the measures and restricted their contact because that's been the major factor in the reduction of case rates but as we exit and as we gradually ease those restrictions we're dealing with a much more infectious strain of the virus that's one of the things that is behind the caution the important caution we have about easing restrictions as we then forecast a future when you're forecasting about the middle or the end of the summer that's a real act of astrology because you can't properly predict on any sensible or reasonable scientific basis what the virus will be doing at that point in time and that's because we need to see what happens through each point of easement we need to see what happens with the behavioural response how we all choose to behave as each of those set of restrictions ease and what then happens with virus transmission so the caution is there because you're into you're into the sort of stuff that you might read in in your call about your star sign but I don't think that's the way people expect the government to behave in a once in a century pandemic that is yet to be finished so we also know that vaccination will help lots of us but that we'll still have a large number of people who won't be vaccinated say by the middle of June because we won't have vaccinated all of our adults under the age of 50 by then that then means there are a range of people who haven't been vaccinated who are susceptible to infection and actually one in four of our hospitalisations are actually from people under the age of 60 so you can see the potential for harm if we go out of lockdown too quickly if the message to the public is you can do what you like now then actually that mixing will mean the resurgence of the virus with real harm being caused so there's a balance to be struck here and I think the responsible evidence-led approach we're taking in Wales is exactly the right way to go about this and I think that's why the public are still so supportive of the general approach that we're taking here in Wales Thank you Adam, I've now got Will Hayward from Wales Online Thank you Health Minister Given you have said that stay home instructions are ending does that mean that people in Wales will be able to drive as far as they want to exercise or visit people at a social distance in 10 days time would someone in Newport be able to drive to Tenby or someone in Wrexham be able to drive to Snowdonia for example? Of course there's more than one Newport in Wales but look I understand the general point you're making Will and we said we may be able to stay at home we're considering what that then means we haven't made choices Will we're thinking about whether we should have a stay local period or whether there'll be travel permitted across the whole of Wales now that's a choice that the government needs to make we haven't concluded that but there are I think very easy to understand arguments why we would want to have a period of stay local before moving to wider travel and of course it's important to remember that at this point in time the restrictions in place in England mean that people shouldn't be travelling from England into Wales we know that the reality of essential travel for work or other purposes is one thing but actually there shouldn't be travel for leisure at this point in time and that's because as a result of rules in England if nothing else so you'll have to wait a few days longer before the First Minister is here to give you the update on the Welsh Government's decisions about the 21 day review that we have yet to complete Just by way of clarification given that you'll be opening some accommodation potentially at Easter would any stay local restrictions only be in place for three weeks and on the health service there didn't appear to be any extra funding in the budget on Wednesday to allow the health services to catch up with the backlog of elective surgeries when they're able to do more routine work is the Welsh Government planning to allocate additional money to the NHS in the coming years to make it possible to speed up the rate at which delayed treatments can be resolved and if so how long do you think this could take could it be years till we work through this backlog? So yes I do understand your first point about self-contained accommodation and whether that's going to be viable if we still have a stay local period as opposed to wider travel and those are active considerations for the Government but as I say we've yet to complete our determination on those and the First Minister will announce that in just a week's time on your point about the NHS recovery task that faces off it's a huge task and I said previously that I think it'll take a full centre term to get back on top of the backlog that's been created pointing that the budget didn't confirm extra sums of money for the National Health Service because we will need more resources not just to cope with the unfinished pandemic we're living through and it's a matter of fact that we're still going to be living with those additional challenges for our health service additional physical spacing between beds so less capacity the reality that the additional infection prevention and control measures including additional PPE requirements for our staff compared to a year and a half ago mean that you can't undertake as much activity so the throughput is less so actually the backlog will still continue to grow for some time now I don't think it's tenable for England never mind the rest of the UK to work on the basis that there isn't going to be extra funding for the National Health Service in every part of the UK to deal with the backlog it's disappointing that that wasn't addressed by the Chancellor in his budget but I do expect that we'll have to be addressed with in the autumn statement or otherwise here in Wales though we're sticking with our commitment to make sure that the health service does continue to have real terms growth but our ability to address the backlog will be affected if there aren't additional resources and as you know we'll have committed to publishing an NHS recovery plan before the end of this month to set out our approach here in Wales to addressing these really difficult issues not just for plan care but for mental health as well thank you Will Dan Wilson from LBC thank you Minister the Welsh Government said yesterday that it was considering allowing indoor care home visits to resume from next weekend it would be one designated person if that happens is there not an argument that perhaps it shouldn't just be restricted to one designated person that that could put a lot of pressure on that one person and that perhaps those who are have had a vaccine it should be entitled under obviously current restrictions to be allowed to see their loved ones well again we've indicated where we want to go that should allow people to plan and get ready if that is the decision that we end up confirming but the challenge is again this point about a step by step approach to easing and we do know that there are still going to be people in our care homes who are potentially susceptible to being infected and the harm that could do in that environment we think that both people who run and work in our care homes understand there's a need to have a cautious approach and I also despite the really heartbreaking stories of both residents and their loved ones outside of the care homes they live in there's an understanding of need and take a properly cautious approach as well so it's why we're thinking about the numbers of people who could visit but also about the fact that we'll be deploying lateral flow tests as well as an additional measure to make sure we can screen a potential asymptomatic cases as well so we'll deal with a renewed approach we hope to get that over the line and if we then see a continuing improvement in the public health situation and the measures we've introduced to help make visiting as safe as possible we can then consider what we might be able to do in the future. Thank you and as you've mentioned several times during this press conference there is an extreme amount of caution ahead of next week's review of the restrictions but given that the First Minister has hinted that the stay at home rule could be eased from next Friday how much of the decisions made over the next week has to take into account keeping the trust of the public you've suggested something could be eased many especially with the positive spin on the figures will be almost hoping that they are so how much of that decision making is based on you having to keep the trust of the public on their side basically well we have to be honest with the public we're honest about the figures about where they are and what that means we're honest about the uncertainty of the Kent strain of the virus that is dominant and the fact that there's even more infectious than the original version of coronavirus we're also honest about the things we are actively considering including the potential to move away from stay at home including the potential for self-contained accommodation in the future but we've always been clear that we have to look at the whole package of easements that are possible and to then understand what all those together will mean that's why the advice my scientific advice and the chief medical officer is so important and the overarching perspective in all of this is not just how we keep well safe but our first priority is to make sure the children and young people can return to face to face learning in our schools and FE colleges and that's what we're looking to achieve so each of the measures have to be balanced against that to see if we still have the room to do that to do it safely and of course we then want to see our children and young people return to school in full measure after the Easter break so we're balancing a range of things and I think it's all nice to tell the public those are the things we're considering to try to give people an indication of where we're going but not to do so for a really long term punt for the future because I think that most people will be deeply sceptical about that and I certainly take very seriously the requirement to maintain good faith with the public here in Wales that's been a key factor in our response to pandemic throughout Thank you Dan, I've now got Nicholas Thomas from the National Thank you Minister You've already spoken about possibility of an NHS pay rise but what are the work is being done now and for the future to encourage people to work in health and social care and to make those jobs attractive and rewarding? I think Wales has a very good story to tell on what we've done to make a career in the health service an attractive one, that's because we've worked with people who already work in the health service some specific points about recruitment and assistance for people on their education and trading journey and we've also listened to what the public have to say as well I don't know if you'll recall but at the start of this term we were in a very different position on GP recruitment we've made steady and sustained progress we're in a really good position now with GP trainees including the fill of our training places during the pandemic that's really positive for us that people want to stay in Wales who are already here but also people want to locate themselves into Wales for the next stage of their career and the potential to stay here on a much longer term basis as well on nurse recruitment, on nurse training the same for midwifery, the same for therapists we've got a very good record of significant growth and record investment in every year that I've been the Cabinet Minister for Health in reinforcing the need to replenish and renew our NHS workforce so I think we've done both we've made our jobs attractive to people across a range of areas and we put our money where our mouth is and we've invested in extra training places Thank you In December some health boards reported staff absence levels as high as 10% and there were reports too that this was having quite a serious impact on frontline services If another wave of infections were to happen what has been done to mitigate any new rises in NHS staff absences? Well this is one of the challenges that any Government will have to deal with not just before the time but at the time as well so the restrictions in the ability of the NHS by the full range of services that we're still living with now are partly because we didn't have the staff to deliver a full range of services but also of course of the cases that were coming in with coronavirus that required treatment in an NHS bed even today our critical care units are over 100% of their normal capacity so we're still dealing with a service that is very full of people with coronavirus even as we move into a better position If we did see another wave at any point whether the summer the autumn or the winter and what we do know is because of the success of our vaccination programme when making such progress at our second doses that means being able to expect more of our staff to be protected protected from harm that doesn't mean though that every single person will be because no vaccine is 100% effective so we still have to live with the fact that some people will still be susceptible to an infection even with our highly successful vaccination programme even with our highly effective vaccines so any minister in the future would need to make a choice alongside the service about any need to either restrict treatment or indeed to introduce new measures for infection prevention and control so there's a balance in these measures and whatever happens the next washroom will still be dealing with coronavirus for some time to come and need to balance how we open up more activity while still keeping all of us and our country safe Thank you Nicholas I've now got Mark Hutchings from FiveLife Thanks very much for those running non-essential retail or closed contact services and trying to plan and second guess ahead of next week's review how much notice will they be given about possible reopening could it be say within 10 days? Well at the first minister we'll be outlining the determination we reach and these are not straightforward choices Mark and I'm sure you and people watching understand we've got to balance the information that we're still getting in on the public health position the position of what's happening in terms of how people are moving and behaving as some of our children have gone back to school that's a factor in the potential spread and then the risk as I've said earlier in a whole package of easements to make sure that we still have headroom to ensure that our top priority is achieved for more of our children and young people to return to face to face learning and then to see what else we may be able to do so that's the approach we're taking the first minister will confirm what we're able to do and there are conversations taking place with representative bodies in those sectors to try to understand the notice they would ideally need and then the reality of our decision making time scales you know some people could start pretty much straight away whereas other people need to get stuck in have their staff put back from furlough so it'll take more time to get up and running and do that in the discussions we have but you'll hear first properly from the first minister next Friday I'm conscious this is a bloke asking another bloke but in recent weeks there's been a various report from women exercising, running or cycling for instance about feeling vulnerable or even being victims of physical or verbal assault as well as being calling it itself to show the need to return to group exercise or reopen gyms as soon as possible well the first thing to make clear is that whether it's verbal or physical abuse it's wholly unacceptable women whether individually or as the rules provide exercising with another person are entitled to do that and to do that without being in fear for their safety that's a very clear and simple message but we did change our rules partly because of the concerns some people had raised about whether they felt safe exercising on their own so you can exercise with one other person now and we asked people to maintain a social distance were we thinking again about what we might be able to do in the next period it would allow some social interaction with social distancing or the potential to do things outdoors those are real factors we're having to take account of we also though as you know have to balance the various different and competing elements of safety so people's safety when they're out exercising but also what it means to our general safety if more people are mixing whether outdoors or indoors and gyms of course is an indoor environment even with control measures there's a risk in their opening as well I don't think you could really sustain an argument that the real imperative to open gyms of safety when exercising outdoors because we've still got to balance the challenge that as good as the figures are and the directs they're moving about 50 cases in 100,000 still shows a real reservoir of coronavirus within our communities and that's why we've got to move cautiously but these are real challenges for us to consider and take on board and real factors that we do discuss across the government and with sectors as we plot our path out of lockdown and hopefully to a better and sustainable future Thank you Mark, I've got Rob Taylor from rexham.com Gathlein Minister, you've likely touched on this but we're aware of a range of businesses starting preparation work for reopening and beauticians and hairdressers taking appointments all in an expectation of a full reopening on March 15th for clarity is that fair expectation and if not does Welsh Government need to be clearer with the signal into sectors about what specific dates mean Well we're working with the sector about this and people who are taking appointments for the future know they're going at risk to do that You've heard the First Minister talk about the desire to have some services open but you also are aware that when we ease out of lockdown previously from the first wave we opened in a phase way close contact services so it was haircuts first and other services afterwards so we're having to consider how we address each of these points in turn and how we try to make sure that in the advice we get we understand the potential impact of every easement as a package and how that affects our top priority of returning children and young people to schools Thank you and the local health board explained on Tuesday about limitations of vaccine supply with about half of their supply being used for second doses only Is the expectation of increases of supply from next week bearing fruit and how substantial will that uplift be? Yes you'll see from today's figures that we did around about 30,000 doses yesterday so we're managing our stocks to make sure this is particularly the case for the Pfizer supply that that's making sure that we have got enough to deliver second doses what we don't want is to end up in a position where we're not able to deliver those second doses because we've actually been too optimistic in the way we deliver our first doses that's a point about managing carefully and efficiently the programme we've got The significant increases that we expect to have through this week you're starting to see in today's figures you expect to see those through the rest of this week we'll then need to make sure of course that in about 12 weeks time we've got enough supply to deliver the second doses for the ones we've delivered as first doses this week so there is still a logistical and a practical challenge in managing our stocks to make sure we're going as quickly as we can but bearing in mind we now have this challenge of first and second doses so I'm expecting that in the figures through this week and into next week you'll see that significant increase in supply being delivered into arms across the country and we then expect to see a levelling down but then actually a more steady state a supply which allows to keep going at a significant pace and meet our expected targets in both mid April and then for every adult in the country to have the opportunity to have their jab by the end of July Thank you Rob, I've now got Alan Evans from Planetly Online Thank you Health Minister The Royal College of Surgeons has said that the overall number of people waiting for treatment in Wales is the highest number on record more than half a million patients they called the figures staggering and devastating and I've called on Wales's seven health boards to start planning for the recovery of surgery in Wales and I know you touched on your answer with Will earlier on saying that the NHS budget didn't go far the budget didn't go far enough to allow for this but shouldn't the money given to other projects sort of be more prudently used and diverted towards this given the sort of urgency and the risk factor of losing lives will given these figures Well the figures are staggering you know we have got a significant backlog built up because of the choices we've had to make to keep the country safe and if we hadn't made those choices and if we have some NHS services being restricted in the reality is we would have seen more mortality and the figures are really upsetting in any event when you consider the amount of people have lost their lives to coronavirus but it could be much much more significant that does mean though that as we hopefully see the end of this pandemic we will then have this significant waiting times challenge and we will have big challenges around the mental health need of our staff and the wider public too so we are already thinking about how we are going to address that and as I said we will publish an NHS recovery plan before the end of this month that will set up the approach we take the operational plans that will then come a little later but that will allow us to give more detail on what we are able to do because as I say the position we see today position we are likely to see is that we are still going to see our ability to treat people to deliver on the sort of pace to eat into that backlog is going to be affected because we won't have seen the end of the pandemic we know we are still going to have additional PPE requirements for our staff so they can't operate the previous levels of output and that means the backlog even if other activity is returning is still going to increase for some time to come that's why I say this is likely to take a full centre term to get on top of it and there are a risk to Manusula of more harm and the potential for irreversible harm for some people who are waiting and that is a very difficult balance to strike but it's one we are going to have to do to make sure we have the best sort of recovery whilst at the same time recognising that our staff haven't had a break and we can't simply expect our staff to run faster at the end of this pandemic to catch up on that really enormous waiting list The Minister may have seen the documentary on the DVLA recently given the large number of cases and alleged inadequate protection in the work list at the DVLA which resulted in the employees suffering both physically and mentally what if any are the repercussions for the most senior managers what a duty of care for the employees and the community at large and will the Minister consider launching a public inquiry or police investigation into potential breaches of legislation and the potential that had to further infect the environment Well I'm not in a position to commence a public inquiry I'm also not in a position to require the police to investigate what's happened and of course the DVLA is a UK government department and so you'll recall that the First Minister has written several times to the responsible Minister with the concerns that we've had about workplace practices and the rules, the laws of Wales have been properly applied in the way that the DVLA was operating as an employer in particular about people's ability to work from home and the distance between people in the workplace and whether that has caused an additional risk and I know the trade union side are particularly unhappy about the way that the employer has operated we have had direct conversations between our contact tracing service public health Wales and the DVLA it doesn't appear there's been some improvement but I fully expect that there will be a need to have a retrospective look at what's happened both to learn lessons to make sure our staff are properly protected but also there's a point here about accountability for senior managers that is an issue for responsible UK ministers but I don't think it's something that they can contract out of and I don't think that this issue is resolved Thank you Alan, that concludes the questions for today, I look forward to talking to you at some point in the future Take care everybody and stay safe


Okay Okay, now we can finally start I guess okay again Okay Shorter version geometry and topology very interesting understanding their interaction Yeah, so I guess the the the tool that I want to explore a little bit in this three lecture is the The the Iraq operators, okay, so and I guess the The way the thing I want to discuss a little bit today is like what the rock operators are and how you can somehow relate Them to these two fields so so we really see some nice interaction today between geometry and topology and Just as a as a warning so here. It's really the analysis of Of the rock operators that comes into play I'll be very sloppy with the analysis I want to focus more on the interactions, so I'll say maybe something which is imprecise, but you know nothing falls, okay? Yeah Okay, so let's start with what the rock operators are so Yeah, so I guess before starting to talk about the rock operators Let me just kind of remind you that the essentially the most fundamental operator is the Laplacian And this is the Laplacian, okay, and well, maybe you're used to seeing the one with the pluses but I Prefer the one with the minuses because this is the positive one. So yeah, I guess we'll you know This is better for geometries in some sense Yeah, in particular, you know, this is you know the the solutions To these equations are called harmonic functions Okay, and they're very important objects, for example, they They come up everywhere like in physics or even you know, for example the the potential in electromagnetism or stuff like that So, you know the study of harmonic function is a very important topic of Study in in geometry Okay. Yeah, so I guess the idea is that you know, this is Very nice. It's a second-order operator in both second derivatives, and I guess the rock 1928 asked asked himself I Guess what what he asked himself is that he wanted a first-order operator D this square is The Laplacian, okay, he was motivated by trying to study the the quantum mechanics of electrons and stuff like that But yeah, I guess from our perspective is just well, can you find some operator that squares to the Laplacian? Okay, and of course, you know, even from a very busy perspective is very nice to have a first-order equation because it's much easier than a second-order equation Okay.  Let's try the simplest thing you can try You know just write a general differential operator first order with some constant coefficients Well, and now let's let's set, you know, this is the equation we want to satisfy This one so let's set it up. So Yeah, so if you write the square of this Well, we're assuming that AI are constant. So so this square turns out to be, you know, the sum of AI squared plus the sum of AI a j the over Okay, so if you want this to be The sum of minus square the Xi square you get a system of equations Which is AI square equals minus one Well, the derivatives commute. So What you get is that, you know, the excited xj is the same as the xj the Xi so and the term does not turn We'd mix the derivatives here. So what you get here is that AI a j plus a j AI is 0 Okay, so this is a very easy simple idea Yeah, so let's see if you can solve these equations somehow. Okay, so Yeah So if n equals 1 Because n equals 1 Well, you know, I just want You know, it's it's very easy.  I guess this we learn in high school But you can do it in complex numbers, right? So yeah, it's very nice because then you can set the number Okay, and then you can get your the the operator you want. So this is called the Dirac operator D So we're happy with this. Yeah, so this is and you can check easily that if you square this you get minus first There you at this good the second there the index one square Okay Yeah, and if n equals 2. Well, I think that's a little more interesting, right because Well, this is saying that H2 is minus h2 a1 Okay, so this you cannot solve, you know If if you work with complex numbers, you cannot solve this just because well complex numbers are commutative and we know What why these implies that they're non-zero? So it seems like you kind of you know, there's no solutions of these such Equations in complex numbers, but the rocket Dirac's idea is like Well, you can solve this if you you look at more complicated algebraic structures and in particular you can look at two by two matrices So you can set Yeah, you can set for example a1 Equals Zero minus one One zero two okay, and they satisfy the equations you want Okay, so then your Dirac operator turns out to be You know a1 b and x1 Yeah, and this is just minus d in the x1 plus i One plus And you can check that these squares to the Laplacian if you you know We so here at the beginning we're thinking of the Laplacian acting on functions But you know it turns out that the Laplacian has a square root if you think of it as having values in C2 valued functions Okay, so if you try to write a square root of the Laplacian on function you fail But it's nice because you know this square equals Laplacian acting Yeah, so I guess the question is like how do you go back here? Yeah, instead of thinking you as a like a function I just think of it as a as a column vector for example. Yeah, so it's a C2, you know more than Vector value function. Yeah, so this is exactly we got so I guess that the idea is that you know You have to move on from functions number value function if you move on with vector value function, then you can find a square root Yeah, exactly. Yeah, and you know just to keep on going. Let me tell you maybe I write them here and they'll be Yeah, just as a reminder, you know, just as point out this these things very look very closely to You know the anti-holomorphic and holomorphic derivative of a function, right? So Yeah, so the Dirac operator in n equals two dimension is very close to holomorphic geometry Okay, so this is more or less, you know depending on your convention. This is more or less okay, so That's some very interesting relation between this natural the Dirac operators and holomorphic geometry Okay, so maybe I'll leave it here. So I guess n equals two. I'll leave it here for later Two zero and then if you want n equals three Well, we go back to the we got the famous Pauli matrices so you get sigma one is I zero zero minus one sorry a one one zero Okay, so this is solution again in two by two matrices you can solve what we want It's eyes. Yeah Yeah, so we wanted to square to minus one So this does the job and then n equals four Can people see here? I guess you can see from there. Yeah, you You get some matrices. Yeah, you need four matrices and what we do is just you look at four by four matrices With blocks. So you get minus the identity Identity Okay, so this is one and the other are obtained by these AIs in some way So, okay, these are try this means transpose conjugate the star Okay, so once you get to four by four things You cannot find solutions in two by two matrices anymore But you know you can solve it using four by four matrices and the four by four matrices have this nice block structure Okay, and you know if you if you look at these maybe at some point You can guess how to get five six and so on you just keep on blocking Any question. Oh, I guess well Oh, sorry. So I guess the original question is why do we want this? Yeah, so I don't want to say silly things because we're not theoretical physics building. I'm not a theoretical physicist I think yeah, so to to to find the relativistic quantum mechanics. You need a first-order equation So he wanted something which was related to the Laplacian, but was a first-order equation So actually his paper original paper for net 28 is very readable. So I suggest it's very nice He's using the same notation we used today essentially, so it's like so it's one of the few a hundred year papers all Paper that still very modern. So the question is like, well, what's the dimension you need? Yes So actually every two every time you go up, you know You multiply by two and then you stay the same you multiply by two you stay the same So I think it's two to the floor of n divided by two or something like that I'll say more formally what the thing is But yeah, just draw the the first few low-dimensional cases just to show you you know the occupator Usually people Introduce them by oh spin bundle clifor algebra. It's something super concrete You just write down the matrices and you know you work with those Yeah, so I won't talk about spin bundles and clifor algebra So Yeah, you can think of yeah, so this is yeah in some sense, you know this this is I wrote down a solution to this problem But of course not the unique one, but this is in some sense the minimal one. So in a specific sense Yeah, yeah, so the question I guess the people in the chat can read that so I don't have to repeat that I have to repeat for them. Okay. Yeah, so the question is like, yeah any operator Which is this is positive because so you can't find Square root in the functional calculus sense But yeah, it's it's much nicer to work with differential operators because those are more close We'll see that that would be much more closely related to geometry. I guess that's the answer. Yeah, but yeah, of course Because it's an operator, you know, which which is positive that Appalachian has a square root as a pseudo differential operator But you know, that's hard to yeah any other question Yeah, let's move on then. Yeah, so let's keep this in mind Yeah, so maybe I'll Yeah, so we we have this the rock operator So, you know, I just wanted to say something super concrete and but this is in our end So let me tell you what what happens on a manifold Okay, so I'll just tell you the formal properties and you should think of it Oh, it's something that you know has these formal properties as and looks Something like this at each point. Okay, so this is we'll go to Yeah, so the rock operators Yeah, so the actual definition. I was mentioning I guess it's pretty complicated You know, I would have to spend probably the fall the three hours just to define the derock operators properly But yeah, let me just tell you like the formal properties of So this is a Riemannian manifold Yeah, and we let's see we choose a spin structure So choose a spin structure suppose Yeah, so I Won't even define what a spin structure is it's something about taking square roots of your bundle Which is you know, it makes sense because here we're somehow taking square roots of your most fundamental operator on your manifold It's a plushion But yeah, let me let me tell you what what's how to check if a manifold that meets a spin structure so This this is if and only if the second stiffer Whitney class of your All the tangent bundle, which is an element in Each two is two coefficients banishes. Okay, so it's a very easy thing to check if your Manifold is spin, you know, for example, if the tangent bundle is trivial, then you admit a spin structure Yeah, and it's equivalent so if And the dimension of your manifold is at least three this is equivalent to Yeah, I'm restricted to this to skeleton Yeah, so this is it so we'll always assume that there is a spin structure. It's not unique So the operator will write out will depend on which spin structure we we choose Okay, so let me just tell you what Formally the Dirac operator is then oh, yeah, by the way, I typed out notes for these lectures and I'll post them afterwards This afternoon, I guess or right after class on my website Yeah, so the the lecture notes are more or less a transcriptional way I read them with some more detail and references and Maybe point out some details that are important, but I don't have I'm not going to say in class Okay, so what what does the speed structure give us? Well, I guess The Dirac operator had this thing that you had the target space, you know We're looking not at functions, but Cm valued functions. So in general on a manifold that will correspond to some kind of bundle Okay, so this gives us the C to the M. Okay, so what is this? Well, this is a bundle. So this is a S over M. This is a hermitian bundle. So it gives you the dimension we're discussing so to the guys if I equals one it's a Like a one-dimensional. So here it's C. So here the target space was C. I guess the same color is C2 C2 and C4 Yes Yeah, and this also comes with a Connection so this is a hermitian Okay, so once you have a bundle, you know, we're not in our end anymore. So you need a way to Talk about how the fibers relate to each other. That's a connection Yeah, and that that's determined by the metric and then you have the Dirac operator And the Dirac operator an operator but D from sections of S to sections of S Yeah, so, you know, this is a little more abstract, but you know really If you get lost you should always think oh, this is the same thing that we saw in our end So we have our operation or operator between section of this bundle in in in our any is just a trivial bundle So it's CM valued functions and then, you know, there's your standard derivative and you know, the rank is what it is Yeah, and you know if you if you're careful in some sense in local coordinates this operator in low-dimension looks exactly like This these operators that I wrote. Yeah So the question is like where is this bundle coming from and the question is like locally it's Essentially exactly what I wrote here The problem is that you to do it globally you need to make it make sense of it globally You need to be very careful and that's where the spin structure comes into play Unfortunately, I won't have time to tell you where the bundle comes from but yeah There's very detailed references that in the notes. So if you're interested you can look at that Yeah, so I guess the bundle is like, you know, if you try to glue this this Yeah, the bundle. So here. Yeah, I always think of M and oh, sorry What does the but the question is like what does the bundle depend on? Yeah, it depends on the manifold And I guess the actually it depends on the metric to and the choice of the spin structure Yeah, so this is an object associated to a spin Riemannian manifold. Thanks Um, okay, let me tell you two key properties of the Dirac operator so two key properties first of all it is a Formally Self-adjoint. Okay. So what does this mean? It means it's essentially you should think of it as a symmetric matrix But if you pick two sections of your bundle, which are compactly supported, for example, if the manifold is compact then So if you do d phi Psi the L2 so, you know, you take the inner product and integrate it over the whole manifold So maybe I'm here. So, you know, S is a Hermitian bundle. So you can take inner product of things So with the node Okay, this is you know, it's a symmetric matrix. So you can bring the D to the other side Okay, it's a fun exercise in integration by parts to show that this is true for the operators I drew Down there and this is also integration by parts then a more important. Well, I guess another very important property is that It is elliptic. Okay, so this is a very important analytical property of the equation I want to find okay. I want to find Okay, but we will use a lot the consequences of this ellipticity So for example, a very important consequence key consequence. For example, is that, you know if d phi equals zero Okay, so if you have a solution to the to the to the Dirac equation. So this is a question here. So This is like phi is the harmonic And the same sense in the same way that if you have function who's Laplacian is equal to a harmonic function This is all I didn't I didn't give the name to this S. This is called a spinner bundle Okay, so the sections are what we call spinners and this is a harmonic spinner. Okay, so, you know D is a first-order operator. So for this equation to make sense You need to be able to take one derivative and that's enough Okay, so this makes sense for example for For a C1 function, okay So, you know if you if you have differentiable maybe with continuous derivative then this equation makes sense The very cool thing is that if this is true, then you're you're automatically a smooth So this is one of the key proper, you know It means that it's a very nice equation You know in general, this is not true You know if you write a random differential operator, it's not true that a C1 solution is necessary smooth But is this true and you know maybe because of what I said this is Shouldn't be too surprising for you. You should compare this Well, this is true for example for the bar equation This is what we learned in complex analysis that holomorphic functions are necessarily smooth Okay, even if you know the definition you really use very few properties and also for the application So harmonic function also have this property and spinners are kind of closely related. So so this is not to surprise Yeah, yeah, yeah. Yeah, I guess if you square it, it's yeah. Oh, sorry. Yeah, so I guess the question is Yeah, so Well, it wasn't the correct one more like a calm like Yeah, is that the the fact that this is true for the Laplacian implies this for the Iraq essentially Yeah, so that's true in our end in on a manifold. We'll see that this square is not exactly the Laplacian It's very close. So yeah from what I say later. That's that's That's true. Yeah Yeah, any other question so now we're you know, this is our Gadgets these are out. This is the tool that will allow us to somehow connect geometry and topology on a manifold Okay, so let me just let me start by telling you how this can be related to Okay, so what another key property of ellipticity? Is that also implies? Another key property that if on a compact manifold KERD. Okay, so this is The file in the space of harmonic spinors is finite dimension Okay, this is not true for the general operator So, you know, if I give you a random differential operator the space of solution You know, there's no reason for it to be finite dimensional But this is true for general elliptic operators on a compact manifold So yes in particular you get some numbers out of your operator Which is some dimension and it turns out that that number which is obtained in this analytic way So it's trying to solve some differential equations Is very closely related to the topology of the manifold. Okay, so Yeah, so we need a little bit more to state what the relation actually is I need to tell you a little more about a little bit of structure Yeah, so now let's assume and even okay, so for the first Sure, I will mostly talk about the and even dimensional case and then we'll switch to our favorite dimension Which is three and then of course we have to go to an odd In the last lecture, but yeah, let's look at an even what happens Yeah, so it's very interesting. So, you know an odd, you know, there's this diagonal term So, but you know if you like that and even The diagonal terms are always zero. So here is also always zero Okay, and it's a bit so the dark operator somehow splits in two parts and you know just and only has off diagonal terms Okay, so you also here if you write down explicitly what it is There also be some how do the composition in two pieces and the operators exchanges the two pieces Okay, so when n is even Your bundle splits There's some natural splitting. So that's the case of our end and actually it holds also on a general manifold that your bundle splits Okay, and your Dirac operator You can decompose it as a block in two blocks zero minus D plus and zero Okay, so you should keep in mind You know the case n equals two that I wrote out explicitly Okay, so the Dirac operator you can write it as these blocks with zeros and two of the other terms and this Yeah, so you have You know sections of this bundle plus and section of this bundle minus and this is plus and this is Okay, and these are called the the chiral. Yeah, so now Because these adjoint Sorry this self adjoint Okay, it's because these are symmetric matrix It means that D plus and D minus are adjoint to each other. Okay, so in some sense, you know D plus five psi Okay, so well, they're not self adjoint because you know, they don't even go to from the same place to the other But somehow they're one of the transpose of the other Okay Yeah, so if you are So and now oh Yes, sorry, this is also yeah, thanks. Yeah, the question is like should we integrate and your answer is yes Yeah, so in particular, you know if you remember from linear algebra So the if you have a matrix the image of the matrix the orthogonal This is the kernel of the transpose Okay, so this in our situation. No this still holds in this situation and we get that the dimension of the co-kernel of D Plus so the everything modular the image of D plus This is the Dimension of the okay, so this is and this is finite Okay, because well the kernel of D minus is containing the kernel D plus so and these are all finite So we can form this quantity I guess definition The index of D plus is index of the plus this is the Dimension of the kernel of D plus Minus the dimension of the kernel Sorry, well, I guess Okay, and this is a natural number Okay, so here we use ellipticity in a very very strong way in general neither of these two numbers is finite But in this case because of ellipticity and you know on a compact manifold So for now on all it was work with compact manifold. This turns out to be an integer. Okay, so what's the what's the magic here? Yeah, so the point here is that These two things These are not invariance spin structure Okay, so if you you know manifold is what it is, but if you change the metric and the spin structure, so maybe Yeah, sorry under changes of G So if you change the metric is not a topological invariance on the dimension of the kernel of the operator and the dimension of the Cockernel the operator. They are not invariance of the many. They're not topological invariance So, you know our goal is to somehow Understand interaction between topology geometry. So these quantities separately. They're not good Objects to study just because if you change the metric you get something different Okay, but the miracle is that this index it is it is an invariant. Okay, and there's a formula for it in terms of characteristic classes okay, so Yeah, so let me write this formula. So this is as very special case of the The idea singer index theorem To the apply to the case of guess this is Theorem due to what here and singer okay, so then under assumption we have so I'm a compact spin manifold Then the index So the index of the plus which is a integer Okay, and it's an analytic quantity because you know to compute it in principle You know you need to solve the Dirac equation find the space of solutions and compute dimensions So this is an analytic quantity. This is analysis can be computed in terms of the Pontiagin numbers Yes, so the question is like what happens with spin instead of spin see what's been see depends on an extra line bundle Yeah, there's a more complicated formula. So I'll just stick with the simplest case. Yeah Yeah, I guess the idea is that for any elliptic operator does a formula. So yeah Sorry, there's someone in the back first. Yeah, it turns out it's the same I read the formula and then you'll show that it doesn't depend on the spin structure Yeah, the question is like does the index depend on the choice of spin structure and the answer is no But there's no a priori reason. It's just the formula doesn't involve the spin structure Yeah, in the case of spin see the question. Yeah, but in the case of spin see structure depends on which spin see structure Yeah, and it's very cool this theorem is very cool because it relates analysis on one side and Here you have topology, you know Pontiagin classes are topological invariant of your of your manifold So somehow this is the way that the thing that will allows us the first step going from topology to analysis And let me just tell you a little bit of formulas. So n equals 2 Well, then the index Is always zero so any equals 2 mod 4 guys or 2 6 10 so somehow nothing interesting happens So if n equals 4 Well, the the index is Minus the first Pontiagin class Of TM so this lives in H4 so we can evaluate it on the top class and Then divided by 24 Okay, and you know for a format if all the Pontiagin class is related to the classical signature So you can also get you know, but here's the book you get to minus signature of M Divided by 8 guys. So the index of the dirac operator Is related to the signature and you know that there's crazy formulas going on So I'll just write the next one if any cost 8 the index is Minus 4 p2 plus 7 p1 square. So this is a class in H operate so you can evaluate it on the fundamental class of M and then you divide by 5,760 Okay, so, you know, it's already very non-trivial that this number here is divisible by 5,760 so you can you can get a lot of interesting divisibility for Pontiagin classes and a lot of cool theorems Just by using the fact that this turns out to be an integer Question that yeah, so the question is like is there a homotope invariance So I always forget the statement that the Pontiagin classes rational contract in classes are homotope invariance, right? That's not because something so yeah, they're ration their homotope. I guess Paul is Okay, yeah, they're homotope invariant The integral ones are not right, but the rational ones are yeah. Yeah, any other question? Oh The question is that why the index defined for D plus well, you can define it for D minus they're adjoined to each other So the index of D minus is minus index of D plus. So yeah, so one of them is the you just need one of them Yeah, thanks. Okay. Yeah, so I won't have this is one of the you know That yes in your index theorem is one of the most important theorems of the last century and I'm not yeah You know, I don't have time to discuss anything here other than the formula that you get so but yeah I'll there's references in the in the notes if you're interested in learning more about it Okay, so this was the link, you know our goal was going from topology to a geometry and somehow we managed using the Index here to get to analysis. So now let me tell you how to get from harmonic spinners Analysis to geometry. Okay, so yeah, so now this brings us back to the beginning of the lecture. So Geometry, okay, so recall the Dirac's motivation Was we want this square equals we want something that squares to the Laplacian, right? and In our end, okay So Dirac the place we started with is that Dirac wanted an operator that squares to Laplacian In our end and yeah, so somehow well we did a lot of steps Right, so we we wrote it down in our end and then I told you oh you can do the same thing on a manifold But not clear now that you know what what this even means and if it's true on a manifold Okay, so in fact, it's almost true, but we I need to tell you in which sense is true So on a manifold. Okay, so on a manifold. What's the Laplacian first of all? Well, the Laplacian You can think of it, you know Laplacian on our end is Minus the divergence composed with the gradient. Okay, so to make sense of these I need to tell you what the divergence and the gradient are on a manifold. Well, so the gradient, you know so this is You know, we have our connection on our bundle. So that's the analog of the gradient That's a connection picks sections of your bundle and spits out one forms with values in your bundle and this is You can think of the connection as the analog of the bundle analog of the gradient Okay, so this is Yeah, I guess again The very nice thing is that the divergence is the adjoint of the gradient in the integration by part sense so You can take You can define simply by thinking this is the adjoint formal adjoint is the analog of minus Okay, so when you integrate by parts you remember there's always a minus sign that pops out Which is now which is annoying and that's why we we Include them in our formulas. So you don't have to worry. So yeah, very very concretely what this means is that So I keep on looking for erasers, but Yes, so so what this means, you know again L2 adjoint means that you know L2 Okay, so it's a you define it by integration by parts and this is True in our end if you replace this by gradient and this by diverges Okay, and in particular, you know this expression we can define the Laplacian So, you know if you do NABLA to go here and NABLA star you go back to the original place you start with so Is our the analog of our Laplacian? Okay, so this is the analog of the Laplacian on your bundle. So this makes out, you know, this here I didn't use didn't use anything about the bundle. It's you know, the spinner bundle This is this you can make this operator for any Hermitian bundle with a Hermitian connection or like, yeah So then this is usually called the the bachner Laplacian. Yeah, so this is the natural Laplacian on any bundle with a connection Compatible with a metric. Yeah, so the key result that connects The analysis of the rock operator with Geometry is the following version of this for a manifold So this is a theorem due to the rockets Yeah, by the way, I didn't I didn't give you the time frame. This is all stuff that happened in the 60s more or less Yeah, so yeah, so this formula is almost true But not quite so there's a you know when you take squares There's always second derivatives and on a manifold, you know unlike a random second derivative I see don't commute on a manifold so you can expect that when you do things There's some curvature term popping out and it turns out that the actual formula is very nice So this is this so you can think of you know, this is exactly the formula But then you have some correction which is given by the scalar curvature of your manifold. Oh, so the question is like is there a nice proof of this theorem I think Yeah, so I guess yeah that you either you just write out I think we talk about with this Otis a lot I think by pure thought maybe one can try to figure out some just symmetries of tensors and stuff one might Figure out that it's related to the scalar curvature somehow and that's the only thing I haven't tried it Yeah, but I guess I don't know how to guess the over four though from that So, you know, am I able to somehow figure out they only depends on the scalar curvature, but then yeah, I guess that's Yeah, yeah, I don't know. Yeah, I haven't tried though. So there might be a way Yeah, so okay, I won't do the computation for you because I don't need to define properly the dirac operators But yeah, so what why is this interesting? I think you know, this is a very You know, it seems like a very innocent formula But I think what's the magic behind this formula So the magic So, you know, we were looking for a square root of the Laplacian So it's not surprising that, you know, so this square nabla star nabla our second-order operators With the sphere with the same Second-order part by design. Okay, so we were looking for an operator that's worth a lot plus in our end So, you know, it's not surprising that the second-order parts are the same But if you take the difference of two operator, which are the same second-order parts, you know You will expect that to be a first-order operator Okay In principle, okay, but the magic is that the first-order term is zero Okay, so this is a zero-thorder term is multiplication by the scalar curvature Okay, now you'll see that, you know in the argument I'll give you that if you had a first-order part and things wouldn't work out nicely at all Okay, so this is really the magic. Okay, so the magic part is some magic consolation of first-order terms Yeah, so s is just the actual scalar curve. It's a function. Yeah Okay, that's it What was the second? Oh, the second question. Well, can you stand over this board? Okay Okay, yeah, yeah, I can Yeah, any other question while I stand in front of this board, I guess I need to erase this board anyway So I guess, well, it's a good place because I forgot to the general statement. So it's a good place to write it Yeah, so I guess We said that the index The plus is zero if and is 2 mod 4 and In general, it's a combina, you know I wrote the formula for n equals 4 and n equals 8 but in general there is this crazy expression in the Pontchagin class It's called the a had genus This is called. This is a hat. This is called a had genus. This is an expression Pi of tm that you can write out if you want Okay, so let me put these two things together I guess it's a I guess it's nice because I'll just erase the middle part and So yeah, I guess yeah, you can think of this somehow relate differential operators so analysis to geometry so you can think of this as analysis Okay, so if we put these two things together, we get a relation between geometry and topology as follows So I guess this is theorem so suppose Mg a compact spin manifold and suppose that the scalar curvature is positive everywhere So they will say positive scalar curvature. Okay Then the index of the plus is zero. Okay, so this is some particular for example a G if It's a four-dimensional then, you know the index of the plus we know that it while I raised it by was the signature Than the signature is okay So for example, you know, this is in general, you know, there's a very a lot of metrics you can study on a on a Riemannian manifold and you know that it is a it's relatively easy to give constraints on the geometry of positive sectional curvature and positive itchy curvature, but scalar curvature is very hard to Study in particular. I think this is was the one of the first thing that the first intense in which you can give topological Substructions to the existence of this positive scalar curvature metric. So for example, you know There's this famous for dimensional life, which is called the K3 surface This is spin and has signature equals minus 16 Then So no PSC metric on the K3 surface. Yes Yes, that was the main What about Cp2? Yeah, that's my next example. So example Cp2 Has signature signature equals one and has has PSC metric Okay, the Fubini study metric. Yeah, so why is it that why this is not a counter example to the theorem? Yeah, let's not spin Cp2 is not a spin manifold. Okay, so these are very sharp theorem. You really need to use spin structures to spin Yes Yes, in particular, you know, if you write let me rewrite this theorem Using the previous one. So this implies that this tells you that they had genus of TM the top part Okay, so this is a topological constraint and this is a geometric constraint Okay, so somehow we managed to relate Geometry to topology using Dirac operators Okay, any question about the statement. Yes, we have a question there Well, the question is like can you have a hat non zero? I guess The question is like well, the the way this is the top class So by Pongare, you know the duo the top class has to be zero I guess because it's a perfect pairing It's like a one-dimensional vector space the top dimensional. Yeah Yeah, I guess Yeah, the lower dimensional the hey had genus really it's it's a lot of classes They're not just and those I this doesn't tell you anything about those Yeah, so this is only tells about the the the combination put drag class in the top So it's a linear constraint One linear constraint on your contrary classes. Well, I guess it's not linear in the contract. Okay. Yeah, so Yeah, let's prove this it's It's it's integration by parts Yes, so maybe I'll keep this here and I'll erase this one I guess let me call this the proof of theorem a Okay, so we show You know we want to show that the index of d plus is zero the index is the dimension of the kernel and the dimension of The co-kernel so, you know the easiest ways to show this is just to show that both those numbers are zero, okay? So we'll show that Care D is Trivial, okay, and then you know this contains care D plus and D minus Okay, and the recall index is the dimension of care D plus minus Okay, so if you show this then this implies that the index of D plus is zero Okay, so suppose it's not zero there exists Phi which is not identically zero such that D Phi, okay, then we can use this formula here. So apply it. Let's apply this formula here. So so d square Phi equals nabla star nabla Phi plus scalar curvature for Phi Okay, but of course, you know, this is D of D Phi. So this is zero. Okay, so now now we'll do our you know, let's take We take I Don't think I need this anymore. So maybe I can move so we have this formula and let's take The inner product now to with Phi. Okay, so we have zero equals nabla star Phi to But now, you know Phi is not identically zero. So it's positive. It's on zero at some point and The scalar curvature is positive everywhere by assumption. This is where we use our assumption So this term is strictly positive. It's an integral and at some point. It's positive. So this is strictly bigger then nabla star, okay, so here is where we use You know, I can't change but it's positive everywhere and now we can use, you know by definition nabla star You can bring it as a nabla On the other side. This is the definition of the fact that nabla star is a joint of nabla So you can bring it to the other side, but without the star Okay, and now this is an inner product of an element with itself. So it's non negative Okay, so we get our contradiction because there's a strict inequality here and have two zeros at both ends Yeah, so I guess this is like one of the first intents like probably the first instance in which People realize that you can't you know if you want to study Geo the relation between geometry and topology or many for you can look at the Dirac operators and you know the the rest of this class will look at some More complicated examples Yes But yeah, let me tell you, you know, these are very nice theorem, but you know, it's well you need assumption that the manifold spin But yeah, of course, you know, it's it's too nice for you to expect to be for it to be sharp in some sense So let me just tell you a very simple example in which this theorem fit. Yeah, so I guess the question I guess the question. So I guess Let's take that and torus just Rn modulo alaris so z to the n and because you know the the matrix on our end the sense So this has a has a flat metric PC Scalar curvature identically is you metric? Okay, you know You keep our together scalar curvature from the sectional currency.  Okay, so can you somehow bump the curvature up at each point a little bit? Okay, so let me just compare these so compare. This is Myers theorem Okay, so this is a classical theorem in a Riemann geometry that doesn't use much technology Like all the comparison But yeah, the theorem is that if you know if you have a If M has metric With positive richie curvature, then the pi one of M Is finite? Okay, so in particular, you know what what you show is that the universal cover is compact So in particular the the pi one is fine. Yeah, and it's very interesting that, you know, richie curvature. So Remember scalar curvature is the trace of richie curvature But somehow this is much more, you know, there's much stronger or some constraints on this on having positive richie curvature and People took people on time to figure out constraints on the existence of positive scalar curvature metrics Yeah, and of course the torus, you know, pi one of torus Infinite, so no no matrix with positive richie curvature. Okay, and you know our our Now so let me right here. So here the tangent bundle of Tn is trivial Okay, you just pick like the basis the standard vector fields on on a ran and they descend to a trivialization downstairs Yes, in particular this tells me that in particular Tn is spin and Tn is spin and It's been and Well, all the pronged shagging classes are zero Okay, so in particular also So they had genus. Okay, so this constraint Doesn't apply to Tn. So the the theorem we just proved doesn't allow us to conclude that The torus doesn't emit metrics with positive scalar curvature Okay, so I guess the idea is that you know, we've been looking at us So I guess this is I don't know as a spoiler next time will prove that the answer to this question is yes And the idea is that this theorem will look at only a single dirac operator But really there's a lot of dirac operators floating around on a manifold. Okay, especially when b1 is positive So let me tell you a little bit It's this obstruction, you know, let me raise it because it's not useful for this more complicated problem Um, so so theorem a Does not answer Okay, so to answer the question, you know a theorem a we look at a single dirac operator But really, you know as the title of the mini course suggests, you know, the title is families of the rock operators So it's much better in general not to just look at the single dirac operator But you can look at some families of twisted versions of that's a question. Oh because Well, the question is like how do you conclude that kernel of D is zero Well, if you have an element in the kernel of D plus If you have an element in the kernel of D plus, that's also in the kernel of D because it's a block matrix D minus D plus zero and You know an element if an element in the kernel of D plus has this shape So then also the you know the yeah, but I guess by the definition of the domain it has to look like this Yeah, thanks. Um, yes. Yes. So the idea to study Families twisted. Okay, so what do you twist the dirac operator with so I guess I didn't really define properly the rock operators So I won't even define properly what twisting a rock operator means. I tell you I just tell you what you can twist the dirac operator with Okay, so So on M. So Consider, you know, we have the trigger bundle on M The trivial bundle and we can pick a Flat connection. Okay, so you have a trivial line bundle and we look at the connections on this line bundle in general There's a lot of them At least if you have some b1. So, you know, these these are in bijection up to up to equivalence With The torus of flat connect. This is a each one Okay, and the correspondence is given by a whole on me around. Okay, so if you have a flat connection, you know You can hold on me part of transport things around loops and you know things will come back a little rotated. So that gives you Whole on me and you know a flat connection on a bundle is determined up to equivalence by its whole on me so we have a lot of Interesting flat connection on this trivial bundle. Okay provided. Well, this is this is b1 of M Yes Yeah, and once you have something like this you can create the twisted the rock operator This is by this connection. This is the twisted the rock operator and this goes from the same bundle Okay, so you we twisted our rock operator by this flat connection be on a line Okay, so we have a big family of these Operators on the torus for example. So the idea is like try to give up take obstruction out of this big family of the rock operators And this satisfy All things we said before you are the standard the rock operator. It's just a little twisted version Yeah, so all the statements. I made it for about D Also holds for DB Okay, here you have to be careful I'm twisting with a flat connection if you twist with a non flat connection and things change But you know if you just a flat connection of things remain. Yes, you will find out next time But the answer. Yeah, I guess the answer is is no Sorry, the question is like is the answer. Yes or no Well, the answer is No, oh, yeah, we kind of proved the existence of metrics using this technique We can prove that things don't exist.  If you remember I started writing down matrices and matrices, you know Yeah, you know this twisted the operator are not that much more complicated Listen simple examples. So let me just tell you the simplest example of this twisted Iraq operator Yeah, so basic example, right, let's say M equals s1 Yes, I guess sorry the question is like up to gauge. Yes, sorry So yeah, I guess I said it Verbally, but maybe let me record it in written form Up to gauge up to isomorphism up to bundle isomorph is there Causified by holomony. Yes There's a question there Yes, the question is that do they all square to the same thing? Yes, because it's a busy flat connection in general if it's not a flat connection You got a curvature of B term too. Yeah, but I guess in this in this mini course We'll always work with twisted by flat connections. So the the the Lickner of its formula wrote also holds for that And you used you why you need to use instead of nabla nabla B. I guess that's the yes So I'm because it's one if you remember the derac operator was very simple. It's I the let's say, you know This is Corian X. This was ID in the X from, you know complex valued function to itself Okay, and this twisted derac operators are not much more complicated. So, you know M. I think of it as R mod 2 pi Z Okay, and then I pick next here Okay, the twisted Well, all connections on on S1 are flat. So we were you know up to equivalence. They have the form the I D in the X plus C Okay for C From C infinity Complex number itself and you need to pick C Real number. Okay, so any of these turns out to be a flat connection. Oh, sorry a twisted derac operator and You know really there will be an S1 family and here we have a Real parameter, but some point they become the same so To to C prime, you know in general they look like different parators But well if you think of matrices there are like you can conjugate one to them So their equivalent does a change of basis that brings one to the other our equivalent C minus C prime is in Z. So this is an exercise in the at the end of the notes if you want there are some exercises that You know should help you grasp What's been going on in class? This is one of the exercises So, you know, we have this family of operators parent tries better real line And they are the same when the parameters the pan, you know, they're different by integer So we have an S1 family of of twisted derac operators Um, yeah, so I will and here maybe I guess I have two minutes. So I'll just Mention a Definition that I forgot to say I guess this fact of having I guess I'll say it a lot of times next time. So I guess it's good to for me to say it so we look at this deraco operator and It has finite dimensional kernel and co-kernel and those kind of parators have a name so and Equivalent like there's a nice change of basis between of them. So if you if you do a Yeah, I guess, you know at the end of the day This they become sorry Yeah, sorry, the the question is like in which sense are these operator equivalents Well, you know that equivalent meaning in the sense of matrices. So there's a unitary transformation That carries one to the other Okay, and at the end of the day, this is the say, you know You can think of this somehow as a some kind of this gives you a family of flat connection on circle And they are equivalent exactly when this is So you have this S1 family of connections. Yeah, but I think yeah the exercise Ask you to think from the point of view of being equivalent as operators Yes, there's another question up there Yeah, I might be no, I think it's pie Z right. Sorry the question is like is it pie Z to pie Z or Z? and I think The question the answer is I don't know, but I think it's Z. I thought about it before writing it, but yeah Okay. Yeah, that's yeah, that's too pie here. So the the Fourier modes are e to the And Z. Yeah, so there was a question there. Oh so the question is that is it obvious that the dark operator is Fredon and No, but it's in general true from ellipticity. So elliptic operators on a compact manifold Are Fredon between the right spaces? Yes Okay, I'm one minute over time. Can I take one minute to write what Fredon means? Okay, so definition But this so next time we've got deep into more No algebraic topology and functional analysis how they relate so this is a Word that I say instead of saying, you know, it's just a word to say find a dimensional kernel a pokernel I'm in a single wall. So so suppose you have H Hilbert space. Okay, so we'll always assume separable for us Okay, and then T from H to H a bounded operator dimension of kernel T and dimension of Cochernel T are both finite. Okay, and we can define its index in general index of T is defined to be so so here it's where I start to lie because the Dirac operator is not bounded and That there's a lot of lies, but you know, it's but yes follow has a question Yeah, so that I guess the question in general if you have the questions I do you need the image to be closed? That's automatic for Hilbert spaces or Banach spaces in general Yes follows from the open mapping Theorem, but yeah in general if you if you look at from operators between random topological vector spaces You need to add the condition that the image is closed. Yes Yes, and also yeah, I guess to be at some point. I lied to you, you know the fact that the the image of the The orthogonal the image of these the Cochernel also used that the image is closed and that I didn't even mention That is true, but also follows from ellipticity. It's a footnote in my notes. Okay. I think I'll stop here


To get a closer look at a single displacement reaction, which is also an oxidation reduction reaction, we will use a microscope camera to see the reaction between copper metal and silver nitrate. A drop of one molar silver nitrate is placed on a microscope slide. A thin copper wire is touched to the solution. You can see the rapid formation of shiny needle-like crystals of silver metal. Notice that the solution turns pale blue due to the formation of copper-tuned nitrate.


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