Good afternoon. You are back with the Vermont house government operations committee where we intend to do a final markup and vote on as 124. I think without, without any further delay, I will invite Betsy and to point us to the document that she'll be taking us through. Alright, hello. Let me just quickly I should have done this also I will send a quick email to all of you containing a potential revised edited draft 4.1 this morning I sent the editors your 3.1 draft the clean copy. And they worked so hard to get it done quickly get it done get that edited quickly the 3.1 that we reviewed this morning. And then what I did in the meantime, in between our break is make the few changes that you had discussed this morning. So let me just I emailed that to you and Andrea will be able to post it soon also so people will be able to look at this draft 4.1 clean copy edited. It's an actual strike all amendment form, except for right now for the purposes of this discussion I have highlighted the six changes that you discussed making to it this morning. So it's okay to proceed. We've got Lauren has a question. Yes, I just wanted to ask Andrea to post that on our legislative documents and handout pages quickly she canceled, because I otherwise I have to leave this screen and go over to, you know, on my iPad, but I've got my laptop here so she could post it there. And I had just sent it to her before the meeting started so that's why there's a little lag in posting it should be there now. Awesome thank you Andrea. Okay, I will refresh that page might just take a minute to go over the interwebs. I think we deep breath for a moment and given that Betsy Ann had about a five minute, well 10 minute break I guess between appropriations and coming back to committee thank you. That's the last one. Okay, got it. Thank you Andrea. Okay. So, this is 3.1 taking the 3.1 clean copy, which had incorporated all the annotated revisions and then adding to it the six changes that you discussed today. So Madam chair you're looking for just a super high level run through pointing out the changes. I believe so. Alright, page one changing the name of the criminal justice training council to criminal justice council in order to reflect all of its duties and powers and then directing ledge council to change that name in during the statutory revision process not having to come back to you with a technical corrections bill. Page two updates to law enforcement applicants and adding in their current law duty description to maintain statewide standards of officer professional conduct through their professional regulation duties. And on page two line 17 that they're also approving education programs that's current. Moving on to page three, the council membership itself would not change as you reviewed it this morning. That would mean keeping on there the commissioner of corrections, adding the commissioner of mental health as proposed by the Senate, then scrolling over to page four, starting online 16 those seven new public members that would be appointed to the governor. No law enforcement connection, not be legislators or employed in the criminal justice system. At least one of them would be a mental health crisis worker. Page five one, a member who's an individual with an lived experience of mental health condition or psychiatric disability, and then two members chosen from among the Vermont chapters of the NAACP from nominations made to the governor from those chapters. Still online 10 governor appointing the chair of the council from among those public members. Online 13 council members are entitled to per diem compensation and reimbursement of expenses as permitted under 32 vsa 1010 from monies appropriated the council. That means you get a per diem if you're not already getting paid to be there. For example, our commissioners who serve on there are a G they already have state salaries it's part of their state duties to serve on this council as this as you would require them to section five at the bottom of page five allows current members to continue to serve out under the remainder of their terms if they're going to be reappointed under the new membership. But here's your first change you discussed this morning on page six line for that the new members of the council have to be appointed by December one, your prior draft said So that's change one page six section six is where I was going to have to adopt rules to identify and implement alternate routes to certification aside from what's provided the academy and then lower down having to be required to offer courses in different areas of the state and strive to offer non overnight courses whenever possible. On page seven. There's that language on line 17 maintaining the as past senate deadline that by July one and next year a level two officer may be able to transition to level three by portfolio experiential learning or club testing without having to restart the certification process. House of probes did discuss the concern that the council has in regard to not having that it stated it did not have the resources to meet this requirement, but so far no one has indicated they requested a change to this date at this time so right now it's leaving it at the 21 deadline on page eight and section eight here is the report back by January 15 from the executive director of the council. Darn it and how did I miss a just or no I did fix it okay criminal justice council. Okay online seven. Their report back on how they're how things are going on restructuring its programs and so the first report back is about how what its plan is to replace some of its overnight training with non overnight training. The second change that you discussed this morning was that they they would have to specifically address in this report back any plans it has to offer training by remote means reflect what you wanted to do there. Okay, and they'll also be telling you online 17 how it's going with that transition for level two to level three. And how it's changing its restructuring its programs to handle that so you will be hearing about that in January. So, if you don't change that date now potentially that'd be a date to change next session, but they do have the rule adoption deadline of July 123 to to adopt the rules it's required to about restructuring its programs. So page nine section nine there's that can Council services being contingent on agency compliance with reporting requirements and other policies required by the Council chapter, not to adopt procedures on how to handle that new requirement. Page 10 is essentially a technical update to statute to reflect current day realities which is that one agency can provide training to other officers and those officers can get credit for that that goes toward their certification. Page 10 section 11 there's that duty to contact a current agency before a potential hiring agency is going to hire the officer they have to get the written disclosure from the current agency about the officers performance there. It's a written disclosure page 11 line 16 is the requirement you discussed this morning that they have to keep that disclosure confidential. Page nine 18 has the prohibition on a CBA between an agency and the bargaining unit for officers employed by that agency from having a prohibition on the exchange of information between the employing agency and another agency about the officers performance there. Page 12 sec 12 has transitional provisions on those new disclosure requirements. A is saying that the requirement to disclose does not apply if there's a binding non disclosure agreement prohibiting that disclosure. And then subsection B is providing that a CBA that already prohibit that's in effect and that prohibits the exchange of information. You don't have to break the contract of those that CBA provision, but that requirement will apply upon the expiration or termination that CBA and to any other CBA that takes effect on after the effective date of that section. Section 13 at the bottom of page 12 in regard to the use of body cameras and a body camera policy and complying with one of your officers are going to be using a body cam so a has a new subdivision one designation but only the one designation is what's new. That contains the language that starting on January one each agency that authorizes its officers to use body cams has to follow the model body camera policy that the council will establish. But as a transitional provision because there will be officers that are using body cams that includes the requirement for all DPS to use law enforcement officers to use body cams the act. 147 slash s2 19 and any other officers might be using body cams the new subdivision to requires until that date in subdivision one the January one 22 each agency that authorizes its officers to use body cams shall adopt follow and enforce a model body worn camera policy established by the leab pursuant to the 2016 act and each officer who uses a body cam needs to comply with that policy so there will be a body cam policy in place it's the lebs and that's what will control until the council. Model body camera policies established and officers and agencies have to start complying with that on January one 22. So that was another one of your changes you discussed today. Section 14 contains that prohibition on facial recognition technology unless the use would be permitted with respect to drones under 20 vs a 46 22, which does authorize the usual facial recognition technology in limited circumstances. There's a definition of facial recognition technology on page 14. Then we get to section 15 which is about the council's duty to professionally regulate officers and here this section is discussing when agencies have to report to the council alleged unprofessional conduct committed by an officer, specifically in regard to category B allegations on page 15. On line eight agency has to report that when there's a credible complaint, not after an agency goes through a full investigative process. Page 16 line five is the requirement for the council to provide a copy of any investigative report it receives to the council advisory committee, which recommends appropriate action to take in regard to an officer who's the subject of that report. Then we get to section 16, which are the law enforcement recommendations these are reports that will come back to you for the general assembly to do further work in this area, as you may choose to do so in the next biennium. This provides that by January 15. Entities have to report their progress on the following topics, including any recommendations for legislative action, with that exception that the criminal justice council shall submit a verbal progress report to the gov ups committees by that date January 15. And any recommendations for legislative action on or before March 15 21 so you're giving them more time to come back to you with recommendations for legislative action. Otherwise everybody else has to do so by January 15. You're going to be getting on page 17 recommendations about officer qualifications from the LAB and the council. You'll be getting recommendations on officer training from the council. On page 18 I just made a terminology update to refer to people with a mental health condition or psychiatric disability to more fully describe the people that experience those conditions. And just noting on my page 18 starting online nine. There's talk about the restructuring of the council and just a reminder that you had eliminated the language about moving the police academy so that no longer exists there. Online 18 for models of civilian oversight. And G coming back to you with recommendations after consulting with entities that include BLS is Center for Justice reform on page 19 line for you'll also be getting recommendations back about reporting allegations of officer misconduct. And a number five access to complaint information that will be coming back to you from the council advisory committee. Body cameras number six online 18 here adding back in the language that the LAB would be required to report any changes it deems necessary to its body. Model body camera policy that it established pursuant to the 2016 act required it to do so, since there will be the requirement from now until January 1 2022 for officers to follow that policy. So we're just rolling on page 20. You'll be getting a report back on military equipment for the council will recommend a statewide policy on officers acquisition of it. All right, page 21, the bill starts to move into the topic of state data collection and analysis. This is the section 17 requirement for GAC to consult with executive director of racial equity, social equity caucus and the CPO and accept recommendations from other entities in order to approve by that date population level indicators that demonstrate the quality of life for black, indigenous or people of color, as they relate to the current law population level outcomes. And once those are approved by the GAC, the CPO would be required to report on them in the state outcomes report. Section 18 is the amendment to the outcomes state outcomes reports statute to provide on page 22 online nine the more language about the intent and the purpose of the outcomes and indicators, which are that outcomes are intended to reflect the well being of all Vermonters and indicators are intended to represent the experience of all Vermonters, including an especially Vermonters who are members of marginalized groups, as they are the ones that are more likely to be the outliers on that indicator data. So if you have that statute there for reference if you get any questions about the current process for approving indicators you can see that on page 23, starting online six, with that GAC authority to do so now. Page 23 line 17 section 19, there is the state granting authority contingent on compliance with the roadside stop data reporting requirements, and now the death or serious bodily injury reporting requirements that are required. When an officer is involved in a mental health crisis and death or serious bodily injury results. The one correction or the one update to this language is on page 23 line 21 you discussed earlier to strike local proceeding law enforcement agencies so it's clear that this applies to all agencies who would like to get a grant. Just a note on that this section has an effective date of January one 2021 because that was the effective date that you had in S2 19 for adding this language so accordingly that would also take effect. These amendments to that language would also take effect on January one. Page 24 section 20 is in regard to the VCIC requirement to have definitions that all officers must use when entering data into their agency system of records. So the VCIC on page 25 consulting with the Vermont crime research group statewide racial justice groups and statewide groups representing individuals with lived experience of a mental health condition or psychiatric disability in establishing and providing training on a uniform list of definitions to be used by agencies when they're entering in crime data into their system of records which might be the court or Spillman and every officer would have to use those definitions when entering their data. Then we get into the LAB repealing it from where it currently is in law putting it where it's supposed to be maintaining the Senate's proposed changes to the membership there be four new members added to it. As discussed there's no per diem or expense reimbursement authorized. Then the Senate, aside from changing the membership or in accordance with changing that membership just updated the quorum requirements accordingly. But you had not discussed making any changes to that page 28 section 23 is just a recodification directive it's just saying that now any references to LAB in 24 VSA 1939 where it currently lives are going to be deemed to be references to 20 VSA 1818 and Ledge Council needs to revise accordingly any reference to the old location and statute section 24 is requiring LAB to include in its annual report. Recommendations in regarding the ways towns can increase access to law enforcement services. LAB has to file an annual report. Then starting at the bottom of page 28 we get into the Department of Public Safety Dispatch issue and section 25 so let's scroll over to the relevant section which is subsection I on page 29. As you discussed in your draft 3.1, you can see on page 29, starting online 15, the elimination of the current law language that allows DPS to charge sufficient charges to recover the costs of their dispatching that they perform under contract. And says dispatch functions that are fully funded under these contracts are authorized under the grant statutory authority. Hold the thought on this because there's a also a related session law section right after, but just a reminder on page 30 that 1873 removal of commissioner. There has been a dead law sitting in statute since 1979, since the Vermont Supreme Court said that this language was superseded, because all governor appointees under the office of governor serve at the governor's pleasure. This language has been outdated pursuant to the Vermont Supreme Court case law. The statute said that Commissioner public safety can only be removed after hearing and only for cause. It is a really fun case to read. If you ever want to read it. Basically, the governor tried to use the new authority to remove the commissioner is pleasure and then the person just showed up the next day saying hey I get to continue to serve. But anyway, it's just a technical correction because it's been dead law. We get to section 26 on page 31 line 16. And this is in regard to that dispatch fee structure authority. So this new language is updating what you had earlier today in your draft 3.1. We had the prior understanding that DPS was in some cases charging for dispatch. Nolan confirmed with DPS that's not happening DPS has these contractual arrangements but it's not charging for them. What it was planning to do was set charges that they were going to start charging municipalities. This language would say instead that in accordance with the amendments to 20 vs a 1871 sub I that we reviewed up above, which eliminated that charging authority from law DPS shall not charge fees in any contractual arrangements it enters into to perform dispatching functions for state municipal or other emergency services until the General Assembly establishes in law a dispatch fee structure for those charges. So it is clear that DPS cannot charge for the dispatch functions it performs until General Assembly establishes a fee structure. Page so can I can I just flag this here so that I don't lose sight of this after we finish and vote on this. I just want to be sure that we get a heads up to the chair of the ways and means committee about this I believe that the language we've put in here. Because it doesn't change or establish a fee shouldn't require the bill to go to ways and means. But since we have had a few snags along that line in the last couple of weeks. I'd feel better if we sent her this particular section and and just gave her a heads up to ask for it if she feels like she needs it but I don't believe that she will. At least possible it's going to get automatically referred by rule just because even though DPS is not charging by statutory authority currently has it could so it would just depend. Whether it has to go there or not but good I good call just to make the chair aware of that. Page 32 line three. Here's the language you already reviewed about DPS having a process to propose to the General Assembly a fee structure it would say that by March 15 of the next year. There are at least three public hearings and consult with the LCT EMS advisory committee, the police chiefs association state firefighters association and local emergency medical services police and fire agencies. I would support by that date to house go box house ways and means Senate go box Senate finance, the department's recommendations for an equitable dispatch fee structure for DPS to charge for dispatching emergency medical service police and fire services and potential funding mechanisms for those charges that do not rely on property taxes. And if the department decides to overrule substantial arguments and considerations raised against the fee structure potential funding mechanisms it ultimately recommends DPS has to include in its report a description of those arguments and considerations and the reasons for the department's decision to overrule. Let's get into the EMS part of the bill on page 33. What's going on here is a substitution of Department of Health for State Board of Health and establishing EMS districts and issuing ambulance licenses keeps rolling page 35 there's that language requirement about ambulance licenses can only be issued to be provided if those services will be provided in a non discriminatory manner doesn't discriminate on the basis of income funding source or severity of health needs in order to ensure access to ambulance services within the licensee service area and Department of Health would be required to adopt rules in order to administer that new qualification scrolling bottom of that page 35 its Department of Health that issues ambulance licenses not the State Board Page 36 section 28 is a requirement for the HRAP the health resources allocation plan to address EMS resources and needs that are identified by the EMS advisory committee which appears later in the bill I'll just pause here madam chair. I remember you had mentioned that health care was going to be reviewing the EMS or at least partially this language. So just putting it back on the radar in case they need further information. Yeah, I did make the chair of house health care aware of this. Again last night in case he had any concerns and and I had told him at that point that we were intending to vote it out in our 830 committee meeting so I'm going to take that as the language the way it is but I suppose if there was some earth shattering change that needed to be made they could offer it as a floor amendment. Page 38 line 10 we get into the language to just make it clear in the statute that you still have to be affiliated with an affiliated agency in order to be EMS personnel. And just due to some of the removal the credentialing requirements there. It's better to from the feedback from DOH it's better to make clear that affiliation is still required even though credentialing is not required, you still have to be affiliated. Page 39 starting online eight is a requirement for department of health to establish at least three levels of EMS instructors and the education required for each right now there's only one. Page 40. We get into that alternative for psychomotor skills testing the hands on testing of your qualifications to be EMS personnel. The national crediting entity that and our EMT requires its own psychomotor skills testing for the higher license types the advanced EMTs and the paramedics but it does allow flexibility for testing the psychomotor skills for the lower license types. And so this would allow psychomotor skills testing for EMRs and EMTs to be accomplished either by the demonstration of those skills competencies as part of their education as approved by the department of health or by the and our EMT psychomotor exam. And we'll just say hands on. We've just got more updates about the affiliation requirement on page 41 top of page 42 is a requirement for department of health to establish by rule and entry level certification for EMS first responders. So this would be the lowest level authority to be EMS personnel. But it could help get more people into providing EMS services. Page 42 line five is eliminating an outdated statute that doesn't need to address transitional provision from certification a licensure that happened multiple years ago. Second is the sunset review where DOH would have to at least once every five years do an ongoing review of the continuing competency requirements that any EMS personnel professional has to achieve in order to renew his or her license and it's modeled off your S233 uniform licensing bill. It contains similar language for example on page 43 where the department has to amend its rules or propose necessary statutory amendments in order to revise the continuing competency requirements that are not necessary to protect the public. On line 10, we've got the EMS advisory committee and adding the new EMS Education Council. On page 44, the EMS advisory committee would need to include in its annual report the mutual aid calls that are going to outside of the area that EMS personnel have to respond to. And on line six, that new requirement for the EMS advisory council to identify EMS resources and needs in the EMS districts and provide that information in the Green Mountain Care Board so that they can include it in their periodic revisions to the HRAP, the health resource allocation plan, which is addressed earlier in the bill. On line 11 is that EMS Education Council, which would be new and it would be established from among the current EMS advisory committee. This new education council can sponsor training and education programs that are required for licensure in accordance with the Department of Health required standards for training and education, and then also provide advice to DOH regarding the standards for emergency medical personnel licensure and any recommendations for changes to those standards. Page 45, there's just an update to the statute that provides to whom EMS training funding can go and it's adding in the new certified Vermont EMS first responder certification that would be added by this bill and then also the current law licensed EMRs, emergency medical responders, so that they could also have access to that funding. Page 46 contains the transitional provisions for these EMS portions of the bill. The rule adoption deadline is July 1 unless LCAR extends it. The ambulance license service non-disc, ambulance service license non-discriminatory requirement begins on July 1. Or by the date DOH adopts the rules to administer that whichever is later because they could get an extension to adopt those rules. C is that language to transition an instructor that's currently licensed. There's one instructor license level now. And this would say that the person get transitioned into the new level and according to the three that will be created pursuant to this bill. So you go into the license level that's consistent with your scope of practice from among the three license levels. D on line 18 is the requirement to establish that new Vermont EMS first responder certification by July 1 of next year. And then E on page 47 line three is the first sunset review that they'll have to do. It would need to be in conjunction with the rulemaking required by this act. So as they're looking at these rules, they'll take a look at those continuing competency requirements. See if any of them need to change because they're not necessary to protect the public. And then finally, we've got public safety planning with your new language about regional planning commissions creating an inventory for its towns of each town's public safety resources that are available to them. And how those resources can be shared on so towns can better understand how those resources can be shared on a regional basis. You discussed bumping up the deadline to be July 1 22 and then the state inventory would have to address mutual aid agreements that any towns have and any public safety plans that they have. And we're talking about public safety resources it's referring to law enforcement fire and EMS and dispatch. Finally your effective dates overall the bill will take effect on October one, except that new council membership would take effect on December one in accordance with the deadline to make those appointments by December one. So that language regarding the contingency for state grant for law enforcement, need to be contingent on complying with data collection that takes effect on January one, as you or language did for as to 19. Excellent. Thank you. So there we have it committee we have a 48 page bill. Jim Harrison. Yes. Thank you, Madam chair I was wondering if maybe ledge console will go through the bill just one more time. I'm kidding. I'm kidding. As always, good job Betsy and you're very. I learned to talk quicker so I could, you would leave me in the dust. Anyhow, as with any bill, there are certainly parts that I think are very positive and parts that I might take issue with. And I really like to support this bill. But I have a an exception and I will just not to the book to belabor the point, but I would throw it out one more time. And that is allowing the governor to appoint the chair of that. Pretty dramatically expanded criminal justice console. I don't really worry about putting a brand new person in from the get go and having all these requirements and studies and reports and policies thrown at them. I just think we're setting ourselves up for a less than perfect outcome. I would encourage the committee to consider making that change and if you would indulge me then I would like to be able to support the bill. We can certainly have that conversation one last time I appreciate that that is a sticking point for you Jim and and I understand that it is. It is a bit of a change. But we did. We did cross this bridge a few times earlier today. Warren, I was persuaded by house comments earlier today that really the chair doesn't have to be the expert in all things, the job of the chairs to be an excellent facilitator of the meeting and to draw other people into the conversation. So I don't, I don't see the necessity of having great experience at the head of the table. They just have to be fair and impartial and run a good meeting. I would support leaving it as it is. Although I understand Jim's concern. I really do. I wouldn't be afraid of having the governor make the appointment, but I like the language as we have it. That's that's all I got. Thanks Warren. Any other committee discussion on that point before we move to a motion, JP. Just renew my preference. From my set a couple times ready I prefer to have the chair elected from the council members themselves. If that's not to be the case then I guess my, my second choice would be to have the governor appoint, but from have the ability to appoint a chair from all members of the council. Again, that's just my opinion and my belief and I'm very strong on that, but thank you. Other committee discussion. All right. I would. At this point, I guess I would entertain a motion to move this bill out of committee. John Gannon. I moved to approve draft 4.1 of S 124. I'll give Marsha a moment to pull out her roll call sheets. Any other committee discussion before we move to a vote. Marsha when you're ready. I am ready madam chair. Dan. Yes. It's Miller. Yes. Ro Wickey. Yes. Leclerc. No. Harrison. No. Gardner. Yes. Classic. No. Cooper. Yes. Ronald. Yes. Colston. Yes. And Copeland hands us. Yes. So the vote is 830. Thank you. Thank you committee for a marathon of. Work and productivity on the bill. And, um, and I am so thankful to all of you for, uh, your engagement in this and for the work that, uh, that you did to improve all aspects of the bill. Um, even if in the end, uh, there was a sticking point that kept you from being able to support it. Um, and, um, and Nolan for your help with, uh, with everything on this bill. Um, Betsy Ann has probably got the bill memorized at this point because we've made her go through it so many times for, for us. Um, but thank you Betsy Ann for your great work on this. Uh, Jim. Thank you. If she had gone through it one more time, maybe I would have seen the light. Um, just a procedural question, madam chair. Um, does this mean we get tomorrow morning off? Yes, it does. Thank you. It means that you don't have to report to committee at 830. You can, um, report to the floor. I believe at 10. Unless there's something else that you'd like to take up tomorrow morning, which, you know, I don't know what to do with that. I don't know what to do with that. This is a hard working committee. There's always next week. If we have other things we want to take up. Madam chair. I have a question. Um, if the member from Chittendon would like to go through the bill one more time, maybe he could be one of the presenters. And could get a better understanding that way. And do lots of extra homework. Evidently he needs it. Obviously he needs it. But I think he's going to be able to do that, smooth as, uh, legislative council is on, um, explaining the various sections of a bill smooth and thorough, I should say he's got a lot of work to do to be able to do that. That is a very, very high bar for him. Yes. Alright. So, um, a couple of other logistics. I believe that this is coming up on the floor on. It'll go over to appropriations. They have taken some testimony. They'll let us know if they need us back. It'll be up during the week next week. However, I do want to just make note to the committee so that we can all manage our expectations, myself included, that with the understanding that the legislature is moving to final adjournment on this 2020 legislative session by next Friday. In the event that there was some significant compromise that needed to be made in order for the Senate to be able to concur with our proposal of amendment, they may bring that back to us in the form of an amendment that we might be asked to make in a friendly nature to our own bill on the floor of the house. I just want to make note of that, that we are going to make sure that the Senate has had a chance to look at it and can live with the changes that we've made in this short time frame. Madam Speaker, we'd be very open to some significant amendments on this legislation. I hear you. There might be other people like maybe two other people who agree with you on that one. This afternoon is going to be a bit of a marathon floor session. John Gannon, are you ready? Do you need us to send you off with some really good wishes for hopefully, what is your final report of the cannabis bill? I truly hope it is my final report of the cannabis bill. I hope all the members of the committee can support the conference committee report. It was great to work with Rob, and Janet, and the senators weren't that hard to deal with. It took a long time to get where we ended up. It'll be interesting how it goes on the floor, but. Just make sure you mentioned the hard-working and talented Government Operations Committee. That will do the trick every time. Well, I just want to say thank you to you, John, and to Rob as well for putting in what is now going on a year and a half of work and thought and preparation into this day, and we certainly hope that it goes well for you, so that this gift that keeps on giving can go be somebody else's baby. I have to tell you, Madam Chair, I wasn't a user before, but I'm highly considering it now. There are certain things that 2020 is bringing us that are challenging, and this one has been a continuous challenge, so thank you, John, for your good work, and we'll be right there behind you on the floor this afternoon. Hopefully it'll be a speedy report, minimal interrogation, and a quick vote. I hope so. It is a fairly speedy report, so I will not keep it too long. Great. We're not going to go through the 105 pages section by section. I got Nelson says thank you. All right. Thank you all committee for your great work, and we will not have committee tomorrow so that we can do our work on the floor of the house and then maybe get some fresh air and exercise and sunshine if the haze has gone away. Any other questions or comments? Oh, I should tell you that Marcia and I are going to report this bill, so I will get version 4.1 to the house clerk's office in order to make that bill go on its merry way in a timely manner so that we can adjourn and get out of here, so thank you Marcia in advance for helping me with that project. Absolutely. All right. If that is all, Jim your hand is still up, but I'm assuming that that's just from before. Great. All right. See you all on the floor in 40 minutes or so.


Just an overview sort of of today's talk.  So what we showed last time, or we sketched the proof of Andre's theorem that BCM algebras exist for any complete local domain. And so in some sense, the point of the today's talk is to talk about the ways in which we can now use them to try and mimic things from positive characteristic in mixed characteristic and sort of look at F singularities from this point of view. So we'll fix the following setup for the rest of today's talk. So let RMK will be a complete local domain. If I forget to say it, usually dimension D. So I'm going to specify again that P is inside the maximal ideal. So there are two cases, either the characteristic P case, meaning equal characteristic P, where the characteristic of R is also P, and the mixed characteristic case where now the characteristic of R is 0. So the first set of singularities I want to tell you about are what are known as BCM regular singularities. And these were introduced by Ma and Schwede. All right, so here we go. So let's say that fix first BCM algebra B. Everything's been perfectoid, let me say that. So we say that R is BCM regular with respect to B, if it's normal. And as Keiichi discussed this morning, also Kuhl-Gorenstein. And the main condition, the map from R to B splits. So there is an R module retraction here splitting off of B. And of course, since R is assumed to be complete, that's the same thing as asking this map here is pure. OK, all right, great. So there's my definition of BCM regular with respect to B. From this, and this is a pretty common trick here, so I can cook up an ideal which will detect whether this holds. And that ideal I want to think of is the test ideal with respect to B. So we'll denote by tau B of R, just the image of the evaluation at one map. All right, so, and this here, now, OK. So I want to think of as the BCM, or what was it, the B test ideal. And why I say that and the motivation behind that is something that will come up later in the talk, all right? So of course, this notation with the tau B, I haven't really used anything about BCM anywhere. So if I write tau sub S of R for any R algebra S, so this is, some people call this the trace of S as well, that also notation is still well-defined, OK? So, but note just from the definition here, we have that, right, of course, this ideal detects whether you have something which is BCM regular with respect to B. All right, so that's what you do if you have a fixed BCM R plus algebra B, right? And then, of course, we say that R is BCM regular, right? Without any qualifications, right? If, well, implicitly again, normal and Q-gorenstein, and it's BCM regular with respect to all perfectoid BCM R plus algebras. So again, I can cook up an ideal that would detect this condition just by intersecting over all of the B test ideals for all B, right? So you might write, right? So tau script B, right? To be the intersection test ideals for all Bs, OK? And we've already seen something which is maybe not so easy to get just directly by looking at this. This intersection here, in fact, stabilizes, right? So the intersection over all of these will be equal to tau B of R for all sufficiently large B because of the domination trick we saw or we asserted about BCM algebras before. So, right, so take any element, right? Or any set of BCM algebras, if you will. I can always find another one so that all of the maps, the original set, factor through to that guy. And when I do this, the traces can only get smaller, right? So it's pretty easy to convince yourself that that will give you some B for which you have a quality. And once you do any larger B, anything bigger than that is obviously also going to be something which has to give you a quality, right? So this is this domination trick of Manjui, right? Is everyone happy with this? Okay, so this is my first sort of class of singularities that you could define. And we'll see in a second why this is interesting, right? Or what the relationship is to the positive characteristic versions but I want to do something now here at the board which, again, maybe in a class is not so great to do if you're trying to take notes but the notes are all online.  So second topic, BCM rational instead of BCM regular, all right? So we say that if I have a fixed B, R is BCM rational with respect to B. If, well, instead of normal and Q-gorenstein we just have that it's comacoli. And I'm gonna tweak this definition here and say not that it splits but that the following map, well, let's do it in two ways, right? So first is maybe the traditional way, the way it was first defined. The induced map on the top local homology, that should be injective. And of course, equivalently, right? So when I state it like that it doesn't really look like it's something that's all that close to BCM regular but if I take the matless dual of what I have over there, what do I get? Well, I can look at the homset, hom B omega R and that comes together again with some sort of trace map here, just evaluation at one, right? And that map is dual to the induced map on local homology, right? So equivalently, this evaluation at one map from hom B omega R back to omega R should be surjective. Continuing with my tweak, I can obviously define an ideal or not ideal but test sub module, analogous again, as we'll see to the parameter test sub module if you will, right? By looking at the image of that evaluation map and that gives me at least some sub module of omega that detects BCM rational. And again, BCM rational in general means BCM rational with respect to all big comacoli algebras if you will, right? And the test sub module with respect to all the bees is the intersection and is the same as the test sub module for all sufficiently large bees, okay? Is everyone happy with my gloriously ornishly decorated tweak? Yeah, thank you, I didn't quite finish my tweak, right? So that's the implicit. Great, so what are some of the easy implications here? And maybe this will start to justify some of the terminology to you already. Well, if you're BCM regular, then in fact, you must be BCM rational. Why? Well, if you're BCM regular then all maps from R to B split, and what I can do is just take the splitting and hom it into omega R and that gives me that the induced map over here is surjective. Moreover, of course, if R is equal to omega R, I am Gorenstein, right?  All right, so if you're Gorenstein then BCM rational is the same as BCM regular. All right, another easy one.  Now I'm gonna be a little sloppy with my notation here. I've only defined this assuming that I put in a B which is perfectoid, right? And all my perfectoids had better be P complete, right? So I'm gonna just omit the P completion from my notation. Let me note that that doesn't really affect anything because I can slip the P completion into local comology or to any of the hams because everything can be P completed, so, right? And so of course R plus regular, so this is a variant of BCM regular with respect to a fixed big comacoli algebra, if you will, is the same as saying you're a splinter that's also Gorenstein, all right? So this is just definitional, okay? So the interesting thing here is you already, even with these implications, run into something which we don't know and is a very important open question in some sense, which is the converse, right? So this is an open conjecture. So but let's just take this, or I'll revisit this in a second so after we've done the next set of discussions, right? So I could ask, right? In positive characteristic we'll see that this is a version of asking that F regular and Gorenstein is the same thing as a splinter which we know is true with the Gorenstein assumption, okay? So my assertion is that that is true in positive characteristic, right? Which is why I wanna make it a conjecture and mix characteristic. All right, but any questions before we move on from the definitions? That's the conjecture. So the conjecture is that asking your R plus regular gives you that your BCMB regular with respect to all Bs, right? And so, yeah. All right, so, okay. So, and again, I'm trying to be as slick as possible to write as many terms as I can on the board. So I've compiled a compositum of as many things as possible to tell you about what these things look like in positive characteristic that really motivate how they fit into the framework of this workshop. So, but without saying some massive list of names, right? So there are at least, you know, I don't know, too many to list here, right? But they include certainly Hoxer, Heuneky, Moshweed, Smith, and the list, Lubesnik, myself, many, many, Bargav, who knows, many people, right? On this list, okay? So, in characteristic P, right? So, assume we're in equal characteristic P, right? So, the first is that BCM regular is the same as strongly of regular, right? And moreover, the test ideal, one of the, this one with respect to all of the BCM are algebras, is just the usual test ideal of R, which, of course, I can write in many, many different ways. So maybe, because we're Kugorenstein, it doesn't matter how you think about which flavor of F regularity you'd like in the same way which flavor of test ideal is known to all be the same. So look at the collection of all the test elements. And of course, it's also known in the setting that you get many other things, right? So, the test ideal with respect to all the B's is the same as the usual test ideal of R, which is the same as the test ideal for R+, which is the same as the intersection of the trace ideals for all finite extensions, which is the same as the trace ideals for all sufficiently large extensions. So R, I guess I should, yeah, whatever. All right, so, and similarly, BCM rational is the same as F rational, and I have a similar string of equivalences for the parameter test submodule, if you like. So tau B omega R is the same as tau R+, omega R is the same as the usual parameter test submodule of R. So, now I can't quite give the same tight closure characterization here, because if I stick in all parameter ideals here, I get the parameter test ideal, which is something slightly different, right? But what I get is the matless dual of, say, the HTMLR module of the tight closure, right? So this is now a submodule of omega, okay? And again, this is the intersection over all of these sort of traces with respect to all finite extensions, or indeed after a sufficiently large extension. And so maybe in this framework here, this is a result of myself, Manuel Blocla and Carl Schwede, but really is a variant on the generalization of the original Equational Lama due to Uniki and Lubesnik, all right? So, so let me start my table over here. So at this point, what this means in some sense is I have now a bunch of notions, which I can sort of match up one by one, right? I guess maybe another thing I should say before I erase everything over here on the left is that many of these equivalences in things like this on these two boards are discussed in my lectures from last summer on test ideals, right? And the set of notes that came out of that as well, right? So, so, right? So we have sort of paired up these notions and see that, well, the definitions of these guys over here reflect at least one of the definitions or characterizations of the guys on the right, okay? And of course, you can see, I tried to pair this up. There's a split, there's a crack in the board right here. Okay, so, oh, what's your question, Holger? Everything, if I use a BCM regular, so, all right, so maybe here up here at the top, all of these are Q. Gorenstein, normal, and Comacali, so I run into no problems. So let's not get into any of the difficult cases, okay? So Holger points out that I am, so the obvious question, Holger's question is, what do I do if I have something which is not normal Q. Gorenstein or Comacali, and the answer is you go to work, okay? So, so I've paired these things up, all right? So, but the point here is, when I have something in mixed characteristic, all right, I can travel the characteristic zero just by inverting P, all right? So, since I showed it right, so, or at least equal characteristic zero, right? So where I'm working now over a characteristic zero field, okay? And the point here is, part of the package of results from Mon Schwede is that if I look at BCM regular or rational, all right? So, I get log terminal and rational, right, in characteristic zero. So, I have a characteristic zero singularity notion which matches up with the usual reduction mod P dictionary and many of the topics being discussed in Keiichi's lecture this morning. Of course, you're gonna ask if the same thing holds. The test ideals and what you would expect to get is, well, the multiplier ideal, I won't say it precisely, and say the multiplier submodule, right? Or grout room inch knight or sheaf, if you will, right? In characteristic zero, right? So, and at this point, these do not exist in literature, right? But, let me say that we're working on it. So, just as a quick idea, I can never remember how the order of the letters is supposed to go, right? So, in work with Batma, Pathakvali, Schwede, Wittechek and Waldron, I said the last two in the wrong order again, right? So, in progress, essentially what we can show is some localization results. This tau B, namely when you invert P again, what you get is exactly this multiplier submodule. And again, these results, without the question mark, so in particular BCM rational and inverting P gives you something which is rational, was referenced earlier this morning, at least implicitly in Keiichi's talks, right? So, this is the essential ingredients together with what, knowing what happens when you go mod P, so the adjunction type statement, that will allow you to take something which is F rational and a single characteristic and then deduce that it is, deduce something in characteristic zero. Okay, right? Any questions? So, the only bad thing about everything I've done so far, we just gave a whole bunch of definitions, right? Is much in the same way that writing down a BCM algebra took us a fair bit of time and actually proving that result. I've also been very impressed by the other lecture series speakers and many of the other speakers about how they remind us that it's important to be explicit and to give examples, right? Of many of the things. So, maybe it's not so easy to exhibit things and show that they are BCM rational, BCM regular. In fact, this is proving statements that look like the direct sum end theorem, of course, took quite a lot of time and machinery, right? So, let me give you at least some examples, right? So, at least of BCM regular, in many cases, these will be Gorenstein's, you know, it's the same thing for BCM rational and of course, these are the stronger ones anyway, so all of these will be BCM rational, right? So, here's the first one, right? So, look at the Fermat in mixed characteristic, if you will. So, look at ZP, power series of joint Y2 up to YN and kill the sum of the mth powers of the parameters, okay? And the claim is that that is BCM regular, if P is sufficiently large and M is less than, M is less than N, right, inside of this, right? So, and if this looks somewhat familiar, right? How do you show such a thing? Well, modulo P, these conditions produced the Fermat mod P in one fewer variable, right? And then, I can use reduction mod P, right? If you like, or Federer or whatever you would like to do to show the corresponding statement that mod P is F regular and then I have to lift this using some sort of a junction result, right? And so, this was done in a paper with Ma and Schwede with Chek and Waldron, all right? All right, so some sort of general statement that assuming everything inside is Gorinstein, for example. If you get, what you get mod P is F regular, you can lift that back up, right? So, there's one example, right? So, here's another one, right? So, ZP join XZ mod X squared plus P squared Z plus Z cubed, all right? So, again, to many of you, this may look like a familiar equation. This is, well, if P happened to be a variable instead, this is, I think, a D4 singularity, right? So, assuming P is at least five, or P is at least seven, right, bigger than five, right? Then, this thing here is, again, BCM regular, right? So, and this as well as a detailed analysis of things for all the rational double points was done in a paper with Javier Kovar-Horohas, Lingchua Ma, Thomas Poulstra, Karl Schwed and myself, right? So, more generally, we showed that if you take a rational double point in mixed characteristic with mixed characteristic zero P, at least, P bigger than five, then what you get is in fact a direct sum end of a regular ring, which is another easy way to check that something is BCM, rational or regular. So, the more general statement here is, let's say you have a Q-Gorenstein direct sum end, and of course I could keep going on. So, again, maybe I haven't put an attribution down here, but this requires some actual thinking, right? So, I also have some sort of Toric-like examples or what have been known as log-regular rings. So, these are in mixed characteristic, something which looks like a WIT ring, a join, completed, semi-normal, strongly convex saturated normal monoid, right? Modulism equations, right? So, something that essentially looks Toric, right? So, and I can tell you more about that later, if you'd like. Okay, so, but, and again, this appears in work of myself with Hanlin Kai, Sangsu Lee, Lingxuan Ma, Karl Schwed, right? So, which I'll talk about later in today's talk, but also generalizes work of Gabber and Romero, okay? Right, so, but the point is, is that even in some of these examples, it requires some real machinery to find explicit examples of BCM regular things at this point, okay? Any questions about this? So, with that, let me change gears, but I want to finish to tell you a little bit about it. So, hopefully you can see how BCM algebras at this point fit into the framework of F singularities from these kinds of tables, right? This related a fair bit more to K.H.E.'s talks, but you could ask about if there's something I can do that will put you in the setting of Ilya's talks. So, is there a mixed characteristic analog of F signature and Hilbert Kuhns? And that's the next piece of the puzzle that I want to tell you about for the remaining lecture, okay? So, let me add to my setup a little bit, right? So, additional setup that we'll hold. So, everything earlier plus apply the Cohen-Gabber theorem to get a nice northenormalization, right? So, find a regular subring so that the extension A to R is modulo finite and on fraction fields this is separable. So, this again came up in Ilya's talks already, right? With, let me write down A very explicitly here. So, we're in mixed characteristic zero P or in characteristic P. So, in characteristic P, then A is just the power series ring. If A D variable is X one up to X D and in mixed characteristic, right? So, A has to be the weight ring on my perfect field K, right? A joint X to up to X D and for this just to keep the notation easy to write down, let's set X one equal P so that X one up to X D are still a system of parameters, right? And the point here is somehow that sort of to tell you what the obstruction that has been around for a while to try to fill in this gap here of what the Hilbert-Kuhn's and F's signature are, right? Everything we've done up here before involved taking bigger and bigger comacole algebras B, okay? And in particular, even the smallest one that seems to work, say in positive characteristic is R plus and my assertion essentially is that we think at this point that it's just too big. Even R plus is just massive in comparison to R, okay? So, and maybe I learned about this from an example of Craig at some point. So certainly R plus is not an aetherian but it's even worse, it behaves in a lot of ways in a very non-geometric way. The sum of primes is prime and has other very strange behavior, right? So maybe the right thing to do or what we'll need to do is to do something which goes sort of in the other direction and I wanna think about the smallest perfectoid I could use. Instead of using bigger and bigger B's let's go the other direction, right? So let's set A infinity to be, right? So this ring right, join all the infinite p-th roots of this is sort of parameters and then complete with respect to p and of course this is the same thing as A perf in characteristic p and the other one I wanna introduce is what I'll denote by R A perf, perfed, right, which is, right? So well, if I look at A infinity we know that that thing is perfectoid, right? If I base change up to R what I get is now, well, certainly finite over A infinity so I can apply this perfed functor of botan shoulder to get the smallest perfectoid guy over this tensor product that both R and A infinity map to over A, right? So and again, maybe it's worth saying this is equal to R perf, right? So point here is that it's an analog of R perf which is somehow much closer and smaller and close to R than R plus was and even in characteristic p. So in order to tell you about this mixed characteristic F signature and Helberg-Kuhn's multiplicity the notion I need to first tell you about is normalized length which is originally due to faultings, right? So and so what he did is he says, well, let's take a module over A infinity and assume that it happens to be MA power torsion, right? So in practice, most of the cases the modules I'm gonna be looking at M kills it. So it could be MA torsion if you like, okay? And to such a module I can associate a number called the normalized length and you wanna do it in such a way that it should sort of mimic what you want length to be, but use the explicit structure for A infinity that I have. So first, we're gonna do this in steps. And so the first step, if M is finally presented, so this is pretty much the only case I know how to do this very explicitly. So we define if M is finally presented, in fact, since this A infinity really comes from adjoining p to the eth roots for all E there's in fact some E so that in order to define M I didn't really need to go all the way to infinity. I could have just joined p to the eth roots for all the system parameters. IE M is isomorphic to or is equal to a base change ME tends to over AE with A infinity for some ME. And in this case, you can set the normalized length to be well one over p to the ED times the length of this module ME that you base change to get your original guy. And the point here is that, well up to the p completion this A infinity is given as the co limit over all the AEs, but more than that the map from AE to AE plus one where I just joined adjoined roots of each of the system of parameters is flat, well locally is free, right? Of rank equal to p to the ED, all right? So once I know that it's easy to see how the length scale if I pass from AE to AE plus one and the scaling factor will give me a number that's independent of how I chose this presentation over here event, okay? All right, so I want to define normalized length. I've told you how to do it for every finite presented module. So the second thing I do is define it for a finitely generated infinity module, okay? And in that case, I just build it up from inside. So, well every finitely generated A infinity module is a co limit of the finitely presented quotients right of this thing. So I'm infinity M, I just set to be the infimum of the normalized length of all of the quotients from a finite presented finitely presented module onto M. All right, so certainly if I was dealing with usual length and I take a surjection then the length of this guy has to be bigger and I force that property to hold. And then three, all right? So if now I know how to do it for all finitely generated modules, well an arbitrary module is the co limit of its finitely generated sub modules. So if M is arbitrary, I set lambda infinity of M to be the supremum over the normalized length of all finitely generated sub modules. And again, if I have a sub module of another module I expect the length to be smaller. So this is capturing what that is. So once I've done this, one checks, right? So, well, this is in fact well defined. The crucial step still comes right down here in checking that this is independent of the presentation in the finitely presented case that you chose. But more than that, what you get is an additive untrue to exact sequences. It really does behave like a length function in some way. So just a couple of examples, although the warning in some senses they're not so easy to come by. So one, if you take an ideal I inside of the regular ring A, all right, then, well, with finite co-length, then the normalized length of A infinity mod I, A infinity, is still just the normalized length, or is still just the length of A mod I, simply because A infinity essentially is flat over A. So all of the AEs are flat over A and I can calculate what happens along all those base changes, right? So it doesn't really do anything if I look at ideals inside of A, right? Two, second example, let's say that you take some non-zero element F inside of A, right? And just for simplicity here, or let's say that the characteristic of A is P, so that A infinity is equal to A perf, then you can check that the normalized length of, well, A perf, module O. Well, now we're in characteristic P, so my element F in A perf has infinite P throats. Oh, do I wanna do still just A, right? So take A perf and kill the ideal generated by those infinite P throats as well as, well, I want all my modules to be M power torsion, so I might as well kill M as well, right? So the claim is if you do that, this normalized length is zero, right? And one of the reasons I write this example down is there are many other properties of normalized length I'm not telling you, right? But one particularly important observation here is, well, to put this in the framework of the last talk, this module I stuck in here is so called F almost zero, right? F and all of its P throats kill this guy.  So I'd love to tell you that that statement was actually true, but the, so, and it is, but which notion of almost you mean there factors into the statement, all right? So let me tell you the theorem here that really puts F signature and Hilbert Kuhn's on that table over there, right? So again, still say you're in characteristic P, all right? So as I said, if you wanna do something where you add to that table and put something in mixed characteristic, often what you have to do is come up with a new interpretation of how you used to do everything in positive characteristic. And so that's what this theorem is gonna do, right? So the first thing is to say, well, if you take ideal I inside of R with finite co-length then the normalized length of R infinite, oh, okay, R perf, mod I R perf equals the Hilbert Kuhn's multiplicity, right? So, and, right, so before I write the second statement, I just wanna point out that this is a statement purely characteristic P. It doesn't use anything that you would didn't have 15 years ago to talk about, right? So, or much longer. So I have R perf, I, Hilbert Kuhn's multiplicity, whatever, all right? So in some sense, this is just a reinterpretation of the definition of Hilbert Kuhn's multiplicity, right? But it is one that allows you to get an idea of what to put on that table over there. So, and in the same vein, let's let I infinity be the non-splitting ideal of R perf, right? So look at the set of elements of R perf so that if you map one to those elements which you get does not split, okay? So then, again, the normalized length of R perf mod I infinity R perf, in fact, equals the F signature, right? So for the experts in the room, I challenge you, go ahead and prove that theorem for yourself. It doesn't involve essentially any of the perfect way of mathematics that I've actually done in the earlier lectures. Exactly, so I just erased it, but you could ask, where's the limit? Where's the convergence? Where's anything? And somehow, right, it's hidden inside of the definition of normalized length. So there is content to this, right? So you have to do some fair work, but it means that the end result is something that's not visible as a limit. It's just a length in some generalized fashion, okay? Not that I know of, right? So I'll say something about the relationship with the usual multiplicity here in a second, but that's the thing I'm going to say in a second. So. And again, so if you go ahead and try to prove this for yourself, I just have a caution for you, right? So our perf is not finitely generated over a perf, right? So you have to really work with the definition to try and figure out what this means. And in fact, tracing through back many of Mellon Craig's proof, come down to finding ways to get around this fact, right? So finding, for instance, some element C so that C times our perf is contained in our joint A perf in the existence of test elements all deal with exactly this problem in some other way, right? So, and maybe I'll say more. I'll put a sketch of this in the notes as well if you'd like to see it afterwards. All right, so okay, great. So we're in the setup for the actual definition of factoid Hilbert-Kuhl's multiplicity and signature. All right, so again, in either mixed characteristic or characteristic P, right? Take a finite Kuhl length ideal and I just set the perfectoid Hilbert-Kuhl's multiplicity. Oh my goodness. EX perfed, not HK, right? To be the normalized length of RA perfed, mod I, RA perfed, and the signature, well, the perfectoid signature of R to be the normalized length of RA perfed, mod I infinity, where again, I infinity defined is defined as to the left replacing RA perfed with R, or replacing our perf with RA perfed, right? So, to justify some of the notation, you'll see in comparison to usual Hilbert-Kuhl's multiplicity RF signature, you see these little X's up here at the top, right? And one of the reasons they put that there is my definitions depend on the explicit system of parameters I chose, right? So, and they depend explicitly on the presentations of A infinity that I wrote down, okay? So I joined all the infinite P's roots of the X's, but not some other system of parameters as you saw, right? If I wanna go and add more, that takes extra work, but then often will make my ring big enough but I don't know what to do, all right? So at this point, we don't know independence of X, right? So let me just finish the lecture series by giving you some of the key results on these guys, get a slightly bigger piece of chalk, right? So, well, and again, the point of these results, a lot of the things I'm gonna write down are hard to show, the paper is quite long, 70 pages or something like that, right? But you'll see that that really starts to mimic what you think of for Hilbert-Kuhl's multiplicity RF signature. So for instance, ex-perft of R, which of course means ex-perft of the maximal ideal, is bigger than or equal to one with equality if and only if R is regular, right? Now I've assumed R as a domain, so, you know, don't get too excited about doing the, about associativity formula on mix and this and things and all that else. I've been just doing the domain case here, okay? All right, so and similarly, Sx-perft less than or equal to one with equality if and only if R is regular, okay? And moreover, the relationship between Sx and Hilbert-Kuhl's multiplicity also carries over, right? So the perfectoid signature is the infimum over the relative perfectoid differences, right? For containment of ideals, right? So this is the conjecture of Watanabe Yoshida, that Thomas and I proved, right? This is the Watanabe Yoshida theorem that Ilya spent some time talking about. So you see the shadows of all the things from positive characteristic. Great, so what's another thing that we know Ilya used today in his lectures? Well, we also know that Hilbert-Kuhl's multiplicity, for example, uses those Hilbert-Kuhl's differences in order to detect tight closure. So if I have a containment of ideals in R or finite co-length, all right? So again, the result is that it's easy to show from the definitions that the perfectoid signature or perfectoid Hilbert-Kuhl's of J is less than the perfectoid multiplicity of I, all right? But the interesting part is that you get equality, again, if and only if the extended full plus closure of I equals the extended full plus closure of J. So this, right? So it was really flushed out in the work of Moniteman, at least the closure operation, not the statement on the perfectoid signature, right? But just to have the definition on the board, right? So X is in this ideal IEPF, if and only if something that looks like a tweak of the tight closure definition holds. So there exists some non-zero element C with the property that C, the one over P to the E, let's say, is in the ideal I, P to the N, R plus for all E and N. Bigger than zero, right? So instead of just looking at IR plus, you add in the powers of P for all N. R is a domain, everything is a complete domain, whole lecture, so yes, it's definitely there. C one over P to the E is in the ideal I, P to the N, R plus times that thing, thank you, right? So C one over P to the E multiplies X into the ideal I, P to the N, R plus, yeah, right? And this is what Ilya referred to before. If Y went up to YD as a system of parameters, right? Perfectoid multiplicity of Y, with respect to the X's, is equal to the standard Euler characteristic just as you would expect for multiplicity or hypocons, right? So, but the warning here is that this is not easy. We have to go to some great lengths in this paper to show this and that's partly because the Y's may have no relationship to these X's over here and it's not so easy to switch them, okay? In particular, let me just advertise some basic open question. You could take any statement about Hilbert Kuhns at this point and try and mimic it for perfectoid signature and we don't really know the answer, right? So, here's something that's open. Well, you'd expect, we know from this statement just by taking a minimal reduction, right? And applying this formula that the perfectoid signature is bounded above by the regular multiplicity or the perfectoid multiplicity is bounded by the regular multiplicity. But I don't know something I do in positive characteristic that this is bigger than one of our defectorial times E of I because this uses something that's really about the Frobenius powers of the ideals, right? So, uses something that I don't really have access to in positive character and mixed characteristics still at this point, right? So, this inequality is open. All right, so, and sort of the last thing I'll tell you, right, so, if in addition R happens to be Kuhnstein, well, again, we know that F signature is positive if and only if the ring is strong enough regular. So, here I would expect the perfectoid signature is positive if and only if R is BCM regular. So, analyzes that as well, right? Okay, so I have a lot more information in the notes, right? So, if you're really interested in particular, let me say, you can also do a tweak on this. You can do a dual F signature variance, so, which is something I have investigated quite a bit to get a perfectoid relative rational signature that will, whose positivity will detect BCM rational instead of F regular, all right? And, again, maybe I should say that one of the big applications of F signature also comes out of the paper. So, we get transformation rules for perfectoid signature under finite quasi-etal extensions. So, in particular, we recover sort of well-known results at this point about with applications of the finiteness of local fundamental groups and finite torsion in class groups. And, in particular, we recover results on computing the perfectoid signature for finite quotient singularities, which in many ways date back to, again, the work of Watanabe and Yoshida and Hunikileski for relative Hilbert-Kunz and F signature, respectively. Okay? All right, so I mentioned the big open question with independence of the X's earlier, but I'll end here with another, again, very closely related to Ilya's talks, right? Pretty much anything in mixed characteristic that has to do with localization at this point still is very hard, right? So, this is one of the reasons why it's taking so many of us to just invert P in those kinds of statements, okay? But if you knew the independence, it'd be very natural to ask for localization compatibility, so I expect the F signature to only go up under localization and Hilbert-Kunz to only go down, right? And then I could ask that this determines the semi-continuous function on spec R, either upper or lower in the two cases, right? And again, at this point, these are pretty much completely out of reach for all of us, right? So, there still is a lot of stuff to do in order to make everything match over in this table, right? All right, well, thank you very much. Joe, we did something, I think there may be some inequalities inside of there, mimicking some of these things, but I have to look again before I can answer. So, great question, right? So, as I said, one of the ideas here is that R plus, or these big comacola algebras that we really have our hands on in some way, are huge, right? So, if those are the things that are making things look like F regular, right? So, versus, so F regular is asking that bigger and bigger B split, you might try to weaken this as much as possible. And so, of course, in positive characteristic, F split is the same thing as saying the map from R to R perf, splits, right? Not R plus, but R, right? So, forget, but R perf, okay? At which point, so the natural thing to ask is, well, can you just ask that you are split with respect to some BCM algebra, right? Or maybe that the map to R, just R perfs in some absolute sense splits, right? So, but at this point, there are a group of us that are working on some of those things and we can show a couple of results, but that's not fleshed out yet, but it's a great idea, right? So, F split characterization, F injective, F nopotent, all these other things, I don't have a way to put them here on this thing. In some sense, these are the interior cases where things should be easy, right? That as I go to the boundary towards log canonical or F split and other things, we really are still kind of in no man's land. Yep, very natural idea, which is the question was, if it's not independent X, what do you do, right? And I don't know, right? So, obviously, it is at least in one case where I use this as sort of parameters, right? But that's the only evidence I really have one way or the other, right? And so, the idea then is that, well, maybe I just take an inf, for instance, okay? So, of course, if I want to characterize positivity, I have to be a little careful, right? Because an inf of positive numbers, unfortunately, need not be positive, okay? So, maybe I have to take a soup instead and then I'm not sure, right? So...


Hi everyone. This video has a bit of a restriction because I have my microphone here and I'm gonna need to hold it because I can't put it in my pocket. This is rocking. So rocking is a good floor-based exercise. I like to use it in warm-ups. It's good for reflexively turning on the abdominals, teaches you how to flex your back, round your back, relearn how to achieve those positions, maintain them, and there's plenty of ways that you can progress it. Today we're gonna talk about level one. This is just rocking, okay? So I'm gonna hold my mic. I hope it doesn't mess me up. Okay, and we're good right here. So I'm just laying down on the ground. Try to show you more of the moneymaker here. My legs are up like this and it just looks like this, right? Okay, so I don't know if you noticed. My first one, I really hit the ground hard. Sorry, I hope that wasn't too loud for you. I hit the ground pretty hard. On the second one I smoothed it out and it was more like a like a half moon, like the a crescent moon was rolling back and forth, like a cylinder was rolling, whereas the first one it was like thud thud. Okay, the goal is to even this out. I want your back to be very gradually curved. I don't want it to be flattened one place bent a lot in another and then flatten the second place. Okay, so that's the goal. Couple ways. How can we progress this? If you find this difficult, grab onto your knees and just rock that way. Try to rock all the way up and back. Okay, if that's easy enough, here's our microphone challenges. We're gonna leave it on the ground and see if that works. You can grab your toes and do something very similar. Hey, it worked. So that extra reach, got it, forces you to close down in the front and open up in the back. So it works really well there. If you want to reach further, cross your hands. I don't even know if I should try this. Got it. You probably hear me so well. So also, next progression after that, try not to hold on. Try to keep your reach really active rather than a grip thing. Okay, so I'm just gonna put my hands on top of my toes when I do it that way. And you'll see I thud it again because the ground's starting to hurt my back because this linoleum isn't very soft. What else can we do? We can do other variations. You can reach more on one side than the other. You can keep like an inside reach and then an outside reach like this. Feel free to experiment, but this is a good abdominal warm-up, especially after a long day. Just roll around. Don't think about anything and use it as five minutes of meditation time.


Let us look at the gondola now, gondola as you know is the box in which we keep the payload and all the other items all the equipment that goes as aeronics if they are not directly mounted on the envelope they are put in the gondola. So gondola can be made from a lightweight corrugated plastic sheet for an indoor airship and the best way to attach it to the envelope would be through a series of ribbons which can be wrapped over the envelope. This is the simplest method or you could actually constructed from some lightweight structure for example in our case we have constructed the gondola from cane. You can see that there is a motor mounted on the gondola in the front actually the gondola has been kept at 90 degree angle. So material choices are different you can use as DPE sheet, you can use taro foam sheet anything that is lightweight but that can carry the loads of the items and provide sufficient volume and location. So the material which we recommend would be either a rigid corrugated plastic sheet it should be spacious to accommodate the various avionics items and the payload. Now on the gondola we actually mount the motors so in some cases if it is a corrugated plastic sheet you can just cut a slot and put it there or you could use various kinds of contraptions to attach it is all up to you. So in our case we have put two brushless DC motors one is for the forward thrust the other is for the downward force which gives you the augmented lift and as I mentioned flexible attachment using ribbons and velcro strips or it could be the ribbons. This allows us flexibility because you can locate the gondola forward and backward to match the CG with the center of buoyancy so it is very easy that way. And also it becomes very easy to assemble and disassemble remember we wanted to quickly assemble and disassemble this airship within about 60 minutes when we did when we go to a place for a demonstration. Looking at fin and stabilizer sizing fins are required on the airship on the back for providing adequate amount of stability and the control surfaces are needed if you want to deflect them to create aerodynamic forces. But for indoor airships actually the rudders and elevators may not really work very well. So what we do in indoor airships normally is that we provide a small motor called as a yaw motor we mount it on the fin usually the bottom fin for ease in excess. We mount it on the fin so that you can get direct side force and this allows you to do very quick maneuvers and if you remember there was a requirement to have spot turns. So that is why it is important to have a yaw motor. Sizing of fins actually might require fairly detailed stability calculations but you know for an initial indoor airship you may not necessarily have the information or the need to do such detailed analysis. You could just use information based on the survey of literature of existing remotely controlled airships. So what we have done is we have looked at several indoor remotely controlled airships indoor and outdoor both and based on that we based on the data we arrived at some area ratios aspect ratio all these parameters in terms of the fin geometry. So you can just use it to get a good estimate for the initial for the fin. So we recommend for SDP fins high density polystyrene fins in a plus configuration and you know we can mount the motor on the bottom fin. So you notice that if you have a heavier bottom fin then you have better static stability but what are these slits meant for? Can you just think about why are we putting slits in the envelope actually by cutting slits in the fin we are going to make it weaker at that location. So then why are we doing it? Think about it. The reason is that air has to pass through them in order to provide the side force. If I install a motor with the propeller on the side of a fin and if I do not give space for the air to go through then how will it actually create the side force. So to give enough space for the air to go through and hence create thrust because thrust is created by pushing the air and the reaction gives you thrust. We will not be able to do that unless we have slits. Let us look at the mass breakdown. So if you look at Avionics mass estimate we have just gone or we recommend that for indoor airship you purely go by information which is available. So the graph of thrust versus weight for typical motors is shown here. There is a company called Ternigy which has got a series of electrical motors and we recommend you use these motors for your indoor airship. You can see that for various thrust values there is a various weight. Similarly the number of cells which are required the lithium polymer battery cells which are required they range from 2 to 3 to 4 to 5 even mind and 10 in some cases. So as far as the motor parameters are concerned one parameter that is important is how much current the motor will draw which is a function of the thrust that the motor produces. And the second parameter is the propeller diameter based on the thrust. And remember that we have to increase the current by 20% because the current will be required not only by the motor but also by the electronic speed controller. So that we should increase the current by 20% here. Similarly we did study of what is the mass of typical cells as a function of their capacity measured in milli ampere hours. So it is a fairly straight line as you can see based on the data for 2 cell batteries you can get an estimate for the mass which varies from about 20 grams for very very small batteries to about 500 grams for an 8000 milli ampere hour battery. Similarly you can go for 3 cell battery similarly you can go for a 3 cell battery which is going to have higher masses. So using the equations listed in this figure you can estimate the mass of the battery as a function of the capacity and the capacity will come from the duration and the current that you have to carry. So if we just summarize the design process basically you have to give some inputs to the process which include the payload that you have to carry in this case was 250 grams. The velocity of the maximum speed of the airship typically 2 meters per second or 5 meters per second operating altitude depends on the location at which you are flying you if you are at a high altitude you have to give a higher altitude otherwise you can give it at a low altitude and endurance which we wanted to be minimum 15 minutes. And what do you get as an output from the methodology you get envelope size if you choose the envelope shape and when you choose the envelope you can get the petal layout and all the required avionics mass breakdown and specifications. And remember friends that airships are buoyant systems and therefore handling them is not a straightforward thing you need to have some ground handling support equipment along with an airship because you cannot just leave them on the ground like you can do it for a quadcopter or for a UAV because gravity will just keep it on the ground in our case it will tend to go up. So therefore you need to have a mast as you can see when you work on the airship whether you are installing some item or whether you are installing the payload someone has to hold the airship.  So here is a example of a small indoor mast that you need to build this one is a mast developed by two interns who came to our laboratory Sayedan Bharatwaj Utsav Bharatwaj and you know they the one on the left is the CAD model that Utsav had created and one on the right is the fabricated mast okay. Now this particular work masks designed for indoor as well as outdoor airships it appeared as a paper in the Journal of Institution of Engineers and I am very happy to share that in the year 2017 this particular paper won the President of India Prize for the best paper in this journal I am very happy and very proud about it. So you can see this is how you attach the airship to the mast and the attachment is done through what is called as nose battens. So in this case you can see that the nose of the airship we have created we have made a small structure this is also made up of cane light weight this is a very important requirement for airships nose battens why do we need the nose battens we need structural strength in the front we also need some mechanism to connect the airship envelope which is a flexible material to the mooring mast which is a rigid material. So for that some kind of a nose battens structure has to be done the one on the left is the small indoor remotely controlled airship the one on the right is actually an outdoor airship where we went for a slightly different kind of a layout but whatever you may do you do need some structure in the front of the airship to give it structural strength and also to connect it to the mast and there is one more purpose of this particular mast. So you can see in this airship which was our second airship way back in 2002-2003 we did not put any nose battens. So therefore I would like to show you what happens to an airship when you fly it without a nose battens. So you can see this is flying right now the wind speeds are quite low I would request you to focus the attention on the nose of the envelope and very soon you will start seeing what happens when you fly an airship without any nose battens. You can notice that the nose is imploding that means the pressure of the dynamic pressure half into rho into V square of the ambient air that dynamic pressure acts on to the nose and if it is more than the internal pressure then the nose tends to go inside. This is called as nose implosion and this nose implosion makes it a flat plate like a structure very high dry and therefore it is not a desirable feature in case of airships. So thank you so much and I hope you have got an idea about how to size an indoor airship. In the tutorials you will be able to do these calculations step by step and be able to understand how we can do it for a particular case. Thank you.


The study investigates the variation in exposure outcome relationships in studies of the same population with different response rates and designs by comparing estimates from the 45 and up study, self-administered postal questionnaire, response rate 18%, and the New South Wales Population Health Survey, computer-assisted telephone interview, response rate 60%. The study found that for highly comparable questionnaire items, exposure outcome relationship patterns were almost identical between the two studies and ours for eight of the 10 relationships examined did not differ significantly. However, for questionnaire items that were only moderately comparable, the nature of the observed relationships did not differ materially between the two studies, although many ours differed significantly. The findings suggest that for a broad range of risk factors, two studies of the same population with varying response sampling frame and mode of questionnaire administration yielded consistent estimates of exposure outcome relationships. However, ours varied between the studies where they did not use identical questionnaire items. This article was authored by John Louisa, Banks Emily, Mealing Nicole M and others. We are article.tv, links in the description below.


What's up everybody Jason for Jasper's case breaks calm another random number blocks to help sell a picatin number six of a Mac Of the basketball sold out this one is for the thunder spurs nuggets rockets sons Clippers blazers and Pacers and again guys you get Two free spots. We're gonna give away So we only sold eight out of the ten everybody gets a random number from zero three nine We go by the first number the left side of the serial number dash There's your example there and again any one-on-ones go to spot one any in all redemptions go to spot zero And they not numbered cards are randomized to the group so again what we're gonna do here guys Just like the last one we only sold eight so we're gonna click the dice roller Randomize only the eight customer names first the top two names get free spots in this break We're gonna add the two spots right here They will use the same dice roller to randomize for the customer names and numbers and then we'll finish off the break So here we go. Good luck. Let's roll and it is a Four to four eight the hard way top two names get a free spot in here one two three four five six seven and eight Boom eight four to four Jesse and David congratulations extra free spots for you guys there Boom, I'm gonna put a little character sign there character sign there now. We're gonna randomize eight times customer names and numbers one two three four five six seven and eight Boom eight times eight times Shawn Maddock at the top done a David and I get this would be for the thunder Okay, see one eight times eight times Now let's go eight here one two three four five six seven eight Boom eight times nine down to six Sean with nine Jesse in all redemptions Stephen with seven Nathan with three David with eight two Jesse with four Gretchen with five David one and six So there you go guys and again We're gonna do one more R&B Dallas Mavericks a couple other teams then in a separate video. We'll start the break guys So good luck everybody


Just put a flower in the top of a road cone. A tiny action completely changed us the way we interact with the space around us. The thing that we needed to continue on with was the very, very seed underneath the student volunteer Rami and beneath the branding, beneath the people, beneath any personalities and egos and everything. It was this fact that after a major chaotic event, a whole lot of people went and helped each other. The crisis and the disaster that's come after the earthquake in Christchurch has been when we've forgotten that. The challenge, our challenge as a country is how we learn from the best of Christchurch, the best of this way of a disaster recovery, the best of every learning and lesson that comes out of it and how do we go back to that place. How do we move this thing from trying to be an organization but actually just to realize it's a movement. There's a one very good seed idea at the bottom of it and that's that people like to help out other people in their communities and you just need to find better ways to make that happen. And so this group of us are really excited to be launching today a concept called Serve for New Zealand. Serve for New Zealand is around doing one hour of volunteering on public holidays in New Zealand.


YouTube has new channel translation features to help you grow and get discovered in other languages. Now you can translate your channel title, this text here, and your about page into any language you want, so if people are searching for your content in that language, it will be more easily discovered. Yesterday we talked about translating your video titles and descriptions into other languages. Click that I up there in case you missed that video because this is part two where we talk about translating your channel titles and about page and here is how it works. In your channel you have this little icon, a little gear next to the subscribe button. If you click that and then click this link you'll be taken to a page where you can specify the language of your title and your description and choose additional languages to translate it to. So for example I am going to choose Portuguese, Brazil, and I'm going to translate the title of freedom which is liberdade in Portuguese. And now if someone changes the language of YouTube to Portuguese and then you can see my channel is now in the Portuguese language, liberdade. That's how it works. How will this help you get more views? It's all about discoverability. If I am from Portugal or Brazil and I am searching in the Portuguese language, I'll be searching for liberdade not freedom and I will now be more easily discovered by changing my title to that local language. You can see I have not localized my video titles to Portuguese, that's something I would still need to do. Will you be using these video translation features and channel translation features for your channel and your videos? Tell me in the comments below. And the freedom family is growing every single day. How big are we today? We haven't played that game in a while have we? Well yesterday we had 161,442,000 subscribers. Today we have 161,904,000 subscribers. We grew by almost half a million subscribers in just one day. Exactly 462,000 subscribers growth in one day. That's amazing freedom family.  Give everyone a big hug for helping each other grow together. Can you imagine trying to grow half a million subscribers on one channel on one day? It's impossible. Even PewDiePie can't do that. But as a family, as a team, together we can. That's the power of working as a team. That's the power of freedom. And who am I? I am George, CEO and founder of, my shirt is back from the wash, freedom. And you've been watching and we will grow together as a family because this is the freedom family. You are part of it. We are all part of it. And we're all growing together. So grow with us and partner with freedom. That play now button right there on the video bar is all you need to click to get started. So that's how you will grow faster if you think people from other languages are watching your videos. People from other countries who speak not English are watching your videos. Go through these steps to translate your content, localize it, ask a friend if you don't know the languages. There's also a paid service where YouTube will connect you with professional translation companies. And you can see that just by using the translations tab as we showed you in yesterday's video, there is a section that lets you buy translations. Now I don't recommend you do that. Of course, unless you have tons and tons of money, but try to find people who speak these languages. YouTube is a community. It's all about everybody helping each other grow together. So I bet if you asked, hey, does someone speak Portuguese that can help localize or translate my channel and my videos, you would probably get people offering to help you volunteering because that is the spirit of freedom. We help each other grow together as a family.


Hello and welcome to the session. Let us discuss the following question. Question says fill in the plans. First part is a tangent to a circle intersects it in dash points. Now let us start with the solution. We know a tangent to a circle is a line that intersects a circle at only one point. So we can write a tangent to a circle is a line intersects. This is a circle with center O and AP is a tangent to the circle. AP intersects the circle only at one point that is P. The required answer is 1. Now we can write a tangent to a circle intersects it in one point. We can fill in this blank 1 so this is our required answer. This completes the session. Hope you understood the solution. Take care and have a nice day.