All right, ladies and gentlemen, I think we still have a couple of open seats at the bottom there for those Looking for seats But without further ado get started. Good morning. Well rather good afternoon. I'm Daniel Iola program coordinator at Berkman client center Just wanted to relay a couple of announcements to you all about art some of our upcoming events on Thursday we have litigating free speech cases in the African regional courts featuring Berkman client fellow nanny Jensen Revent law so that will be in Houser and There are next lunch and talk will not be next Tuesday, but the following Tuesday November 15th And that's the end of ownership with professor Aaron presanoski of Case Western Reserve Law School so Without further ado I before I introduce our Introducer of our speaker. I just want to remind you all that this production is live webcasts for possibility It'll be available on YouTube within the next couple of days. So be mindful of what you say and Yeah, that's or not that that's an option as well. So now Thanks everybody, thanks so much for being here I'm gonna be really brief and then hand it off to Pala just by saying that I run the clinical program here at the Berkman client center We work a lot with the ACLU of Massachusetts and Pala first came her work first came to my attention through that connection to ACLU The data for justice project that she's going to be talking about today is an outgrowth of the work that she's been doing at the ACLU I am the ACLU of Massachusetts over the last year as a Ford Mozilla open web fellow And I've seen her right about her development of a data visualization framework I think we're going to hear a lot about that and about more broadly about the themes of data science in the public interest and Without further ado, I'll just hand it off to Pala. Thanks And to be clear, we'll save plenty of time for questions at the end We have a couple microphones so we can make sure everything is captured So just at the end we'll we'll hand those around and do some Q&A Thank you Can you hear me? Okay? Yeah, cool. So thanks all for coming. I'm honored to be here talking to you Especially in this, in this topic that is very interesting in my opinion, let me just put my password So the topic today is public interest in the data science and I want to start by saying that Well with this quote, which is able lawyers have to to a large extent allow themselves to become Agents of great corporations and have neglected their obligation to use their powers for the protection of the people This quote started the public interest law And I think if we change lawyers and we use technologies the same applies So this is what I'm going to talk about today And but first I want to define what is public interest, and I think it's very simple It's to affect change in public policies in the interest of the Commonwealth and to advocate for the underrepresented and Build this or cheer for the underdog so to speak Then what is data science? Data science is to track knowledge and insight from information and data and then it's the intersection of math statistics research information science and computer science and If we take that into consideration, it might be very simple to explain what are what what the public interest data science is so What happens like the motivation of why I applied for these fellowships and why I'm working on what I'm working on is because I was wondering what will happen if you put data and technology in the hands of civil liberties organizations human rights activists media outlets Governments so my background is a little bit in the government in the open data movement and The open source software movement So I I have lived like my experience is Just in a few words is that I will be here if it were not for open source and the internet and the open web so I think that Data and technology see change my life. It can also change society and I think we are going to See that more in the future So now that we recognize that there are civil liberties organizations of human rights activists that need data and technology But why what what can you can we change? Oh? Sorry, let they want me to move here. Yeah Okay, let's they need to control this Okay Yeah, this is better. Yeah, I'm fine But I have a mic here Hey, hello. Hello. Hello. Hello. Okay So What do what do they need data for and I think it's We can affect change in liberty and justice and equality Especially transparency and accountability and I think these are the main things that we can Effect change in and I think that's the key for like a new society So But who else can data help and I think lawyers journalists policymakers and advocates and activists can benefit from data and technology, but Maybe this is like a very hard Like they need the skills They need to gather the skills for them to have an effect on their causes and that's basically why I think this is very important so There I think also that intersections of data and liberty for instance or data and justice and data and equality can have very important impact in in privacy or surveillance or Or ending the war on drugs or the mass incarceration Phenomenon and then we can also through data and technology and Or at least shed light in biases Improve access to justice for a lot of people, especially the underrepresented people So I'm gonna Present the data for justice project, which is a collaboration Well, it's enabled by the Ford Foundation the ACLU of Massachusetts and Mozilla Foundation It started as my as the project of my fellowship like the open web fellowship that I was in 2015 and It became now Like my fellowship got extended at the ACLU and now as far as a Bergman Center fellow I'm trying to absorb Bergman's knowledge into this project So the data for justice project aims to empower empower lawyers and advocates to make fact-based and data-driven decisions and Do their work like advocacy? To improve access to justice in their communities by providing them and decision makers with answers to questions such as how isolation bias policing Race and poverty relate to each other What we're seeing here, and I'm gonna talk about that More in depth enough in the following minutes is this is Boston, Massachusetts What you're seeing is the green color is people of color Obviously the greener the color the more percentage of the higher the percentage of people of color live in that area and the red dots are Drug arrests So this is like only simple example of how data and data visualization in this case can show that basically The water on drugs is racially charged So I'm gonna talk about the process And how We try to implement this process at the ACLU and I think the first thing and it's The most important is how do you how do you get the data? We have here and Lambert which is One of the lawyers at the ACLU and she's very Like this is her topic like how do you get data? And I think freedom of information acts are the key But in many cases especially coming from the ACLU they get denied like so in one of these cases that I'm going to talk about we asked for police records and We waited for a year and a half and they wouldn't give it to us so we sued and also it's very important to Don't give up on getting the data because you want to change This problem, so we stood and we won and then After that I normalize the data and that's all that's my work like the The things involved are my work. I normalize it and process it and analyze it and visualize it But that's not it. It doesn't end there. I think we also have we have cave here, which is an advocate at the ACLU and their work is to socialize it and inform change and then we also have other departments that are that Their work is to make it last so if we use it as a flowchart It's okay. You identify a cause And if you have data well yay data If you don't have data, which is very common Do you know who who has data if you don't know? Well, you need to research and If you do know who has it and it's it's it's it open data If it's not well, you need to go through the legal channels, which are which are the ones I mentioned mentioned foias and lawsuits and Then you get the data and after you get the data This is my work. This is data science and you do you develop Measurements and indicators and you research and measure and you ask your question is this good is this good Is this good until it's not and when it's not you can design solution or or methodology to solve or improve this problem with community engagement and This is where Kate's work Starts and Yeah, and I think it's very important also from the beginning to involve communities in the solution. So it's more like a Co-designed solution because it's very easy for us That have the data and the skills to say oh, I'm able to solve the problem by myself and I'm going to reach out to communities with the solution and I Think that's not how it should work. Especially when you are talking about people Affected communities should be involved from the beginning of the and that's what the ACLU basically does but then if you want to change these things you need to also involve all the actors and Stakeholders, so you also need to co-design with them and you need to Create run workshops and do This co-designing for the solution I think it's very important that the government for instance, so it's also involved which is the hard part because in many ways You are saying to the government or to the public officials that their work is not done as good as it could be Right and especially in the case of the criminal justice system so You have The your workshops and do you have a solution? Do you have an idea that can be implemented? Do you have like practical? or empirical Solution that can be implemented and if you do well Yeah, you can relax a little bit and then make it last make it Through the legislative means or to other means to make it permanent I think it's also very important to think about that, but it's like a separate process and it takes longer and longer and longer so This is how it looks like all the this is the that I call it the data process and It's like an abbreviation and this is how I look the process of the data for justice project So I'm gonna speak about the projects like the particular projects we've been working on Here so we have three projects right now, which can be accessed the first one Is the do can case how many of you are familiar with the hand? Okay. Oh cool, so I'm gonna go inside it independent But we also have a stop on frisk data sets and the marijuana regulation That is coming in the ballot in November 8 So as you can see it's basically Work between the legal department and the advocacy department And they lies in the intersection of them of both. So the do can case And the do can was a chemist that working in a crime lab in Massachusetts and she was In charge of performing the drug tests for contraband or seizures she falsified Evidence and Thanks to data and thanks to the work that the ACLU we have come to the number of more than 24,000 cases so these are 24,000 people that didn't have access to justice that the that their rights were violated because the The I forgot the the work Okay, so yeah, the rights were violated because of this Action of the chemist.  So The impact is that a lot of people were in case incarcerated because of her and Some some say that there that there's no more people incarcerated because of her But still the impact is that some people were deported others have bad records that can also affect In other jurisdictions. So for instance, there's a case in Florida where a man is facing life in prison because He committed a crime or in Florida and the prosecutors Find out one of these cases and that will and that is the his third strike. So That that can happen Also One of the key of the key Things to know about this case is that prosecutors say that these are bad bad people that they were like plodding the the streets with drugs and in fact only six sixty two percent. I mean in fact 62% of the cases are possession only so it's Consumers instead of distributors that are affected by this So only thirty six thirty seven percent as our distribution. So we have some interesting tables So this table talks about is the division of the twenty four thousand and forty eight forty four hundred eighty three cases divided by county And also divided by adult cases and juvenile juvenile cases As you can see Essex and Sophoc are the top two clients of this this this lab and then This is the the distribution of The type of cases so sixty two percent are possession and distribution No distribution and thirty seven percent are other our distribution and one percent is other cases Which are like conspiracy and others that were not possessed possession or distribution Then We have this table that shows the percentage of cases with adverse disposition From 2013 and the percentage What percentage they represent in the total of the cases? So we have some counties that are that almost 30% of their drug cases were affected by this for instance It's I think it's then took it or what is this not Norfolk Norfolk Has 30% of their cases tainted and Sophoc has 32% of its cases Tainted that's a lot of people So yeah You can access these these Are screenshots from the affidavits I submitted that we that I worked with the lawyers at the ACLU So I have the links That I can send it to you because they are like very interesting and it's this only like a light Resume of this case So, yeah, I think that we can actually use data to change these 24,000 people Lives and I think it's very powerful Then another example of this work is and then I we can discuss and all that It's the yes and for which is the question for If you say yes to the to if you answer yes on the question for in the November 8 ballot You are validating the marijuana legalization well regulation and taxation in Massachusetts And the argument we are using and we are using data to prove it is that marijuana related police interactions are racially biased and Because they only happen mainly where people of color live and you can see that it's It's again that the same the same pattern Green color is people of color and the red dots are where arrests happened since August 2015 to date So, yeah, there is a I think a very clear pattern and I want to show you how this looks So I'm gonna Got no, sorry I'm gonna come here So this is an advocacy campaign that can be accessed by This website Yay Okay, so let me just marijuana policing destroys lives every week in Boston Even though Massachusetts overwhelmingly passed marijuana decriminalization in 2008 Right now an average of 13 people every week are arrested or harassed by the police because of marijuana Just One arrest can permanently change one's life for the worse causing barriers to employment schooling and housing the data shows the majority of police Interactions and arrests for marijuana offenses occur in neighborhoods where mostly people of color live Let's explore the data from the 2014 census and the data produced daily by the Boston Police Department 46% of Boston's population is white 25% of Boston's population is black and 29% of Boston's population are other people of color as you can see the city is racially segregated white people of color white people of color Now, let's look at where police harass and arrest people because of marijuana This is where marijuana incidents happened last week and This is where marijuana incidents happened since August 2015 Do you notice a pattern? white people of color white People of color we can help end racially biased policing of marijuana vote. Yes on four so I think that's This is the one of the examples of what data technology can help To do this type of arguments so Let's see How does this work and I'm going to talk about my tools Oh, I can't find a mouse Okay, I think it's here. No Oh You won't come. Okay. Here it is. Okay. Oh, did you get the URL it can be accessed by your phone and all that Share it so This is my work. These are my tools I do a lot of data cleaning and normalization and I think a lot about data structures I love like I consider myself like a systems architecture architect and also a data architect more than a that data scientists I do a lot of structures thinking and I do a lot of processing. I use tons of Python and PHP and SQL. I've been working on SQL for almost 17 years, so I know it very well and that enables me to leverage these skills to Actually make it very easy for us to it to output this process and Process data to other tools like a stata or are which are also things like I use and then also, I think one of the main skills is GIS which is the the things I use for and I think they are crucial for Translating all this data to where people live or where people is So that's why I love post keys. I think it's very important Tool and then to visualize it to tell a story. I think It's also very important and to do that I developed this framework, which I'm not going to talk about that much But I developed a data visualization framework that enables me to do that kind of pieces Like the one of marijuana legalization in very little time and with a lot of information being displayed and Analyzed so then its name it's ant which he which stands for augmented narrative toolkit and it in it be it's built on Already developed tools like the tree and a score magic and jQuery and all these other tools I don't want to reinvent the wheel. I just want to implement road to make it easier for people to To use these very complex tools like the tree so my aim by developing ant is for you to be able sorry for you to be able to Put a map or a chart in a web page as easily as you will than image Which is like an a standard HTML Tag so my theory behind that is that if you enable people to use HTML To do data visualization HTML and CSS. I think that we are going to start to see a data-driven Internet and I think that's that can be very very cool But then what the visualizations are cool But I think the most important Thing it's the average we want to have impact on people's lives I mean the technical challenge is there and it's kind of cool to solve it but I think that if The work I've been doing doesn't have an impact on people's lives I think I wouldn't be happy about it. Like I will be only half happy So I think outreach to communities outreach to the authorities or the lawmakers. It's very important too and so for instance, there's a case where One of these visualizations got the attention of the Urbano project which is a youth of color workgroup in Jamaica Plain and what they asked the their fellows is to create this To do something with it with this data and what the kids did. I think was like a Amazing thing was a they painted the map that I just showed you In a mural in Jamaica Plain and they marched with that mural Through all these neighborhoods and I think It's very low-tech, but I think That's like the that completes a cycle of data Because that specific those data points that those specific data points went from The asking for the data in the foyer then suing for them for for it Then it got processed and we reach out to communities and the communities did something that then impact in in other types of stakeholders, and I think When I saw that I cried I was like, oh, this is this is how things should be working. It's not about the technical challenge. So My conclusions are that data and tech doon don't do anything by themselves people do Data science is very cool. Very very cool. It's a technical challenge But it's only a part of of a team with a common goal I wouldn't be standing standing here if it were not for my co-workers a co-workers at the ACLU that are kick-ass And also the intersections are very important exploring law and data it's it's very cool exploring data and Inequity is also very important. So yeah data and tech in the hands of the public interest organizations Working to reduce inequality can have an impact on people's lives Thank you So if anyone has questions Okay Okay, thanks. Hi, I was just saying that I love this project and all the work that you're doing with your colleagues and My question is about in the beginning of the presentation in your flowchart You talked to there's sort of a parenthetical statement where you said you could also involve the community that's most affected earlier in the process For example when in data gathering or in identifying the data sets that are needed And I know that Urbano project that you just referenced they did a project like that where? They were looking at transportation Inequality in Boston and they had young youth of color from Boston keep journals of their transit times like to school to work that type of thing and they created visualizations of Transit inequality by race and geography in Boston. So I wonder if you could talk a little bit more about Yeah, what are what are the? What are the challenges that are keeping more? Data scientists or people who are doing the type of work that you are from engaging Communities that are most impacted at different points in the process Why does it happen so infrequently because I think it's hard It's it's expensive and Yeah, I don't know if it were not for me being enough in a paid fellowship I wouldn't be able to do it because raising the money to compete with Facebook for the ACLU is a challenge and Then Ritching out is so it's a full-time job. Like it's it's not very easily done and you need to have like a strict credibility And all that you need to have connections to the communities and I think that only through the collaboration with With organizations you can do it like community organizing for reals. I think that's that's the main challenge So data scientists are very Like technically skilled, but they it's they're very far from in I'm generalizing of course what from community organizing Great great project and the visualizations are compelling, of course I wanted to so a lot of these data are very personal and private there is data about race and criminal convictions and location and I'm wondering what's done in terms of Privacy in the data sharing process. Do you get the raw data? Do you get like? to just perform queries and what's the privacy Cover for this well So in a look and case I had access to private information like personal information that that includes Names and social security numbers of people Like that were charged in these cases These data seem pounded and it's like I use security measurements to keep it that way and That's one case the other maps are open data So there they've been through a process of the anonymization and So for instance, if I showed you This So these are not where exactly happened. These are like agreed So agreed that this I think half a mile So if an incident happened there, it goes it goes to that point that enables us to to remove the identification layer so people or for instance So so people don't can't say where something in particular happened so that they don't have the repercussions of being the anonymized Pella I my name is Kate Crockford. I work with Pella at the ACLU I just wanted to actually to Pallas horn a little bit more than you did To tell you all the significance of having a data scientist on staff for the legal department in the midst of the bridgeman Litigation so the bridgeman litigation is the drug lab case We got the disposition data through that litigation as Palla said some of it was impounded We filed a motion asking the court to unseal the non personally identifiable information aspects of that data which they did and then Palla was able to perform this really important analysis which Revealed things like the fact that 62% of these cases were possession only which nobody knew before that So that was reported in the press as a major finding We that was like a major part of our brief one of our briefs in the case and then You know that one in three cases in Suffolk County was tainted by this scandal I mean these are these are really important pieces of our legal arguments in The bridgeman litigation that we would not have been able to make had we not had power on our staff So I just want that to be very clear and also I would just like to tell you I don't know if you know this But the prosecutors filed a motion last week that pretty much dis is Palla really hard and says Which I think is great because it means you're doing the right thing people are powerful people hate you already so fantastic They said They said things they effectively said I mean if you wanted the bridgeman litigation is really insane I encourage you to read about it because it'll put your head in a spin Cycle so one of the things they said about Palla is that It was not the intention of the court when they unsealed this information for anyone to actually look at it They really said that the prosecutor said that in a brief to the court Yes, the court unsealed this information But the purpose of unsealing it wasn't so people could understand what was going on Okay, and we're really upset that this meddling Palla via real person you bothered to look at it So just so you know she's really making an impact. It's really valuable from the ACLU to have Scientist working on information like this. Thank you Is there a is there a page on the web anywhere where you? Link to all the tools that you use to do this So that other people can do the same thing and follow your methodology. Yeah, we have a github page I mean all this except for the Durkan case, which is like a court case and I'm going litigation all other things are public in github Yep, so you'd have Data for justice Yeah, so you Will be able to see the code for these which is in the give this a github page, so You will be able to see this The methodology that I need to update and this is like this describes the The process to get that from the census data to all the police data and all that so and there are other pricks so Yes we know what happened to the person who falsified the Data, but the the question is what happened to the people were in jail So some are Maybe I'm can answer that question in particular. I'm sorry to put you on the The response of the legal system was to recognize immediately that the people who were serving time At that moment when this was discovered had to be have an opportunity to challenge their convictions and get out and in Pretty short order the court system mobilized and had Special magistrates appointed special hearings all of this by the way to follow up on Cade's point was challenged by the Commonwealth Oh, you can't vest those retired judges with judicial powers to admit people to bail or even to have hearings Where they make recommendations to a real superior court judge about releasing people on bail but the Litigation that's going on now and who knows I mean I think probably the court system did and a good First cut job and figuring out who was still incarcerated and doing something about it. The problem was as Paula has described That Then a certain amount of inertia lassitude Set in what about the 24,000 other people? What what are the remedies going to be for them?  You know, we just can't handle that So the point of the bridgeman litigation bolstered by what Paula has been able to demonstrate is You know, we have to do something about it the burden of these Tainted convictions shouldn't fall on the people who got them. It should fall on the people who went about Prosecuting them and then imposing them. So that's the issue that's going to be argued in the court next week Is I mean that's semi an answer. I mean it divides the world of duke and victims into People who were incarcerated where the system sort of responded But not fully and people that the system At least with, you know, Commonwealth versus John Smith the Commonwealth was kind of willing to let those go where the People in the bridgeman case Our organization the public defenders have said, uh-huh. Uh-huh. You got to pay attention to them too. Yeah I've been conducting hosting a discussion in the context of a Harvard ex-mooc called jewelry X Which began discussion just this week of the issue of yes on question for and the most provocative Hello Yes, the most provocative question has been the The way a way how to think about the fact that The leadership of the Commonwealth in the person of the governor the Attorney General and the mayor of Boston have been absolutely opposed to the yes on for and The question posed that was provocative is is there a relationship between The uniformity of the opposition of law enforcement Emphasized in this morning's paper by the support of the prosecutors for the opposition police opposition to yes on for Is there a relationship between cannabis reform and black lives matter and You've answered that question Your your work is such a beautiful Demonstration of it. Thank you that it is remarkable That black lives matter as a movement hasn't cottoned on To cannabis reform as a sister issue. I mean look here. We are We're not a black lives matter group and yet we can see it absolutely clearly It's a curiosity. It's a But thank you very much for the work you've done. It's extraordinary. Thank you. Thank you well I've been reading Kathy O'Neill's new book weapons of math destruction Which is a terrific look at how algorithms are impacting Vulnerable communities around the country. Oh, sorry. So it's a book called weapons of math destruction It's written by this woman Kathy O'Neill Who was at D Shaw and as an implied math person? And I was thinking about that and about your Brandeis quote from the beginning and I'm wondering if you can talk about The origins of your passion for social justice as a data scientist and what we can do in the data science community to get Data scientists excited about this. Do we need some sort of formalized training the way we do pro bono for law students? Okay, so To answer that question I will have to tell you like my background, which is I'm Mexican 32 years old and when I was 15 I started working well I learned to code when I was 12 and I only was able to do that because I Had access to the internet and I started talking to people that were Coincidentally Also developing the chats we were using So that I don't know that picked my curiosity So that's how I learned about Linux back in 1997 Then I learned the about C the language So fast forward three years. I started working as a web developer in Mexico when I was 15 and then Fast-forward four years. I was the sysadmin for Mexico's president's office And I was in charge of securing the web servers and enable enabling Like a development platform for the Mexico's president's office So I started reflecting on what had enabled me to do that because at that moment I recognized that I didn't have like peers my age Doing the work I was doing So The answer was open source and openness So I think that and that's when I decided that Through openness and open source. I could actually enable people more people to do the work I'm doing and now fast forward 12 years and I'm doing that. I'm I've been working in governments. I've been working with sexual and reproductive Organizations freedom of expression organizations And all this type of work are only trying to demonstrate that openness and open source can actually change people's lives and now I think my My goal is to inspire inspire other people to do just that So, yeah, I think that's That's it. I Thank you for the Ballot question on for now. It does bring up a point of You know what in data analysis like this? There's the legal aspect that you dealt with and and certainly You know, that is very apparent now. What is the what's the social impact on this type of data analysis where? cannabis reform is One part is the legal aspect of it. The other part is the effect on the community as a whole What kind of data exists for gateway drugs? In in other communities where it has been decriminalized That's what The social impact would be is there any kind of work along those lines because the the ballot question and I'm speaking as a as a Loan voter out here is flawed because it deals only with the criminalization, but it is not It's it's actually about the long-term social impact that can happen in this community What has what is the data that exists elsewhere? Yeah, do you want to answer that okay? Sure, I can try I Think I hear you saying that if we tax and regulate marijuana more people will smoke marijuana. Is that what you're saying? asking about the data itself must exist not only in the United States, but in in neighboring countries where The decriminalization has taken place. So that's what I'm asking about is that the data analysis has a social impact Right, I guess I'm asking what kind of data Data about use or data about to use yeah, and whether whether in in terms of the Where the cannabis is a drug that One shouldn't be bothered about you know as opposed to to heroin or crack or something like that Well, this is a little field of the subject I think but just very quickly there is data from Colorado that shows that Basically marijuana use rates don't go up significantly at all after legalization Also, we have past decriminalization in Massachusetts So it's already not a crime to possess under an ounce of marijuana for personal consumption And we haven't seen measurable increases in use since then what we have seen however is a Jurassic decrease in the number of people who are arrested for marijuana possession in the state So clearly as Paula's data shows The problem is not over and unfortunately the reality in Massachusetts what Paula's map shows very clearly I think is that Really only people of color are facing any kind of law enforcement interaction because of marijuana today in Massachusetts So Decrim had the effect of making it possible for me to smoke weed and not have to deal with the police But not for my black and brown neighbors So that's really the the issue that this map brings to the fore and and so if you're uncomfortable with those racial disparities in Marijuana, you should vote yes, it's my view Yep, yes, thanks for your work. It's fantastic I have two comments one black lives matter has looked at this issue, but they're not narrowing it to cannabis use only number two The racialized and gendered aspects of this are really troubling People are not looking what's happening to black women Okay, they get criminalized at a higher rate even compared to black men So it would be helpful to take a look at that population segment of the population as well And finally has anybody looked at environmental justice there are environmental justice communities right here in Boston that no one Has done anything about from decades. Thanks Thank you. Thank you for your talk. Um, my question is about How do So you're doing work and I know you're doing good work and you're reputable and you're working with a very reputable organization So the graphs and charts I see that you're producing. I can be Personally trust as reliable I'm wondering if there's any as we're getting into an era where we're using a lot more sort of visualization data if there's any kind of ways to Evaluate whether something is Valid as opposed to propaganda say even if it looks valid and I mean we can just see like how people think people can spin things in all sorts of ways and a lot of us might maybe can't necessarily look at the code and Or anything so I'm wondering do you have any ideas or are there any organizations that will evaluate sort of the data That's being put out in visualization form to tell you if it's Is trustworthy that's a great question and I don't know how to answer and yeah, we are Basically appealing to our reputations, you know, like yeah, so That's also why the code is there, but I recognize that not all the people can read the code and Basically, of course in the Brisbane litigation one of the arguments of the DA's Was that it was not peer reviewed like the my affidavit wasn't And yeah, it is not right so But that doesn't mean that the But to Contrast that I think what I'm doing is not an analysis Right, like, you know, I'm not trying to I'm not using AI or Deep learning or anything like that. I'm trying just to describe things and I try to keep it Like like that. It's like this table is brown and the computer is gray And if you put them together, it's a you know stuff like that. It's just description That way, I think I can convey the message very easily without trying to To like involve more More thinking or more confusion. So, yeah, I don't know. It's it's so complex. Yeah, but I think thanks for asking Thank you so much. This has been amazing I wonder it's clear that there's still a great deal of work Valibable work to be done with this data set and I look forward to hearing more especially in light of next week's Court proceedings, but I'm wondering if you've identified like, what's the next big data set you're going to go to work on that might produce Marvel's real also. I'm sure there are many out there. Yeah, one of them Like I developed tools to explore data one of them is like a sunburst like a peak pie chart but and they like In last in how do you say like an expanded pie chart, you know, like a sunburst That's the name of it. And what I use it for I can show you is For instance with the payroll of Boston's Boston's City employees, yeah Table So we have I already imported the data sets This is the payroll and these are their columns. We have the name of the employee the department It's their title and then we have all these Income categories and the zip code So if we want to see divided by department We can see that Boston police department and Boston fire department have a big chunk chunk of the payroll like the employees. These are only counting and counting employees But if we if you want to compare it to Yeah, in terms of money, let's see Total no way to restart because Okay So we have this and then if we want to do total earnings and we say Some and it's money and we do this and now we have to this is the total earnings and This is the size of the department. So you can see that Like in proportion BPD is Getting more money, but if you want to divide it by I don't know Title it can get very complex, but it's a lot of data You start to explore these type of things and now let's focus on Oh Is that for example Boston police officers who work in in West Roxbury get way more over time Than police in other parts of the city, which is so weird It's like what's going on in West Roxbury that all these cops have like huge Hugely greater percentage of overtime than police elsewhere. So that's the kind of you you know unique insight that these tools allow you to see that You wouldn't be able to see elsewhere. No police are people who work there. This is open data Yeah, this is only for the seat of Boston open data. Yeah, so now we have this which is Boston Police Department only and then by title which is police officer and then by zip code so this enables you to see that the thing that Kate was saying which is that in this in this zip codes like the police officers make significantly more money than In other zip codes Yeah What I mentioned before like even before thinking About data is the the mission like that. I consider this being a mission. Oh, sorry mission driven Data advocacy so that that's how I see it. It's like we have a mission which is prove that the Interact in the police interactions are racially biased And how do you which data do you need? So you start to ask these questions through research? And in many times you will find that there are government Departments that have data a lot of data, but And this is one problem I faced in my one one of my previous jobs, which was In Mexico City's government that we had a lot of data But there was no impact of that data because there was no translation From that. I mean first they don't know They don't know who is going to use it And second they don't actually do the outreach for people to use it So it's it's like a I don't know like a nega chicken problem. I think but if you put a mission on it and You can start to make all these diagnosis of oh, I need this data set and And then I need to ask this person for what this column means and then I need to Do that, but yeah, it's it's very specific and very mission specific things. Yeah, it's there's no explorer and there's no like It's complex Thanks so much for the talk I'm a journalist in South Africa and a lot of my work is bringing sort of information Complicated information to sort of communities. I was wondering what was your interaction like with journalists? Did you bring them on early? Did you work with them in that way or was it a case of it's finished and then handing it over? What was that sort of process like we haven't had that much interaction with journalists yet? Because I mean we It's I love to to have more more Contact with them, but it hasn't happened and I really don't know the ways to do it and also that The little contact we've had this like they won't publish the maps. They are only going to write about it You know like especially here that it's my understanding that understanding that Boston's Media is very specific in its in its actions and how how they handle cases Did they feel that they want to have had control over the whole process to publish it like it's almost like they don't want to I don't know I I mean it's also through that I'm working on an advocacy organization Right, so it's it might be also that that they want to replicate my findings and they But they need the skills to do it. So Yeah, it's it's like a Catch-22 what I love journalism and I think law and journalists are that the like the the key of our society, but yeah That's great. I think we'll probably leave it there. Do anything power for this


What's up everybody, Jason here for Jasper's K-Sprakes.com. We just sold out 2021 Panini Immaculate Basketball 1 box break round in teams number 30. Number high-risk high reward 30 spots over we get to 1 random team So the first 10 customers bought in straight up and then from Michael down to Robert got those from Prism pack filler number 194-195 So good luck everybody Rollets and we got ourselves a three and a five eight times. Good luck one two three four five six seven eight times Alex down to Carter I do actually get the break credit though Rex. I just won't be able to bind to breaks We get store credit Why do you think we're in it for? It wasn't just for bragging rights Customers that work here would still get credit. It's just store credit ha ha Oh, sorry. That's not what I was supposed to do. Let's go eight times here one two three four five six seven eight Nets down to the Oklahoma City Thunder All right, so Alex with the Brooklyn Nets Cleveland for Ryan John with the Pacers Nick with the Celtics Allen with the Knicks Nick with the Blazers Allen with the Hornets dusty with the Wizards Nick with the Pistons Allen with the Clippers Warriors Ryan with the Bucks Ryan a Ryan Bidwell with the Bucks Ryan C with the 76ers Brian with the Bulls dusty with the Lakers Alex with the Raptors Nick with the Heat dusty with the Grizzlies Jonathan with the Kings Michael with the Magic Ryan Bidwell with the Hawks Nick with the Timberwolves and Sons Allen with the Mavericks Robert with the Jazz Christopher with the Nuggets Dusty with the Pelicans Carter with the Rockets Sammy with the Spurs and Carter with the Oklahoma City Thunder So if anybody wants to trade guys, I'll give you guys a quick trade window So Derek just got two spots in that contenders mixer filler We're all just got three and NT first off the line All right, guys, let me know if there's any trades if I'm closing it Try one of the closing guys Here's the magnet. Good luck round team 30 Siakam to 49 for the Toronto Raptors Toronto Raptors as Alex Kaolari another one for the Raptors. There you go We have Kyrie Irving to 99 Brooklyn going to Alex All right, we should have autographs now Next one is Lamar Stevens to 99 Cleveland going to Rhine C We have a Duncan Robinson 4 out of 10 past in presence Miami Heat going to Nick Evan Garnett 2003 all-star game MVP 9 out of 25 Oh Yeah, the only reason why people are still able to come into our shop is because we had the gate open for our Our mill carrier list. She had just was getting here But it was a little kid. They're really probably like a seven-year-old kid and once we say he's like, oh Then he's just like But it's like bro. You guys know we're we're close it. We close us six All right KG Nick Smith. There you go, man So Kevin Kevin Garnett Robinson Lamar Stevens Kyrie See how come there you go. Just like that guys. That was the break And our number 31 is from a fresh case. If you guys want to get the next one going jazby's case breaks.com


And then I start to work on 3D flights and imaging, which is actually another dimension of imaging. Let's think about this way. If you want to understand the life, right, you need to understand two things. You need two kind of information. One is spatial resolution, right? Basically, you have to see the details or you have to see the object. And the other dimension is you want to see the process, you want to see things alive, right? Because if you for any super resolution technique, or most super resolution technique, what do you have is a very high-resolution image of a dead sample, of a fixed sample, a dead stuff. At the moment, on a plane, that's the most of time you can get. I mean, how much you can interpret this life, a living process from that information, not enough, right? And then you will need, you will want to have the image alive, dynamic 3D process alive with high resolution. And that's a different challenge. And that is a project I was working on when I was in Eric's lab and since that, actually. So the differences between like something as a very thin slice of some tissue over one moment in time, versus the already dead, versus something that's alive, like a Drosophila fruit fly, and being able to image while it's alive over time with the brain or something. Right, you can think about that a little bit, or you can say, let's say you are having an HD image, like a 4K image in your hands about a football game, or you prefer to say, let's say, to watch the game at maybe 480 pixel resolution for an hour. One play, just one play versus the whole game. Exactly, so basically actually you want both, right? You want the high resolution live video of the entire process, but at that point of time, 2010, there were several good live image techniques. I like that sports analogy a lot, it's a really good one actually. Sure. Seeing one play of the game and trying to make some inference about how the biology works, versus watching the whole game in high definition. Exactly. That's a really good analogy. So in the end, we are collecting information of the process live process, right? So basically, that was the motivation of my project, how I started it.


Good morning guys, happy Monday. We'll come back to another vlog. It is 9.30 right now, which is really, really late for me. I was planning on working out at like seven this morning and I don't know what happened, but I just like literally couldn't get out of bed and I just woke up like 20 minutes ago. I really like starting off my Monday mornings like very productive and getting a workout in and whatever to set myself up for the week, but like I don't know what happened. I literally just couldn't wake up. Normally I would like shame myself for this and be like really mad that I did this, but not so bad. It's also really gloomy and it's raining like literally it's like eight days straight. So maybe that has something to do with it. I barely have any left of this. I'm getting my grocery store this morning. It'll have to do until the new groceries get here, but I do have my glass straws that Emily Di Donata told me to buy. I was influenced. Change is the entire experience I have to say. I do an updated best Amazon purchases because I have so many. Hi guys, I'm Rina here. My name's Kinsey. I'm a lifestyle blogger. I'm 23, 11 Dallas, Texas. Subscribe if you guys haven't already. I'm wearing my skims pajamas right now. I love these. I got them on Friday and my weekend in my life vlog. I need them in every single color. Like they are my new favorite pajamas. And then this robe, I actually like cannot say enough good things about it. I was worried it was gonna make me like too hot, but no, this is literally like what I wear around my house. Like I actually wear the full matching set and rows and stuff as we know. All right, so I need to edit and get some work done on my computer. So I'm gonna do that for the next few hours. I have a hair appointment at 1.30. Thank God, I really need that. So I'm gonna take you guys along with and show you guys like what I do and what I ask for. If you guys are in Dallas, my sister does my hair. You guys should go to her as well. She's in Dallas, stop. My new curology package came in. As you guys know, I've been using curology for probably like six months, I wanna say. There's so many reasons I love it. I've shared actually that my skin is just so clear and glowing. Basically you go on to the curology website and you take a quiz and you get a custom formula comes monthly, which is so nice because I love when things just show up in my house when I have to go to the store. I wash my face, this is the curology cleanser. I have a new one. I actually ran out so I need to go this way. And then I do the moisturizer. So, so easy and really simplifies your skincare routine. This is what I put on every night. I had some hormonal breakouts recently. Put this on, woke up the next morning, gone. I talk about them all the time. I love them. If you guys want to try out curology, you guys can get your first month free. All you have to do is pay for shipping and handling, which is $4.95. That is a really incredible deal. I will have the link down below. I know so many of you guys use it now. I use it all the time. So many of my friends in real life, they get calls from the guys about their skincare because they're using curology like I'm not making this up. I love my products, especially that they come to my house and I don't know what to go to the store. So since it is Monday, I'm going to record the intro and ads for this week's podcast episode, which is actually a really fun one. It's our like hot girl summer predictions and like plans for the summer. If you're trying to like get excited for summer or just like listen to me, Maddie, my sister and Dom talk for an hour. It's a really good one. I think you guys are really going to like it. It's pretty funny. But I'm going to do that. Send those in. My groceries are about to be here. I have 127 emails to like, I don't know what is going on right now. I really, really just, I don't, I don't know, I don't know. Hi guys, welcome back to the I Love You So Much podcast. I'm your host, Kenzie Elizabeth. Oh, look who just showed up with the merch. Love it. Groceries just arrived. I sent in the podcast episode for this week. You excited, Dom? I'm so excited. Okay, we have some groceries. It's what I get every single week. So it's nothing exciting, but, oh, I have so much salmon. I have so much salmon. I'm bringing back salmon. Oh, yeah, Sam's here too, guys. Do you want avocados or avocados toast? Okay. I'm going to make it on bagels, guys, because I need to get rid of my bagels. Okay? Everyone out here with that? One thing that matters is the best stuff ever. I'll make that for dinner. Does anyone want coffee? I'm not listening. What coffee do you want? What do you want, Dom? I'm actually good. You just said that so aggressively and now you're just changing your mind, right? No? You said yes. Oh. I did? Yeah. I'll replay it in the vlog. Are you sure? Yeah. I have hot and iced. And lattes. Okay. Oh, thanks a lot. I'm losing my mind. Okay, I guess. I guess. Thanks, Kensington. I guess. It really means a lot that you would think of me. Breaking news. My icebreaker. It works. Is the Waterwork Maker? I don't know, guys. I've had a lot. Wait, no, no, no. It's this. Wait, no way. You're joking. So, we haven't used that in months because we thought it broke. Why would you think that? Because it did break, but now- It wasn't working.  I hand shoveled these from the- You did that, but I just wasn't questioning it. Hand shoveled. Guys, I packed my bag for the salon. Besties. This tote is available on shop.com slash gearmedia. Something like that, I'll have it linked below. Great news. This couch is getting picked up tomorrow, but I'm such a great supportive BFF. I'm wearing twin flames. I always do. So cozy. One time, I was driving down Greenville in Dallas and we saw a girl wearing a twin flame hoodie. Isn't that so cool? I know, it was like so cute. We yelled at her, but I don't think she heard us. Anyways, I'm heading to the salon. I think I'm probably gonna stop at Starbucks first because- You don't say, everyone. Do you think everyone should pick one up? Yes. Hi, how are you? No, well, not very well. Good. Thank you. Seriously, the Starbucks order that has taken the world by storm, this matcha one. It's supposed to rain for like eight days straight, but it actually feels good outside. So I'm like really living for this, even though it was like gloomy this morning. All I listen to is either Jack Harlow or the Yee-Haw Bops 2.0 playlist. Like there's really no one between, oh, Olivia Rodrigo as well. Good for you. Best song of 2021. Yes. Thank you so much. Have a good day. Best drink ever. Here we are, everyone. Maddie, so what are we doing to my hair today?  This is my sister. You guys can come here. Global hairdressing salon. Global group. Global group hairdressing salon. I always just say global. Something's about the other day and I think I feel really uncomfortable. Global? Just the more period we can sit there. Honestly, the name was just so long, but look, I can't remember it. They said it was so long and I was just like, huh?  She's really nice. She's rude to me sometimes, but she's nice to everyone else, so worry. Hair is finally done. It feels so good. The color looks so much better. Wow, amazing. There you go. I got a lot of work done on my computer, so I'm feeling productive. Even though, you know, today's Monday and I should have done a lot more, but it's fine. You know that feeling when you get your hair done and you just feel like a different person? That's how I feel today. It really is like one of the top, okay, comment actually below, top three best feelings. Top three best feelings. When you've been laying out all day, you go shower and you're getting ready with your friends to go out. Second, a new haircut. Third, summertime, you're in the car with your friends, you're blasting country music and singing. There we go. Comment your top three feelings below. It is 4.30, I was there for three hours. I am ready to leave and get home. I still have work to do, unfortunately, because I've just been really irresponsible today and I'm really hungry, so we've gotta figure that out. Also, a question, what do you guys like to see in the vlogs? I'm home, I came home to some packages. I have my bottoms from 4.37. Guess what? This is a chocolate brown bikini. It's literally called the Kinsey Bikini. I mean, Kinsey Expresso, whatever, and it's my favorite color, so the bottoms are gone. The bottoms are back in stock, so I finally got them, but the top, you can wear like nine different ways. So of course I got that. I have a package from Recess. Okay, they have this calm, cool, collected powder. Wow, I love Recess drinks. Got some more Recess drinks. These are the best, I'll have my code down below, but they're like hemp drinks, so they're really good for like calming you down. They're Recess Mood. They're Mood-enhancing magnesium and adaptogen-infused sparkling water. So these are a little bit different than the normal ones. Go listen to Mood at Florida, Elizabeth. I got new mats right downstairs, loving an adult. I got the dog's new bowls, only Coco's came in, but I really love these on Etsy, and they were legitimately like $200. I couldn't believe my eyes. Okay, this is so much cuter for dog bowls. Love this. I kind of want to move them to my house, but I don't know what I'm gonna do. Beautiful, beautiful, there we go. Okay, I finished editing the vlog. I'm gonna go work out. I have my cute octore voice-it note here of our sets. I've already filmed this clip when I'm re-filming it. It's really cute. Even my grandma complimented me, so it's mega approved. Also a question, guys, which vlogs do you prefer? Do you prefer where I'm just home doing productive stuff? I feel like vlogging in a work day is so boring. Do you prefer when I'm out with friends? What vlogs do you like? What do you want to see in the vlogs? I'm really at a loss, especially because my life has become very social, so I really don't have to do, you know? And I really only like having friends in vlogs that have been in vlogs since the beginning for the most part of people I'm with every day, because I know that you guys don't really care if it's random people that you aren't connected to, but Sal's been in vlogs since we were 15, 16 since I started my channel. So let me know. It's just like, I'm a very... The other thing is I'm just a very social... People are always at my house, you know what I mean? You're always just... I'm always with people. Or I go to bed at 8 p.m. per year and I read 100 books. But I'm actually still reading 100 books. I'm on my 39th book of 2021 right now. My p.m. I'm 40th, actually. It's not insane. It's because I haven't watched a show. Here it is, Sal. Here it is, what it is. My podcast with Water and Whiskey, Whiskey and Water. I always mess it up. They're really cool. New podcast episode that I guessed it on. We ended up talking about reading like pretty much the entire time. It was actually a really good episode, so check it out. All right guys, so I just got out of the shower. I worked out, did laundry, I cleaned up some more, and I am actually about to go meet up with my sister. So I'll show you what I'm wearing. It's very much so like a Monday night look. Tank top, aloe leggings, my soothies, socks, and this just whatever these things are called. What were we calling them again? Shackets from H&M. I would prefer to be wearing a white tank top, but I also just self-tanned. So that's not gonna happen. I hope you guys enjoyed today's vlog. I know I have so many questions, but like what do you guys wanna see in the vlogs? How long do you guys want the vlogs to be and how often do you guys want me to post? Love you guys so much. I hope you guys enjoyed today's video and I'll see you guys soon. Bye. We underestimate how much people like us in the first interaction. So while you're walking away thinking like oh my god, I totally bombed, very thinking, and she was cool. And so it starts there with we're really hard on ourselves about the experience and if we fumble.


So, you know we are what we are looking at in algebraic geometry is trying to look at things which are which have which have an intrinsic meaning ok. So, intrinsic means these are certain definitions and properties that you make or define which depend only on the object and not on other extraneous factors. So, for example you know what we are worried about is we are worried about varieties alright and then of course the varieties could be you know they could be affine or quasi affine or they could be projected or they could be quasi projective and when you say when you say these things you are trying to say that the variety is either sitting as an irreducible closed subset of some affine space or some projective space or it is sitting as an open subset of such a set ok. So, when we define variety we are already thinking of it as sitting inside somewhere either an affine space or a projective space. So, there is this and there is this fact that you are making it sit into inside something which gives a certain ambiguity ok because you can you for example if you take the line the line can sit inside any affine space in any way ok. So, you can think of the line just as a1 can also think of it as a one of the coordinate axis in a2 ok or even any other line in a2 and you can also think of it as a line in 3 space a3 ok but in any case a line is a line ok. So, what is interesting about the line is that it is a line alright and what is extrinsic about it is the way you are putting it inside you are thinking of the line just as a line or you are putting it thinking of it as a line inside the plane a2 or you are thinking of it as a line in 3 space which is a3 and so on. So, these are so you know what you must understand is that when we when we start working in algebraic geometry start building the theory of varieties we always start by embedding your object into some affine space or projective space even to define a variety you have to think of it as sitting inside some affine space or projective space as an irreducible closed subset or as an open subset non-empty open subset of such an irreducible closed subset. But then finally what we want to really analyze and define and analyze and study is not I mean we want to define and analyze and study only the properties which are intrinsic to the variety we are not interested in they should depend only on the variety and not the way in which it is embedded ok. So, the fact for example you know this saying that the line is one dimensional does not depend on whether the line is being caught of as a line a1 or whether it is being thought of as a line in a2 which is a plane or whether it is thought of a line in 3 space a3 whatever be the space in which you embedded it is a it is still one dimensional. So, you see we say that dimension is a very intrinsic property because and in fact our definition of dimension we have seen that we have given the definition of dimension in two ways one is as a topological space we have given we have defined the dimension of variety to be topological dimension and we also proved that you know this dimension is the same as if the if the variety is an affine variety then it is the same as the cruel dimension of the affine coordinate ring of the variety ok. So, the moral of the story is that you know the definition that the dimension is the topological dimension is a very intrinsic definition it just depends on the topology of the variety it does not depend on anything else ok whereas the definition that the dimension is the cruel dimension of the affine coordinate ring namely the coordinate ring of polynomials on the variety ok that ring the dimension cruel dimension of that ring is the dimension of the variety is another definition but that somehow seems to depend on that ring because that ring is it depends on the way in which the variety is embedded ok if the affine variety is the affine variety could be embedded in so many affine spaces in so many ways as an irreducible close-up variety and in each of those cases you will get a coordinate ring the rings of polynomials ok but thankfully for affine variety is we have seen that the coordinate ring itself is an invariant it is an intrinsic thing namely if you take two affine varieties then they are isomorphic if and only if they are affine coordinate rings are the same so which means that no matter you in which in what way you embed the affine variety the moment you say affine variety you are considering an affine variety is isomorphic to an irreducible close subset of some affine space it could be an irreducible close subset in an affine space of any dimension ok and the dimensions could be different the like the same line being embedded in a plane or being embedded in three space ok so the ambient the bigger affine space in which your variety is embedded as a close-up variety that could be different but still the ring of polynomials on that variety they are all the rings are all isomorphic the affine coordinate rings are isomorphic so we express this by saying that the affine coordinate ring is a is an intrinsic object that is defined connected with the variety ok now so you know this is what I am trying to say we are trying to it is very important that we begin by using extraneous or extrinsic things ok but then finally we try to find out what are the things that are intrinsic ok so in that list comes the affine coordinate ring of an affine variety ok which is intrinsic then of course the other thing that I introduce was the local ring at a point the local ring at a point is something that is that is defined in a very intrinsic way ok it is just defined using regular functions ok and the point is that if you change the variety up to isomorphism then the local ring will also change only up to isomorphism ok namely if you have a variety and a point with an isomorphism carrying this variety to another variety and this point going to another point then the local ring of the original variety at the given point is isomorphic to the local ring of the target variety isomorphic variety at the image of the point that is gotten by the image of this point ok so we say that the local ring is also something that is very intrinsic namely if you change the variety up to isomorphism the local ring will also change up to isomorphism ok. So what we are going to talk about so you know in that direction I want to introduce another important ring associated with the variety in fact this is a not just a ring it is a field it is called the function field of a variety ok so I want to introduce that and they want to tell you that I want to introduce it in a very intrinsic way and then show how to compute it for fine and objective varieties ok. So let me put the title of the lecture as the function field of a variety the function field of a variety and so what is it that we are going to do so you know the idea is very similar to looking at the way in which we got the local ring at a point ok so you know what we did is see if so we start with x a variety ok and of course which means you know it is to be a fine or quasi-fine or projective or quasi-projective and we have a point and then you know how do you define the local ring at a point of the variety. So normally what we did was if you have a point small x in capital X then what we did is we define O capital X small x this is a local ring of x at x to be O capital X small x tilda modulo and equivalence relation ok and how was this equivalence relation and what was this tilda O capital X small x tilda these were just you know pairs of functions which were regular in some open in some Zariski open neighbourhood of the point x. So you know this consists of pairs of the form u, f such that f is in O u and x belongs to u. So you are just looking at functions which are defined in an open neighbourhood of the point x and which are regular alright and you took such two such pairs and then you know what is the equivalence relation the equivalence relation was that two such pairs will be identified if they define the same function on the intersection ok. So you know u1, f1 is equivalent to u2, f2 if well f1 restricted to u1 is equal to f2 restricted I mean u1 intersection u2 is equal to f2 restricted to u1 intersection where of course we you know you have to remember that x is both in u1 and in u2 so x is in u1 intersection u2 so u1 intersection u2 is a non-empty open set so it is in fact dense ok and you are just saying that f1 restricted to u1 intersection u2 is equal to f2 restricted to u1 intersection u2. In other words these two functions can be glued together to give a bigger regular function on u1 union u2 ok. So in some sense in the sense of complex analysis we say that these are two function elements which are direct analytic continuations of each other ok. So now of course you can weaken this condition you could you could just say that f1 restricted to w is equal to f2 restricted to w but w is an open neighbourhood of the point x which is contained inside u1 intersection u2 that is also enough because you know we keep always we prove this and we keep using this all the time that two regular functions if they are equal on a non-empty open subset then they have to be equal everywhere ok this is true always right. So now you know so we got this local ring like this alright and of course we have seen we have seen that this how to compute this local ring ok we have seen how to compute this local ring and what it is in the affine case what it is in the what it is in the projective case we have seen it alright. But now what I am trying to say is if you want the function field it is very simple what you do is you just remove this restriction of concentrating attention at a point ok just do not worry about the point and the beautiful thing is that you if you do not concentrate attention at a point you go global ok you get something global. So what you do is put kx to be kx tilde to be a set of all pairs u,f such that f belongs to ou and now there is no restriction on u, u inside x as an open subset open non-empty ok. Of course here also whenever I say f belongs to ou I am I am it is implicit it is understood that u is an open subset of x ok u is a Zarski open neighbourhood ok fine. So now you see so what is the difference between this and this the difference is that here you are only taking open neighbourhoods of the point small x but here you are simply taking any any non-empty open set and you are taking a regular function of that ok. So you just drop attention to a point and lo behold what you get is you repeat the same procedure what you get is not a local ring you get a field ok. So now what you do is again you define the same thing you define same equivalence relation u1,f1 is equivalent to u2,f2 if and only if f1 restricted to u1 intersection u2 is equal to f2 restricted to u1 intersection u2. Of course for this I mean you need not require it on all of u1 intersection u2 it is enough to even require it on open non-empty open subset of u1 intersection u2 and you must always remember that this very important fact that by definition or varieties are all irreducible and therefore you take any non-empty open subset of a variety it will continue to be irreducible it will be dense and any two non-empty open subsets of a variety will always intersect ok. So this is the problem the Zariski topology the problem is that the open sets are huge ok they are dense their closure is the whole variety I mean of course non-empty open sets they are huge and any two of them will intersect right. And well so it is just enough to require that f1,f2 coincide in an open subset of u1 intersection u2 ok non-empty open subset of u1 intersection u2. So this definition this equivalence is same as equivalence here only thing is you are not focusing attention at a point alright and then you put kx to be kx tilde mod the equivalence ok. So of course when I write mod equivalence I mean equivalence classes ok. So this is the set of equivalence classes this is also set of equivalence classes. Now the beautiful thing is that just like in this case the set of equivalence classes gave you a local ring ok you got a commutative ring which is a local ring which had a unique maximal ideal which was represented by functions which are regular in a neighbourhood of x and which vanish at x ok and in fact the representatives the local ring namely the equivalence classes were called germs of functions ok. So this is the set of germs of regular functions at the point x and every germ of a regular function is represented like this it is represented by a regular function on an open neighbourhood of the point ok. Now here we call the elements here well they are kind of they are called rational functions ok they are called rational functions and basically they are to be thought of as functions are regular on an open set ok that is how you should think of rational functions. Rational function is nothing but a regular function on an open set it is not a regular function on the whole alright but it is a regular function it may be a regular function only on an open subset which may be a proper open subset ok so that which means that there could be points where it is not regular ok where it where you cannot think of where you might extend the function but it may not be regular right. So let me write that down elements equivalence classes of kx are called rational functions they are called rational functions so they represent they well they represent they are represented by a regular function on an open subset alright and further you see this kx this is actually this is actually a field kx is a field and why is that so because you see you see you take two elements in kx so you know so you know I have this I have this map kx tilde to kx this is a quotient map it is a surjection so that is why I am putting this double arrow head and this is a set of equivalence classes this is mod equivalence ok and what we are doing is you are taking if you give me a rational function u1, f1 you send it to its equivalence class which I will indicate by putting a square bracket ok and well if you give me another rational function u2, f2 I will send it to so rather let me use u, f and v, g so this will go to well v, g round bracket followed by a square bracket the square bracket indicating the equivalence class here and then how do you add these two guys it is very very simple you add them like this you give me two rational functions you just add them on their intersection because you know as I told you we keep always using the fact that any two non-empty open sets will intersect so you and we will intersect on the intersection both functions make sense so I could add them so I can do this I can take this equivalence class and define this to be the sum alright and then the same way I could define product I can put product here and I can put product here ok so depending on what your either sum or product can be defined like this and now I want you to check it is easy to check as you must have done in this case that this definition of addition and multiplication is independent of the choice of representatives because I have defined the sum when I define the sum of two rational functions I am actually taking the sum of the representatives and I am I am taking the sum of the representatives on the intersection on the intersection of the domains where the representative functions are defined ok. So and similarly when I do for the product I also use representatives but then whenever you know you use representatives you have to make sure that your definition is correct it is well defined so you have to check ok so check that check well defined as that is a simple exercise that the same kind of exercise as you would have undergone for this case of you know for the local ring. So this will make kx into a commutative ring in fact you will have that kx is commutative ring of course it will be commutative ring because addition and addition of functions and multiplication of functions is point wise and that is commutative because if they are the functions are taking values in the base field k which is commutative alright. So it is a commutative ring with 1 to be with 1 being the given by the pair x, constant function 1 and 0 element given by the pair x, constant function 0 ok. So it will be a commutative ring with unity and with this as 0 element alright and in fact it is in fact not only that it will in fact be a k algebra because you know a regular function multiplied a regular function multiplied by a scalar is also a regular function because scalars are of course regular function they are part of a constant regular function ok. So kx is a k algebra of course you know in our discussion we always fix small k to be an algebraically closed field where we are working that is where we are do studying our varieties. So and this kx is a k algebra ok all functions take values in small k all regular functions take values in small k and well now the point is the following the point is well in this case after you took if you took open neighbourhoods of a point and then you define the sequence and when mod this sequence you got a max you got a local ring you got a commutative ring which was also a small k algebra but it was a maximal it had only one maximal ideal and therefore it became a local ring but the point is in this case you do not get the only maximal ideal you will get a 0 and so you will get a field. So why is this a field to show that this is a field I will have to tell you that every non-zero element here is invertible ok. So if so you know if you take u,f not equal to 0 element the 0 element is x,0 in kx ok. So then you know what I want you to understand is that you see you take 0 take z of f ok see z of f inside x will be a closed subset ok. So what I want you to understand is that whenever you take a variety and you take a regular function on it ok and you take the set of the locus where it vanishes ok that is always that will always be a closed subset ok. So in fact you know I should say it is a closed subset of u in fact because f is defined only on u it should not write x here it should be careful right.  So u itself is a variety f is a regular function on u ok and if you take the 0 set of f namely set of points in u where f vanishes that is going to be a closed subset of u ok because actually you know roughly the idea is that the closed subsets are defined by vanishing of polynomials right in the affine case it is in the affine case they are common zeros of bunch of vanishing of bunch of polynomials and if you are the projective case then it is common zeros of a bunch of homogeneous polynomials ok in any case it is just vanishing of polynomials that gives the 0 the closed subsets for the risk it is a policy. And therefore if you take a regular function on a variety u is also a variety and f is a regular function on that then locally f is a quotient of polynomials and therefore looking at zeros of this regular function is like looking at locally the zeros of the numerator polynomial because locally f is a looks like a numerator polynomial divided by denominator polynomial and zeros of f will be just the zeros of the numerator polynomial ok and therefore essentially you are just looking at zeros of polynomials and therefore again you are going to get a closed set ok. So this is a closed subset and mind you z of f you know cannot be u ok z of f is not equal to u because if z of f is u that means f is identically 0 on u but then if f is identically 0 on u then f is equal to the 0 function which is identically 0 on x and that will contradict the fact that these two are different elements of the function field ok. So z of f is not equal to u because otherwise you will see that u, f is the same as u, 0 and this is the this is equivalent to x, z which is a contradiction ok. So I should say contradiction it is a contradiction to the fact that these two are different alright. So that means 0 z of f is a proper closed subset of u so it is complement in u is again an open subset of u.  After all a regular function is a locally of the form quotient of polynomials where it will not vanish is where the numerator polynomial does not vanish. If the numerator polynomial does not vanish ok already when you are writing the quotient denominator polynomial cannot vanish, if the numerator also does not vanish then its reciprocal the reciprocal of these two numerator and denominator will give you also a rational we will again give you a regular function at that point locally ok. Therefore wherever a regular function does not vanish its reciprocal is also a regular function so what happens is that you will get this pair u minus z of f which is an open subset of u proper it is non-empty open subset of u mind you it is this is actually mind you it is dense in x itself ok it is a non-empty open subset of u and u is non-empty open subset of x so this is a non-empty open subset of x it is reducible it is dense ok it is a huge open subset ok and on this I have the function 1 by f 1 by f is certainly a regular function on that ok. So this this element if you multiply it with the original element u, f what you will get is just x, 1 you will get this ok by the definition of multiplication alright because after all 1 by f and f are going to multiply to give you 1 wherever f does not vanish ok and wherever f does not vanish is certainly is a huge open set alright therefore so what this tells you is that whenever u, f is not the 0 element mind you this is 0 element in kx so whenever something is not 0 it has a reciprocal there is something with which you can multiply it to get 1 this is 1 this is the 0 in kx and this is the 1 in kx ok. So here also when I write 1 equal to x, 1 I mean this is the 1 in kx and this is 0 in kx ok. So what I proved is that every non-zero element is I have a commutative ring in which every non-zero element is invertible so it is a field. So the moral of the story is that kx is a field and is a field extension of k ok because it is a k algebra so you know it is an over ring of k it contains small k as a sub ring and whenever you have a larger ring which is a whenever you have a when the smaller when you have a ring extension and both of fields you it is actually a field extension ok. The smaller ring is k it is a field the larger one that we have defined and constructed is the is also a field extension right now and this is called the function field it is called the function field of x ok it is called the function field of x and what is so special about the function field of x. So the answer to that geometrically is the following that the function field of x will give you all the information that is true on a large open subset of x ok. So this is the philosophical importance this kx will contain all the information about x which you can find on a large open set of course any non-empty open set is large ok. So you take any general open set next and you take properties that are valid on a general open subset of x then all those properties they will be captured by kx ok. So kx geometrically captures what is happening on a large open set of x ok. So kx captures captures the geometry on a general geometry of ok. So the point is you see what I want you to understand is both in the local ring case and in the function field case you are just looking at pairs of functions ok I mean you are just looking at pairs which consist of a regular function defined on an open set ok. Here you are only look concentrating on open sets which contain a given point and you get the local ring and the local ring contains all the information in a neighbourhood of the point ok all information in a neighbourhood of that point how the variety behaves in a neighbourhood of that point all that information is controlled by this and captured by this ok. Whereas if you want to know what is happening on a general open subset of x that information is captured by kx ok. So this is a you know if you want to study a general open subset of x if you have a geometry question about a general open subset of x ok then what you have to study is this if you have question about what is happening at a point of x then the object you have to study is this the local ring ok. So these are two extremes ok and I will tell you how they are connected we are going to see how they are connected of course they are all connected by to each other and essentially they are all I mean everything is done by commutative algebra ok. So now comes the following question so you can ask you can ask how are these things connected. So the first thing I want to tell you is that you know we have maps so before that yeah we have maps we have natural maps oh oh x oh x x ok there are maps like so what is this map let f be a regular function on x you just send it to since f is a regular function of x then I have the pair u x, f and its equivalence class will give me a point it will give me an element of the local ring at this point x because the elements of the local ring at a point are simply germs of regular functions defined on the neighbourhood of the point and since f is a global regular function it is also a regular function at that point so I can take its germ so this is a germ of f at the point and well you what I can do is on the other hand you know this is this map alright mind you that I can also send a map from here to here namely if you take u, g let me write v, g what is this v, g? g is regular function on the open set v which contains a point x ok this is the germ of v this is the germ of g at the point small x ok and of course g this open set v may be a proper open set g may not extend to a regular function on the whole of x it may be restricted it might just be defined on this open set ok now but anything here I can further send it to the same thing here mind you I am using the same notation but the point is this equivalence is the equivalence in local ring this is the equivalence in the function field ok sorry but I am using the same square bracket. Now the point is that of course you know there is also a map which goes like this directly and that is simply that is simply going all the way to x, y which is just the composition of these two maps this will also go to x, y where again this square bracket is the equivalence class here and this square bracket is equivalence class here ok alright so I have maps like this now the fact is that these maps are well the fact is that these maps are injective k-algebra homomorphisms ok. So these are injective k-algebra homomorphisms ok these are injective k-algebra homomorphisms ok so of course they are k-algebra homomorphisms namely they are ring homomorphisms which are k linear ok so that is no problem it is very easy to see the point is the point is the injectivity. So the injectivity comes like this you see that is also something that uses again this fact that you know if a regular if two regular functions are equal on an open set then they are equal everywhere ok of course that open set has to be a non-empty open set of course because you need a point to test you need points to test whether two functions are equal actually ok. So well you see you can see why this is why this is injective is very very clear because you know if x, f is 0 ok then it means that f vanishes in a neighbourhood of small x ok but that means f is equal to the 0 function in an open set containing small x ok but then if you use the fact that two regular functions if they are equal on a non-empty open set they are equal throughout. So if f is if small f vanishes in a neighbourhood of small x then it vanishes everywhere that means f itself is 0 ok that will tell you why this is injective and why is this injective that is also literally the same argument if you take a v, g germ of a function g which is regular in this open neighbourhood v about the point small x if it goes to 0 here it means that it is equal to 0 on some non-empty open subset of v ok but in any case that will also force it to be 0 everywhere in fact if this goes to 0 here it will tell you that g is the 0 function on v ok and the 0 function on v extends to the 0 function on x ok and therefore this will itself be 0 alright. So the moral of the story is both these ring homomorphisms have 0 kernel so they are injective ring homomorphisms and what is the moral of the story the moral of the story is the beautiful thing that the regular functions sit inside the local rings the local rings sit inside the function field this is how the you know so the idea is that somehow when you are looking at it topologically you have the whole your you have the whole variety x you concentrate attention to a point you get a larger ring ok. If you have the whole variety x you have o x which is a regular functions on x ok now you concentrate attention to a point you get a larger ring ok that is a local ring at that point ok which is larger than the ring of regular functions and then on the other hand if you concentrate instead of concentrating at a point if you concentrate on an open set you will get a much larger ring which is in fact a field and that is the field of rational functions. Now what I want you to understand is that you know in all these things nothing will go nothing will change if I replaced x by an open subset u ok. So in fact what will happen is that in fact if u is an open subset of x is open and small x is in u then you have we have a commutative diagram we have equalities in fact o x to o x x I mean at the level of local rings and at the level of function fields you are not going to have any difference so you know so I will put a hook arrow saying that you are considering this as sub rings what will happen is this will be the same as o u x is o u and this will be the same as k u and this will sit inside ok. So if you the function field depends only on an open subset of x so the function field of variety will not change if you replace x by an open subset so k x the function field of x if you replace x by u non-empty open set then k x will be the same as k u by going to a non-empty open set you are not changing the function field and that is just reflection of the fact the function field captures informational on a general open set so if you go to a general open set the function field does not change ok and the local ring at a point also depends only on a neighbourhood of the point it does not depend on the ambient variety. So whether you are considering x is a point of the open set u of capital X and calculating the local ring of small x at the of the variety u or whether you are calculating this local ring of small x with respect to the bigger variety ambient variety capital X you will get the same result ok. So the local ring also depends only on a neighbourhood of the point ok and of course the fact what is what is what is happening here what is happening here is that you also get an inclusion like this you also get an inclusion like this because every regular function on x can be restricted to get a give you a regular function on u and the restriction maps restrictions of functions they are all injective because of the same old reason that if a regular function if two regular functions coincide with an open subset non-empty open subset then they have to coincide everywhere or other way of saying it is if a regular function vanishes on open subset then it is identically 0 it will vanish on the whole variety. So you see this is what happens if you if you go to an open subset and why these two are equal is something that you can easily you can easily you can get this equality in a very very easy way namely what you do is you know you and any element of kx is of form u,f where f is a regular function on u and you take the equivalence class and what do you have to do it is very very simple I should not use the same u let me use v,g or even better let me use w,h w is an open subset of x h is a regular function on w then I have this pair w,h and its equivalence class is the rational function here ok and what will I send it is very simple you see w is a non-empty open subset of x, u is a non-empty open subset of x so they intersect so I can make sense of h also on the intersection that is what and that is the element I am going to send it to. So I am simply going to send it to w intersection u,h restricted to w intersection u I take this class and then I take its equivalence class and that is a element I am going to send it and in principle I should write this as a map and you can see that this map will be a k-algebraism of some but then I put equality because actually as far as the functions are concerned I am simply restricting the same function so that is why I put equality there ok if you think of them as functions then you can you can put equality here and you can put equality here ok. So if you think of them really as functions ok of course you have to remember that here you should think of representative as functions on a non-empty open set ok and here you must think of representative as functions on a regular functions on a non-empty open set which contains the points molex that is the only difference ok. So you have this nice diagram alright and now what I have to do is to tell you what this kx will be if x is an affine variety what this kx will be if x is a projective variety ok we need to understand what that is ok we have already understood what the local rings are when x is an affine variety or a projective variety ok they are given by suitable localizations ok we have to repeat this extend this kind of argument to the function fields ok. So let me just state the result then we will probably prove it in the next lecture here is theorem if x is an affine variety then kx is isomorphic to quotient field field of ax which is equal to o x ok. So if x is an affine variety then you know we have we have defined ax the affine coordinate ring of x and we have proved that that affine coordinate ring is equal to the ring of regular functions ok and this equality being thought of as functions ok we put this equality because we think of a function here also as defining the function there and conversely ok and what if x is projective if x is projective then kx is let me put y whether I do not confuse if y is projective then ky is just you take the homogenous coordinate ring of y ok the homogenous coordinate ring is defined the same for a projective variety is defined the same way as affine coordinate ring is defined namely you take the ring of polynomials of the ambient space that you are considering and then go modulo the ideal of functions that vanish on the variety in this case of course you will have to take all whom you have to take polynomial ring in of the affine space over the projective space in which this variety y is embedded and go modulo the ideal of y which will be homogenous ideal and then you take this and then you know what you do is you take the 0 ideal ok that is a prime ideal and this is localizing at 0 and then I will put another 0 here saying that you are taking the degree 0 part of the homogenous localization of sy ok so this is degree 0 part of the homogenous localization sy so here is the formula for the function field of affine variety and a nice formula for the function field of a projective variety ok and you know if you try to write out if you try to write out this kind of diagram including the you know the local rings and the quotient fields then what will happen is that you know if you take O x small x that is going to sit inside kx and you know this kx is just ax quotient field of ax ok q represents a quotient field of ax that is kx that is what this first part of the theorem says this is isomorphism and we have already seen that this is nothing but ax localize at mx where mx is the maximal ideal in ax corresponding to functions that vanish at small x so I should put small x here so mx is equal to ideal of functions vanishing at x ok we have already proved this and we have this inclusion ok and you must recall a fact in from commutative algebraics very easy if you have an integral domain ok then it is quotient field will be the same as the quotient field of any of its localizations ok the quotient field of quotient field or field of fractions of an integral domain is the same as the quotient field or field of fractions of any of its localizations. In fact the quotient field is the largest localization ok it is a localization where you have inverted everything except the only element which you cannot invert which is 0 and 0 is a prime ideal ok and so you are localizing at the prime ideal ok so this q ax is just ax localized at the prime ideal 0 ok this is what it is you are inverting everything outside 0 right because localization at a prime ideal means invert everything outside the prime ideal in this case invert everything that is not 0 that will give you the quotient field. So this is the picture you get for the affine case ok and you will get a similar picture for the for the projective case if you take o y y small y point of capital Y where capital Y is a projective variety this is going to sit in side ky and what is going to happen is that the function field of ky will be just sy localized at the prime 0 degree 0 part and this will contain sy localization at my and degree 0 part where here my so this is of course this diagram commutes this diagram commutes this isomorphism of the local rings has already been proved ok and this isomorphism of the function fields that I have to prove and here again my is equal to all those ideal of ideal of homogeneous elements of sy that vanish at small y ok. So this is how you get it you get the local ring the quotient field the affine case and this is the local ring and the quotient field in the projective case ok so we will prove this in the next talk.


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