Why don't we make a project to analyze how people are watching it? Grace Hopper's Celebration of Women in Computing coverage continues in a moment. Live from Houston, Texas, it's theCUBE, covering Grace Hopper's Celebration of Women in Computing. Welcome back to theCUBE's coverage of the Grace Hopper Conference. We are live in Houston, Texas. I'm Rebecca Knight, your host. Today our guest is Amanda Ghisharu. She is the co-founder of Tech Republic Africa, also an ABI winner. This is the Anita Borg Institute winner for Change Agent. This is for your work in giving girls and women in Kenya opportunities in the tech industry. Thank you so much. Thank you so much for having me. I'm so happy to be here. So talk to me about your career. You actually started your career at the UN and you started in development and then moved on to Google. So I did start my career in development and that was because I always knew since I was a child that I wanted to do something to make the world a better place and just bring positive impact and change. So I started off in Italy working for the World Food Program around food security, specifically in communications and fundraising, just to make sure that programs are sustainable and were able to scale. And then from there, I actually moved on to do my masters in communications to learn more about how I could effectively raise funds for nonprofits and social enterprise. And then from there, I returned home to Kenya and worked with the United Nations Development Program around several aspects of development, from governance to energy, food security, conflict. So I got a more high level look at, you know, the entire, I guess, landscape of development issues that we do have within the region. And then at that point, I felt that, you know, international civil service was a bit slow and a bureaucratic. So I thought, you know, I want to go to a company that'll bring positive change, but it's a bit more fast moving. And that's how I ended up at Google, actually. Yeah. And so you're at Google, but you also have this real part of you that, as you said, wants to make the world a better place. Yeah. So what did you see at Google that made you? Yeah. So at Google, it's very much, I guess, a startup type of environment. They're always looking to innovate and with their innovations, they're trying to make a positive impact for the world. And specifically at the time when they had opened up the offices in Africa, they were trying to demonstrate to the local population how the internet could make an impact in their everyday life in a more positive way. So that really was attractive to me because we do have challenges within the country where we have a high mobile phone penetration. It's at 88%. But there's a lot of ignorance around what you can do with that phone beyond messaging and calls. So for me, I wanted to be part of that narrative and help people understand you could use the internet to access information and knowledge is power. So I was working in product marketing, consumer awareness. It was less around corporate, let's make money kind of consumer marketing and more around positive social impact. How can we show the local people how they can use the internet to make their lives better? And so as you said, 88% penetration in Kenya and yet some few people really understood the power of this little device they were carrying around in their pocket and in particular women. Yes. So the problem you were trying to solve in founding Tech Republic Africa. Yeah. So what we looked at, like I said, while I was at Google is digital literacy, right? Do people understand how to use the device and the internet? And while we're going around the country trying to help people have this understanding, I realized it's really deeply rooted in our education system where people are learning, I'd say, industrial age type skills, whereas right now we're in the artificial intelligence aid. And if you actually graduate from our Kenyan education system, you're not able to complete globally and the world is now global. We have to look at ourselves from that perspective. And women and girls are particularly at a disadvantage because they do have challenges accessing even basic education. So accessing STEM education is completely out of the question. So I left Google to start Tech Republic Africa with their support as well. And we're looking at how we can reach kids in rural areas as well as, I'd say, disadvantaged urban areas and educate them on STEM with a skew towards CS and tech because what's happening is when they do graduate from our local schools, the skills are equipping them to be more employees and less entrepreneurs and innovators. And that's led to a very high level of unemployment because people come out and are looking for jobs as opposed to starting their own initiatives. So with our program, we don't only teach them STEM skills, we also teach them soft skills and business skills so that once they complete the program, they're in a position to start their own initiatives and create jobs and grow the economy. I want to talk about the results that you're seeing but I also want to ask about the parents because so much of this too is parents either seeing opportunity for their young daughters or not seeing opportunity and really needing to be persuaded that this is a viable path for their daughters. How much outreach do you have to do to parents too? Yeah, that's actually the first point of contact because they are, like you said, the decision makers. So we actually do run workshops and awareness at schools and within the universities that we partner with to educate parents. They are also part of the system in the sense that they went through the same education system. So they also have that same level of lack of information on STEM and the opportunities that exist currently. So we do have workshops, two-day workshops, so we literally take the parents through a taste of the program, help them understand the opportunities, expose them a bit more. And we also use a lot of peer-to-peer marketing. In Kenya, that seems to be much more effective and credible when I hear from another parent my students went through this particular program and this is how they benefited. So that's work for us as well. So what are some of the challenges that you see for women getting this education? What are some of the biggest hurdles? Yes, I'd say that Kenya and maybe the wider part of Africa is still very patriarchal. So what you find is women entering the workforce have to work twice as hard because they have to be taken seriously, they have to gain credibility and also they have to face the brunt of, I guess, this social conditioning that a woman's place is in the home and having children and taking care of the home unless in the corporate world or even as an entrepreneur building her own business and creating her own path. So that's really been a challenge for these girls to get them to understand that. As much as women are perceived as that, we are the backbone of our society. If you actually look at a typical Kenyan household, you'll find that a mother could be at home just looking after the children but at the same time maybe she has a small side business where she's selling vegetables or she's making clothes or she's baking cakes and telling them so they actually are entrepreneurs. They just don't label themselves as that and you need to tap into that particular, should I say, will to do whatever it takes to survive and take care of their families and innovate because they are innovating in their own way. And so you have these women who as you said are small business owners because they are making things and then selling them, also running their households, making decisions probably that affect their children and their husband. Is there a sense that things are changing in the sense of becoming a less patriarchal society? Do you sense an underlying change? So from a policy perspective at the government level there has been a big push towards empowering women more. So there are policies that are recently established that allow women or rather enable women to have whole more positions in government. And of course that's very impactful generally for women across the board because they are in a position to make decisions that will make things easier for women to get into business, into positions of power influence within the government or the corporate world. So there is that change. And then you'll find because of programs like ours as well as other institutions who are working towards educating women, girls are feeling a bit more empowered and starting to realize, hey like actually I have a position of influence like you said within the household and I can be able to take this further and have positions of influence within the corporate world or within governance as well. And you are a nonprofit right now. But you said that you may be moving a little bit toward the social enterprise model. Can you talk about that? So generally if you look at the world right now and the recent economic recessions that have taken place it's become a bit more challenging for nonprofits especially small nonprofits to access sustainable and consistent grant funding. So as a result a lot of nonprofits are shifting towards the social enterprise model that allows them to carry out some commercial operations to allow them to be more sustainable as opposed to being dependent exclusively on grants. Yes, so that's what we're looking at right now just reconfiguring and restructuring our business so that we are sustainable. And if grant funds do come in, yes, it'll be supplementary but not the main source of our funding for us to run our programs. So let's talk a little bit about Grace Hopper. We are here in Houston, Texas the biggest tech conference, the biggest technical tech conference for women. What are your impressions? What are your thoughts? First it's amazing like the energy in here 15,000 plus delegates coming together just to understand more about the tech world opportunities for growth in their careers and understanding like cutting edge, innovations taking place. It's amazing. I'm learning so much. This is actually my first time here. So I'm so grateful for the opportunity. And it's been an amazing first day. This is a second day. I'm learning a lot. I wish I could actually be in all these different workshops like in multiple places at once. Yes, I wish you could divide yourself and go to multiple places. Exactly, because there's so much to do. It's a great experience. I hope to be back here next year as well just to keep learning and growing. And I've met so many interesting people not only just from a career perspective and challenging myself as a woman and what I can do but also from a perspective of understanding where my country could go and could be within the global tech sphere. I'd love to talk a little bit about the election right now too and just sort of the backdrop that this conference is taking place against. I mean particularly you live in Nairobi so you live many, many miles away from what's taking place here in the US. But does it seem, you're talking about this patriarchal society in Kenya and we'd like to think that we've moved on, that we are a more progressive, socially aware culture. But how do you see this election and what are the impressions that you have being in Kenya watching Donald Trump and Hillary Clinton? So in Kenya we're watching very closely. So in my country our celebrities are politicians. So we're very much into the political sphere, the arena and what goes on there. So we're watching very closely. And for me it's interesting to observe that at this juncture it's very like there's a political revolution I guess you could call it because it's unexpected that the candidates that actually have made it to this point in the race are here. Hillary has all this experience and then Trump comes from the corporate sector and it's a very, I'll just call him an interesting character. Do you have an apprentice in Kenya? We actually had an apprentice of Kenyan origin here in the US who made it into that show. So Trump is not like, he's very familiar to us I would say. It's just unexpected that he chose to go this direction and that he's made it this far. That was very unexpected. For Hillary it isn't as unexpected obviously because she has this long running career. But for us it's, I don't know, it's the only one I can use is unusual. And we're watching very closely. It's a mild word to describe this. Yes, I'm trying to be diplomatic. I'll just say unusual. Some would say entertaining but I'd say, I wouldn't call it entertaining. I'd say it's very concerning because I've been following the debates and I've been looking at how it's less about policy and development and moving the country forward and more about individuals and personality and mudslinging in some sense. Well particularly for Kenya. So we have President Obama who is on his way out who has a Kenyan father, really a son of Kenya. I mean, how does that, how do you feel to watch the first African-American president exit office? Wow, I mean it's sad for us because we're always saying that's our son, that's our own, that Kenya helped actually like raise him in a sense.  So he's done a lot of work, especially in international relations and the work that he's done in Africa. I recently actually just completed a state sponsored program called Tech Women and it's selected men from Africa, Asia and the Middle East and brings them Silicon Valley and places that different companies for a month long mentorship where we'd learn more about what local companies are doing here within the tech world and that's something that he started. So it's been amazing what he's done. I just hope that whoever does make it into office is able to continue and build on the work that he's done. Great, well thank you so much. Amanda Ghisharu, thank you so much for joining us here at theCUBE's coverage of the Grace Hopper Conference in Houston, Texas. We will be back after the short break. Hi, I'm John Furrier.


Please stop being illegal. I'm allowed to do this. No, you're not. You're literally not allowed to do this. You're literally not allowed to do this. You're literally not allowed to do this. You're literally not allowed to do this. Box, go for the box. Death girdle going on. You wanted the hardwood. You got the hardwood. What up, what up, what up? Listen, you just made my February. You know that? Oh, yeah. Happy February, brother. Thank you, thank you. Up is going well. So, Cora, where's your dreads at? Well, you got to get those locks, man. You know what? You know what? You got to go to the Bay Area today. I actually should. Oh, hell, you have 49ers. Yeah, shit, that's actually who I'm rooting for today. Oh, same here.  What are you wearing? Oh, yeah, this is new. Oh, yeah, you ain't seen this yet. You like it? You look as snazzy. It actually fits you well.  There you go. How's that? That was not.  Can I get that hat? Apparently, I'm the only one that's got one of these. I asked my tailor to see if we can get one for everybody. Man, I put a hat on like that, and I got daps.  And it's February, man.  Some of us have seniority here. OK, Andrews? Yeah, you just got privilege here, Mr. Cora. That's what it is. Yeah, it's because I'm on duty about 600 hours a month. You know what?  Here, let's see what's going on here.  This place in the world, whoop, whoop, senior buns. Yeah, this place has gotten a hell of a lot better, believe it or not. How's it? How's it? Because this is the first place we're going. Usually, the first place we're going to go to. I'm second. Well, I haven't been here in days. You know what the third place is, Mr. Cora? Can you guess it? Can you guess where we're going next? We're going to the hospital.  We have to go to the hospital. We gotta go to the hospital. I haven't done that. I need to go to the hospital. Yeah, I know. We've got to go to the hospital. There's nothing new. Yeah, yeah, they can all go to the hospital. Police, you guys need to go to the back for the other thing. Oh, excuse me. Hold on a second. Mr. Corwood. Yeah. Mr. Corwood. Don't you ever lie to me again and say this place has changed. Hold on. There's only three people in there instead of 30.  And this place has changed.  Is Little Texas a thing yet? No, but I see I wanted to talk to the coach today. It's still a twinkle in an eye. It's still a thought. No, look, here, I'm going to send you an email right quick. All right. It just says cool. No, that was from earlier. Look at this.  Is that Tim? Wait, what is Tim? Tim, what are you doing? Walking.  I walked from the hospital all the way to the bank.  Hey, are you a part of the LSMC? I guess so. They told me to.  OK, what's your call? Hello, hey. Are you OK? No, we get there was he ran to our car and got ejected. He ran into your car and got ejected? Well, no, like we were making a corner. He was making a corner and we hit each other.  So what happened here? We ran into the side of our car with his front so it can't fall. We were going to the intersection and he's rammed right into the side of our car. Yep, as we were turning down the street. Going like 100 miles an hour. We checked it too. There's 100%. There was no one in there. And then we ran around looking for him being good pedestrians and found him all the way over here. I'm telling you, y'all got to tell all the group six drivers if they're driving like that. I've been taking it easy on group six workers and everybody because I was like, well, I mean, y'all getting no damn felonies because I don't want people losing their jobs or nothing.   And everybody's been real soft with group six workers. That ain't happened no more.  All right, I was taking it easy on everybody. Yeah, yeah, yeah. All right, let's see you guys sign up for the championship.  This is cool. These patches are pretty cool. Does that mean there's actually coming to real life, as you would say, or? I don't know what you're talking about. I mean, is this going to happen is what I'm asking. Is it going to be real life? We'll see. We'll see what happens. It's fine, Mr. Cornwall. We'll catch him in about an hour. It's OK. He's going to crash into something else and get ejected again. Yeah, possibly.  See, this is a different Mr. Cornwood and Tim, this is a FTO trying to be perfect liberal. It's not. Oh, my Lord. That's not a bad thing. We pulled over somebody in. We pulled over somebody in an 18 wheeler that put a turbo in it. Oh, corn would I knew it would have been. I know, but they're yelling for 77.  I know when you're going to throw this in there. Should we hold on one second? By one, three, two, it's funny. Three at the state is seven. Was this where it was? That's an 80 and the boardwalk. Yeah, look at that. Crossing South Rockford, Eastbound. The speech came back for it. Well, you got to play so we can find it.  Then where the car go?  I mean, nobody was looking at it anymore. So, you know, we're an 80, we're an 80, we're an 80. Sorry, we're trying to catch up. You got to get six times Eastbound. San Andreas have right, southbound Elgin Avenue. Damn, we took the prime spot.   You just drive, Mr. Cronin. Yep, we just drive. Heading Eastbound speeds are approximately 60 miles an hour. Miraculously, there's no traffic on the freeway. Now, 120 miles an hour. This SUV with no doors is going 120 miles an hour. Southbound reports a freeway. Crossing over to the opposite lane. We're stopping. Oh, I'm going down. I'm going down. Somebody is jumping off of the. This lady jumped off of the freeway.  Yeah, see, this is the point where you're just like, yep, not today. Opposite lane to travel. We're going back in that lane. Hey, there's a motorcycle that she got onto. I'm dipping from that. I'm coming back to you. What did these people do? Is BBS approved? Oh, yeah. Now that they've done that swap, yeah. Left. Stand by.  What is the, uh, wanton this? What did they do? So I went to a suspicious activity call and they were on top of a rooftop. They saw me. They jumped down. They got into a vehicle and dipped before I could even like, they knew I was there. They ran away with everything that we know. I told you I saw someone dismantling something. Hey, can you stay with her? Can you watch her? We need right here right up to the hospital. What happened? This is your fault. What happened? Wait, what happened? Someone on happen. These motherfuckers. What state stay with her, please? And, uh, what, two people on this? Why are we lighting this?  I'm taking it out. I'm taking it out from the secondary, you know. We'll, you know, they, they. Why are we shooting them? Because, because, because they, because they stopped. They stopped? This is not, this, this. All right, we'll see secondary, uh, eastbound innocent. I'm good about this stuff. That's not a shooting situation. I'm, I'm all about the shoot situations. That's not a shooting. They ran somebody. I know, I know, I know. Just, just trust me on this. There's still eastbound innocent. Hey, keep in mind that they did run somebody over.  All right, time for, we can get aggressive with them now, by the way, with pets. Uh, we're out of the populated area. Okay. Sounds good. I want you to bring it on back. Holding laughter on the wrong road here. Westbound. Actually, I didn't, I didn't think they could turn it. And they car that big that much. So I sent it. Hey, I was impressed. Correctly, traveling to travel now with Westbound Olympic Freeway. What's this? This is bitches calling it. It's so good for algae. Uh, trespassing. All this for, oh, Westbound. So we're doing all this for, for a fine? Is this lap on the wrist? No, just, Andrews, do you want to, do you want to do policing today or not? Cause, cause this is our, Andrews, this is our job. Everything that we do today, we're complaining about. They're doing all of this for fucking, for this. It literally every single thing that we're doing, you're going to, okay, do you want to do traffic stop? Do you want to do, do you want to stop a robbery? Do you want to find out who murdered somebody?  This is what we do. We got hired here for a reason. Mm-mm. I thought, um, Rooka, what's the status of the lady who got hit?  Is, um, is there? What's your beef corn whip? I'm, I'm trying to do work. And I'm trying to focus and try, we're trying to do our damn job, you know? Oh, we got to get this prepared, probably. Let me see this. You've changed. What do you mean? This is what we do. This is what we do. Look, I have seven daps. Okay. I get three more. I'm fired. I understand that. I get that. Game plan. I'm shooting the next guy as. No, that's not what I'm saying. I'm shooting like that. Also, we're not allowed to shoot out of our cars. I did that once already and I got told not to do it again. So then what do we do here? What do we do?  That's a hard, we need to stop this person. They need to go sit on time out for a little bit. That is, that is a hard one right there. That's not a, you know, give them a warning for running somebody over after fleeing from us. Because if we would have ran somebody over, we would get what?  Remember what just happened to you back in the day, Corwin? No, you're right. You know what? Actually, you are right. I don't think there's like a rule or law for stopping this and getting in trouble for it. Like what would be the problem? All right, Mr. Corwin, I am here for the ride. Anyway, everybody ready to see the chiefs lose? Am I right? Oh, we're chasing. Oh, come on, traffic. I feel like that's the car we're chasing. No, it's a. Got a repair to some shop. In the car, please, ma'am, in the car. This guy in front of the other, the other ma'am.  He's driving away. That the guy that. Is he, is he, is he, can you stay with her? Is he stay with her? See, we're not complaining. But we went to our favorite place, see your buttons. And yeah, God, I want to talk to the judge.  Hey, stop, stop, stop, stop chasing her. Stop chasing her. You wait here. Stop chasing her. Wait here. Stop chasing her. Wait here.  Oh, good luck. Sir, can I see your ID? Can you turn off the car and step out of it, please? Oh, God, it's it's it's Andrews. Yeah, I'm going to keep driving then. The other car was flagged in 60. I didn't see them leave. We're going to go back, we're going to go back. What, the car that was chasing them was stolen? Yeah, the other car was stolen. Chad, the other car was stolen. Yeah, the other, so the other car was stolen. That one was not flagged in 60. It did not belong to either of the people that said their car got stolen. So we're going to go back here and pick up Miss Fan. Hello? Did you lose that guy? No, I didn't. Get to the right side of the road. Get to the, go get on the right side of the road. What's your name? Future.  Some guys are borrowed at the province. No, it's not. It's a car that you stole. Okay. No, is it? Yes, it is. Yes, it is. Yes, it is. Yes, I will help you get your car, but I'm doing. Okay, okay. Southbound in front of.  Don't drive through there. That is the emergency vehicle area. And this is a motherfucking emergency, bruh. Yeah, I know. That's why I'm my lights on, the sirens. We're stopped at the emergency entrance. Listen, this can go on a two-way corn ball. This is trying to give us stipulations. He calls corn with corn ball. Hey, if you want the hard way, I'll give you the hard way. Or the hard way. If you want the hard wood or do you want the corn wood, you get to decide. You listen to what I say.   All right, well, we have a little wager here. Cornwood took. So we're back on in this pursuit here, going through Davis Avenue area, just driving through the park, running over. Hey, that's cheating. Westbound, Innocent, Southbound, now Roy Lewenstein. He's going through the parking structure now, almost hit a group six worker. He is on Roy Lewenstein going Southbound. Westbound on Carson Avenue. We get 77. Northbound through the alleyways off of Carson Avenue. Kind of sounds like he wants tea and biscuits. Going Westbound through the parking structure here of the courthouse over Davis Avenue into a rundown strip mall. Oh, he's just 10-50ed into an invisible thing. And he's screaming.  Get out of the car. Get out of the car. Get out of the car.  Get out of the car. Going Westbound. You're being illegal. On Innocence Boulevard, he's crashing. Please stop being illegal, sir. Three, six, catching. Please stop being illegal. Sound like a baton. Please stop being illegal. I'm allowed to do this. No, you're not. You're literally not allowed to do this. You're literally not allowed to do this. You're literally not allowed to do this. You're literally not allowed to do this. You got the deck girdle going on you wanted the hardwood. You got the hardwood Sounds like Eli. Oh, we just ran somebody over You went through the car wash area here, we have no visuals. He didn't fact lose us. It's okay I'm gonna find him.  So, Mr. Corkwood, you know, I apologize for being a Debbie Downer all the time Not sure if we went left or right.  So I'm sorry for being a Debbie Downer.  What do I apologize for for changing on me for being excited? How did I change? I didn't change. I'm doing my job. Okay. Like I said, I'm sorry. I'll be the bigger man. It's okay. You're my brother. And for the sake of, you know, the law, you're my only brother. I have the vehicle again. I had that same vehicle that we were chasing. Cornwood got me vanilla unicorn. Is it future? It's future Hendricks in there about pink dreads. Yeah. We go west bound, currently off road, standby, not heading northbound alleyway, standby. We're underneath the Olympic freeway.  Coming back out onto, I believe this is power. Stop being illegal. Stop being illegal. All right, 1080 with this vehicle again. Going behind the Legion square bank coming out onto Vespucci Boulevard, heading westbound. If you break, check me one more time.  Right.  I got it ready.  If you break, check me one more time. Can you believe how my car is smoking? Can you believe this shit? Chal 2 480. Chal 2. Cornwood hitting it from the back. This is probably why our car is smoking also. Oh, yeah. That's a wrap. Damn. Yeah, I was definitely. This was a poor decision that our cars are so much less durable than a civilian car that they can't do that. You can't do that. That's complaining. Let's come on. Let's see the positive. Oh my Lord. Why? Let's see the positive of this. There are no positives. Do normal patrol on it. I need your feedback on how it drives, what it needs, and there is a report called New PD Vehicles. I want to get your feelings about it. Okay. No. Okay. You can write it. You can write it. She just asked you. No, she had a spider on her knee. And I tried. Are you good? I don't know.  No, she had a spider on her knee. I mean, at least help her pick the bike. I was kicking the spider off. Thank you. Thank you for getting that spider.  It was poisonous. I couldn't have lost my leg. Yeah.


Okay, so let us continue our discussion about you know rings of functions of varieties okay, so let me recall the picture as it was explained in the previous lecture. So on one side we have the that is the geometric side we have if you want affine varieties and on the on the commutative algebraic side we have the so called rings of functions on affine varieties. So let me recall you see an affine variety is supposed to be a closed irreducible close subset of some affine space okay and affine space is of dimension n is just kn k cross k cross kn times given this a risky topology alright and by the risky topology of course if you recall it is just the topology that declares closed sets to be common 0 loci of a bunch of polynomials okay in the appropriate number of variables. So here on this side you have close subsets irreducible close subsets of affine space so standard examples are affine space itself is irreducible of course we are always working with k an algebraically closed field k is an algebraically closed field okay if you give if you take the affine n space then the ring of functions for affine space is a of an which is the polynomial ring in n variables over k okay so there is this association that takes to every affine variety swing of functions okay and of course the functions we are interested in are polynomial functions right and more generally if you give me a closed subset of an affine space which is also irreducible this is what a general affine variety is defined to be an irreducible closed subset of some affine space and for such a irreducible closed subset what is the ring of functions the ring of functions is defined to be the ambient ring of functions ring of functions sorry the there is a ring of function of the ambient space the largest space in which x sits in the ring of functions on this divided by the ideal of that subset and the ideal of this subset mind you is the ideal of functions in the ideal of polynomials which vanish on all of x okay and you know that that is prime because x is irreducible alright and therefore this quotient is an integral domain okay because you are going you are taking the polynomial ring and you are going model of prime ideal okay basically you are going model of prime ideal so the result the quotient is an integral domain and you see this closed this irreducible closed subset here shows up as a quotient here this is a quotient and this is just the you are just going mod ideal of x okay and well see the point is that what is actually happening is whatever is happening here is on the geometric side is an exact image of mirror image of what is happening here on the commutative algebraic side and algebraic geometry derives from this translation basically okay so that problems on geometric problems can be translated into commutative algebraic problems and vice versa and you can and a problem that you cannot solve here may be you can solve it here and the other way around okay. So the first thing I want to tell you is of course on this side it is affine varieties on this side it is rings of functions on affine varieties and I am saying that there is some kind of a bijection so there is also a map that is going on this side about which I mentioned the existence of but I did not tell you what it was in the last lecture I would like to tell you what that is and that is the that is the so called max spec so I will come to that and these two put together will tell you that whatever is happening here and is exactly a mirror image of what is happening on this side okay and so let me recall see for example if you have that X is a geometric hyper surface that corresponds to on this side X being X being and a hyper surface in the in the commutative algebraic sense okay and in fact on this side I should not write X actually I should write something about its ring of functions and the fact that X is a hyper surface in the commutative algebraic sense is the fact is that X is a hyper surface in the commutative algebraic sense is something that is defined by a single equation it is a locus defined by a single equation and so I of X so the condition is I of X is generated by single polynomial F and so X is just the zero locus of single polynomial and since I of X is prime okay it will force that F has to be an irreducible polynomial and it has to be irreducible non constant polynomial okay so that is the condition for hyper surface in the commutative algebraic sense okay and X being a geometric hyper surface is what is the geometric meaning of being hyper surface the geometric meaning is via dimension so it means that dimension of X is actually one less than the dimension of the ambient space the bigger space so it is n-1 okay where n is the dimension of the largest space and of course here by dimension I mean the topological dimension alright and we saw last time that these two are equal and that was basically an application of both the Krull's how TDL such and also the important theorem that you know and that a noetherian integral domain is a U of D a unique factorization domain if and only if every ideal every prime ideal of height 1 is principle okay so you know for both these theorems we do need that the ring involved is noetherian okay Krull's principle ideal theorem says that if you take a noetherian ring and you take an element F in a noetherian ring which is not a unit and it is also not a zero device okay then every minimal prime ideal that contains this element has height 1 okay that is Krull's principle ideal theorem and so the geometric content is that you know while both these theorems I tell you that these two are actually equivalent okay and the other beautiful thing is that you can so you can ask the question of when the ring of functions are an affine variety is a unique factorization domain and the answer to that is that you know you take any affine variety and look at its ring of functions the condition that it is a unique factorization domain is equivalent to saying that every geometric hyper surface in that variety namely every co-dimension 1 irreducible subset is defined by a single equation okay that is the geometric translation of the fact that every prime ideal of height 1 is principle okay the prime ideal of height 1 condition tells you that the dimension of your locus goes down by 1 okay that is it has co-dimension 1 it goes down by 1 from the dimension of the bigger space in which you are considering it okay and so you know therefore the geometric content is that if you have a unique factorization domain I mean when are rings of for which affine varieties are the rings of functions unique factorization domains they are precisely the affine varieties for which every co-dimension 1 irreducible subset is defined by the locus of a single equation that is what that is a geometric content okay so this is one particular example so you know I still I want to say a little bit more going from this side to that side. So you see let me give you two examples one is the let me take the two plane okay so the two plane there are many avatars of the two plane so for example I can take well I can look at a2k this is a two plane okay and if I look at the corresponding you know ring of functions I will get the afa2k which is just k if you want x1, x2 okay it is a polynomial putting in two variables then I can also consider you know two plane as if you want the xy plane in three space that is also after all the two plane only that you are considering it in a bigger space. So you know for example you can take xy plane in a3k that means you know you are looking at three dimensional space k cross k cross k you are calling the coordinates as xyz if you want and then you are looking at the xy plane that is also the plane two plane anyway and if you look at the ring of functions on this you will get a of you know well let me let me write that a of xy plane in three space what am I going to get I am going to get the ring of functions on a reducible closed subset is defined to be the ring of functions of the ambient space which in this case is the affine three space so it is a of a3 okay modulo the ideal of that irreducible subset okay that is the definition. So it goes by modulo you have to go modulo the ideal of the xy plane and if you see you see what this will give you is you will get a of a3 is of course k xyz because we agreed to call the coordinates on this three plane as xyz okay and it is xy plane you are interested in and what are the functions that vanish on the xy plane it is the z coordinate where xy plane is simply written as z equal to 0 and so it is actually the ideal of xy plane the xy plane is just the ideal generated by z okay and so it is the polynomial ring k xyz modulo z which is actually isomorphic to k xy so and so you see if you watch these two are one and the same geometrically they are it is just the two plane here it is a two plane considered as the whole space and here is a two plane considered as the xy plane okay and see what is happening on this side what you get is these two are isomorphic you get that the ring of functions are isomorphic this is also polynomial ring in two variables that is also polynomial ring in two variables so they are isomorphic okay. So let me give you one more example in fact let me take more generally you can take some plane a more general plane in n space okay if you take a two plane in an okay and if you associate to it the ring of functions a of the two plane in n space okay what will happen is you will still find that it is still isomorphic to polynomial ring in two variables it will happen so what is it that I am trying to say I am trying to say that geometrically you are looking only at the two plane whether you are looking at the two plane completely the two plane itself or whether it is embedded as a as the as a plane in three space or whether it is embedded as a plane in n space no matter how you look at it this definition of the affine coordinate ring the ring of functions on an affine variety this definition gives you isomorphic rings it does not give you something different and the fact that you all the three cases give you up to isomorphism the same ring is an indication of the fact that you are looking at the same geometric object you are looking at only the plane okay. So the moral of the story is that the ring of functions okay is go easy it really captures the object that you are looking at it is not going to change just because it is not going to distinguish between a two plane in three space or a two plane in n space because the two plane is always a two plane there is nothing different geometrically about it the only thing is the way it is embedded right. So the moral of the story is this gives you one rational as to why you should study rings of functions the ring of functions that is a geometrically it is an intrinsic object when you take an affine variety okay intrinsic means up to isomorphism it depends only on the affine variety not on where you see it okay. So that is one of the justifications of studying rings of functions and affine varieties. So I should now tell you that you know there is another common name in the literature people also use the word co-ordinate affine co-ordinate ring okay so people instead of saying rings of functions and affine varieties people just use the word affine co-ordinate ring so this is the word that you will often see in the literature okay. So the moral of the story is that so you see so you can expect what does this isomorphism here correspond to here what it is telling you is you know you should expect that this should be isomorphic to this and this should be isomorphic to this and this isomorphism as varieties as affine varieties is precisely what is getting translated on that side as isomorphism of their affine co-ordinate rings that is what you should expect and that is exactly what happens in fact this association is actually an equance okay I will tell you how it is an equance how strong it is on this side if you put if you take only isomorphism classes of affine varieties okay then on this side you should put isomorphism classes of such rings affine co-ordinate rings okay then this is a bijective association because if you change the affine variety up to isomorphism then its affine co-ordinate ring will also change only up to isomorphism okay that is one point then the other point is if anything that you that happens on this side has a meaning on that side so a close sub variety corresponds to a quotient okay suppose I do not take the set of isomorphism classes of affine varieties on this side and there also I do not take the set of isomorphism classes of affine co-ordinate rings okay then if you give me a closed sub variety like this this closed sub variety will automatically correspond to a quotient so you see this closed something being a closed subset of the irreducible closed subset of another thing that will this is called a closed immersion okay the word use is a closed immersion some affine variety being thought of as a closed irreducible subset of some other affine variety okay this closed immersion reflects on this side as a quotient okay so quotients here correspond to closed immersion okay it does not stop there isomorphism here correspond to isomorphism there that is exactly what we saw here okay more what do open subsets correspond to okay the beautiful answer is open subsets here will also correspond to ring homomorphism here in fact there will be K-algebra homomorphism and you know what they will be localizations so beautiful thing is closed immersed will correspond to quotients open immersion namely the inclusion of an open inclusion of a variety a variety being sitting inside another bigger variety is an open subset okay and this is something that I will explain soon okay that will show up there as a localization of rings okay. So and this is a very beautiful equivalence of categories so I should only tell you one thing what is what are the objects here so what are so if I say they are just coordinate rings of affine affine coordinate rings of affine varieties that means I am giving a definition here which actually depends on this side but if I want to give a definition only on this side then the correct definition is these what you should put on this side is finitely generated K-algebras which are integral domains okay. So what you should put on this side okay so on this side so let me write that down somewhere here I am a little short of space but anyway let me write it so let me write it here finitely generated K-algebras which are integral domains so what does this mean this means they are algebras of this form K of some polynomial ring K of y1 etc ym modulo p where p is a prime ideal let me write it below so that I can modulo p where p is a prime and you know how it is going to go from here to here okay do not worry about this max spec for the moment but you know what you know the object on this side for which this is the affine coordinate ring you know what it is it is actually am okay whenever you write a quotient I told you it is the affine coordinate ring of the 0 locus of the denominator which has to be a prime ideal considered in the ambient space for which the functions are the numerator so the numerator here is a polynomial ring in m variables that is a ring of functions on am okay and p is a prime ideal there therefore it is 0 locus will be an irreducible closed subset of am and that therefore defines an closed sub variety of am okay m dimensional affine space over K and each ring of functions will be precisely this so you know this will correspond you know if I draw a arrow like this this will correspond to well let me write it here it will be some this will correspond to z of p as an irreducible closed subset of am that is what this corresponds to okay. So on this side what I am putting is so called finitely generated K algebra which are integral domains and the definition of that is actually it is simply a polynomial ring in finitely many variables over K modulo a prime ideal I mean this prime ideal comes because I want the quotient to be an integral domain so you have to go modulo a prime and finitely generated is because it should be quotient of a polynomial ring in finitely many variables because the yi's they generate the polynomial ring in m variables therefore their image is in the quotient namely the yi bars okay the cosets yi plus p which are the yi bars in the quotient ring they will generate the quotient ring as an algebra over K okay so that is why this is a finitely generated algebra over K which is an integral domain and conversely every such algebra is defined to be something like this okay. So what I am trying to say is we have an equivalence of actually categories okay we have an equivalence of categories on this side you have affine varieties and morphisms between affine varieties on that side you have these finitely generated K algebra which are integral domains and the morphisms are K algebra homomorphisms and it is an equivalence of categories okay I will explain all these things soon but there is one thing I should say at the outset I have been saying morphisms I said isomorphisms of varieties I said morphisms of varieties so that is something I have to define I have to define what is a morphism between one morphism from one variety to another okay so that is going to come very soon but I am saying that this is the general picture in which you get a complete equivalence of categories and in that sense everything that is that you see here is actually a mirror image of that and vice versa okay. So all this tells you that studying the affine coordinate ring the ring of function on affine variety is a very important thing because that captures everything okay whether you look at it as coordinate rings of affine varieties or whether you look at on this side the affine varieties it is one and the same because of the sequence. So this is why I am saying all this I am saying this is why you can say that studying rings of functions on affine varieties is important and it also kind of you know it also justifies in a way Felix Klein statement that you know the geometry of a space is controlled by the functions on the space okay. So you know somehow these functions they control the geometry mind you we had only affine n space which came from kn and all we normally know about kn is only that it is an n dimensional vector space over k but we forgot the vector space structure and the geometry came in by looking at polynomial functions and using them to define the Zariski topology. So you see the even the topology on this side was gotten by functions on that side you must understand and the beauty is the functions on this side completely capture the topology okay. In fact not only the topology the most deeper statement is that everything the functions on this side completely capture the geometry on this side. So geometry at the lowest level is the topology then the next level is more deeper properties like the manifold theoretic properties, smoothness, regularity and things like that okay. So everything on this side the geometric side is captured by the competitive algebra which is the ring of functions okay. So all this is to tell you why it is so important to study rings of functions okay. So I will elaborate on all this whatever I said very soon but now what we will do is I wanted to you know okay so let me tell you about this max spec thing so let me explain this max spec that is the arrow that goes from this side to that side alright. So you know so let me again repeat the following thing. I will show later on that if you have two affine varieties on this side and you have a morphism that will give rise to a morphism in the reverse direction of their coordinate rings and conversely. Conversely if you have two coordinate rings of two affine varieties and you have a k-algebra home morphism that will give rise to a morphism in the other direction okay that this will be a pukka equivalence and under this equivalence a closed subset will correspond to a quotient and open subset will correspond to a localization okay this is what will happen. You can see the morphisms are going in the this is an arrow reversing equivalence because you see from X to An if you have closed embedding okay then you have from the functions on An to functions on X this is just restriction of functions which is seen as a quotient okay. So this is what will happen and when if you have U affine variety U sitting as an open subset of An what will happen is from An to A of U you will get a home morphism which will be actually localization of this ring at a suitable multiplicative subset in the sense of commutative one. So everything on this side will translate to something on this side and conversely okay but all that needs to be proved we are going to prove it okay but I am just telling this to you in at the outset to tell you that why it is so important to study rings of functions on affine varieties okay which are of this form finitely generated K algebra which are integral domains. Now I will go one step further what I am trying to do is to go one step further and say that suppose I forget the picture on this side suppose I completely forget affine space I forget the points on affine space I completely forget everything. Suppose I have only the functions suppose I had then from the functions I can get back my space okay that is what this arrow from this direction to that direction is all about that is just from the functions I can cook up the space that means the whole space is already here in the functions okay and that is a much stronger proof of the veracity of the statement of Felix Klein that the geometry of the space is not only so you know what he said is geometry of a space is controlled by the functions on the space but more seriously if you give me the functions then I can give you the space I can cook up the space from the space of function from the if I know the functions on a space then I know the space I can consider the space from its functions okay it is a very very deep thing so that is what this max spec does and that is what I want to explain. So what I will do is I will start with something that you must have seen in a course in commutative algebra but nevertheless it is not very difficult to recall these things so you know you start with R a commutative ring with 1. So what you do is you look at spec R this is called the spectrum of R in fact it is called sometimes it is also called the prime more explicitly as the prime spectrum of R and this is supposed to be the set of all prime ideals in R okay so what you do is you look at the set of prime ideals of R okay and regard each prime ideal not as a prime ideal regard it as a point so this square bracket script P is supposed to tell you that you think of here you are thinking of the prime ideal P as a point in spec R when you think of it as a point in spec R you are not thinking of it as a prime ideal inside R you are thinking of it as a point on a space okay. So the spectrum is just the points which correspond to prime ideals okay and then so you should not confuse this with equivalence class because normally you know you put a square bracket around an element to say that it is equivalence class of that element so you should not confuse it with that there is no equivalence class it is just to distinguish between the prime ideal as a subset of R and the prime ideal as a point of spec R they are different okay so my space is spec R now the spec R will become a topological space okay how so there is a Zariski topology on this just like you have a Zariski topology on affine space you have a Zariski topology on spec R and in fact you can say both ways this is inspired by that and that is also inspired by this okay it depends on where you start they are equally one the same so let me define this Zariski topology on spec R is given by taking sets of the form Z of I okay so define to be equal to a set of all P in spec R such that P contains I where I in R is an ideal to be close sets okay so you see the Zariski topology is defined like the Zariski topology is defined as the to be it is defined by specifying a collection of close sets what are the close sets the close sets correspond they come out of ideals ideals of R you take an ideal I of R immediately you can write down the close set Z of I okay which you should think of as a 0 set of I okay if you compare with the Zariski topology on the affine space and the 0 set of I is all those prime ideals it is all those points which correspond to the for which the corresponding prime ideals contain I okay this is the way it is defined and you can check that this satisfies the condition for a topology namely sets of this form satisfy the axioms for close sets in a topology and you get thus spec R becomes a topological space okay and then you see you can also check you can also check that you know if you have if you give me a given a ring homomorphism F from R1 to R2 okay so R1 and R2 are commutative rings with one and F is a ring homomorphism from R1 and R2 then the map spec F going from spec R2 to spec R1 which is given by give me a prime ideal of R2 and simply send it to its inverse image which is a prime ideal of R1 okay. If you give me a ring homomorphism from R1 to R2 then it induces a map in the reverse direction on the spectra on the prime spectra and that is simply given by pulling back prime ideals because you know the inverse image of prime ideal in a ring homomorphism is again a prime ideal it is not true for maximal ideals but it is true for prime ideals inverse image of maximal ideal need not be maximal but inverse image of prime ideals always is a prime ideal okay. So what will happen is well the fact is you get this map that is not the point the point is this is now a map of topological spaces see it is a map between two topological spaces the most obvious thing we can expect is that it should be continuous and it is okay so this map is continuous so this is another fact that you can that you should have come across in a course in commutative algebra if you have not you can it is very easy to check okay and in fact what happens is if you look at so let me maybe I will put a star here I will put on the star here then here is so it is like a bulleted list so but it is of bullets I am using stars so here the third thing I want to say is that what about what about a close subset like this okay what about a close subset like this so the so what I want to say is that z of i so if you take consider r to r mod i the canonical so I am writing c a n for the canonical quotient map from r to r mod i okay then if you look at speck of that which goes from speck r mod i to speck r okay because you know I have already told you if you have a ring homomorphism from one ring to the other you are going to get a map on the spectra in the reverse direction in particular you take the ring homomorphism to be the canonical ring homomorphism that you get from a ring to its quotient then the corresponding map on spectra what will be this will actually identify speck r mod i with the close subset z of i in speck r so what will happen so identifies speck r mod i with z of i so mind you z of i is a subset of speck r by definition it is a close subset of speck r because of the risky topology I have defined on speck r and speck r mod i the image of speck r mod i under this is identified with that okay so in other words z of i is also spectrum offering the model of the story is that the subset close subset z of i also is a spectrum so every we define a close subset inside a spectrum and it turns out that every close subset itself is a spectrum can be identified with the spectrum what is the spectrum it is spectrum of the quotient by that ideal okay and in some sense you know you must think of it like this you must think of it think like this if r is thought of as the ring of functions on speck r okay abstractly then r mod i is a ring of functions on z of i which is identified with speck r mod i okay so you see r is a ring of functions on speck r alright functions in what sense is something that has to be made formal more precise but do not worry about it r is a ring of functions on speck r then what is the ring of functions on z i it is r mod i okay because speck of r mod i can be identified with z of i okay and there is one more thing you see what is the situation with respect to localization so two particular kinds of ring homomorphisms are quotients and localizations okay and so what happens with localization consider r to S inverse r l which is a localization of r with respect to a multiplicative subset S inside r so if you remember in commutative algebra even a ring r we define what is meant by a multiplicative subset a multiplicative subset is a subset which does not contain 0 and which for which which is close under multiplication so if there are two elements in that set then the product is also in that set okay and we can in commutative algebra form the localization of r with respect to the multiplicative subsets which is just inverting the elements of the multiplicative subset so S inverse r which is the localization of multiplication localization by the multiplicative subset sometimes it is also written as r bracket S inverse okay so it can be thought of as a polynomial it can be thought of as polynomial ring in r in as many variables given by reciprocal of elements of S modulo the natural relations that come because of the elements of S being inside r okay and that is the reason why one writes it writes S inverse r as r square bracket S inverse but essentially it means that you are just inverting S so you know the simplest case is if you take r to be for example if you take r to be integers and you take S to be the complement of the complement of 0 okay then S inverse r will just be rational numbers more generally if you take an r as an integral domain and S is the complement of 0 which is a prime ideal okay the complement of a prime ideal is always a multiplicative subset and localization by that is always going to give you localization and in this case complement of 0 if you localize the integral domain at 0 at the complement of 0 then you are going to invert that means you are inverting everything outside which is different from 0 and that means you are forming the field of fractions so forming the field of fractions is a very special case of localization localization is a generalization of that you have seen this in multiplicative algebra but the fact is that if you take this localization map okay then what does it correspond to in the spectrum at the level of the spectrum then spec l will go from spec S inverse r to spec r okay you will get this map. Let me take a particular case put for example S to be just the multiplicative subset given by a single element r r squared where r is in r is not nilpotent okay I do not I want r to be the element r in r small r in capital R to be nilpotent because I do not want some power of it to be 0 if some power of it is 0 then the multiplicative set contains 0 and I do not want 0 on the multiplicative set there are some books which say that 0 is allowed in a multiplicative set but then the convention is that the localization becomes 0 ring and nobody wants to study the 0 ring for as far as geometry is concerned so we always never want 0 to come inside the multiplicative set that is the reason why I do not want small r to be nilpotent okay and then in that case we normally write S inverse r as r localize at small r okay we also write it as r bracket small r inverse some people write it as r of 1 by r okay these are various notations okay and the fact is spec and spec of if you calculate spec of rr what you will get is you will simply get all those prime ideals the points corresponding to prime ideals such that the prime ideal does not contain small r okay and this so you will get this identification okay and this is just compliment of z of r in spec r so you see spec r has a Zariski topology as I have explained and then there is the ideal generated by small r okay it is ideal generated by single element r and you take the 0 set of that ideal what is what is the 0 set of the ideal by definition it is all those prime ideals which contain the ideal r but a prime ideal contains the ideal generated by r if and only if it contains r itself so this is z of r precisely has for its points those prime ideals which contain small r and what is the compliment it is all those prime ideals which do not contain points corresponding to prime ideals which do not contain small r and that is precisely spec rr capital r of small r and how do you get this identification this identification is provided by this map the image of this map the image of spec l in this case identifies spec of S inverse r with this subset which is an open subset of spec r okay because it is the compliment of a closed set. So the moral of the story is you know the ring of functions on the compliment of the locus defined by a single element okay namely the ring of functions on the locus where a single element does not vanish is precisely localization by that element okay which is if you think of small r you know if you think of small r as an element of capital r and if you think of r as the ring of functions on spec r then small r is a particular function and what is z of r you should think of z of r as the locus where r small r vanishes okay and what is its compliment is compliment is should be the locus where small r does not vanish but where small r does not vanish if small r is a function that does not vanish 1 by small r should also be a valid function. So it in other words it tells you that in this locus where small r does not vanish which is an open set the functions are simply given by r of 1 by small r namely its functions are of the form some function of some element of r divided by some power of small r which is intuitively correct and in fact you get it here okay so this identification is via l in this case. So the moral of the story is the ring of functions corresponding to a closed subset is given by an ideal is given by the quotient ring and the ring of functions corresponding to an open subset a so called basic open subset where a function does not vanish is simply given by inverting that function that is what it says okay. So this is the background from commutative algebra okay and we can use this to go from this side of the diagram to the other side of the diagram which I will continue with this in the next lecture okay.