a lot of fun, because we're asking the question, I should say, are you an influential nonprofit? And this is a big, big issue for so many of us. If we haven't met yet, I'm Julia Patrick, CEO of the American Nonprofit Academy. My intrepid co-host, Jared Ransom, is going to be off for a couple of days. We'll be rejoined by her next week. Again, we want to thank all of our presenting sponsors who are with us day in, day out as we move into year three of the non-profit show, which blows my mind to say that. But yes, we're moving in to year three, 450 plus episodes. I'm giving you a prom-com shake for that right now. Whoa, that's amazing. Thank you, Mary Ann. That's the best. And you know what? If you missed our live episode, you can see us on Roku, YouTube TV. I should just say YouTube, Amazon Fire TV, Vimeo, speak into your smart speaker and say the non-profit show, and it will freakishly come up. Okay, Mary Ann Dersh, you are back with us. Woo-hoo, CEO. I should say Chief Excitement Officer. That's why I sit so close to Palm Homs. Well, that's the way to be. That's definitely the way to be. Here's Mary Ann's communicating how to get all to Mary Ann. But okay, let's dive into it because we're talking about influence. And in the green room, you were giving us your idea of what influence means. So if you could start there and then talk to us about what the traits are. Yeah. So first, I'll say influence is like not, I feel like the word gets kind of a bad wrap lately.  And so then influence my definition and the definition that I work with is getting people you have no authority over to do what you want them to do. And so in the nonprofit space, we have so many people we have no authority over that we need them to do things. And easily it's the ability to easily enroll people in your vision. And a lot of times in our world, we kind of feel alone. You know, like we're carrying this organization on our backs and trudging through. But the traits of an influential nonprofit is that that is no longer the story that you feel that connected and the support of a community. And you feel that, you know, you are not in it alone that you have a army of support around you. You know, Marianne, that's a fascinating thing to frame it that way. And I would have never framed it up that way myself. But it seems to me that that isolation and that sense probably leads to burnout, it leads to job dissatisfaction. And think about when you are supported and engaged in that way of being like, yeah, we're in a team, we're going through this together. It allows you to move through tougher times. I mean, so it's a healthier thing. 100%. Like, and like the organizations that I work with, you know, during COVID, they felt the love from their board, they felt the love from their donors, they, they asked and God, you know, that's the thing about influences. When you have influence, you can easily ask for and receive what you want. And a lot of times, you know, with that asking, you know, asking for money, asking for support, like in that can feel really overwhelming. And so mastering the art of influence is being able to easily ask for what you want and then and have people respond. And you're right, you know, that, that I think isolation is a big issue for us, like for the community in general. And, you know, and the ability to then have that feel that support, like that is so like, like the quote I use all the time is, we don't give from the depths of the well, we give from the overflow. And a lot of people are giving from, that's a roomy quote, right? Like a lot of people are giving from the depths of the well and the influence when you have influence, you're giving from an overflow, you're giving from this energy rich place. Wow. Okay. I love this concept because it seems to me, Mary Ann, this is almost like a foundational approach to how we manage our nonprofits, how we look at our nonprofits, how we look at our relationships with staff, community donors, leadership and all that. But the bigger question is maybe how do we get there? How do we build this, this influence when we now understand it with a different lens, that it's not manipulative? Yes. Right. So that again, I'll start with what it's not, which is we don't convince, push, pitch, sell, that's not, that's not what we do. And we've been taught, this is how you get money, you have to pitch, you have to convince. And that actually causes people to get defensive and push back. And so the influence, let me just say this before I get into the how, it's sort of like one of the things I say a lot is people hate to be sold, but they love to buy. And what I mean is I didn't want to be sold because that's pressure. And then you get defensive. And when you get defensive, you go back here. And when you go back here into your instinct, into the base of your neck, the back of your brain, then your, your ability to have reasonable thoughts to take into information decreases. It diminishes exponentially. So we want to keep people open and open to new information. And that's by like, by buying, they're choosing, right? They're choosing every step. You know, and go ahead, what are you going to say? No, I love that you said that because I can put, I can see that with the donor. Writing the check might not be the best part or making the commitment, but then the engagement and the joy of, as they say, the joy of philanthropy, you know, that you have invested in that is where it's at. I've never heard anyone say it that way. Yeah, you're good for us. Yeah, you're welcome. And you're right. You know, people really don't talk about this in a sense of like, you talked about it being the foundation. And that's why I do this work is because what I've seen over so many years is like, how come some messaging and marketing work for some organizations and dinner for others, you know what I mean? It was all good stuff. But why would it stick in some places and that and others? And it was really around their internal communication. Because if you're not in turn and communicating effectively internally, you won't be able to do it externally. And I, and so my work started to move more and more towards internal communication. And, and, and my internal communication, how do you get better internal communication? You develop better leaders, leaders to yourself and to each other. So in the sense of being, how to be an influential nonprofit is really turning the lens to yourself. Right. And when I say turning lens to yourself, say, okay, I'm, I, our habit, I feel like this is what I've seen is to say, okay, if only my board would do this, if only my donors would do this, right. And then I would be happy or I would be successful when, when influence turns the lens to you and say, okay, how am I contributing to this? How can I, right? Because I, I can't expect other people to show up differently unless I'm willing to show up different. Right. I want to, I raise my game first, right. And influence for me, it's all about attraction, right. Like we're all master attractors. This is what I truly believe we're all master attractors, whatever we put out is exactly what we get back. And so you can't put the same thing out and expect a different thing back. So I just teach people really how to do it is we're willing to send out exactly what we want back. And that's how we're going to get it. So if we want, you know, support, we need to be supportive. Does that make sense? Is this? Yeah. And I think it's, it's for me, it's modeling. I think it's, it's like, and this is going to sound really crass, but it's like, you know, going to the doctor and the doctor says, you need to stop eating potato chips and stop smoking and stop drinking. And then you see them outside in the parking lot with a bag of potato chips, a cocktail and smoking, which is like, you know, that's extreme. But do you know how I'm hearing it? Is it, you got to walk the walk and, and to look internally and say, what does that mean? Maybe that's the starting point. That is, I can't believe you said that, that is so perfect because that is one of the things about alignment, right? Being in alignment means what you say and what you do are as close as possible. And when you see that doctor, right in the parking lot, like, don't be like it, and then they're doing it like, wait, come on. So the alignment is what I say and what I do is close as possible. And that builds trust and trust is the foundation of anything, including influence. And, you know, sometimes our organizations will say, you know, we support strong families. Well, like, yeah, our clients, but then we're working our staff so hard that they can't take care of their families. That's not that right. Like it's alignment is we've support strong families all across, not just these families. And that's what sometimes the culture can be crazy making because you're like, wait, why doesn't that sit well with me? Oh, okay, that we're out of alignment. Because what we say, what we do aren't as close as possible. So what can we do to put ourselves in into alignment? And we were talking earlier about like, like showing up with confidence. But that's how we create the confidence is like, I understand who I am as a person, as a leader, and as an organization and the value that I bring. That's the like, that's like the first step to influence. You know, I love it. And I think what you just said is a magical way to reframe this, because I think a lot, I think in the nonprofit sector, there's a lot of judgment, you know, and you could talk about that from the, you know, white savior approach to I'm going to help those people on the other side of the tracks to all these things. And I think you're right. I think that's also part of this great resignation where you do have staff, but they're burnt out and they're like, what the heck, you know, we talk about these mental health issues and behavioral issues, but it's only really for our clients and not for our teams. Yes. But I love that you brought that up. That's fascinating to me. And that's another aspect of how to build influence is like you were saying, like, like you were saying, being the model, right? Like, I'm the model for the change I seek. And I, in our culture, and especially in the nonprofit, we have to sacrifice ourselves to make other people happy. When really the opposite is true, we when the more that we care for ourselves and then and then we then we become the model and and again, because what you put out is what you get back. And so we become the model and we become the thing that people gravitate to and want to be around. And and then that becomes very powerful. And it's all about like the work I do, you know, is like that roomy quote, you know, we want to give from the overflow, not the depths of our well and the influence, right, that's you have that comes from that overflow.  I'm not saying this is not rainbows and uniforms. I'm just saying like, but like their approach and their culture is one of possibility and problem solving and boldness. And that's what we want, you know, like, Oh, yeah, I want that. Right, that's what I'm saying. Like, are the major law of our universe is like attracts like, you know, that's that's the end. So if if if we use that, okay, so that's influence like, what am I putting out there? Well, okay, so what else can I put out to get back what I really want? And that's that's the truth. And it's not about changing other people. It's starting with yourself. Wow. Okay, this is like, I'm just so intrigued by this because I love this conversation. I think it's the right time of the year to have this conversation. You know, Marianne, sometimes with a nonprofit show, we get these amazing gas and concepts that come through. But at some points, especially towards the end of the year, I know that with my team, I'll say, you know what, this is going to be a topic that frustrates people because they're at the end of the year. And they're trying to get much and it's it's going to be it's not going to be a positive, it's going to be a negative. But what I love your what you're saying is that this is the time to start that this is the time to start working on this and refining this mentality and this approach, so that as you we get into that heavy season and we're ending, you know, we're moving through the quarters of the year, that we're more we're armed, you know, we're better for it. Let me say something that's going to blow your mind. Okay. I'm going to get I'm going to get like, hold on. Okay. So time is a really an illusion and all time lives in our head. The past and the future live in our head. Really the only thing the only thing we have is this present moment. And anytime we think of the past, like, you know, or worry for the future, we're in our head and we're not fully present. So really, all we have is this moment. So I look at it like this, if you want tomorrow, which never really comes, if you want tomorrow to be different, then you do something now. Right. Okay. That makes sense. Like, oh, and next month, I'm going to get I'm going to do this. Well, next month, right, you know, like, but okay, so what, like, like just really focusing in on this is like, there's no perfect time to make stuff better. Right. Like, oh, when we're going to like get all this organized, and then we're like, it's just like, this is all we have this moment. And so anytime I'm like, I'm like, ruminate like past regrets or future worries, that means I'm in my head and I'm not present. Okay, I needed to hear that.  I needed to hear that for many a reason. And I think you're right. I think you're absolutely right, especially in our sector. Yeah, I think it's really an important thing. Now, I want to ask you this next question, because our time is blown by and you've already given me so much to think about. But how do we know when we're successful with this concept?  I like, and I know we're like overly cautious or or centered on measurements and everything. But how do we intuitively maybe or we are we can actually measure this out of way? So I, you know, for me, it's, it's really about the mindset, either a scarcity or an abundant mindset. All of these concepts I'm talking to you about are rooted in abundance, that they're the infinite, the universe is infinite. And there is infinite amount of resources, and there's infinite amount of time, because like we see time is linear, and that's not really how time works.  Yes, there's difficult times, and we know we can find a way through. That's that because we know that there's resources out there. And so much of our I feel like our fundraising gets hard, because we're afraid to release people that aren't the right fit, because we're afraid we won't find somebody else. But if I'm rooted in abundance, that means I can let you go because I know there's somebody better for me. And so like, so for me, that is the most basic shift is one from a scarcity mindset to an abundance, or, you know, like a scarcity fixed mindset, nothing will ever change, blah, blah, blah, to a growth mindset, an abundant mindset. And like, we may not exactly know how we're going to do this, but we know we can get it done. We just always, you know, and that that is that's when the possibilities open up and then the solutions kind of flow to you. And for me, I think the biggest thing is, and that's this is the thing you can measure is, is, is when the money, the people, the energy starts to flow to you, right, like, like people are coming to you saying, I want to be a part of you, tell me what's going on, that's when you know it starts working, it takes a little bit to get there. You know what I mean? There's some, it's a process and it takes some consistency and it takes a lot to turn the yacht around, you know, this isn't something that's going to happen. You know, you can start shifting and then, you know, and then the organization will slowly shift with you.  I'm in 100%. It's really, it's an interesting thing. I learned this great, this phrase from that great Terri Axelrod, the woman who started raising more money. And she said, I remember learning this from her almost 20 years ago, you know, the concept of bless and release. If something doesn't work or it's not working for you, it's okay to say, all right, this isn't working. And so we're moving on. And that then you free yourself from. Yes. I just did a webinar yesterday and one of the things I taught them was like the 10 words to never, never tolerate again is let me think about it and get back to you. Because you've said it and I've said it and we don't mean it. And they don't mean it, but we're saying it to be polite because we don't want to say no or we don't want to feel rejected. And that's right. Like, so the bless and release, if I can go, you know what, hey, whatever you decide is totally okay. Because, yeah, and I can hold space where I would really love this gift. And then I can hold space for and if it's not the right fit, I'm okay. So there's, there's this, I think there's this for influence for me, it's this underlying value of trust, right? Like just trust that that that your donors will show trust that you're, you are supported trust that you have value. I love it. I think this has just been amazing. I want to talk about a few more things and our time is drawing close. I mean, it's shocking how fast this conversation has gone. First and foremost, here is Marianne's information. She's got a great website and you can get a lot of different information. And I think it's just so positive and supportive and giving for at least for me, Marianne, it, it almost articulated things that I kind of know intuitively, but I needed like a reframe or just like an injection of, yeah, this is the mentality. This is where I need to be. So check out MarianneDurch.com. I want to also make sure that everybody knows you have a podcast, the influential nonprofit, and you can find those on that site as well. So it seems to me, this is not a one and done Marianne, right? I mean, this is like a coaching kind of like reframe of our approach. And so I love that you have these tools that we can get back to because this is not a conversation that everybody's having. I know that's, that's, this is why I, that's what I saw the gap in the market was we were trying to solve mindset and communication issues with strategy, right? Like my board doesn't trust each other. Okay, let's get a plan. No, like, no, we need to build a trust. It's okay if they don't trust each other, like we can, but let's work on that and then we'll create a plan and we want to kind of avoid the drama or avoid difficult conversations. And that was, that's a high price to pay. And people just don't have the tools. And that's where I really saw the need was, you know, was to help people navigate and become better leaders of themselves and take more ownership of their own outcomes instead of looking outside for everyone else to solve it. I love it. And I think that's super magical. One other thing I want to dig into is this is such an interesting thing, the donor attraction forum. It's your second annual, it's coming up later in the month. Talk to us about that and share with us what this, how this looks and how it works. Yeah, so this is a two day event. I have gathered so many amazing people to talk about donor, attracting donors the different facets of it. So from using advocacy to attract donors to copy, how to, how to craft your copy and messaging, how you, how to use video to attract donors, you know, like, and I assembled out using email, like how to better because I think that's a big thing that nonprofits and that's a, that's an inexpensive tool that we could use so much more effectively to attract donors. So how do you, how do you bring people into your organization? And we're really going to cover a bunch of stuff. And I have just some wonderful speakers to just talk about different ways to attract donors. So the whole, the whole schedule is on there. It is a free to register. So there's no just go ahead and register, you can pop in. There's a VIP thing you can buy on the other side if you want to get the recordings, but it's totally free for you to pop in. I'll be speaking more in depth around influence and growing your influence as well. Like some of the concepts that I shared with you, I'm going to go in a little bit deeper, especially like around managing rejection and objections, you know, navigating those kinds of things, because one of the biggest things of influence is being able to ask for and receive what you want and need. And what keeps us from asking is because we're afraid we're going to hear no. So how do we navigate that? So I'm going to be talking about that. And it's just, I host it. So I'm there. It's not like, it's like you and I talking, right? Like, you know, and so I'm hosting it the whole time. So and I ask questions and you can ask questions. And it's just such a really, really cool experience. Great. And it's donorattractionforum.com slash Marianne. Yeah, look it out. I think it's a really great how you had a huge number for your first one, didn't you? Yeah, like we had 800 people registered. Our goal is 1500. We have almost 500 now. So and what, you know, we have Amy Fazio, Bethany Snyder, Michael Hoffman, you know, like all of these people that are experts in attracting donors in different areas. And I'm really excited about this. And I'll just share like this VIP package, which is which you can purchase. But after the event, I do a workshop with anybody who gets that. And we I teach you how to figure out what ideas will work best for your organization. So like, you can just take all of this in and then be like, okay, because sometimes we go to things we're like, Oh, I need to implement that. I need to implement that. And you're like, and then none of it happens. Right, right. Well, it becomes frustrating.  And to your very point, you have to articulate an influence. Yes, people that weren't there to buy. And that's not like selling Oh, I hate with this really good idea. It's like, it's really about listening to their needs. And then whatever your idea is is going to meet their needs, which is different than how we typically learn to get our ideas across. Right, right. Oh, my gosh, well, this is going to be great. I'm going to be there watching and participating in that, that, you know, looking for new ideas. And I think, too, a big part of this courageous communication is that we need to be supported or reminded, I mean, you know, touch base with some of the things that we know, but we forget, or we lose our way. And so I think something like this is just so powerful to help you get on track, maybe even start a new track. So I'm going to be really excited to see how people respond and what what the feeling is, because we need to be doing better. And we can be doing better, but we need the tools. And I think this is one of them. So Marianne, thank you so much for doing this for our community. Thank you so much. Oh my gosh, it's been a lot of fun. Again, I'm Julia Patrick, CEO of the American Nonprofit Academy. Sherry Ransom will be back with us next week. She's enjoying some time with her family in the South. So how cool is that? Again, we want to thank all of our presenting sponsors. Without you, we would not be here having these great conversations as we have today with Marianne Durge. It's been a lot of fun, Marianne. And I suspect we're going to have you back on, because I love your energy. Oh, thank you. And I love the things that you say. I think it's really powerful. And it might be fun to have you back on in a bit to find out what your take is and the pulse of what you're seeing with these Q1 and Q2. Because it's a different time, but it's going to be the same time. We're going to be having to deal with things that we're dealing with. So best of luck. You're going to rock it by the end. Thank you. Oh, this is the end of the month. This was the fastest 30 minutes ever. I know. I know. Believe me, it goes fast for me, too. It really, really does. Hey, everybody, check out MarianneDurch.com. I think you're going to find a lot of really cool things we were talking about. And then, you know, see about finding some time to pop in to that two-day workshop. I think it's going to be a great value. And again, it's free. So that's when you register, you'll get the lineup so you can say, you know, oh, I really want to hear this person or this person. So you'll, you'll, you'll can figure out the timing and you can, yeah, pop in for a little or all, whatever you want to do. I love it. I love it. I love it. Well, hey everybody, we end every episode with this little mantra. Stay well so you can do well. We'll see you back here tomorrow. Thank you, Marianne. Thank you.


Hungry Jen online school will be for a purpose of helping people to discover, develop and be deployed into God's harvest field. We will be having them on Monday nights. You will be able to join us online as we are teaching our local members who will become ministers. You will be able to be a part of that. We will have a team that will be directly working with you, helping you to go through this training. You will be able to do it from the comfort of your home. You will also be able to catch up with school work as you are maybe perhaps not able to make it live. It will have three main tracks. The first track will be life class. It will be about seven weeks. And then after that there will be the destiny training which will be about 12 weeks. And then after that is the ministry school where we will go deeper into the practices of ministry and that will be about 12 months. There is a cost attached to each track. The reason we don't want to start with the ministry school right away is because our desire is to lay a foundation of proper disciplines and pattern of thinking before we start teaching people on the equipping parts of what to do in ministry. This school, all of this, these three tracks are for everyone.  Our goal is not just to fill you with information, you want to fill you with revelation and give you also practical application where you can live this out in your own life. So sign up today and we will reach out to you and give you next steps you can take so we can see you in our online training program.


Hi everyone, Joe from jaspyscasebreaks.com coming at you with that 2019-2020 Panini Contenders Optic Basketball 2 box pick your team number 28 with the Pelicans Randomized to someone in this break if you bought at least two teams. I think after 28 break I think most people know the drill by now big thanks to these folks right here for getting into the action appreciate it So now let's sort by your first names Right here Matt. You've got four teams two entries Matthew shiro Mookie bets yard Nick Stover with six teams three entries Oliver with two teams. That's an entry rodrigo two teams one entry Ryan shackle did five to well six four count still still two entries still two entries And zoom in with eight teams. That's four entries Let's copy those names in there So the name on top after three times pelicans one two And three It's matt Lieber. There you go matt After three times you get the Pella cans Not the Pella cans Pella cans Well, let's see what the Pella cans can do with a new coach Got a new coach out there They've got why am I blinking on the coach? Oh, yeah, right stan van Gundy That could be interesting. How long has he been out of coaching? I feel like he's still been doing a lot of announcing following the game He's not like he's doesn't seem like he's out of touch right think he gets the Wasn't out of hoops for too long I don't think his his piston stint is indicative of his coaching talent It might be good an old voice good for a young team A couple good drafts, maybe a free agent or two Zion losing a little bit of lbs is If you get really interesting What is that? Is that the is there two base cards down there than the autograph? I think so, all right All right, there's Robert Parrish winning ticket For the Celtics matt Where does chris paul end up? I feel like he's going to get moved from oklahoma city This goes to zoom in and okc. I heard people talk that he could be an interesting force In milwaukee, I think milwaukee can somehow make that move There's vince carter season today. I mean if chris paul can a healthy chris paul, especially in the playoffs Can do what he did with that young okc team imagine what he can do with be honest Timberwolves with this one zoom in With the carlithian towns and behind gordon hayward I've heard some talk about gordon hayward. Maybe getting dealt We've got nicola meli Palakins matt lever. We're looking for his teammate. Sorry nicola nicolo nicola meli And gordon hayward Celtics matt lever. All right next box It'll be a lot of interesting changes in the uh In the mba as always a lot of fun player movement possibilities Autograph on the bottom, I think so Bradley beal I think mba gmming must be the funnest of the gm jobs Your zack levine It's not like if you be an nfl gm.  Yeah, there's a lot a lot of players. You got a draft Baseball is kind of a long draft too And as a gm you're trying to put together minor league teams and minor league contracts and A lot of layers there. That doesn't sound like fun It was honest, but mba gmming I guess that sounds like fun. All right and behind anthony davis champion Is new york nix Alfred payton mba ink that goes to ryan shackleton And the new york knickerbockers All right, so nothing too crazy here ryan. That's 13 out of 125 nothing too crazy here, but We've got another two boxes in the store right now that we can run back on jazby's casebreaks.com So check that out And i'll see you next time for the next one of these jazby's casebreaks.com. Bye. Bye


Good morning everyone. I am Dr. Ashwati Sunil from Osmania Medical College, Hyderabad. My topic for the day is a role of MRA in the evaluation of cellar and supracellar lesions. It means to use magnetic resonance imaging as a modality to identify various lesions that arise from cellar tersica and supracellar lesion with histopathological correlation wherever necessary to document the incidence of optic nerve and cavernous sinus involvement. Introduction. The cellar and supracellar lesion is an anatomically complex area where a number of neoclastic, infectious, inflammatory, developmental and vascular pathologies can occur. Magnetic resonance imaging has superseded other imaging techniques as a modality of choice for the evaluation of cellar and supracellar lesions. Methodology. Study design and place of study. A prospective study was carried out in the department of neurodiagnosis and imaging at Osmania General Hospital, Hyderabad. Study subjects. 35 patients age ranging from 5 to 65 years with suspicion of cellar and supracellar lesions were referred from department of neurosurgery Osmania General Hospital, Hyderabad. Duration of study one year. MRA was performed for all patients using GE 1.5 Tesla scanner. Patients were placed in supine position in the MRA table and typical MRA protocols were done. Inclusion criteria. All patients with clinical suspicions of cellar or supracellar lesions. Exclusion criteria. Patients with already treated cellar or supracellar lesions. And patients con with contraindications to MRA. Coming to the cases. Axel and flare image is showing hyper intense lesion, which is showing intense homogenous enhancement post-contrast study. And there is associated widening of the cellar. Deletion on coronal post-contrast D1 weighted image. There is infiltration of bilateral cavernous sinus with complete encasement of bilateral internal carotid artery. With associated no narrowing of internal carotid artery. Deletion is also involving bilateral optic nerves. This is a pituitary macrivedinoma. Axel T2 weighted image showing T2 iso intense lesion with few hyper intense area. T1 weighted image shows multiple hyper intense area within the lesion and shows blooming on GRE. On post-contrast study, heterogeneous enhancement of the lesion is seen. This is a case of pituitary macrivedinoma with apoplexy. T1, T2 and flare hyper intense lesion which is predominantly cystic and occupying the supracellar region and causing compression of the third ventricle. On post-contrast, heterogeneous enhancement is seen. No restricted diffusion is seen on DWI. This is a case of craniofaryngeoma. A T1 iso intense T2 iso to hyper intense lesion seen in the supracellar region with no restricted diffusion. On post-contrast, homogenous enhancement is seen. This is a case in supracellar meningema. T1 hypo intense T2 hyper intense lesion suppressing on flare with restricted diffusion on DWI. This is a case of epidermal cyst. T1 hyper intense lesion in the supracellar region which is heterogeneously hyper intense on T2 showing blooming on GRE and no restricted diffusion. A few T1 hyper intense 4K are seen in the circle spaces in high right frontal parietal region. This is a case of ruptured supracellar epidermal cyst. A T1 T2 iso intense thickening of the pituitary stalk which on post-contrast shows intense enhancement. This is a case of lymphocytic hypophycitis. T2 hyper intense lesion is seen in the supracellar region which is involving the pituitary stalk and left optic nerve. On post-contrast study, remenhancement is seen. Multiple remenhancing lesions are noted in visualized brain parenchyma. This is a case of supracellar tuberculoma. A well-defined pedunculated mass arising from the tuberocytarium which is projecting into the supracellar system. And the lesion is iso intense to gray matter on T1 and T2 weighted imaging with no restricted diffusion. This is a case of hypothalamic hamartoma. Results. Out of the 35 cases evaluated, 41% were pituitary adenoma and 23% were craniofaryngeoma. We also got two meningioma, two epidermoid, two partial mpcellar, one radkake left cyst, one ruptured demoid cyst, one lymphocytic hypophycitis, one hypothalamic hamartoma, one supracellar tuberculoma and one isianurysum. Age distribution, 25% of the cellular supracellar lesions were found in 36 to 45 year age group with pituitary macrodenoma being the most common pathology. A bimodalase distribution was found in craniofaryngeoma, which included 5 to 15 years and 55 to 65 years. And it was commonest in 5 to 15 years age group. Sex distribution. Female predominance was seen constituting 62.8% of cases and pituitary adenoma was the most common lesion in both males and females, followed by craniofaryngeoma. Optic nerve compression was seen in 54% of cases, 42% of the optic nerve compression were due to pituitary adenoma and 32% were due to craniofaryngeoma. Bilateral optic nerve involvement, which is given by grey, predominated, followed by a right sided involvement, which is given by blue, with only one case of left sided nerve compression, which was seen in supracellar tuberculoma. Kevin Asain's invasion was seen in 10 cases, 90% of which were due to pituitary macrodenoma. 45% of pituitary macrodenoma had bilateral involvement, 33% had a right sided involvement and 22% had only left sided involvement. Histopathological diagnosis was available for 28 cases and MRI could accurately diagnose 96.4% cases. One case reported as subapendymoma, third ventricle, turned out to be craniofaryngeoma. Conclusion, pituitary macrodenomas are the most common cellular supracellar lesion. Most commonly affected age group is 35 to 45 years and most commonly affected genderist female. Craniofaryngeoma, most commonly affects 5 to 15 year age group. Optic nerve involvement and Kevin Asain's invasion are common in cellular supracellar lesions, most commonly by pituitary macrodenoma. MRIs accurately diagnose most cellular supracellar lesions. These are my references. Thank you.


into it. So, what do we remember from here? Indian English is different if we look closely at it. Indian English is different from English in India. It was not that the use or the introduction of English in India came about only in the 18th century or in the 19th century. Henrik Mario Ninh in his book English in India points to the date 1600. In fact, the last day of you know 1600 and says that the granting you know by the crown of England of the charter to the merchants to deal with the ease and eventually the ease India company. It really should be the correct date to talk about the beginnings of English in India or the English language in India. So, 1600 is the date that is set by Mario Ninh and many scholars if at all we are to point to you know the coming so to speak of English in India. So, it came with once the charter was granted. So, it came with the company. Now, let us go back again to Braj Kachru and let us you know look at how he talks about the four stages of the language in India and he says that the four stages are these first. When we talk about how English came into India, we have to refer to the missionary efforts. We need to refer to the Christian missionaries right who on the evangelical mission also used you know also brought the Christian ideals and the Christian texts. So, the first stage Kachru says is the stage of exploration and this stage of exploration was largely by the missionary and their efforts. It is also you know it is also followed the missionary effort was also followed by the missionaries themselves to learn to learn the native languages. It is an error to think that the Christian missionaries only brought the you know they sought to spread the English language here. After a certain time they realized that if in their zeal in the evangelistic zeal if they did not learn the native languages themselves then it would be after a point of time when they are impossible to spread Christianity. I can give you an example from Assam for instance. It is said that the first Assamist dictionary was by Miles Bronson who was a Baptist missionary who came to spread Christianity in Assam. So, the very first dictionary of the Assamist language in the North Eastern part of India was not by a native was in fact by you know by Miles Bronson and his colleagues who very successfully brought out the Assam you know the first Assamist dictionary. So, it is really as I said the exploratory part as professor Kachru says exploratory part must be you know it must be realized that the Christian missionaries were responsible for bringing in English and later on to switching source as it were to the native languages. Then Kachru says that the next phase is the phase of implementation. In the phase of implementation he famously refers to Macaulay's minute on education. Macaulay's minute on education that was submitted to the government was you know is considered a famous or an infamous if you will you know minute on education where he laid out the policies and the strategies behind the need for English education you know to cultivate as he calls you know group of people who would be Indian in colour, but in taste they were to be raised as as raised as Englishmen. The idea here was to introduce the English language as well as inter-education as well as create a class of you know people who were called the babus for instance who would play important part in at the clerical clerical level in British administration. So they would then need to be trained in English language and in certain values of Christianity and western life. So the third phase as mentioned by Professor Kachru is a phase of diffusion where he says that the British control was paramount and the diffusion of English language grew to a great extent during this time and finally is the implementation. The implementation of all the educational strategies and policies that were had the hitherto being forwarded by various you know by various bureaucrats in the British government. So what then are the four stages, the four stages are A the exploratory stage begin with beginning with the missionaries then was the phase of implementation you know coming with Macaulay's famous minute of 1835 followed by the diffusion phase of diffusion of British control and finally of implementation of their policies. So this is how the growth beginning and growth of English in India is to be seen along with socio political and socio political and evangelist developments. Now a very important scholar here in you know in the delineation of English studies really of you know the political imperatives behind English in India has been a scholar named Gauri Vishwanathan. Now some of you may be aware or you know they may be familiar to you particularly her doctoral dissertation I think which went on to be published as masks of conquest. So Gauri Vishwanathan's book is one that I would recommend to you to be read if you really have to understand the policies and strategies behind you know the introduction in this case of not just the English language but of the literary study in India. Now you would be surprised to know if you do not know already that the study of literature as a discipline was or the study of English literature as a discipline was not something that came or was not something that began in England. The study of rhetoric was very important at the university of say Cambridge for instance. The study of philology, the study of language, the study of rhetoric these were the disciplines that were being taught. Surprisingly English literature as a discipline was introduced in India first before it was introduced in England. So when we raise this question what was the need, what is this anomaly here that the literature of a certain country is first taught you know or first forwarded as part of the curriculum in a colonized country. So Vishwanathan very beautifully very successfully brings out what was behind the scenes or what the policies were of these and I am now going to quickly read from masks of conquest and the theoretical you know impetus comes from the concept of hegemony that was given to us by the Italian Marxist scholar Antonio Gramski. Hegemony is according to Gramski not something not something that is necessarily imposed on us. Hegemony is something that we can the person who on whom it is imposed may also very willingly adhere to. So she says here cultural domination works by consent. Now this is the important point. So domination is usually understood as an act of power and act of coercion. So following Gramski Vishwanathan says that cultural domination does not have to be you know only one of power and act of power. Cultural domination works by consent and can and often does precede conquest by force. Power operating concurrently at two clearly distinguishable levels produces a situation and she refers to Gramski here where Gramski writes the supremacy of a social group manifests itself in two ways as domination and as intellectual and moral leadership. So they are two sides to hegemony. One is the usually understood sense of domination but also in the guise of leadership of intellectual and particularly moral you know leadership this hegemony may in fact be consented to by those who are dominated because it is seen as something that is desirable. In fact because it is a desire sometimes being created by those in power as something that is to be desired by the population. So let me again read this. Cultural domination works by consent and can and often does precede conquest by force. Power operating concurrently at two distinguishable two clearly distinguishable levels produces a situation where Gramski writes the supremacy of a social group manifests itself in two ways as domination and as intellectual and moral leadership. It seems clear that there can and indeed must be hegemonic activity even before the rise to power because there has to be a level of consent rise to power and that one should not count only on the material force which power gives in order to exercise and effective leadership. So this is precisely what according to Gauri Vishwanathan happened in India. It was not an act of power but it was also a leadership projected as something that would give one intellectual and moral upliftment. So therefore English was used to as a tool or English literary study was used as a tool so as Vishwanathan says to mask colonial exploitation. So in the name of providing a quote unquote better education to the native population in India what was going on was actually colonial exploitation and the second point is in the name of stability. So the two one is giving you enlightenment and the second is creating a veneer so to speak of stability and not just creating a veneer of stability I should say also to ensure administrative stability. So these were the two things one masking according to Gauri Vishwanathan masking colonial exploitation and the other is to provide administrative stability in India for which the English language and English literature in particular was seen as highly desirable. Do you follow? Next if you look at the words of McColley and I had referred to the strategies of behind the introduction of English in India English studies. McColley says that we need to create a class this is from his famous minute we need to create a class who may be that is a class from Indians we need to create a class who may be interpreters between us and those whom we govern a class of persons says Indians in blood and colour but English in taste in opinion in morals and in intellect. So it was a very clear agenda really that there would be a class created from educated Indians who would be the go between so to speak they would be interpreters between the British government and the native Indian they would be interpreters and even though they were Indian in blood they would be really be Sahibs or the Brown Sahibs as you know they were called who would whose taste would be formed whose intellectual moral whose intellectual moral taste would be so formed. So that they are attuned to English values and English culture. So this as Vishwanathan and several other scholars have pointed out that this was a deliberate attempt and a deliberate strategy in the history of English in India. As scholars have pointed out they were really you know you could clearly see two sages you could clearly see two sages in you know in the subject matter so to speak of English in India. Now one period is from the from 1830 to 1850 and the impetus here was on clearly on Christian imagery. This was the time when as we noticed the missionaries were encouraged. So the impetus was on teaching Christian imagery through Christian texts and a general English you know to sort of to celebrate English ideals and English standards. So this was the first phase of you know first is behind introducing English as a language and English literary studies in India that was of you know trying to show the Christian religion as a religion that was better than you know there are so many tracks really if you look at you know that were derived that would derived Hinduism call it a superstitious you know superstitious religion for instance and Hindus as steeped in superstition. So in the first phase we found that Christian imagery was used a lot and English ideas in general. But what happened was from 1950 onwards was realized that the missionaries you know and the proselytizing you know done by the missionaries were being resisted by the Indian population in many pockets and it was seen as not really no more desirable thing to do. In many books in for instance in many history books on the history of missionaries in northeast then part of India you will also find you know when you look at the exchanges of documents or official documents between the missionaries under British government you will be surprised to find that you know there are a number of times they have been you know they have been almost at loggerheads with one another. One trying to say that it was important to proselytize important to convert the population into Christianity and the other putting a stop to it and saying that it was inimical or sometimes at odds with government policy. So then from 1850 onwards we see that impetus is more secular than Christian. So the stance here in of English in India was by the British government was secular in nature and where was the focus? The focus was on commerce. The focus was on commerce, commercial and administrative goals. So movement away from Christian ideals of English ideals of texts on to do with Christianity and Christian imagery to a desire to educate the Indians in matters commercial and administrative. So this is a clear policy change that we find in the history of English in India referring again quickly to Macaulay's you know how many of the British saw their literature as being you know far above the literature of India for instance. Macaulay for instance is known to have famously known to have said that and I quote him I have never found one amongst them who could deny that a single shelf a single shelf of European library was worth the whole native literature of Asia and Arabia. So it was again not really it was not not just by matter of strategy and policy it was also sought to be so by the British themselves and it is also you know part of you know their goal to show to the Indians that the books or the literatures, the cultural ideals of the British were far better and superior than those of India. Therefore if we may say that English was introduced in India as a tool first of civilization, a tool of light, a tool of enlightenment. English was used as a medium for understanding science and technology and also the industrial revolution's technological impact was also one of the points that we cannot ignore here. So the entire if you look at it the whole picture that was created was as you said India was not civilized and hence had to be civilized with a first Christian and then later on you know science and technology ideals of science and technology and the fruits of the industrial revolution. So we also have here following the same same idea some a few words by Charles Grant for instance who said that the true curse of darkness is the introduction of light.  The communication of our light and knowledge to them would prove the best remedy for their disorders. So these are some of as I said may call a Charles Grant etcetera these are some of the ways in which they had expressed this point. In Sri Lanka then and Ceylon also in 1827 we find Sir Edward Barnes who laid the foundation of a Christian institution saying this to give a superior education to a number of young persons who from their ability piety and good conduct were likely to prove fit persons in communicating a knowledge of Christianity to their countrymen. And if you look at other you know the history of other colonized places and this a similar kind of history would definitely also emanate again the history of enlightenment, history of you know you say commercial and administrative gains for instance you know behind the spread of a language. Now the English reading from Gauri Vishwanathan, the English education of 1835 proposed by Governor Gen. William Bentink on Macaulay's advice made English this is very important 1835 a date that we had mentioned you know go with the implementation phase was that the English education member this the English Education Act made English the medium of instruction in Indian education. This is 1835 an important date here with the formal institutionalization of English as a language of instruction the stage was set for a new direction to Indian education. But as the next chapter that is in Vishwanathan's book will elaborate Bentink's resolution was not as revolutionary in the introduction of a new language the English language was already being taught in India even before 1835 as an endorsing a new function. So, it was new in the sense that it endorsed a new function and purpose for English instruction in the dissemination of moral and religious values. So, this is one of the points that has been highlighted by the author. Further again she says the history of education in British India shows that certain humanistic functions traditionally associated with literature for example, the shaping of character or the development of the aesthetic sense or the disciplines of ethical thinking were considered essential to the processes of socio political control by the guardians of the same tradition. So, along with the fact that language had to be used language had to be interpreted by the local population there were these other ideas as mentioned here clearly which is it was shown as if English and English education would help the native to shape character. If one was seen as or one was told as one was superstitious one was in darkness and in need of enlightenment also showed that this the portraiture was such that you know English and English education would help you to shape your character your personality. Also then it help you develop the aesthetic sense they help you understand it help you understand the beautiful the sublime and the good for instance further the disciplines of ethical thinking that it also make you an ethical person. So, if this was the projection it was quite readily taken in by many from the educated middle class. So, English was sought to be as even it is today English was sought to be sort of the passport to higher development of oneself and higher moral and aesthetic you know ideal than one should one should aim at. Then she says the tension between increasing involvement in Indian education and enforced non interference in religion was productively resolved through the introduction of English literature. Significantly the direction to the solution was present in the charter itself one of whose section empower the governor general in council to direct the sum of not less than one lakh of rupees shall be annually applied to the revival and improvement this is the revival and improvement of the teaching of English literature in India and to the encouragement of the learned natives of India. Those of you who are a literal studies would also know that one of one of the opinions of writers is that literature is one a literary text is one which you read not just for pleasure. Good literary text is supposed to also give you certain values and through the through the certain positive characters characteristics for instance of say the protagonists for instance or certain undesirable traits in for instance again the certain undesirable qualities in the villain for instance. So, these literary text are not just text from which you learn the language or which you just enjoy reading. You also inculcate and many studies in cognitive studies of literary perception also do reveal that a reader is sometimes finds herself emulating the characteristics. So, what better way to sort of tutor the Indian mind in the certain ideals than to introduce English literary text. This is one of the reasons as we mentioned earlier why English literature as a discipline came in the colonies first rather than in the mother country. Then Vishwanathan says increasingly there was less patience with the policy of conciliation. The initial wave of euphoria over the literary treasures of India rapturously described as so new so fresh so original. So, unlike all the antiquated types and models of the west with which the mind was aroused and enraptured had by the 1820s given way to caustic criticisms of a systems of learning. Minters minute this is this is talking about an earlier history minute of 1813 favoring the revival of oriental learning was harshly criticized for not making the slightest effort to introduce in whole apart by implantation or engraftment the improved literature and science of Europe embodying as these do all that is magnificent in discovery ennobling and in truth and elevating in sentiment. So, it seems that when like in many cases when the British first came in contact with the literatures of the orient say literature in India they obviously, they found such literature exotic they found the many differences in ideals the many differences in the way of expression as something that was very appealing at first. So, this very appeal of the Indian text to the to the British was later on discouraged and these texts were shown to be inferior to western text not perhaps only because they really saw it as inferior, but also because it was at odds with the policy of introducing English. So, we I will end this lecture by again coming back to professor Braj Kachru and referring back again to our lecture on the alchemy of English. If this has been a history the history of English in India has been a history of political management of hegemony as we saw through Gauri Vishwanathan of hegemonic impulses not by coercion, but by the creation and manufacture of consent. What is its status today? The status today as we saw in the lecture on the alchemy of English is that it is really a that English has created you know a new caste if you may use the word here by analogy English has created a new caste of people who in India who who are enjoying the fruits of of of a modern technological world. On the other hand the language by its true attitudinal neutrality can also erase certain markers of oppression of inequality. For instance, certain markers of caste that are present in Indian languages are not there in English. So, English can be a great leveler today in India and and at the same time as it also creates a particular or very very privileged class or as Kachru says a caste of people right who who enjoy the fruits of liberalization and modernization. So, the alchemy of in the alchemy of English if you recall Kachru had said that competence in English and the use of this language signify a transmutation like alchemy and added potential and added potential for material and social gain and advantage which is really available to let us you know face it as Kachru says available to a very few. Knowing English is like possessing the fabled Aladdin's lamp which permits one to open as it were the linguistic gates to international business technology science and travel and this is precisely what is happening in India today. Even as we speak right now the percentage of population in of the population total population in India that knows English and reaps the benefits of knowing English is enormously low. So, this has created a disparity among the population itself. So, this is something that we need to even as we celebrate the use and the great potential of the language to take us very far particularly in international scenario we have to understand the other side as well. This also Kachru says has created as we saw in that lecture created a sort of linguistic schizophrenia. What is this linguistic schizophrenia that is still present in contemporary India that he talks about is that in anti English circles we find surprisingly that there is one policy inside the home and another outside this is linguistic schizophrenia in case of English. You will often find as Kachru said many policy makers or many public figures in their public speeches talking about the need for us to learn in the vernacular language to send our children to schools where the medium of instruction is in the native language. But on the other hand if it is to be believed so these very personalities do not hesitate to send their own children as Kachru puts it to English medium schools or to places of higher education in the European or American world or the western world in general. So, there is also this kind of linguistic split in the country. So, this has created as he says a new cast of English using speech fellowships across cultures and usage a shift of power from the traditional class structures of India to and the new cast has developed an important social change giving users power for mobility and social advancement fine and I would like to end this discussion today by referring to the to a well-known writer named Raja Rao the author of the famous Kanthapura who says that as long as the English language is universal it will always remain Indian. We shall have the English language with us and amongst us and not as a guest or friend but as one of as he says as one of our own of our cast our creed our sect and our tradition this obviously would again take us back to you know to a lecture like world English is for instance where we have our own variant of English. English is here as Raja Rao says in our country no longer as a guest no longer as a hitherto colonial tool of mass exploitation and commercial administrative advantages English has been Indianized as many would say English has been Indianized and we say see this in the many in a creative and linguistic so to speak experimentation that has been done by poets writing in India writing in English by novelists I said I mentioned earlier Arundhati Roy for instance in the god of small things and many other writers who have created a specific kind of English which is no longer English English but is Indian English and thank you for being with us see you in the next module. Thank you.


Survival Difficulty, Class 1, Safe, Secure, Minimal Entity Count Level 1 is the second level of the back rooms. Description Level 1 is a large sprawling warehouse that features concrete floors and walls, exposed rebar, and a low-hanging bog with no discernable source. The fog often coalesces into condensation, warming puddles on the floor in inconsistent areas. Unlike Level 0, this level possesses a consistent supply of water and electricity, which allows indefinite habitation by wanderers, providing that appropriate precautions are taken. It is also far more expansive, possessing staircases, elevators, isolated rooms, and hallways. Grades of supplies appear and disappear randomly within the level, often containing a mixture of vital items, food, almond water, batteries, tarps, weaponry, clothing, medical supplies, and nonsensical objects, assorted car parts, boxes of crayons, used syringes, partially burned paper, live mice. Lace in a catatonic state that has been injected with unknown substances, shoelaces, loose change, bundles of human hair. The crates should be approached with caution due to their contents but are a valuable resource. In addition, crude paintings and drawings with no apparent origin or meaning appear on the walls and floors. They are known to change in appearance and disappear when not in a direct line of sight or when unlit. The light fixtures within level one are prone to flicker and fail at inconsistent intervals. When this occurs, supplies are liable to vanish inexplicably and hostile entities may appear unexpectedly. These entities rarely attack in groups and tend to avoid light in large gatherings of people. It is advised to carry a reliable light source and sleep holding whatever items you do not wish to lose. Level one is home to the hub. Bases, Outposts, and Communities MEG, Major Explorer Group, Base Alpha. This is the first main base of the MEG. About half of the group's team live here, mainly housing explorers and supply outposts shared with the BNTG. The base is close to one of the entrances of level one, heavily guarded against threats such as entity attacks. BNTG Traders Keep. Traders Keep is a large city-like complex. Around 412 inhabitants, non-honstyle, possesses rudimentary societal infrastructure. The main supply hub for wanderers. Traders Vault is set up in the storage hall in level one, keeps communications and friendly relations with the MEG. Tom's Diner, run by Tom, a former chef before no-clipping into the back rooms, has plenty of food and storage, resupplied by the BNTG every few months. Tom is an amiable and optimistic person, and his diner serves as a small social hub. Entrances and exits. Entrances Get here from level zero by no-clipping. This level has many entrances, opening ordinary doors in many levels has a good chance of leading to level one. Additionally, climbing one of the trees in level 11.1, the end of the fifth event in level 92, and completing the King's Trial level 94 are all reliable ways of returning to level one. Exits You can leave level one by simply continuing to explore the back rooms. Over and down any extended hallway often leads to level two. Finding holes in the wall will lead you to level 19. Where the ceiling has paintings on it, no clip through and you will be at a party. Additional Material Recovered Documents The following notes have been recovered from various locations in level one. No apparent source has been found. The artist lives in me. The artist dies in me. Don't you stay for the after party.  The blue bird beckons you, soothes you, you'll feel safe here. Black Queen to H5, and that's Checkmate. Watch the clock, the final prophecy draws near.


This lecture is part of an online course about Lie groups, and we'll be discussing the question of the relation between Lie groups and Lie algebras. So in the previous lecture, we show that if you're given a Lie group, you can get a Lie algebra. And the question is how much of the Lie group does its Lie algebra actually capture? So for example, we can ask the following questions. If two groups G and H have the same Lie algebra, or at least isomorphically algebras, are they isomorphic? In other words, does the Lie algebra determine the Lie group? And we will see the answer to this as no. Secondly, if we've got two groups G and H, suppose you've got a homomorphism from G to H. Then it's easy to check that you get a homomorphism from the Lie algebra of G to the Lie algebra of H. I mean, it maps the tangent space of the identity of G to the tangent space of the identity of H, and it's not difficult to check this preserves the Lie bracket. And we can ask, is the converse true? So let's put a question mark there. If you've got a homomorphism between Lie algebras of Lie groups, does this give us a homomorphism between the Lie groups? And the answer is in general, no. Or suppose a special case of this, suppose G is a subgroup of H. Then the Lie algebra of G is a subalgebra of the Lie algebra of H. We can ask, is the converse true? So if we have a Lie algebra that's contained in the Lie algebra of H, does it come from some closed subgroup? I guess I should have said here G is a closed subgroup of H. None closed subgroups can be very weird. And generally, when you talk about a subgroup, you're very often talking about a closed subgroup, not always. And the fourth question we can ask, does any Lie algebra come from a Lie group? So in this lecture, we're implicitly assuming that all Lie groups and Lie algebras are finite dimensional. And for a dimensional case, it gets a lot trickier. I think I forgot to say the answer. This question three is no. And the answer to question four actually turns out to be yes for once. So what we're going to do is going to go through and give some examples to sort of illustrate what's going on. So the first question, you remember, said that if two Lie groups have the same Lie algebra, are they isomorphic? This is obviously false because G has the same Lie algebra as is connected component. So in general, if a group isn't connected, there will be a connected group with the same Lie algebra and the groups aren't isomorphic. But we should ask the question for connected groups. So suppose G and H are connected and have the same Lie algebra, which is a sloppy way of saying their Lie algebras are isomorphic. Well, are they the same? Well, no, because we can do things like we can take the reals, maps onto the circle group S1. And these both of Lie algebras isomorphic to the reals. Another example might be the group SL2 of R maps onto the group PSL2 of R. And the kernel is, consists of the elements plus or minus one. And again, these two groups have the same Lie algebra, but they're not isomorphic. For instance, this one has a non-trivial center and this one doesn't. And as I mentioned in an earlier lecture, what's going on is that these pairs of groups are locally isomorphic. So that there's a neighborhood of the identity of this group that looks like a neighborhood of the identity of this group. And if two groups are locally isomorphic, they have the same Lie algebra. So locally isomorphic implies the same Lie algebra. In fact, the converse is also true. The two groups have the same Lie algebra. They are at least locally isomorphic, but this takes a bit more work, so I'm not going to cover it this lecture. And you can get lots of examples of locally isomorphic groups by taking G modulo, a discrete closed subgroup. So that's what's going on in these two examples here. The discrete closed subgroup is the integers, and here it's just plus or minus one. And if you mod out by discrete closed subgroup, you're obviously going to get a locally isomorphic group because this doesn't affect what's going on near the identity. So in fact, discrete closed subgroups of G are quite restricted. So suppose G is connected, then any normal discrete subgroup is in the center of G. So in particular, it has to be abelian. And you can see this as follows. So let's fix some gamma in this closed normal subgroup. Let's call this closed normal subgroup gamma. And we look at the map taking G to the commutator, G to the minus one gamma to the minus one G gamma. And you notice the image is in gamma because we can write this as G to the minus one gamma to the minus one G. And this is in gamma because gamma is normal and we'll multiply it by gamma, which is also in gamma. So here we've got a map from G to gamma. And G is connected and gamma is discrete. So the image is a single point. On the other hand, if we take G to be the identity, then this is the identity. So the image must actually be the identity of G. Well, that just means that little G commutes with gamma for all G. So any normal discrete subgroup must actually be in the center. So the normal discrete subgroups we're quotienting out by here, I guess that should have been normal, are actually quite restricted. Well, so let's have a look at the next question, which says that you remember if you've got a homomorphism of groups, then this gives rise to a homomorphism from the Li algebra of G to the Li algebra of H. And we want to know, is the converse true? And again, the answer is no. So here we've got a homomorphism of groups from R to S1. And if we look at the Li algebra, it's just R to R, and this is actually an isomorphism. So there's an isomorphism of Li algebras back the other way, but there's no homomorphism of groups from S1 to R corresponding to this. Well, what's going on here is that this group here, S1, is not simply connected. So we recall that simply connected means that any loop inside the group can be sort of contracted down to a single point if it provided you keep a sort of point fixed. In other words, we can take a loop and sort of pull it down like that. And obviously the real line is simply connected. If you've got any loop, in other words, the image of a circle in the real line, it sort of might look something like this. And you can obviously just contract it down. On the other hand, if you've got the circle group S1, there's a pretty obvious loop that just goes round S1 that you can't contract down. And it turns out that if G is simply connected, then a map from Li algebra of G to the Li algebra of H induces a homomorphism from G to H. Again, we don't really have enough technology to prove this quite yet. But as I said, this lecture is just giving a few examples. So the third question says that suppose G, suppose you've got a Li algebra contained in the Li algebra of a subgroup of a group H. Is there a closed subgroup corresponding to L? And the answer is not quite. We have the following funny example. Let's take G to be a product of two copies of S1, which is just R. You can think of as R2 modulo the lattice Z2. And what I mean to do is to take L to be the multiples of X alpha X beta for X in R, where alpha over beta is irrational. And we can think of G as being a sort of square with the top and bottom identified. So what you should do is you should think of the top of this as being identified with the bottom and the left hand side identified with the right hand side. And now if we tried finding a subgroup corresponding to L, we can find such a subgroup. It's just a subgroup of all multiples of alpha and beta. And the subgroup sort of looks like this when it comes up there and then it comes up there. And it sort of goes on like this. And the problem is the subgroup of all points of the form X alpha X beta is dense in G. So it's not closed. And so in some sense we do get a subgroup, but if a subgroup isn't closed inside a group, then you run into sort of funny problems trying to take its Lie algebra. So there's sort of a subgroup corresponding to this subalgebra, but you have to be a little bit careful about it. Finally, we can ask the fourth question says that does any Lie algebra come from a Lie group? Again we're doing the finite dimensional case. And here the answer is yes, but it's actually a little bit tricky to prove. Let me say a bit about how to prove the easy case where the center of the Lie algebra L is just zero. So what's the center of a Lie algebra? The center is the set of a group. It's the set of group elements that commute with everything else. For Lie algebras, commuting with something corresponds to the Lie bracket being zero. So this would say that x, y equals naught for all y or rather the set of x such that x, y equals zero for all y. And if the center of L is none zero, then L can be then written as a subly algebra with the n by n matrices over r where n is equal to the dimension of L. And we do this by actually identifying L with n by n matrices. So linear transformations of L correspond to elements of this Lie algebra. And what we want to do is to get a map taking from each element of L to some linear transformation of L. This will give us a map from L to the Lie algebra of gln. Let's call this map rho. And rho of an element a is the linear transformation such that rho a of b is equal to ab. And we need to check that this is a homomorphism of Lie algebras. So we want rho a b to be rho a rho b minus rho b rho a. Well in order to check this we need to see that abc is equal to abc minus bac. And this sort of looks more or less like the Jacobi identity with possibly a few signs. And these signs come from the fact that you sometimes switch the orders of a and b. So this follows from the Jacobi identity and anti-symmetry. So this gives one interpretation of what the Jacobi identity is. It's saying the natural action of L on itself is actually a homomorphism of L to the Lie algebra of n by n matrices. Now we've got L as a sub-algebra of n by n matrices. And you can show from this that L actually corresponds to a subgroup. Although doing that, well you need to use something like the exponential map that we'll be covering in a later lecture. So I'll postpone this for the moment. You notice the fact that L is a sub-algebra of m and L follows from the fact that the centre is equal to zero because anything in the centre of L would have imaged zero under this map here. So that's why we need the centre of L to be zero for this easy proof. If the centre of L isn't zero then it's still true that L is a sub-algebra of some of the algebra of n by n matrices but it's rather more difficult to prove. Again we might do that in a later lecture. So let's have a summary. So we've seen that Lie algebras don't quite correspond to Lie groups because there are the various problems. However, Lie algebras are very similar to simply connected Lie groups. So we saw that most of the problems that arose that Lie groups not quite correspond to Lie algebras were related to the fact that the group was either not connected or not simply connected. So what I'm going to do now is to say a little bit about the relation between Lie groups and simply connected Lie groups. Well, if the group isn't connected then the group of components can be any discrete group and we can't really say much about that. So what we're really going to do is compare connected groups to simply connected groups. So in other words, suppose we had somehow managed to understand all the simply connected groups. How can we find all the connected groups from that? Well, as we said, if we've got a simply connected group then we can form another group by taking the quotient of G by gamma where gamma is discrete and normal and as we saw earlier this implies that it's actually in the centre of G and this is now a connected Lie group. And we can do the converse. If we've got any connected Lie group we can actually construct a simply connected Lie group such that our connected group is a quotient of this. And I'll just recall how to do this. So suppose H is a connected group. We put G to be the universal cover of H. And let's recall from algebraic topology what this consists of so we fix the identity E is a base point of H and G is the set of all homotopy classes of paths in H starting at E. This means a map F from the unit interval to H with F of 0 equals E. So if we've got a group H like this and point E then a typical path would be some path like that. As I said, these are homotopy classes of paths. These are homotopy classes fixing F0 and F1. So what we do is we take the path and we fix the beginning and end of the path and we're allowed to move the path provided we keep the start and end points fixed. So this path here is considered to be homotopic to this path here and these are all going to represent the same element of the universal cover. And the basic result of algebraic topology that this universal cover is simply connected. And it's also a group. So in order to see it's a group let's just take H and if we've got some path starting at E going to some point here and some other path going to some other point. So this goes to a point G and this orange path might go to a point H. We can sort of just shift the path going to H by multiplying it by G everywhere. We take every point of this path and multiply it by G. So this now goes to a point GH. And you can easily check that this is a well-defined operation and in fact gives you a group structure on the universal covering space. And furthermore this group is locally isomorphic to H sorry to G. Again this is a general result from algebraic topology and works for any space. It's nothing to do with being a group. So we've got a map from a simply connected group on to our group G which is a local isomorphism. So these have the same least algebra but this group here is simply connected. Incidentally the group we quotient out by is actually you can check it's actually isomorphic to the fundamental group Pi 1 of the group G we started with. I mentioned earlier that a discrete normal subgroup like this here has to be abelian so this sort of suggests the slightly surprising result that the fundamental group of any Lie group is always abelian. In fact you can check this directly as follows. So suppose we've got two elements of the fundamental group of G. So an element of the fundamental group of G will be a path from the unit interval to the group G such that the image of beginning and end of the path are both the identity of G. So suppose we take one element of the fundamental group and we take another element of the fundamental group. So this will now give us a map H mapping to G and again the images of the beginning and end of the path will be in G. Now we can compose these two paths in two different ways. So we can multiply this element of the fundamental group by this element or we can do them in the other order. And in order to show that the fundamental group is abelian we've got to show these two maps a homotopic. In other words we've got to kind of fill in this square here which will give us a homotopy between this path in G and this path in G. And that's quite easy because we can just map any point x, y in this square to H of x times f of y where we multiply these two elements in the group. In fact you can see in order for this argument to work we don't actually need G to be a group we just need it to have a nice continuous product with an identity element. So fundamental groups of Lie groups are particularly easy to deal with because they're abelian. So anyway let's see some examples of this. Well the simple example is one we've had before if we take G to be the circle group S1 then it's universal covering space. It's just the reals which sort of sits over in a sort of helical pattern like this and so on. So this would be a copy of R mapping to S1 and the fundamental group of a circle is of course just C and you can see the kernel of this map from the reals to the group S1 is again just isomorphic to Z. Well for a more interesting example let's look at the general linear group in two variables over the reals and we want to know what is the group of components and what is the fundamental group Pi1. And to do this we recall the Gram-Schmidt process which is sometimes called the Iwasawa decomposition. Iwasawa decomposition is something that works for more general groups. So the Gram-Schmidt process you remember is a way of turning any base of R2 into an orthonormal base and the Iwasawa decomposition writes the group as a product of a certain compact subgroup and a certain abelian subgroup and a certain nilpotent subgroup and in these three cases the nilpotent subgroup is going to be these matrices and the abelian subgroup is going to be these matrices with diagonal entries greater than zero and the compact subgroup is just going to be the group of orthogonal matrices. And you can see that writing gl2 of R as a product of these three groups is really the Gram-Schmidt process because multiplying something by this nilpotent subgroup is really making a base orthogonal by adding a multiple of one vector to another. Then multiplying the base by this diagonal matrix is just multiplying each basis vector by a positive real which you can use to make the matrices all have norm one. And finally what we're left with is an orthonormal base and any two orthonormal bases are related by a unique orthogonal rotation so it's an element of the orthogonal group. So the Gram-Schmidt process is just a way of saying that every element of the gl2 of R can be written uniquely in this form. And now we can figure out what the fundamental group is because these two spaces are both contractable. So in fact this is isomorphic to the reals and this is isomorphic to the product of two copies of the positive reals and they're both contractable. So the components and the fundamental group of the general linear group are the same as the components and the fundamental group of the two-dimensional orthogonal group. And this is much easier to work out because the orthogonal group of R has two components one of which is SO2 of R, which is just the rotations and the other is the reflections of R2. So we've got two components. We can also work out the fundamental group. Let's just notice that the group of components of gl2 of R which is usually noted by pi zero is just isomorphic to z modulo 2z. So it's order 2. And pi1 of gl2 of R is isomorphic to pi1 of SO2 of R. Well the group of rotations of R2 is just isomorphic to a circle. So this is just the fundamental group of a circle which is isomorphic to z. So the fundamental group of gl2 of R is actually z. So we have a simply connected group. Let's call this gl2 of R plus and put a twiddle over it. So this means you take the identity component and this means you take the universal cover. And this maps to gl2 of R and the quotient is a group of order 2 and the kernel of this map is z. And it's kind of really quite difficult to describe gl2 of the universal cover of gl2 of R in terms of matrices. The problem is this is not isomorphic to a group of matrices. In fact, we will see later when we look at the representation theory of gl2 of R plus that any map of this group on a finite dimensional vector space must vanish on this kernel z. So it really factors through gl2 of z. So this group is kind of hard to get hold of. You can't write down explicit matrices for it. At least not in finite dimensions. You can if you allow infinite matrices. Similarly, the group sl2 of R also has a universal cover with an infinite center z. And these covers are also a bit mysterious. The only one that I've ever seen turn up anywhere is the so-called metaplectic group where you take the double cover of sl2z. So you can take a cover with fundamental group z but you can then quotient out by 2z and get a group that's just a double cover. And this double cover of sl2z actually turns up in the theory of modular forms. It's very closely related to modular forms of half integral weight. But again it has no faithful finite dimensional representation so it's a little bit tricky to handle. So that's looked at the fundamental group of gl2. Now let's take a look at the fundamental group of gl3 of R. Well as before it has the same group of components and the same fundamental group as O3 of R. We can again do the Gram-Schmidt process and see that gl3 of R is homotopy equivalent to the orthogonal group O3 of R. So O3 of R has a subgroup SO3 of R, the special orthogonal group of things of determinant, plus one of index two. So pi zero of gl3 of R and pi zero of O3 of R, which is a group of order two. So it is two components which are of course the matrices of positive determinant and the matrices of negative determinant. Pi one of gl3 of R is isomorphic to pi one of SO3 of R because we can forget about the other component. And this pi one of SO3 of R is not at all obvious. It turns out to be another group of order two. One way to see this is to use quaternions. So you recall that quaternions are the group of the ring of all matrices a plus bi plus cj plus dk. And it has the multiplicative group has a subgroup S3 which is the set of a plus bi plus cj plus dk such that a squared plus b squared plus c squared plus d squared equals one. So obviously a sphere. This is a group and it maps on to the group SO3 of R. And the way it does this is we can think of SO3 R as being the rotations of the vector space bi plus cj plus dk. And the way S3 acts as rotations is if we've got some quaternion g here it acts on a vector v as g of v is g v g to the minus one where this vector is g. So we can define a homomorphism from the group S3 to rotations of three dimensional space. You can easily check that its image does in fact preserve the obvious quadratic form. And the kernel of this map is just the matrices plus or minus one. So what we've got is we see that the group S3 is a double cover of SO3 of R. And the group S3 is of course simply connected. And so the fundamental group of SO3 is the kernel of the map from its simply connected cover which is just a group of order two. So you see that the fundamental groups of lead groups can sometimes be a little bit surprising. I mean you probably wouldn't guess if you didn't know that the fundamental group of GL3 was Z2 and the fundamental group of GL2 was the integers. So by the way, the fact that the fundamental group of orthogonal groups often has order two is a special case of the construction of spinor groups that we'll be doing later. And in fact S3 is one of the simplest examples of a spin group. Okay, so next lecture we will be discussing the exponential map that gives another relation between the lead algebra and the lead group.