Hello everybody, welcome back to a second channel video. I'm your host, Gumby. All right, today we're going through the subreddit again. Full disclosure, I am a little bit worried to go through the subreddit today because I've gotten some tweets saying that the subreddit is in shambles right now and people said that there's people fighting on the subreddit so I'm a little bit worried to look but I haven't been on it in a few days so I figure I should probably go on there and check it out so we're gonna sort by top, we're gonna sort by month. Hopefully that will avoid any negative posts. The first time I met Danny was horrible. Okay, not a great start. I just want to start off by saying this is no disrespect to Danny. I'm posting this so you understand meeting YouTubers isn't always going to be a good experience in no way is this Danny's fault. Is this real?  Let's use my critical thinking. Do we think this is real? Honestly, it does sound kind of real at the beginning but it has a lot of upvotes and awards. I don't know why it would get awards if someone was just like, I met Danny and he was just kind of awkward but I don't know, let's check it out. Anyways, here's my story with meeting Danny. So basically me and my friend work at the Bombard Theater in Kentucky. Usually we don't work backstage when a show is on but my manager led us knowing how much we wanted to see the tour. Me and my friend were just backstage goofing around when I noticed Danny glancing over at us. I don't know what I'm about to do in this story but I just want to defend myself and say they were backstage goofing around. So, I mean, what would you do personally if you saw this? If this was your tour and you saw two of your fans goofing around backstage, would you not go ballistic on them? Would you not have them booted from the show? I'm just kidding, obviously. I flash him a quick smile and wave. I didn't think much of it until I noticed how angry Danny looked. He then started sprinting at me and screamed, get back to work. Okay, so this is fake. He grabbed my shoulders and threw me up into the air onto the stage. Okay, so this is real. He picked me up and started kicking me around like a soccer ball. Curtis and Drew screamed, oh no, Danny, not again. Luckily, some considerate Greggs climbed onto the stage and carried me off. Thank God that was fake, dude.  That would be the worst day of my life but luckily this is fake, all right? Don't believe this. Don't believe this for one second, dude. This did happen and I started beating them up but Curtis and Drew actually joined in and helped. Yo, look at this fan art, dude. This is awesome. This looks just like me. I guess before I got a haircut. I got a haircut just today so my hair's a little bit shorter. I look naked now but back when I had long hair, this was me. I look just like Danny. I'm censoring for privacy reasons. Even though you're censoring your face, I'm just gonna come out and say I don't see it. The arm looks nothing like my arm. This arm hair is like way too dark to be my arm. All right, look at this. So your arm looks nothing like my arm is basically what I'm trying to say. Can we get some more arm doppelgangers in the subreddit? Except not actually though because I feel like that'll just take over the subreddit if I ask for that. No arms, please. Oh no. My dad watching Danny with other dads. Yeah, I didn't really think about this when I asked people to show their dads on the subreddit. You know, not everybody has a dad but this is pretty tight, dude. Your dad is kinda swagged out right now. Very cool little case that you got him in. I had a lot of complaints about the last dad with his cable management. This dad, honestly, I'm gonna give him a 10 out of 10. Not gonna lie, you fitted him out pretty well. He's carrying on in luxury. This is actually pretty funny if you don't get sad. Yeah, I'm not sure if I'm like, if the person who posted this is okay with like laughing about it. I guess they posted this so I assume that they're cool with it. Yeah, okay. They're responding with jokes about him being cremated. So sorry for your loss at any rate. Very swaggy urn and a 10 out of 10 dad. Danny and Danful. Oh, Danny, Dan and Y and Dan FL. The Dan FL is also the Dan football league. It's like the NFL but the Dan FL. So you could have made it dad as well but this is nice. This was funnier in my head. Yeah.  But you know, I appreciate it nonetheless. You could have made Danil, D-A-N-I-L and done my home state of Illinois. And then it would have only almost been like Daniel. I feel like that would have been better, honestly. Danny and Danil.  That's pretty much all Illinois is actually. It's the bean and cornfields.  I don't know why. You've dreamt about me multiple times and every time I'm in your dream I'm insane. Am I the new hat man, dude?  Actually, that'd be pretty sick. I haven't seen any influencers.  Mr. Beast has like burgers now. I want to enter like the dream space. Not the dream SMP but like people's dreams. I want to become a new recurring character in people's dreams. You got the old hag. You got hat man. I want people to start seeing Danny when they have sleep paralysis. Hey guys, LOL, it's me Danny. So hey, ha ha, it's me, LOL.   I'm literally not even joking.  Later on Twitter.  The fuck? Oh, and Curtis is here too. Guys, I don't know what's going on with Danny. He won't answer anybody. He's gone missing. Wasn't the other post that I talked about on here where someone had a dream about me, leaving like cryptic messages. It was like, follow the cat or something? Okay, yeah, please find me, follow the cat. Okay, so here's the new dream lore, all right? I'm going to start being a recurring character in people's dreams now, but instead of like sitting on your chest or standing in the corner of people's rooms, like the old hag and the hat man, I'm going to get lost and people have to find me. So maybe if you have sleep paralysis, I'll be kind of like hidden in your room and there will be clues and you have to find me. If someone could add me to the villains wiki, actually that would be sweet and we can just kind of create some lore. And then maybe people will dream about me.  And I'm running down the street.  It's funny now, but in the dream it was really realistic and everyone was super freaked out and worried. Yeah, I'd imagine that in the dream that would be like really worrisome. Thank you to your dream self for being concerned for me though.  I got to stop watching this guy. And you were like, I'm worried about Danny. That's, that really says a lot about the character of your dream self.  If like it became a thing amongst like sleep paralysis culture. I don't even know if that's a thing. Let's make it happen guys. If it's not a thing, let's start it. Just another day in the Toby McGuire subreddit. No dude, get out of here with that. This is not Toby McGuire subreddit. Get him out of here. Why does this have so many awards? Wholesome award, this is not a wholesome. Giving this poser awards on my subreddit. Getting him one of the top voted posts. Get this thing out of here. This is my dad with a Danny Gonzalez balloon he made. That's sick. I mean like the fact that he like got kind of the general shape of my hair is like really cool. He didn't just like do any old standard like balloon person. I feel like he actually made it look like me on purpose. That's kind of insane. It looks just like me actually. Can we get a side by side real quick? It looks like me. Epic kind of scary though. Yeah, this is kind of scary. I like that my hands kind of look like hooves or like white coffee beans. It does kind of look like something. And this is no disrespect to your dad because it's very cool. But just being the nature of like kind of looking like a doll. This kind of looks like something that would appear over your bed at three in the morning. And thus I am going to have to also get this added to the villains wiki as the evil counterpart to my sleep paralysis self.    I already know what this is from this picture. My friend found an article from 2012 that's a cast list for Hello Dolly at Danny's Old High School. And he had like a lead role killer dance moves. Dude, I don't even want to watch this man. I know that it's weird. He always gave off theater kid vibes. Yeah, well I've also said that I was in theater in high school so maybe that's why I gave off those vibes. Didn't he say he was in theater in his Reddit video? Yeah, there you go. Oh wait no, didn't he say he wasn't in theater in his Reddit video? When did I say that? I don't think I've ever said that before. I was in theater in high school. I was in theater and speech. Those were the two things that I did in high school. I would never deny that. What are you talking about? Why are you trying to start fake rumors about me? Toes slurper? He wasn't expecting to read that username and for it to be so nasty. Yeah, guys, if you see somebody commenting on my subreddit with a name like Toes Slurper, do not trust anything they say, all right? He should have never denied he was in theater. Dude, I didn't deny I was in theater. I never did that. LOL, I was there for this production during the intermission Danny came out on stage and called everyone who didn't get up to use the restroom a very naughty turnip.  The top comment under that is I truly can't tell if this is a joke or not. And honestly, I can't either. Yeah, this was literally 10 years ago.  Yeah, this was literally 10 years ago. I would have no memory if that was true or not.  This is where that IMDB acting credit comes from. Yeah, my leading role in my high school performance of Hello, Dolly. That's sort of how I got into the industry. Let him into Juilliard, thank you. Actually, this is the only appropriate place to comment that and I do appreciate that, thank you. Why does he look exactly the same? Cause I am the same, all right? I'm the same person. The stiffness of his body makes me cringe, no offense, Danny. Yeah, that's just kinda how I was back then, dude.  Where's Danny, but I made an actual Danny character to find, he's in there and he needs some critical thinking skills to be found.  Whoa, you did make a little Danny. Oh, cool. So I gotta find this fool who's holding some kind of cane slash bat device. Shit. Oh wait, is that me? He's wearing a Greg hoodie. Wait, did I find myself right away? He's holding a cane. I think that is me. Did I literally find myself that fast? Jake, if you're editing this, don't even edit that down.  That is me right there. I've got a cane, I've got a blue hoodie. I feel like I found that in literally like five seconds. That was absolutely mental. My critical thinking skills are off the charts. This is insane.  They're so adorable. Yeah, we are. Look at me, dude, I'm straight up asleep. God damn, that's such a cute picture too. Look at us. Two little peas in a pod. He looks so drunk. Dude, I'm literally asleep in this picture. Danny, you have to explain this picture. I need an explanation. What's going on here? Honestly, I just took this picture in my room one time and it's been my profile picture for literally like six years. Maybe I should get a new profile picture. I don't know. I don't even really do my hair like that anymore. I also like Photoshop this really poorly. I don't know if anyone's ever noticed. I didn't like cut out the background all the way. I think there's like little parts in the middle of my hair where you can see the background. I've honestly never been a huge fan of this profile picture. It just kind of happened and I just kind of stuck with it. Maybe one day I'll change it. Who knows? Didn't Curtis change his? Or was that just a joke? Yeah, Curtis, I think edited his face to have his mustache and his longer hair on his already existing profile picture. So maybe I should just edit this. Maybe I should just copy Curtis. That's kind of a good idea. Maybe I'll just make Curtis's profile picture in my profile picture. Okay, honestly, I'm not seeing any of this like arguing going on on the subreddit. Where is this? I like this, POV, you are the meme. I like that.  Oh, is this a little video about my new couch? Okay. I guess I'm curious to see what people think about the new couch in the comments. On the last video I posted that had the new couch, people were pretty into it. It's a nice couch. Look at it. Nice leather, very comfortable.  When I need to get some work done, I'll go sit back there. It's kind of nice. Now I've got multiple places to sit in here. I feel like it really makes it feel more homey. But I also, I understand I didn't address it or anything.  And for your information, I didn't get rid of the IKEA shelf, okay? In fact, I can show you. It's just over there. So here's the couch and it's just right over there, okay? So it's still happy. It's still got stuff on it. It's, everything's fine. All right, let's see what people are saying about it. In time of political confusion amongst us, Greggs, let's lift our heads, join hands and focus on what really matters. What we've loved, what we've lost. And before we lose anyone else.  Is this person referencing what people were talking about in my Twitter mention? What political confusion amongst Greggs? Very upsetting that the shelf was replaced with no explanation, ceremony or mention. It just vanished. A ceremony for a shelf? There's no, no ceremony is necessary, all right? The shelf is happily, happily over there. It's by the window now, all right? It's getting some nice sunlight. I put some guitar stuff on it. So whenever I'm playing my guitar, I'm over there interacting with the shelf, okay? The shelf is fine. Also it's a shelf. Here's my shelf. This is a little storage thing from Ikea. Why is it making me kind of emotional, dude? It's just a shelf. Getting teary eyed over here, dude. Make this look like Danny's search history, no wrong answers. Danny Gonzalez first name? How to slap someone with legal side effects of living like the liver king. Can you eat raw liver?  How to get critical thinking in my subreddit. How to get LaCroix to sponsor me? That's pretty good. Except now, honestly, I don't even want him to sponsor me anymore. You know what? I'm happy to just drink whatever freaking sparkling water I want. Honestly, I've never felt so free than I do now that I can drink any brand of sparkling water. I can switch it up anytime I want. So this is honestly for the best, all right? I can't be tied down to LaCroix unless they wanted to fork over just a little bit of cash. But you know, it's fine. How many nipples do rats have? I specifically do remember looking that up when I was filming the rat tattooing videos. Like I know they have many nipples, but I need to know for sure how many they have. All right, I gotta find wherever this political discourse is. Maybe I gotta go to top within the past week then. That's when I've been getting these DMs.   Oh, we got a dad in here though. My dad bought these cord concealers just for Danny. That's a, yeah, okay. Honestly, personally, I would prefer the cables to go in the wall, but that's definitely better than whatever this dad had going on. These cables are everywhere. So I'll give you like a seven out of 10 on the cable management. Pretty good dad though. Also, you can get cable runners that are like the same color as the wall so that they blend in a little bit better. So if you don't wanna go into the wall, I would at least recommend that. Maybe just paint it or something. Go landlord style and just fricking put a slab of paint over that. I don't know what the Greggolution is and at this point I'm afraid to ask. Okay, so is this what people are talking about? A Greggolution? Greggloution? Is it really Greggloution? That just makes it sound like you're saying like Gregglotion wrong. Danny has become a dictator so we started Greggulation. Okay, there's three different spellings of Greggolution in here. There's Greggloution. Greggulation. As in congratulations. And Viva-la Greggolution, which is what I feel like it should be. So people are rebelling against me.  That's all I can say. So we started the Greggulation to fight against his iron fist rule. We want back upvotes, awards, Lacroix, freedom for the nutcracker guy, et cetera. We still want him to be king. We want more equality between us, Danny. Dude, what is going on? What is this? Come on, guys. What? Let's sort by new here so that we can see what people are posting about. Okay, literally like every post on the subreddit is about this now. Little Nutcracker guy doesn't deserve this. He has a beacon of joy, a presence in our life. He does not deserve to be used as this war's pawn. You guys are hurting him. People are creating political spectrums for you to be on.  What is the Greggolution? Okay, this one's spelled differently also. Greggolution.  That's what this is all about. And they think the nutcracker's being treated unfairly. Dude, this guy wants to be here, all right? I try to get him to leave. He's the hardest working motherfucker I know, all right? He clocks in early, he stands there all day, and he doesn't leave. He's the last one to leave, all right? I try to get him to leave. He likes it, and he's paid very handsomely. Nutcracker, blink twice if you're being held captive.  He's fine. In terms of the awards, I understand that, you know what? There is some good content in the subreddit and people should be rewarded for those things. I don't even know how to respond to this. This is so strange. Take the awards. I don't care. I don't need them. I don't need them. Take the awards if you want to. Can we just get the subreddit back to normal, please? I'm a little worried about you guys. Danny, encourage people to think more critically. They do it and gave me fun puzzles. They used too much of their critical thinking skills and started a Gregg-Greggolution.  I guess I really created a monster, didn't I? You guys are too smart now. There's like a whole culture in here that I don't even understand. How am I like out of the loop in my own subreddit, dude? Here's what we're gonna do, okay? We're gonna, I'm making a post on here. It's gonna be called the Gregg-Yuloh. The Gregg-Gregory-Regulation is over. Please stop this nonsense. You can have the awards. I will keep the nutcracker. He is fine. I promise. Post. All right, well, I guess that fixes that. The rebellion is over. Thank you guys for watching. Goodbye.


Okay, thank you for joining today I just want to know like how many men in a room like can you just like I think is I was guessing like a dozen, but Maybe less right. I thought this impressive because I've been to so many events so many tech event was absolutely the opposite sometimes So I think it's great that they were able to bring like so much Woman in a room today and thanks also for having some some men joining I want also to Enphasize this a little bit on the fact that they bring so many amazing woman today So I've like five in the panel. I know this is very challenging And I've organized some event before and you need to invite woman is very very difficult There's not that many at this level. I have some numbers to later to mention that But it changes like a lot of women are doing so many things. So it's very difficult for them So it's why I want to say thank you for joining us today So like maybe we just start with like one or two minutes each of you to just introduce yourself what you do in your in your companies and Yeah Okay, I'm Susan Jen. I'm the chief marketing officer for Asia Pacific for IBM and Based in the US, but I've been in Asia for a total of about nine like nine years And it's the best place on the planet Particularly Singapore may I add So hi everybody, I'm a Sunita core. I'm the managing director for Asia at Spotify. I'm Michelle Simmons I'm general manager for Microsoft's Southeast Asia new markets. So I'm responsible for Consumer and commercial business in nine countries in Southeast Asia Hi, I'm no pride your coupon and I'm with Google. I work a lot with absent game developers to help them scale up that business Hi, good morning. My name is Rosanna and I'm a lead business analyst in ThoughtWorks. So what I really do is a lot of talking Mainly to ThoughtWorks being a consulting company. I talked to the clients. What are your requirements? What do we need to build and I talked to the developers. This is what we need to build I bridge the gap and Mostly, I hope to share to you guys today is really just just to share a bit of an information I've worked in the tech department for For for all my career, but actually didn't graduate with a with a tech degree. So So that and another thing is like I'm a mother of two with one infant So I guess, you know, that's that's what I wanted to share with you If you can pick up anything on that if I can help on that area amazing Thank you. So the thing is I mentioned before when we look at leadership in a check on all your companies How many women on the boards of your companies, right? So usually it's between two to three Women on board of each of those company like IBM, Microsoft, Spotify Even if ThoughtWorks, so ThoughtWorks, I just look at the C level because there's no board I mean it was difficult to find it's not the listed company and Spotify It was quite interesting because they're maybe planning to go for an IPO and recently They just bring like two women on the board and before it wasn't any woman on the board apparently So this is interesting when you just look at the board level, but when you look at some numbers, so for example, I found for ThoughtWorks because they got in a while recently so they Describe how many women at each level. So if you look at overall in the company is 40.5 percent of women So you're close to the 50-50, right? But when you go to mid-level is is 46.2 percent and then when you look at senior level is only 30 percent And when you go to executive level is 23.8 percent So the higher you go the less woman you have in the organization is the same thing with Alibaba So Jack Ma was also very supportive of women But it's the same thing on his organization and I think you get only one woman on his board So it's less than than any of those companies today So my question would be like what would be your advice for women to be able to take those leadership role and and be able One day to reach this kind of sea level or or board of those companies like how can we see more women going to those positions? Well, we do have a female CEO You you are the only company IBM is the only one here and across the four 500 is like Around like 20 30 companies with a female CEO, but IBM is true We see the pictures and there aren't very many One thing, you know She just did a very unusual Interview that I would refer people to with Steve Kramer where she talked about her personal life and her challenges and it's really very inspiring when you look at someone who's come to the level of CEO and She came into being the leader of IBM in a period of time where the business suddenly became very challenged and a lot of it was personalized to her but Watching her has been really inspiring to me because and she talks about you know What helped her to do that and what she brought out of inside her which is basically not to let people characterize you as what they think You are but to be who you are and be authentic and genuine in yourself And I think that's in them, you know, that's was very inspiring to me even to and I wanted to share that Okay, so so you mentioned ThoughtWorks, right? So ThoughtWorks We do have a CTO CTO that is female. That's Rebecca Parsons So we're quite proud of that fact that our CTO is a female and here in Singapore our MD Who's actually sitting there is Jesse. So she's so she started she became the MD last year, right? So personally here in Singapore for me being employed in Singapore I was like yay a female MD and apart from that the the Asia Pacific MD who's Ange is also a female. So Compared to the other previous companies I've worked for I would say like ThoughtWorks is Doing much better in that aspect. So that's a very very personally very happy about that But speaking generically I do see that the stats that you mentioned they're very very true and So how do we you know, how do we solve that problem? I guess it's a very very complicated problem that that we should tackle both in many different ways. So personally from my experience When I was really really young just starting out in my career, right? I've never really felt the difference of Men and women in the workforce because I was single and I didn't have a family. I'm like What do you mean like equality in the workplace? What's it like? I feel totally fine. I don't feel any difference with my male colleagues, but things have changed my first my actually the the the The main change that I felt was really when I had children When I got married little difference, but when I had children it's like I had a male boss and and After I came back to maternity from maternity leave I was like, you know, I'm really looking for a woman mentor somebody that I can talk to about the the Challenges that I'm facing and him being a male my male boss He was really really really supportive, but he was not able to to Understand a lot of the things that I'm experiencing and that was like an aha moment for me like, you know This is this is that's probably why At the you know the entry level Maybe it's 50 50 men and women, but as as women age, you know, because they are you know They're further down in their night like like they have start having families They start having kids and they have a lot more to juggle. That's when they start, you know opting out of the workforce so For that problem. They're both internal and external Forces at play so internally, you know, they did mention that Being there's the imposter syndrome if you should look at, you know, how many female leaders they are I said, oh, who do I aspire to not much? So that's an internal problem external problem as well like because of these changes because for example now you you have children There's you finally realize that society expects very different things from you as a woman or you as a man So and you suddenly realize oh, there's so much more responsibility So there are also external factors that need to be addressed to make the I guess, you know The expectations more balance for men and women. So so it's a very intricate problem And I believe that in general our society is both our society Changing the external factors and really helping women with the internal factors. It's progressing But it's not something that happens overnight. So it's a long process and we have to continually do it Again and again, we may not all of a sudden see a lot of changes But we have to pursue so that you know at the end we finally reach equality that we're aiming for I would add Speaking specifically about why we don't see more women at the board level in the C-suite I think that what one of the things that Susan talked about earlier is how women actually speak to the the opportunities their their Business life their work life and the challenges at that level And so I think there are more and more women who are speaking out and and sharing those stories We also we have a female CFO Amy Hood And she's also very open about the challenges that she faces about balancing Work and the demands of work with her family life. So I think that that's a good start in the right direction But I also think that Earlier in the career if women aspire to that level we need to be very thoughtful about the types of career choices that we make And I think that's often not part of the discussion. It's typically a Male-female diversity discussion, but there's also if we are aiming for C-suite There's typically a need for diverse diversification of experiences as well that comes into the discussion I think just to pick up on on one word which was which was balance I think you know balancing anybody's life is it doesn't matter what what gender you are and We are at a point Professionally just as a as a world where Conversations are changing and I think taking gender off the table is incredibly important I'm not saying women use a lot of things as an excuse But when you can balance things out As an individual or as as a company That's very very important and one thing that I've definitely learned at Spotify, which Really helped to do that. I mean, you know working for for 22 years I can't remember when we didn't talk about gender inequality But what we did at Spotify about two years ago is we changed that conversation Not from a woman's perspective, but from a man's perspective And the first step we did was we extended six months maternity leave and also applied six months Paternity leave and I very recently had an interesting conversation with our head of strategy and operations who's a guy and just had a little girl and sat with me and You know almost basically was quite uncomfortable with the fact that he had to take six months off work And so with that conversation changing it evens out the playing field and it evens out The conversation and the balance and I think that's where the next generation needs to take that I think this is very interesting because there's several companies who are looking to also offer the same Duration for men and women for paternity or maternity leave But I think there's some numbers already on some companies where they found out that even if they're offering very few men Are really taking it or they would just take a small part of it or none at least So how can we really encourage because I think it's maybe also some social pressure for men like if you're a man with a Baby in a street and people see you like every week that kind of the thing you don't work, right? They would don't think you're an important leave for six months So they may get some social pressure from their friends, etc. So how can we also encourage like this side? How can we see like more men doing those kind of things? Yeah, it's interesting you say that my husband stays at home He's been at home for nearly 14 years So he faces that I think that perception we lived in Korea for three years where a stay at home Father is basically unheard of So I think there are some social pressures But at the same time I would argue if Men choose not to take that time, you know when I took I had three children within 39 months And so that was a lot of maternity leave in a very short period of time But I chose to because of that to go back to work as at 12 weeks For each of those that was my choice other people at Microsoft took more time And so I think that we need to enable Men and women to make those choices, but make those available and also make them acceptable In the workplace and so there's societal aspects as I mentioned that I think stay at home and do face but there's part of The more the more men that you do have taking paternity leave staying at home.  I think it's all going back to looking at the quality of people work and being supportive of one another's and Advocating for the cause that having diversity at your table helps the company make better decisions Help the company, you know, like to have stronger cultures being more inclusive in your leadership I think that's very very important when it come to company culture that Allow us to do that and allows all individuals to thrive to thrive And I think also I mean today the discussion is about a lot about women But when we talk about diversity is like far beyond only woman is about age is about politics is about LGTB, etc, right? And I think at Google also they have like some specific benefits some time based on also like LGTB Can have some specific benefits that maybe other people don't have not because they want to discriminate But because they want to offer what people need so I think is also very important about this inclusion on the company I think it's exactly because you know, there's different groups in and We approach and we think about employees and Recently I helped one of my team member who was going through his now her his discovery and it's not easy, you know, there's It's not going through traditional life and having having a family and and having a baby but that's also a different type of diversity and By being inclusive then you encourages your employee to be happy and when they're happy they just contribute more and and creating that kind of environment where you allow people to be themselves and To bring that true self to work. I think it's very very important Yeah, I think also when when you look at the unicorns were being built in the US They mentioned like 51% of them have at least some foreign funders So I think it's also quite interesting really to have like this diversity and I mentioned is you go beyond like just woman and man I know like all your other companies are doing things to promote diversity and inclusion So three out of the five today. It's really the listed one of a section about diversity and inclusion and their HR page For spotify and tout work is more about on the blog. There is some mention on those points So what do you think you can do to support like more diversity or what your organization is doing to support more diversity and inclusion? at your office, I Think the culture points are really important one and and we tend to think of culture as being either driven by the Organization or driven by the leadership and those things are there, but you know to your point We all create the culture and one thing that made a remarkable change For us in IBM was moving from Moving from cubicles to a totally open space and That includes everybody so it was a little uncomfortable for me at first to sit at this long table That included, you know people two three four levels below me's Administrative assistance, etc. But the change was remarkable I mean once you get over the fact that you can't talk too loudly or the Italian guy, you know has an annoying accent But it changed the way we work and everyone's opinion becomes part of you know that becomes the culture We create the culture. It's not given to us and that's when Things really start to happen in the creativity and the personalities and all of that actually make it much more than when everyone's sitting in There, you know little space on a headphone I would add one of the things You know Microsoft has had a commitment to diversity and inclusion But one of the things that we've done over the last couple years is focus on unconscious bias And I know there are a number of companies that are that are doing that because then you start to really Understand not just too often. I think diversity and inclusion becomes a numbers discussion Which doesn't actually change behaviors It doesn't change the way it doesn't change one's thinking and so by focusing on unconscious bias We actually start to break down Your your your thinking that you may not be aware of what are you bringing to work What are you bringing to your team your organization your workspace? Both positive and negative and it was interesting. I was just talking to someone at Microsoft last night a co-worker and he was mentioning he said I thought it was there was a picture in the news and it had President Trump and it had the CEO of Apple the CEO of Microsoft and the CEO of Amazon and his perspective not being an American was how incredible is it to see a gay CEO a foreign CEO and also another CEO who has you know a different background as far as ethnicity and I Didn't think of that at all coming from the US. I'm a US citizen My bias was just oh look there's some technology leaders meeting with President Trump And and so there was interesting his So not necessarily a bias, but the perception based on what I grew up with Versus what he grew up with he was more in tune to that diversity sitting at that table And I think I love the The thought of you know diversity not being a number and just hiring the best person for the job And one thing that's that's definitely on on our minds at Spotify for us diversity is is the generation gap You know you've got You know people like us who I mean I haven't you know had my own office since like 1996 and But then you've got this whole wave of Of a workforce that's coming in that doesn't want to be at their desk at nine o'clock in the morning, which I Struggle with right because I'm just you know eight o'clock at my desk You know I clock my my hours my days very structured And everyone's like well, you know, I'm gonna come in at ten thirty and But it's it's that it's that sharing it's that sharing of different generations and Again, you know when even when it comes to performance reviews, which I know is like a ooh, but My generation is very very used to very structured once a year. I will sit down with you I will judge you and I will tell you what your next year is gonna be like and I remember when when I first started at Spotify four years ago that was massive pushback on that where Everybody I mean the average age at Spotify is about 23 So I got massive pushback saying you cannot sit there and tell me once a year how I'm doing and I thought I think that's really fair. So, you know in in our Asia office. It's immediate feedback We've got two development talks a year. And so by the time we have our review There's so much, you know data to fall back on and it's a very healthy conversation and so That whole diversity with age is where we both or three of our generations can learn from each other You mentioned like you when you are you're focused on finding the right person, right? But when we look at numbers like HP of a studio about that like a man may apply for a job when he don't Have all the skills so and he may be on only 60% of the skill he will apply Woman will wait until they reach 100% of the skills, right? So the thing for your job you may get maybe more men than women applying So even if you select the best person but sometime I feel like recruitment could be a bit passive You just put the job offer and you wait for people to apply But how can you find also maybe other top talent who can allow you to have like maybe a more diverse range of choice So I have a team that is fairly diverse and I've actually sought out people who And there's a number of women who in my organization who perhaps didn't have the hundred percent of the skills But they were bringing something unique to the table to add to the diversity and thinking within the team And so I do think it requires From a hiring manager perspective from an MD perspective to actually be looking at the whole Organization and having discussions around talent and being thoughtful about seeking out People who bring different skills who can can question that's one of the things that I'll often ask New hires to do is to challenge the thinking I've been at Microsoft 16 years I want people to challenge how I'm thinking about the business Of course in a respectful way But I want them to challenge my thinking and others thinking and I think that's a great thing that new hires bring And so I actually seek that out How do you do that in Asia getting people to challenge you because for example I went I'm from Europe, right? So French would challenge a lot When I was student in China professor didn't expect us any questions, right? You go in the US people expecting asking question. So what do you get in Asia people coming and challenging you? It is difficult But I think it's it's seeking out specific feedback from individuals Providing opportunities for them to perhaps work on projects or to give feedback to others. So it requires a lot of different approaches in order to do that and I do think and it's a generalization I hate making generalizations, but in Asia one tends to rely more on their expertise and so Bringing them in to challenge based on their expertise. I've found also successful I've got a few points to add on that. I think it's really important to be intentional about your hiring so I've been at Google for a little over nine years and have my fair share of hiring people to the company and The hiring process that we have is not the hiring manager Interviewing the person and say hey, you go ahead and we'll go with you but we actually putting together a panel of you know diverse interviewer on the on the panels with you know Someone from the the function someone from cross function Cross-functional and we're looking at different dimensions from you know Can they do the job to what they fit the company culture do they have the Leadership skills that we're looking for because we're not hiring someone to be in that box forever But someone that can grow with the organization and I think also looking out for I love you know looking up for all of those unconscious bias and I think when you hire people just think is this person best for the job and Sometime when you thinking too much about gender you can also fall in the trap of being very you know positive discrimination and Hiring people and they're coming in and then think oh am I your diversity trophy? So that's you don't want to get into that So it's all about hiring the best people for the job But giving people equal chance to be successful in the company by stripping out of your biases and giving people equal footing to to allowing them to be successful and on how to build culture and and How to actually encourage just people to speak up so at different points and I'm actually from Thailand But been in Singapore for about 10 years I've had different People from different nationalities on my team And it's it's very colorful and the type of conversations that we've had on our team You know I've had Filipino I have all of the South East Asian nationalities Some you know French Lebanese from the US Malaysia And all over the place of Australian Dutch and when you put everyone on a table Then you can notice hey the Asian were very quiet And if you hire if you manage to hide that you know the introverted in an Asian one Oh the introverted Asian one they even walk quieter or the Japanese one the discuss between themselves before they you know Han over the official point of view So I think it's then Later, so then it I think it come back to if you are the one leading that organization or leading that team Are you providing people with? Are you being conscious and aware of the types of you know how people process information how people think? How people make decisions and if you know that you have a mix of introverted and extroverted on the team They make decision differently then create an equal playing field let them know the agenda way in advance so that they can form their Opinion and come to the meeting prepared And not let the extroverted just take over the meeting and then call people out when they talk too much and then allow the other Person to talk but don't call them on the spot if you know that that you know preferences and I think the other point. It's extremely important is people are not going to speak up if they don't feel safe Psychological safety is probably the most important thing that you want in your culture People will not speak up people will you know if they if they feel that they might be retaliated If they feel that hey if I give feedback to you know like someone on this table and all my career will be in jeopardy They're not gonna speak up So I think it's all about going back to then are you are you walking? Walking the talk if you welcome feedback if you want them to and I think people people know If you're genuine or not and if you're being authentic and you know that And they know that you have the best intention of taking the feedback and making their lives in a company better. They'll speak up So I want to touch on two things from the perspective of ThoughtWorks first is in terms of hiring So in ThoughtWorks, we really try to do to get creative with our hiring process So instead of the traditional interviews.  Oh, tell me about this experience. How about this one? Instead of doing that question and answer we do You know your job mostly you will be doing this. How about we do it together? Let's simulate it Let's do it together. That's pair. That's when we really are able to tell You know how that person will potentially do in a job as opposed to somebody who really do know just practice You know, these are the I looked at glass door. These are the potential questions and then I've nailed them down But really you know that person may not necessarily Be able to perform the job. He just really he or she just really you know prepared for the question So that's vastly different. So we really try as much as possible to to do things together to do case studies And we have quite a lengthy Like interview process lengthy not in terms of like it's harrowing again and again, but we really try to get to know the person So, you know, it's what during hiring. It's not really just The potential employee putting the best foot forward, right? It's also the company, right? We want to show that we are really we're a really good company And you know, here's how our marriage might work, right? Because it's it's a marriage, you know You know, the potential employee with a company both have to be happy But it has to fit together has to be that chemistry. So on both sides, we want to get to know each other So so that's for the hiring and in terms of diversity, right? One of the things I really love about Thorworks is that diversity is not just on our blog. We really celebrate diverse diversity in such a way that you know For example, there's the pick-not event that's coming up in July one although, you know Singapore government has been strict that this year you cannot Yeah, PR in Singaporeans only you foreigners cannot attend You know our office before that that was announced our Thorworks Singapore has really really been excited to be part of this one for this event But the fact that they they made that restriction this year a lot of the employees were really really sad that they couldn't join So let's say you know how about let's do work around. Let's do some sort of mini party before the event So so even when we're not officially allowed to join the event There's a lot of things going on, you know as work around to still celebrate diversity just very specifically We had two sessions of ask me anything where in the first session there was a Openly gay person like you really ask me anything so in the office right pigeon hole You know you can ask me anything about my experience being a gay man blah blah blah and you know Come if you compare that to other companies who say yes, we like diversity, but that's about it You know we put posters, you know we we send newsletters But this one is like technically they asked me anything session is not really anything related to consulting to technology, right? But but we do it in the office because we really you know, we celebrate diversity It's really important to us It may not be directly related to you know your client you're you know how you do your coding But we celebrate the different people in our office and we really want them to feel and understand to feel Safe and understand each other so so that so that's the just just wanted to really share the events that we're planning for I think that I will ask my last question for today Jack my mention recently that the secret sauce of Ali Baba is having a lot of women And he mentioned like because women kind of care they don't only care about themself, but they care about their family They care about other people Which means like a lot of things to do so my question would be how do you balance your work and and personal life? How do you manage? I mean your kids you just mentioned like you're traveling a lot. What how do you do that, right? Mostly imperfectly But but recognizing that and living with it before we were talking about I had a business meeting once where the nanny failed to Show up, so I just took my daughter with me on a plane to Boston. Those were the days when you could do that She'll still remind me that I missed three of her birthdays in a row because I had a meeting that was always the week of her birthday, but on the other hand she's lived in five countries and She's traveled around the world and she she you know You get the balance from the feedback from other people and you do the best that you can and be authentic about it And in the end you'll probably be terrific. I think Yes, there's there's a lot of chat about about this because you know women come across as having a bit more empathy perhaps But as I think back on on Spotify, I do admit I'm about 80 percent manager 20 percent mama bear But it's just it's that's who I am and so across my entire life whether it's my family or work or You know other things that that make up my life. It's it's my life just like any other human being and it's the same with With lots and lots of guys that I know where they've got their families and they've got work and They've got other parts of their lives as well, and I think again it goes back to To leveling that that playing field and and you know to to your point about you know men tend to You know go for jobs where they may be 80 percent qualified and women feel like they they need to be 100% qualified. I stopped doing that 15 years ago and My philosophy has been fake it till you make it So you because there will be so much that would get you in through the door And then as you learn on the job and everybody learns on the job That's gonna get you to that hundred and then that hundred and twenty percent and maybe on this point I will say usually try to skip the HR go to the hiring manager Maybe the one able to understand the what you can do even if you don't have done the same job at the competitive company before For me, I'd say First is prioritization. I prioritize those things first that are most important So whether that's working out first thing in the morning because I'll never get to it if I wait till the evening or whether that's You know, I I tend to work Hard during the week and I try to focus weekends on family time But the second thing I would say is I'm unapologetic so I think you said imperfectly and Certainly all of us are imperfect and I'm not Apologizing for that. So if I can't do something from home or there's there's something that I'm not able to do because of my travel schedule or whatnot, I mean my kids are resilient and They tend to understand I've you know, I grew up with my father traveling Monday through Friday I've traveled the majority of my career and You figure out how to balance it and I don't beat myself up for it While I don't have kids, but I have witnessed an amazing, you know like group of female leaders within my my company and I've had some as my mentor and sponsors and I think it's it's just a Touched on two things. One is discipline and Being very, you know, like structured in how you choose to spend your time and getting your priority, right? So if you can't travel on your, you know, child's birthday or you know, you can't get an any so being Forward about that. I think the team what just around you. They are supportive network and so so being very disciplined and your priority and the other one is on Imposter syndrome, which I thought was touched on multiple times So the research say that about 70% of people regardless of gender have experienced that At least once and it's just how you cope with it And I think what happened is you start it when you start having self-doubt and you're not, you know Recognizing that you get the success that you get because you earn it and you start thinking oh, you know what? It's it's it's just luck. I'm it's just luck. Someone gonna find out I'm fraud and when you start going down that route It's spiral down. I think I spent my first four years thinking I'm gonna get fired Every year like oh, they promote me again. I'm so gonna get fired. They're gonna find out I Think at some point you need to stop doing that because if you keep doing that you just spiral down so find your support system and Believe in yourself and if and when you have doubt Shut down the success that you've done validate yourself And use that as a extra push to then say you prepare more you you know You know more about the subject and put your hands up for opportunity All right for me personally work life balance is something that's very close to my heart. I Feel I feel for that topic almost every day Well like literally every day. So so how it is for me right now is I'm actually technically based in Thailand right now, but my family's in Singapore. So to balance both sides I travel Monday to Friday. I'm in Thailand Saturday Sunday. I'm here in Singapore with my family And it's very important to me For me to be here in Singapore every weekend because I have two kids one being just 11 months old And I'm also breastfeeding. So every every Friday slash Saturday very early in the morning actually Bring a big pack of frozen breast milk for my baby because you know, I want to give the best for my baby It's it's like I'm gonna do this not gonna fail as a mother. So So I often get that question like how do you manage to do that? Aren't you exhausted? the short answer is yes, I'm exhausted very very exhausted and There is no magic pill answer to say oh if you do this if you do this here here are the tips Everything will be fine. So it helps that I really really enjoy my job So when I am in Thailand, you know, I am there hundred percent. I'm enjoy I enjoy whether doing I get my I get fulfillment out of it and when I'm in Singapore I get you know, I try to maximize the time with my family as much as possible but that being said it's again, it's not One perfect answer say just follow this one. Everything will be okay. I Have good days and I have very very bad days as well and Exactly what they're saying, you know, if you have a bad day Just, you know, maybe give it a give a good cry and then pick yourself up do it again tomorrow and you know find Strength from the things and the people you love and just power through and in the end, you know Hopefully everything will work out fine and then you know, yay I've made my family happy and my career as well. Yes. So so there's no one straight answer And it's something that you know continually we have to balance again and again. Thank you very much Hello, thank you so much Wow, that was that was a lot of great conversation to take in as an emcee I'm supposed to usually summarize a panel. I don't know. I was taking notes and I kept getting longer and longer and longer And I to be honest that was amazing. So thank you Michelle Sunita, Rosanna, Susan, Nopra and Arnaud for starting off our morning with amazing conversations About gender equality about how each tech company has been so forward thinking but in their own ways and just kind of sharing with Us your personal stories your challenges and your own words of wisdom at the end. So thank you so much again


Alright, so what we are doing now is trying to understand what the spherical derivative of Meromorphic function is okay. So well you know the reason for all this is this idea of spherical derivative is important to study the topology of families of Meromorphic functions okay. See the reason is that normally you know there is a relationship between see basically we are interested in compactness okay and you know compactness is the same is strongly related to sequential compactness okay which is that you know given any sequence you have at least you are able to find a convergence of sequence okay and of course if you are worrying about Euclidean spaces then you know compactness is the same as closed and bounded but then you know for saying things like bounded you need a metric and so on and so forth but you know if you are working with spaces of holomorphic functions or analytic functions the point is that you know you will have to work only with normal convergence you will not get uniform convergence okay you will get uniform convergence only on compact subsets and then it is had it been only uniform convergence then you could have taken the soup norm okay and you could have used that to define a metric but the point is that you do not have uniform convergence you have only uniform convergence only restricted to compact subsets that is called normal convergence. So it is not so easy to think of a metric alright but then you still want to think of compactness and compactness is kind of related to sequential compactness and so then you know this is also connected with uniform boundedness it is connected with equicontinuity okay and it is also got to it is also connected with for example if you want boundedness of the derivatives okay so these are a bunch of interrelated results okay on the topological side this is the so called Arzela Ascoli theorem okay and on the holomorphic side or on the complex analytic side it is the so called Montel theorem okay which we need to prove okay and so you see the somehow we want to do it not only for analytic function we all do it for meromorphic functions because you see we have to worry about meromorphic functions because that is the these are the function that you need to study families of such functions to get to the proof of the Picard theorems which is what our primary aim is okay and since you are worried with meromorphic functions the problem is that you know they are not always different they are not differentiable everywhere I mean they are not analytic everywhere if you go to a pole at the pole of course the function goes to infinity so you cannot differentiate the function at the pole it is not differentiable because it is a singular point basically okay so your usual derivative will not work your usual derivative will not work at a pole so what will you do the method is that you introduce a spherical derivative because spherical derivative is something that will work even at a pole that is the whole point okay. So I want to tell you in general why we are getting so worried about why we are making so much noise about the spherical derivative because see that is what we need that is the thing that will work even for meromorphic functions it will work even at poles okay whereas ordinary derivative you cannot think of at a pole because it is a the moment you say pole it is a singular point and at a singular point derivative does not exist so you know you are in lot of trouble that is the reason we introduce a spherical derivative. So I was so let me continue with what I was telling you last time I was trying to tell you that the spherical derivative if f is a if small f is a meromorphic function as you can see here so let me use a different color for the moment say so if you have this f which is a meromorphic function on a domain D in the plane then there is this spherical derivative okay f hash of z and in fact this is spherical derivative in absolute value mind you I have put 2 times mod f dash of z by 1 plus mod f z the whole square so it is a non-negative real valued function okay normally a derivative should be derivative of a complex valued function should again be a complex valued function but this is not exactly a derivative which is complex valued it is actually positive real non-negative real valued and it is actually so you should think of it as absolute value of the derivative okay so we will when we say spherical derivative I mean absolute in absolute value okay and why is it that we are interested in this absolute value because it is a scaling factor you see so you see what is happening there you see the point is that you know as I was telling you last time see if you look at this if you take this function w equal to f of z which is a transformation from the z plane to the w plane okay then you know you assume it is an analytic function okay and assume it is not constant so then what happens is that you know if it is a non-constant analytic function then the image of any open set is open so if I start with this open set D here which is supposed to be for example in this diagram the interior of this dotted boundary okay then the image of that which is f of D is an open set okay and if I take a curve gamma inside D the image of gamma will be f of gamma okay and this f of gamma is now going to be a curve in the image which is f of D which is open and you know if I calculate the length of gamma it is given by the formula namely integrating mod D z okay because mod D z is the infinitesimal version of the Euclidean distance which is mod z1 minus z2 okay so you take if you take 2 points z1 and z2 on the complex plane then the distance between them is given by mod z1 minus z2 if these points are very close to each other you can call 1 point as z1 you can call the next point as z1 plus delta z then mod z1 minus z2 will become mod delta z and you replace delta by D to get the infinitesimal version so you get mod D z so mod D z is just the infinitesimal version okay it is called you may also call it the you know the element of arc length which you have to use to integrate okay and if you integrate the curve over the arc length you will get the length of the curve okay and of course it is very important that the curve needs to be rectifiable okay it should be a curve which has finite length okay and that is why we always put a condition that we work only with contours and contours are you know they are continuous images of a closed bounded interval which are in fact piecewise smooth and in fact piecewise continuously differentiable okay and for such curves for such contours length will always be a finite quantity and so you have this so the point is that you integrate mod D z you will get the length of this gamma alright now on the other hand what happens if you by the same philosophy you know if you integrate mod D w here over its image which is f of gamma you should get the length of f of gamma which is the f of gamma is the image of gamma and ref which is the image curve but you know but here the variable of integration w is f of z so if you integrate mod D w I mean if you substitute for w f of z then mod D f z will become mod f dash of z into mod D z okay and you see the difference between this formula here and this formula here is that there is this extra factor of mod f dash of z okay so what it means is that if you simply integrate over mod D z you will get the length of the source curve if you integrate if you multiply it by the modulus of the derivative you get the length of the target curve the image curve so the point is that the extra factor you have to put in the integrand to get the length of the image curve is the derivative the modulus of the derivative okay. Now in the same way what is happening going to happen is the following thing if you take f to be if you take the function f to be a meromorphic function on D okay now what you are doing is your see you have the complex plane okay and you have this domain D inside it and you have this function f and this is well I am now thinking of this function as a function into c and infinity mind you it is a meromorphic function so I am thinking of it as a continuous function into the extended complex plane because I define the value at a pole to be infinity okay and with that it is continuous and how do I actually think of this so you see that is this that is this this is identified via the stereographic projection to the Riemann sphere okay which I think I should put it as S2 S2 two sphere okay centered at the origin in the in three space real three space radius one unit okay and with the north pole being identified with the point at infinity okay. Now you see what you do is you essentially think of this look at this function okay if you look at this function think of the function as a map from the domain in the complex plane onto the sphere okay so geometrically whenever you are thinking of the Riemann whenever you are thinking of the extended plane think of the Riemann sphere geometrically that is how you should think of it okay so this function f think of the function f as going into the Riemann sphere so you know if you want you know its abuse of notation I will still call this as f okay in fact I should it is f followed by this stereographic projection so in principle I should call give it some other name but I will still call it as f because I am thinking of this as an identification I am thinking of the stereographic projection as an identification okay I am identifying the extended plane for all practical purposes with my Riemann sphere now what happens now what happens is that you see on the one hand you have the complex plane okay and you have this domain in the complex plane okay so this is my domain in the complex plane and the variable here is z and then on the other hand I have the sphere I have I have the Riemann sphere okay and what is happening is that you know if I now take if I now take a small if I take a if I take a curve gamma here okay in the complex plane in my domain and take its image under f okay then what will happen is that the image of this curve will also give me a curve here on the Riemann sphere okay now so this will become the curve f of gamma and how will I get the length of f of gamma okay now think about what I just told you some time ago to get the length of the image curve you have to integrate over you have to integrate over the original curve with the you with the original you know with the original infinitesimal element of arc length and then you have to scale it by the modulus of the derivative okay but now you see what I am doing is I am actually trying to get the I am trying to get the length of the image curve it is the length I am getting is actually the spherical length see it is the length on the Riemann sphere and the length on the Riemann sphere corresponds to the length on the on the extended complex plane given by the spherical metric see the only different with the only point you have to remember is in my target the metric is not the Euclidean metric it is the spherical metric okay I have to use the spherical metric I have to use the element of the spherical metric is what I have to integrate so what is the length of f of gamma under the spherical metric so I will put L sub s to just to indicate that this is length of spherical metric what is this is just I have to integrate over f of gamma okay I have to integrate over f of gamma I have to integrate over modulus of d z sub s I will keep putting the sub s to just emphasize that I have to integrate over an element of the spherical arc length but what is the element of spherical arc length you see so in fact I think I should not put yeah so it is important that my variable let me call this variable as W okay and I should be careful inadvertently you make mistakes like this see I am integrating over f of gamma so my variable should be in f of gamma variable of integration and that has to be not z z is in the source z is on gamma whereas W is what is on f of gamma so you know if this should not have been z I should correct this this should be d w sub s okay it is an element of arc length spherical arc length with respect to the variable W on the sphere alright but W is f of z so this transformation is given by W equal to f of z okay and the only funny thing is that this W is now on the sphere mind you W can take the value of the north pole which is corresponding to W equal to infinity that is also allowed now okay because we have allowed values in C union infinity extended plane alright now you see but what is this what is this what is this element of spherical arc length I told you that last time that the element of spherical arc length is actually 2 times mod d W the usual Euclidean element of arc length divided by 1 plus mod W the whole square this is what the element of spherical arc length is and see the you know the reason why you got this is if you want as an aside let me write that down you know and use a different color so you see so what happened was that if you take the spherical distance between 2 points W1 and W2 okay then that turned out to be 2 times mod W1 minus W2 by square root of 1 plus mod W1 the whole square into square root of 1 plus mod W2 the whole square this is the spherical distance between 2 points W1 and W2 on the on the extended plane or the Riemann sphere okay this is the spherical distance it is actually the in fact I should not even say this is not spherical distance sorry this is actually the this is the chordal distance in fact okay yeah this is still not the infinitesimal version sorry so this is the d sub c this is the chordal distance so you take 2 points W1 here and W2 and you join them by this chord okay it is a line segment it is a line segment in 3 space joining those 2 points on the Riemann sphere but you see mind you those 2 points are actually points on the extended plane I am simply identifying the extended plane with the Riemann sphere so I am still writing W1 and W2 where actually I mean the stereographic projection of I mean the stereographic projection of W1 and the stereographic projection of W2 okay so see if so this is the chordal distance this is the chordal metric and what is this metric this metric is a metric in R3 mind you this sphere this Riemann sphere is sitting inside R3 this is this is inside R3 okay and in R3 I am simply measuring the distance and I asked you to check that this is a this is an exercise for you to check that oops it is an exercise for you to check that this is the distance formula okay I asked you to do that you should try you should do it if you have not done it so far. Now you see this is the chordal arc length if I want the spherical what is the spherical arc length is this this arc length and what is that arc length I take the great circle there is only one big circle on the on the Riemann on the sphere which passes through these 2 points and that circle with these 2 points 2 points in a circle determine a minor arc and a major arc okay and you take the length of the minor arc that is the definition of spherical distance. So if I want the if I want the spherical length okay and I want the infinitesimal spherical length that is that is what this quantity is this dW sub s is the infinitesimal element of spherical length and for that what I will have to do is I will have to bring W1 and W2 very very close okay and as I bring W1 and W2 very very close the chordal distance will approximate the it will come close to the spherical distance okay. So what I do is you know in this in this calculation in this formula what you do is you put W2 is equal to W1 plus delta W okay and then and then you write it in such a way that you only allow you only worry about delta W and do not worry about delta W whole square delta W whole cube higher order terms because you think of them as being very small and negligible and then you change the delta W to dw. So what will happen is that this will this thing this formula as W1 tends to W2 okay this formula will give you this formula. So you see in the numerator instead of W2 if you put W1 plus delta W the numerator will become 2 delta W and this both of these quantities will become equal to root of 1 plus W2 so you will get the square of that which is 1 plus mod W the whole square okay. So each of these quantities will become square root of so the first the second term will become square root of 1 plus W1 plus delta W mod the whole square okay and as delta W tends to 0 you will get this quantity. So this is the this is how you get this infinitesimal element of spherical arc length and integrating over that integrating that over the curve f of w f of omega should give you the length of f of omega the spherical length of f of omega which is what we are interested okay but then in this you plug in what W is see your W is fz so plug fz inside that if you plug fz inside that what will you get you will end up with well okay let me go back to the other color that I was using yeah so what will I get I will get well I will get integral since I have changed I have made a change of variable w equal to f of z now the variable becomes z and z now varies over gamma so I will put gamma here okay and if I now calculate this mod d fz will become mod f dash of z into dz so what I will get is 2 mod f dash of z mod dz by 1 plus mod f of z the whole square and what is this if you look at it carefully what is this this is just 2 times sorry this is just integral over gamma oops this is just integral over gamma 2 mod f dash of z divided by 1 plus mod fz the whole square this whole thing multiplied by mod dz so what are you getting you are getting length of the image curve and in the spherical metric of gamma under f is this formula and go you go back to what I was telling you some time ago if you want the length of the image curve you have to multiply by the modulus of the derivative okay if you simply integrate you will get the length of the source curve but if you multiply by the modulus of the derivative you will get the length of the image curve so you see what this tells you this tells you that if you want the length spherical length of the image curve f gamma under f you will have to multiply by this quantity here and that quantity therefore has to be the absolute value of the spherical derivative therefore the spherical derivative so this is what we call as the so this is what is called as f dash of z this is called the spherical derivative of f okay and mind you this is I should write it in bracket it is in absolute value this is an absolute value alright because it is a so you must remember you know go back to your first course in complex analysis when you take f dash of z okay if you take a point z0 if you take f dash of z0 where suppose z0 is a point where function is analytic so the derivative exists you take f dash of z0 what is the what is the geometric significance of f dash of z0 see the modulus of f dash of z0 is the scaling factor it is locally the factor by which an image is scaled if you take some if you take a small square containing z0 a very small square containing z0 and you take its image under f you will get something that looks like a square okay okay and its area will be you know its length will be scaled by mod f dash so the modulus of the derivative is a scaling factor the argument of the derivative is a rotating factor it is a factor of rotation okay but the argument of f dash of z0 is the angle by which the tangent rotates okay if you have a source point and you have a curve passing through the source point z0 and you have tangent at that point now you take the image curve which will pass through f of z0 the image point and you take the tangent there the difference in the angles that the tangents make with the x axis is precisely argument of f dash of z0 okay so the geometric meaning of f dash of z0 is that the modulus of f dash of z0 gives locally at z0 the magnification factor and the argument of f dash of z0 gives locally the rotation factor this is how geometrically the map f behaves locally and you know if the derivative f dash is not 0 then you know it is conformal you would have studied this in a first course conformal means you know it will preserve angles between curves okay so for example if you take something like a something like a square okay its image will be something like a distorted square alright if you take something like a circle its image will be something like a distorted circle you can expect it to be like for example something like an ellipse or something like that okay and this is of course if you consider it sufficiently small okay and at a point where the derivative does not vanish okay and this is why it has got so many applications to engineering because of conformality so you see the why I am saying trying to tell you all this is that the modulus of the derivative is the magnification factor and therefore multiplying by multiplying the infinitesimal arc length by the modulus of the derivative always gives you the length of the arc length of the image curve and that is what is happening here you see this is the multiplication factor this multiplication factor is therefore called the spherical derivative okay now I want to tell you a few things few very very interesting things in this integral see first and foremost the amazing thing about this is that you know I told you f is a meromorphic function okay f is a meromorphic function so you see f could have poles okay f could have poles of course they are isolated but f can have poles and the beautiful thing is your curve gamma you see your curve gamma can pass through those poles okay now that is the amazing thing see normally when you do integration you never tried the integrand is always supposed to be continuous okay when you integrate an analytic function or for example whenever you do Cauchy's theorem or you know you do argument principle in all these things when you want to apply you always make sure that the contour does not pass through any singular points it cannot pass through poles it cannot there are cases when you are doing the logarithmic integral in the argument principle you will assume that the contour does not pass through any poles and also through any zeros okay you do not allow the contour even to pass through zeros because you are integrating the logarithmic derivative which is f dash by f and if there is a zero then denominator f will have a zero then you cannot integrate it so in all these things that you have been that we have been doing so far we always make sure that the contour on which you are integrating does not pass through any zeros or poles it should never pass through poles of course but also not through zeros if you are trying to apply the argument principle but now mind you we are not we are dealing with meromorphic functions they have poles and my point is now the contour gamma can pass through as many poles as you want it will not create any problems okay it will not create any problems for this integral because you see that is a matter of calculation that you have to understand I will tell you roughly suppose your contour gamma passes through some poles mind you that can be only finitely many such poles on the contour because you see the set of poles is anyway an isolated set by definition of pole is isolated okay it is an isolated similarity so the set of poles is an isolated set and if you take the set of poles lying on gamma it is an isolated set it is an isolated subset of a compact set mind you gamma is compact any contour is compact it is closed and bounded okay because it is actually continuous image of an interval closed and bounded interval so it is closed and bounded it is compact and you know any isolated subset of a compact set is finite because had it if it were infinite it will have a limit point okay and that limit point will not be isolated okay therefore what will happen is that you have if gamma passes through poles of f mind you f is air meromorphic it could have poles if gamma passes through poles of f it can pass through only finitely many poles because gamma is compact and what happens at a pole see nothing happens to the integrand at a pole it is bounded that is the beautiful thing that is why this integral is valid even if gamma passes through a pole or several poles that is because you just imagine suppose f has a pole at z0 okay then in fact you know we can write this down suppose so let me write this down suppose f of z so let me rub this and probably go down a little bit so that I have more space suppose oops suppose f has a pole at z0 which is lying on gamma this will create this will not create any problems this will not create any problems why because you see in a small disc centered at gamma sorry centered at z0 you see f of z will look like g of z by z-z0 to the power of n where n is the order of the pole okay now you calculate you calculate this quantity okay notice that and of course you know g of z0 is not 0 okay g of z0 is not 0 and of course this is how a pole locally function looks locally at a pole of ordering alright and g is of course analytic okay g analytic at z0 okay you have this now you see just look at this expression that I have written about if I take f dash if I take the derivative the derivative will continue to have a pole of one more order if I differentiate this gz by z-z0 power n if you want using quotient rule then I will get a z-z0 to the n plus 1 of the denominator so what I will get is I will get a pole of higher order okay of order greater order and if you go to the denominator the denominator will have a 1 by it will have a mod gz the whole squared by z-z0 to the power of 2n and as z tends to z0 you will see that the numerator tends to I mean this whole quantity will go to either 0 or to a finite value okay because what is actually happening is that as z tends to z0 z-z0 is going to go to 0 so f is going to infinity alright but the fact is that the denominator will go to infinity faster than the numerator because the denominator contains f squared f squared has a pole of order 2n whereas the numerator has a pole of order only n plus 1 okay therefore the denominator will go to infinity faster than the numerator as a result the integrand is bounded so the point is that this integral is valid even at a pole that is the big deal there so this integral is your gamma can pass through poles of f there is no problem and geometrically also you should believe this because if gamma passes through a pole of f its image will pass through the north pole on the Riemann sphere after all at a pole of f f is taking the value infinity and that corresponds to the north pole on the Riemann sphere so after all what you are saying is that the image curve is passing through the north pole of the sphere how does it matter it is not going to affect it is no the north pole of the sphere is in no way different from any other point on the sphere okay so the moral of the story is that shows that the L s of f gamma is well defined even if gamma passes through poles okay so this formula is always works it works even for a meromorphic function it even works if gamma passes through poles no that is a very very important thing okay and then there is so this is something that you need to know and there is another fact that I expand into in the next lecture the fact is that if you take the reciprocal of f you see f is a meromorphic function then 1 by f is also meromorphic function and the beautiful thing is that if you take each spherical derivative you will get exactly the same as a spherical derivative of f. The reason is because of the fact that the spherical distance is invariant under inversion I told you the inversion on the complex plane translates to a rotation of the Riemann sphere about the x axis and it leaves spherical distance is invariant therefore the spherical derivative is also invariant if you invert the function okay so this is another important fact that we use and the advantage that you can replace f by 1 by f is that whenever f has a pole 1 by f has a 0 so you can reduce from the meromorphic case to the analytic case so you can still work with analytic functions you see this is the advantage of having spherical derivative okay so spherical derivative does not distinguish between f and 1 by f and the advantage of moving from f to 1 by f is that you can read what is a pole for f becomes a 0 for 1 by f and 0s are very friendly more friendlier than poles okay in the neighbourhood of a 0 you can usually you apply usual analytic function theory because the function is after all analytic okay so that is the advantage so that is another motivation for having a spherical derivative okay so I will stop here.


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