and welcome to Gallery Works. We have a special treat for you today. We have glass artists and award-winning famous, internationally famous glass artists, Beth Lipman with us today. Hello Beth. Hi. Thank you for coming. Thank you so much. Thanks for having me. Oh, it's a pleasure. And I am just, you know, I saw an article in the newspaper about you a few weeks ago and I read it and I thought, who is this gal? I've got to meet this gal, you know? And the fact that she lives right here in, you know, Sheboygan where I live and everything, you know? And so then you were so gracious, you know, to set the date with me and everything. And then since then, to find out that you've won this huge award, a $50,000 grant. And you're one of 52 people out of how, what was it, like 2,000 or something like that? It actually wasn't that many, but it was about, it was a little over 300 people are nominated nationally and then those nominees apply. So you have to be nominated and apply to be selected. That's fabulous. That really is. It's pretty amazing. I'm still pinching myself, yeah.  How do you juggle all that? Yeah. Well, I actually, I have a super supportive husband. He is the studio and business manager for my art. So we're working, we're both working full-time, basically in the studio. I don't know, I wouldn't be able to do it without him. His name is Ken Sager. And so we split the days, half the day I'm in the studio making my work and then the other half of the day he is doing everything besides making the work. So whether it's studio maintenance or packing and shipping or even making some components for the work. And then whenever one of us is in the studio, the other one is with the kids because they're under three. So are they twins? Yes. So I'm sure once they start preschool, we'll have a different schedule, but that's what we've done for the last two and a half years. It'll be a little bit easier once they start preschool. I can just see that that could be a little, breathing very, little breathing. Yeah, yeah, yeah. And now tell me, how did you get started with this particular form of art? Well, I went to art camp when I was 15 and I took a glass blowing class. It was called Horizon's New England Craft Program and the process looked interesting to me at the time. I was a teenager, I was very artistic growing up.  So my parents were both very supportive of me looking towards the arts or practicing art when I was young. And when I said I wanted to go to this program, they were really enthusiastic and sent me there. And there was a variety of different things being offered at that camp and I just happened to choose both glass blowing and also textiles. And that really kind of piqued my curiosity and I ended up going to art school for glass and fibers afterwards. So. And now the article that I read didn't mention anything about fiber work. So you have drifted over to the glass and have stayed with that. Yeah, although I have to say there's a lot of my process that I think is inherently more kind of aligned with a fiber process. I make all of my work on templates, brown piece of paper that become templates and that idea of kind of assembling a pattern and constructing in that way from multiple parts is pretty historically, yeah. Yeah, I see. It's historically more of a fiber way of approaching making. So it's there. It's just kind of deeply embedded in the process. Yeah. And you have been at John Michael Kohler Art Center in residence, right? Yes. And was that once or twice? I was there in residence in 2003 in the pottery. And yeah, there was really a life changing moment for me. That's how I ended up here in Sheboygan Falls because in 05, the position overseeing that residency program became available and I took that position and moved back here. Oh, really? So I worked as an arts administrator for over 10 years, essentially. I've only recently been practicing my work full time, so. Well, you have done some really major things. The Norton Museum, I believe, and many others. And you know, tell us a little bit about that.  What happens, does the museum call you up and say, say, I saw something of yours? Yeah. Yeah. I would probably recommend that most artists don't contact a museum and say they want to do a commission. As you know, Kitty, it doesn't really happen. Right. But I was, so the new director at the Norton Museum of Art, Hope Alswang, was the director of the RISD Museum in 2008 and I was invited to do an installation there at that time. And when she left the RISD Museum and landed at the Norton Museum, I just kind of kept in touch with her and she called me out of the blue and said, oh, you know, we'd love for you to do something. We have an amazing still life collection. So the process took probably over a year when it was all said and done of just having a dialogue, visiting the site, seeing what they had in the museum. I have a primary, my primary gallery is Heller Gallery in New York. I also work with Katie Tompkins projects in Rhode Island. So Heller Gallery negotiated the contract. And so I had kind of a general idea of what I thought I might do for them. But when I really actually visited the site last year, almost exactly a year ago, actually, well, you know how inspiration comes. It's like this thing that almost gets channeled in from the outside or something. So usually when I go to visit sites, when I'm looking to do something site-specific for someone, I almost feel like I have another sensory thing that's happening that I just am open to, you know, I know that I'm there for a focus. Exactly. And I've been fortunate in that I haven't been left high and dry to date, that I usually get some understanding of what needs to happen when I visit that site. It's the same way with a painter. There's a point where the canvas actually tells you what it needs. And as long as you're open to that, it's a deep sense of a connection, a spiritual connection with your work and the creative process that makes that happen for you. I know it's really wonderful. It's hard to describe as well, but I think it just takes a lot of solitary time. I think a lot of puttering around as well, kind of allowing the problem to percolate in the peripheral vision of what's going on in your life. And it's like when you forget something and you can't remember it in that moment that you needed. And then after a period of time it comes to you, I think it's the same. Right, exactly. It's inherent in the creative process. I totally agree. And I've been really fortunate in that the last several years in my work, I've had many opportunities come to me that allow me to evolve and change as an artist. And I'm being given the opportunity to do this and being compensated for it. Following my muse now I'm able to have, thankfully, a number of people that have responded to my work that are giving, affording me the opportunity to kind of continue to create work. Right, and so now you're at that point where you really feel validated as an artist. I do, especially the last several years. I feel like, OK, what I'm doing is worthwhile. Certainly getting the USA grant, United States Artist Grant. It's a fellowship, essentially. It's just even more than the money, it's a form of validation that what you're doing matters, especially in living in this society where there is really not a lot of condonement from the government, that the arts matter or anything like that, that it's really... Well, there's not... There's more private support than public support. I personally feel that things seem to be so impermanent. Everything seems to be so in the moment that's like, oh, well, that's so five minutes ago, or something like that. I mean, I don't know that, and that kind of bothers me, but I think that really good art transcends that. And that's why it's important that the artists continue to create that really fine art, too. Transcend that throwaway, whatever is the latest fad-type mentality that I think we as Americans a lot of us have, just my thoughts. No, but I think it's also completely in keeping with the age of computers as well and social networking and everything is a nanosecond now, every single thing. Everything's just moving so fast. Yeah, so when you make something, it is permanent. It does have a permanence. And we need that. In the fleetingness of life or in trends, as you said, that kind of thing. Right, I agree.  These are components, some components, for an installation that you had at the studio. Because naturally, your installations, you can't bring them in there. I could, but it would take a bit of time, more than a half hour of time. But can you just take these pieces that you have on the table and move them around a little bit and just give us an idea of your thought process, if you can verbalize that? Sure.  They're made with a variety of different techniques. Sometimes I work with different people as well to create works. So not only am I the one making components, but other people help me. And I think that broadens my vocabulary, actually, in the making. So I amass a certain number of objects. And then I start to reduce. And like I said, I work on a table with a blank piece of paper. And I'll just start to put together things that it's a very formal exercise. I try to arrange things in a way that begin to form a dialogue with the different objects.  The still life genre has been around since the early 1600s. And even prior to that. Yeah. And I also think that it is really in keeping with our practice in the domestic realm as well, in the interior. It's very much about that nesting thing. So it might take me months to really resolve a composition, but I essentially move things around a lot in my studio practice. I call it puttering around. I'll come in and basically I'll listen to what the work needs. And I'll just try to figure things out, see how they fit together, what the relationship is in between different objects is interesting to me. This kind of opening that happens and this is a broken. I use things that are broken as well. I am essentially non-judgmental about the objects. So the objects, for me, are really kind of capturing a moment in time with whoever made those objects. So it is not in my place to take this object and say it's good or bad. It is an object that exists. So there's a place for everything in my work. And I basically figure out where things are supposed to exist in relation to other things. So this is a broken goblet that I know is going to have exactly the right place in some composition. It's going to really click together into a puzzle in the way that it absolutely is a sentence. You know exactly what I'm talking about, obviously, because you also practice. So these are artichokes. These are solid sculpture. What I do, what I will say about when I work is I do try to keep in mind very common, formal parameters. If something is heavy and solid, I'll try to pair it with something that's lighter and more delicate. There needs to be height, varying heights within the objects. They need to relate to one another. Some things are very thin and delicate. Other things hold a different kind of weight. So this is not my most ideal. Absolutely, absolutely. Yeah, so for instance like this, and I also try to play with almost the impossible, like what happens if this knife were to be balanced within this bowl of bananas in some way. So the impossibility of the composition is also kind of important to me. And that's also in keeping with the still life tradition, where you're seeing perhaps objects on one table that would never have existed on one table during the 1600s. And I think the parallel that I think about now in relation to that idea is just the amount of access that we have in our lives, and what we choose to collect and own, and what it says about us is all very interesting to me. So the work tends to be pretty excessive. So anyway, I could do this the entire time essentially. So I'll stop now. So it just, yeah, I try to give myself enough time, whatever the deadline is, that I try to work backwards with enough time to know that I need that time to have things come to me, which is what we were just talking about. I think it's just fascinating. And you must have a kiln and blow the glass. But the thing that I'm curious about is do you ever work with colored glass? Oh, yeah. I rarely do. I find, because I'm working in historically decorative material to begin with, that the color for me is a little bit too decorative, actually. And also when you add color, even if it was a transparent color into the material, it does change the way that your eye moves through the material, as well as having the material reflect back at you. So it's luminosity, I think. Yeah, there's something. For me, the clear is really the essence of the object, or the essence of the idea. I do also work in black and occasionally white. But usually the black is really kind of in keeping with Victorian decorative arts, which I'm really quite interested in. So for me, it's a very more funerary or kind of a morning decorative art, when I'm using the black, it is heavily symbolic in that way. Although I will also say in the Asian cultures, white is also a symbol of mourning and death. So that's also very interesting in keeping with the still life tradition of Vanitas as well. So yeah, I don't really use color. I don't have a need for color. I think that's marvelous. I think that's marvelous. You're the only glass artist that I have ever seen that works with clear glass. Not that I've seen that many, but I know quite a few people that work with glass. And I just think that's marvelous. I mean, I can just, I think it's luscious. I'd like to show some photos of installations, of completed installations that you've done, and have you tell us about them, OK? This first one, what can you tell us? Well, I just finished it. Month ago, it's called One in Others, and it is a commission for the Nord Museum of Art. It's going to be installed at the end of January and be on view through the spring there. And this, what you kind of can't see in the image, but the piece is actually essentially sitting on a casket. So the casket is to my dimensions. So I consider this a composite portrait between myself, the Nord Museum, and their permanent collection holdings, because a lot of the objects found on this composition are in reference to a lot of the still life paintings in their collection. Additionally, the Nord Museum is built on an early settler's grave site. Oh, my. And there is a man in the basement that was a pineapple grower. So the entire piece is covered with pineapple plant parts. So that's the pervasive kind of overall message. No, it's not a message. It's like a persona. It's the thing that's consuming the composition. So yeah. So that's what that is. Yeah, it's beautiful. And now this one here, this really touches me. I think this is just absolutely gorgeous. This is Pitcher with Vine. I also just completed this piece this year. And in fact, this is the shot from the back of this piece. You enter the piece from the front where the vine just barely creeps over the front of the table. But several years ago, I started really studying kudzu vine and also the kudzu plant, which many people know is growing invasively in this country. And so I'm interested in the manifestation of the natural imbalance or a moment of excess within nature that is somehow connecting with the still life, which also talks about a certain abundance or excess or can. Not everyone chooses to work in that way within the still life tradition, but that's something that has been a thread in my work for the last 12 years or something. So at this point, these two ways of being kind of imbalanced or excessive in the world come together, the natural and the human made. It's beautiful. Thank you. And now, is this installed someplace, Bride? It's in my barn right now. But it will be installed actually in Oklahoma this coming summer. Wonderful. Yeah, the Oklahoma Museum of Art in Oklahoma City. And Bride is a piece I made last year. And what you kind of maybe can't tell from the image, but essentially, there is a certain amount of order and formality at the very top of the work. And then it kind of descends into chaos towards the bottom of the work. So it's beautiful. Yeah, beautiful. And now, this one is of the black glass that you were talking about earlier. Yeah, this is candlesticks, books, flowers, and fruit. And this piece actually is going to the De Young Museum. They just recently acquired it. And where is that? San Francisco. Wonderful. So it's marvelous. Yeah, I'm very excited about it actually. How many museums are you in? I don't know. Not so many that I've lost count, but I don't know. Yeah, I would say maybe 10 to 15. That's marvelous. And out of the United States also, right? Yeah, I've had exhibitions outside of the United States. And some of my work is in private collections abroad, but there's no permanent collections outside of the States that I'm aware of. I'm thinking of it at this point. Yeah, I'm sure. And now this one, this is one of my favorites right here. I love this composition. Oh, yeah. Yeah, this is called candlesticks. Now, this is a print on plexiglass. So in 2007, I started creating compositions for photographs. And so I'd create the composition. I worked with a photographer, Rob Quinn, who is basically my hand. And we take photographs.  So I'll either destroy or recycle the glass within it. So there are additions of eight. Oh, OK, OK. Well, that's a little different then. The print on the plexiglass, that's a rather unique approach, I think. Yeah, I don't know what gave you that idea. One of those things, once again, I was sitting in dialogue with a curator about one of my pieces. And it was, we were looking at the photograph and admiring the photograph. And I kind of had this aha moment. Oh my god, it would be amazing to just do compositions for photographs. And then I do, I have those like aha moments every couple of years. You know, a year or two might go by and oh my god, I could do, you know, this is important. And then I just like, you know, then it's just the perseverance, the tenacity to keep kind of exploring those ideas. And we're almost out of time. Real quick, where is your next installation going to be? Oh, OK. Well, it would be at the Norton Museum. One and others is the very next one. And then also that following that is Art Pump Beach, which is an art fair. So my work will be on view there as well. And you also have something at the Milwaukee Art Museum right now, right? Yeah, I do, actually. That is up as we speak. It's called the Tool at Hand. And how long is that going to be there? I believe it's going to be there through April. Marvelous. I'm going to see it. I tell you, all of you go down into the Milwaukee Art Museum and see this woman's work. It's just fabulous. If you could just see it and look at it up close, you'd be amazed. Once again, I thank you so much for being on the show. And so this is Kitty Lynn Klisch for Gallery Works. We've been in the studio today with Beth Lipman, glass artist or extraordinaire. Don't miss the next show because it'll be another great one. Bye-bye for now.


So, let us continue with what we were doing. So, this is lecture 3. So, we defined so recall that tau is a subset of the power set of R 2 consisting of subsets u which satisfy this property star and recall what property star was if x is in u then there is epsilon positive such that S epsilon ok. So, instead of x I should write a comma b is point in u is completely contained. So, let us check that tau satisfies the conditions defining topology. So, the first condition is phi and R 2 should be in tau. So, as in the example of the standard topology on R. So, phi is in tau is true is vacuously true because there are no points in the empty set and similarly R 2 is in tau is also clear because if we take any point in R 2 we can always take a square of side length 2 for any a comma b in R 2 we can take S 1 a comma b this is obviously contained. So, obviously the first condition is satisfied. So, let us look at the second condition the second condition requires that if we take finitely many elements in tau then the intersection in the intersection u i is in tau. So, to check this. So, once again so let we take a point a comma b which is in this intersection and the same proof as in R since a comma b is in u i and u i satisfies star there is epsilon i positive such that this S epsilon i of a comma b is contained u i right. So, we have our u i may be some set like this and we can take a point a comma b here and there is some epsilon i this S epsilon i this is u i right. So, we take epsilon to be equal to the minimum of epsilon 1 epsilon 2 up to epsilon n and clearly S epsilon a comma b is contained in S epsilon i a comma b right and. So, therefore, S epsilon a comma b is contained S epsilon i a comma b is contained in u i for all i. So, this shows that S epsilon a comma b is contained in the intersection right. So, thus the intersection satisfies this property star and therefore, thus the intersection u i is in tau. So, therefore, tau satisfies condition 2 and let us quickly check that it also satisfies condition 3. So, condition 3 was given a set i for each i in i an element u i in tau and then we need to check that u i is also in tau. So, once again we let a comma b we need to check that the union satisfies this property star. So, let a comma b be an element in the union which implies that a comma b is in u j for some j in i which implies that there is epsilon positive such that the square of silent epsilon around a comma b is contained in u j as u j satisfies is in tau and. So, satisfies this property star and which implies that S epsilon a comma b is contained in u j which in turn is contained in this union. So, thus the union satisfies and so is in tau. So, this shows that tau also satisfies the third condition to define the topology. So, therefore, tau defines a topology and our sixth example which I will leave it as an exercise is the standard topology. So, this is very similar and let me just give some hints how to do how to define this. So, let us take a vector x let us write a vector in R n as x. So, this is a n double and we define once again this subset S epsilon of x to be those y in R n such that the absolute value of x i minus y i is less than epsilon ok. And once again we define this property star in the same way.  So, for every this is this property vector x in u there exists epsilon positive which once again epsilon may depend on x such that this subset S epsilon x is contained. And we let tau be the collection of subsets u contained R n which satisfy and I will leave it as an exercise that show that tau defines a topology ok. And so, this is one exercise and I want to give another exercise. So, before I write the exercise let me just. So, in the example of R 2 or even R n. So, in the example of we could we can ok first define. So, let us write like this. So, define for x in R n epsilon positive define the subset B epsilon of x as this is those y in R n such that summation i could 1 to n y i minus x i square is less than epsilon. So, this is open ball of radius epsilon around and we can define.  So, this is this property for every x in u there is epsilon positive which may once again depend on this point x such that this B epsilon of x is continue ok.  So, then there are two exercises here one show that tau prime defines a topology and two show that tau. So, this is tau in example 6. So, example 6 is this one. So, the standard topology on that we define is equal to. So, what is what do I mean by the second exercise? So, both tau and tau prime are subsets of R n show that these subsets are the same I am sorry are subsets of the power set of R n. So, show that tau is equal to both the subsets are the same. So, this the second exercise let me make a remark exercise 2 above shows that the same topology can be defined in different in different ways. So, we have seen quite a few examples of topological spaces now. And we want to give more examples. So, we want to explain now explain some tools or some methods which we can use to define topologies on topological spaces. So, for that so next we want to explain some ways we can use or some methods we can use to define topologies. So, our aim is to construct a large collection of topological spaces and study the topological properties. So, first we have to construct a large collection of topological spaces. So, we are headed in that direction. So, let us give a definition before we proceed a very basic definition of open sets. So, let x comma tau be a topological space. So, a subset u of x is set to be open if open in tau if u belongs tau ok. So, if we have a set a set x and we have given a topology to x. So, subsets of x which are in tau will often be called open subsets. So, whether u is open or not depends on the topology we have given to x. So, with this so let us define a basis basis for a topology. So, let x be a set let tau be a topology on x. So, a collection B. So, B is a subset of tau is called a basis for tau if it satisfies following two conditions the following condition that is not true. So, given any u in tau and x in u there is an element w in B such that x belongs to w and w is containing u. So, let us see an example of a basis we have already seen it in a form, but let us just make it more explicit. So, example let x be the real line let tau be the standard topology. So, let B be the collection of intervals a comma b. So, first of all we claim that every interval a comma b is open in the standard topology. So, what do we mean by this statement we just we simply mean that that is every interval a comma b is in tau. So, this is an easy check it is yeah we already did it in the example of 0 1. So, you can do this you can try this exercise. So, we have it is the same we showed that the interval 0 1 satisfies property star and the same way. So, hint use the same method or I do not know idea that we used to show that 0 1 satisfies property star to show that the interval a comma b satisfies. So, recall that the standard topology on the real line was defined using the property star and we showed that 0 1 satisfies the open interval 0 1 satisfies this property star. So, the same proof will can be easily modified to show that this interval a comma b also satisfies property star ok. So, the conclusion is that thus B is indeed a subset of tau ok and we want to check that we want to check that B satisfies this condition. So, let us check that. So, suppose so, we claim. So, let me write as a claim yeah. So, claim B is a basis for tau. So, let us prove this claim.  So, we take. So, let w be the interval x minus epsilon comma x plus epsilon then x belongs to w w is an element is an interval and therefore, it is an element of the set B and clearly w is contain u right. So, this shows that B is a basis ok. So, similarly so, similarly the collection let me call it B 2 consisting of those sets S epsilon a comma b where a comma b is a point in R 2 and epsilon is positive form forms a basis for the standard topology ok. So, this can be checked in the same way that we checked for R and once again similarly the collection B n this is the collection of those S epsilon x where x is an element of R n and epsilon is positive forms a basis for the standard topology. So, we have seen we have defined the basis for a topology and you can familiarize yourself with the definition of a basis and these three examples how these three examples of standard topology on R and R 2 and R n and the basis for these and in the next lecture we will see two important properties that these basis have that a basis has and using those two properties we shall explain how given any subset B of the power set which has those two properties we can use that set B to define a topology. So, we will end this lecture here.


This is the Fandal Hurry Up, my name is Jim Son. It's here today to tell you about wide receivers I think are undervalued in 2021 for season long fantasy football. One of those guys is Justin Jefferson. The reason I like Jefferson is because I want wide receivers who get a lot of targets and Jefferson, we know we'll get those. In games he played last year alongside Adam Thielen, Jefferson had 27% of the team's overall targets and 44% of the deep targets. That's kind of like a Calvin Ridley type of target distribution and it came in an efficient passing offense. Now Jefferson goes into year number two. If you're worried about passing volume in this offense, don't be. Over the final six games, Jefferson hit double digit targets five separate times. He is currently going at the start of round three or the end of round two. I think he would be viable as a mid to second round pick as a guy with a lot of upside and potentially being a first round pick next year. So to me, Justin Jefferson, despite going pretty high, still undervalued in fantasy. Another guy I like this year currently going as the wide receiver 28 is Odell Beckham Jr. The Browns last year, we saw the offense pick up after Beckham got her, but that's not because Beckham got her. It's because Baker played really well as he got more familiar with that new system and the offensive line played well too. Beckham in those first six games before the injury had 25% of the team's overall targets and 35% of the deep targets. If he gets that in an offense, that'll be as efficient as it was in the second half last year, he could explode. I've got him as a wide receiver 20 this year. Going at wide receiver 28 to me, there is a lot of value in Odell Beckham in his second year with Kevin Stefanski. Finally, a bit deeper on the board, I like Mike Williams a lot. He is currently the wide receiver 40 for this year, but I've got him as the wide receiver 33. And a lot of that is because I want low cost exposure to the Chargers offense. They've got a rebuilt offensive line. Joe Lombardi is there to be their new offensive coordinator and it's a second year with Justin Herbert. All those things are things that make me think this offense is gonna score a lot of points. And if they do, Williams should benefit. He was a deep ball maven last year. He had 30% of the team's deep balls. If we look at just the games where Kenan Allen was healthy. In those games, Williams was volatile, but a lot of that was because he didn't score touchdowns. With that body type, you do expect those touchdowns to come. So to me, Mike Williams, a guy to target if you want to get low salary exposure to this Chargers offense, which I very much do. That's gonna wrap things up for today. Back next week, though, to talk about undervalued tight ends for 2021. We'll see you next time here on The Fandal Hurry Up.


Proud lay, we hail. New York City, where the American stage begins. Here is another program with a cast of outstanding players. Public service time has been made available by this station to bring you this story. As proudly we hail the United States Army. The story is entitled, Pursuit in Jade, Time the Present, Place Japan, The Action, a visit to Niko for the annual parade of the samurai, which results in some surprising experiences, romantic and otherwise. Our first act curtain will rise in just a moment, but first... Young man, be honest with yourself. Have you reached a standstill in life? Is each day just like any other? Are you worried about your future? And most important of all, are you feeling dissatisfied with yourself and your personal development? Well, if this description or any part of it fits you, it's just about time you investigated the opportunities waiting for you when you enlist in the United States Army. Every man in the Army has a skill, and more often than not, the Army taught him that skill in one of its fine schools. The Army offers an interesting present and a secure future with plenty of promotions along the way. And above all, the Army molds you into a man, a man whose family, friends and country are proud of him. If you think you can measure up, stop in at your nearest recruiting office and see if you can qualify to wear the mark of a man, the uniform of your United States Army. And now your United States Army presents the proudly we hail production, Pursuit and Jade. I must be crazy. Getting up at six o'clock in the morning to come up here all the way from Tokyo to see this darn parade of samurai. Oh, wait a minute, what's that? Mister, you dropped something. Oh, go ahead and let's have a look at that. Hey, hey, you dropped this box. Hey, excuse me. Stay back, please. Oh, policeman, that man, the samurai on horseback in the parade, he dropped this. I'm trying to give it to him. What can you say? This box, man dropped it. What can you say? Oh, darn it, you don't understand. Oh, okay. Okay. Gosh, what a madhouse. Corporal O'Neill, what are you doing up here in Niko? Sergeant Gorman. What's the matter? I don't look that terrible, do I? No, no, I just wasn't expecting to see anybody I knew, especially anybody from the office. Taking in some of the local color? All right now, I think it's taking me in. What a crowd. Gee, you look different in civilian clothes. I didn't recognize you with that big hat. Is that a compliment? What do you think? Say, I was just on my way to a tea house somebody told me about. I thought I'd take a breather. How about coming along? There's no fun drinking tea by yourself, you know? Sounds wonderful. Well, come on. Let's see if we can fight our way out of here. Hold on to my arm. I don't want to lose you now and I just found you. Come on, let's see. Come on, let's see. Come on, let's see. I've gotten so I almost like this stuff. Sitting on the floor, soft oriental music. I guess I've been in the Far East too long. It does have a certain restful effect. And anything's nice getting away from the office. You can say that again. You know this is my first Sunday off in two months, O'Neal. You couldn't call me by my first name, you know. After all, we're not on duty now. Okay. What is it? Dottie. Oh, Dottie, of course.  Okay, O'Neal. A Dottie. More tea? Oh, don't you feel my cup? I'll lose face. Lady's supposed to do that. I thought that was only in the dives along the Ginza. No, no, that's an old Japanese custom. Okay, I approve. You can light my cigarette too, O'Son. While you're at it. Immediately, Master. Anyway, I can try out my new Japanese lighter. Oh, for heaven's sakes, I forgot. Why, did you leave it home? No, no, this box. What about it? A man in the parade dropped it, and I tried to give it back, and the policeman didn't understand, made me go back in the crowd. I forgot all about it till you sneaked up on me, and just now I opened my purse, got my lighter, and I found it. Well, open it up. Oh, you think I should? Why not? I don't know. Okay. Looks like a jade necklace. Kind of pretty beads, aren't they? Pretty beads? Do you know what these are worth? Nah. Are they real jade? Yes, real jade. It's only worth about $5,000, that's all. $5,000? Holy smoke, you sure? Of course I'm sure. Well, how do you know? Well, my father's a jeweler back home. See, I've never seen anything nearly as good as this one, but I know a little about it. Enough to know it's very good jade. Anything else? More tea, please. Thank you very much. You know, we'll have to take this right down at the police station. You sure it's real? I'm sure.  No, look, you can see when the light hits it. Gosh, I've never seen anything so beautiful. Oh, it is pretty. You see how dark and translucent it is? Almost like emeralds. Yeah. I never dreamed jade was so expensive. Well, you see, the color and the way each bead is matched or what determine the value. Uh-huh. Sure is worth a fortune. Kind of makes me nervous just to hold it. I'll put it back. That was wonderful, but I guess we'd better be going, huh? Yeah. I saw the dread getting out in that crowd again. Must be breaking up by now. Anyway, we have to take the necklace to the police. Well, let's go collect our shoes and get going. Okay. I beg your pardon. Would you mind stepping into my office, please? Who are you? George Henson at your service, Sergeant. I am, in a sense, your host. I own this house. Well, we're just leaving. We have an appointment. What was it you wanted? Just a few moments of your time, please. Come in, and I will explain. Okay, darling? Well, I guess so. Now we can talk. Please be seated. No, thanks. My house boy brought me very pleasant news. He tells me that the young lady has something which belongs to me. What do you mean? The necklace. Well, how did he know? He started displaying it to the sergeant when he came to serve you. He knew that it had been taken from me this morning. Obviously, there could not be two strings of jade so much alike. That's true. It's not very likely. So now we can conclude this little interruption and you can get on with your appointment. Would you be so good as to hand me the box, Madame? Don't do it, darling. How do we know this guy's on the level? We don't. She found that necklace. It was dropped by a man in the parade this morning. A samurai warrior on horseback. How could it have been you? A samurai? Was he wearing armor and a visor? Yes, but... Oh, although I am an occidental, I have resided here for so long that by courtesy I am asked to participate in the festival. Well, we were going to take it down to the police station and let the owner reclaim it there. Oh, no. You must not do that. Why not? Because, Sergeant, you don't know how these native police operate. There will be forms to fill out. They will question you. There will be delays. I assure you it will be most uncomfortable. They may even hold you while they check into your story. No, you must not subject yourselves to that. Well, I'm sorry, sir. We just couldn't take the chance with anything as valuable as this. Come on, Dolly, let's go. Okay. I will make it well worth your while. We're not interested. I hope you will not regret this, step-sargent. Gee, I haven't even seen a cop for the last 15 minutes, let alone a police station. We must all be celebrating the festival, too. Well, anyway, I'm glad to be away from that tea house and that cheerful Mr.... What's his name? Me, too. If that's his real name, I'll eat my hat. Me, too. Sure, a creep. Wasn't that funny the way he didn't want us to go to the police? Yeah, it seems so. Of course, he may be right about the way they operate. Maybe we're suckers for going to them ourselves. I'll feel better about it, no matter what. Well, I'll feel better about it when we get rid of the door-gone thing. It's beginning to make me nervous, too. She must be late. It's almost dark. Well, we drank a lot of tea. Hey, isn't this something, though? These Japanese streets. I get a kick out of it every time I go down one. There's a movie set. Expect Dr. Fu Manchu to step out of any doorway.  If you move, I will be forced to shoot. Well, it's not Fu Manchu, but I'll bet this one could do real fine imitations of Boris Karlov. What's the idea? The lady has something on her purse which does not belong to her. Oh, he smoked out of you. Everybody in Niko must have seen you pick up that package. Police, hand over the box, and you will not be hung. Okay, kid, give. I have to. Well, the man has the advantage. Come on, come on. He may get nervous and let that thing go off. All right. Thank you. Beautiful Niko, city of shrines, the annual festival and parade of the samurai. And as the sun fades away in the western sky, we leave all the happy little thugs behind us. Except that the sun fade about an hour ago. You're hitting it on the head. Well, they got it. Now, come on, kid. I can search the police station on the double. When they hear what happened, that guy Hanson's going to have some explaining to do. So, you see, as soon as we could, we came here to report it to you. Mm-hmm. Ah, so, so, so, so, so. Well, what are you going to do about it? I do. Ah, so... You see, this guy Hanson's at the bottom of it all. When he saw that we wouldn't give the necklace to him, he simply had somebody who works for him, kidnap us and take it. You may be right. Though... Though what, sir? Mr. Hanson is highly thought of member of this community. His reactions against such a one could jeopardize my office. However, you seem to be most direct in your story. Come along. We will go and talk with Mr. George Hanson. Right. Inspector Motakawa, what a pleasure visiting our fair city for the festival. I will have my boy prepare a room for you and your friend. Ah, Hanson-san, a great pleasure indeed, but fear I come on business. Business? You know what it means. Yes, you do. You've got somebody working for you to take the jade necklace back when you couldn't get us to give it to you this afternoon. Necklace? What necklace? Indeed, madame, I am most baffled. But, please, we will begin preparations for the dinner. Mr. Hanson, please, must ask you a few questions. When Sergeant and Lady visited your house this afternoon? This is some joke, Motakawa-san. For a moment I really thought, silly, I know all business. It is no joke. But, my dear inspector, as you know, my office has a small concealed window here, and from it I can and do watch everyone who enters my house. It is only good policy to keep out the undesirables. Furthermore, I was here all afternoon. Eh, just a moment. I will call my boy. You see, inspector, it is as I say, I was here all afternoon. I've never laid eyes on these people before. They've never set foot in my house. You are listening to the proudly-behaved production, Pursuit and Jade, will return in just a moment for the second act. High school seniors, ensure a secure, well-paying future by preparing for it now. The United States Army's Reserved for You program will guarantee you a classroom seat in an exciting Army technical career course before you enlist. You'll get top-notch training and on-the-job experience while serving side-by-side with America's finest young men and women. The choice is wide open, and it's all yours to make. High school graduates can take their choice from more than 100 interesting courses, everything from atomic technician to welding, the fact-fifth, booklet Reserved for You tells you all about this program. You'll learn of many other fine Army benefits, too, like regular pay increases, promotions, exciting travel assignments, and unbeatable leisure time activities. Get in on the swing. Get your free copy of Reserved for You by visiting or writing your nearest United States Army recruiting station. You are listening to Proudly We Hail, and now we present you the second act of Pursuit and Jade. Neil, Donny. Don't scare me like that. What's the matter, Jim? Colonel Smith blowing another gasket? Listen, I've got an idea. Well, wait, Lance, this call, huh? Tokyo Motor Pool, Corporal O'Neill. Yes, sir. I always call it five minutes to five. Yes, Major. Right. Send immediately. Sandcar 37, Main Entrance, Pershing Heights for Major James Morgan, J-1 section. Right. Thank you. Now, for the brainwave. Now, listen, you know you said the Jade necklace was a perfectly matched one? Mm-hmm. And Hansen, or whatever his name is, said they weren't too like it? I'm fascinated. Go on. I'm serious. Wait and see. Anyhow, Jade is very distinctive, and the value of a perfectly matched necklace is Neil if it's broken up, right? Yes, yes, yes. I'm with you this far, but I don't see what you're driving at. Well, just this. If there aren't any others like it, and if they can't break it up without it's losing all its value, then the thing is around somewhere. Elementary. So? So. Sooner or later, that necklace is going to be on the market. Well, so what? Hansen's got it back, so he'll sell it. Yeah, but don't you see, he doesn't own it. If he sells it, it connects him to the people who took it from us, and that makes him a fence. I think he was taking delivery on it when you picked it up by mistake. That dropping was on purpose by an accomplice. If he had accidentally dropped it as he claimed, he would have discovered it when he got home, and he'd never have known exactly where. Mm-hmm, yes, that's logical. It could have happened anywhere along the parade route. So you and I are going to make things unpleasant for Hansen's son yet. Oh, Jim, we can't follow him around till he produces it to make a sale. We got jobs. Well, he doesn't even live in Tokyo, and Nico's four hours from here by train. Don't forget. The odds are with us that it'll wind up in Tokyo. This is the biggest city. All the important wealthy merchants live here. Tokyo's the number one market for it. Mm-hmm, sure. And there's six and a half million people to buy it. He could sell it to anybody. But he won't. Why not? Because my sweet, he's going to sell it to us. To us? To us. Come on, here's the place. Now, let's see how good an act we can put on. Try to get that sort of half-asleep look in your eyes that people get when they're engaged. Half-asleep? Now, you know. All right, I'll try. Okay, here we go. Good afternoon. Konnichiwa. Konnichiwa. We're looking for a necklace. Maybe a jade necklace. It's an engagement present. Oh, so this way, please. This tray. This one is the best, of course. Only 36,000 yen. 100 bucks. How much you got anything better? Better than this. Very fine color. Very good match. Ah, it's too light. Look how opaque it is. That is the best I have in stock. How much would you like to spend? Would you like to spend? Perhaps I can get something? $5,000. I do not believe there is such a thing on the market in all Japan today. Well, that's what I'll pay if I can find the right one. Can you find one? I can try, but I think not. Okay, I'll keep trying myself and come back again next week. See what you can do. Very well. Konnichiwa. Next week. I'll be back. Don't forget. Sigh, subtle. I have nothing like that. Okay. See if you can get one. I'll be back. Come on, children. I have contacted my brother Osaka. He has nothing like this what you are looking for. All right, keep looking. If there was such a thing on the market, I would have it. All right, thanks. Gosh, I don't know, Jim. You know, it seemed like such a good idea, but we've been looking every day for two weeks now from five until the shops are closed. We must have covered every juror in Tokyo that possibly have or get the thing. Yeah. And even the places we've gone back to have been no dice. I know it. I thought sure when they'd go out to try to find a really good jade necklace, we'd turn it up. Sure. Because in order to make a sale of that kind, you'd think they'd go to almost any lengths. And you know the Japanese grapevine and how things get around. Well, it must have been our act that didn't convince them. Oh, I don't think so. I'll tell you what we'll do. It's almost eight o'clock. Too late to go to mess. I'll buy some chow. I think we deserve a nice, quiet dinner with some soothing music. Well, I... Poison, Nekatsu Hotel. Ayako. It's been a wonderful evening. I almost felt as though we were celebrating something. Well, I wish we were. But I don't know. Looks like we're going to have to give it up. I don't know. Maybe we'll think of something else. I better go in. OK. Uh, Daddy. Yeah? Uh, it has been wonderful. I just wanted you to know. Yeah. Yes, it has. Well, I'll see you tomorrow. OK. Uh, sir. Uh, Sergeant. Sir. Not tonight, boys. No money. No money. Oh, I'm not the beggar, sir. This is our moment of conversation. Yeah, what about? About Jade Necklace. What do you know about a Jade Necklace? Well, I know. Yeah. Take this card, sir. Hey, hey, come back. Oh, this was the address on the card. I don't think there's anybody home, Jim. I'll ring again. Oh, gosh. The door just slid open without anybody there. Yeah, kick off your shoes and put on these slippers. OK. Now it's closed by itself. Jim, I'm scared.  I think you were right. It's just push button operated. Nothing to be afraid of. OK, open it. Try to open it. I bet you can't. How about we can't get out? OK. This way, sir. Come on. We've got business to do. Good evening. I suppose you're still going to stick to your story of never having seen us before. Hello, Sergeant. I recognize you plainly. That was a motherly clever trap, you said. Pretending to want to buy an expensive jade necklace. I must say I fell into it. Pretty neat, huh? Unfortunately for you, yes. What do you mean, unfortunately for us? Inspector Motakawa is going to be very interested in knowing about this, I think. Yes, indeed. But of course, he never will. He's got a gun. Yes, I do. Boy, son, take these people upstairs. I'm afraid they will be our guests for a short while. You can't do this. Be careful, Jim. Remember what you told me about people with guns getting nervous? I guess you're right. Quite so. I'll give the gun to the boy here. You'll regret this. I think it's more likely that you will, if a corpse can feel regret. Can't even move. Tied my legs to the chair, so tight have I even wiggled. The rope's cut into him. Well, you've got to try. It's awfully quiet downstairs for a long time. Wonder what they're up to. The car drove in again. That was the boy's son coming back and the guy who looks like Boris Karloff. Probably went for shovels to dig our graves. I don't think they'll bother. I've got one knot here I can just barely reach. I think in time I might get it loose. Trouble is, I don't know how much time we've got. Jim, do you think they dare kill us? Don't talk. Try to get loose. I'll bet they're trying to think up a way so it'll look like an accident. What a prize Dope I was not to think it might be Hanson himself tonight. I never thought he'd take the chance. Well, I guess he never thought we'd turn out to be the buyers. At least they're moving around. Oh, no, if that's better than having him quiet. Jim. Yeah. Do you smell something? Jim, do you smell anything? I smell smoke. Yeah, yeah. They've set a fire downstairs. There they go.  I'm trying to get loose, Dotty. This house will go up like kindling. There's nothing but bamboo and straw. Oh, I've got my legs free. You have? Yeah, that won't do much good, will it? Maybe. We've got one little chance. What? Can you get up? Yeah, I think so. Well, come over here. The boys don't bother to empty my pockets. Even with your hands tied, you may be able to reach in. I've got a knife in my right hand pocket. Okay, Jim. We do have a chance. Come on, hurry. I told you the place would go in a hurry. Okay, cut. Now, if you hold it in your hand, I'll lean up against it. That would be easiest. Easy now. Now, don't cut yourself. Okay. There. You're loose. Yeah, now.  There. Okay, come on. That way's out. I think that was the stairway just fell anyway. Try the window. Here, here. Go. If we crawl along this light, we can reach the trellis there. All right, you go first. You're lighter. If it caves in, I'll jump. Yeah, I won't cave in. Can't have you breaking an ankle. We've got a lot to do tonight. Once we get out of here. All right, Inspector. Now we want to hear the whole story. You promised as soon as dinner was over to tell us. Very well. But first I must light excellent cigar provided by intrepid Sergeant Gorman. Ah, so so. Now, you remember last night. I ordered roadblocks. Yeah, sure. As soon as we told you what happened. Into the roadblocks walks Mr. George Hansen and friends like lambs into fold. Did he have a necklace? I blush with ease with which we accomplished mission. A necklace was indeed there, also certain other items. A same being sent to jeweler for examination. Other items? Just so. Official appraisers, honorable opinion that we have recovered entire loot of a new eye company robbery three months ago. Oh, I read about in the paper. Oh, that's why Hansen couldn't afford to have us take the necklace to the police. He knew they were looking for it and would surely recognize it. Have duties now elsewhere and must depart from charming company. Must express great gratitude for assistance rendered in this matter. No, we didn't do anything. No. The relentless pursuit in which you engaged brings up a thought in poetic Japanese mind. We'll entitle this case in closed file. Pursuit in jade. Have small word of advice for Sergeant Gorman. Old proverb say when egg is hatched, chicken need not sit longer. I hope I can soon return hospitality. Good night. Good night. Jim? What in the world did he mean about the chicken? Oh, nothing, nothing. Donny, I was just thinking. Yeah? Well, I thought... I mean, uh... I'm sort of used to being engaged to you. Do you think, uh... Do I? Gee. Did you know that you can get ahead, enjoy your work, have security and a steady job, all when you enlist in the United States Army? Why, you'll get your first promotion when you've been in only a few months. There's a wide variety of jobs available, but most important, you'll be helping to preserve the national security of our country. Why don't you check at your local recruiting office and let them tell you how you can make the most of your life in the United States Army? This has been another program on Proudly We Hail, presented, transcribed, in cooperation with this station. Proudly We Hail is produced by the Recruiting Publicity Center in New York for the United States Army, and this is Mark Hamilton speaking, inviting you to tune in this same station next week for another interesting story on Proudly We Hail.


Okay, thank you. Okay, so. All right, so let's go back to where we stopped. Let me actually also go back a little bit to touch me the space right I, I spoke about touch mirror space right. And so let me, let me remind you what it was. Let's say types of x right this was the set of all complex structures on x module the equivalent selection that I called till the zero, which meant that which was saying that two complex structures were equivalent if one was the pullback of the other via a morphism which was isotopic to the identity. Right. Okay. And then we had complex, right, which was the set of all complex structures, just modular differing morphisms so we don't ask anymore that they are isotopic mister to the identity and we, we said that this was. This was the quotient of tish by the action of the group of components of the group of the few morphisms of x right. And in between here, we have our module I have marked. And then we have our scalar manifolds and gamma. Okay, so. So these these these two, this M gamma, what can happen is that sometimes the touch mirror space has got actually infinitely many components, it could have finally many components but it could have infinitely many components. But the M gamma usually I mean has finally many components and then the comp. Often people hope that the complex really has that only one component but that's not necessarily the case of course we don't we don't we don't really know necessarily. Anyway, so this this. And the other thing that. So, so M gamma is the quotient of tish you know by some discrete group action right which is a sub, you know, which is some group of G.  Complex structures right, and so the main difference between the touch mirror space and then gamma is the basically the automorphisms of the lattice right so you could, you could have some. You know the manifold which will act trivially on the homology lattice right so then those guys would give you the same marking, but they would not necessarily give you the same point of touch mirror space right so. Okay. So that's how it works, and we have the period map, we define the period map. And then we have this modular space of marked complex structures right. Again module sum equivalence relation. Let me write that one like this. It's just to work just to differentiate between that and the other one. Um, so then. Okay, and then we have the period map right so the period map we define the print map from and gamma. But we could compose it with the map to from types. And then we do the period domain which I think I called that Q. Q gamma, right, this was a quadric right. It was a real open sub manifold of the other complex quadric inside the projectivization of our lattice cancer would see. Okay. And this was a period map, the gamma right. And what we what we also saw was that so this and gamma is not necessarily house door but it is a complex manifold so this map here is is again a local isomorphism. So, both touch with our space and in gamma are complex manifolds but neither of them is house door so in gamma is a non house door complex month. Q gamma on the other hand is house door and simply connected. So what, um, so what what do people do so did the, you know, where which is clever idea what was it he said well okay this guy is not house door.  Okay, so we he you know he introduced what he called you know the. I don't know the house door if you're like I don't know. I don't remember exactly what what they call these things but so you can you can sort of you can make a new. A new version of and gamma which I will call and gamma s s for separated right. The period map will factor through this right this guy is here is the period map and m gamma s is now house door.  So you identify the points. The infinitely near points of m gamma and by that what do I mean so I mean that so a point P is infinitely near to Q. If every neighborhood of P contains Q and vice versa. So, um, so you can you can construct a new space which is house door and the period map factors through it because Q gamma is house door right. So then you have a new period map from this house door visualization right and and so now what do we have this m gamma s now is a house door complex manifold. And we also know that the period map we we saw this before this PS is a localized morphism right the differentials are injected locally. So then, so we would basically almost get a Tarelli if we could show that the period map is surjective so if we can show. That's the theorem if you like, that's the main part of the result of the of the proof of Tarelli is that PS is surjective from any component from any connected component and gamma s to to Q gamma. So if I restricted to any connected component I get. I get a surjective map now combined with the fact that that PS is a local isomorphism, you get as a corollary is that PS induces. In fact an isomorphism between components. So maybe I should say isomorphisms plural induces isomorphisms between components and gamma s and Q gamma. Okay. So that's modular surjectivity. And this uses of course that that Q gamma is simply connected right and then that the map is et al the PS is et al and Q gamma is something connected if you have something that's surjective. It's going to be a nice morphism on every connected component right. So now there was, there was a question last time about you know what exactly is the difference between M gamma s and N gamma so. So in other words, when do two, when do two hypercalia manifolds give you intermediate points of the marginalized place of marked hypercalia manifolds right. And so there's a almost a complete answer to that. Let me see. Okay, so. So this is mostly due to high brash right. So first of all, there's the following result. So, if, if I have two marked complex structures. So x, the, and, let's say, x prime. The prime are two infinitely been two infinitely near points of. And gamma, then x and y are by more morphic. And in fact, you can also say something a little bit more you can also say something about the period point right. And the period point of. x phi, which is also equal to the period point of x prime, the prime is contained in a particular type of hyperplane, which is the, the period domain q gamma intersected with the park of some element of the lattice. And the lattice is a freeze, a free, finally generated a billion group right. So, so we we take here the park with respect to the quadratic form right on the lattice. And then we intersect, we intersect that herb with q gamma so we get, you know, we get a hyperplane inside to gamma and you're saying that the, that these all of these non separable points belong to hyperplanes of this in the period domain. Right. Okay, so this is one direction. So, again, do the hybrids in the other direction, which tells you that which says the following. Sorry. There's a question, yeah, go ahead please. In the previous statement of what is why. What is why. Do I have a why. Oh, no, I meant, I, it's expansion. Okay. Yeah. Sorry about that. It's not why. Is that good. Sorry. Now it's all good. Okay, thank you. All right. Okay, so, um, so let's take us one moment. So the other, so the proposition which was which kind of goes in the other direction, right. So, um, we have the following, which is again due to progress. If, if again, if now x and x prime are compact hypercalium manifolds that are birational right so let or bimeromorphic so let x and x prime be compact hypercalium and bimeromorphic. So again, basically it tells you that they give you non separate points of the, of the module I've marked manifolds so so that that's the result precisely right. So, there exists families of complex manifolds. So x to s. And, x prime to us. Actually, let me, instead of calling it as mega should call it these for this, that would be better. x to be and expand to be over the same complex disk be such that number one. Now that the, you know, it's just a complex this so it has an origin right so x zero which is the fiber over zero is isomorphic to x, x prime zero is isomorphic to x prime. So these are the central fibers, and there exists a birational map a bimeromorphic map if you like, from x prime to x, which is of course which of course commutes with the two projections to the disk, which is an isomorphism over the minus zero. And, I didn't give the name, but a map between x and x one. Okay. Let me just rewrite that a little bit like I have a bimeromorphism that I will call here. Let me call this one little app, right. And then I will call this other one. Okay, so I have, I have a bimeromorphism between x and x prime right, then I'm saying that that bimeromorphism is basically induced from a bimeromorphism between two families over some this on such that. Well, that's it. So, which is a nice woman over over this minus zero and induces X zero. I'm just saying that big app restricted to X zero is equal to. So, so it's it's so what is it telling you then that it's pretty much telling you that what you're doing when you pass from, when you pass from M gamma to M gamma s, you are identifying pretty much the bimeromorphic. Okay. You know, the thing is that this is this. The thing that's complicated here is this group of components so we don't know. We don't know how many components this M gamma s has so you were asking about this Namika example right. The Namika example is just, you know, is just cool is just with kumars so you know you just you know we introduced the kumar the generalized kumars for complex. Tori of dimension two right so this is what you get here is that he what he what he produces here is that he takes. He takes a complex torus and then he takes just its second kumar which is a four fold, and then he takes the second kumar of the dual torus. And then he shows that they are hodge isometric so they have the same h2 and that with the same hodge structure and the same quadratic form, but they but you know but but but they don't necessarily have to be birational to each other. They don't necessarily have to be bimeromorphic. So it's a very simple example. It's a little bit strange because, you know, you take a torus and it's do normally, you know, a lot of the time, I mean they're just kind of belong to the same component of the marginalized space of complex Torah. So, so it is, you know, understanding these. These components is not is not so easy. Let me also just quote another result of progress for this which is kind of nice. Well, it is nice. He has another nicer nice results. So, it's a finite nest result right so. It's fine and this would be now interesting right so you can say okay you know that the bimeromorphic guys go to the same period point, but how many non bimeromorphic guys will go to the same period point and this is, this is again this this kind of tells you a finite limit. Okay, so that that's nice anyway. So if we fix. Given given a, let me see how should I say this. Given a period point, and let me call it, I don't know little x in, in q gamma. So this is knowing already that the period map is subjective okay so I'm going to use the fact that. So, let's say we already know, assume we already know that the period map is subjective. So the activity of the period map. Okay, so assume you already know that then then what do we get we get the set of the set of hyper killer complex structures with image, I should say marks I guess with image. No, actually that doesn't really follow here. Sorry, just one second, let me think for a second here. Does the marking change anything. Okay, let me let me restrate this in a different way. So that I'm certain that it is correct. So the set of hyper killer complex structures on our differentiable manifold X. With a fixed arch structure, H two of X Z is so then is so non empty the non empty part as I said that's the subjectivity of the period map right now this is the interesting part consists of a finite number of bimeromorphic equivalence classes. Okay, so what you're saying here is that. If you identify all the bimeromorphic guys then you have only finally many hyper killer complex structures with a fixed heart structure on the H two right so it is it is a pretty strong result. Of course unfortunately there's no effective balance on this finite number. Obviously, any kind of bound would be would be very interesting right. Alright, so. Okay, so we have this. Are there any questions before I move on. I would like to do is say a few words now about the subjectivity of this period map or there any questions before I move on to that. For this proposition. So do we need to fix the deformation type like we, we fix the connect component of the motorized space. Always the finance like says the also the finance for the deformation types. Yes, you are fixing you are fixing the, the differentiable structure on M right. So you're throughout all of this, we had we have a fixed see infinity manifold compact. For all of this yeah we are we are fixing that yeah. So, can I ask a question. Right. So, so we are not anymore on the modular space of mark the paper killer manifolds right, as we are. No, for this one no we're not know. So the marking I mean understanding how the marking. That's a different thing that has to do with the lattice right so how many different markings can you have I mean then you're looking at the automorphisms of the lattice. Okay, okay. That's a separate question. And the can be expressed in terms of types Mueller space, the statement, I mean. Oh this statement here. Yes. In terms of the space. I mean, well, well, you see the cashmere space could have many many components I mean there are examples of. So, okay, so the cashmere space sits above everything right so already these fibers here can be infinite. Okay, okay, okay. Yeah, there are examples where it is infinite actually. Okay, okay. Thanks. Okay, any other questions. Okay, very good. So, so now let's see so now, as I said I'm going to say something about the subjectivity of the period map. This is where biscuits clever tip clever trick right. So what does it. It uses what what we call what people normally call twister lines, which actually knowing how cause very much more aptly he calls them twister conics, which I think it is my makes a lot more sense. These are actually impact really conics. So, so there are two ways of thinking about these. And let me let me start with the first way so if you let's go back to the to the period map right so we have this lattice gamma with with its quadratic form q gamma right and then we had this big q gamma which was the period domain right sits inside the actual quadratic which sits inside the project realization of gamma. And so we'll see what this was a period domain. Now, for a for a hyper cannon manifold, the signature of q gamma, after I cancer it with our you know after I extend the scalars to our is of the shape you know three. I write it as B two, but B two here is the rank of gamma. And I write it as B two because usually it's going to be the second betting number for the hypocritical right. Right. So this is the signature. What this means is that then you have.  Each time. I'm sorry. Yeah, that's what we do. So, we can choose each time we choose over three dimensional real plane inside you know gamma tensor are positive for q gamma. So this is a twisted comic.  Now I don't want to call it. We call it something else. I don't know. Yeah, maybe I'll just call it PR. Let me call it projectivize it so they call it. Okay, let me call it FR then what I can what I'm going to do I'm going to take FR I'm going to answer it with see you know then this will be inside gamma and so it will see then I'm going to projectivize it so I'm going to say to say that P is the projectivization of this tensor with C. So this now is is just a P to write inside P of gamma. And so it will see. And now you can intersect it. So P intersected with Q gamma this is by definition. This is a twister coin. Okay. All right, and you can show it's not it's not that difficult. I'm sure that Q gamma is what we call twister path connected. What that means is that any two points of Q gamma can be joined by a path by a sequence by a connected sequence of course, of twister connects. And this is this is you know this is very elementary, you can find the proof in holy breath for Vaki seminar, seminar talk. So, so now you have this and, but then the, the, the trick is that you can you can produce these twister conics in a different way, and that's that has to do with twister families so that's the next thing that I would like to you explain twister spaces. So let's go back again so let's say that suppose that X is hypercalor. So G is now my hypercalometric right then there then then we know that we know that there is an S two of the Taylor complex structures on X right. We already saw that there exists. Okay, Taylor complex structures killer with respect to G of course. And then this gives us a whole S two of Taylor complex structures right so for any ABC in S two in the, in the unit sphere in our three. Right. Lambda equal to AI. Okay, so now we're going to define the, oh, and, and, and we recall also the caliform so recall that the caliform associated to land up with let's call that omega sub lambda, and you know we define omega by its action on a pair of vector fields right so those are my two dots my two vector field, and the way that it acts on vector fields is by taking G of lambda acting on the first vector field and then putting in the second vector field right so. So this gives us. The caliform right for each of these complex structures, and we have a family of these X lambdas right of compact killer manifolds. And so with this, we can define the twister space right so definition, the twister space. Let me call that, I don't know what am I going to call it. Yeah, I'm not sure, because I've used so many, all of these things well okay let me just call it curly X like I always do it's not exactly the same as the previous definitions that we had but. So X, it maps to P one of x g so the twister space is specifically associated to our given my particular manifold right is. So as a differentiable manifold is just a product. So X times P one, but. So as a same thing the manifold, and but it has a particular complex structure happened there. Okay, there we go with the complex structure. Actually it's really it's really an almost complex structure right meaning that it's only defined on the tangent space with the almost complex structure. So once I'm going to define this on the tangent spaces so the tangent space to the product is the direct sum of the tangent spaces to the two factors. And I will say what linear operator I have here as my almost complex structure. So it sends a pair of vectors to, of course, on one factor on X we want the complex structure lambda. That's what that one's going to be. And then on the other side we're just going to put the complex structure of P one. So it's the most natural thing that you could come up with. And this is integrable, meaning it comes from a real from an actual complex structure by a result of hitching. Linsprong, and rosec. So. All right so you have this, you have this fact this this product structure, you put on every fiber you so you see that the complex structure of the fibers is moving right so what you have here then is just. It's just a you know you have your X over P one right, the complex structure and for each each time you take a point of this P one which we think of. Also as S two right, so each time you take a lambda here, the fiber over lambda. It has complex structure. Lambda. Okay, so it's a very nice simple notion. And then you can you can map this way over the period map because P one is simply connected you can just you can just put a marking. Because P one is simply connected you can trivialize the the H two is along the fibers right you can identify H two of all the fibers with the H two on one fixed fiber right so it has a you know you can just define markings right. You can choose markings, consistently on all the fibers right. And we can choose consistent markings, the fibers to get the period map P from P one to the period domain Q gamma, which sends lambda to the point. Sigma X lambda. Sigma is a generator of H two zero right so where Sigma X lambda is a generator H two zero. Sigma X lambda which is equal to age zero or omega two of excellent. Alright. Okay so you get a you get a period map and then you can you look at it you can look at the image of the P one under the period map, and then you have a lemma, which I'm not sure who this is due to. I suppose I mean you know once you knew that this was a holomorphic family, you know, then probably, I mean I spoke hitching car hit the instrument rosec could have done this right or maybe somebody else did it before. I'm not sure. But the lemma says that the image, or the image P of P one is a twisted chronic. So this is now basically what gives you the surjectivity of the period map because. So the corollary is corollary of this, and the two surpass connectivity is that the period map from M gamma to Q gamma is surjective on any connected component of M gamma. Okay, and you see here. I'm saying on any connected component because if you look at your family, you can take any marking or of any fiber and extend that to a marking to markings of the whole family right so you can. So this this this just this work perfectly fine that's where you get you get your surjectivity. So what you do is that you know that there are. You know the map the period map is a local isomorphism right so you just take an interior point of Q gamma, which is in the image of the period map right and then. You have a point which is anywhere else in Q gamma you just connect that point to a point in the in your interior ball right via a chain of twister conics. And then you know that all of those twister conics are images of twister spaces, why the period map right so then that will give you the surjectivity. Did I. Remember you long sorry well yes but you can take the break now maybe or. Oh yeah yeah yeah I will I will take the break now yes was there a question or no. Okay, so so we will stop now for a. So I missed just one, like the lines of the previous slide. Previous slide this one. Yes, to get to exactly that. Okay, I can I can make the slides available if people wish. Maybe I should have actually done it sooner I'm sorry about that. I could have. Maybe I will. Can I email the slides to someone you could, if you wish you could post them on the, on the website. Yes, of course you can. Well you can email them to me if you want but also you can just email them to any of the secretaries, and they will post them so. Okay, I'm not sure who that would be that the usual address let me just give it to you the. Okay, thanks. This is the official email address of the school so then. Okay, all right, thank you. Yeah, or if you prefer just email them to me and then I'll. Yeah, I can that that's fine I can. You can send it to that data to SMR three six point all right. Okay so we'll stop now for five minutes until 11 o'clock.


What's up, everybody? JC over at jaspyskatesrace.com, 223 Panini Luminance Football, one box break, ran on teams number three just sold out. And again, guys, we'll randomize the names and teams, give away four free spots in this filler to increase the chances of winning the two NFL kickoff spot mixers later at the end of the break. So dice roll here, names. We'll click the dice roll, randomize the names, top four get spots, we'll paste the four names, we'll do the break as well. And then at the end, we'll have a new dice roll and an original list right there with the winners to give away those spots, guys. So roll it, five and two seven times. One, two, three, four, five, six, seven. Seven times, seven. Zach, Chris, John and Chris, so top four got free extra spots. Now this list here will be used later for the giveaway. Now we'll randomize the names and teams for the box break seven times. One, two, three, four, five, six, seven. John, John and Chris, two, three, four, five, six, seven. Patriots and other backers. All right, well, again, it is a filler, so no trades, but I'll give you guys a quick minute to see your teams. Slap hands, slap hands. Here we go, let's do the break. Kenneth Walker, Relic, Luminosity. Justin Fields, looks like a really cool insert. We have a Kubrick up to 50. We have a numbered Jalen Hyatt for the Giants to 200 and Saquon Barclay Prestige. Leo Herbert, Adrian Amos to 100. Shutter signatures. Herbert at 35 and Aiden O'Connell. Will we be seeing Aiden O'Connell at some point this year? History says that Garoppolo doesn't play the whole season, so this guy is the backup. Raiders are that one, that's gonna go to Chris. I told Joe I'm calling it. If the Raiders somehow are in a chance to make the playoff wall card and Garoppolo goes down or something, Tom Brady's gonna come out of retirement and be the first owner quarterback in the NFL. Marvin Mims, and Will Loves. Mims is a, all righty guys, there you go. Took a miss in a minute. What is it like to say that, Josh, just because he's like a part owner of the Raiders. And if you can't tell me if like Garoppolo goes down late in the season and like there's a chance to make the playoffs or they make the playoffs, you don't think he'd come back and play? It's a win-win for him. So there you go, all right. Switch scenes and let's do it. Last two spots in the NFL kickoff mixer. Me giving away here, guys. Roll it. Four and a six, four and a six, 10 times. Here we go. One, two, three, four, five, six, seven, eight, nine, seven, and I clicked the fast. I clicked the fast, sorry, I didn't wanna see it. All right, guys, so archer number three, I know you're gonna be extra sad and mad at me, but it's not my fault. It's not my fault. Chris, it's not my fault, it's not my fault. But you did have multiple spots, so I can't say you're out of it just yet. All right, first one in. Chris, whoo, saved one there. And at number one, second spot in. Chris, little sweat. There you go, man, you got both. So, pasting your names there, and this would be one, three. All right, appreciate you guys coming up next is the mixer, mixer, mixer.