Kevin Knudson: Welcome to My Favorite Theorem. I’m your cohost Kevin Knudson, professor of mathematics at the University of Florida. I am joined by cohost number 2.

Evelyn Lamb: I am Evelyn Lamb. I’m a freelance math and science writer in Salt Lake City. So how are you?

KK: I’m okay. And by the way, I did not mean to indicate that you are number 2 in this.

EL: Only alphabetically.

KK: That’s right. Yeah. Things are great. How are things in Salt Lake?

EL: Pretty good. I had a fantastic weekend. Basically spent the whole thing reading and singing, so yeah, it was great.

KK: Good for you.

EL: Yeah.

KK: I didn’t do much. I mopped the floors.

EL: That’s good too. My floors are dirty.

KK: That’s okay. Dirty floors, clean…something. So today we are pleased to have Chawne Kimber on the show. Chawne, do you want to introduce yourself?

Chawne Kimber: Sure. Hi, I’m a professor at Lafayette College. I got my Ph.D. a long time ago at University of Florida.

KK: Go Gators!

CK: Yay, woo-hoo. I work in lattice-ordered groups.

KK: Lattice-ordered groups, very cool. I should probably know what those are, but maybe we’ll find out what they are today. So yeah, let’s get into it. What’s your favorite theorem, Chawne?

CK: Okay, so maybe you don’t like this, but it’s a suite of theorems.

KK: Even better.

EL: Go for it.

CK: So, right, a lattice-ordered group is a group, to begin with, in which any two elements have a sup and an inf, so that gives you your lattice order. They’re torsion-free, so they’re, once you get past countable ones, they’re enormous groups to work with. So my favorite theorems are the representation theorems that allow you to prove stuff because they get unwieldy due to their size.

EL: Oh cool. One of my favorite classes in grad school was a representation class. I mean, I had a lot of trouble with it. It was just representations of finite groups, and those were still really out there, but it was a lot of fun. Really algebraic thinking.

CK: Well actually these representations allow you to translate problems from algebra to topology, so it’s pretty cool. The classical theorem is by Hahn in 1909. He proved the special cases that any totally ordered Archimedean group can be embedded as a subgroup of the reals, and it kind of makes sense that you should be able to do that.

KK: Sure.

CK: And then he said that any ordered abelian group, so not necessarily lattice-ordered, can be embedded in what’s called a lexicographical product of the reals. So we could get into what that is, but those are called Hahn groups. They’re just huge products of the reals that are ordered in dictionary order that only live on well-ordered sets. So this conjecture, it’s actually a theorem, but then there’s a conjecture that that theorem is actually equivalent to the axiom of choice.

KK: Wow.

EL: Oh wow.

CK: Right?

EL: Can we maybe back up a little bit, is it possible to, for me, I really like concrete examples, so maybe can you talk a little bit about a concrete example of one of these archimedean groups? I don’t know how concrete the concrete examples are.

CK: No, they’re just really weird ways of hacking at the reals, basically, so they’re just subgroups of the reals. Think of your favorite ones, and there you go, the ones that are archimedean. And as soon as you add two dimensions of ordering, it’s even more complex, right? So the classical example that I work with would be rings of continuous functions on a topological space, and then you can build really cool examples because we all understand continuous functions, so C(X), real-valued continuous functions on a Tychonoff space, so T-3 1/2, whatever.

KK: Metric space.

CK: The axioms so you have enough continuous functions. So Gillman and Jerison in the 1950s capitalized on a theorem from the 1930s by Gelfand and Kolmogorov that said that the maximal ideals of C(X), if you take them in the hull-kernel topology, are isomorphic to the Stone-Čech compactification of the space that you’re working on. And so if you have a compact space to begin with, then your space is isomorphic to your maximal ideals. So then, just build your favorite—so C(X) is lattice-ordered, if you take the pointwise ordering, and then since the reals have a natural order on then, you pick up your sups and infs pretty easily. So there you’re starting to touch some interesting examples of these groups. Have I convinced you, Evelyn?

EL: Yeah, yeah.

CK: Okay, good. So they’re huge. You have to have some complexity in order to be able to prove anything interesting about them. So then there the Hahn embedding is pretty obvious. You just take the images of the functions. There’s too much structure in a ring like that, so maybe you want to look at just an ordered group to get back to the Hahn environment. So how can you mimic Hahn in view of Gelfand-Kolmogorov? So can we get continuous functions as the representation of an ordered group? Because the lex products that Hahn was working with are intractable in a strong way. And so then you have to start finding units because you have to be able to define something called a maximal sub-object, so you want it to be maximal with respect to missing out on some kind of unit. And so then we get into a whole series of different embedding theorems that are trying to get you closer to being able to deal with the conjecture I mentioned before, that Hahn’s embedding theorem is equivalent to the axiom of choice.

EL: Yeah, I’m really fascinated by this conjecture. It kind of seems like it comes out of nowhere. Maybe we can say what the axiom of choice is and then, is there a way you can kind of explain how these might be related?

CK: Yes and no.

KK: Let’s start with the axiom of choice.

CK: Yeah, so the axiom of choice is equivalent to Zorn’s lemma, which says that maximal objects exist. So that’s the way that I deal with it. It allows me to say that maximal ideals exist, and if they didn’t exist, these theorems wouldn’t exist. You use this everywhere in order to prove Hahn’s theorem, so that’s why it’s assumed to be possibly equivalent. This isn’t the part that I work on. I’m not a logician.

KK: So many things are equivalent to the axiom of choice. For example, the Tychonoff product theorem, which is that the product of compact spaces is compact. That’s actually equivalent to the axiom of choice, which seems a bit odd. I was actually reading last night, so Eugenia Cheng has this book Beyond Infinity, her most recent book, good bedtime reading. I learned something last night about the axiom of choice, which is that you need the axiom of choice to prove that if you have two infinities, two countable infinities, you want to think [they’re the same], it’s countable somehow. If they come with an order, then fine, but if you have two, like imagine pairs of socks, like an infinite collection of pairs of socks, is that countable? Are the socks countable? It’s an interesting question, these weird slippery things with the axiom of choice and logic. They make my head hurt a little bit.

CK: Mine too.

EL: So yeah, you’re saying that looking at the axiom of choice from the Zorn’s lemma point of view, that’s where these maximal objects are coming in in the Hahn conjecture, right?

CK: Absolutely.

KK: That makes sense.

CK: That’s kind of why I drew the parallel with this theorem about C(X), these maximal ideals being equivalent to the space you’re on. Pretty cool.

KK: Right. Because even to get maximal ideals in an arbitrary ring, you really need Zorn’s lemma.

CK: Right. And there’s a whole enterprise of people working to see how far you can peel that back. I did take a small foray into trying to understand gradations of the axiom of choice, and that hurts your head, definitely.

KK: Right, countable axiom of choice, all these different flavors.

CK: Williams prime ideal theorem, right.

KK: Yeah, okay.

EL: So what drew you to these theorems, or what makes you really excited about them?

CK: Well, you know, as a super newbie mathematician back in the day, I was super excited to see that these disparate fields of algebra and topology that everyone had told me were totally different could be connected in a dictionary way. So a characteristic of a ring can be connected is equivalent to a characteristic on a topological space. So all kinds of problems can be stated in these two different realms. They seem like different questions, but they turn out to be equivalent. So if you just know the way to cross the bridge, then you can answer either question depending on which realm gives you the easier approach to the theorem.

KK: I like that interplay too. I’m a topologist, but I’m a very algebraic one for exactly that reason. I think there are so many interesting ideas out there where you really need the other discipline to solve it, or looking through that lens makes it a lot clearer somehow.

EL: And was this in graduate school that you saw these, or as a new professor?

CK: Definitely grad school. I was working on my master’s.

KK: So I wonder, what does one pair with this suite of theorems?

CK: It’s a very hard question, actually.

KK: That’s typical. Most people find this the more difficult part of the show.

CK: Yeah. I think that if you were to ask my Ph.D. advisor Jorge Martinez what he would pair, he is very much a wine lover and an opera lover. So it would be both. You’d probably see him taking a flask into Lincoln center while thinking about theorems. So he loved to go to Tuscany, so I assume that’s where you get chianti. I don’t know, I could be lying.

KK: You do, yeah.

CK: Yeah, so let’s go with a good chianti, although that might make me sound like Hannibal Lecter.

KK: No fava beans.

CK: So we’ve got a chianti, and maybe a good opera because it’s got to be both with him. It’s hard for me to say. So he comes up to New York to do an opera orgy, just watching two operas per day until he falls down. I sometimes join him for that, and the last one I went to was Così fan tutte, and so let’s go with that because that’s the one I remember.

EL: If I remember correctly—it’s been a while since I saw or listened to that opera—there are pairs of couples who end up in different configurations, and it’s one of these “I’ll trick you into falling in love with the other couple’s person” that almost seems like the pairs being topology and algebra, and switching back and forth. I don’t know, maybe I’m putting ideas in your mind here.

CK: Or sort of the graph of the different couplings, the ordered graph could be the underlying object here. You never know.

EL: An homage to your advisor here with this pairing.

CK: Yeah, let’s do that.

EL: Well I must admit I was kind of hoping that you might pair one of your own quilt creations here. So I actually ran into you through a quilting blog you have called completely cauchy. Do you mind talking to us a little bit about how you started quilting and what you do there because it’s so cool.

CK: Yeah. Of course I chose that name because Cauchy is my favorite mathematician, and as a nerd there would be no other quilt blog named after a dead mathematician. So I am a little mortified that when you google “Cauchy complete,” as many students do, mine is actually the first entry that comes up on google.

KK: Excellent.

CK: I don’t know what that means, but okay. So yeah, when I applied for tenure, which is kind of a hazing process no matter where you are, no matter how good of a faculty member you are, I really wanted to have control, and you don’t have control at that point. And so I started sewing for fun, late at night, at 1 am, after everything kind of felt done for the day. I never imagined that I’d be doing what I’m doing today, which is using quilting to confront issues of social justice in the United States, and they’ve been picked up by museums and other venues. It’s this whole side hustle out there that I kept quiet for a long, long time. And then once I got promoted to full professor I came out of the closet.

KK: Were you concerned that having a side hustle, so to speak, would compromise your career? Because it shouldn’t.

CK: Yeah, I think something so gender-specific as quilting, something you associate with grandmas. At the end of the day, the guys I work with, I must say half of my quilts have four-letter words on them, you know, the more interesting four-letter words, so as soon as my guys saw them, they were totally on board with this enterprise, so I didn’t really need to be into the closet, but I didn’t want anybody to ever say, “Oh, she should have proved one more theorem instead of making that quilt.”

EL: Yeah.

KK: It’s unfortunate that we feel that way, right? I think that’s true of all mathematicians, but I imagine it’s worse for women, this idea that you have to work twice as hard to prove you’re half as good or something like that?

CK: Do we need to mention I’m also a black woman? So that’s actually how I was raised, you need to do three times as much to be seen as half as good, and that’s the way that I’ve lived my life, and it’s not sustainable in any way.

KK: No, absolutely not.

EL: But yeah, they are really cool quilts, so everyone should look at completely cauchy, and that’s spelled cauchy. There’s a mathematician named Cauchy. I actually have another mathematician friend with a cat named Cauchy, or who had a cat named Cauchy. I think the cat has passed away. Yeah, and I actually sew as well. I’ve somehow never had the patience for quilting. It just feels somehow like too little. I like the more immediate gratification of making a whole panel of a skirt or something. You do really intricate little piecing there, which I admire very much, and I’m glad people like you do it so I don’t have to.

KK: Sure, but Evelyn, you don’t have to make it little.

CK: You don’t.

KK: I’m sure you’ve seen these Gee’s Bend quilts, right, they’re really nice big pieces, and that can have a very dramatic effect too. But yeah, the intricate work is really remarkable. My wife has done a little quilting, and she always gets tired of it because of the fine stuff, but then she’s a book artist. She sets lead typing in her printing press by hand, and that’s fine, but piecing together little pieces of cloth somehow doesn’t work.

CK: It seems more futile, you take a big piece of fabric and cut it into small pieces so that you can sew it back together. That is kind of dumb when you think about it.

KK: Well, but I don’t know, you’ve got this whole Banach-Tarski thing, maybe.

EL: Bring it back around to the axiom of choice again.

CK: You guys are good at this.

KK: It’s not our first podcast. Well this has been great fun. Anything else you want to promote?

CK: No, I’m good.

KK: Thanks for joining us, Chawne. This has really been interesting, and we appreciate you being on.

CK: Great. Thank you.

EL: Thanks.

Kevin Knudson: Welcome to My Favorite Theorem. I’m your cohost Kevin Knudson, professor of mathematics at the University of Florida. I am joined by cohost number 2.

Evelyn Lamb: I am Evelyn Lamb. I’m a freelance math and science writer in Salt Lake City. So how are you?

KK: I’m okay. And by the way, I did not mean to indicate that you are number 2 in this.

EL: Only alphabetically.

KK: That’s right. Yeah. Things are great. How are things in Salt Lake?

EL: Pretty good. I had a fantastic weekend. Basically spent the whole thing reading and singing, so yeah, it was great.

KK: Good for you.

EL: Yeah.

KK: I didn’t do much. I mopped the floors.

EL: That’s good too. My floors are dirty.

KK: That’s okay. Dirty floors, clean…something. So today we are pleased to have Chawne Kimber on the show. Chawne, do you want to introduce yourself?

Chawne Kimber: Sure. Hi, I’m a professor at Lafayette College. I got my Ph.D. a long time ago at University of Florida.

KK: Go Gators!

CK: Yay, woo-hoo. I work in lattice-ordered groups.

KK: Lattice-ordered groups, very cool. I should probably know what those are, but maybe we’ll find out what they are today. So yeah, let’s get into it. What’s your favorite theorem, Chawne?

CK: Okay, so maybe you don’t like this, but it’s a suite of theorems.

KK: Even better.

EL: Go for it.

CK: So, right, a lattice-ordered group is a group, to begin with, in which any two elements have a sup and an inf, so that gives you your lattice order. They’re torsion-free, so they’re, once you get past countable ones, they’re enormous groups to work with. So my favorite theorems are the representation theorems that allow you to prove stuff because they get unwieldy due to their size.

EL: Oh cool. One of my favorite classes in grad school was a representation class. I mean, I had a lot of trouble with it. It was just representations of finite groups, and those were still really out there, but it was a lot of fun. Really algebraic thinking.

CK: Well actually these representations allow you to translate problems from algebra to topology, so it’s pretty cool. The classical theorem is by Hahn in 1909. He proved the special cases that any totally ordered Archimedean group can be embedded as a subgroup of the reals, and it kind of makes sense that you should be able to do that.

KK: Sure.

CK: And then he said that any ordered abelian group, so not necessarily lattice-ordered, can be embedded in what’s called a lexicographical product of the reals. So we could get into what that is, but those are called Hahn groups. They’re just huge products of the reals that are ordered in dictionary order that only live on well-ordered sets. So this conjecture, it’s actually a theorem, but then there’s a conjecture that that theorem is actually equivalent to the axiom of choice.

KK: Wow.

EL: Oh wow.

CK: Right?

EL: Can we maybe back up a little bit, is it possible to, for me, I really like concrete examples, so maybe can you talk a little bit about a concrete example of one of these archimedean groups? I don’t know how concrete the concrete examples are.

CK: No, they’re just really weird ways of hacking at the reals, basically, so they’re just subgroups of the reals. Think of your favorite ones, and there you go, the ones that are archimedean. And as soon as you add two dimensions of ordering, it’s even more complex, right? So the classical example that I work with would be rings of continuous functions on a topological space, and then you can build really cool examples because we all understand continuous functions, so C(X), real-valued continuous functions on a Tychonoff space, so T-3 1/2, whatever.

KK: Metric space.

CK: The axioms so you have enough continuous functions. So Gillman and Jerison in the 1950s capitalized on a theorem from the 1930s by Gelfand and Kolmogorov that said that the maximal ideals of C(X), if you take them in the hull-kernel topology, are isomorphic to the Stone-Čech compactification of the space that you’re working on. And so if you have a compact space to begin with, then your space is isomorphic to your maximal ideals. So then, just build your favorite—so C(X) is lattice-ordered, if you take the pointwise ordering, and then since the reals have a natural order on then, you pick up your sups and infs pretty easily. So there you’re starting to touch some interesting examples of these groups. Have I convinced you, Evelyn?

EL: Yeah, yeah.

CK: Okay, good. So they’re huge. You have to have some complexity in order to be able to prove anything interesting about them. So then there the Hahn embedding is pretty obvious. You just take the images of the functions. There’s too much structure in a ring like that, so maybe you want to look at just an ordered group to get back to the Hahn environment. So how can you mimic Hahn in view of Gelfand-Kolmogorov? So can we get continuous functions as the representation of an ordered group? Because the lex products that Hahn was working with are intractable in a strong way. And so then you have to start finding units because you have to be able to define something called a maximal sub-object, so you want it to be maximal with respect to missing out on some kind of unit. And so then we get into a whole series of different embedding theorems that are trying to get you closer to being able to deal with the conjecture I mentioned before, that Hahn’s embedding theorem is equivalent to the axiom of choice.

EL: Yeah, I’m really fascinated by this conjecture. It kind of seems like it comes out of nowhere. Maybe we can say what the axiom of choice is and then, is there a way you can kind of explain how these might be related?

CK: Yes and no.

KK: Let’s start with the axiom of choice.

CK: Yeah, so the axiom of choice is equivalent to Zorn’s lemma, which says that maximal objects exist. So that’s the way that I deal with it. It allows me to say that maximal ideals exist, and if they didn’t exist, these theorems wouldn’t exist. You use this everywhere in order to prove Hahn’s theorem, so that’s why it’s assumed to be possibly equivalent. This isn’t the part that I work on. I’m not a logician.

KK: So many things are equivalent to the axiom of choice. For example, the Tychonoff product theorem, which is that the product of compact spaces is compact. That’s actually equivalent to the axiom of choice, which seems a bit odd. I was actually reading last night, so Eugenia Cheng has this book Beyond Infinity, her most recent book, good bedtime reading. I learned something last night about the axiom of choice, which is that you need the axiom of choice to prove that if you have two infinities, two countable infinities, you want to think [they’re the same], it’s countable somehow. If they come with an order, then fine, but if you have two, like imagine pairs of socks, like an infinite collection of pairs of socks, is that countable? Are the socks countable? It’s an interesting question, these weird slippery things with the axiom of choice and logic. They make my head hurt a little bit.

CK: Mine too.

EL: So yeah, you’re saying that looking at the axiom of choice from the Zorn’s lemma point of view, that’s where these maximal objects are coming in in the Hahn conjecture, right?

CK: Absolutely.

KK: That makes sense.

CK: That’s kind of why I drew the parallel with this theorem about C(X), these maximal ideals being equivalent to the space you’re on. Pretty cool.

KK: Right. Because even to get maximal ideals in an arbitrary ring, you really need Zorn’s lemma.

CK: Right. And there’s a whole enterprise of people working to see how far you can peel that back. I did take a small foray into trying to understand gradations of the axiom of choice, and that hurts your head, definitely.

KK: Right, countable axiom of choice, all these different flavors.

CK: Williams prime ideal theorem, right.

KK: Yeah, okay.

EL: So what drew you to these theorems, or what makes you really excited about them?

CK: Well, you know, as a super newbie mathematician back in the day, I was super excited to see that these disparate fields of algebra and topology that everyone had told me were totally different could be connected in a dictionary way. So a characteristic of a ring can be connected is equivalent to a characteristic on a topological space. So all kinds of problems can be stated in these two different realms. They seem like different questions, but they turn out to be equivalent. So if you just know the way to cross the bridge, then you can answer either question depending on which realm gives you the easier approach to the theorem.

KK: I like that interplay too. I’m a topologist, but I’m a very algebraic one for exactly that reason. I think there are so many interesting ideas out there where you really need the other discipline to solve it, or looking through that lens makes it a lot clearer somehow.

EL: And was this in graduate school that you saw these, or as a new professor?

CK: Definitely grad school. I was working on my master’s.

KK: So I wonder, what does one pair with this suite of theorems?

CK: It’s a very hard question, actually.

KK: That’s typical. Most people find this the more difficult part of the show.

CK: Yeah. I think that if you were to ask my Ph.D. advisor Jorge Martinez what he would pair, he is very much a wine lover and an opera lover. So it would be both. You’d probably see him taking a flask into Lincoln center while thinking about theorems. So he loved to go to Tuscany, so I assume that’s where you get chianti. I don’t know, I could be lying.

KK: You do, yeah.

CK: Yeah, so let’s go with a good chianti, although that might make me sound like Hannibal Lecter.

KK: No fava beans.

CK: So we’ve got a chianti, and maybe a good opera because it’s got to be both with him. It’s hard for me to say. So he comes up to New York to do an opera orgy, just watching two operas per day until he falls down. I sometimes join him for that, and the last one I went to was Così fan tutte, and so let’s go with that because that’s the one I remember.

EL: If I remember correctly—it’s been a while since I saw or listened to that opera—there are pairs of couples who end up in different configurations, and it’s one of these “I’ll trick you into falling in love with the other couple’s person” that almost seems like the pairs being topology and algebra, and switching back and forth. I don’t know, maybe I’m putting ideas in your mind here.

CK: Or sort of the graph of the different couplings, the ordered graph could be the underlying object here. You never know.

EL: An homage to your advisor here with this pairing.

CK: Yeah, let’s do that.

EL: Well I must admit I was kind of hoping that you might pair one of your own quilt creations here. So I actually ran into you through a quilting blog you have called completely cauchy. Do you mind talking to us a little bit about how you started quilting and what you do there because it’s so cool.

CK: Yeah. Of course I chose that name because Cauchy is my favorite mathematician, and as a nerd there would be no other quilt blog named after a dead mathematician. So I am a little mortified that when you google “Cauchy complete,” as many students do, mine is actually the first entry that comes up on google.

KK: Excellent.

CK: I don’t know what that means, but okay. So yeah, when I applied for tenure, which is kind of a hazing process no matter where you are, no matter how good of a faculty member you are, I really wanted to have control, and you don’t have control at that point. And so I started sewing for fun, late at night, at 1 am, after everything kind of felt done for the day. I never imagined that I’d be doing what I’m doing today, which is using quilting to confront issues of social justice in the United States, and they’ve been picked up by museums and other venues. It’s this whole side hustle out there that I kept quiet for a long, long time. And then once I got promoted to full professor I came out of the closet.

KK: Were you concerned that having a side hustle, so to speak, would compromise your career? Because it shouldn’t.

CK: Yeah, I think something so gender-specific as quilting, something you associate with grandmas. At the end of the day, the guys I work with, I must say half of my quilts have four-letter words on them, you know, the more interesting four-letter words, so as soon as my guys saw them, they were totally on board with this enterprise, so I didn’t really need to be into the closet, but I didn’t want anybody to ever say, “Oh, she should have proved one more theorem instead of making that quilt.”

EL: Yeah.

KK: It’s unfortunate that we feel that way, right? I think that’s true of all mathematicians, but I imagine it’s worse for women, this idea that you have to work twice as hard to prove you’re half as good or something like that?

CK: Do we need to mention I’m also a black woman? So that’s actually how I was raised, you need to do three times as much to be seen as half as good, and that’s the way that I’ve lived my life, and it’s not sustainable in any way.

KK: No, absolutely not.

EL: But yeah, they are really cool quilts, so everyone should look at completely cauchy, and that’s spelled cauchy. There’s a mathematician named Cauchy. I actually have another mathematician friend with a cat named Cauchy, or who had a cat named Cauchy. I think the cat has passed away. Yeah, and I actually sew as well. I’ve somehow never had the patience for quilting. It just feels somehow like too little. I like the more immediate gratification of making a whole panel of a skirt or something. You do really intricate little piecing there, which I admire very much, and I’m glad people like you do it so I don’t have to.

KK: Sure, but Evelyn, you don’t have to make it little.

CK: You don’t.

KK: I’m sure you’ve seen these Gee’s Bend quilts, right, they’re really nice big pieces, and that can have a very dramatic effect too. But yeah, the intricate work is really remarkable. My wife has done a little quilting, and she always gets tired of it because of the fine stuff, but then she’s a book artist. She sets lead typing in her printing press by hand, and that’s fine, but piecing together little pieces of cloth somehow doesn’t work.

CK: It seems more futile, you take a big piece of fabric and cut it into small pieces so that you can sew it back together. That is kind of dumb when you think about it.

KK: Well, but I don’t know, you’ve got this whole Banach-Tarski thing, maybe.

EL: Bring it back around to the axiom of choice again.

CK: You guys are good at this.

KK: It’s not our first podcast. Well this has been great fun. Anything else you want to promote?

CK: No, I’m good.

KK: Thanks for joining us, Chawne. This has really been interesting, and we appreciate you being on.

CK: Great. Thank you.

EL: Thanks.

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