Evelyn Lamb: Hello, and welcome to my favorite theorem, a math podcasts where we ask mathematicians to tell us about their favorite theorems. I'm Evelyn Lamb. I'm a freelance math and science writer in Salt Lake City, Utah. And I am joined as usual by my co host Kevin. Can you introduce yourself?
Kevin Knudson: Sure. I'm still Kevin Knudson, professor of mathematics at the University of Florida. How are things going?
EL: All right. Yeah.
KK: Well, we just talked yesterday, so I doubt much has changed, right? Except I seem to have injured myself between yesterday today. I think it's a function of being—not 50, but not able to say that for much longer.
EL: Yeah, it happens.
KK: It does.
EL: Yeah. Well, hopefully, your podcasting muscles have not been injured.
KK: I just need a few fingers for that.
EL: Alright, so we are very happy today to welcome Ursula Whitcher to the show. Hi, can you tell us a little bit about yourself?
Ursula Whitcher: Hi, my name is Ursula. I am an associate editor at Mathematical Reviews, which, if you've ever used MathSciNet to look up a paper or check your Erdős number or any of those exciting things, there are actually 16 associate editors like me checking all the math that gets posted on MathSciNet and trying to make sure that it makes sense. I got my PhD at the University of Washington in algebraic geometry. I did a postdoc in California and spent a while as a professor at the University of Wisconsin Eau Claire, and then moved here to Ann Arbor where it's a little bit warmer, to start a job here.
EL: Ann Arbor being warmer is just kind of a scary proposition.
UW: It’s barely even snowing. It's kind of weird.
EL: Yeah. Well, and yeah, you mentioned Mathematical Reviews. I—before you got this job, I was not aware that, you know, there were, like, full time employees just of Mathematical Reviews, so that's kind of an interesting thing.
UW: Yeah, it's a really cool operation. We actually go back to sometime in probably the ‘40s.
KK: I think that’s right, yeah.
EL: Oh wow.
UW: So it used to be a paper operation where you could sign up and subscribe to the journal. And at some point, we moved entirely online.
KK: I’m old enough to remember in grad school, when you could get the year’s Math Reviews on CD ROM before MathSciNet was a thing. And you know, I remember pulling the old Math Reviews, physical copies, off the shelf to look up reviews.
UW: We actually have in the basement this set of file cards that our founder, who came from Germany around the Second World War, he had a collection of handwritten cards of all the potential reviewers and their possible interests. And we've still got that floating around. So there's a cool archival project.
KK: I’m ashamed to admit that I'm a lapsed reviewer. I used to review, and then I kind of got busy doing other things and the editors finally wised up and stopped sending me papers.
UW: I try to tell people to just be really picky and only accept the papers that you're really excited to read.
KK: I feel really terrible about this. So maybe I should come back. I owe an an apology to you and the other editors.
UW: Yeah, come back. And then just be really super, super picky and only take things that you are truly overjoyed to read. We don't mind. I—you know, I read apologies for my job for part of my day every day for not reviewing. So I’ve become sort of a connoisseur of the apology letter.
KK: Sure. So is part of your position also that you have some sort of visiting scholar deal at the University of Michigan? Does that come with this?
UW: Yeah, that is. So I get to hang out at the University of Michigan and go to math seminars and learn about all kinds of cool math and use the library card. I'm a really heavy user of my University of Michigan library card. So yeah.
EL: Those are excellent.
KK: It’s a great campus. That’s a great department, a lot of excellent people there.
EL: Yeah. So, what is your favorite theorem, or the favorite theorem you would like to talk about today?
UW: So I decided that I would talk about mirror theorems as a genre.
UW: I don't know that I have a single favorite mirror theorem, although I might have a favorite mirror theorem of the past year or two. But as this kind of class of theorems, these are a weird thing, because they run kind of backwards.
Like, typically there's this thing that happens where mathematicians are just hanging out and doing math because math is cool. And then at some point, somebody comes along and is like, “Oh, I see it practical use for this. And maybe I can spin it off into biology or physics or engineering or what have you.” Mirror theorems came the other way. They started with physical observation that there were two ways of phrasing of a theoretical physics idea about possible extra dimensions and string theory and gravity and all kinds of cool things. And then that physical duality, people chewed on and figured out how to turn it into precise mathematical statements. So there are lots of different precise mathematical statements encapsulating maybe something different about about the way these physical theories were phrased, or maybe building then, sort of chaining off of the mathematics and saying something that no longer directly relates to something you could state about a possible physical world. But there is still in like a neat mathematical relationship you wouldn't have figured out without having the underlying physical intuition.
EL: Yeah. And so this is, the general area is called mirror symmetry. And when I first heard that phrase, I assumed it was something about like group theory that was looking at, like, you know, more tangible, things that I would consider symmetric, like what it looks like when you look in a mirror. But that's not what it is, I learned.
UW: So I can tell you why it's called mirror symmetry, although it's kind of a silly reason. The first formulations of mirror symmetry, people were looking at these spaces called Calabi-Yau three-folds, which are—so there are three complex dimensions, six real dimensions, they could maybe be the extra dimensions of the universe, if you're doing string theory. And associated with a Calabi-Yau three-fold, you have a bunch of numbers that tell you about it’s topological information, sort of general stuff about what is this six dimensional shape looking like. And you can arrange those numbers in a diamond that's called the Hodge diamond. And then you can draw a little diagonal line through the Hodge diamond. And some of the mirror theorems predict that if I hand you one Calabi-Yau three-fold with a certain Hodge diamond, there should be somewhere out there in the universe another Calabi-Yau three-fold with another Hodge diamond. And if you flip across this diagonal axis, one is the Hodge diamonds should turn into the other Hodge diamond.
UW: So there is a mirror relationship there. And there is a really simple reflection there. But it's like you have to do a whole bunch of topology, and you have to do a whole bunch of geometry and you, like, convince yourself that Hodge diamonds are a thing. And then you have to somehow—like, once you've convinced yourself Hodge diamonds are a thing, you also have to convince yourself that you can go out there and find another space that has the right numbers in the diamond.
EL: So the mirror is, like, the very simplest thing about this. It’s this whole elaborate journey to get to the mirror.
EL: Okay, interesting. I didn't actually know that that was where the mirror came from. So yeah. So can you tell us what these mirror theorems are here?
UW: Sure. So one version of it might be what I said, that given a Calabi-Yau manifold, with this information, that it has a mirror.
Or so then this diamond of information is telling you something about the way that the space changes. And there are different types of information that you could look at. You could look at how it changes algebraically, like if you wrote down an equation with some polynomials, and you changed those coefficients on the polynomials just a little bit, sort of how many different sorts of things, how many possible deformations of that sort could you have? That's one thing that you can measure using, like, one number in this diamond.
UW: And then you can also try to measure symplectic structure, which is a related more sort of physics-y information that happens over in a different part of the diamond. And so another type of mirror theorem, maybe a more precise type of mirror theorem, would say, okay, so these deformations measured by this Hodge number on this manifold are isomorphic in some sense to these other sorts of deformations measured by this other Hodge number on this other mirror manifold.
KK: Is there some trick for constructing these mirror manifolds if they exist?
UW: Yeah, there are. There are sort of recipes. And one of the games that people play with mirror symmetry is trying to figure out where the different recipes overlap, when you’ve, like, really found a new mirror construction, and when you’ve found just another way of looking at an old mirror construction. If I hand you one manifold, does it only have a unique mirror or does it have multiple mirrors?
KK: So my advisor tried to teach me Hodge theory once. And I can't even remember exactly what goes on, except there's some sort of bi-grading in the cohomology right?
KK: And is that where this diamond shows up?
UW: Yeah, exactly. So you think back to when you first learned complex analysis, and there was, like, d/dz direction and there was the d/dz̅ direction.
UW: And we're working in a setting where we can break up the cohomology really nicely and say, okay, these are the parts of my cohomology that come from a certain number of homomorphic d/dz kind of things. And these are the other pieces of cohomology that can be decomposed and look like, dz̅. And since it’s a Kähler manifold, everything fits together in a nice way.
KK: Right. Okay, there. That's all I needed to know, I think. That's it, you summarized it, you're done.
EL: So, I have a question. When you talk about like mirror theorems, I feel like some amount of mirror symmetry stuff is still conjectural—or “I feel like”—my brief perusal of Wikipedia on this indicates that there are some conjectures involved. And so how much of these theorems are that in different settings, these mirror relationships hold, and how much of them are small steps in this one big conjectural picture. Does that question make sense?
UW: Yeah. So I feel like we know a ton of stuff about Calabi-Yau three-folds that are realized in sort of the nice, natural ways that physicists first started writing down things about Calabi-Yau three-folds.
When you start getting more general on the mathematical side—for instance, there's a whole flavor of mirror symmetry that's called homological mirror symmetry that talks about derived categories and the Fuakaya category—a lot of that stuff has been very conjectural. And it's at the point where people are starting to write down specific theorems about specific classes of examples. And so that's maybe one of the most exciting parts of mirror symmetry right now.
And then there are also generalizations to broader classes of spaces, where it's not just Calabi-Yau three-folds where maybe you're allowing a more general kind of variety or relaxing things, or you're starting to look at, what if we went back to the physics language about potentials, instead of talking about actual geometric spaces? Those start having more conjectural flavor.
EL: Okay, so a lot of this is in the original thing, but then there are different settings where mirror symmetry might be taking place?
EL: Okay. And I assume if you're such a connoisseur of mirror theorems, that this is related to your research also. What kinds of questions are you looking at in mirror symmetry?
UW: Yeah, so I spend some time just playing around with different mirror constructions and seeing if I can match them up, which is always a fun game, trying to see what you know. Lately, what I've been really excited about is taking the sort of classical old-fashioned hands-on mirror constructions where I can hand you a space, and I can take another space, and I can say these two things are mirror manifolds. And then seeing what knowing that tells me, maybe about number theory, about maybe doing something over a finite field in a setting that is less obviously geometry, but where maybe you can still exploit this idea that you have all of this extra structure that you know about because of the mirror and start trying to prove theorems that way.
EL: Oh, wow. I did not know there is this connection in number theory. This is like a whole new tunnel coming out here.
UW: Yeah, no, it's super awesome. We were able to make predictions about zeta functions of K3 surfaces. And in fact we have a theorem about a factor of as zeta function for Calabi-Yau manifolds of any dimensions. And it's a very specific kind of Calabi-Yau manifold, but it's so hard to prove anything about zeta functions! In part because if you're a connoisseur of zeta functions, you know they are controlled by the size of the cohomology, so once your cohomology starts getting really big, it’s really difficult to compute anything directly.
EL: So, like, how tangible are these? Like, here is a manifold and here is its mirror? Are there some manifolds you can really write down and, like, have a visual picture in your mind of what these things actually look like?
UW: Yeah, definitely. So I'm going to tell you about two mirror constructions. I think one of these is maybe more friendly to someone who likes geometry. And one of these is more friendly to someone who likes linear algebra.
UW: So the oldest, oldest mirror symmetry construction was, it's due to Greene and Plesser who were physicists. And they knew that they were looking for things with certain symmetries. So they took the diagonal quintic in projective four-space. I have to get to my dimensions right, because I actually often think about four dimensions instead of six.
So you're taking x5+y5+z5 plus, then, v5 and w5, because we ran out of letters, we had to loop around.
EL: Go back.
UW: Yeah. And you say, Okay, well, these are complex numbers, I could multiply any of them by a fifth route of unity, and I would have preserved my total space, right?
Except we're working in projective space, so I have to throw away one of my overall fifth roots of unity because if I multiply by the same fifth root of unity on every coordinate, that doesn't do anything. And then they wanted to maybe fit this into a family where they deformed by the product of all the variables. And if you want to have symmetries of that entire family, you should also make sure that the product of all of your roots of unity, I think multiplies to 1? So anyway, you throw out a couple of fifth roots of unity, because you have these other symmetries from your ambient space and things that you're interested in, and you end up with basically three fifths roots of unity that you can multiply by.
So I've got x5+y5+z5+v5+w5, and I'm modding out by z/5z3. Right? So I’m identifying all of these points in this space, right? I've just like got, like, 125 different things, and I’m shoving all these 125 different things together. So when I do that, this space—which was all nice and smooth and friendly, and it's named after Fermat, because Fermat was interested in equations like that—all of a sudden, I'd made it like, really stuck together and messy, and singular.
UW: So I go in as a geometer, and I start blowing up, which is what algebraic geometers call this process of going in with your straw and your balloon, and blowing and smoothing out and making everything all nice and shiny again, right?
UW: And when you do that, you've got a new space, and that's your mirror.
KK: So you blow up all the singularities?
UW: Yeah, your resolve the singularity.
KK: That’s a lot.
UW: Yeah. So what you had was, you had something which is floating around in P4. And because we picked a special example, it happens to have a lot of algebraic classes. But a thing in P4, the only algebraic piece you really know about it, in it, is, like, intersecting with a hyper plane.
So it has lots and lots of different ways you can vary all of its different complex parameters on only this one algebraic piece that you know about. And then when you go through this procedure, you end up with something which has very few algebraic ways to modify it. It actually naturally has only a one-parameter algebraic deformation space. But then there are all of these cool new classes that you know about, because you just blew up all of these things. So you're trading around the different types of information you have. You go from lots of deformations on one side to very few deformations on the other.
KK: Okay, so that was the geometry. What's the linear algebra one?
UW: Okay, so the linear algebra one is so much fun. Let's go back to that same space.
EL: I wish our listeners could see how big your smile is right now.
KK: That’s right. It’s really remarkable.
EL: It is truly amazing.
UW: Right. So we've got this polynomial, right, x5+y5+z5+w5+v5. And that thing I was telling you about finding the different fifth roots of unity that we could raise things to, that’s, like, a super tedious algebraic process, right, where you just sit down and you're like, gosh, I can raise different parts of the variables, like fifth roots of unity. And then I throw away some of my fifth roots of unity. So you start with that, the equation and the little algebraic rank that you want to get a group associated with it.
And then you convert your polynomial equation to a matrix. In this case, my matrix is just going to be like all fives down the diagonal.
UW: But you can do this more generally with other types of polynomials. The ones that work well for this procedure have all kinds of fancy names, like loops and chains of Fermat’s. So like Fermat’s is just the like different pure powers of variables. Loops would be if I did something like x5+y5+z5+…, and then I looped back around and used an x again.
UW: Or, sorry, it should have been like, x4y+y4z, and so on. So you can really see the looping about to happen.
And then chains are a similar thing. Anyway, so given one of these things, you can just read off the powers on your polynomial, and you can stick each one of those into a matrix. And then to get your mirror, you transpose the matrix.
EL: Oh, of course!
UW: And then you run this little crank, to tell you about an associated group.
UW: So getting which group goes with your transposed matrix, it's kind of a little bit more work. But I love the fact that you have this, like, huge, complicated physics thing with all this stuff, like the Hodge diamond, and then you're like, oh, and now we transpose a matrix! And, you know it’s a really great duality, right, because if you transpose the matrix again, you get back where you started.
EL: Right. Yeah. Well, and it seems like so many questions in math are, “How can we make this question into linear algebra?” It's just, like, one of the biggest hammers mathematicians have.
EL: So another part of this podcast is that we ask our guest to pair their theorem, or in your case, you know, set of theorems, or flavor of theorems, with something. So what have you chosen as you're pairing?
UW: I decided that we should pair the mirror theorems with really fancy ramen.
EL: Okay. So yeah, tell me why.
UW: Okay. So really fancy ramen, like, the good Japanese-style, where you've simmered the broth down for hours and hours, and it's incredibly complex, not the kind that you just go buy in a packet, although that also has its use.
EL: Yeah, no, Top Ramen.
UW: Right. So it's complex. It has, like, a million different variants, right? You can get it with miso, you can get it spicy, you can put different things in it, you can decide whether you want an egg in it that gets a little bit melty or not, all of these different little choices that you get. And yeah, it seems like it's this really simple thing, it’s just noodle soup. And we all know what Top Ramen is. But there's so much more to it. The other reason is that I just personally, historically associate fancy ramen with mirror theorems. Because there was a special semester at the Fields Institute in Toronto, and Toronto has a bunch of amazing ramen. So a lot of the people who were there for that special semester grew to associate the whole thing with fancy ramen, to the point where one of my friends, who's an Italian mathematician, we were some other place in Canada, I think it was Ottawa, and she was like, “Well, why don't we just get ramen for lunch?” And I was like, “Sorry, it turns out that Canada is not a uniform source of amazing ramen.” That was special to Toronto.
KK: Yeah, Ottawa is more about the poutine, I think.
UW: Yeah, I mean, absolutely. There's great stuff in Ottawa. It just like, didn't have this beautiful ramen-mirror symmetry parents that we had all
EL: Right, I really liked this pairing. It works on multiple levels.
KK: Sure. It's personal, but it also works conceptually, it's really good. Yeah. Well, so how long have you been at Math Reviews?
UW: I think I'm in my third year.
KK: Do you enjoy it?
UW: I do. It’s a lot of fun.
KK: Is it a permanent gig? Or are these things time limited?
UW: Yeah, it's permanent. And in fact, we are hiring a number theorist. So if you know any number theorists out there who are really interested in, you know, precise editing of mathematics and reading about mathematics and cool stuff like that, tell them to look at our ad on Math Jobs. We're also hiring in analysis and math physics. And we've been hiring in combinatorics as well, although that was a faster hiring process.
EL: Yeah. And we also like to, you know, plug things that you're doing. I know, in addition to math, you have many other creative outlets, including some poetry, right, related to math?
UW: That’s right.
EL: Where can people find that? Ah, well, you can look at my website. Let's see, if you want the poetry you should look at my personal website, which is yarntheory.net.
There's one poem that was just up recently on JoAnne Growney’s blog.
EL: Yeah, that's right.
UW: And I have a poem that's coming out soon, soon, I’m not sure how soon in Journal of Humanistic Math. Yeah, it's a really goofy thing where I made up some form involving the group of units for the multiplicative group associated to the field of seven elements and then played around with that.
EL: I'm really, really looking forward to getting into that. Do you have a little bit of explanation of the mathematical structure in there?
UW: Just the very smallest. I mean, I think what I did was I listed, I found the generators of this group, and then I listed out where they would go as you generated them, and then I looked for the ones that seemed like they were repeating in an order that would make a cool poem structure.
EL: Okay, cool. Yeah. Well, thanks a lot for joining us. We'll be sure to share all that and hopefully people can find some of your work and enjoy it.
KK: Thanks, Ursula.
UW: Thanks so much for having me.[outro]