Kevin Knudson: Welcome to My Favorite Theorem, a math podcast for your quarantine life. I'm Kevin Knudson, professor of mathematics at the University of Florida. And here is your other host.

Evelyn Lamb: Hi, I'm Evelyn Lamb. I'm a freelance math and science writer in beautiful Salt Lake City, Utah.

KK: Yeah.

EL: How are you, Kevin?

KK: I'm okay. I had my—speaking of quarantines, I had my COVID swab test this morning.

EL: How was it?

KK: Well, you know, about as pleasant as it sounds. But yeah, I'm sure you've been to the pool and gotten water up your nose. That's what it feels like.

EL: Yeah.

KK: And then it's over. And it's no big deal. I should have the results within 48 hours. It’s part of the university's move to get everybody back to campus, although I don't expect to go back to the office in any serious way before August. But this is late May now for our listeners, who will probably be hearing this in December or something, right?

EL: Yeah. Who even knows? Time has no meaning.

KK: Hopefully this will all be irrelevant by the time our listeners hear this. [Editor’s note: lolsob.] We'll we'll have a vaccine and everything. It will be a brave new world and everything be fine.

EL: It’ll be a memory of that weird time early in year.

KK: That’s right. The before times. So anyway, today, we are pleased to welcome Michael Barany. Michael, why don’t you introduce yourself and let us know who you are and what's up.

Michael Barany: Hi. So I'm a historian of mathematics. I'm super excited to be on this podcast. I feel like I've been listening long enough that the Gainesville percussionists must be in grad school by now.

KK: No. One of them is my son, and he just finished his third year of college.

MB: Okay, yeah. So older than he was anyway.

EL: Yeah.

MB: Yeah, so I’m a historian of mathematics. I'm based at the University of Edinburgh, where I'm in a kind of interdisciplinary social science of science and technology department. So I get to teach students from all over the university how to think about what science means when you step back and look at the people involved and how they relate to society, how ideas matter, how technology's changed the world, all that fun stuff that gets people to really rethink their place in the world and the kind of things they do with their science.

KK: That’s very cool.

EL: And I know some people who are historians of math will get a degree through a math department and some get it through a history department, I assume. And which are you? I always wonder what the benefits are of each approach.

MB: Yeah, that's great. History of mathematics is a really strange field. It’s actually, as a field, a lot older than history of science as a field, and even older than history as a profession.

EL: Huh.

MB: So history of mathematics started as a branch of mathematics in the early modern period. So we're talking like the 1500s, 1600s. There are always debates about what you classify as this or that. And it started as a way of trying to understand how mathematical theories came about, how they naturally fit together. The idea was that if you understood how mathematical theories emerged, you could come up with better mathematical theories, and you could understand the sort of natural order of numbers and the universe and everything else that you want to understand with mathematics. And then more toward the 19th and the 20th century, there are all these different variations of history of mathematics that branched out of fields like history and philosophy, and philosophy of science and history of science. So my undergrad training was in mathematics. My PhD is from a history department, but from a history of science program in that department. But it's possible to get a PhD in history of mathematics from a mathematics department, it's possible to sort of straddle between different departments. And it makes it a really rich and interesting field. Mathematics education departments or groups sometimes give PhDs in history of mathematics. And they really use the history for different purposes. So if your goal is to make mathematics better, you're taking the perspective of someone doing it from a mathematics department. If your goal is to become a better educator, then you can use history for that in a math education context. I tend to do history as a way of understanding how things fit together in the past and trying to make sense of social values and social structures and ideologies and ideas and how those fit together. And that's the approach that that you come at from a history or history of science perspective.

KK: Very cool. And How did you end up in Edinburgh of all places?

MB: Well, so the academic job market is bad enough in mathematics, right, but in history of mathematics, in a good year, there may be two to three openings in history of science jobs in general. So that's the cynical answer. The more idealistic answer is Edinburgh has this really important place in the sociology of science. In the 1970 s and 80s especially, there was this group of kind of radical sociologists in at the University of Edinburgh who sat down. It was called the Edinburgh School of the sociology of scientific knowledge, which is known for this sort of extreme relativism and constructivism view of how politics and ideology shape scientific knowledge. And I did a master's degree in that department many years later, in 2009-2010, sort of getting my feet wet and starting to learn that discipline. And that approach has been really formative for me and my scholarship. And so it was an incredible stroke of luck that they just happened to have an opening in my field while I was on the market. And I was even even more lucky to have the chance to go there.

KK: Wow, that's great. I’ve always wanted to go there. I've never been to Edinburgh,

MB: It’s the most beautiful city in the world.

KK: Yeah, it looks great. All right, well, being a historian of math, you must know a lot of theorems. So the question is, do you actually have a favorite one? And if so, what is it?

MB: So my favorite theorem is more of a definition. But I guess the theorem is that the definition works.

KK: Okay, great.

EL: That works.

MB: Which, actually—saying what it means for a definition to work is actually a really hard problem, both historically and mathematically. So it's interesting in that regard. Ao the definition is the definition of the derivative of a distribution.

KK: Okay.

MB: So distributions, as you’ll recall from, from analysis—I guess, grad analysis I is usually when you meet them.

EL: Yeah, I think it wasn't until grad school for me at least.

KK: I don't know if I've ever met them, really.

MB: So distributions were invented in 1945, more or less. And in the early years, actually, people were saying you could teach this as a replacement for your basic calculus. So the idea was, this would be something that even beginning college students or even high school students would be learning. So it's interesting to see how they have people pitched that the level of a theory or the the relevant audience, and that's part of the story, too. But in earlier stages of one's calculus education, you learn that there are functions that are integrable but not continuous; continuous but not differentiable; differentiable but not continuously differentiable, and so on. And so a big problem is how do you know something's differentiable when you're studying a differential equation or trying to prove some theorem that involves derivatives. And distributions were the kind of magic wand that was invented in the middle of the 20th century to say that's not actually a problem. Basically, if you pretend everything's differentiable, then all the math works out. And when it really is differentiable, you get the correct differentiable answer, and when it's not, then you get another answer that's still mathematically meaningful. But it's sort of your magic passphrase to be able to ignore all of those problems.

So a distribution is this replacement for a function. Where functions have these sort of different degrees of differentiability, distributions are always differentiable and they always have antiderivatives, just like functions do, but every distribution can be differentiated ad nauseam for whatever differential equation you want to do. And the way you do that is through this definition—my favorite definition/theorem—which is you use integration by parts. So that's a technique you use in calculus class, too, as a sort of trick for resolving complicated integrals. And distributions actually don't tend to look at the things that make the calculus problems challenging or interesting, depending on what kind of student you are, or what kind of teacher you are. So you set them up in a way where you don't have to worry about boundary conditions, you don't have to worry about what the antiderivative things are, because you're working with things where you already know what the antiderivative is. And the definition of distribution uses this fact from integration by parts that you essentially move the derivative from one function to another. So we don't have an exact way of saying functionally what the derivative of a distribution is. You can still say if you multiply it by a function that's super-smooth and over a bounded domain—so you don't have any boundary conditions to worry about, and so you always know how to differentiate that—if you multiply that by a distribution, and take the integral, then if you want to take the derivative of that distribution, integration by parts says you can instead throw in a minus sign and take the derivative of that smooth function instead. And so using that kind of trick, of moving the derivative onto something that is always differentiable, you can calculate the effect of differentiating a distribution without ever having to worry about, say, what the values of of that distribution are after you’ve taken the derivative, because distributions are often things that don't have sort of concrete values in the way that we expect functions to have.

EL: And I hope this question isn't very silly. But when you think about integration by parts—you know, if you took calculus at some point and learned this, there's the UV, and then there's the minus the integral of something else. And so for this, we just choose a function that would be zero on the boundary, and that would get rid of that UV term. Is that right?

MB: Exactly? Yeah. So the definition of distribution sets up this whole space of really nice smooth functions. All of them eventually go to zero, and because you're always integrating over the entire domain, and it's always zero when you go far enough out into the domain, those boundary terms with that UV in the beginning just completely disappear, and you're just left with the negative integral, and then with the derivative flopped over.

EL: All right, great. So if anyone was worried about where their UV went, that's where it went. It was zero. Don't worry. Everything's okay. Yeah. Okay. So what is good about this? Or what do you like about this?

MB: Yeah. So I think this is a really interesting definition from a lot of different perspectives. One thing that I've been trying to understand in my research about the history of mathematics is what it means for mathematics to become a global discipline in the 20th century, so to have people around the world working on the same mathematical theory and contributing to the same research program. And this definition is really helped me understand what that even means and how to understand and analyze that historically. So we think, well, you know, a mathematical theory or a mathematical idea is the same wherever you look at it, and whoever's doing it. As long as they can manipulate the definitions or prove the theorem, it shouldn't matter where they are. But if you look historically, at actual mathematicians doing actual mathematics, where they are makes a huge difference in terms of what methods they're comfortable with, how they understand concepts, how they explain things to each other, how they make sense of new techniques. I mean, learning a new mathematics technique is actually really hard in a lot of cases. And so the question is, how do you form enough of an understanding to be able to work with someone who you can't go and have a conversation with over tea the next day to sort of work out your problems? And the answer is, basically, you use things like this definition and take something you're really comfortable with—integration by parts—and give it a new meaning. And by taking old meanings and reconfiguring them and relating them to other meanings, you make it possible for everyone to have their own sorts of mathematical universes where they're building up theories, but to interact in a way where they can all sensibly talk to each other and develop new ideas and share new ideas. So that's one of the things that that's really exciting about that the definition to me.

One of the other things is sort of how do you know what the significance of the definition is? I mean, a lot of people early on said, isn't this just like a pun? Isn't this just wordplay? Quite early on, when Schwartz was sharing this definition, and some people were getting really excited about it. Some people said, well, you know, it's a cool idea. But isn't this just basically integration by parts? What's new? What's interesting about this? And the history really shows this debate, almost, between people with different kinds of values and philosophies and goals for mathematics, for mathematics education, for the relationship between pure and applied mathematics, where they take different ideas of what's really going on with this definition. Is it something that's complex and difficult and profound and important in that way, or is it something that is utterly trivial and simple, and therefore really useful to people who may be, say, electrical engineers who are trying to work with the Heaviside calculus, and need some sort of magic way to make that all add up? And what made distributions and this definition really powerful is it could be these multiple things to multiple people. So you can have mathematicians in Poland, or in Manchester, or in or in Argentina come to these very, almost diametrically opposed views of what it is that's significant or challenging or easy about distributions, and they can all agree to talk to each other and agree that it's worth sharing their theories and inviting them to conferences, and reading their publications, and they can somehow all make a community out of these different understandings.

KK: I’ve never thought about the sociological aspects in that way. That's really interesting. So the theorem that basically says that this definition is a good one. Is that a difficult theorem to prove?

MB: So there are a lot of different parts. It’s not—I guess it doesn't even boil down to one statement.

KK: Yeah, sure. Yeah, that makes sense. Yeah.

MB: So there's the aspect that when you're dealing with a function, but dealing with it using the distributions definition, that anything you do is not going to ruin what's good about it being a function. So anything you do with a distribution, if you could have done it as though it were a regular function, you get the same answer. So that's one aspect of the theorem that sort of establishes this definition. Another aspect is that distributions are, in some sense, the smallest class of objects that includes functions where everything that is a normal function can be indefinitely differentiated. So that's one way of arguing that distributions are sort of the best generalization of functions, and this competition—I mean, there are a lot of different competing notions, or competing ideas for how you can solve this problem of differentiating functions that were circulating in the 1930s and 1940s. And distributions won this competing scene, in part by the aspects of the theorems about the definition that show it’s sort of the most economical, the simplest, smallest, the best in that sense. And then you have all the usual theorems of functional analysis, like everything converges as you expect it to; if you start with something that's integrable, you're not going to lose interpretability, in some sense.

EL: So this might be a little bit of a tangent, and we can definitely decide not to go down this path. But to make this really concrete—so when I think of a distribution, the example I think of—it’s been a while since I've thought of distributions actually, is the Dirac delta function. I naturally just call it a function, but it is really a distribution. And so this is a thing that, I always think of it, it's something that you can't really define what its value is, but it has a convenient property that if you integrate it, you get 1. Like, its area is 1 even though it's supported on only one point, and it is infinitely tall. And so zero times infinity, we want it to be 1 right here.

MB: And magically it turns out to be 1.

EL: Yeah. And basically, if you decide that this function, this distribution, has this property, then things work out, and it's great. Was that before or after Schwartz? Did this definition—was this kind of grandfathered into being a distribution? Or was it the inspiration?

MB: I love how you put that. Yeah. So this, this phrase that you said at the beginning, we call it a function, but it's really a distribution. I mean, that's evidence of Schwartz’s success, right? The idea that what it really is, what it fundamentally is, is a distribution rather than a function, that's the result of this really sort of deliberate—I mean, it's not it's not an exaggeration to call it propaganda in the second half of the 1940s by people like Laurent Schwartz and Marston Morse and Marshall Stone and Harald Bohr and all of these far-traveling advocates for the theory—to say, you think you've been working with functions, you think you've been working with measures, you think you've been working with operator calculus if you're an electrical engineer, for instance. Or you think you've been working with bra and ket, with Dirac calculus for quantum mechanics, but what you've really been doing ultimately, deep down without even knowing it, is working with distributions. And their ability to make that argument was part of their way of justifying why distributions were important. So people who had no problem just doing the math they were doing with whatever kind of language they were doing, all of a sudden, these advocates for distribution theory were able to make it a problem that they were doing this without having the kind of conceptual apparatus that distributions provided them. And so they were both creating a problem for old methods and then simultaneously solving it by giving them this distribution framework.

So, they did this to the Heaviside calculus, which is about 50 years older than distributions. They did this to the Dirac calculus, where the Dirac function comes from, which comes out of the 1920s and 30s. They did this to principal value calculus, which is also an interwar concept in analysis. Even among Schwartz's contemporaries, there were things like de Rham currents, which were—had Schwartz not come along, we would all be saying the Dirac function is really a de Rham current rather than a Schwartz distribution. But then there were even things that came after distributions, or sort of simultaneously and after, that Schwartz was able to successfully claim. Like there was this whole school of functional analysis and operator theory coming out of Poland associated with Jan Mikusiński. Where Schwartz was—because he was able to get this international profile so much more quickly and effectively—he was able to say all of this really clever research and theorems that Mikusiński is coming up with, that's a nice example of distribution theory, even though Mikusiński would have never put that in those terms. So a huge part of this history is how they're able to use these different views of what a distribution really is to sort of claim territory and grandfather things in and also sort of grandchild things, or adopt things into the theory and make this thing seem much bigger than the actual body of research that people who considered themselves distribution theorists themselves were doing.

EL: Okay. And so I think we also wanted to talk a little bit about—you mentioned in your email to us, I hope I'm getting this I'm not getting this confused with anything—how this theory goes with the history of the Fields Medal.

MB: Oh, exactly. Yeah. So this was a really surprising discovery, actually, in my research. I didn't set out—the Fields Medal kind of became one thing, one little bit of evidence that Schwartz was a big deal. I never expected in my research to come across some evidence that really changed how I understood what the Fields Medal historically meant. And this was just a case of stumbling into these really shocking documents, and then having built up all of this historical context to see what their historical implications were. So Schwartz was part of the second ever class of Fields Medalists in 1950. The first class was in 1936, then there's World War II, and then they sort of restart the International Congresses of Mathematics after the war. And Schwartz is selected as part of that second class. The main reason he's part of that class is because the chair of that committee is Harald Bohr, who is the younger brother of Niels Bohr. Actually, in the early 1900s, Harald Bohr was the more famous Bohr because he was a star of the Danish Olympic soccer team.

KK: Oh!

EL: Wow!

MB: He was a striker. His PhD defense had many, many, many more soccer fans that mathematicians. He was this minor Danish celebrity. And he went on to be a quite respectable mathematician. He had his mathematics institute alongside his brother's physics institute in Copenhagen. And during the interwar period especially, he established himself as this safe haven for internationally-minded mathematics in this period of immensely divisive conflict among different national communities. And because he kind of had that role as this respected figure known for internationalism, he was selected by the Americans who organized the 1950 Congress at Harvard to chair the Fields Medal committee. And Bohr, shortly before being appointed to that committee, had encountered Schwartz in a conference that was sponsored by the Rockefeller Foundation and took place in Nancy in France, and he was just totally blown away by this charming, charismatic young Frenchman with this cool-sounding new theory that seemed like it could unite pure and applied mathematicians, that could be attractive to mathematicians all over the world. And so Bohr basically makes it his mission between 1947 and 1950 to tell the whole world about distributions. So he goes to the US and to Canada, and he writes letters all around the world, he shares it with all his friends. And when he gets selected to chair this committee, what you see him constantly doing in the committee correspondence is telling all of his colleagues on the committee what an exciting future of mathematics Schwartz was going to be.

So the problem is, then sort of the question is, what is the Fields Medal supposed to be for? And they didn't really have a very clear definition of what are the qualifications for the medal. There was a kind of vague guidance that Fields left before he died. The medal was created after John Charles Fields’ death. And there was a lot of ambiguity over how to interpret that. So the committee basically had to decide, is this an award for the top mathematicians? Is this award an award for an up-and-coming mathematician? How should age play a factor? Should we only do it for work that was done since the last medal was awarded? A long time to consider there, so that didn't really narrow it down very much in in their case. And they go through this whole debate over what kinds of values they should apply to making this selection. And ultimately, what I was able to see in these letters, which were not saved by the International Mathematical Union, which hadn't even been formed at the time, they were kind of accidentally set aside by a secretary in the Harvard mathematics department. So they weren't meant to be saved. They just were in this unmarked file. And what those letters show is that Bohr basically constructs this idea of what the metal is supposed to be for in a strategic way to allow Schwartz to win. So there's this question, there's this kind of obvious pool of candidates, of outstanding early- to mid-career mathematicians, including people like Oscar Zariski and André Weil, and Schwartz's eventual co-medalist, Atle Selberg. And they are debating the merits of all of these different candidates, and basically, Bohr selects an idea of what the Fields Medal is for, to be prestigious enough to justify giving it to this exciting young French mathematician, but not so prestigious that he would have to give it to André Weil instead, who everyone agreed was a much better mathematician than Schwartz, and much more accomplished and much more successful and very close in age. He was about five years older than Schwartz.

KK: He never won the Fields Medal.

MB: And he never won the Fields Medal, right. And so what you see in the letters from the early years of the Fields Medal is actually this deliberate decision, not just by the 1950 committee, but I was also able to uncover letters for the 1958 Committee, where they consider whether the award should be the very best young mathematicians, and they deliberately decide in both cases that it shouldn't be, that that would be a mistake, that that would be a misuse of the award. Instead, they should give it to a young mathematician, but not a young mathematician that was already so accomplished that they didn't need a leg up.

EL: Right.

MB: And that was my really surprising discovery in the archives, that it was never meant to crown someone who was already accomplished, and in fact, being accomplished could disqualify you. So Friedrich Hirzebruch in 1958, everyone agreed was the most exciting mathematician. He was in his early 30s, sort of a very close comparison to like someone like Peter Scholze today. So already a full professor at a very young age, with a widely-recognized major breakthrough. And they considered Hirzebruch, and they said, No, he’s too accomplished. He doesn't need this medal. We should give it to René Thom or someone like that.

EL: Yeah. And, of course, people like me, who only were aware of the Fields Medal once they started grad school in math—I wasn't particularly aware of anything before that—Think of it as the very best mathematicians under 40 because it has sort of morphed into that over the intervening decades.

MB: Yeah. And one of the cool side effects is now you can now put an asterisk next to—Jean-Pierre Serre is known to brag about being the youngest-ever fields medalist. But the asterisk is that he won in a period when it was still a disqualification to be too accomplished at a young age.

KK: Yeah, but he still won.

MB: He did still win. He’s still a very important mathematician.

KK: You sort of couldn’t deny Serre, right?

MB: Well, they denied Weil, right?

KK: They did. But I think Serre is probably still—Anyway, we can argue about— we should have a ranking of best mathematicians of the ‘50s, right?

EL: I mean, yeah, because ranking mathematicians is so possible to do because it’s a well-ordered set.

KK: That’s right.

EL: Obviously in any field of life, there's no way to well-order people. I shouldn't say any field. I guess you can know how fast people can run some number of meters under certain conditions or something. But in general, especially in creative fields, it's sort of impossible to do. And so how do you choose?

MB: That’s what I love about studying the sociology of science and technology, is that you get these tools for saying—you know, even in fields like running, we think of sprinting as this thing where everyone has a time and that's how fast they are. But look at all of the stuff the International Olympic Committee has to do for anti-doping and regulating what shoes you can wear, like there are all of these different things that affect how fast you are that have to be really debated and controlled. They're kind of ultimately arbitrary. So even in cases like that, you know, it seems sort of more rankable than mathematics or art or something, and you can tell a great sprinter from someone like me who can barely run 100 meters. But at the same time, there are all of these different social and technical decisions that are so interrelated that even things that seem super objective and contestable end up being much more socially determined.

EL: Yeah.

KK: Yeah. All right. So part two of this podcast is you have to pair your theorem with something, or your definition or whatever we're going to call it your distribution, whatever it is.

EL: Yeah. If you treat it as a distribution, it’ll work fine.

KK: That’s right.

MB: Exactly.

KK: So what have you chosen to pair with distributions?

MB: So what I thought I would pair distributions with is a knock-knock jokes.

KK: Okay.

MB: So I did a little bit of research before coming on here, and I basically found there are no good math knock-knock jokes. I mean, someone please prove me wrong, like tweet at me. And yeah, tell me tell me.

KK: Are there good knock knock jokes, period?

EL: Oh, definitely.

MB: Yeah. So I did come up with one that sort of at least picks up on some of the historical themes. So Knock, knock.

KK and EL: Who’s there?

MB: Harold.

KK and EL: Harold who?

MB: Harold is the concept of a function anyway?

That's the best I could do.

EL: Okay.

MB: So why knock-knock jokes? They involve puns. So you're talking about shifting the meaning of something to come up with something new. They're dialogical: there’s a sort of fundamental interactive element. They sort of make communities. So sharing a knock-knock joke, getting a knock-knock joke, finding it funny or groan-inducing, tells you who your friends are, and who shares your sense of humor. And yeah, they fundamentally use this aspect of wordplay to to make something new and to make something social. And that's exactly what the theory of distributions does and what that definition does, just sort of expand your thinking. And they're also sort of seen as kind of elementary, or basic. It's kind of like a kid's joke.

EL: Right.

MB: It’s this question of distributions as this fundamental theory, your basic underlying theory. So I think it sort of brings together all of those aspects that I like about the definition.

KK: You thought hard about this. This is a really thoughtful, excellent pairing. I like this.

EL: Yeah, I like it. I'm trying to figure out what is the analogy to my favorite knock-knock joke, which is the banana and orange one, right, which is classic.

MB: It’s the only one I use in real life.

KK: Sure.

EL: It’s a great one!

KK: Yeah.

EL: Fantastic. But, like, what distribution is this knock-knock joke?

KK: The Dirac function, right? Excuse me, the Dirac distribution.

MB: Yeah. Aren't you glad I didn't say the Dirac distribution? Yeah, no, it's the only one you actually use all the time. Yeah, the Dirac distribution, or there's that theorem that any partial differential equation can be resolved as the sum of derivatives of these elementary distributions. That's your go-to ubiquitous, uses a pun, but uses in a way that kind of makes sense and is kind of groan-inducing, but also you just love to go back and to use it over and over and over again.

KK: Right.

EL: Nice.

KK: I think back in the 70s—dating myself here—I had a book of knock-knock jokes, and it actually had the banana and orange one in it. I mean, it's like, this is how basic of a book this was. So I might be ragging on knock-knock jokes, but of course, I had a whole book of them. So anyway.

EL: Oh, they're great. And especially when a child tells you one.

KK: That’s right. That’s what they’re there for.

MB: The best is when you have a child who hasn't heard the knock-knock joke you’ve heard 10 million times, and you get to be the person to share the groan-inducing pun with the child. I mean, that's how I imagine Schwartz going to Montevideo and explaining distribution theory, like the experience of sharing this pun and having them go “Ohhh” and slapping their forehead. There's this cultural resonance, to introduce something that you immediately grasp. And yeah, that's a really special experience.

KK: Yeah.

EL: So at the end of the show, we like to invite our guests to plug things, and I'll actually plug a couple of your things because we've sort of mentioned them already. You had a really nice article in Nature. I don't remember, it was a couple years ago—

MB: 2018.

EL: —about this history of the Fields Medal, focusing on Olga Ladyzhenskaya, who was on the short list in ’58 and would have been the first woman to get the Fields Medal if she had gotten it, but it was really interesting because it touches on these things about how the Fields Medal became what it is thought of now and how they made that decision at that time. So go read that. And you also have an article about this distribution stuff that I am completely now blanking on the title of, but it has the word “wordplay” in it, and you probably know the title.

MB: There’s “Integration by Parts” as the title.

EL: Okay.

MB: And then there's a long subtitle. So this is the thing any historian does, is they have some kind of punny title and then this long subtitle. I think one of the reasons I empathize with the theory of distributions is, like, this is how I think as a historian. I come up with a pun, and then I work out how all of the things connect together afterwards. You see that in all of my titles, basically, and papers, That's not that's on my website, mbarany.com, and the show notes.

EL: Yeah, we'll put those in the show notes. We'll link to your website and Twitter in the show notes. And yeah, anything else you want to mention?

MB: Yeah, so if you want all of this math and sociology and politics and stuff about academia and the values of mathematics, then my main Twitter account at @mbarany is the one to follow. If you just want sort of parodies and irreverent observations about math history, then @mathhistfacts is my parody account that I started in August, but the key to that is that behind every thing that looks like it's just a silly joke is actually something quite subtle about historical interpretation. And I always leave that as an exercise to the reader. But I do try to—this was my response to, you know, St. Andrews has this MacTutor archive of biographies of mathematicians that has hundreds and hundreds of mathematicians, these sort of capsule biographies. And they have these little examples, or these little summaries, like so-and-so died on this day and contributed to this theory, and it’s just kind of morbid to celebrate them for when they died. But then even the one that makes the rounds every year on Galileo's birthday, so Galileo is actually one of the—not Galileo, Galois. Galois is one of the few people who actually has an interesting death date, whose death is historically significant, and there's a Twitter account that tweets based on on these little biographical snippets, and does it for his birthday rather than his death day and then says, like, “Galois made fundamental contributions to Galois theory.” So this was my response to that account, those tweeting from these biographical snippets saying there's there's more to history than just when people died and what theory named after them they contributed to, and tried to do something a bit more creative with that.

EL: Yeah, that is fun. I felt slightly personally attacked because I did just publish a math calendar that has a bunch of mathematician’s birthdays on it, but I did choose to only do like a page about a mathematician on their birthday rather than their death day because it just seemed a lot less morbid.

MB: Very sensible. There are some mathematicians with interesting death days. So Galois, Cardano. Cardano used mathematics to predict his death day, so it's speculated that he also used some poison to make sure he got his answer right.

EL: Yikes! That’s a bit rough.

MB: But yeah, there are a few mathematically interesting death days. But yeah, I mean, birthdays are okay, I guess. I'm not super into mathematical birthdays anyway, but better than death days.

EL: Yeah. I mean, when you make a calendar, you've got to put it on some day. And it's weird to put it on not-their-birthday. But yeah, that's a fun account. So yeah, this was great. Thanks for joining us, Michael.

MB: Thanks. This was super fun.

On this episode of My Favorite Theorem, we were happy to talk with University of Edinburgh math historian Michael Barany. He told us about his favorite definition in mathematics: distributions. Here are some links you might find interesting.

Barany’s website and Twitter account
His article “Integration by Parts: Wordplay, Abuses of Language, and Modern Mathematical Theory on the Move” about the notion of the distribution

His Nature article about the history of the Fields Medal
Distributions in mathematics

The Dirac delta function (er, distribution?)

The Danish national team profile page of mathematician and footballer Harald Bohr