Welcome to "On A Tangent", where we tell the stories behind the mathematics. In each episode, we meet a different early career mathematician from Mathematics Münster, and learn about their research, their path towards mathematics, and their hopes for the future. We explore the many different shapes that mathematical research can take, the early memories that led people towards the subject, and try to understand a little bit better the voices that make the mathematical community of today.

The podcast episodes are available on this webpage, on Podigee and on popular podcast platforms such as Spotify, Apple Podcast or Deezer.

A new episode is published every last Monday of the month.

About the host: Simone is a doctoral researcher in model theory and occasional science communicator. He likes all things related to stories and fiction.

Episode 3: Paradoxical Sets and Ice Cream, with Azul Fatalini

In this episode of On A Tangent, Simone is joined by Azul Fatalini, a doctoral researcher in Set Theory. We learn about how the Axiom of Choice transforms the universes it holds in, how logic is the mathematics of mathematics, and what’s the quickest path to ice cream.

Link to Azul's website

• Transcript

  Simone: Welcome to On a Tangent, the podcast, where the main characters are the stories behind the mathematics. My name is Simon and in each episode I am joined by a different early career mathematician from Munster to learn about their paths towards mathematics and the hopes for the future. This episode, coming out during Pride Month, I am joined by Azul, a doctoral researcher in set theory. We learn about how the axiom of choice transforms the universes it holds in, how set theory is the mathematics of mathematics, and what's the quickest path to ice cream. I hope you enjoy this episode. (...) So, Azul… Welcome to the podcast.

  Azul: Hi. It's nice to be here.

  Simone: Yeah. Thank you for joining us. So as you know, in the podcast, we sort of explore the stories of our guests. And so I would like to start from your past. So what is your earliest memory of mathematics?

  Azul: Actually I love that question. So when you told me you were going to do this podcast, I ask you what type of questions you would ask. And you told me this example, and I was so glad because I have this story and nobody ever asked me this. And I didn't even know this until you ask me them. So I'm super happy to tell this. So the thing is, when I was a kid, in my hometown, the map of the city is a grid. And there is this main square in the city centre that is a square. So it's 100m times 100m, but it also has a cross in the middle. And one of the things we did with my mom a lot, was walking from my house to the city centre and going to have an ice cream in the corner of the square. And to do that, the best way was to cross the square through the diagonal, right? And most of these, most of the blocks in my city don't have this, it's just that part that you can cross in diagonally. And when I was a kid, I was always thinking, so first, if it was really better, I think, that one I could see, like, I could just look around and see that it was shorter than doing two blocks of 200m. But I always wanted to know precisely…

  Simone: How much better.

  Azul: How much better it was, because it seemed to me that it was more than one block, but less than two.

  Simone: So better than two.

  Azul: But yeah. But not like so much better. So… how much. And I was obsessed with this and I was trying to estimate. But it was super hard.

  Simone: Getting faster to the ice cream.

  Azul: Of course. How much faster are we getting to the ice cream?

  Simone: Exactly.

  Azul: But there was always trying to do this, and in my mind estimation was like 1.5. I didn't even know decimals I think, at this point. But like half.

  Simone: Half. Yeah.

  Azul: But I never could answer if the diagonal was less or more than 1.5. And this was just something I had in my mind for a long time.

  Simone: Until many years later in school, you suddenly found out?

  Azul: Yeah. Yeah, definitely. It was much longer, and I was not even living there anymore. It came the answer ten years later.

  Simone: And do you somehow think that this way… Well, this is in a way, a very geometric problem then sort of connects to what you do nowadays in your research. Have you always been a very visual thinker or a geometric thinker?

  Azul: I would say so. When I was in a Math Olympiad in high school, my favourite branch of problems was geometry. So Euclidean geometry. So in that regard, yes. But I would say I'm terrible at visual thinking in the sense of… It's very hard for me to imagine figures. Okay. In that sense I would say no, but because I cannot imagine so much, then I would draw things. I mean, I like to compensate this lack of imagination with like doing diagrams and drawings to help myself.

  Simone: And so… I just sort of already went into this topic: what is it, that you do nowadays in research?

  Azul: So I'm studying set theory. That's the branch. It's inside logic, as you know.

  Simone: Of course. We finally have a logician on the podcast.

  Azul: Yes. I mean, it's easier to tell you, right?

  Simone: Of course. But for our audience…

  Azul: Yes. So I study problems that are related with the axiom of choice and the subsets of the real numbers. So somehow the objects are very much like normal math, let's say. And there are some geometrical objects that I study. But from the perspective of the theory, because the questions are related to how much axiom of choice do you need to construct these objects or not, which is essentially a set theoretic question.

  Simone: Yes. And is this how you explain it to non maths friends? If they ask, of course.

  Azul: Of course not. I mean they don't know what axiom of choice is.

  Simone: And why set theory is important for people I mean.

  Azul: I mean, I don't even know if a mathematician… I'm not sure mathematicians know why set theory is important in general, I think. Yeah.

  Simone: Except for the fact that, I mean, yes, this is the foundation.

  Azul: That's it. Not mathematicians. I usually say I do something in logic. Logic is a word that people know, right? And I mean, of course they don't know what logic, mathematical logic means, but there is a concept of logic, an idea. And then if they ask me more, I say, okay, there is a branch of logic that is set theory that studies more or less the foundations of math and the structure of how math works. So there are these meta mathematical questions that set theory takes care of.

  Simone: I think I usually tell people we do, like what linguists do to the real world, like, people study the language in which people communicate and logicians study the linguistics of mathematics.

  Azul: Yeah, it could be.

  Simone: In terms of funding, probably it's a similar parallel.

  Azul: In terms of funding?

  Simone: We get as much funding as the linguistics people.

  Azul: Yes. I'm not sure I don't have any idea of this.

  Simone: Yeah. But I imagined it’s not much, as all humanities people.

  Azul: I mean, I agree on what you're saying, but in particular set theory I like more to simplify as the math of math. I think because you're not using linguistics to study, you know, the language of real life, it's like a different thing. Right? Linguistics is different from real world or, or the language that is spoken. But the theory is itself a part of math. And I think that's one of the most beautiful things.

  Simone: And the most confusing.

  Azul: And it's super confusing. But this thing… I mean, math can study things that are things of math. And when you say this to people, even if they are not mathematicians, it is like, what? How do you do that?

  Simone: Yeah. And I do remember my first set theory class, there is all the sort of paradoxes which seem to stem from the fact that you actually talk about set theory when you do the set theory of math. I mean, when you do the maths of maths, you're in particular studying the maths. I mean, you know…

  Azul: Yeah, there's this self-reference. Self-reference at the beginning.

  Simone: Yes. I mean, it is confusing. I mean, it's also something to explore.

  Azul: Well it's definitely both for me. The fact that is so weird in a sense, in comparison to other branches that don't have this feature makes it more interesting.

  Simone: And so your research deals with sort of, let's say, normal mathematical objects, which if another mathematician said this to me, that it would be offensive, but we can say it. Yeah. I mean, you know, because if somebody told me, ah, yeah, we do the normal mathematics, you do the logic, I would be a bit offended. But I can say, you know, the natural mathematical objects that you find in nature and in nature, well, you know, in nature, like in, in mathematics, I mean, mathematical nature. In mathematical nature. Exactly. And you often hear about how the axiom of choice builds these sort of paradoxical objects. Is this the kind of things you're interested in?

  Azul: Yeah. So my PhD thesis is called paradoxical sets and the axiom of choice. That's definitely the object. And yeah, I agree this about the normal math and not normal math.

  Simone: But would paradoxical sets be considered a normal mathematical object.

  Azul: Yeah, I think so. I mean, for example, you have this example that I think is one of the most well known. I think that is the Vitali set. So the standard example of a non measurable set in the reals. And this is I mean this is part of the course of real analysis. This is just something most of people would have seen at some point or Banach-Tarski that is so famous. And, well some people don't like actually Banach-Tarski, like it's weird. Right. But for example, the Vitali set is really a part of understanding of measure theory.

  Simone: No, we don't work in the, … what is it? The Solovay paradise where all sets are measurable?

  Azul: Yeah, exactly. So, I mean, but of course, when you start asking a theoretical question, then the object kind of doesn't satisfy the same role as if it is because you're not studying the object itself, but rather also or in my case, which are the axioms you need to construct, or how consistent is this existence of this object with some other set of axioms or some other objects? So, even if the name is right the same, like Vitali set, then what you're doing with it is very… different, you know.

  Simone: You’re not studying the properties of it, but rather what, what its existence says about the universe you find it in.

  Azul: Yeah, somehow. Where does it lie? In the map of things.

  Simone: And I know you're not meant to play favourites, but what is your favourite paradoxical set?

  Azul: What is my favourite paradoxical set? Let's see. Well, I think if I have to pick one, it would be the one I thought more about. So there is this theorem in ZFC. So using the axiom of choice. That R3 can be partitioned. So the space can be partitioned in circles, so only with the border. And these circles can be taken to have radius one. So they are all the same radius. But still, you can cover R3.

  Simone: Which is, I think you the first time you hear it, like of course I can do this. And then you mentally start placing circles one into the other and then very quickly get to the point where you don't know where to place the next one. Right?

  Azul: Exactly. I mean, you can put more and more circles, right? Because each one is kind of small in inside the space. But how would you end up completing everything over there?

  Simone: Yes.

  Azul: Yeah. Like, imagine you put a lot of circles and like it's dense in R3. But there is some space that you have to fill. If you have like some isolated points, for example, then of course you cannot do it. So even if you have like a full circle, but a few points are not already taken, then you cannot put a circle again.

  Simone: So this means you did it wrongly from the start?

  Azul: Yes. I mean at some point there was something that you did that didn't allow to continue the procedure. Right. But there is a way to do this process, wisely so that this doesn't happen. And this is a theorem of ZFC. So you really need the axiom of choice. Well, less than that, but some form of choice. Some part of choice. Yeah. I think this is my favourite. It’s nice that you can tell people you are studying something like this.

  Simone: Because they can visualise it.

  Azul: They can visualise it is just Euclidean geometry. So I can even tell my family about this if they are patient enough to hear this.

  Simone: Yeah. Or to look at the pictures at least.

  Azul: Yes.

  Simone: And do you feel like because this feels a lot like one of those problems in number theory where the statement is very easy. But then the math behind it is very complicated. So I presume in a similar way the problem can be stated very easily. But then proving it is not Euclidean geometry is actually.

  Azul: Yes. So the proof of this theorem is not well, of course it has some Euclidean geometry because the object is. But the main tool is doing transfinite induction on the cardinality of the reals, whatever that is. But the point is you do this induction, but it's longer than the natural numbers, but even more. You keep going after you did countable many steps, and then you keep going. Yes. And then you keep going on and you keep going until you reach the cardinality of the reals, which is the same cardinality of all the points in the space that you have to cover. So this tool is not from Euclidean geometry. It's set theoretical. So the existence of this paradoxical set already has a proof that is set theoretical.

  Simone: So you said you need less than choice. So maybe something like a, well, ordering of the reals or dependent choice, whatever this means for the non logicians. But then maybe in the vein of what we said before, from the existence of this set, just so you look sort of look at some universe where the set exists. Can you then say anything about what axioms are true in this universe? If, for example, is some amount of choice still true?

  Azul: Right. So that's exactly the questions we were trying to think of while my PhD was happening and so basically the answer is no. There is I mean, you could have this set and basically no choice. I mean. This, you can formalise this. For example, there is this concept of countable choice. Doing the choice countable many times. And we got the result that there is a model of the theory in which you have this set, so this partition of the space in unit circles, but you still you don't even have countable choice.

  Simone: And I guess for the non logicians or maybe even for the non set theorists, the striking idea is building a model of set theory because it's kind of feels like building a universe of but we are in one. So it's a bit confusing.

  Azul: Of course.

  Simone: If you start thinking about this philosophically you can go on for hours, I guess. But yes. How does one build a like a universe of set theory or model of set theory?

  Azul: I mean, I think that's the full thing of what a theory does, building models of set theory that satisfy the things you want. So how do you build the model? Well, first you assume there is one.

  Simone: Which is already… Yeah.

  Azul: And then from there, I mean, there are many techniques. So one way to build a model is build a smaller model, inside the one you have. So from all the sets you have in your model, let's take only the definable sets which have some definition and then you'll get something in principle, smaller. It could be strictly smaller. It could be all of it, depending on what you're doing. But there is also a way to construct bigger models. Okay. So there is this technique that is called forcing that allows you from a model and some other elements, construct a bigger model than the one you started with. So those are the two ways people do it.

  Simone: I think I remember in my set theory class, it was described to me as… So in the model, there are people and they believe in some sort of like entity. They know sort of how this entity looks like. They know some things, you know. Like so in the first thing you have some properties encoded, but then you sort of look up at the sky and this entity doesn't exist. But then like in the level above, the entity is there and looks down on the people and sort of has an effect on them. So it's a bit like religion, which I don't know. I mean, you do have a lot of cardinals in set theory.

  Azul: I do think it's a bit like that. So I don't know if you know this. Is it a book or…, I think it's a book.

  Simone: The higher set theory?

  Azul: No, no. Nothing to do with this. Okay. Flat something.

  Simone: Flatlandia.

  Azul: Flatlandia. Yeah. Is it English?

  Simone: Flatland.

  Azul: Maybe. Flatland.

  Simone: Yeah.

  Azul: Yes, flatland. I didn't know it in English. Sorry. So in flatland, there is this people that cannot. Well, actually, did I read this book?

  Simone: I'm sure I read it in school, like, when I was very young.

  Azul: Yeah, but. So the point is, if you are in flatland, you cannot see three dimensional things, right? You're a circle. Yeah, or a square or something. But then a sphere in flatland is like a circle that moves and has different radius. Right. Because depending on the section of the sphere, the circle changes. So somehow from flatland you can see this three dimensional things as another thing which is not three dimensional, right. Because you cannot but you can guess what it is somehow and can imagine what three dimensionality is by all the sections, for example. And this is more or less what forcing is. Yeah, it's a simplification of course. But what forcing is about. So from the original model you can kind of imagine what the exterior model would be like.

  Simone: And so from your research you're doing nowadays, let's maybe look into the future a bit. What is something you would like to see prove in the next ten years by somebody? Not necessarily by you, but… some questions you would like to see answered?

  Azul: I don't know. I think I would like… so I have this maybe vision about what I want to do in. I don't know the future very abstractly in general at some point with math, and it is trying to connect a set theory with, yeah, like subset of the reals or things that are like normal math now.

  Simone: That you find in nature.

  Azul: Yes, I am not comfortable with it.

  Simone: I think I often say, you know, model theory does logic on objects found in nature. This is what I usually say.

  Azul: Well, yeah, I don't feel it is in nature. Right. So, I feel uncomfortable with that concept. In theory the real numbers, for example, are not the real line. If you tell any mathematician, imagine the real numbers, they probably imagine the real line or maybe they just say pi or something. But the set of reals, which is the set of reals? They will draw a line. And for the set theorists usually it is not. For example the power set of the natural numbers or all the functions from the natural numbers to the natural numbers or things like this. Like there are several sets that we consider them all as the real numbers. They are all bijection with each other. So for most of the questions, this is the same if you care, for example, only about the cardinality of that set, then they are all the same. But for the objects I'm looking at, for example circles in the space. You are thinking about the real line. I mean, the space is three times the real line. So you have to look at the reals as the real line to just approach these questions. So what I would like to be solved is the things I'm curious about, of course. Which is this abstract idea of some theoretical questions that are related to the reals as the reals of the mathematicians, which is the real line.

  Simone: Mhm. To do a more geometric version of the reals.

  Azul: So yeah, but not only, I mean, even if you consider it for example the reals as a field, it's also something that is not… I mean, the power set of the natural numbers is not a field.

  Simone: It’s just a set.

  Azul: Yes. So even if you ask more algebraic questions, this also has to do with this view that I have, that we have a gap there.

  Simone: Okay. Well thank you very much for the insights!

  Azul: Thank you so much.

  Simone: See you in the corridors.

Episode 2: Springs and Memory Alloys, with Mert Bastug

In this episode of On A Tangent, Simone is joined by Mert Bastug, a doctoral researcher in PDEs and Calculus of Variations. We discuss Mert’s first meeting with mathematics, how we can understand materials with the help of PDEs, and where they might take us next.

Link to Mert's website

• Transcript

  Simone: Welcome to "On A Tangent", the podcast where the main characters are the stories behind the mathematics. My name is Simone, and in each episode I am joined by a different early career mathematician from Muenster to learn about their paths towards mathematics, and their hopes for the future. In this episode, I am joined by Mert, a PhD student in PDEs and Calculus of Variations, to find out about what mathematics can say about memory alloys, why they're not the same as memory mattresses, and how mathematicians think about their work. I hope you enjoy the episode!

  Simone: Hi. Welcome. And thank you for joining us.

  Mert: Hello. Yeah, thank you for having me.

  Simone: So let's start a bit further in the past. And so let's start from your very first memory of maths. So what's the first time you've seen maths enter your life?

  Mert: Well, the first time. I mean, I guess I could surely mention all those times I would draw on numbers or learn how to count. But I think my first moment where I consciously thought about a math problem was when my dad asked me a simple counting problem. But to me, at the time, it didn't look simple at all because it involved very large numbers. Actually, I can talk about the problem. It was about some number of horses all needing some horse shoes and each horse shoe requiring some number of nails. And then one would need to tell the total number of nails one would need. And the total came up to something in the millions which, as a child in kindergarten or maybe elementary school, it was ... I didn't really grasp this.

  Simone: A lot of zeros!

  Mert:  Right. I mean, somehow… Yeah. Yeah, definitely. And I don't even think I attempted to solve the problem. It was just to me that defined what a difficult math problem was. So that's as far as I can remember, right.

  Simone: So from somehow seeing what a difficult math problem was when you were a child, now you come to dealing with difficult math problems now, in your life.

  Mert: Yes, totally.

  Simone: So what is it that you do? How do you explain it to your friends? How do you how would you have explained it to yourself as a child?

  Mert: Oh, myself as a child, that would certainly be difficult, but, I mean, I guess I would use some tricks. Now it is okay to lie to others when you say you're doing something in math when you're not actually doing it. For me personally, the kind of math that I'm interested in ultimately goes back to problems in physics. So if I want to talk about math, well, not want to, but maybe if people ask me to explain to them what I do, I will start with some physical motivation. In my case, it would be something related to understanding materials. More specifically, what happens to materials when you try to bend them or when you just in general try to deform them? What sort of shapes are possible or why do they bend the way they do? Or, if they retain their form after you've let go of them, why that happens? I mean, okay, these are certainly not all questions that I try to deal with, but yes, if I'm trying to give people some sense of what I do, then I will freely talk about other things as well.

  Simone: I mean, somehow it's always helpful, right, when you can appeal to some physical motivation or so. There are any particular materials, I mean, that you actually work with or…

  Mert: Me personally, no. But I know. Well, yeah. So different people, of course, maybe they try to understand. I think they're called memory shape alloys.

  Simone: Are these a bit like memory mattresses, like you know, the ones where you sort of you lay down and over time, they sort of take your shape?

  Mert: I guess not really. Yeah, I would say these are a little more different than memory mattresses. They actually retain their form again. Maybe if you heat them or you expose them to some other external condition.

  Simone: So it's like those videos on TikTok of these materials that you bend over in strange shapes, and then the moment you sort of heat them up, they go back into, I don't know, a line or…

  Mert: Right, right.

  Simone: Maybe this goes back to earlier than TikTok, I mean… on YouTube in the early 2010s or something. This is magic of these alloys. Right. So this is the physical motivation. Right. So this is what you tell people. But what is the maths that you actually do for this. Or I mean I'm assuming you don't do experiments.

  Mert: No. Well I don’t. I’m sure some people out there do. The part of mathematics that deals with these questions is related to calculus of variations and also partial differential equations. In calculus of variations, one looks at minimisation problems. Now the quantities that one tries to minimise typically come from geometrical considerations, but as in my case, also from physical problems.

  Simone: So these materials you were referring to at the beginning.

  Mert: Exactly. So if you look at a material, then the shape that material wants to assume, will actually be the shape that minimises some sort of quantity, and this can be some sort of elastic energy or some other type of energy that one might want to consider.

  Simone: So, the natural state of this material is the one where this energy is minimal. And I think if everybody has ever played with a spring, you know. You extend the spring somehow. This is, you feel that there's some tension and you let it go. There is none.

  Mert: Yes. Right. And also, when you apply force to these materials, they will still try to minimise the energy in the best way they can. But now there will be some external condition, which is the force you're applying to it. Right. And in partial differential equations one looks at again, maybe physical or geometrical models of something. And one tries to understand how that model changes over time or over space and these are then described in terms of some equations that involve derivatives, which is where the name differential equation comes from.

  Simone: So somehow these in this case you're again thinking about models of materials and how they change over time when you give them some input.

  Mert: Right. And actually the two areas are related because it turns out that minimising something, amounts to solving a particular equation. So in other words, to every minimisation problem you can associate well, in most cases at least a partial differential equation. Yes, but I mean, if I wanted to talk about math to my friends or my past self say, I don't think I would necessarily mention the actual work that I do, but I would just try to give them a feeling of what doing math is like. That's why I often also mention, um, number theory, because, well, naturally, numbers are the objects that everyone can relate to mathematically. And I might also mention famous open problems from number theory, because in the end, no matter what it is in math, you're doing, um, you're trying to solve a problem that hasn't been solved before. And it's just the nature of the problem differs from area to area. But the motivating force, which is trying to understand something that hasn't been done so far, is the same in each area, I would say.

  Simone: And somehow you write model theory. Sorry, number theory professional deformation, there's all these open problems which are somehow easy to state, but then require years and years of difficult or difficult maths to be solved. Right? I mean, like Fermat's Last Theorem, where you would say, okay, I want to solve this, this equation with integers, and then, okay, everyone can more or less think, okay, how do you solve this? And then it takes, I don't know, eight years of Andrew Wiles to do this, which is of course the striking fact.

  Mert: Um that's right.

  Simone: So so you use often number theory as an example. Do you think somehow number theory and what you do then actually feel the same when doing them. Do they, do they sort of. Let me let me try to explain. Do you when you do maths on your on a daily basis, what is your guiding intuition for example? Last time we had Alex, who had a very geometric intuition about things. Uh, in this case, do you also somehow imagine your materials actually somehow evolving in space visually? Because, for example, some mathematician I can give the example of myself, I'm not a visual mathematician. It also doesn't help that my field is not visual at all. Um, but in other areas. In GR we had the discussion last time. It's a very visual area. You can imagine the geometric configuration. Does this also help in your problems? And do you think it's somehow, um, do you also think in this way or is it or is there some alternative way of people think about problems in PDEs or in, in calculus of variations?

  Mert: Well, um, I would say that I am also quite a visual thinker. Um, now. For me, it's difficult to apply my visual thinking into problems directly, but it's always helpful to draw a sketch of what I'm doing. Um, just so I have something that I can work on. Uh, even though the methods that maybe I require in solving a problem don't necessarily appear in the picture that I'm drawing. Um, I mean. So to answer your question, whether geometric thinking helps in, um, say, solving a differential equation or maybe more broadly, and in the two areas that I mentioned. Um. I would. Well, I would first have to say that I don't think I'm qualified enough to answer it, but my impression is that most of the time in analysis the proofs come down to some smart estimates or some smart, um, identities that one should recognise or, somehow rather the than the intuition about geometry often is useful the intuition about the quantities that you actually computing with uh or some standard tricks.

  Simone: I mean, because I always have this, this clash when I talk to people who maybe do differential geometry or this or analysis where they there's a lot of estimations and computations going on. There is no computation going on in what I do on a daily basis. And somehow I always struggle a bit to imagine how that feels like, because there is hardly any computation you can do to solve a theorem or to prove a theorem in model theory, you usually have to maybe give some structural argument or appeal to some standard tricks. Um, so somehow what you're, you're saying is, is there is a more quantitative, maybe intuition that people have rather than geometric.

  Mert: Right. Um, I mean, I would imagine, I would imagine that the situation changed actually, as one, gains more experience, um, because as I've said, the, the problems that one deals with, um, well, at least in my case come from physics and there maybe not the geometrical intuition, but the physical intuition plays a greater role. Um, now, certainly if one has a mathematical proof of something, that proof doesn't have to relate to the physical world, but. It is usually the case that the proof is based upon some physical intuition that the author that the mathematician had. And um, for me so far, it has been more or less possible to work without the physical intuition. But I think for everyone wanting to work in an applied field that is, um, quite an important tool that one should have in their arsenal.

  Simone: And so I guess then the natural question is how did you come to an applied field? I mean, what brought you to working in analysis, but not just analysis, but applied to some actual concrete problems like materials.

  Mert: Right. Um, well, actually, analysis was the first math subject that I wanted to learn. Um, and this was back in high school when I was trying to think about what to study in university. And, um, somehow math, um, became, for me, the most reasonable choice. And then I wanted to, before even starting university, see whether I would like it. Um, and then I looked online. I searched what people recommended, and I found this one book that was, um, on analysis. And as I studied more and more from this book, I came to notice that analysis was, for me, full of, um, nice ideas and geometrical intuition, at least at the level I was learning. Um, and then when I got to university, I didn't want to immediately cling onto this one subject, but rather keep my options, um, open. However. Yeah. Due to having a sort of head start in analysis, I felt more comfortable working with it. And ultimately, in my case, it turned out that professors and analysis were more pleasant. Uh uh. Well, they gave more pleasant lectures to attend. Um, and around the same time, I was also getting, um, interested in the idea of seeing how math could be applied to real life, because at a certain point, the math curriculum at the university can get quite abstract. And I noticed that there was a sort of a certain rewarding sense I felt whenever I saw math being applied to a real world problem. Or at least I could find. I found it that for me, it was more interesting to feel attracted to the problem, knowing that it was coming from something physical.

  Simone: There is a natural satisfaction that comes from the fact that you are actually having an impact, which I think is something that maybe people who do very theoretical maths struggle a bit with. Certainly I can say, I mean, or, um, that your math does not have a direct impact. And it's very common to say, well, yes, but I don't know. Alan Turing developing the computer. His theoretical maths also did not have any impact on the world. And see now what happens. Well, okay. Yes. But I don't think we can use exceptions as our motivating examples. And I understand somehow it's, something that Alex was saying is, you can motivate yourself a lot through the fact that you're dealing with a concrete problem that actually has an impact.

  Mert: But I would actually like to argue in another way. For me, of course, it's helpful to know that what I do might be impactful in real life for the development of science or whatever you can call it. But I would say the, the selling point for me was that. I have a more immediate way of seeing why this problem is interesting. Now you have problems in, say, algebraic geometry that require a lot of background information just to be able to understand. And I'm sure if I could understand those problems, I would find most of them interesting as well. It was just the amount of sophistication necessary to appreciate problems from analysis was lower. And it is not to say that the problems have lower quality.

  Simone: Of course not. As we've said before, right? Is this number theory problems that you can state to a child, then they require all the work in the world to be solved. Right. And somehow, in the same way, just because you can appeal to the physical intuition, it doesn't mean that the problem is less interesting or less worthy or less hard, right? I mean, yes, sure. Of course we don't want to discriminate in any way. So what are your hopes for the future of your field now then somehow, what is something that you would really like to be to see solved or to see addressed?

  Mert: Well, I have to admit that I am still quite fresh in my field. So I don't feel qualified, to pick a problem that would be interesting. Now I can say, for example, that if one thinks of materials in a broader sense as, things that are made of continua, such as maybe fluids as well, then the biggest open problem at the moment would probably be the problem relating to Navier Stokes equation. But again, even though. This problem is quite popular. I don't feel quite invested in the problem yet. So it is difficult for me to pick a particular subject. But I've always had certain thoughts about the future of mathematics. You hear often that there isn't a unified theory for partial differential equations, and that each equation is by itself, very special and requires different methods to be dealt with. I mean, this is natural when you think that for every physical system that you can describe, there is a differential equation that would maybe associated with that system. So we cannot hope to maybe find a unified theory for the whole field of partial differential equations. But then I'm reminded of the history of math and how calculus came to develop at a time when there were lots of problems, all requiring a different methods to be solved and how calculus built the ground for, well, not necessarily a unified treatment for each of them, but at least a language to describe all those problems then. I always find it exciting to think what could happen in the future in the field of partial differential equations or more generally, in analysis. In general I would be curious to see something like a new revolution happening, a new kind of calculus appearing and forcing us to think about the things we already know in a different light, but in a way that opens up new avenues of research. Of course, I don't know what that would look like, but it's always, for me at least, interesting to ponder about the possibilities, because we tend to think that what we know is the edge of knowledge and things will remain the way they are. But often we're surprised by advancements in our knowledge.

  Simone: Well, thank you for the lovely insight. And we'll see you in the corridors.

  Mert: Thank you. See you.

Episode 1: Oranges and Eclipses, with Alex Tullini

In this episode of "On A Tangent", Simone is joined by Alex Tullini, a doctoral researcher in General Relativity. We discuss Alex’s journey towards mathematics, going through oranges, eclipses, and the odd similarities between a career in mathematics and surgery.

Link to Alex's website

• Transcription

  Simone: Welcome to "On A Tangent", the podcast where the main characters are the stories behind the mathematics. My name is Simone, and in each episode I will meet a different early career mathematician from Muenster to learn about their stories, their paths towards mathematics, and their hopes for the future. In this episode, I am joined by Alex, a PhD student in General Relativity, 'GR' for short, to learn about eclipses, cosmic censorship, and surgery. I hope you enjoy the episode!

  Simone: Hello, Alex and welcome. Thank you for joining us.

  Alex: Hello, Simone. It's really nice to be here.

  Simone: So you're a PhD student in general relativity. So when you go home and your friends ask you, what do you do? What do you tell them?

  Alex: So first, I take a very deep breath. Actually, I think that as a mathematician, being in general relativity makes this job of sharing what I do easier, because it's one thing to say, oh, I'm a model theorist, and I study this and that.

  Simone: This is a personal attack to myself.

  Alex: It is. I mean, it's another thing to mention, say, a black hole, which is much more part of pop culture. And that helps me in some sense. It also saddens me because I realise that I have to be imprecise in order for them to receive the message. But okay, what I typically tell them is that I try to study through math the stability of black hole solutions to Einsteins equations, and this is kind of easy to get through because then people imagine, oh, black holes, there's these things that are described using math and okay, these things are hard, but I can do something with this math. But it creates an idea in their mind typically. Then the next thing they ask me is, so do you make experiments, which I think is something unique to people in my field, I don't think people ask you about experiments.

  Simone: No, definitely not.

  Alex: Okay, then I have to explain to them that what I do cannot really be experimented, but it's more connected to questioning the consistency of a theory. And I do think that what I do, the message really arrives to say my family, my friends. So I think I feel like I'm in a privileged position.

  Simone: I was once reading a message, I think maybe on Twitter by a professor who was saying, you know, everybody learns what a black hole is in high school or what the DNA is, right. Nobody learns what a manifold is in high school, which might be trickier, but somehow it is a bit easier if you can explain it with the pretty pictures.

  Alex: Yeah, and the physical phenomenon maybe is maybe easier to relate to. I think the weird thing is that they don't realise that what I do is still very abstract, and they think of it as being, um, enjoyable in a way that is not really what I go through. Um, there's much more technicality involved, but there's no hope of passing it through, of really creating a picture of this type of technicality. So I just accept that I'm going to mention interstellar and they'll be happy and very excited.

  Simone: Having a mainstream reference for it is a very good attack point.

  Alex: Yeah. I'm almost as lucky as a physicist, let's say, even if I don't really do what they do. I mean, I don't have the abilities to do what they do.

  Simone: So this is you now. Let's take a step back. What is the earliest moment you can remember in your life where maths entered the picture?

  Alex: Um, so I'm afraid I might be unrelatable to the general public when I actually answer this question. But I have to be honest, math has always been with me, and that for whatever reason that I couldn't really understand, among all the things that I like, it was my greatest interest and the first memory of it I that I have, which might seem also unrelated, but to me it was a moment of mathematical interest and understanding was when at some point we were waiting for an eclipse to happen. I was home with my mom and dad, and they just told me, the eclipse is going to happen. And I had some trouble understanding what it what it meant, um, because I couldn't picture in my head the kind of the geometric configuration that would allow for the eclipse to happen. And so I asked my dad to explain to me, and we went, you know, over his big bed, and we took some balls, I think an orange and, like, maybe a lemon or something. And my dad, like, put them on the on the bed and said, okay, this is the solar system, this is the moon, this is the Earth. And what we're going to witness today happens because of this geometric configuration. Right? And for me, that was my first approach with math. And I think it also in some sense it is coherent with the sort of interests I've had later, which were more connected to geometry.

  Simone: I was going to ask, do you think somehow this is what brought you to a field where maybe geometric intuition or imagining visually what's going on is important?

  Alex: I don't know if it brought me. I mean, I don't see it as a cause, rather I see it as a maybe as a symptom of something already there. Maybe I don't have that story of a little kid thinking about numbers, but I was always intrigued by geometry, by a perception of distance, by just configurations of objects in space. And this definitely is something that I also witnessed in the choices I made later during university and uh, and everything that led me to where I am right now.

  Simone: So how did you arrive where you are? How did you choose what to do?

  Alex: Um, so as I said, I was always drawn to it. It almost felt like a choice that was made by someone else. And that for whatever reason, I always felt like, oh, this is actually what I like the most. Then I also liked other things very much. I liked philosophy very much. I really wanted to go into that. I liked ancient languages very much. Part of me wanted to study Latin and Greek.

  Simone: Did you study them in school?

  Alex: I only studied Latin in school, unfortunately. But I made this choice because I thought, okay, if I want to study math, and it is certainly what I want to know the most before anything else. wI also believe that there is a right time to do it, because I felt like my brain would be better off studying math at an earlier age, and then maybe other things at a later age, then the opposite. And so this made me confident that what I felt was right for me, which was to study math, was also objectively, in some sense, the most optimal choice\'85 in terms of long term plans, even if it meant sacrificing other interests that I maybe had. And sometimes I wondered, okay, could I try doing something else? And something which may seem unrelated, but that I was very interested in: I've always wanted to be a surgeon, just because how cool is it? Okay, this is going to sound weird, but to actually see the inside of a body.

  Simone: It's certainly something, right, you don't see every day. It's a bit of a mysterious thing.

  Alex: And also, okay, the idea of having that role where some problems that are very hard to solve, they maybe can be solved through surgery. And somebody that has, you know, studied for years has a special set of skills. They can do that for you. And, um, I try to imagine myself into that career. But then I realised I would have suffered too much the loss of the mathematical knowledge, whereas I could have, you know, accepted the loss of, say, the surgical knowledge, or the knowledge that I've had to sacrifice because of course, there is only one thing you you can reasonably master. I mean, unless you're specially gifted or something.

  Simone: Or you have a lot of free time, and somebody at home that takes care of everything. So this is how you came to math. But how did you come to general relativity then? I mean, you told me you've always been geometrically inclined.

  Alex: Yeah, but I would say I came to GR because at some point I really took into my own hands my journey. I studied in Italy, where, as you know, we tend to have less freedom than in other countries. Which okay it can be both good and bad. It certainly allowed me to see a lot of different math, which was good. But then I reached a point where I realised that even if I was very much interested in geometric problems and certain type of problems specifically, there was something about GR and black holes because of course, I live in the same world as anybody. And as we said before, GR is much more part of pop culture. And of course this has an effect on me too. And at some point I realised, okay, maybe I like it. This is kind of the base, you know, the bottom line, it's that I like it, but it's also something that it is probably easier to maintain interest in because of course, studying math is hard, right? Sometimes you're like, it's the equivalent of walking into a dark room. Um, and at least for me, I thought it was a good move to pursue an interest in something that, um, I can much more easily renew my interest for. Because in some sense, it's at least for me much easier to renew my excitement for black holes than it is to renew my excitement for the homology groups of positively curved manifolds.

  Simone: And also, how cool is it that you can go and tell people "I work on black holes", right?

  Alex: Yes, yes. I mean, let's not hide that. There is, at least for me, an influence in my choices given by the fact that I like to be the person that does this, other than I like to just do it. And of course it is. It is cool. I mean, I like to think of myself as a person that has this interest but at a certain point either you're lucky and the professors and the people around you in the university where you study actually share those interests, or you have to create a path for yourself. And this is what I did. I did it a specific time. I don't think you can do it at all times, like when you're much younger, of course, it's much harder. It certainly would have been too hard for me. But it also was somehow natural because I reached a point where I had a I felt like I had a decent understanding of all the tools needed to pursue this, this interest. And I realised, who are the people that could help me get into this field? And even if they were not in my home university, I looked for ways to get in touch with these people. And I don't know, for whatever reasons, the planets aligned again.

  Simone: Like in the eclipse.

  Alex: Like an eclipse.

  Simone: The orange, the lemon...

  Alex: And somehow I've ended up doing something that that I like very much, which was not the only option here, I want to say again, but it's the one on which I bet.

  Simone: So you moved to Switzerland first and then here.

  Alex: And then I came here, yes. Because there was some previous student of a professor that I had that was somehow between what I was doing before, which was a geometric analysis, and GR, which is what I'm doing right now. And this person was a bridge. But most importantly, moving away from my hometown, which was actually not my hometown, but my home university. It allowed me to meet other people that also made a bridge. And, you know, most of the times, unless you meet the people, you don't even realise that certain areas of research exist. This was my case. I didn't realise that it was possible to study black holes using this set of skills that a mathematician has, which is the set of skills that I have at my disposal. Until I went to Zurich and I met a person that was doing this. So I think overall, other than believing in yourself and kind of for once not doing exactly what you're supposed to do, more than that, you also need to be a bit lucky right, to meet the right person that tells you yes, you can do this. And then you realise, oh, I could do this, and then you do this. Um, so it's a series of factors and. Okay, luck is always a factor.

  Simone: So this is the now. This is why you do it. So what is the future ahead of you that you would like to see? What is a theorem that you would like to see proven well by yourself maybe, or others of course, in the next ten, 20, 30 years.

  Alex: Among all the things that I initially looked at when I started exploring this mathematical general relativity field, there is something specifically that really caught my attention, and that's something called strong cosmic censorship conjecture.

  Simone: Which is a very metal name.

  Alex: I mean, it's very cool. It's very easy to sell. And I mean, they sold it to me effectively. Yes. I mean, reasonably, this is something that is probably gonna take on a hundred years, maybe they won't even be enough because also my field is relatively young compared to other fields, but in essence, what I like about it, and I think this is like further proof of how much more easily I can communicate what I do if compared to other people. Basically, this conjecture says that the theory of general relativity is deterministic, and which you would say, oh, why? Why shouldn't it be deterministic? Well, it turns out that this is not something to take for granted because, okay, in very layman terms, there are some black hole solutions to Einstein equations that present something that's called Cauchy horizon, which I won't go into the the reason why it's called like that, but in essence, it's like a geometric locus where the theory at times appears not to be capable of predicting uniquely what happens to an observer or a person or an object passing this horizon. And this conjecture that I mentioned, the strong cosmic censorship conjecture, in essence, it says that although you can write down solutions of the Einstein's equations which display this behaviour, they are not generic in the sense that if you consider the initial data that give rise to the solution and you perturb them, then once you apply this perturbation, you no longer end up with a spacetime or a solution that that has this feature.

  Simone: So somehow the solutions that don't work like we expect, we expect physical theories to work deterministically, right? Yes, we put in the initial data and we know and we predict what's going to happen. Yes. So the solutions that don't behave like this are just an exception. So they just happen. And then but if you move a bit away from that then it works just fine.

  Alex: Yeah. Like trying to balance a picture upside down. Right. There is an equilibrium point. Ideally you could balance it and put it up on the wall. But you know the moment you perturb it a little bit, it's just going to go down and eventually it's going to stabilise along the other reasonable equilibrium point that you can use to put a picture up on the wall. And this is the same idea. And also I feel like this is the best representation of what it means to do general relativity. As a mathematician with the skills of a mathematician, I don't have the ability to do what the physicists do. But you know, with our set of skills, we can investigate questions such as this one, which they are really about the consistency of the theory. They are really giving meaning to the words determinism of general relativity. And they're doing it in a very reliable way, because I'm really, you know, taking the equations and rigorously trying to prove that the theory is deterministic.

  Simone: And actually predicts our reality as we expect it.

  Alex: Yes, yes. Um, so this would be very nice. Um, yeah. I think whoever puts the last brick on this collective effort is certainly going to be a famous man. Uh, but I don't know how long it will take. I would be happy if I gave my contribution. But not necessarily for the glory of it, which I confess for sure. And for the first years of my journey in math it was part of the motivation. But then I think, certainly once you entered the PhD, what I really want for myself is to have given some contribution and at the same time have a certain set of skills and of knowledge and of understanding that allows me to sit with my peers and give something to them in conversation, right? What I envy is not, say, the list of awards that professors have or may have. I mean, in part, we all envy that, right? But what I envy on a daily basis is the fact that they, they can navigate their department and talk to other people, and they have conversations where there's an actual like exchange between them. They don't just have to sit there and take as much as they can, which is what you just have to do at the beginning, right. Somehow it's also a necessary step, I mean, to have these conversations, right? Because nobody will prove the strong cosmic censorship conjecture or any other big conjecture on their own.

  Simone: Math is necessarily a collective effort that we all have to do together.

  Alex: And then we need to be able to interact and communicate. Yes. I feel like this is in the end, what once you're when you're inside, when you're doing your PhD or doing some research, or maybe if you're just a master's student, you realise that really what gives you the feeling of having owned your spot inside the community? It's really just this, you know, sitting down, having a coffee after lunch and being able to contribute to a conversation. It's not, I don't know, the list of pieces of paper that you've earned. And really it comes from the human interaction, which is something that I only discovered and touched with firsthand once I started doing my PhD. And maybe, maybe this is something that professors and researchers and everybody in charge should try to pass on to students while they're studying, maybe to boost motivation, because I think people think of math as a very lonely career. Unless maybe you're at the top of the top.

  Simone: Yes. Because we have these images, right? We see movies or in or in on also the stories we hear. I mean, of Wiles proving Fermat's Last Theorem alone for eight years.

  Alex: Yeah, and okay, this can happen. But actually the majority of the work is not like this, right? I mean, it's more about really sitting down, having a coffee, discussing what's going on. Slowly. Brick by brick is anybody would do in their generic work like office work. You have your colleagues, you work together on something and you collaborate. If anything, we're luckier because we are actually working for ourselves. And so yeah, I think that is the best part. That is what I also what reasonably I think keeps us motivated because one does not just blindly pursue this career in hope of one day maybe receiving an incredible piece of recognition. It really is a daily life to daily department life that fuels you and it allows you to move forward.
  
  Simone: Yeah. Also because right, you cannot force maths to work.
  
  Alex: No, sometimes it doesn't work.
  
  Simone: Right. Uh, and so sometimes you, you go and go and go at the same problem for, for infinite time and then you still don't get results because that's how it goes. So yeah, that's true.
  
  Alex: I mean, motivation can hardly come from the success because you cannot control it. You cannot predict it. You have to find a way to make the process, if not enjoyable, because maybe this is aiming too high, but sustainable. So a good balance of positive emotions and negative emotions otherwise I mean you're not going to go anywhere because I feel like it's just against the human nature. And it's weird that the narrative that that people receive from the outside is of people somehow going against the human nature, which is probably the reason why those that are depicted, maybe movies are this very, um, um, this genius people or this very, um, uncommon people in some sense. And I feel like this is what makes them uncommon in the eye of the, of whoever is watching. But really, this is far from the reality of the department, or at least of the departments that I've inhabited and navigated.
  
  Simone: Well, thank you very much for this lovely chat.
  
  Alex: See you soon around the corridor.