In this episode of our Voices of Mathematics podcast we talk to Julian Sahasrabudhe, Professor of Combinatorics in the Department of Pure Mathematics and Mathematical Statistics (DPMMS).
Julian Sahasrabudhe and his collaborators have made headlines for a number of recent breakthroughs, including remarkable results in sphere packing, which you can imagine in terms of packing oranges into a very large, higher-dimensional box, and Ramsey theory, which involves finding order within systems which, at first sight, appear completely disordered. His outstanding research has been rewarded with several prizes, including a Whitehead Prize in 2024.
Julian is also one of the invited speakers at the International Congress of Mathematicians (ICM) in July 2026. Taking place every four years, the ICM is one of the highlights of the mathematical calendar: it celebrates the diversity and excitement of modern mathematics, and also sees the award of some of the subject's most prestigious prizes, including the famous Fields Medals.
We talked to Julian to learn more about his path into mathematics, a brief introduction to his work in the field of combinatorics and how it offers partnerships with other areas of maths, and the advice he'd give on how to make progress when wrestling with hard problems. And we also heard from Julian on what he's most looking forward to about attending this year's ICM.
The podcast is hosted by Marianne Freiberger and Rachel Thomas, Editors of Plus, from the communications and outreach team at the Mathematics Faculty.
To find out more about topics mentioned in this podcast see:
- Our 2024 article exploring Julian Sahasrabudhe's work when he won the Whitehead Prize.
- The podcast mentions Fields Medallist Maryna Viazovska's work on sphere packings - find out more in this Plus article.
- Meet all three of the Cambridge speakers at the 2026 International Congress of Mathematicians in this article.
You can listen to the podcast using the player above, and you can listen and subscribe to our Voices of Mathematics podcast through Apple Podcasts, YouTube, Spotify and through most other podcast providers via Podbean. (The podcast is also listed without the transcript on the Maths Faculty website.) The full podcast transcript is available below. The transcript was created using AI to generate the text from the podcast recording, and was then sense-checked and edited for readability and accuracy.
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00:00:15 Julian Sahasrabudhe: Also, I think the people that are in mathematics and have been able to make new ground on old things, one really has to sit and admit to themselves just how deeply lost you are, and yet stick with it. Try to find a new angle.
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00:00:40 Marianne Freiberger: Hello and welcome to Voices of Mathematics, the podcast from the Mathematics Faculty at the University of Cambridge. I'm Marianne Freiberger.
Rachel Thomas: And I'm Rachel Thomas. And we're from the Communications and Outreach team here. The person you just heard is Julian Sahasrabudhe, Professor of Mathematics in the Department of Pure Mathematics and Mathematical Statistics, otherwise known as DPMMS, at the University of Cambridge. Julian was talking about frustration, this inherent yet weirdly fun and satisfying aspect of doing mathematics. And later on this podcast, he will tell us more about this and what his advice is about frustration for high school mathematicians. He's also going to tell us how he thinks artificial intelligence might or might not help, and explore the role of forbidden relationships in his field of study, which is combinatorics.
Marianne: Yeah, and the reason we talk to Julian is because he will give a talk at the upcoming ICM, that's the International Congress of Mathematicians, which will take place in July in Philadelphia and it's one of the biggest maths conferences there is, thousands of people.
Rachel: I think there's regularly been 4,000, 5,000 when we've been in the past, and they happen every four years in different places around the world. And there's a competition every year as to who's going to host the next one, just like the Olympics.
Marianne: Exactly. And on top of that, some very prestigious prizes are announced at each ICM, in particular the Fields Medal.
Rachel: Marianne, you met up with Julian on one of the very hot days at the West Hub in Cambridge.
Marianne: Yeah, that's right. That's close to the Centre for Mathematical Sciences, where we're all based. And I met up with him in the run-up to the ICM, so he's going to tell us about the ICM in this podcast. But while we were setting up the recording equipment, he told me that his path into mathematics has actually been very long and winding, as he called it. So I asked him to explain more.
00:02:41 Julian: Of course, I have to say that as a Canadian, I dreamt the dream of every Canadian, which is to become a professional ice hockey player when I was young. That's where I started. But that quickly fizzled out when I realized I didn't quite have the athletic ability of some. But yeah, then I got into music and I was a drummer for many years and studied essentially to be a professional drummer, jazz drummer. And then I got sick of that and I had a friend who was very interested in biology and biophysics and this kind of thing. And that's kind of what ultimately, just chatting to him, gave me, in fact, I still talk to him quite a bit, but so that got me thinking, okay, I want to be a biologist. And then somehow I just accidentally became a mathematician instead.
00:03:36 Marianne: So you're going to be talking at the ICM. How do you feel about that?
Julian: I guess I feel different things. I think it's an honour, of course. It's a lot of work to do sort of anything in mathematics, and it's sort of mostly frustration, actually! One has lots of ideas that basically all of them don't work out. So when you do have success, I mean, it's very nice to be recognised for that success. And I think the last few years, me and my collaborators have done some stuff that I'm quite excited about. So I'm really happy to have that opportunity to share it with others. That being said, I actually don't quite know what the ICM will be like. I've never actually attended it before.
And it will be, I think it'll be quite interesting because it's, I suppose like maybe from the outside, math might seem like a rather small discipline where everybody knows everybody. But really, it's actually a very vast discipline. And if you sort of throw a random mathematician into the room with me, we might actually have very little in common beyond our undergraduate education, so the very basics. So it will be interesting to see a little bit of the mathematics, what I can get of the mathematics from other areas, but also to see a little bit of the society aspect of different areas. Of course, I spend quite a bit of time in my own areas, I mean, hopefully adjacent areas as well to expand, but one's reach is only so... it's limited. And of course, one gets some sense from colleagues and such, just from being at the same institution or places I visit quite a bit, I have some sense of other areas. But it will be interesting to see the kind of, maybe such a big event for all areas.
Marianne: I was talking to someone involved and she said one of the major functions of the ICM is actually to keep that track record of the entire field through the years. [Julian: Right, right] Because every time they have to think about which are new areas that we actually need to put into the schedule and things. So is that part of what you're interested in as well, that you can see the big picture?
00:05:46 Julian: Yeah, it would definitely be good to see a little bit of the bigger picture. I suppose as a person that's ambitious, I'm also interested in trying to communicate to as much of that bigger picture as I can. So if one sees trends in other areas that maybe are relevant for the way that I can think about things or I mean... the most exciting thing is when one sees that in another area that people are stuck on something that's actually something that... similar to something where we've either stuck or had some success, then maybe that's some sort of avenue. Also, more recently with computer science, machine learning, these sort of newer fields, high dimensional statistics, getting involved. I mean, I'm definitely keen to learn more about what's happening in those newer fields. Although I have to say I'm not completely sure those are newer fields, but they're certainly newer to me.
Marianne: [Laughter] Now the field, or the section that you're in, so the schedule of the program for the conference is divided up into sections.
Julian: That's right.
Marianne: You're in the combinatorics section. So for somebody who might not know yet, how would you describe combinatorics?
00:06:58 Julian: Yeah, I suppose combinatorics itself is a little bit broad. So let me maybe narrow it to my sort of, let's say, half of combinatorics, which is extremal combinatorics or sometimes probabilistic combinatorics. So sort of similar communities. And there we're interested in these very structures. Yes, maybe just giving like a phenomenological sort of point of view of the field before saying anything more specific because we're interested in structures that are quite loose. So, in some sense, there's very little underlying structure to the things that we study. In fact, maybe at first blush when you think about the objects that we study, that there shouldn't be anything interesting that you can say about them.
Marianne: Give me an example.
Julian: Yeah, so let me give an example. So one of the foundational or one of the most studied objects in combinatorics are graphs, or for a popular audience, maybe better known as networks. These are just very simple objects to define. You just take a collection of points, so some finite set, and every pair of points is either connected or not connected. So just a binary relation on/off between every pair of points. Are they connected or not? This is sort of very different than other fields when this basic object is incredibly easy to define. I could give you 1000 examples off the top of my head.
So let me even give some. perhaps a real world quote unquote example. So for example, you could imagine a group of friends, sorry, a group of people rather, at say a college or something like that, as the points or the nodes. And you could just connect every two nodes if those two people are friends. Right? So let's just assume every two people are either friends or not friends, and this is a symmetric relationship, which one could disagree with. But let's just say that's true, then automatically we arrive at a graph. So in some sense, this is a very like floppy object. There is, you can just immediately define it like this. Or you can imagine as another graph, every web page on the internet is a node, and then two web pages are connected if there is a link from one to the other. This defines a graph. So. And we're interested in proving theorems about this graph. So proving statements that hold, say, for all graphs. And interesting, you can say quite a lot about these objects.
So for example, one example I like to give is just a very first kind of baby example of a theorem you can prove here, is you might ask, okay, how many connections can I put in this graph before I'm forced to have a triangle? So it's a triangle, just three points or three nodes that are all mutually connected. ABC, such that A is connected to B, B is connected to C, C connected to A. How many connections could I have before I'm forced, no matter what I do, to have a triangle?
Marianne: So in a social network, that would be how many people do you need before you have three... three frends?
Julian: Yeah, so the other way around. So you fix the number of people, say, and ask how many friends can you have before you have a triplet of friends. And so it turns out the best thing to do is sort of split people up, put half the nodes on one side, half the nodes on the other side, and now create all the friendships across this divide. And if you think about it for a second, this won't have any triangle in it. But the interesting business, I mean, this is now ancient history, proved in the 50s, that this is the best possible, in the sense that if you have any more connections than this, you're forced a triangle, no matter what you do.
So, in some sense, this is some sort of very simple toy example. In fact, it was, it appeared in sort of one of these ‘Intelligencer’ magazines in the 50s. But really this leads to some very interesting mathematics when we go to different structures, when we forbid different structures, and look at different settings.
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00:11:30 Marianne: So what Julian was talking about here is called Ramsey theory, and in a sense it's about the idea that once a system is large enough, so for example there's enough people or connections in a social network, you can actually always find some sort of order in it, such as a triangle of friends. But there's also another famous combinatorial problem which actually won someone a Fields Medal at the last ICM in 2022.
Rachel: Yes, that's right. That was Marina Viazovska. And her work that she was recognised for starts with something called the Kepler Conjecture after the famous 16th, 17th century polymath Johannes Kepler. And I think the problem was originally phrased in terms of stacking cannonballs. So you can imagine you have a bunch of cannonballs that you want to stack in a pile on a ship and you want a way that's going to make the best available use of space. You want to fit in as many balls as possible in a given space? And what arrangement should you choose? What's the most optimal way to pack them? And actually, the answer to that is already obvious to lots of people, including greengrocers, because it's the same way people stack oranges in the shop. It's hard to describe in words, but we'll put a picture in the show notes. It's something that you will have seen before. And although we all knew the answer to that problem, the fact that it was the most optimal way in terms of using up the most space, that wasn't proved mathematically until the late 1990s by a mathematician called Thomas Hales. And it took not just traditional mathematical approaches to it, but it also took the help of a computer. It was one of the first famous computer assisted proofs.
So, Marina Viazovska got her Field Medal for solving an analogous question to the Kepler Conjecture, but this time in higher dimensions, so many more than our three dimensions. So, here's Julian again making a link between what he talked about before, Ramsey theory, and sphere packings, that is arrangements of spheres like oranges, like cannonballs, in space. And just to note, if you're doing this in two dimensions, the spheres now are flat discs.
00:13:40 Julian: And yeah, so we talked a little bit before about sphere packing. And actually, this sort of fits into this kind of context. And this is certainly the way that I approach it. So, you could imagine one defines a graph. Now, this is an infinite graph with the points in space. So, let's just start with the plane, say. So, I have this table here. And I could imagine I connect two points on the plane. If I can't put two circles, say, of radius one on those points without them overlapping, the graph is now the conflict graph. I can't put these two radius one circles.
Marianne: So you connect them if you cannot.
Julian: That's right.
Marianne: So basically if they're too close together.
Julian: Exactly. So this gives you the forbidden relationships. And what are you doing? So, now in the setting of graph theory, what are we looking for, say, to find a sphere packing or packing many distinct discs in this space is the same as finding a bunch of points in my graph that don't have any connection between them.
Marianne: Because when you have a disc packing in a two-dimensional case has discs not overlapping each other, so when you have a disc packing, the centres of those disks will be nodes that will not be connected in the forbidden relationship.
Julian: That's right. Exactly. So, in some sense, it seems a bit silly because, okay, I just translated the one problem into this other language, so what? But it turns out that this is quite a... it's actually a valuable perspective. So something like Ramsey theory, which I've spent quite a lot of time thinking about, actually, is exactly the study of trying to understand large independent sets in graphs that forbid certain structures. And what are you doing for the sphere packing problem? You're trying to find a large independent set in this specific graph. Really, we're interested in large dimension rather than in small dimension, the example, where we don't quite forbid structures, but there's some kind of structure given by the geometry. So now I kind of want to think that as some sort of constraints put on this combinatorial object and sort of study it in the abstract like that. And this was our perspective.
00:16:05 Marianne: So, is that what you're going to be talking about, that kind of work at the ICM?
Julian: Yeah, I'll be talking about some kind of interrelated results that are kind of from this perspective. So, I guess me and my collaborators have been kind of developing this perspective that really comes from probabilistic combinatorics, extremal combinatorics. And we've been able to connect it to some different problems, both in sort of classical problems in combinatorics like Ramsey theory, but also this sphere packing problem, also this question on Littlewood polynomials in harmonic analysis. And that's one of the things that's actually really exciting is, okay, I do combinatorics, but I'm really looking for interactions with other areas.
Marianne: And what got you into that? Why are you interested in these kind of problems? What do you find fascinating about them?
Julian: I guess just the appeal, one of the main appeals is that they're all sort of very simple to state problems. I think a lot of combinatorics minded people will state that as a reason that they're interested in... I certainly subscribe to that view. A lot of the problems are sort of simple enough I can tell a high schooler that's done some math course that I can tell them essentially ... any of my recent papers the main result of it, for example, I really like that aspect of it. That okay you work very hard, you use quite advanced technology and so on, mathematically speaking. But what comes out at the end of the day, you could tell to a high schooler at a party and they'll understand. I think there's something just very aesthetically appealing about that.
00:17:46 Marianne: And since you're talking about high school, and you earlier talked about that, you know, doing maths can be frustrating or there's no frustration... So, what advice would you give to younger mathematicians, like to people who are at high school and are thinking about going to study at university? And then there is this aspect of frustration, but what advice would you give to somebody?
Julian: Oh yeah, I think that's a great question and maybe even a better question in the context of recent technological tools that maybe alleviate some of the frustration in mathematics. But I think, I mean, myself included, and everyone I've worked with has benefited a huge amount from sort of sitting with the frustration in mathematics. Also, I think the people that are in mathematics and have been able to make new ground on old things. One really has to sit and admit to themselves just how deeply lost you are and yet stick with it. Try to find a new angle. I think sometimes a lot of our barriers are actually cognitive in a way. Like we're stuck on certain things because there's something just holding us back from going forward, just believing that we can push through and do it, or that yet another tweak on this idea will require, okay, it's also, it's not so simple. It's a complicated game. We have finite time. We can't check all possible routes. But sort of... at least being unafraid of a certain amount of frustration is very important.
Marianne: Resilience.
Julian: Yeah.
Marianne: And so when you said talk about recent technology. So, do you think AI is going to take some frustration out because it's going to do more of the dog's work, like the checking through lots of cases? And do you think?
Julian: I think it's something that we're definitely in the middle of right now, trying to understand exactly how it fits into the picture of mathematical research. And I don't think a lot of people have a clear (at least the people I'm talking to) have a clear vision of how it fits into mathematical research. Certainly, one is able these tools to take out certain time-consuming aspects of the work. But then also you have to be wary that in doing so, in removing these grunt tasks, are you also actually removing these tasks that bring yourself to be more familiar and understand more deeply what goes on?
And so for example, myself, I quite like actually writing a first draft of a paper. So often how we work is we spend a lot of time, sometimes a huge amount of time at the blackboard talking to each other about how to solve a problem. And, you know, one builds up a whole way of talking about a problem, all these techniques, all these heuristics. Okay, but then when one really thinks they have a solution or maybe a big part of a solution, one goes and sits down and tries to write it down properly. And I actually very much enjoy this task, even though it is maybe grunt work and something that you might ascribe to an AI. Because I feel, once I've done, I’ve spent that time with it, I really understand that each little thing that you can adjust. And I feel much more comfortable when I go back to the next conversation and be, actually, we can turn this whole thing on its head and forget all of this now that I've carefully worked through it. I thought I needed these 15 assumptions, but only really there was these two assumptions and now we can fly with it. So, I think it's, we're just on the verge of, I mean, everyone's just recently experimenting with these things and I don't think we know totally how it will fit into the workflow. But definitely frustration in mathematics probably will continue.
00:22:10 Marianne: All right. Well, you mentioned just now also an important aspect is to talk to people and stand at the blackboard and stuff. So now you're a member of DPMMS, the Department of Pure Mathematics and Mathematical Statistics. So how has that contributed to your success as a mathematician, to be in that community, to be in that place to work?
Julian: I think it's a wonderful place to work and it's a very stimulating place to work. I think we have a great group of people there. I have wonderful students that I really enjoy to talk to and a wonderful postdoc. So I've really benefited from Cambridge and the scientific community here.
I think that actually... so I did my Part 3 Masters here many years ago and I just remember arriving here and being just blown over by the... I mean ... for many of my colleagues who came from also very prestigious institutions, they maybe didn't feel quite as strongly as me but I wasn't coming from such a background and it was just overwhelming how much interesting mathematics I could learn. And that's really continued to this day, having really top-notch research, top-notch people coming through town all the time.
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00:23:45 Marianne: So that was Julian Sahasrabudhe talking about the upcoming ICM as well as many other things. And Rachel, you and me, we're going to be there, aren't we, at the ICM?
Rachel: That's right. We're going there very soon in a few weeks, going to Philadelphia in July to meet with as many of the thousands of mathematicians as we can and report back on who wins the big prizes in mathematics.
Marianne: Yes, we will. And that takes us to the end of this podcast.
Rachel: If you've enjoyed listening, please recommend us to a friend or rate and review wherever you're listening. Thanks for listening and bye for now.
Marianne: Bye.
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