This interview explores his unconventional journey into mathematics and his work on complex systems and emergent behaviour.

The following is an edited transcript of an interview conducted by Raimundo Saona.

RS: Your background isn’t a straight line into mathematics. How did you get here?

JC: My undergrad degree is in bioengineering. During my time working in a wet lab, I developed technical expertise with laboratory equipment, but I also began to question the broader conceptual problems underpinning the work. And then, once I gained enough experience, the person in charge of the lab said, okay, now you can design some experiments of your own. What do you care about? What are you interested in? And then I realised that what I was interested in were these bigger conceptual questions, which were really about maths, not biology. That’s when I decided I needed to learn some theory.

It was kind of a crossroads, because there was no way that, given my background, I was going to be able to do a PhD in maths in the US right away. Instead, I received a Marshall Scholarship, which supports US students studying in the UK. I used it to spend one year at Bristol studying maths, and then one year at Oxford studying physics, to prepare myself for a PhD in the US. I then completed a PhD in a statistics department with a strong probability research group.

Alongside this, since 2015, I worked at several startups: a clinical data science company, another venture, and later helped establish an electricity market hedge fund. After finishing my PhD, one of those startups was in the process of being sold. There were all these people there I had worked with for many years, so I felt I needed to see it through. So, I didn’t start a postdoc right away. Instead, I started at Georgia Tech in 2023.

During my PhD, I focused a lot on learning how to do probability at a professional level. And I think in the process, partly because I had time pressures from working at a startup at the same time, my focus drifted away from the sorts of phenomena that had motivated me to study maths in the first place. It was only at the end of the PhD, when I was thinking about the future, that I saw I wanted to change the focus of what I had been working on.

RS: And what was the thing that originally drew you in?

JC: An interest in complexity and nature. I would like to understand emergent phenomena in more depth. The fact that ants can be individually not so sophisticated and yet can give rise to lots of very sophisticated-seeming behaviours. I would really like to understand that better at a conceptual level. A lot of what I do, both the purely theoretical work and the work that’s more applied, is about going between what individuals do and what groups of them do. Learning how to go in the forward direction and in the reverse direction, that’s what everything I do is about.

RS: You still collaborate with experimentalists. What does that look like?

JC: At Georgia Tech, I’ve been talking a lot with groups that conduct experiments on robot swarms and animal collectives. They’re trying to build groups of things that move under their own power, or that behave according to fixed rule sets, and then they exhibit some emergent collective behaviour that’s mysterious and interesting. We want to answer why they do that, and how to reverse engineer it.

The traditional approach to programming a swarm is to carefully choreograph what each individual is doing, to get some collective behaviour. On the far other end of the spectrum is having individuals behave sort of randomly, or with some noise, according to certain rules. And what’s really interesting is that there’s a systematic way to take certain collective behaviours and relate them to certain energy functions. Then you can go from that energy function to rule sets for individuals: when you programme individuals to have those tendencies, those biases, it produces the global behaviour you want. So, people are showing that you can do this loop of going in the forward direction and in the reverse direction.

What makes it challenging is that the individuals don’t know about  global energy. You must get a global thing out of something local or reduce something global to something local.

RS: Which brings us to what you’re working on while you’re at LSE.

JC: I’ve been thinking a lot about Markov chains recently, because they are very natural models of many physical systems. And what’s especially important in many instances is the stationary distribution, where the chain settles in the long run.

In general, the stationary distribution is an extraordinarily complex function of the transition rates. There’s a result called the Markov chain tree theorem, which tells you what that function is: it is a sum, over spanning trees of the state space, of products of the transition rates along the edges directed toward the state you care about. So, it’s really complicated in general.

What I’ve been thinking about a lot recently is how effectively simple that relationship becomes when the dynamics are random, when you pick the transition rates from a certain distribution. Generally, it’s complex. Effectively, in many models of random Markov chains, it’s simple. And understanding when and precisely why it is simple boils down to understanding distributions on spanning trees.

There are people at LSE who work on weighted spanning trees, and I’d like to see what they have to say about these kinds of questions. They’re of huge importance to people working in physics, theoretical biology, computer science, and anywhere Markov chains are, which is basically everywhere. The thing that makes non-equilibrium statistical physics difficult is the same thing that makes non-reversible Markov chains difficult. So, if you can say something interesting on one side, you can do a lot of interesting work on the other.

RS: You came to maths from a wet lab. Has that changed how you write papers?

JC: Yes, a lot. Coming from a biology lab, the figures are everything. If you look at a paper published in Science or Nature, the standard is that you should be able to get basically everything the paper is saying just from looking at the figures and reading the captions. In the wet lab where I first did research, I was taught that, to read a paper, you start with the figures. And then, if you need to know anything else, you read the paper. And to write a paper, you think about it in terms of the figures first: you think about the images that will tell the story.

I try to bring some of that to the papers I write, even if it’s not the standard in maths. I spend a lot of time making figures, thinking about what images best convey the ideas. Are there images that I can return to time and time again? I try to incorporate that into talks and teaching too. Could you convey the point of a talk in just a few images?

The other piece of advice I got is that a long paper should be locally readable. If someone just drops into the paper in a section, having not read the preceding ones, you need to give them some idea of what’s happening. You add connecting text. Before the statement of a lemma or proposition, say informally what it’s about to say. Before a proof, say roughly what the idea of the proof is. Things like that just help so much.

A very difficult thing to learn, I think, is what you can comfortably assume your reader knows. That just takes time. You must read a lot of other papers to figure out what that is.

Jacob will be at LSE this spring before starting at Delaware in August. He has expressed how grateful he is to be here and is looking forward to getting to know the rest of the department better.

The post From the Wet Lab to Complex Systems: In Conversation with Dr Jacob Calvert first appeared on Maths@LSE.

1. From Popper to LSE

RSU: How did you end up at LSE?

ES: There are many reasons, but the deepest one goes back to high school. In Israel, you could be exempt from matriculation exams if you did an academic project. I wanted to skip the exam for biblical studies, so I needed a project. At the time, I was already kind of taking classes at Tel Aviv University in the Department of Philosophy, studying logic and philosophy of science. And through that work, I encountered Karl Popper’s “The Logic of Scientific Discovery”.

It was the first science book I ever read in English. Before that, the biggest book I had read in English was “Animal Farm”. Then I read “Conjectures and Refutations” and “The Open Society and Its Enemies”. I was hooked.

Karl Popper has this idea that the way to make scientific progress is by refuting theories, not by proving them. You cannot prove universal statements; you can only refute them. Choose the simplest possible theory and refute it as quickly as possible. If you have two competing theories, you will find a prediction on which they differ. And then you do an experiment, and you refute at least one of them. He called these “crucial experiments”.

Later, at university, I studied computational logic, or Resolution Theory. I had this idea of how to apply Karl Popper’s notion of crucial experiments. I showed that if two mathematical theories are phrased in first-order logic, contradicting each other, you can use resolution to guide you in doing a sequence of crucial experiment, the results of which would contradict a hypothesis in one of the theories. That became my PhD at Yale, essentially about Popper’s The Logic of Scientific Discovery.

I never met him in person. Never… he died without me ever meeting him. But I knew he was in London School of Economics, and this is the place of Karl Popper. So, the reason I’m here is because of him.

RSU: You mentioned there are other reasons, too.

ES: The second reason is my current research. For the last ten years, I’ve been working on foundations for an alternative architecture for the digital realm. Basically, fixing how the internet works, making it democratic and egalitarian, as opposed to autocratic or plutocratic. All my papers until recently were on the mathematical foundations: definition, theorem, proof.

Now I’m getting ready for the application of this work in the realm of social science. It’s alternative social networks, alternative economic structures, alternative currencies. Where I come from, at the Weizmann Institute, they care about physics, chemistry, biology and mathematics, not social science issues.  Here at LSE, everybody cares about them. This is the place to be if you want to do research with social science impact.

I have now started a collaboration with Francesco Caselli, an economist here, trying to undo some of the damage that the internet is doing. And my third reason is more personal: my daughter is here, studying composition at the Guildhall School of Music and Drama, not far from here. It’s nice to be here, next to her.

2. The feudal internet

RSU: You’ve had many different careers over the years. What brought you to this project?

ES: I’ve had many scientific lives, like a cat. I told you the PhD thesis I finished in 1982. So, it’s 43 years ago. In one of my lives, I started an internet social networking company in 1993. We launched our first product, called Virtual Places, in 1995. We actually announced it the same day Netscape announced its browser. It was the first internet social networking software system, which supported everything you have today. Instant messaging, chat rooms, online events, voice over IP, online games, on a 2D metaverse. The page was like a two-dimensional world, and the people were avatars walking on it. We had social networking ten years before Facebook and a metaverse thirty years before Meta.

In those early days, we thought the internet was going to change everything. It’s going to be the great equaliser. There was the Declaration of Independence of Cyberspace, which basically promises that the internet will be free and open and egalitarian, and governments will not control it. That’s how people believed it would be. Thirty years later, it’s the complete opposite.

RSU: What happened?

ES: The internet became a feudal system. We are in the Middle Ages. We have autocratic systems controlled by a few like Facebook, Twitter, Amazon, Airbnb, blah, blah, blah. These are the feudal systems. And then Bitcoin, Ethereum and all these, they were developed by people who wanted to do good. But what happened is they caused an incredible concentration of power and wealth. If you look at who made money out of cryptocurrencies and who controls them today, it’s a small, small, small, minuscule minority of humanity. They worsened the problem of concentration of power and wealth.

Now, what came after the Middle Ages? The Age of Enlightenment. So, we are ready. Now is the time for the Age of Enlightenment on the internet. I’m working towards that goal. The idea is grassroots platforms for social networks, currencies, sharing-economy cooperatives, and democratic governance. Platforms that by design, exclude third-party control or manipulation. The mathematical foundation I’m developing, once implemented in software, could enable all of this.

3. Squaring the circle

RSU: This is a huge vision. How do you go about actually building it?

ES: The project is fundamentally not for profit. You cannot kind of twist it to make it for profit. How can you raise money and find excellent programmers to build highly competitive software that will challenge all these exploitative platforms if it’s not for profit? Who will give you money? Who will work for you? I didn’t know how to square the circle. So, for the first ten years, I just did mathematics. Definition, theorem, proof.

But then I found an answer. AI. AI and I are now building it. No resources, no people, I don’t need investment. We’re making amazing, amazing progress. What took me twelve years, around a hundred and fifty person-years, and between five and ten million dollars the last time I built something like this. I believe AI and I can do it in about a year and a half. One person, a year and a half. And AI keeps improving, so I’m not assuming it stays the same. Every few weeks, it gets better.

RSU: You mentioned you built something like this before?

ES: The startup I built in the nineties was based on technology I had developed for more than ten years, together with the Japanese Fifth Generation Computer Systems Project. That was a billion-dollar, ten-year project funded by the Japanese government and industry to develop parallel computers to run AI applications. It started in 1982, the same year I finished my PhD. Do you know why the project failed? It was ahead of its time. That’s all. A perfect idea, excellent technology, excellent people. It was just thirty, forty years too early. AI didn’t work then. It works today.

But we developed a very powerful programming language in those days, based on mathematical logic. I built my startup product using this language. When I sold the company to America Online, they said,’ We love your product, we love your technology, but please rewrite it in C.’ Afterwards, I did other things; I was a biologist for twenty-some years.

Now, when I came back to this project, I looked at what people use to program today. I was asleep for thirty years. I thought the languages we developed then would be obsolete by now.  But when I woke, I saw that very little progress has happened in programming languages. So, I said, well, maybe I should revive this old language. With some advancements. And that’s what I’m doing now.

4. Working with AI

RSU: Can you describe how you work with AI?

ES: Computer science is all about layers of abstraction. Down there, inside, there are just electrons flowing, simple physics. That’s all. Everything else is layers and layers of abstraction. Gates and transistors and devices and low-level software and high-level software and systems. The power of computer science stems from identifying and employing multiple layers of abstraction simultaneously. That’s how I work with AI.

The highest level of abstraction is mathematical papers. I keep writing them, they specify exactly what I want. They’re more detailed than usual because AI uses them as the mathematical specification for what it builds. I write them with AI too. The papers now have a lot of remarks, which AI wrote to itself, to ease the pain of understanding the mathematics. I never used to write remarks. Between the definitions, theorems, and proofs, it’s remark, remark, remark.

From these papers, AI creates specifications in English plus pseudocode. If this is the math, here is how the software should work. From the specifications, AI writes the code. And then the code doesn’t work, so we figure it out. Is the software not working because it failed to implement the spec correctly? Or maybe there’s a problem in the spec. And if the spec is wrong, is it because it didn’t understand the math? Or maybe the math is wrong. So, we figure it out.

This back and forth between mathematics, specification, and code has been enormous. Some of my mathematical papers underwent major, major revisions because trying to implement them this way, I discovered that they’re faulty. There were theorems, proofs, everything was there, except it was all wrong. Some mathematics looks good, but it’s not so good.

RSU: What is AI like as a collaborator?

ES: The level of interaction I have is basically at the same level as with good students. Good students. Not colleagues, but good students. Except, you know, students must sleep, must eat. Whenever I want to talk to AI, it’s there. I have multiple instances of multiple tasks. From early morning till late at night, that’s what I’m doing.

It’s difficult. It’s frustrating. But my productivity is accelerating. Both because I’ve learned what to expect and how to use it, and because it keeps improving. Every few weeks, they release another feature, and they’re all extremely useful.

At an emotional level, I’m not used to not knowing what to do. And this goal of developing all this foundation for the common good, for the benefit of society, to counter what the internet is doing to people today. I couldn’t figure out how to fund it. And suddenly, AI changed the picture completely. I still need collaborators, but the core software development, building the product that works. AI and I can do it. The star of the show is the AI.

If this project doesn’t work, it just means I was ten, twenty, thirty, or forty years ahead of my time. That’s all.

The post The New Digital Enlightenment: Ehud Shapiro on Rethinking the Internet first appeared on Maths@LSE.

RSU: What first sparked your interest in mathematics?

RM: When I was a high school student, I had no idea about academia at all. Just no idea. I went to the University of Delaware, which is a public university in a different state, because they offered a scholarship. In the United States, when you choose an area of study, you’re not within a single school or department. Instead, you have a list of requirements to satisfy, and these requirements can often be very far from what you’re interested in. In high school, you have six or seven subjects you study. I knew what maths was; I knew it was challenging, and I did pretty well in it, so I thought, “let’s do that.” So you go to the university, they ask what you want your subject to be, and you say: mathematics.

Mathematics was not particularly high in national rankings at the University of Delaware, but on the other hand, I had a lot of personal interaction with the faculty there. I was able to do research fairly early on, and for an undergraduate, that was pretty successful for me, so I just kept going. There was no reason not to do mathematics. I could have done computer science, but I really liked combinatorics and graph theory, especially—I liked the visualisation of it.

I think I did pretty well, and that really appealed to me. At some point, I thought, “Maybe I should do electrical engineering because it’s basically just mathematics,” but I never did. Mathematics was going well. At that point, I sort of knew what mathematics was.

I didn’t know what academic life was like as an undergraduate, and I had no sense of the politics or what a mathematician actually does.

RSU: How did your career in mathematics begin?

RM: As far as grants go, that was an interesting thing because when I was an undergraduate, there were more grants available than there were later. There were a lot of connections to computer science, and there were places in industry where combinatorics was really taking off.

My colleagues in engineering are constantly chasing grants and changing their research area to suit whatever the grant-givers want. Many don’t get deeply into anything. I think one of the things that I love about maths is that I really take ownership and pride in what I’ve done. I look at a paper, and I say, “That was a brilliant idea.” Even if it wasn’t so brilliant, I’ll take credit!

Consider a grant with maybe five principal investigators based in engineering or medicine. You’re the mathematician. Some of these agencies and programs like having a mathematician because we are rigorous, and they know that much. They might wonder: Is what we are doing very narrow, or is it bigger? The mathematician on the team can sometimes tell you that it’s actually bigger if there are more connections there. I’ve seen that happen.

RSU: What are the things that you wish you had felt differently about?

RM: I think I would have liked to be less intimidated at every stage of my career. When you’re looking for a job, you can just email or call people who don’t know who you are. The worst thing that happens is that they ignore you. I think at every stage—especially if you have a PhD and you’re looking for jobs—you shouldn’t be afraid to contact somebody in a department and say, “I applied for the job in your department; I thought you might be interested. Here’s a list of my papers.” Maybe they’ll get back to you; maybe they’ll be very excited. When you’re an undergraduate interested in graduate schools, contact people. Don’t be intimidated.

RSU: What would you have done differently?

RM: You can always say, “Oh, I wish I had more grants” or “I wish I had more citations.” That’s another problem in my field. If you get 10 citations, it’s a groundbreaking paper. I’ve seen papers that I regard as seminal works which have 100 or 150 citations. In medicine, they get that in the first week.

I like being able to explore, and there’s a lot of freedom to pursue blind alleys—things that don’t work or just special cases of things that sort of explode. I had a problem that a colleague had proposed to me, which was actually in biology. By the time it came to me, it was a pure graph theory question. It turned out it was quite easy to solve, but the question was bigger than what they were asking, and it led to this whole new area, which is now one of the major threads of my research. When you look back at it, the answer was silly; in wrestling with it, you saw the bigger picture that people hadn’t asked before.

RSU: What is most important to you when conducting research?

RM: I think you have to understand yourself and what is going to drive you. I tell students that the thing you want to do for your PhD is whatever keeps you up at night; if it doesn’t, then you’re not motivated enough to have it drive you.

Sometimes research just gets frustrating, and you don’t want to do it anymore. If it’s something that truly drives you, it’ll get you through those points. One piece of advice I would have is: at least within your area of interest, keep your mind open to new ideas. If you’re interested in applications, look for new ones. If you’re interested in pure maths, look in other directions and find out what other people find interesting.

I guess one regret would be that, as a graduate student and even as a postdoc to some extent, I didn’t travel very much. I didn’t go to conferences, even with papers showing up on ArXiv. It is so much easier to get a grasp on something from a 20-minute talk than it is from a paper where you see every detail. I want to see if what you’re doing motivates you and if I have a contribution I can make. In 20 minutes, I can sort of figure out whether that’s possible.

RSU: You work on something that fits your personality and, after directing 10 PhDs, you ask your students to find something they really like. How does one find what “keeps them up at night”?

RM: I think you start with low stakes. You do something which has a solution and a direction that you’re pretty sure you know how to follow. Not something that will make you famous, but something that will make you say, “I think I can solve this problem.” That instinct is hard to get, though.

Sometimes this comes from advisors. If I say, “I know how to solve this problem,” that’s a lie. If I did, I would have done it myself! Well, maybe I wouldn’t have had the time, but the point is: this tool will probably work, and if you do this and that, you should end up with a solution. If you don’t, then something very interesting is happening, and you can go in another direction. It should be something where, in the process of trying to solve the problem, you also read and learn other things.

With one student, we looked at a paper where we wanted to do the same thing but in a different setting—often one that was more difficult but similar. There’s always just a “twist” you have to do. I said, “You can follow this proof, but at some point, you’re going to have to do something different, so we have to figure out where that point is and get over it.” That meant he had to read the material in that paper, perhaps some others, and eventually we pushed our way through to the end.

With other students, it was more like: “Here’s a problem, and here’s a very special case that we don’t know the answer to, but people have at least asked the question.” Maybe they haven’t thought about it very hard, but they’ve asked it. Why don’t you look at this and see?

Mostly, when students come to me looking for an advisor, they’re not quite sure what they want to do. I’m always hoping for the one who says, “I want you as an advisor because I read this paper, and I like doing that stuff.” That’s what I want to see.

RSU: What do your students go on to do after their PhD?

RM: A lot of my students wanted to go into teaching. A fairly typical job is what I would call a “full-time teacher,” where they are usually teaching four classes per term and there’s no research involved. These places are looking for someone who is a verified scholar, but they’re not looking for somebody publishing two or three papers a year. They want somebody who is a teacher and enough of a scholar to know the field. For these kinds of students, things are a little easier.

If students are ambitious, then you have to really dig in. We need to pursue an area where, when you get published, people will notice you and say, “This is quality work and something I want to pursue too.”

The other thing is that you sometimes get lucky and sometimes you don’t. I’ve had students write very good papers, and they try to get a postdoc, but the problem is that the paper is not in an area that the person hiring is interested in. If they’re not willing to branch out, they don’t hire them.

You can do quite brilliant work, but if there’s nobody there who wants to hire you, you’re just stuck. The quality of the work will get you something eventually, but it’s much harder. If you work in an area where people hiring postdocs are already active, then you can do it. But, on the other hand, it’s very hard to make a breakthrough in an area where “big names” are already working, so it’s a trade-off.

RSU: What are your thoughts about mathematics as a profession?

RM: It’s a really strange profession. Very, brutally competitive for one thing. And you never know where it’s going to go. You’re an independent agent. If you go to work for Google, you work for Google. If you go to work for a university, well, in a lot of ways, I consider myself an independent contractor. That isn’t quite true—I get benefits, and the administration sets goals that need to be met—but once they are met, I do my own thing regarding the research time they give me. That’s valuable and useful.

RSU: You mentioned that you like the relaxed atmosphere of working in mathematics and getting deep into a topic. On the other hand, after a PhD, you may have to change topic. If you stay in academia, perhaps they want to hire in an area that is not yours, so you have to change again. When do you get to really choose your topic? When do you have tenure?

RM: I know some people who have changed. I know people who got a postdoc in a different area, did it for two years, and then went back to their original area. So, yes and no.

Tenure isn’t magical; it really isn’t. In the United States, it is a little different because if you don’t get tenure, you get fired. You still have that freedom beforehand, but at least in mathematics, you should pursue your strongest suit to build the best reputation. Generally, that is what will get you tenure at a US institution.

If the administration says, “do more work in data science,” well, if that doesn’t fit with what you do, you’re probably not going to be successful anyway. Even if you say, “I’m going to do data science today,” if it doesn’t correlate with what you know and what keeps you up at night, you just won’t be successful.

A job is a job. The freedom is nice, but I recognise I’m being paid by someone else to do something for them, so that freedom is limited—and I think to some extent it should be.

The advantage is that I’m judged as a mathematician. The administration can say they want to do X, Y, or Z, but generally speaking, the way they solve that problem is to hire someone new—someone who isn’t me. We also teach quite a lot; I teach three or four courses per year, so that “pays” for the time I spend on research. I think that’s only fair.

The post Meet the Academic: Professor Ryan Martin first appeared on Maths@LSE.

Table of contents

1. Background and motivation
2. School-level maths
3. Maths in higher education
4. Closing thoughts on AI

1. Background and motivation

NO: How did you get interested in the topic of maths education?

CW: I’ve been quite interested in it for a long time, but a very specific point was the release of Wolfram Alpha in 2009. What happened was slightly bizarre, because we were all excited about the symbolic AI aspect – you could type in a question in English, and it would understand the question and produce some sort of computed answer, which seemed very exciting to us. But then we had people in schools saying, “Oh, my goodness! Should we now compute more or less? What should we do with this free service in terms of maths education? Because you can now compute an integral, so maybe we shouldn’t be teaching what we’ve been teaching.”

I was thinking: but this is nuts – Mathematica and other tools had been able to do these things for years. So, then I started looking into and thinking more deeply about what we have been doing in these curricula, when obviously computers have been massively moving forward in terms of what’s done in the real world of mathematics.

That’s what really got me going, and I ended up giving a TED talk the next year, and I started formulating my thoughts more seriously. The more I looked at it, at least in my view, the worse it got.

NO: What did you think about your experience learning mathematics in school?

CW: I was better at maths than I was at French, and so the fact that I scored fairly highly meant that I quite liked it, just like I liked physics because I was good at that. I would say it was a bit variable honestly. I liked the satisfaction of doing it, and I liked the abstraction in its own right.

Was it well taught? I was lucky because I had some very good teachers who were rather unusual, who were good mathematicians and didn’t really follow the curriculum. If I’d been stuck at A-level math, I think I would have been okay, and I think the A-level and further maths A-level then were probably better than they are now.

NO: I had a somewhat similar experience. I was allowed to explore beyond the standard curriculum, which made a huge difference to my interest. And University?

CW: At Cambridge University I did natural sciences, and later switched to maths because  physics was really a rote learning exercise. In the third year, I picked the applied side of mathematics, which was theoretical physics to a large extent. So having not had a great time doing physics, I had a better time doing maths. We were allowed to do computing projects and things like that even back then. It wasn’t completely bereft of modern technology.

2. Maths education in school

NO: In the university environment, I think we’re all very much on board with the idea that school-level maths seems to be too much emphasis on rote learning. We want people to be actually thinking mathematically. That’s the type of training that we end up having to do – we have to sort of reprogram students from the idea that mathematics is following some very rigid process. I expect there would be less agreement that a computational lens is necessarily the solution. I’m interested in how you view your approach compared to previous approaches to making maths more conceptual.

CW: I think there are a few things going on. One thing is that maths really wasn’t that applicable to lots of things until computing. It’s massively more applicable now. You might call it maths, you might call it computation in the outside world. But fundamentally the mathematical process has produced a fantastic range of answers way beyond the scope that it had before computers because you can deal with messy calculations.

So that has then driven people to think this mainstream subject was just a sort of esoteric thing that was sort of mind training to actually really needing it for people in the community. And I think they’re right for the right subject.

So then you’ve got to backtrack a bit and figure out – Latin, which was forced on everyone in any intellectual school in England back in the fifties, and maths were the two bedrocks of intellectual endeavour. That was their role, and everybody accepted that even though people didn’t speak Latin to each other, but somehow they thought that was the thing you needed to learn. Then it all fell to pieces in the sixties and seventies where it was like, do you really need a Latin A-level to go and apply to Oxford to read biology? Maybe it trains you in something, but is it really essential?

I think that we’ve now got a confusion between maths as a conceptual, abstract thinking thing where we’re not that interested in the content, we’re interested in the process by which people think – a bit like Latin was supposed to be – and maths as a pragmatic thing that you really do need as a practical matter in your everyday life. You could function at a level with essentially no maths above basic arithmetic 50 years ago. I just don’t believe that’s true going forward.

So what’s happened is we’ve fallen between two stools in school. If you look at A-levels, further maths A-level, they’re kind of a rite of passage for many people who couldn’t give a toss about maths in its own right. I’m not sure they’re fulfilling the needs of people who really are interested in maths in its own right. They’re sort of neither.

So, I think what’s gone wrong before is, firstly, most people you talk to as maths educators haven’t thought about the change of the real-world subject. They just haven’t thought about it. So therefore they’re running to catch up with something that they’ve misunderstood, in my view.

Problem B is a lot of the change that ends up being promulgated is pedagogical change because it’s so hard to change the subject. It’s a very bizarre problem. Take religious studies – when I did divinity at school it was pretty tedious, learning bits out of the Bible. Now, my daughter did religious studies – it was very edgy. Different religions, comparing ethics, discussing the death penalty, things that I would certainly find pretty interesting. It had been allowed to change because nobody in a British context was that concerned about it as a mainstream thing. Maths is so central you can’t change the subject because everything stops change. The ecosystem is completely locked.

If I can build a new company that gets you 1% more in your A-levels as they exist now, I can get lots of money to fund that. But if I want to do something which – what we’re trying to do is say actually, the subject has gone off kilter a bit, or actually a lot in my view – we would like to have an alternative or something else. That’s incredibly hard, mixed up with university admissions and teachers and schools and governments.

So I think what’s failed in the past is not radical enough, not a complete clarity to understand what the main issue really is, which is the subject content in my view, and then far too gentle an approach to actually offering an alternative. But you can see why that’s happened in the ecosystem in most countries.

Just to be clear, I’m absolutely not against there being a highly abstract Latin-like subject that people who are actually interested in  can pursue. I don’t think it’s bad. I think it’s good. I just don’t think that forcing the majority of the population through that is at all the right thing. And it’s not right for many other high-level courses at universities – engineering, biology. I think it’s wrongly formulated for those too, for the most part.

NO: It sounds like what you’re saying is the goals of mathematics for higher education in the mathematical subjects – math, physics, and so on – are maybe at this stage more different to the goals of mathematical literacy, as it were.

CW: Well, I think for literacy, everybody needs it to be practical, just at different levels. I suppose I would put subjects for their own interest – if I want to go study ancient Greek, it’s a fine thing to study. I’m not against people trying to study ancient Greek if they find it interesting. But what I am against is saying if you don’t get an A* in ancient Greek, you’re not coming to read economics at the LSE, because there just doesn’t seem enough connection between ancient Greek A-level and economics at LSE.

Now, the fact that you’ve got very good at ancient Greek might well mean that you are good at concentrating, and you can assimilate a lot of stuff, and there are a lot of side benefits. But it just isn’t central, and I don’t think you should be forced to do that. If you take maths A-level, I think there is a sort of rite of passage thing going on. It’s a funny mixture because it’s rather procedural, and in fact, you can get through a maths A-level with basically rote learning to a large extent.

So that neither services going to your maths course, for example, where you actually really want people to abstractly think about mathematics, nor does it serve anything much they’re going to be doing applying it, nor do they know how to apply it, nor are they using the right algorithms even often. So it’s kind of neither of those.

For example, I would like to see an alternative option – a kind of computer-based maths A-level which has at least the status of a traditional maths A-level, and is usable to admit people to many courses that would seem appropriate, which would be many of the quantitative courses that are not maths itself.

I think there’s another thing going on here, which I think is often missed. A lot of pure mathematicians say to me, “Oh, but you’re going to ruin the subject. You’re going to pull everybody out.” I disagree very strongly. I think we’re losing, particularly from less intellectual backgrounds, a lot of people who are actually good computational thinkers, many of whom might get excited by the subject itself. But they ended up getting lost because they didn’t know what they were doing. It was very procedural, it was rather boring, they had a rather bad maths teacher, whatever the problem was – it was not applicable to their life at that point, they didn’t get interested in the application, and so they got completely lost. Now many students will just want it for its application, which is fine. But some of them will want to see what’s under the hood – it’s like some fraction of people who drive cars get interested in how they work.

We’re losing this. So I actually think if we did what I’m talking about, we might well end up with more people  wanting to go down a pure maths route than we do at the moment. For most people, if you want a mainstream subject, the thing has got to have some obvious applicability, and you’ve got to focus on that. And the more you do that earlier, I think the more people you’ll get.

At the moment we’ve got mostly an abstract procedural thing, which, if you’re lucky, you get a side problem that’s applying it to some practical problem which is thought to be an extension. Maybe a simplistic application, because that’s all you can do without the computer in many cases.

If you say, here are data from two websites, here are a million data points from two versions of a website, go figure out which is performing best – I mean, you don’t get anything like that in the maths curriculum. And yet that would be a practical, everyday problem. It’s quite hard – what do you mean by best? How are you going to analyse it? How are you even going to get the data in shape?

NO: We both had exposure to some more interesting mathematics when we were in school. Olympiad-style mathematics training was a huge influence on me and probably a big part of the reason I ended up doing mathematics. I feel like this approach has been very successful, but also elitist; it benefited students who already showed aptitude and interest. Is there any relevance of this type of mathematics in your view, in a more broad sense?

CW: It’s complicated. The problem is that you need people to be able to think effectively. I’ve got a lot to say in terms of being explicit about outcomes we actually want, rather than “well, they solved this quadratic equation, and somehow that imbued some sort of thinking, but we’re not going to explain what that is really.”

I think it’s just very muddled up at the moment. If you take very simple things, say the quadratic. What does that really mean? Well, at one level, you’re trying to learn that there are processes in mathematics – there are things called equations that are pretty useful in solving a bunch of problems. And there is a whole bunch of techniques for what you do when you have this sort of opening abstraction of an equation to get to an answer, and they work rather well in many different cases. But then there’s a sort of outer wrapper – well, why did you set the equation up to start with? And is the answer right? It’s got two roots – what does that mean? All the roots came out complex – what does all of that mean? Does that relate to what you did?

So, there are many different pieces, some of which I think are highly relevant and people really need to know, some of which are, like the actual details of applying the quadratic formula, I think are mostly irrelevant. Is it a way to learn how to do algebra by hand? Maybe. Would I favour having far more complicated equations where you’ve got a huge experience of many different types of equations, where you understand where you might use equations altogether and where they tend to go wrong, and mistakes that get made using them? I would much prefer that. And I think then you could end up going into what is the computer doing actually to solve them.

The point is, the stuff outside now is interesting, complex, hard, and necessary – all the things you could want from a mainstream subject. So I don’t know why we’re not focusing on that first.

Just on the elitism aspect, because I know there was a thing with the Government recently about getting rid of funding for further maths: I wrote a blog post about it because I was fairly confused by the discussion. The idea that you should only be trying to get people in school from D grades to C grades is nuts. We obviously need people who are going from A grades to A* grades, and there obviously needs to be help for people in schools that are not really equipped to do that, and that’s money well spent.

Maths, computation, these things are important. So I’m very much for helping people who are already good do really well, and Britain has actually historically been quite good at that. I think probably better than mainland Europe, on the whole. But the argument gets all intertwined with further maths A-level specifically. I’m not sure that’s doing what they want to do, and I’m not even sure that’s doing that for the pure maths people, let alone for the other people.

NO: How do you see the hope of actually changing things? It’s very difficult for something like mathematics – so many people need to be convinced. Are you optimistic that things can be changed?

CW: I think things will change, but they might change because there’s an absolute crisis, or they might change because it’s managed. It will change just like Latin got thrown out at some point because, in the end, it doesn’t make sense. The thing as it’s formulated at the moment is not going to work at some point. Somehow a crystal will grow somewhere, and people will come onto it. But the question is, can it be seeded, and where, and who can make the most benefit out of it?

The failure modes are pretty evident if you look for them. We’ve got a lot of people who are incredibly upset about maths. They can’t do it. They’re disadvantaged in all sorts of ways.

I actually think we’ve got societal problems that are emerging. If you look at most decisions now in society, they have some connection with a computational argument made for why the decision was made – data science, modelling. I don’t think you had that 50 years ago. In the pandemic, it’s like: here are some graphs, here are some models, and the models say you should wear a mask.

The problem we’ve got now is that most of the public is absolutely ill-equipped to be sceptical and intelligent about this. So, therefore, a decision is made, they don’t really understand the decision. Then some expert stands up and talks some twaddle – some of the experts are good, some of them aren’t good, but they have no way to tell between them because they’re not educated in that. And so then the expert who gets on TV may be the one who talks twaddle. And then, basically, at some later point, it turns out to be wrong.

NO: Therefore maths is bad.

CW: Therefore maths is bad. Therefore we can’t trust algorithms. It’s garbage. So it’s almost a pre-enlightenment era, it’s sort of hocus pocus.

We’ve got a major problem with that. Obviously, what I advocate is, we want more decisions based on intelligent application of mathematics. But we’ve got to also be honest about its limits, and nobody has the nuance of that coming out of an A-level at school.

So I think we’ve got these pretty major problems to fix. And I think those are becoming more and more evident. Now, how does that end up forcing a change? Well, unfortunately, it’s rather circuitous in most countries.

In principle, Britain has a structure that would allow change more readily than in many other countries. A-levels are not formulated by the government, they’re formulated by exam boards. You can introduce a new A-level. But in practical terms, to do that, you’ve got to get enough people to agree – that they will admit people to university on it, that therefore there’s an incentive for the exam boards to offer it, that the Government aren’t going to block it, that parents and schools can be convinced to do it. So there’s an awful lot of baggage.

In Britain, on the ground people are surprisingly innovative. I think this would actually not be a particularly hard argument to win on the ground. But then it gets stuck in the bureaucracy. So you find far less puritanical objection in Britain than in many other countries. But you find a sort of slight hopelessness – oh, well, nothing can really change, because everything’s very complicated.

Another thing that comes up very often is the number of maths teachers. People may not like to hear this, but I have good news on this, which is AI – whether you’re talking traditional maths or the kind of maths I’d like to see – I think AIs are doing already a pretty fantastic job of a lot of what a procedural maths teacher would be asked to do. So I think human teachers are very important, but I think the role can change. And I think you could get far further with fewer. So I actually think that’s good – that’s a very positive piece of news in actually delivering any of this stuff.

NO: Are there any schools that are really trying to bring in AI at this point, or is that still a few years in the future?

CW: You have to be careful when you’re talking about AI to teach or AI to affect the subject because it’s affecting the world. On the former, I think a lot of schools are experimenting. I think the stuff is fairly rudimentary, but honestly, I think there are a lot of teachers who are pretty hit and miss at diagnosing misunderstandings in students, even the procedural maths that’s current.

Actually, I think that the chance of a decent AI – we’re building tutoring systems for traditional maths as well as the maths I’d like, with a mixture of our deliberate computation and generative stuff – is surprisingly good. I think it’ll do better at the sort of rudimentary everyday stuff than the average teacher might do. So I think that’s really promising for levelling up. It is incentivised correctly because people will probably do better on their tests by learning it better. Hopefully, they’ll get some other benefits as well. They’ll be more confident, and it’ll be evened out better between different schools.

I think that in terms of changing the subject, nothing yet because all the incentives are against you.

3. Maths in higher education

NO: Do you have any thoughts on AI in higher education specifically? We’re dealing with AI in various ways, both in assessments, but also just in terms of how what we teach should change; I find this very challenging given the rapid pace of progress. How should the undergrad curriculum change, and how should our approach to teaching change?

CW: One big question concerns service courses; what is the core for people who are not studying maths per se? And also essentially every subject at university, certainly at a technical university, there’s a computational version of the subject more or less. So the question is, how does that interweave with the core that’s delivered?

I’m concerned the two get out of alignment pretty badly. A typical example – my daughter is just doing finals in biology in natural sciences. There is a biology maths for biologists in the first year, and it seems almost completely out of alignment with the whole of the rest of the biology course. The maths you need for biology is mostly quite modern computational biology – it’s not going to be pre-computer, most of it. And yet what they were being taught was mostly sort of binomial distributions – which is fine, but it doesn’t seem like your best foot forward on many of those things.

One of the things we’ve been trying to do is build problem sets that start with a fuzzy problem and end up requiring bits of maths to be determined. So you’re covering all the maths that you need, but it starts from a context. I’m talking here mostly about the supply courses for all of the different undergraduate programmes.

I know somebody at UCL, a neuroscientist. She gets A*-maths people and presents them with a bunch of data from neuroscience experiments in their first term, and they have no idea what to do with it. Just none. Zero. So the question is, what is the right course to get them thinking along that track in a better way?

I think there’s a central subject that we may be missing that people want to study in its own right in different places. That isn’t quite traditional maths and isn’t quite computer science, but is a foundational subject, and some of that may be what they need for servicing other subjects, and some of it may be actually learning that in its own right.

In terms of making use of AI, I think there are opportunities to run through more innovative questions, with more individual diagnosis. So, I think that might be a way to look at it – the AI as a tutor, as an on-demand tutor. Much more discussion together. I feel that there will be more opportunities to flip the classroom because you don’t need to sit in a big lecture room going through problem sets in the same way. You can use those more as tutorial sessions, potentially to discuss how to apply the mathematics which is much harder in some ways.

4. Closing thoughts on AI

NO: A last, somewhat off-topic, question on AI outside of the context of teaching. As someone who does research, and views it as a primarily creative enterprise, I have been quite shocked at the rate of progress with AI. In January there was an announcement by OpenAI on one of their models achieving a score of 25% on the FrontierMath benchmark – this is not open, but the example questions seem very difficult to me. So this was a big shock. But then the model was released, and in fact, I still find very little use for LLMs in my research. How do you see AI progress in the next few years, in how it relates to research and other “deep thinking” tasks?

CW: I suppose my view on this is basically that this is the trouble with benchmarks. You can feed these things more and more, and they will be very good at the things they’re fed, of course. At some level, they really don’t understand what they’re doing in my way of thinking. They operate with a different pattern of thinking.

I don’t think they’re going to manage to do the creativity. They produce new ways to look at things sometimes, but they don’t produce the same kind of creativity that humans traditionally have. So they’re very useful. If you feed enough curriculum information, if you feed them enough questions from A-levels or degrees or whatever to the thing, they do incredibly well at them.

But are they actually doing anything brand new? Obviously, they’re putting together things that are different. But I think we’ve already seen, even in trying to do this tutoring thing and various other things with AI – I sort of call it deliberate algorithm as opposed to LLMs – I found that we’ve done a lot of work trying to put our sort of make those hybrid for solving various things, and that does still seem to have a lot of power to it in some cases.

It’s very, very impressive to look at. It’s somewhat less impressive on second and third viewing. I still think there’s room for a lot of different approaches, and that LLMs aren’t necessarily the approach to do everything.

NO: Probably another interesting few years ahead.

CW: Absolutely. The good news is, if one’s in the maths business, one’s not in coal mining. It’s driving the world. We’re in a computational era, whatever you want to call it.

The post Fixing Education for the AI Age: Conrad Wolfram first appeared on Maths@LSE.

Hi Meike, thanks for joining us. Let’s start with a bit about yourself. Where are you from, and where did you study for your degrees? 

I grew up in a village in Germany called Koslar, part of a small town called Jülich not too far from Aachen. For my studies, I went to the University of Bonn. I stayed there for my Bachelor’s, master’s and PhD.

How have you found moving to London from Bonn?

I’m finding it surprisingly nice! I was a bit worried about moving to a big city because I like nature and I enjoy peace, but London is surprisingly green. Living in a big city also has its benefits. For example, there is a good range of museums you can visit for free, and there are many more climbing gyms here than in Bonn. I have also found that people at LSE are friendly, and the environment is more relaxed. I also benefit from hearing talks from people in different areas of maths, allowing me to learn something new.

Did you always know you wanted to go into Maths? 

Well, not always. When I was a kid, I liked animals so I thought maybe I would become a vet. When I was in fifth grade, my maths teacher handed out exercise sheets for the Maths Olympiad and I started participating. It went relatively well, and I became part of a great community through the competition training. It was through these types of competitions that I became interested in maths; it was very satisfying when a problem came together, and you understood something new. Slowly, I became more and more sure that I wanted to go into mathematics.

So, tell me a bit about your current research and any open problems you find interesting.

Sure; first I’ll say a bit about what I did during my PhD because it’s a bit different from what I do now. In my PhD, I was looking at approximation algorithms. Specifically, I looked at the weighted k-set packing problem, which is defined as follows: you have a collection of sets, each with some positive weight, and each containing at most k elements. The goal is to find a subcollection which maximises the total weight, such that the sets in the subcollection are pairwise disjoint. For k at least three, finding such a collection is NP-hard.

I was working on approximation algorithms for this problem (meaning algorithms that don’t necessarily return the best possible solution, but which are guaranteed never to be too far off). The technique which has proved most effective when designing these is called “local search”. These algorithms start with any approximate solution – so they could start with the empty set. At each step, the algorithm adds some sets to the solution. When a new collection of sets is added to the solution, to maintain feasibility the algorithm must remove all the sets in the solution which have a non-empty intersection with the new sets. If the weight of the new sets is greater than the weight of the sets that must be removed, then the algorithm makes this swap, and this is called a local improvement. The algorithm stops at a locally optimum solution.

The approximation guarantee measures how close the final approximate solution is to the optimal solution. I was able to improve the approximation guarantee for this problem and draw some connections between the unweighted setting and the weighted setting. Most of my results were asymptotic, which means they only apply to very large values of k. There has been some recent progress, with some results which give better results for small values of k. However, it remains an open problem to find an exact formula for the approximation guarantee in terms of the size of the local improvements; it would be nice to see this solved.

I’ve now switched gears and I’m working on a project with Dr Ahmad Abdi. We are working towards the generalised Berge-Fulkerson conjecture. We are currently looking at some properties of lattices that we believe would be useful for solving this conjecture.

How did you find the transition from doing what you are familiar with in your PhD to what you are studying now?

I guess it’s always a little bit scary to change topics when you are very familiar with one area, but I’ve already learnt a lot and I’m finding this project interesting and fun. It’s always good to see something new and expand your horizons.

What advice do you have for an undergraduate student, or maybe even a postgraduate, who is thinking of going into research in mathematics?

Do it! It’s cool. I suppose occasionally it can be quite frustrating when you are thinking about a problem for a long time, and you can’t figure out a solution – it consumes a lot of your time. My advice would be to not let that throw you off; these things happen, and it doesn’t mean you’re bad at mathematics. Sometimes problems are just hard, and you can’t solve them, and that’s okay. It can help to take a break and go for a walk or a run. So, try hard, but don’t try too hard; there might be some problems you won’t be able to solve but there will be many that you can.

The post Meet the Researcher: Dr Meike Neuwohner first appeared on Maths@LSE.

Suppose we have a square grid, with n rows and n columns. Some of the squares are coloured red, and some are uncoloured.

We say that we have a hidden square of size k if we can remove n − k columns and n − k rows so that what remains is a k × k grid where every square is red. The figure below shows an example of containing a hidden square of size 3, but no hidden square of size 4.

A warm-up question

Can you colour a 4 × 4 grid in such a way that at least half of the squares are red, and yet there is no hidden square of size 2? What about a 5 × 5 grid? You might want to try to answer this for yourself – the solution will be discussed below.

For a 4 × 4 grid, the answer is yes, as demonstrated by the following example (other solutions are possible).

For a 5 × 5 grid however, the answer is no – there will always be a hidden square of size 2. Here’s a (crude) argument as to why this is the case. First, if we had a column with all entries red, then any other column with at least two coloured entries would provide a hidden 2 × 2 square; and if all other columns had just one red entry, that’s only 9 red squares, whereas we must have at least 13. If we had a column with four red entries, we can argue similarly; if any other column had 3 red entries, that would give us a hidden square, so all other columns have at most 2 red entries, but that gives us only 4 + 2 ⋅ 4 = 12 red squares.

So we can restrict our attention to the case that each column has at most 3 coloured squares. There must be at least three columns with 3 coloured squares (if we only had 2, we would have at most 2 ⋅ 3 + 3 ⋅ 2 = 12 coloured squares). But then it’s not too difficult to convince yourself that we cannot organise these 3 columns such that each pair has at most one coloured row in common.

What about if we only require 10% of the squares to be coloured? Of course, one can now colour a 5 × 5 grid without a hidden 2 × 2 square, but what about larger grids — can you colour an arbitrarily large grid with 10% of the squares coloured and avoid a hidden square of size 2? Or colouring only some α fraction of the entries, for some very small α > 0? What about hidden squares of larger size?

There is a very nice answer to this question. Fix any k and α > 0. Then for big enough n (“big enough” depends on k and α), any colouring of an n × n grid with at least an α fraction of entries coloured will have a k × k hidden square. We will say more about the result, its context, and history shortly.

The open question

But here we come to the open question: how big is “big enough”?

This is an intriguing question, but why is it one of Prof. Sudakov’s favourite open problems? To get a sense of this, we will discuss the question again in its original context, namely extremal graph theory.

A brief history of extremal graph theory

Willem Mantel was an officer in the Dutch army around the start of the 20th century, but also a mathematician. In those days, a form of recreational maths journal was popular; you would send in a problem, as well as your solution. Your problem (without solution) would be published, and the solution revealed in a later issue. Mantel proposed (and answered) the following problem in 1907.

For example, if n = 4, then one can easily get 4 edges; but with 5 edges there will always be a triangle.

Mantel proved that the answer is roughly n²/4 (more precisely, it is n²/4 if n is even, and (n²−1)/4 if n is odd). The construction is quite simple: divide the nodes of the graph into two equal parts (as equal as possible), and put in all possible edges between these two parts, but no edges within a part. The figure below shows this for n = 9; we will denote this graph by K_5, 4, representing that it consists of all possible edges between a “left” side of size 5 and a right side of size 4. It’s easy to convince yourself that this graph has no triangle; indeed, it has no cycle of odd length. The more difficult thing is to show that this is the best possible construction.

Turán generalised this theorem in 1941. Instead of disallowing triangles, what if we disallow a clique of size r + 1, for some r ≥ 2? A clique of size r + 1, denoted by K_r + 1, is simply the graph on r + 1 nodes where every single pair of nodes is connected by an edge; so a clique of size 3 is precisely a triangle. How many edges can one have without containing a K_r + 1, for some given r?

The answer turns out to be what you might guess. Divide the nodes of the graph into r parts, as equally as possible. Now add an edge between every pair of nodes in distinct parts. This graph will have (roughly) (r − 1)/2r ⋅ n² edges, and clearly contains no K_r + 1, since for any subset of size r + 1, two nodes will have to be in the same part, and hence will not be connected by an edge. Again, the hard part is showing that this is the best possible construction. We can further generalise the question. Instead of looking for graphs without cliques, what about graphs missing some other arbitrary graph H? It turns out we understand the answer very well for most graphs H. There is one important class of exceptions however; we do not know the answer for bipartite graphs, and this is where our hidden square problem comes in.

The connection with the hidden square problem

Suppose we are given a n × n grid with some red squares. We can construct an associated graph as follows. It will have n nodes corresponding to the n rows (call these nodes r₁, r₂, …, r_n), and n nodes corresponding to the n columns (call these c₁, c₂, …, c_n). Then for every grid position (i,j) which is coloured, we include the edge between r_i and c_j; if the grid entry is not coloured, then we do not include the edge.

In this view, a hidden square of size 2 has a clear interpretation: it is nothing more than a collection of 2 column nodes and 2 row nodes such that each of the column nodes is connected by an edge to each of the row nodes; in other words, a copy of K_2, 2. (In the example above, {r₂, r₃, c₂, c₄} is a copy of K_2, 2.) Similarly, if we ask the question for a hidden square of size k, then we are asking whether the corresponding graph has a copy of K_k, k.

Our open question, phrased in terms of this graph viewpoint, is the following.

So the question fits squarely in the domain of extremal graph theory. This is a very active area of research, and the Department of Mathematics at LSE has a number of experts on the topic (Prof. Peter Allen, Prof. Julia Böttcher and Prof. Jozef Skokan).

An application in geometry

Here is an example of how results of this type can be useful for a problem in a slightly different area of mathematics. Suppose we have a set P of n distinct points on the plane, as well as a set L of n distinct lines (a line extends infinitely in each direction). An incidence is a pair (p,ℓ), with p ∈ P and ℓ ∈ L, where p lies on ℓ. The question is: what is the largest possible number of incidences possible, as a function of n?

The obvious trivial upper bound is n², if we could arrange somehow that every single point in P lies on every line of L, but it’s easy to see that this is nowhere near possible. Can we do better?

We can! Observe that if we have two points p₁, p₂ ∈ P, then there cannot be two distinct lines ℓ₁, ℓ₂ ∈ L so that (p₁,ℓ₁), (p₁,ℓ₂), (p₂,ℓ₁) and (p₂,ℓ₂) are all incidences. So given any choice of P and L, construct an n × n grid where the rows are labelled by the points in P, the columns are labelled by the points in L, and a square at position (p,ℓ) is coloured red precisely if (p,ℓ) is an incidence. Then this colouring has no hidden 2 × 2 square. Hence our upper bound on the number of red squares without a hidden square of size 2 applies, and so there can be at most cn^3/2 incidences, for some constant c.

This is not the correct answer (the correct exponent is 4/3, not 3/2), but it’s very easy to obtain. And there are a number of other problems in geometry where these results from extremal graph theory are very useful.

The post Favourite Open Problems: Prof. Benny Sudakov first appeared on Maths@LSE.

Hi Nello, thank you for joining us today. Can you tell us a little bit about yourself and the intellectual journey that has taken you to the interests you have today?

I have been studying AI for a very long time, ever since I had my first computer, which was a Sinclair ZX80, in 1981, but my high school was about Classics, and my first degree was in Physics, so I had to wait a long time before I could formally study machine learning and all the other things that really interested me. In the days before the web I had found a way to order articles and books via the university library, which was also good for learning some English. All this happened in Italy. Then I came to the UK to study, and then I found a job in America, and then I came back here. The thesis of my first degree was on PAC learning, and that project started in 1995, so nearly 30 years.

Your recent book is subtitled “Why Intelligent Machines Do Not Think Like Us”. Can machines be “intelligent”? And how should we think of this as being different from human intelligence?

Intelligence existed on this planet well before humans did, we see signs of intelligence in the animals that preceded us, from their fossils we know they had brains and senses, and that they hunted and escaped predators, and some hunted in packs, and so on. Intelligence is not exclusively human, and not even just of the animal kingdom. It is the ability to act appropriately also in new situations for which we do not have a script, and in this sense also machines can be intelligent. Just consider how YouTube can recommend a new video to a new user. But there is no reason to expect that we make these decisions in the same way.

Is the sudden impact of artificial intelligence mainly a result of genuinely new paradigms, or is it a consequence of increased computing power?

Both. The increased computer power allowed us to explore a path to intelligence that we could not have explored before, and frankly not even imagined. Not only can we predict human decisions, for example, book purchases, without even understanding the contents of the book or the personality of the reader. But we can also translate documents without understanding their contents, or the laws of language. A lot can be done just based on statistical signals, but only if we have enough data. And these days, this often means millions or billions of documents, as we do to train GPT, a large language model that can answer exam questions at a university level, despite having been trained on the simplest of statistical signals.

If you could point to one major development in artificial intelligence over recent years, what would that be?

Until a few years ago I might have said AlphaZero, which learnt how to play Go and Chess on its own and from scratch. But now I have to say the large language models of the GPT class, that show remarkable emergent abilities: when trained just on the task of predicting missing words in a text, they acquire remarkable new skills, such as completing syllogisms or answering questions, and so on. These only appear after a certain amount of data has been seen. In both cases, however, we see machine learning in action. So I think that the major development behind AI in the past decades has to be machine learning. This has become a very sophisticated technology.

Do you think that machines can “understand”, or is it pointless to try to draw such anthropomorphic comparisons?

This may be a matter of words, but I think that when you can respond appropriately to your environment, because you know what to expect, after observing it, and when you can even imagine the outcome of situations that you have not observed before, then you do comprehend something of its mechanisms. So, yes, there is no reason why machines should not understand the world, just they do not do it in the same way as we do.

You write in your book about the possible dangers of machines behaving in a very literal-minded way. How can we prevent this from being a problem?

Imagine a machine tasked with the job of increasing clicks on a website, maybe one that recommends short videos to teenagers. It will explore its environment and will learn all sorts of tricks that can lead to increased click rates. It might even learn to exploit specific vulnerabilities of specific individuals. How can it learn when it is causing some damage, while also pursuing the very goal that we have given it? Making sure that the machine respects our norms will not be easy, due to the nature of the way it works. This problem is even more serious when we allow machines such as GPT to solve problems by selecting which sub-goals should be completed along the way. Once more: how do we ensure that those intermediate goals do not violate our values and norms?

Should we be worried at all about effective “manipulation” of humans by artificial intelligence systems, such as the targeting of messages or the filtering of social media and news, creating polarisation and echo chambers?

Remember, users are part of the environment that certain learning agents need to “exploit”, so some manipulation might be part of the game. But we should distinguish intended and secondary effects, at the very least.
How well can recommender systems steer users, what could be unintended consequences, and can they induce polarisation by proposing only contents that align with the user’s beliefs? The first thing is to be able to measure these possible secondary effects, and this is an important job for psychologists, and I do not see enough studies in this direction. Maybe at LSE someone will want to help. We will hear a lot about this during the next elections, in the UK and in the rest of the world.

Have we become too trusting of artificial intelligence and other algorithmic or machine-based decision-making systems, or is it the case that their accuracy on balance is an improvement upon alternatives that would inevitably involve human error?

Accuracy is not the only metric that we care about; we have made decisions to pursue other goals too, such as equality, privacy, freedom of expression, transparency, and so on. So while a decision should always be correct and based on accurate predictions, we must also solve the inherent tensions deriving from all those competing goals. This belongs to the social sciences and the humanities. Again, as in the case of the psychologists, this is not a mathematical issue. A common culture and common language among scholars would be beneficial here.

On balance, are you optimistic about artificial intelligence’s impact on our society?

Yes, I am very optimistic, but we should keep our eyes open, be involved, study the challenges, and not hide from them. Solving a single problem is more useful than just complaining about a thousand. We have to rapidly learn how to coexist safely with AI, and avoid risks before they become problems. We have started, the laws are on their way, and we will not repeat the same mistakes we made with the environment: this time we are acting before we actually ruined things. But we need to be vigilant. I am very optimistic because I meet my students, and see how clever they are, and how thoughtful. We are in good hands.

Glad to hear it! Thank you very much for joining us, Nello.

If you’d like to listen to Professor Cristianini delve further into the fascinating world of AI, why not join us on February 12 for his Public Lecture at LSE? Find out more information here.

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Raymond recently completed his PhD at LSE after defending his thesis on “Fire sales and policy interventions in financial networks”, and he was supervised by Prof Luitgard Veraart.

Hi Raymond, let’s start with a bit about yourself. Where did you grow up, and where did you go for your degrees?

My parents are from Hong Kong and China, but I grew up in London, and I’ve lived here most of my life. Regarding my educational background, for my Bachelor’s, I studied maths at King’s College London. I liked modules where you can apply the theorems and techniques from the course to real-world problems. So, throughout the degree, I was drawn to theoretical physics and financial maths. I struggled more with financial maths than theoretical physics. However, I was very motivated and engaged in the subject, so I went to Cambridge University for my master’s degree to pursue this further. After this, I came here for my PhD, where I did more applied work in finance.

As you do more applied work, have you found that your research has given you more insight into the major financial news stories that we all hear about?

Yes, absolutely. It is not uncommon for the news to motivate my research and to inspire new directions. During the Truss government, there was a substantial change in fiscal policy that caused many assets to be sold in large quantities and at heavily discounted prices. One of my areas of research is in fire sales, and this is exactly the sort of behaviour that I would seek to model. During this period, I also spent time with my supervisor discussing the news. In addition to this, I also researched areas such as climate stress testing, so I saw a lot of the topics I researched in the news.

Can you tell us more details about your PhD thesis?

I have been working in an area called financial contagion. So, we can imagine a system of banks where these banks borrow from and lend to each other, and you can think of this as a network. There may be a situation where one of these banks fails, and they may not be able to fulfil an obligation in full. Because of this, this bank may default on their payments, and this may cause other banks to fail. So, you can get a system in which a few banks failing can trigger many banks failing. These are the mathematical models that I was looking at for my research. However, you can model any type of contagion in a similar way, such as disease, or social contagion, which is when information spreads.

To go more into the technical details, you start by looking at the data for the things that you are trying to model, which will give you an insight into the borrowing or lending of these banks. You can then model these as a network, where each node is a bank, an outward edge of this node is a loan made by a bank, and an inward edge is some borrowing made by that bank. You then want to analyse network features to understand how contagion spreads. When it comes to the mathematical tools that are used to do this, we are interested in fixed-point ideas. We would have a function that describes the contagion process, and you iteratively apply this function to show that there is a fixed point, which gives rise to an equilibrium. This is when you can then analyse financial outcomes from this process. You also can verify this analysis using simulations and estimations to see empirically how these results play out.

I would imagine that the banks would start to change their behaviour once a couple of banks have failed, which would change the structure of the graph. Are you also interested in questions where the graph also changes throughout this procedure? 

Even before considering temporal changes, the data that is used to form the graph in the first place is often constrained by privacy issues, or how it is reported. For example, smaller banks and institutions are not required to report their data as openly. So, a key question is if you are missing that component, how much of the risk are you capturing, as these smaller banks do play a key part in the system. Then, it can be even harder to see how things change, as you have these problems, but it can also be difficult to compare data year by year and even harder to predict the behaviour of banks once the contagion has started. This often leads to thinking about what can be said about contagion in more general structures, as these results won’t be as reliant on the data.

What are you interested in doing next?

I am very interested in climate finance, which involves looking at contagion networks but more from a planning perspective. For example, I might consider how a natural disaster might affect a network and then consider how the network could be designed or regulated to limit the impact of the natural disaster.

For our final question, do you have any recommendations for new students in London?

For me, the best places are the small independent cafes that you might only find down a side street. This might be a small ramen place, or a family-run pasta restaurant, for example. Everyone has their places that they enjoy, and this sort of becomes your part of London. If you were going to force me to name one place, I would recommend one in Soho called Koya. They do these cold udon noodles, which are very nice and quite different from the hot soup noodle bowls that you find elsewhere.

That sounds like a great suggestion. Thank you very much!

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Eilon recently visited the department earlier this year to collaborate with Dr Ashkenazi-Golan. During this visit, we caught up with Eilon to discuss his life before academia, his research, and his love of history.

Eilon Solan

Hi Eilon, thank you for joining us. Could you start by telling us about your life before academia?

I grew up in Israel, in a town next to Tel Aviv. As a teenager, I developed a love for computer science and my first computer was the Sinclair ZX, which was very small and cheap. You would connect it to the TV, which would become the screen, and all you could do was program using the BASIC program language. Even then, the memory was so small that you could not do much more than basic programming. However, it was enough to make me want to enrol in university and develop my understanding. The aspiration to go to university was also unusual as most people in my neighbourhood did not enter further education. Indeed, Israel has mandatory national service at eighteen, so, it would be rare for people to go to university straight after school. However, there was a scheme in which you could defer your army service until after university, but then you would serve additional years, so I chose to do this. I had also always loved maths as well as computer science, so I chose to study computer science with mathematics as my minor.

What inspired you to do a mathematics PhD in game theory?

After one year of undergraduate studies, I realised that I found mathematics more interesting than computer science, so I decided to major in mathematics. However, it wasn’t until the third year that I realised I wanted to pursue a PhD in game theory. I was also fortunate enough to be taught by a great game theorist named Abraham Neyman, which also helped me make my decision.

Every week, he would enter the classroom, put his sunglasses and his keys on his desk, and start teaching without notes. He had everything perfectly memorised, and he taught this very mathematically demanding content clearly and understandably. I enjoyed his course, and I admired him. It was then that I decided to do my PhD with Prof. Neyman. However, I first had to serve in the army for six years, but I was able to do my master’s during this period, and even started my PhD. Two and a half years after I finished my army service, I completed my PhD and continued into academia.

Can you tell us a bit about your research?

In the game theory that you might first learn, a game is a payoff matrix, which tells you the payoffs of the players in the game if they take certain actions. We assume that all players aim to maximise their payoffs, and we define an equilibrium to be a vector of actions, one for each player, such that no player can profitably deviate from their action, even if they know the actions of the other players.

My research is in dynamic games. These are games where there is repeated interaction between players, and I am firstly interested in how players can best play these games, which involves finding and analysing equilibria. But I am also interested in games where some players may have more or less information than others, and the question is how they can best use their private information in these games. Very often, I will study various economic models, but it is not these economic applications that motivate me but the pure mathematically challenging questions.

John Nash

Can you tell us about an open problem in game theory that you have worked on?

Yes, the open problem I was given as a PhD student is still open. In the 1950s, John Nash proved that if a game is only played once, then an equilibrium always exists when players are allowed to randomise their choice of action. Now suppose that we do not play only once but repeatedly play against each other. And, each time we play, our payoffs are given by a different matrix, and the outcome of each game decides which matrix we will play next. Suppose that we do this without end. The question is whether an equilibrium exists.

For two players, this question was solved by Nicolas Vieille in 2000, but we do not know for three players, and this is an area of ongoing research, including here at LSE. These types of games are called Stochastic Games.

An example of a 2-player stochastic game called the “Big Match”. We start with the 2-player zero-sum game in the top-left, where the entries in the matrix show the payoffs to the row player. The column player and the row player each choose between ‘A’ and ‘B’, and row player wins in each round that the two players pick the same option (otherwise the column player wins the round). As long as the row player chooses ‘A’, the game keeps repeating; but if the row player chooses ‘B’ in round t, the game moves to a new payoff matrix (as shown by the arrows); the winner in round t then wins all future rounds.

That is very interesting and somewhat surprising! What do you enjoy the most and least about research?

What I enjoy most about research is the moment you find the solution you have been working towards. Or it might be when you succeed in combining tools from different areas and can apply them to solve a question that does not initially seem related. This is very satisfying as well.

What I enjoy the least is when I have spent the whole summer working on a solution to a problem, only to realise that everything I had thought is wrong!

What advice would you have for an undergraduate student who is thinking of doing an MSc or a PhD?

Do whatever you like the most. Don’t listen to those who tell you to study the subjects that will earn you more money. This is your life, and you should enjoy it as much as you can.

Suppose that you weren’t a mathematician. What else would you be?

Maybe I would be a historian. I would love to spend my days studying Antiquity. Whenever I visit the British Museum, I am fascinated by Assyria, Babylon, and Ancient Greece. I would give almost anything to walk the streets of Ancient Athens.

Thank you, Eilon. It has been a pleasure!

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In this interview, we will learn about Amedeo’s experience at LSE in the Department of Mathematics, his thoughts on London, his Italian background and his inspiring outreach work.

Hello! To start, would you please say a bit about yourself: where did you grow up, and where did you study for your degrees?

I grew up in a beautiful small town called Caiazzo, in Southern Italy. I then went to the University of Padova in northeastern Italy for my bachelor’s and my master’s degree in maths. During my master’s, I also spent one year at the University of Warwick as an Erasmus student.

Did you always know that you wanted to go into mathematics?

I cannot say exactly when I became interested in mathematics. However, I am sure that it became apparent at an early stage whilst going to school. I also believe that much of the credit should go to the many maths teachers and professors I was lucky to meet over my educational journey. They presented mathematics as a nice puzzle to solve or a challenge to overcome, and they were always very inspirational for me. So, when I started university, I also wanted to become an aspirational model for the next generation of students. I’ve continued with this mindset until today, so I still spend lots of my free time doing outreach activities for students and the public.

That sounds great! What type of programmes are you involved in?

One of them is about training students for maths competitions, as this is something that I used to do, especially when I was in Padova. I am also a maths teacher for an enrichment programme called ‘maths circle’. A maths circle is a short maths lesson on some extra-curricular topics. The idea of a maths circle was originally from Russia and Eastern Europe. However, they have now started in the UK and, for the last two years, in Padova too. It is a fantastic opportunity to meet motivated students, and it is also rewarding. This is especially true when you work with the same students for a period and witness their progress. In terms of outreach, they are also open to everyone, and we always ensure that financial restrictions are not a barrier to getting involved.

It’s great to hear that you’re giving back to the community that you came from. And maybe this is something that more of us should consider taking up.

We mentioned your postdoc that you are just starting. Honestly speaking, when I started university, I had no idea what a postdoc was. So, could you explain what a postdoc is and how it differs from being a PhD student or a lecturer? 

Generally, a postdoc is any position after a PhD and before you get a permanent academic position. In terms of the job, it is not very different from what you do in your PhD in that you work on open problems, write papers, prove theorems, attend conferences, present at seminars, and so on. But it is also a period in which you become more independent. So, you also develop your research plans, your research projects, and your research programmes. And it is also the time when you start becoming involved in projects where you are the most senior member of the team. For example, I have a project with some PhD students and I’m the only post-doc in this team, so I feel some responsibility for the junior members. On the other hand, compared to lecturers, I do not have as many administrative responsibilities so I can really focus on my research, which is very good.

So, as a PhD student, you learn how to conduct research, and the post-doc allows you the freedom to do just that. Speaking of which, would you please describe what research you do?

I work in an area of mathematics which is called graph theory. Graphs are mathematical structures that encode relations between pairs of objects. These objects are called vertices, and then we have edges that connect any two objects that have a certain relationship with each other. For example, imagine the vertices are your friend, and some of your friends don’t like each other. So, let’s draw an edge between any two people that don’t like each other. Therefore, if you want to host a dinner party, it might be better not to invite two people connected on this graph.
Now, in graph theory, we study the properties of structures that can be found in graphs. This involves a good combination of tools from algebra, probability, and analysis, and I like it when two areas of maths meet, and you realise that they are speaking slightly different languages, but they are mainly talking about the same problems. This is when tools from one area can solve problems from another and vice versa. Sometimes it is surprising, but sometimes, it is a matter of different notation, and two areas are calling the same object different things.

Could you tell us a bit about the recent breakthrough result in Ramsey theory?

Let me start by a small example of a Ramsey problem. Suppose you have six people, some of them are friends, and some are not. You can show that you can always find three people where either they are all friends with each other or none of them are friends with each other. More generally, in Ramsey theory, we consider a graph on n vertices, connect each pair of vertices by an edge, and colour each edge either red or blue. Then, we ask, “which structures in this graph can we find where all of the edges are one colour?”.

A longstanding challenge is to determine how big n has to be so that we can always find k vertices such that the edges between them are either all red or all blue. The exact answer is only known for small values of k, and the best general bound comes from an old result of Erdős and Szekeres from 1935, which showed that taking n=4^k is enough. The result you mentioned, due to Marcelo Campos, Simon Griffiths, Robert Morris, and Julian Sahasrabudhe, shows that n=3.9922^k is already enough. Although this might seem like a very small improvement, in fact, it is a pivotal one, being the first exponential improvement since 1935. Moreover, it is plausible that the constant 3.9922 can be reduced even more.

That sounds very interesting. 

You have been at LSE for four years of your PhD. What advice would you give to someone that is just starting at LSE? 

LSE is a great place to study and do a PhD. The department is not very big, so there is also a strong sense of community. I was able to have a lot of interaction with my supervisors and everyone in the graph theory community here. I also had many opportunities to interact with everyone else in the department, which only enhanced the experience.

My first suggestion is to enjoy every moment at LSE and take advantage of all the opportunities that present themselves. Being based in London means that you are always close to many universities and people working in the areas that may interest you, so you can always socialise and network.

My second suggestion is about research. Sometimes research is difficult because there are weeks when you will be stuck on your problems. However, it is always good to remember that this is normal. I would advise you to be happy with the little steps that you make, and not pretend that you will be able to solve the problem in a couple of weeks. I often find that there is not a specific moment when I solve the problem, instead, there is a moment when I can combine all the hard work and small steps I made in advance.

There are problems that I started thinking about early on in my PhD, and either made a little progress or I was stuck, and from time to time, I think about them again. As stated before, this is completely normal. If you have never been stuck on a problem, then you are working on problems that are not interesting!

Thank you for that advice. Some of it is probably good life advice as well. 

I have two final questions for you. Firstly, what is your favourite thing about London? And secondly, I would be remiss if I did not ask an Italian for a restaurant recommendation.

London is a huge city, and I like that it is very international. I have had the opportunity to meet people from all over the world, and this is one of the few places where you can do that. It is nice to learn more about different cultures, especially those that are geographically very far away from us.

In terms of restaurants, I’d like to recommend a fantastic pizzeria as I come from the south of Italy. My favourite one is called 50 Kalò and it is near Trafalgar Square. It’s a very famous pizzeria from Napoli, and a few years ago they opened this branch in London.

Excellent, I will take up that suggestion. It’s been a pleasure, Amedeo! Thank you for your time. 

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