MIT, for example, has the luxury of having the world expert on just about anything physically within a mile of, like, your dorms. And so we're designing all kinds of programs that, for example, in the School of Engineering, try to bring the students in for research experience, for hands-on learning, for going into industry and seeing what things look like, and trying to build the undergraduate experience more out of that and a little bit less out of exams and problem sets. Now, there remain a lot of questions. For example, assessment is a big challenge for us. What we observed in the last year or two is that our students are getting nearly 100% on all their homeworks. Maybe it's because MIT students are super creative.
Can I just say, so I took the train from New York to New Haven, quite a lot, and there are a lot of Yale students, watching them on their laptops right now. They're all on mostly Claude, actually, writing essays or, like, at least doing the basic research via chatbot.
Hello, and welcome to another episode of the Odd Lots podcast. I'm Joe Weisenthal and I'm Tracy Alloway. So Tracy, obviously one of the big concerns is, like, job loss. What are we going to do in the future? What are humans going to be better at? And then what are we going to do all day? What are we going to do all day, etc. And we really don't know the answer, I think, to any of these questions yet. But I am aware that, you know, obviously the models are getting very good at math, and at least some mathematicians perceive, you know, a state of existential angst about this and certain, like, crises, perhaps some feel about, like, well, what are we doing here? These problems that I devoted my life to solving and, you know, the model solves it in a weekend, you know, like, is this one's potentially coming for many of us.
Yeah. Math is an interesting one because you can tell that a lot of the AI companies themselves have an interest in math, right? Like, they see math as a sort of holy grail for, I guess, proving that the models are able to reason. Like, it's the thing they need to do in order to say that they've achieved AGI.
Yeah, I think that's the way it's couched. And it also makes good headlines. So even though I have no idea what the Navier-Stokes problem actually is, I don't know, even if I'm pronouncing that correctly, is it Navier or Navier? Okay. Navier-Stokes. I see the headline and I'm like, oh, a model has solved the Navier-Stokes problem, as if I know what that actually means and the significance for the mathematics profession. And it's also funny because, like, I downloaded the paper and it's like, you know, one paragraph and I have no idea, you know, like, I can't even begin to parse the output. You know, another fascinating aspect of this is just, like, it would not have been intuitive to me, say two years ago or three years ago, that a language model would be so good at mathematics. In fact, at one point they weren't very good at math at all. Like, very, you know, go back to, like, I don't know, GPT-3 or GPT-3.5, if you ask them to count, like, the number of letters in a word or something, they would get it wrong. And weirdly, they still, like, struggle with certain things like that. But it's not obvious to me, like, okay, you scan all of the text on text, you know, language, etc., and yet somehow that generalizes into this extraordinary math capability that, to me, is very surprising in this conversation.
Can I just say before we start, I would also just like to learn what advanced mathematics actually is, because everything I know about it, which is nothing, comes from, like, watching Good Will Hunting and just seeing, you know, a bunch of formulas on a chalkboard. And people get really excited when they start crossing off certain parentheses and stuff like that. I have no idea what it is. I have no idea how it differs from pure maths of, like, two plus two equals four. And so I would just like to come away in this conversation with a better understanding of what it is overall and how it actually fits into it.
Absolutely. So, like, I was thinking along the same lines, like, okay, for someone who, like, devotes their life to, like, advanced theoretical math and, you know, other things like advanced theoretical physics that don't necessarily have, like, obvious industrial applications, etc., who is the type of person, what are they hoping to achieve? Why is there still interest in, you know, some of these things that perhaps the applicability is not necessarily so obvious? But then, like, obviously, you know, like, what is the role of the mathematician at this point? And then actually specifically, it's like, okay, across the universe, this is actually the thing I'm very curious about, which is that, like, you know, like, my son studies math, right? Or my kids study math and, like, hopefully not theoretical math. Not just yet. No. Yeah. How old is he? My daughter is ten. My son is seven. You know, they take math, etc., but, like, setting aside, like, what the future holds, I'm very interested in just the present because it's like you have all these math departments at universities all around the country. And, like, maybe some are excited about AI, etc., but, like, what do you do today in terms of, like, rethinking pedagogy? Because it must be, or I would assume it's pretty different in 2026 than, like, those same departments in 2021 or something like that. Absolutely. And, like, how are professors, teachers, etc., thinking about just the sort of process of teaching math right now? It's just, like, just sort of, like, very straightforwardly, how is it affecting their jobs? I used to be really, really good at, sort of, mental math, like times tables.
All right. This is a question I have too. If you are an advanced mathematician doing theoretical maths, how often do you actually use a calculator? I'm curious, it's going to be my first question, actually.
You know, there was Jensen Huang. I don't know if you read the transcript of his interview with Ezra Klein a couple of weeks ago. But he said something like, oh, maybe it's just not... He is a little kind of a weird interview. He said something like, maybe, like, knowing your times tables isn't going to be an important skill in the future or your handwriting. Which is, like, kind of weird, like, I still want, you know, my kids to know. It raises a very important question, which is you can go to a model and have them verify a proof for a theorem or something like that. But if you don't understand how you actually got there and how the model is getting to its verification, then what are we actually learning here? Right. And then there's the question of, like, you know, people learn a lot in the process, right? So it's like even presumably in just the, you know, people work on some of these problems for years and years and maybe they never arrive at any proof. But intuitively, you would think there is something, maybe it's exercising a muscle or something, or sort of other benefits to just the process of learning. How does that change anyway? I have a lot of questions about this, and we really do have the perfect guest. We're going to be speaking with Justin Solomon. He's associate dean of engineering education at MIT and very interested and involved in math education and thinking about the intersection of math and AI. So, Justin, thank you so much for coming on Odd Lots.
Tracy sort of already suggested it, but set aside AI for a second. What is it that people who end up going into sort of, like, advanced math or advanced theoretical math, like, who are these people? And, like, why? Like, what are they hoping to do?
Yeah. That's true. I feel like the typical caricature that you see is maybe this character in, what, Good Will Hunting. Incidentally, they're always writing math on windows, which is very impractical.
Yeah. That's true. It's great for a camera. Yeah. But that aside, I think it's an intensely personal thing. So, to be clear, my personal career, like, I'm a professor of engineering, is in an area of math called applied math, which sometimes is applied and sometimes not. That's a discussion for another day. But essentially some people go into math because they want to see it translate into applications or to bring insight into the kinds of things we do every day, whether it's predicting markets or physics, what have you. Other people in areas of math, sometimes you call them pure math, I don't love that terminology, like pure versus applied, but in other areas of math, it's almost just for the joy of the game or understanding the universe around you. All of these are totally legitimate reasons and true intellectual pursuit for the sake of intellectual pursuit. But along the way, I think you develop a lot of really important reasoning skills and skills that support work in all kinds of different areas.
And how often do advanced mathematicians, whether they're in, like, applied maths or purely theoretical maths, how often do they use a calculator?
Okay. Yeah. No, my impression again is everyone is doing this by hand on chalkboards and windows. I mean, there's, like, lots of Greek symbols and letters and variables and things.
But no, I'm terrible at arithmetic. I've got a calculator out for real time.
Oh, yeah. This gives me hope, actually, that maybe I could understand some of this, even though I'm pretty bad at it.
Oh, that's my take as well.
Okay, so what's the difference between a math problem then and a math proof or a theorem? These sort of big picture things that we're talking about AI models actually doing.
Now, that's a great question. And it's really important to dig into exactly what the models are doing and not doing. And who deserves credit, by the way. But essentially in mathematics, it's all about verifying. In some sense, verifying true facts is one part of the puzzle, but it's also all about figuring out what are the interesting problems to study and what insight they bring to the broader world. So in some areas of mathematics, there are all kinds of different questions you could ask. And some of them are super uninteresting, but you could ask a computer to prove them. And others maybe tell you something about a physical system or about a system of equations or something like that, that bring broader understanding. And so this exercise of proof writing is all about sort of writing a mathematically convincing argument that tells mathematicians, so there's a human aspect, which is an important one, that a certain fact is actually maybe reality, right? It aligns with logic. There's not some flaw hiding in there somewhere.
And where do your ideas for a particular mathematical proof or theory come from? Because again, I'm going to come at this from, I guess, the movie angle, but I think about A Beautiful Mind, right, in the famous bar scene. And was John Nash, right? That was his name. And he has this, like, big realization while he's at the bar with these women, and it ends up turning into a famous mathematical theory. Is it like, where do the ideas actually come from for the rules that you are trying to prove in maths or these big picture ideas?
This is a great question. So sometimes in mathematics, there's some really natural questions out there in the universe that people have been asking for a long time. So, for example, these days in the news, we hear about some of these really high-profile math problems that have been around forever. There are other types of mathematicians that are more theory builders, where they're the ones asking the questions rather than answering someone else's questions. Yeah, there are definitely different approaches that are out there now. When it comes to where these questions are coming from, some of them are inspired purely mathematically. And other ones maybe come from a very practical setting. Maybe somebody observed some phenomenon out there in the real world, and they're trying to explain it with the smallest model possible. Now, one thing I should clarify is that math is a really social exercise. I think that's something that people don't necessarily understand, especially because these movies really play into this almost great man theory of mathematics. But that's just not how it works. There's this huge community of people that are working on important problems. They're communicating with one another, collaborating. And I would say, for example, from my own research, almost everything that our team has worked on comes as a team effort. We're all sitting in the same room, we're at the blackboard writing computer models, what have you, and that's where the insight is coming from. So it's not just sitting in your ivory tower. It's being out there in the world observing things happening and talking to your colleagues.
Before we get more into the sort of AI question, what is, I've actually been curious, what's applied, what is the difference between applied math and pure math?
Yeah, it's a great question. So there's sort of a divide in the math world, but I think it's fuzzier than maybe it sounds when I describe it here. So roughly, there's an area of mathematics that you might call applied math, where the goal is to either derive theoretical results or even come up with computer algorithms that explain phenomena out there in the real world. So, for example, in applied math, maybe you work on algorithms that simulate the weather, and behind the scenes is a really complex mathematical model. And it's all about making sure that your model is correct and efficient and well-behaved, all that good stuff. In pure mathematics, sometimes the motivation is a little bit less attached to a particular application, and more just attached to the math problem itself. So something like Fermat's Last Theorem or something right there where he's like, okay, I forget what it is exactly, because that would not be applied math. Right? So Fermat's Last Theorem and many things I think in number theory fall into that category. But also many times these things bleed into one another. So for example, a lot of modern developments in cryptography came from areas of mathematics that we used to think of as extremely pure and abstract.
Oh, interesting. This is also Good Will Hunting, by the way.
But it all goes back to... That's right, that's right.
You worked at Pixar before, many years ago. Was that applied maths, working at Pixar?
Oh, I love that question. The answer is absolutely yes. So, movie studios are a great example of a place where I would say state-of-the-art in certain types of mathematics are taking place. So when you watch a typical movie with different movie effects, there's cloth, there's smoke, there's all kinds of fluids, you know, rigid bodies flying around. All the dynamics of those are governed by things called partial differential equations. And coming up with the right algorithms for simulating those, and by the way, making them so well-behaved that an artist doesn't have to think about math, is a really challenging mathematical problem. I typically tell my students that if you're looking for really challenging math problems, you don't have to make them up. They're out there in the real world. And that's a nice example.
What's the Navier-Stokes?
Okay. So Navier-Stokes has been in the news a lot lately, and it's been great for those of us with a little bit of background in differential equations.
Okay. We're getting lots of media. Getting lots of media inquiries.
That's exactly right. Yeah. We have our 15 minutes of fame here.
It'll die fast, but we'll enjoy it for now. Right.
So the Navier-Stokes problem is one that comes from the world of fluid dynamics. So the Navier-Stokes equations were this model, they go back quite a long time, that explain the motion of, for example, liquid in a cup. So like both of you guys have coffee cups here on the table. And if you, like, shake the coffee cup up, you know the initial velocity, you know what's inside of the cup, then the Navier-Stokes equations are trying to predict, like, where the fluid moves over time, right? Like where is the velocity, vorticity, all these different things. Now the Navier-Stokes equations are really complicated. I mean, when you look at your latte, when you start in the morning and you see, you know, the milk making these cool shapes and so on, there's all kinds of complex behavior that can happen. And so the Navier-Stokes equations, although they're easy to write down, they're very hard to say a whole lot about. And so there is this big open math problem, which I think is what you guys are referring to because it's been in the news, which involves existence and uniqueness of solutions. So the basic idea here, I think actually my undergrad differential equations professor years ago explained it to me in kind of a funny way. He sort of said, you know, all you got to do to make $1 million is prove that water doesn't explode. And that's roughly right. I mean, with a lot of asterisks, essentially, what we didn't know is if you know the initial configuration of your fluid and you let it kind of go under its dynamics over time, then whether it exists for all time and doesn't have some really crazy turbulent behavior, or if, for example, maybe it blows up or it develops a singularity or something like that. So this was kind of a yes or no question, like prove or disprove. And I guess the recent news was actually a counterexample. So in other words, it showed in some finite amount of time, with an asterisk, which is with a very crazy spoon, if you start your fluid just the right way, you really can create this totally singular and frankly nonphysical behavior.
So proving in math is like one of these things. Like I think I, you know, like the formal proof, the two plus two equals four. It's like a nine-step thing, isn't it? Right? Like if you, or like sometimes eight, sometimes eight, but like, can you explain, like, what a proof is? Like if I know two plus two equals four, it seems obvious, but like what is it explaining? Like this, what is a proof such that that would say that to prove that two plus two equals four is something that requires eight steps, right?
This idea of proof writing, I think, is actually, you can think of it as almost the result of an identity crisis among mathematicians, maybe 100, 200 years ago, where it used to be a lot of math was drawing pictures and kind of working in the same fashion that you're talking about, kind of intuitively. But at some point, people realized that they wanted to make sure that the field had really strong foundations. And so mathematicians will agree on a certain set of maybe logical principles or rules. I would say that's more the domain of philosophy, precisely what those should be. And then the idea of proof writing is to build up from these kind of basically like we agree that these five facts are true to a much more complicated fact by combining them in different and creative ways. And so theoretical mathematics oftentimes is all about, I don't want to just say mixing and matching different facts, but coming up with smart ways to justify mathematical ideas built on very simple principles and steps from logic.
So when it comes to AI currently, is the role more in, I guess, proof checking versus proof generation or idea generation? Like what are the models actually doing at the moment?
Well, they're sort of doing it all and they sort of do nothing. So, right, the current AI models, in terms of the types of mathematics they're doing, I don't think it's quite right to say that it's just proof checking. And in fact, actually, they're quite bad at proof checking. And one of the revolutions in, I would say, the last year or so is the realization that actually what you have to do is to provide your model, like your large language model, with a second module that can check the AI's proof. And so for example, there's a programming language called Lean, which is basically just a sort of codified version of these mathematical rules. And these days what people will do is they'll put in their conjecture, so like the thing that they want to prove or disprove, and then ask the AI system to generate a proof, but then they'll say, rather than just writing it in human-readable language, which is the substrate that mathematicians use to communicate with each other, maybe in addition to that, they'll actually produce this Lean code so that there's a separate piece of software out there that can check that this proof truly is correct. So actually the correctness part of it is usually deferred to another piece of software, which isn't AI. But somehow because of the interaction between the AI tool and this proof checker, that's what's created a lot of progress that we can believe in a little bit more than we did in the past, if that makes sense.
This I have heard of only because, you know, I've been reading the news. I've heard of this thing called Lean. And it sort of seems extraordinary to me. If you explain it a little bit, it seems extraordinary to me that there is this, I don't know, is it software or this framework such that these extraordinarily complex problems or solutions, I guess, is the answer, can be dropped into this, and it's capable of, like, yes, this is a valid... But it's also kind of crazy that you need basically a translator for it. Right? Can you explain a little bit, like, how this works, how Lean works?
Perhaps I can explain a little bit. To be clear, I'm not an expert in this particular corner, but so roughly, essentially, as we all know, the things that come out of AI are unreliable. All of your listeners should know. And that goes for mathematics as well. And also we think of it as this area that is extremely reliable. Everything should be true or false. Now there are gray areas, but certainly we try to avoid that. So this programming language Lean, what it does is it has a few simple maybe axioms that are built in. So math things that we sort of agree are the starting points for writing proofs, like the steps of basic logic. And then what happens is that you write a theorem in this Lean language. So rather than in English, you write it in this programming language. And then it's kind of cleverly designed so that once the software checks that that theorem is true, it can get added to the library of sort of true facts. And then you can build it up. So you make this more and more complex edifice out of these simple steps. And it's been really impressive what they can do. I mean, some of the modern proofs that you're seeing, especially the AI-generated ones, are like hundreds of thousands of lines of code. I don't think a human could check it. That also, by the way, means that we're trusting that someone wrote that software correctly. But, you know, if you believe that Lean is good at checking proofs, then it provides a lot of certainty that the things that the AIs are writing down are actually correct.
Devil's advocate question. But why is it important that we need to understand the actual problem and the proof? Because if I'm an engineer and I don't know, some model somewhere comes up with this really new, interesting mathematical theory and I can apply it to my day-to-day work, is that not enough that, you know, like, I can use it? It seems to work for the most part. Maybe it's been rigorously tested in a physical environment or whatever. Why do I need to understand everything that's gone into it?
Oh, that's a fantastic question. So let's see. My research group works a lot on simulation algorithms. So like I have some physical system and I develop a technique for predicting what it would look like in the future. And we spend a lot of time writing theoretical results to verify that our algorithms are correct, they behave the way we expect and so on. And that's absolutely critical because without that, you really can't trust what your software is producing. So, for example, if I'm trying to say yes or no, is this building going to collapse during an earthquake? That's right. And all of that is built on mathematical models, techniques like finite element modeling and so on. They're not too exciting outside of my universe. But you're also trusting these things to do really important engineering calculations and things that, sure, like one way to justify your software is that your building hasn't collapsed yet. But that's not a whole lot of, you know. Yeah, I hope I can trust a little bit better than that.
You know, my dad was a physics professor, and I have memories of watching a video of the famous Tacoma Narrows Bridge collapse. You know that video? Really? Yeah. And like, that was, like, for him, like this, like... Yeah. Yeah, I watched that several times in my household.
Our engineering students have to take a lot of ethics courses that are pretty fraught that way.
Yeah. Let's start with, like, I mean, there's a couple of avenues I want to explore, including just the process of teaching and how it's different in 2026 versus 2016, given the new technology. But maybe give us a lay of the land for, like, the math community. Yeah, because I know there are some people in despair, I believe, very anxious about what this means for the future of, like, math as an intellectual discipline, etc. And then others seem to be very enthusiastic about the opportunity of human and AI exploring new ground. But what are the sort of conversations that you're in these days with fellow mathematicians?
Oh my gosh, things are changing by the day. And really, it's a huge collection of individuals. Every single person has their own opinion here. So I can tell you my view of the world. If you talk to somebody else, I'm sure they'll have a very different one. Math is an academic discipline. It also shows up in a lot of applications, as we mentioned. But every aspect of the work is changing from how people produce mathematics and choose problems to work on, to how mathematics is sort of measured in terms of productivity. When we talk about hiring faculty, for example, like what are the standards to evaluate a person's body of work and what did the human contribute? I think are really critical questions to, like, why I do this exercise at all, maybe leave it to a computer. There is all kinds of identity crises that are happening. Now among mathematicians, there are people that are maybe theory builders, and there are people that are like theorem provers. The theorem provers, I think, are especially under threat at the current moment where, like, maybe you've been working on one problem and it's well-defined and you've known about it for, you know, 50 years and it's just this crowning, you know, question that you want to answer about the universe. And these AI tools are sort of at least good at sort of cherry-picking the ones that are somehow close to being proved and taking that last mile in some cases or another. You know, other people disagree with that characterization.
What do you mean by that when you say they disagree with that? You know, like exactly how interesting.
For example, a mathematical proof is a very human judgment. As is what problems you can and cannot, as is what problems you should or should not actually go about working on in the real world. So, there are a lot of true facts. It doesn't mean we have to write them all down, essentially. And because of that, this is definitely a human exercise. To a large extent, mathematical research is about humans communicating to other humans, right? It's not just about verifying true things about the universe, but bringing insight and helping people understand systems and so on. So I think you mentioned an identity crisis. I think what's happened is, and this is reflected, there's a really nice write-up by Terence Tao, who's a famous mathematician, that kind of talks about how there were a lot of different sort of signs of mathematical progress that used to be correlated with one another. So, for example, if I wanted to judge whether a mathematician has quality research or output, maybe I look at where they're publishing, the number of things that they're writing, the interest of the result, whether they're earning these prizes and so on. And now these things would have been at odds. So, for example, the AI companies have figured out, as you guys pointed out, that, you know, proving some big theoretical result maybe is an interesting press release right now. So, you know, there's an economic knowing how to do the right prompt. And then there's a big question which is, okay, so, you know, at least in the sort of cartoon version of a mathematician, you sit in a room, you think about a really hard problem, you write down a proof that's subtle and challenging. Now, as you say, maybe you're the guy that wrote down the prompt. If the prompt, for example, was prove or disprove Navier-Stokes equation, did that guy really bring a whole lot of insight to the problem? Is an interesting question, and one that I think we're really grappling with today.
Yeah, I want to ask more questions about credit, but just on the models themselves. Does sycophancy exist when it comes to math prompts the same way it exists for other, I guess, language-based queries? Does the model go like, oh, what an interesting problem, you're on the right track? Like maybe, you know, look, I'm just judging from my email box, I think the answer is yes.
And so I'll tell you a funny anecdote. So all the time I get emails from people that are looking for an expert to maybe verify, you know. And what I've noticed lately is I get a lot of emails from outside folks that sort of, you know, they've been playing with their AI system, they've proven a new result. And they told them it was really important. And they need a professor to validate. Which is absolutely fascinating. In fact, for fun, I played with some of these tools and put in a few open problems in my own domain. Interestingly, they did not get solved. So apparently my corner of math is safe at the moment. But the first thing that I did prove was, from my perspective, a pretty superficial thing. And then I had a little footnote on the bottom, I think, like, as far as I can tell, this doesn't appear in the literature. Like, would you like me to convert this into the template for a journal? You know, it was an interesting... I think it explains a lot what's going on in my email.
So you can imagine a future where, I guess, they're like, you're sort of overwhelmed with new proofs, and the bottleneck becomes the ability to actually verify them with humans.
That is a problem right now. And in fact, it's not just in mathematics. It is across academia. So to give you an example, actually, interestingly, I would say the academic field that's been affected most by this phenomenon you're talking about is actually AI, machine learning itself. So, in that particular corner of academia, research is driven by conferences. So you write your research paper and there's a sort of annual cycle where you submit and it undergoes peer review, and the whole community spends like a month or two reviewing each other's papers, sort of like, you know, you pass a paper to the left in class and grade, it's sort of a very complicated version of that. If you look at the number of submissions to the machine learning conferences, it is wild. So in the last ten years, I think about ten years ago, it was maybe one or two thousand. The deadline for ICLR, the International Conference on Learning Representations, one of the big ones, I think it had 60,000 submissions. And if you think about it, it completely breaks the academic system the way that we're used to thinking about it. We don't have bandwidth to, never mind, just check all of these proofs, even open up the PDF. And so you're then relying on an extremely superficial review process. There's many anecdotes of, like, high school students that have somehow gotten involved in peer review because, you know, they had a project once and the system decided they were in. You know, which is frustrating, you know, to build an academic career. Essentially some of the standards of quality that we're used to in the field, you really can't rely on for exactly the reason you identified. The way Terence Tao put it is that it used to be we were in this era of proof scarcity. You know, that mathematicians, it was really hard. And it was this very bespoke object that took a lot of craft work and training to produce. Now we're in this era of proof abundance. So now we have to kind of think about different ways to do our job and to do mathematics in a productive and interesting way.
Yeah, I think a lot of fields are dealing with very similar issues.
In which we used to have like certain, I mean, even take writing, for example, we used to have these certain heuristics that we maybe like shortcuts that we use. It's like, okay, if this person can construct a series of well written sentences, we might know spelling errors.
Yeah. Like how I never make spelling errors.
I mean, we might start, you know, we start from this area. Okay. This person might be worth like reading. They might have something. You know, we all have to like operate with these cheats and shortcuts. And now, like obviously, producing text with no spelling errors or anything is available to literally everyone. So it's like we have to go a step further, etc. I think it's like, you know, all these like heuristics, you know, speaking of like everyone now, like thinking they can come up with a proof. Do you remember last year, Travis Kalanick, he was the Uber founder. He was on the All In podcast and he was talking about how he was doing vibe physics with Grok, and he felt that he was, quote, getting pretty damn close to some interesting breakthroughs.
I just like doing vibe physics. What's crazy is he could just hire a bunch of physicists to like verify the research, and then you'd have a real sycophancy problem.
So I work in education. One of the things that we really rely on is somehow the friction in the technical process. I mean, you mentioned writing is a fantastic example. How do you learn how to write? Well, you do a lot of writing. And now it's very difficult to convince our students to sit down and write an essay because they have these AI tools. They can do it. But then, you know, writing a five paragraph essay was never the goal. It was to teach you how to think. Mathematics is similarly experiencing it very similar.
So let's talk about the field of education itself. I mean, are the students very anxious right now? I mean, we talk, what's like, let's start there. What's the vibe amongst students? And they're thinking, you know, especially like maybe grad students who started their career before the idea of AI generated. The thing is, suddenly they're in grad school, right? What's going through their minds these days?
Yeah, the landscape of both graduate and undergraduate education is changing a lot. Now in the graduate school world, for me, that would mean PhD, in other words, that maybe, you know, master's degrees and so on. There's sort of divides into maybe professional education, preparing you for a job, and then maybe academic style, like PhD work. In the PhD world, of course, it's being strongly affected by just the larger trends and changes in the research universe. And then when it comes to professional degrees, the main question we have to ask is what is the right thing to teach our students in the moment? So, for example, I'm in a department that teaches computer science. The field of computer science is changing by the day, right? It used to be, for example, we talked about programming is this core skill, thing that everybody should learn from day one. I would argue that's probably still the case, the same way that maybe learning to write and be literate is a key part of being, you know, a person in the society, even if you're not doing that every day. But, you know, jobs as an actual programmer are becoming more rare, right? We see a decline, for example, in enrollment in computer science for the first time in like 20 years.
Since you mentioned computer science and, you know, unsolved questions. What is P versus NP? And I can answer all your math questions.
Just all these questions I have stored up in my head. Right. And we should come back to. I don't think I actually told you the end of the Navier-Stokes story. So yeah, go back to that. But yeah.
So P versus NP and Navier-Stokes are examples of very famous mathematical problems that have been identified by the Clay Math Institute as sort of this benchmark for progress in modern mathematics. Right. So there are open questions we don't know how to answer. P versus NP is a computer science problem in the sense that it's basically asking are all problems solvable in polynomial time. Meaning can your computer solve, or all is a little strong. But there's certain classes of problems that we know are solvable pretty efficiently. So for example, your computer can sort a list of numbers really fast. The other problem is that we don't know how to solve efficiently. But that's different from saying we can't solve them efficiently. And so that's the sort of fundamental question in that universe. Like do there exist problems that are fundamentally the known unknowns and unknown unknowns equivalent of the computer.
