5/25/2021Milind Hegde
Saintsing:Hi, you're tuned in at 90.7 FM KALX Berkeley I'm Andrew Saintsing and this is the graduates, the interview talk show where we speak to UC Berkeley graduate students about their work here on campus and around the world. Today, I'm joined by Milind tech day from the department of mathematics. Welcome to the show Milind. Thanks, bye. So I guess to get started, um, how did you get involved? How did you end up deciding to go into math? Why are you a mathematician today?Hegde:Right. Um, I think when I was a kid, it was a few more of a, a science thing, uh, generally. Um, so my, I want to be a researcher maybe quite early, like seventh or eighth grade or something. Um, mainly cause I just, I just read these popular science books, which does seem exciting. And I didn't know maps, I think at that point I wrote a few popular math books as well, but not, not really. Um, it was only in high school. I think that I started learning about more, more advanced maths, some chocolates and stuff. And I started seeing some more of the depth in the subject. And I think that excited on some level on seeing that there's more happening here, that I could actually kind of do myself on some level.Saintsing:You mean? So like in high school calculus that,Hegde:Yeah, just generally, you know, public till high school math is in some sense, quite boring. You just keep doing these kinds of calculations, which are, you don't know why it's important and you know, there's no real patterns there, but after a point that you come to the part of subject where you can see some patterns it's oftentimes pull, they don't, they don't tell you the patterns, but you can kind of see some patterns. Um, and so for me, kind of seeing the glimpse of these patterns was very exciting and then realizing that there's a lot more richness in the thing where you can find, so I'm using these word patterns to just describe general mathematical kind of properties or structures or needs or something like that. Um, and so seeing that there's a lot more of this French cause there, which you can kind of explore yourself was very appealing at that time.Saintsing:And then, so did you go to undergrad focused on math right from the beginning?Hegde:So, uh, not, well, no. So I think my major, about two years in, but I went to a school where basically you pick from a science major. So it's either physics, chemistry, biology, math, or maybe environmental science, I think. And so I found myself to be really bad at the other lab work. And so I must prefer theory for sure.Saintsing:You mean, you tried to like go into the lab like pipette and it was just like notHegde:Happening. Yeah. I, I, it's not that I was disorganized. It's just that I didn't, like, I felt like I'd rely on certain machines to do jobs correctly. And if the machine failed, I just didn't know what to do, what not for kind of on your own. And so if you, if something fails it's, it's on you. So once is that's bad. Uh, but in another sense, you're not beholden to something else working for you to be able to do something, but that's what you like about math. Yeah, exactly. And you'll get at the research level, you'd go out, you rely on some theorems, which you don't know how they're approved, but it can break. You know, he can't be that you're trying to do something and then the machine fails and then you have to wait for somebody else to fix it. Cause you don't know what's really wrong with it. That was my experience as an undergrad in these labs,Saintsing:How is it to be a mathematic researcher? You have like a particular area of math?Hegde:Well, I worked in, in probability theory, um, just, just sort of stuff, studies, mathematical notions of randomness. And so, you know, there are many years of math is number theory. There's algebra, there's, you know, Fremont analysis, a lot of areas of math. Um, and you don't really need to know most of them to pursue any quicker one, right? It doesn't, which feels somehow more international than others, but even within a field, you then have to specialize into sub topic. So it probably could do it. For example, you have to know quite a lot of, of general theory about probability, but then those were established like 30, 40 years ago. So when you get to stop this happening today, you would go into a specialization within property theory. So what I do is this area called stochastic growth models and what are those? Right? So, so as the, as the name says, it's, it's, it's a model of growth of which will task some of those to random incident stochastic.Hegde:Uh, so for example, there are many real world examples, which are not actually what we study. We study models on that in a very abstract sense, but, uh, but the real world examples are, you know, so you can imagine, you know, for a biological example, you could imagine you have a bacterial cell initially and I get this completely wrong. So please forgive me if I do those, those to begin with. And then, you know, it grows, uh, it divides and it grows, uh, at the colony grows outwards. And so, you know, there is kind of the, the thing we're interested in is the way it grows. So for example, one question might be as the colony grows outwards, it's roughly circular, uh, or roughly kind of yeah. Basically circular. Uh, and so you have on one little, you have a rate, the bloat outwards, how big, how fast the radius grows of the circle.Hegde:Um, and then it's not actually a circle. It has, it has kind of these fluctuations at the boundary, um, away from this kind of nice smooth shape and you get these kind of rough fluctuations. And so one other aspect of interest is to understand the nature of those fluctuations. How big are the fluctuations? Do they have some kind of distribution and things like that? So basically you, you were interested in the growth of an interface between two things in this case, uh, that blue me and the outside. Uh, but also in other settings, for example, a more physically set in physics, the example would be you have some kind of crystal, um, you've if maybe you maybe initially it's kind of a flat surface. And then from this flat surface, the crystal kind of grills upwards into, into some medium, and then you have a boundary between the crystal and the medium, and then which is, which has also grown with time. Um, and so just like the bacterial cells talk back to colonies, uh, raises increasing here, the height is increasing of this crystal interface. Uh, and just as you have fluctuations at the boundary of the actual colony, here are the fluctuations of the, of the interface around the kind of straight line, um, that it's kind of averaging to B.Saintsing:So you're talking about like two, like completely different things, right. But you're saying that the models you're looking at are good for both of these.Hegde:Yeah. Because somehow there's a belief that many, many types of growth, uh, have similar behaviors in the sense of how fast they grow outwards and the fluctuations around a average shape. And so even though, as you said, he's a bit very different physical phenomenon, still the idea is that the same mathematical models kind of capture important aspects of boats handy. The, the models themselves are quite abstract. If you, if I, if I tell you the model, even not say, oh, this is definitely, you know, describing an extra colony or something, but yes. Yeah, right.Saintsing:But you're more interested in the theoretical model then you, the real world applications of it,Hegde:Right? This, this, this, this belief that the same kind of model described by a wide variety of, of phenomenon. It's also kind of, there's, there's a, there's a mathematical aspect of it where many, many different models, uh, which are kind of different in the details, but it's an elevated for models. Still give you the same behavior in some sense. So, you know, you might actually have different if you look at the actual precise details, but you might have a different model for the, the car crystal relative situation and the, um, the bacterial colony relative situation. Uh, but the idea is that nevertheless, you've lived through different models. They events the same behavior in a microscopic sense. So the simple example, which has maybe people more familiar with, would it be, um, is more classical area probability, honestly, the central limit theorem, which is that if you have many, many kind of random factors, which are in which are involved in, uh, some, you know, some bigger, some phenomenon, then that phenomenon will, will display a bell curve.Hegde:I know, to kind of make sense. Uh, so for example, you know, you look at, look at heightened population, he started by gender in the Heights and population. Uh, and then you see this kind of distribution of, of, of Heights, which is just a bell curve because there's some other lot of random inputs in the genes or something, which, which gives you an independent structure. And the important thing is that no matter what is the nature of the, of the input, random input, you get the same thing at the end when you get the same bell curve. So this is what's called universality in this area. And so the idea is that similarly in, in random roads models, many may different models exhibit the same kind of universal features, not the Belker, but some other structures, but nonetheless that's the same for everything. Right.Saintsing:That makes sense. I got you. What are you trying to do though, actually with the model, I guess, like what, what is pushing the boundaries and knowing the theory of this model, um, me maybe, or like definitely in the context of your own research, and then I guess generally kind of what, what aren't mathematicians pushing in these, uh, theoretical models generallyHegde:In this area of ultimate goal is actually to prove some form of universal. Self-esteem some form of saying that no matter what model you have, you get the same kind of behavior across all the months. And that's the ultimate goal. We're very far from that in done. So if I, if I can go back to the central limit theorem example, uh, so we, now we know that so many in general, how you put in the randomness, we get to say bell curve out, but in the beginning, and I think maybe in the 17 hundreds, this was known for exactly one situation. Uh, and this is the [inaudible].Hegde:So if you tell us, uh, yeah, it's a point like a, I mean, a million times some absurdity of number, and you look at the number of heads and you do this many times, uh, you'll see the number of heads is in roughly half, but it's not always going to be, has it been, have fluctuations around, around that task? And you see how many times is it? Is it, is it two more than half times is 10 more than half and you kind of eat plop this and you get developed. And so the very first situation where we knew the delicate horizon wasn't disjointed, and the reason was you could write down a formula for the number, for the probability of getting any number of pants you wanted. So you can say, okay, probably getting a hundred heads out of 300 tosses is exactly this number.Hegde:And so you go, you could get this exact formulas and then you could, you could analyze those formulas to say, okay, 90 that's where you see this bell curve coming up, but these farmers don't hold for other situations for you have different forms of randomness that aren't so simple, like acquaint tusks. And it took a long time to overcome that difficulty and develop a theory which could handle all kinds of randomness. And so now within stochastic models, we're kind of in a similar situation where he had a few nice models where you can write down fullness for things exactly you can handle them and you can get your limiting behavior, which is, which is universal, but you can't expand beyond those models because the farmers don't hold beyond those models. And so part of my research has been to kind of step towards a universality, tried to move away from models, which are, you know, exactly solvable because they're can exactly write down formulas for their solutions. I can try to move away from exactly solving models towards it's like a more robust approach to proving universality. That's kind of the short description of what I'm trying to do.Saintsing:Okay. I guess, like, what would that mean? If you couldn't write down a formula? Like what, what would you be doing then?Hegde:It's hard to say what you'd actually be analyzing without the formula. So there's a part of the problem. You know, you have to invent something which captures important aspects of the model without requiring you to be able to write down something exquisitely about it. So in the case of the, um, the central limit theorem, the bell curves, uh, this, the thing that was invented was the four-year transform. So if we were friends for me, you could write down for random burglar what it is, but maybe you can't actually calculate it. But the way to present the limit there was to use this object. What you kind of came out of nowhere. Nothing. If you look at the, you know, the point is you don't really see any 40 transforms involved, uh, but you could still introduce the transforms that you can analyze that thing to get the Velcro. And so there's no kind of analogous object, which we can say, oh, if we understand this thing very, very well, then we can prove universality in show category. So it's a good question. I mean, I don't have a succinct answer except that as part of the problem to know, to what to look at,Saintsing:For sure. I see. So, but that's kind of, um, something like the 48 transform is what you're moving towards with the goal to,Hegde:In a very analogous sense. Right. But yeah, something which could play the role. I mean, I don't know if there is such resistance, not such a sense that there's such a single object because somehow this situation is a much more complex situation than a developer, but somehow something analogous to, in terms of its role, the, for the customer would be something you could try to get towards and hedging. My, what I'm saying is simply because there are way more things happening here. So it's hard to, hard to imagine that you have to capture all of it with one object. Um, but maybe several things like, like the furniture transform,Saintsing:Right? For sure. You're almost done with your dissertation, where have you pushed these boundaries? Like what, what have you found on these problems?Hegde:So, because we only have a few models where we can say anything, which would be exact, is all available models. Right now the general strategy has been to work with these models, but to write down assumptions, which are kind of halfway abstracted away from those models. So your regular was obviously what you know, to be true in those models and no other models, but what you still hope to be true in other models, even if they're different, even if the models themselves are different. And so the idea has been to write down two assumptions, which, you know, to be true in these malls and then derive consequences of those assumptions using methods, which don't require formulas. Okay. So in a way to kind of move past, take limited is that it's also will input. And then from then on use only more robust techniques to drive new prediction or new and new results,Saintsing:Be able to walk it, like tell us an assumption, or would it be like too theoretical?Hegde:Oh, the assumption. So, so in the central limit theorem, you know, if you take, if you toss the coin and times and can be like any number that it's known, that the, the, so you're going to be around and by two, that's your number of heads typically, but given our fluctuations around that, and we know that the size of the fluctuations is, uh, is, uh, is, uh, is around the square root of N. Um, so this is a very important aspect of the central limit theorem, but your fluctuations around me are of order the square root of the number of trials in stochastic role models. It's very different. You don't see a square root or her fluctuations. You see if you have pen, uh, it decides your system is N not the end point doses. Then you would see fluctuations of order. And to the one, this is just the reasons why I see one third, basically, you see, you see the end of the one-third hundred. So it's a smaller amount of fluctuations. And this is known only for like a few models. The fact that it's entered the one-third, this is a very basic fact, but you not only for these kinds of very few models, so one assumption it's exactly that the fluctuation of our order enter the one-third nice.Saintsing:Okay. So you're saying like, um, cause you have these models, there are some models that we have for stochastic growth that we already know. And so since we have them, we can analyze them and we can say these fluctuations are under the one-third, but you want to be able to like, say essentially the goal is to say like, this is just a fact for all, even if you don't have the models, what is the thing that you do to get from that assumption to like the research that's pushing it.Hegde:Okay. So maybe now maybe I could tell you a model, hopefully, without getting too much technical stuff, but basically it it's pretty simple actually. So suppose you have a grid. Okay. 2d and the grid size is, and like N squared number of points in the grid at Eddie's corner, in your grid, you have one cortex. Okay. Just like maybe you can imagine some dots, you associate to each verdict or random number. Okay. And that's going to be like a award if you go through that. Perfect. So you can't even really imagine it like in like a, you know, a city chronicity. So you swing print and you've got to start at the bottom left corner of your square. You're going to go to the top right corner by going at the people, you can go to the right and you can go up at each intersection, as you, as you go from you, can't go back.Hegde:You can't go. You can't put down, you can't go left. You can only go up. And right. These are, these are the allowed ways of movement in this grid. And depending on which path you take, you're going to pass through these, these, these intersections would each have a random number associated the reward of that intersection. And you collect the rewards as you pass from here to here. So then you are told reward based on how you went through the, through the grid, which is the sum of each reward yet you actually went through. And now we're gonna, we're gonna look at the best way to get from bottom to top. [inaudible] the maximum reward. And this is the model we want know, we'll go out to know what is the best award and from what, from bottom to top. And we want to know, uh, what is the, the shape of the path, which achieves this maximum best reward and notice because the, because the rewards are random, the total reward, the total Alaskan award is also random. And also the, the best path is random. And so here you can see the end of the one-third fluctuations of what it means in this context is that the maximum reward, which is random, it's going to fluctuate around some mean value by end of the one thing, because it's not your life. Yeah. So this is sorry to make sure I didn't miss it.Hegde:So, and in fact, so you can have different types of randomness at these intersection reward founds, uh, and it's known for exactly two types of randomness that it should be under the one-third fluctuation. So it's known for two cases of the randomness at the intersection values that the rent for the one third, but you expect with basically any can randomness, not just these two, you have the same order fluctuations. Um, and so we, as you, we have that order of fluctuations and we don't assume anything about the nature of the randomness at the vertices. And now you can, you can now hope to move beyond the, the exactly solvable the models by looking at the paths, the paths you see, they care less about in some sense, very vaguely. They care less about the details of the manuals and because they are the best, perhaps they have certain properties, which you can then try to analyze using those properties and the idea that anything you derive with your assumptions and the, and, uh, understanding the past will apply to any other model where you can verify the assumptions. And so this is kind of trying to get, get past formulas because the paths are kind of geometric objects, which, which they're, they're in a formula, but they should be, uh, more G their behavior should be more generally, uh, consistent for different types of randomness.Saintsing:So you are kind of generating a bunch of possibilities based on this model that, you know, and then you can, I guess, have this kind of like a big matrix or something, and then you can look at it with other models and kind of variable. Yeah.Hegde:Yeah. So in this case, the other model would just mean different distributions of randomness at the, at the intersections.Saintsing:Right. And so you would just kind of keep verifying, like, if, if at any point this matrix falls apart, then you would say, well, this assumption or this, I guess the, whatever you were trying to prove with that particular thing would say, that's not pushing us towards the universality. Right, right,Hegde:Right. Yes. That's a very difficult job, which is completely out of, out of, you know, it's beyond army Trek now is to verify that assumptions that I mentioned actually hold for other federal randoms. So we're sidestepping that very difficult issue and saying, okay, let's suppose we knew that then let's use, uh, uh, let's, let's, let's, let's make arguments about the maximizing paths, which are more easy to understand. And, and we'll see what we can get from the next month. YouSaintsing:Need to have like a pretty high throughput computer, basically, to do a lot of the research you need to do.Hegde:So we don't, we don't actually do simulations because I don't know why actually. I mean, it can be useful. So I sometimes do simulations, but my computer is not balanced, crazy fast computer. So usually it just takes a long time, but typically, typically Matthews does not in most innovations. So really it's just a question of paper and,Saintsing:And they're like solving by hand a lot of the time, like, yeah,Hegde:It's all, it's all by hand,Saintsing:Like research time, like when you do research, you're just like sitting at a desk, like writing out equations patients,Hegde:You were going to approve something. And so you'll have, you know, kind of vague ideas of how you might try to prove it. And so you'll, you'll, you'll, you'll roll some picture of, see what you're saying makes juristic sense for maybe you'll actually try to write down as of the proof, and then it might be an equation. It might be, might be just kind of some written like a verbal argument. Uh, you'll get stuck somewhere. And then you think about how you're going to get past that, which maybe you'll just sit there and look at the ceiling, or maybe you'll go for a walk over for walks is quite helpful. Um, the picture people have of mattresses is not that accurate. It's not inaccurate. It really is a lot of just sitting and thinking about stuff or going for a walk and thinking about stuff or reading papers to see what other people, what other people have tried. But it really is a very like low, low electricity job. Yeah. Um, and I should say that, you know, this is something that if it's all not just telling you, you'll be sitting at your desk and you'll think you have an idea currently very often, um, and what you do at that point, cause you go for a walk because you know, you've got to, you've got to appreciate, you're going to enjoy the feeling of having an idea before you try to write it down and see that it's wrong.Hegde:So you really have to go, you know, it usually your idea's wrong, it's wrong. And so you've got to really take advantage of the time when you do it to get an idea before he's musical 50 again.Saintsing:Right. So there are there conferences like academic, math conferences, what is a, what's a math talk? Like, like what does somebody do in a math talk?Hegde:So I have heard from my friends at other other disciplines that math talks kind of break all the rules that they've heard of about how to give a talk. Um, there were basically no pictures for the most part, maybe a few bit depend on depending on the topic, but often the no pictures. Um, and it's usually just like a bunch of bullet points of text explaining the model, explaining the argument, uh, maybe some equations to explain right there. I don't what the actual proof is, but typically, yeah. So it's one person, a slide show and the first two slides within minor, again, it's just to set up the model and then you set up the question, maybe explain what people have done in previous work. And then you're trying to give an idea of the proofs, but it's very different from front that talks to my friends that I've gone to that and I'll put a disciplines and it's way more way more just like sentences on the, on the slides.Saintsing:Well, so now you're all you're moving on after grad school, you're going to be done with grad school. So like what, uh, are you going to go into an academic career?Hegde:Yes, I'm going to, I'm going to try starting a post-doc um, after this. So, uh, so actually Berkeley has this place called MSRA, uh, mathematical sciences research Institute, which is right next to an LBNL on the Hills. And so, uh, NSRA has a, has a program in math every semester, uh, and I, a different area of map. And what happens is researchers from around the world who work in that area, come to MSRR and think about stuff together. And so next semester, the program and MSRA is in my particular area of math. And so I'm going to be a post-doc there for six months because everybody's going to be there, the works in migraine. Um, and then after that I'll be at Columbia. SoSaintsing:Like just a lot of time, everybody's like Sharon whiteboards talking about where is that? Like, kind of like just math, like that's, that's what you envisioned for your career. I mean, there'll be times when it's like you writing on a desk on times writing on the whiteboard, but that's basically yeah.Hegde:Yes. That's about the, that's about the span of it. Nice that as you expand beyond, you know, pure math, but within, within your math, that's definitely the majority of it.Saintsing:Yeah. If you love math, like that sounds like,Hegde:Yeah, exactly. If you're really okay with, before the pandemic started, I would just be sitting in my office the entire day. So this is not that typical, but my two opposites did not come to my office very often. So often I'm just sitting there for the entire day working on something and still independent. I started, it was the same thing, except I do it at home. Now you have to like math to be able to do it, I guess.Saintsing:Yeah. You talk about people needing to love math. How often in your life do you encounter people who just like don't like math and then I guess I'm interested. Cause I always hear people talk about, you know, not being able to do math, which is, I definitely get people not liking math, but the idea that people can't do, math always seems so odd to me. Cause it's like, I mean, you know, you can write down stuff on a paper. Like what do you, what are your thoughts about, um, that, uh, mindset?Hegde:Yeah, definitely. I think they, you know, they don't like mentally, they can't do math and stolen. I personally often think that it's, it's, it's definitely to do with the fact that the math we're taught as kids or students is not that interesting. At least that's my view, mainly because I was going to say before, it's, it's rather wrote, you know, um, you have to figure out the area of triangles, whatever, where you have to compute some integral, you don't really, you don't really see that there's an idea involved in these competitions. So there's this, I think this is a little bit that does it, that there's this essay [inaudible] traditional, which I think is controversial among metrics and by what did they do with the conviction or not, but it's called math, which is a moment where he just lemons that math is taught, not like art, basically.Hegde:So an art, you know, people do, you know, when the kids did, they were kind of awkward, just kind of free paint with their fingers and whatever. Um, it's very fun. There's no real holes to it. Um, and maybe later they, they they're, they're taught some skills, you know, how to, how to paint their evil. Excellent. You have to do about like drawing circles and so on about, about what throughout the thing, you were aware that there are these great works of art around, um, which, which you can look at and you can appreciate if you think, okay, this is what I want to do. This is a step towards that in math, you know, there are also these great works of art that they're doing these kinds of theorems in that there's complicated theorems, but kids don't know that kids are just told, okay, just multiply, learn these notification tables or, you know, do these types of [inaudible] a triangle, like which numbers pick or things like this. And there's no real motivation. Like why would you, why would you care about this in the first place? Nobody knows. I think one reason people don't like math is because this is the impression they have of that. There's no real ideas involved. There's no larger story about what's uh, why people came to these questions. Why do I want to answer them? What's interesting about them. Yeah.Saintsing:That's so, yeah. That's interesting. Yeah, because like, when you talk about art, um, I get that, you know, kids can, it's kind of like a space for exploring and in math, I, you know, that there should be space for exploring, I guess, but also math is, you know, I'm kind of also thinking about it in terms of like learning English or learning your native language in school, you know, you have to like learn these building blocks, like system. And so math has that need for that system. And I don't know. Yeah. It's funny. Cause then, you know, learning the times tables, like, I don't know, is that something that's necessary,Hegde:But I guess that's the issue. If you do need these skills later, you know, to do a map later, but you know, for example, you know, spelling is annoying, grammar, things are annoying, but you read books and you, and you think, oh, well this is, this is what I've gotten out of that stuff that I can appreciate, you know, stories in books, um, which again is not really available. And if, if you're doing math basically until undergrads, I'd say, um, maybe, maybe [inaudible] or something, but for the most part until then, it's really kind of mysterious why this is interesting. Right.Saintsing:I got you. So it's like, if you were learning English, but they just gave you like words and they never actually gave you a book. Yeah, yeah. That does make it sound bad.Hegde:Well, so my personal view is that if it was taught differently, maybe, you know, there are certain skills as you pointed out about modifications that what you do have to do, uh, and maybe, you know, it's hard to, same as everybody can actually enjoy learning that study uniform stuff later. But certainly I think you build a belt better appreciate why this is interesting if it was taught at different times, but that's hard to teach. It's my head start to say as an easy thing to do, unfortunately,Saintsing:We've run out of time for the interview. It has been so much fun talking. Today we've been speaking with Milan Heggerty from the department of mathematics about, um, his research on stochastic growth models. Thanks so much for being on the show, Milind. Hegde:Yeah. Thanks for having meSaintsing:Tune in, in two weeks for the next episode of The Graduates.