Gresham College Lectures

The Mathematical Vision of Maryam Mirzakhani

June 06, 2023 Gresham College
Gresham College Lectures
The Mathematical Vision of Maryam Mirzakhani
Show Notes Transcript

In partnership with the London Mathematical Society.

The first female Fields Medalist Maryam Mirzakhani, left an astonishing mathematical legacy at her untimely death in 2017. This talk will explain the lasting contributions of her work to our understanding of the world, and give a glimpse into Professor Mirzakhani's imaginative and hands-on approach to mathematics. 

This lecture will be delivered by Professor Holly Krieger who is the Corfield Lecturer in Mathematics and the Corfield Fellow at Murray Edwards College, University of Cambridge.


A lecture by Holly Krieger recorded on 24 May 2023 at Barnard's Inn Hall, London.

The transcript and downloadable versions of the lecture are available from the Gresham College website: https://www.gresham.ac.uk/watch-now/lms-2023

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Before I begin, I do want to, uh, give a little bit of thanks to some of my mathematical colleagues who, who directly or indirectly helped me prepare this talk. Um, the first is my two PhD advisors, my former PhD advisors whose names will come up in this talk. Um, Rammy, Talu Besh, and Laura DeMarco, uh, who are both good friends of Mary's and who have, um, communicated to me over the past couple of decades the experience of working with her and knowing her. Um, the others are through their writings. Um, Anton so, and Kurt McMullen and Alex Wright, among many others have written very beautiful pieces, um, on Miriam's work that, uh, perhaps are not as accessible as I hope that this lecture is <laugh>, but inspired a lot of, um, the directions that I decided to take when I was deciding what to tell you today. Okay. Alright. So, oh, and of course, thank you to Gresham College for hosting me and the London Mathematical Society as well. Alright, so we are here today to discuss the work of one of mathematics brightest modern stars, Maryam Meshk. Uh, Maryam was an Iranian mathematician whose deep and creative mathematics received the highest accolades in our field. Um, and a warm and enthusiastic woman whose life and work were cut short at just the age of 40. Uh, when she's come to breast cancer, what I would like to do today is to communicate to you some of Miriam's work. Um, really what I'd like to share with you all is the appreciation that I feel <laugh> when I read Miriam's work, and to give you a sense of the excitement and the enthusiasm that she felt while she was working. So to pass on to a, a little bit of her mathematical legacy as best I can. Um, so Miriam's work considered the connection between two subfields of mathematics. The first is geometry, which is familiar to us basically from nursery <laugh>, right? So even in, in nursery, we learn things like circle or triangle, these geometric objects, but we also learn some, some more, uh, some deeper notions in geometry, like bigger and smaller or, um, sort of <laugh>. So these are the geometric ideas that go into this field. And geometry, of course has innumerable applications outside of mathematics, right? Physics, art, architecture, biology, and so on. Um, but it also connects to many other subfields inside of mathematics, like number theory, the study of the whole numbers and prime numbers, um, or algebra. So the second subfield that, uh, involved, that Miriam's work involved is the field of dynamics, um, which is roughly speaking, it's the study of, uh, a system which evolves according to some fixed rules, usually simple rules to describe. So what you should have in mind here, for example, is sort of the basic laws of motion des uh, describing the movement of planets in the solar system, right? This is a type of dynamical system described by somewhat simple rules in this talk, our simple dynamical systems. Um, however, there will also be more complicated dynamical systems like walk from a mathematical object, you know, to one that has similar mathematical properties. Okay? So Miriam's work focused on a family of mathematical objects known as reman surfaces, and she used dynamics to understand the surfaces themselves as well as some more sort of metadata, things like understanding the collection of all such surfaces. So what we're going to do is we're gonna take a tour of two of her most important contributions in that direction. But I actually wanna start with a quick warmup problem that she herself considered in her teenage years. Um, mostly it's interesting in and of itself of course, but I also want you to sort of see how even though her career was cut somehow, sorry, what do I wanna say? I wanna say the warmup problem will help you sort of understand how mathematics was treated in her work as a, an experimental and, and visual science, uh, which is perhaps not how we think of mathematics. Okay? Um, Maryam was born in Tehran in 1977, which was just before the cultural revolution. Um, this was fortunate in a sense that there was enough sort of educational stability by the time she left primary school that a lot of educational opportunities were available to her. Um, she attended a very high quality and prestigious school for girls in Tehran. Um, and by, by the seventh grade, uh, it was clear that she, she had really a standout talent in mathematics. Um, with the support of her school principal, she and her friend Roy Beheshti, um, were the first women permitted to participate in the, the preparatory national mathematical competition and, and join the national team for what's called the International Math Olympiad. Okay? So the Immo, the International Mathematical Olympiad is, is a competition held every year, uh, that is a competition in problem solving for pre university students, and it really draws immense mathematical talent from around the world, <laugh> countries around the worlds. Um, and it is somehow famously difficult <laugh> despite drawing together some of the best trained and most talented mathematicians of their age. Um, well, you can see I have <laugh> the points distribution from zero to perfect score here is not looking so good at the high end <laugh> for the the 1995 imo, for example. And so doing very well. So achieving a gold medal here at the imo or even better obtaining a perfect score is something which is very rare. Um, here's the picture of the 95 team. You can tell which one she is <laugh>, um, and scored a rather astonishing two gold medals, uh, one point off of a perfect score in her first performance and a perfect score in her second. Um, and I think it's also worth knowing that, uh, being a woman accomplished in mathematics and science in Iran at the time itself is not so unusual. Um, while gender segregation was absolutely a feature of education there, um, there were quite a few opportunities still available to women working in science at the time. Um, and as Maryam herself mentioned in an interview in 2008, um, she, she had to explain while she was living in the US quite a few times that she was in fact allowed to attend. Well, she ni rather humbly says a university. In fact, it was the best university <laugh>, uh, in Iran. Alright, so I'd like to start with this. This warmup example comes from a summer course that Maryam attended as a teenager, was held at the university for gifted math students. So this is, uh, a, a memory from Ramin my, one of my two PhD advisors that I mentioned in a very nice collection of memories and the notices of the American mathematical Society in, in memory of Miriam's, um, life and work. And, uh, what he mentions is that in this summer course, one of the topics that the professor, um, had talked about was about decomposing graphs. And it was, it was considered the first difficult case of a general problem. And he had asked the students to find examples where this was possible, this mathematical question could be answered, offering a dollar for each new example. And by the time of the next lecture, Maryam had in fact not just found some examples, but found an infinite family of examples along with ruling out infinite families of examples. So necessary and sufficient conditions for decompos ability. Uh, I didn't choose to include it in the zoom because I was sort of running out of work, but Ramin includes the joke that Maryam was obviously smart but not smart enough to only dole out one example a day. Alright, so I'd like to discuss this problem a little bit because it gives us a wonderful glimpse into the sense of exploration, um, and persistence also that Maryam, um, indeed really any mathematician must use to approach a new problem. So graphs, it was a question about graphs. So graphs of course are just visual tools that mathematicals mathematical mathematicians use to model relationships. Okay? So, um, graphs are models for a huge array of relations and processes. So say network analysis and neural networks are certainly popular applications these days. Um, there are applications in biology, I have here a network of protein, protein interactions you can see in the middle, um, as well as um, say computational neuroscience or social science and so on. Um, but all of these fields are sort of using graphs to, to encompass a simple idea <laugh>, which is that it's a visual mathematical encoding of the relationship between objects. So here is a hopefully friendly example and hopefully familiar example. Um, you might have thought I should have chosen group B for my example given the nature of the talk. But I am American and I didn't wanna escalate any tension, so I <laugh> chose groupy. Um, so the round robin stage, the group stage of the world cup is something that we can represent with a particular type of graph. So here I've got Groupy with Spain, Costa Rica, Germany, and Japan. An edge of the graph. Okay? So this is our visual encoding of the relationship between objects, right? Two teams are playing a match if a line connects them. Um, so here I think of each team as somehow a point in my graph. These points are known as vertices in graph theory and the lines between them are called edges. Alright? So this property, so what is around Robin, around Robin is a, is a type of competition where each team plays the other exactly one time, right? What that means for my graph is that each vertex is connected to the other by exactly one edge, okay? This property for a graph is what it means for a graph to be complete, okay? A graph is complete if each vert, each pair of vertices is connected by exactly one edge. And in graph theory. So putting this particular graph aside for a moment, one of the ways that we study graphs and try to understand them is by breaking them into smaller pieces, okay? This is totally a, a universal strategy in mathematics, right? We study whole numbers by breaking them into their factors, particularly their prime factors, right? We will see more examples of decomposition later, but in graph theory, what you might hope to do is to say take your edge set and decompose it into some simpler collection of edge sets. All right? So decomposing a graph can help understand the structure and the properties of the graph. So here, for example, I have my group stage round robin and each of the eight groups, actually of course we know <laugh>, the graph should look exactly the same for each of the eight groups, which is that each team plays the other. That is a complete graph on four vertices and sorry, and this decomposition can help us if for example, we want to understand, I'm not sure how good my colors look from different angles of the room, but for example, if we wanna go ahead and schedule the matches, right? And we need them subject to the, to the restriction that no team plays two matches on the same day, right? So what I'm gonna do in order to do that is I'm gonna color the edges of my graph where each color represents a day for the match. And so if I can color my graph in such a way that no vertex has two attached edges of the same color, then that means no team is playing two matches on the same day, right? And this decomposition, although it's sort of simple, especially since I started with a graphic that looks very simple, <laugh>, this decomposition allows me to promote this sort of very basic way of achieving that on this small graph to the larger graph that it, it, um, is part of the decomposition for, okay? So that's just a, uh, a sort of basic example, but it gives us a hint of the utility at least of graph decomposition. Okay? So what was the actual question that Maryam considered? So what was the actual question that Maryam considered? So the actual question, in fact I will just read the title of the resulting paper and then explain the words one by one <laugh>. So the actual question, in fact I will just read the title of the resulting paper and then explain the words one by one <laugh>. So was about decomposition of complete tripartite graphs into five cycles, okay? We've already talked about what decomposition is. So what is a complete tripartite graph? Why do I say it that way? Well, complete tripartite is slightly different. So this will take a little bit of explanation. A tripartite graph is one whose furies, whose points can be grouped into three groups, okay? I have indicated that hereby color, which I'm hoping these colors are sort of different enough from each angle that <laugh> most everyone can see them. Um, so the vertices are grouped into three separate groups and the edges of the graph are completely determined by the, the vertex colors, which is to say that two vertices have an edge between them exactly when they are different colors. That's all. Okay? So here I've got three groups of vertices. And I don't have any loops, right? Nothing is attached to something of the same color. I just have one edge between every pair of vertices, okay? This complete tripartite graph, I've got two blue vertices and one each of green and red. And so I have all the possible edges there except one connecting the two blue dots, okay? And so on. And if I've done my color incorrectly <laugh>, you'll see that that's actually exactly what I've got. This is a table here of a lot of complete tripartite grafts with small numbers of vertices. Okay? So that's what it means for graph to be complete and tripartite. Um, you can imagine this being useful by the way for modeling like a causal kind of relationship if they're sort of A and then B and then C <laugh>, um, these tripartite graphs can come in handy. Alright? So what does it mean to be a five cycle then? Well, it's exactly somehow what it sounds like. If you start at a vertex and you travel five edges and you end up back where you started, you have formed a five cycle. So here's a three cycle, a four cycle, and a five cycle. And the question that was asked was, is it possible or can you find examples because I'm sure he already knew it was possible. Uh, of tri part, complete tripartite graphs which can be decomposed into five cycles. Okay? So let's stare at this table for a second. I mean, just to take a very naive try, <laugh>, I definitely cannot decompose this first one into a five cycle, right? I've only got three edges, after all I need a five cycle and I can't reuse an edge for a decomposition. Okay? Generalizing that idea. It doesn't take you long or much experimenting at all to see, especially when you have this picture in front of you that in fact we need the total number of edges to be divisible by five in order to have a five cycle decomposition. Okay? So this is what's called by the way, a necessary criterion mathematically, right? It's necessary for a graph to have edge number divisible by five if we have any hope of having a five cycle decomposition. So here are the two graphs in my table which have edges, which I, which if I count them, uh, there's a total which is divisible by five. Okay? This graph has five edges, this graph has 15 edges, okay? Okay? Alright, so the question is, here's at least two candidates for graphs which might have a five cycle decomposition. We know none of the rest can because they don't have the right number of edges. And so if you try and think about is it possible to have five cycle decomposition say of this first graph? Okay, I know it doesn't look like a cycle, but maybe we could kind of like weave around <laugh> and then back where we started. If we start say at the green vertex and we travel around, it doesn't take long to see that we're going to run into trouble <laugh>, right? So if we try and cook up a cycle, we're going to end up back at the green vertex eventually, and there's only three edges coming out of the green. So once we leave again, we're never allowed to return, right? You can try starting at a blue, but actually the red and green vertices are still gonna obstruct what you're trying to do because when a cycle comes into a vertex, it has to leave it again. Another way of saying that is that in order for a graft to be decomposed into five cycles, one of these complete tripartite graphs, each vertex has to have an even number of edges connected to it, right? There's another necessary criterion for you. This graph clearly fails cuz the green and the red have three edges connected. You can check that this graph succeeds, it does satisfy that criterion. We've got I guess six edges up here connected to the red vertex and four for each of the others. In fact, you can, if you are persistent and do enough experiments, cook up a five cycle decomposition of this graph. Alright. So these kinds of experiments and deductions allowed Miriam to make progress as a teenager overnight or maybe two nights. I don't know how often the lecture met <laugh> to construct these infinitely many fam or sorry, these infinite families of examples for this decomposition. And so while this problem is in some ways elementary and, and very different from her, her post university mathematical work, um, notice how different this is than the kind of mathematics you encounter in school, right? There's no algorithm that tells you what to do. It's not like when you learned how to differentiate and you learned a rule and then you applied at a hundred different times.<laugh>, right? This is very little memorization. There's no technique. You already know how to solve the problem. There's persistence to draw <laugh> some of these crafts in the first place, right? And to run these experiments to try and figure out what the obstructions, what the necessary and sufficient conditions might be. Um, but there's also a sense of play and visualization and experimentation. And as a professional math mathematician, Myam had a an extremely impressive technical ability that was, you know, thanks to her tenacious work and immense talent. But she also had a very strong sense of the visual and the experimental. And this is really a, an amazing sort of, uh, <laugh>. It's amazing that we can already see that in her work as a teenager. Alright, are you all warmed up?<laugh>? Because it's gonna, alright, so Miriam's work after university was focused on, um, remounts surfaces. So surfaces are geometric objects and what defines a remount surface is that if you look closely, if you put a remount surface under the microscope, then it really looks like the plane that we're all used to. In fact, there's something a little stronger. That's true. So here I have a picture by the way of the earth's surface, which is certainly a reman surface. So keep in mind I'm taking like just the surface of the earth is sort of hollow earth is a reman surface. If I zoom in, if I take a map, which really zoomed, probably I wanna do even better than this, this zoom in on London or zoom in on this block <laugh>, then it's something which really just looks like sort of the flat normal Euclid plan that we're used to working with. But remont surfaces have an additional feature, which is that, um, there's a type of independence about whose map you're looking at. Okay? So do you know these like tourist maps of London that are sort of like cartoons and the, the, the iconic things are really large and the streets don't match up to reality, anything like that? So do you know these like tourist maps of London that are sort of like cartoons and the, the, the iconic things are really large and the streets don't match up to reality, anything like that?<laugh>, something like that where if I took my surface and I have these little patches that look like reality, but they don't quite match up like that, that's not a reman surface, okay?<laugh>, something like that where if I took my surface and I have these little patches that look like reality, but they don't quite match up like that, that's not a reman surface, okay? A reman surface is one that not only has these little patches that look like reality, but they all join up in some kind of sensible way. A reman surface is one that not only has these little patches that look like reality, but they all join up in some kind of sensible way. Okay? Okay? Alright. Alright. So remont surfaces, if all you're interested in is shape, right? So remont surfaces, if all you're interested in is shape, right? So if you allow yourself to stretch things and, and pull things and skew things, um, if they don't have any boundaries, right? So if you allow yourself to stretch things and, and pull things and skew things, um, if they don't have any boundaries, right? So I know that feels weird because we're used to thinking of earth as a three-dimensional object, but again, we're only thinking about the surface of the earth here. So I know that feels weird because we're used to thinking of earth as a three-dimensional object, but again, we're only thinking about the surface of the earth here. There's no boundary to, to really talk about. There's no boundary to, to really talk about. Then they really only come in certain types. Then they really only come in certain types. They're essentially totally determined in terms of shape by how many handles they have, how many holes they have inside them. They're essentially totally determined in terms of shape by how many handles they have, how many holes they have inside them. So here's one with no holes, here's a donut or Taurus with one hole, here's two holes and so on. So here's one with no holes, here's a donut or Taurus with one hole, here's two holes and so on. Okay? Okay? This number, number of holes is known as the genus of the surface holes or handles however you prefer. So if you have <laugh>, I don't know if anyone other than mathematicians has heard this joke, but if you've heard the joke about a topologist not knowing a coffee cup from a donut <laugh>, I mean I guess yeah, this is what they mean. That's what both of those things are, right? If you are a topologist, if all you care about is shape, then all that matters is the number, number of handles that a Vermont surface has. Okay? However, we don't just care about shape, we care about geometry and reman surfaces do come with notions of geometry. And when I say geometry, what I mean is a mathematical description of how to measure ideas of distance on the surface or length, um, angles or curvature. Um, all of this kind of mathematical description in a way that is intrinsic to the surface. Okay? So <laugh>, here's where it gets really tricky. What does intrinsic to the surface mean? These are very nice pictures, however, they are wildly misleading when it comes to the geometry of the surface in all but one of the cases, okay, so for this sphere, we see this sphere here as I've got it in the picture, as you think about it, sort of sitting in three dimensional space, that picture is a very good representation of what the geometry that we can put on this sphere looks like. Okay? If I wanna measure the length from one point on this sphere to another one, all I need is a good bit of string <laugh>, right?<laugh>, right? The curvature of the sphere sort of curves outwards. That's real <laugh>, that's an accurate description of the geometry of the sphere, okay? However, that is not true for these other figures. So for anything of genus higher than zero, the picture is misleading when it comes to ideas like distance or curvature. Okay? Curvature in particular is the one I wanna focus on. So this fear has positive curvature and sort of bulges outwards, okay? Instead of being flat or bulging inwards. One way that mathematicians characterize this that you might have encountered if you, if you learn about these non Euclid geometries is that if you draw triangles right, if you wrap rubber bands around your sphere and make a triangle out of them, the inside angles of the triangle will add up to more than 180 degrees, okay? That's a characterization of having this kind of curvature, positive curvature. This Taurus, this donut, it looks like in some places it has positive curvature. In some places it has neg negative curvature may be flat somewhere in between. That is not true <laugh>, this is not a good representation of the actual curvature of the Taurus. The Taurus is actually flat. If you draw lines, straight lines on the Taurus and you measure angles inside of a triangle, they will add up to 180 just like it would if I drew it on this floor. Okay? We will understand a little bit more why that's true later in the talk, okay? The hardest to understand perhaps is the case of genus 2 34 and so on. Okay? When our genus is at least two, our surface turns out to be negatively curved. So the geometry we can put, the types of geometries we can put on the surface, um, are all negatively curved. So people often describe this as sort of a, a saddle picture because that helps you sort of visualize what the geometry might look like. In particular triangles have angles that add up to less than 180 in one of these negatively curved spaces. That's true everywhere on this surface of genus two or more. Okay? So the geometries we put on this surface, it's just this abstract description of how to measure distance, length and so on. But the curvature for example, has these features which are not well represented by these pictures, okay? So you might reasonably ask why did I not draw you better pictures?<laugh>, um, these pictures do an okay job of some things. Um, so for example, these pictures don't do a terrible job <laugh> of representing distance, okay? They do a very bad job of representing curvature. They don't do a terrible job of representing distance. Um, and if all this makes your head hurt, I mean sort of rest assured that that's also true for professional mathematicians <laugh>. In fact, there's basically an entire subfield of geometry is about good models for these kinds of remodel surfaces. However, it is in fact Ethereum, it is known that you cannot come up with a nice good model that actually represents the geometry and then put it into our Euclidean three space. And that's why I can't draw you better pictures here, okay? Alright. All right. So each type of visualization has its strengths and weaknesses. We'll see a couple more throughout the course of the talk, okay? So not only do reman surfaces admit geometries, these notions of distance or length, but reman surfaces in genus greater than zero can admit lots of different geometries, in fact infinitely many different geometries. So for example, here's a genus two surface and maybe in some geometries the handle is very skinny and in some geometries the handle is very fat. Okay? So there are different types of geometries that I can put on the same surface. And so how can we possibly understand them, right? We wanna classify objects as mathematicians. How can we understand all these different geometries that we could possibly put on these surfaces? Well, the answer is the same as it was for understanding hole numbers and the same as it was for understanding grass, which is the idea of decomposition, right? And if I told you I'd like you to take this Gina's two surface and give me some easier to understand surfaces, what you might do is grab the nearest pair of scissors <laugh> and cut this surface in half, right? Let's cut it into two genus, one pieces. You would be exactly on the right track, okay? That is a very intelligent way to decompose my surface. However, <laugh>, you have just introduced a boundary to my surface, right? So all of a sudden I have this curve, which is a boundary for my surface <laugh>, please forgive my mess here. Remember these things are hollow, okay? So we would sort of see the back of the interior of the other surface, which I am not capable of drawing, okay? But, so the point is if we do this kind of decomposition, then we introduce a boundary to our reman surface. And so if we're gonna use this technique to understand reman surfaces, really we need to understand reman surfaces where we allow these kinds of boundaries, okay? And so this is a lot of what Miriam's work focused on in the case of these hyperbolic geometry. So these surfaces of genus at least two. Um, so it turns out not to be a problem to allow us to consider boundary curves on these surfaces. In fact, all of these surfaces are built out of a particular subsurface known as a pair of pants <laugh>. But by the way, I, I asked, I often ask my students about, um, like Americanisms and if they will embarrass me in front of <laugh> crowds here. Um, I asked my students about this and because of course I, I know the, the difference in American English and British English, English for pants. And um, they told me first of all that it would be fine, but second of all that, uh, they had an anecdote of a Cambridge professor. I don't know if this is real, I don't know if even one of my colleagues who's watching this online or something <laugh>, but of a Cambridge professor who was so unwilling to say pair of pants while he was teaching the geometry class that he called them T shapes <laugh>. I don't know if that's like t for trousers or tea cuz I, it does not look like a tea to me. I guess you could sort of topologically think of it as anyways. Um, but I found that a much more embarrassing idea than just saying pair of pants <laugh>. So there you go. So there you go. Um, alright, so these surfaces are built out of pairs of pants and um, there's, right, so right, I did wanna make the, the point that if you were sort of topologically inclined, you could imagine sort of inflating one of the legs of these and deflating another leg and see that what you actually have here is a sphere with three little discs missing, okay? Um, alright, so these surfaces are built out of pairs of pants and um, there's, right, so right, I did wanna make the, the point that if you were sort of topologically inclined, you could imagine sort of inflating one of the legs of these and deflating another leg and see that what you actually have here is a sphere with three little discs missing, okay? Or three punctures as we might call it. Okay? So it turns out that these are good choices for building blocks for the collection of all reman surfaces, okay? They can always be decomposed into pairs of pants like this. But here's the thing, if we want this decomposition to remember to respect the geometry that we put on the surface, we can't cut just anywhere, okay? We need to make good choices of cuts. We need to make these sort of straight line choices of cuts so that everything sort of matches back up when we try and glue the surface back together from its decomposition, okay? And these straight line cuts are what are known as geodesics. So an analog of straight lines that we're used to in our Euclidean world. Alright? So what is a geodesic in particular closed geodesics are what we're gonna be talking about mostly in this talk. So a geodesic is a walk that you take on the surface with no acceleration, okay? No turns, no speeding up, no slowing down, and it's closed if you end up back where you started. Okay? So what do I mean by no acceleration? If I asked you to stand up and walk in a straight line, you would not walk off into space off the edge of the earths <laugh>, right? Despite the fact that the earth is curved, you would walk along the surface of the earth, right? It's a straight line with respect to the earth's surface. And similarly, even in hi genus, there's a notion of walk in a straight line with respect to the surface, okay? So even though it might not look like a straight line to us from the surfaces perspective, it's a straight line. So these are also sometimes called sort of length minimizing pass. Um, that's true in a slightly tricky technical sense if you sort of infinitely zoom, okay? So geodesics on a sphere, for example, the closed geodesics on a sphere are the great circles OneSphere. So these lines of, of longitude, for example, on a Taurus, things get a bit more interesting. So there's this beautiful sort of freely available movie of a lady bird following a geodesic on a Taurus. So this is a closed geodesic on a Taurus. So let me start the movie. So here we see our donut, our Taurus and the lady bird will appear and choose a direction and then walk along the geodesic, walk the straight line in that direction. And as you can see, if, if we we're just sort of thinking in our Euclid space, you might expect her to just sort of cut off the top of the, the donut, right? But that's not a straight line from the geometry's perspective, what she's doing is a straight line from the geometry's perspective. So here you see that closes up. So that's an example of a closed geodesic on the Taurus. Okay? So already you can see that from our perspective, these might be a little tricky to describe <laugh>. Okay? Alright. So understanding a surface is really somehow tightly bound because it's under understanding a surface comes down so much to understanding these pair of pants, decompositions, which have to be made along, cuts from closed geodesics is really tightly bound to understanding its closed geodesics. Um, um, so Miriam's PhD work focused on precisely describing these things essentially, um, from various sort of angles. Um, so she uh, completed her PhD at Harvard, by the way, in 2004 under the supervision of Kurt McMullen. And, um, I, I sort of already highlighted this accomplishment and as a teenager, uh, she accomplished something which any professional mathematician would tell you is much e even harder than two gold medals at the Immo. And a perfect score, which is that her thesis was comprised of essentially three papers that went in the top three journals, international journals of mathematics. Um, I will not tell you like how that compares, for example, to my own thesis work <laugh>. Um, very impressive indeed. Um, but so what did Maryam discover about these geodesics? So I'm gonna focus on, as you might see from sort of the words involved in these titles. I'm gonna focus on one of these three topics because it's somehow the easiest to get a feel for the, the, the friendliest to describe. So let me first start by analogy. I wanna explain to you how one might go about counting an infinite set, okay? And if you're familiar with this notion of sort of accountability and things like that, that's not what I'm talking about <laugh>. Um, what I mean is, is counting a set in a way where we decompose it into sort of finite pieces. Okay? So here's a set of infinite things in mathematics, namely the set of prime numbers, okay? There are infinitely many prime numbers, however we can count prime numbers. That is they become a finite set if we impose a length bound. So if for example, I'm only interested in the prime numbers with a hundred digits or less or fewer <laugh> at most a hundred digits, then I can actually write down and count how many there are, right? So here I've <laugh> hopefully written down correct counts for pri the how many primes there are of 1, 2, 3, and four digits. In fact, it's well known now in a statement known as the prime number theorem, exactly how we should expect the number of primes of length of say l digits to grow. Okay? So if we impose a length bound, which unfortunately mathematicians don't love base 10, this is really in base E So I'm, I'm I'm being a little bit unfair, but I hope you'll forgive me for the sake of exposition. If you impose a length bound, like I only want my prime number to have at most so many digits, then as that so many increases, we have a good description of how many prime numbers there are, okay? And in fact that growth is is not quite exponential, okay? So it's almost exponential, but with a, with a moderating factor. And perhaps the most interesting interpretation I should say here is that we know how many numbers there are if we forget about being prime or not, we know how many numbers there are with say a hundred digits or a thousand digits. And one of the consequences of the understanding of this growth pattern is that prime numbers become rarer as numbers become longer. Okay? So this can be a really sort of useful heuristic describing how many elements of an infinite set have bounded length. Okay? So the amazing thing is that it's possible in fact, even though every hyperbolic surface has infinitely many closed geodesics, they can still be counted if we impose a length bound, right? I told you the whole idea for these surfaces that we had a notion of length, okay? And so we can certainly describe the length of a closed geodesic and if we impose a length bond, then we can count them in the same way that we can count prime numbers of maximally say a thousand digits or a hundred thousand digits. Okay? And here's the sort of spectacular phenomenon that we see. There's a type of prime number theorem for geodesics of hyperbolic surfaces. So if we count closed geodesics on a hyperbolic surface and we bound the length by some number L, then we can understand exactly how the number of of closed geodesics on the curve grows. There's an additional assumption here that I'm not retracing a path over and over again. That's what primitive means, okay? And the growth is actually exactly the same as it is in the prime number theorem. Okay? So the surprising thing here perhaps is not that these two growth matches, um, the really shocking thing, or at least what should be the really shocking thing is that this growth rate didn't care what surface I started with. It didn't even care how many handles my surface has. It's totally independent of that information. Okay? So this is a rather strong counting theorem for for closed geodesics, okay? What Maryam considered was a much more difficult, very easy to say out loud um, um, question. So she was interested in counting not all closed geodesics, but rather simple closed geodesics. A geodesic is simple if it never crosses itself. Okay? So here is a simple geodesic on a genus two cx here is a non simple geodesic on a genus two surface, okay? And in a sense of course these are even more, these simple geodesics are even more basic building blocks <laugh>, right? And it turns out that they're much harder to find on hyperbolic surfaces. So let me explain a little bit why this is a tougher problem. So even though it's not a hyperbolic surface, I'm gonna think about the Taurus cuz it's a good example for explaining this. So the Taurus I mentioned before has a flat geometry and here's where it actually comes from. So we can visualize the Taurus as actually a square if we identify parallel edges of the square. So top and bottom left and right, okay? Um, I'm meant to have a piece of paper out to do this.<laugh>, if you have a piece of paper you can imagine taking it and folding it up to identify the left and right sides that would give you a cylinder if you take your cylinder and wrap it around itself. So that's what I'm doing here. Then you actually get this donut a Taurus, okay? So this is where the Taurus geometry comes from. Um, this is sometimes called like a video game geometry because you have this feature where if you somehow hit the left edge, then you merge back from the right edge and so on. Okay? Alright, so the thing is that in this square model, a geodesic really is a straight line path. Okay? So here's that simple geodesic from the last slide. Under the identification of the sides, all these red dots are the same point on the Taurus, okay? And so the description of simple geodesics on the Taurus is exactly the description of straight lines in this square model. So here's a better way to look at it perhaps if you don't wanna think about sort of like this union of segments, then you can think about taking lots of copies of the square, just repeat it over and over and over again. And then this straight line or this sort of pieces of straight line path truly becomes a straight line segment, okay? And if you wanna do this to think about all the possible judaics on the Taurus, then you need to take infinitely many copies of the square. Alright? So using this we can re we can translate our question about counting closed geodesics of the Taurus into counting integer points, right? So going back here, these were the vertices of my infinitely repeated squares integer points inside of a plane. And bounding the length just means that the length of the segment is bounded. So why they live inside some disk in the plane. And so it's the same question as counting integer points, that is to say coordinates with integer X and inter integer y um, inside of the inside of the disk. It's counting simple closed g dess in the Taurus however, is a more complicated question. So you can show that it actually comes down to what's called counting what's called primitive. This is a different primitive, if I'm afraid integer points in the disc in this sense, well, sorry, it's actually the same primitive, but I'm gonna give it a different definition in this sense. Primitive means that when I hit that, when my, I take a segment like this and I hit my integer point, that's the end of my segment. It's the first integer point that I ran into on my segment, okay? Sometimes people call this visible integer points cuz if you were to look in a straight line, there's no other integer points blocking your way. Okay? Alright, so it turns out the second thing is a much harder problem. So counting integer points in a disc is basically the same as computing the area of the disc, right? You just associate say, I have a point here, just associate it with the, the unit square that it's the right upper right corner of okay? That gives a pretty good approximation of counting integer points inside the disc counting primitive integer points inside the disc. This visibility thing, this primitive that I mentioned is really the same statement that the two integers defining the point have no common factors, okay? They're cPrime and the probability that two integers have no common factors is actually a relatively sophisticated piece of mathematics compared to counting computing the area of a disk. Okay? So that's, to give you an idea, of course we're gonna work in a hyperbolic world, but the Taurus really captures a lot of the difficulty that in fact new mathematics has to come in <laugh> in order to translate to the simple closed geodesic counting problem. Okay? Alright, so now we're in a position to understand one of the significant results of Miriam's thesis work. So she provided the first precise counts of the number of simple closed geodesics on a hyperbolic surface. Okay? Um, so by precise I mean that they were known to have this sort of polynomial growth, meaning that it's very rare for a geodesic which is closed on a surface, a primitive closed geodesic to be simple, okay? But she gave precise values for these, these counting and how we should expect, um, this function to grow. Okay? Alright, so notice in particular, by the way, this growth rate is polynomial, this growth rate is basically exponential. This is telling you that the probability of geodesic is simple as somehow decreasing as the length of the geo Jessica gets large. Okay? So more striking than even the result I think was the methodology, which was rather than considering this counting problem on a single surface, which is what her result was about, like take a surface. And here is the counting formula she considered, she solved this problem by considering the same question across all possible geometries at once. Okay? So she did this by sorting the geodesics into groups which are really somehow topological objects which don't really see the geometry. And to give you an idea of what the, what the sort of flavor of her method is, let's go back to counting integer points in the plane, right? One way I could have counted these things is, so I'm gonna just use this, uh, sort of random choice of grouping is to split these points into groups. And if I understand the groups very well and can count those, then I can count the total, right? So here I've split the integer points of the plane into groups according to parody. So if both coordinates are even then the point is colored red, if they're both odd, they're colored blue and if there's one of each, they're colored green, okay? And the key feature here is that this coloring the density of each collection is independent of where in the plane we look. Okay? There's a homogeneity to this kind of coloring. And what that means is that if I take some transformation of the plane, which preserves the coloring of the points, then that transformation will not change the density of say the green points, okay? Or the blue points or the red points. And so all I would need to t to count points inside of say the image of this disc, which is an ellipse under the particular transformation I chose, is to understand how the notion of length changed when I applied that transformation. Okay? Alright. So this is a sort of toy model, although very <laugh> actually pretty reasonably descriptive of the technique that Maryam used in this, in this approach, okay? She implemented this idea by splitting geodesics into groups. So a finite collection of colors if you like, um, known as mapping class orbits. And she showed that in fact they have this kind of homogeneity property that, that it doesn't matter so much where you look to understand the density of each group in the bigger space. Okay? Alright, so here is a fun application. So let's think about a genus two surface. I mean, sorry, when I say ma application, I mean as a mathematician. Here is some more fun maths for me to tell you. Um, so here is a fun application, uh, to, to understand the topological types, the shapes of geodesics that can occur on these different geometries. So geodesics can either cut my surface up or not, right? So in a genus two surface, there's sort of two types of geodesics, the ones that cut it into two genus, one pieces and the ones that don't cut it up, right? Like these yellow curves, if I were to cut along them, I would still just have one piece. The red curve, however, would split me into two pieces. So the one, the yellow ones which don't cut my surface into two pieces are called non separating. And the red one is called separating. It follows from Miriam's work that the probability that a geodesic a simple closed geodesic like this is separating, is independent of the geometry that we put on the surface, okay? In fact, you can write it down <laugh> and it turns out that the probability, so it's an asymptotic result. So you wanna assume your geodesic is kind of long, but the probability that a long, simple closed geodesic is separating is very low. So in particular, it's 48 times more likely to be non separating than separating, okay? Alright? Which is not something people thought you should be able to <laugh> get your hands up. But just to put it in some context, alright, so the technical heart of Miriam's early work, this PhD work was understanding, uh, this space of all possible geometries that you can put on a particular surface. So mathematicians call this a modularized space. So it's an abstract idea, but here's a very concrete example. A modularized space is basically a map, okay? Each point in the space corresponds to a mathematical object instead of a place. So for example, the modularized space of all geometries that we can put on the Taurus looks like this pink region, okay? And maybe over here we have a Taurus with pretty chubby handle <laugh>, and up here we have a Taurus with a skinnier handle, okay? Each point in this region corresponds to a different geometry, uh, more or less that we can put on the Taurus, okay? We study these spaces because they help us understand the objects they contain. Um, they're often interesting themselves, but also they help us understand what does it mean for, for two touri geometries to be like close to each other, right? So to how to deform these objects. And that leads me to talking about the second piece of Miriam's work that I wanted to understand or to communicate to you. Um, so after finishing her PhD at Harvard, Maryam went to Princeton, um, under a clay fellowship and then, uh, accepted a professorship at Stanford University. And in that time she began a, a sort of longstanding and very celebrated collaboration with Alex Eskin. And um, what they were interested in studying was a different type of modularized space, uh, which keeps track of objects known as translation surfaces. And um, what they were interested in studying was a different type of modularized space, uh, which keeps track of objects known as translation surfaces. So these objects in mathematics arise. So these objects in mathematics arise. One of the ways they arise is about thinking about billiard dynamics. One of the ways they arise is about thinking about billiard dynamics. And so I wanna describe a little bit of that to you now, um, and maybe here I should say that I should have mentioned in my thank yous, a thank you to Diana Davis who made my earrings, which are pentagonal billiard dynamics, which we will see in a, in a moment. And so I wanna describe a little bit of that to you now, um, and maybe here I should say that I should have mentioned in my thank yous, a thank you to Diana Davis who made my earrings, which are pentagonal billiard dynamics, which we will see in a, in a moment. Okay? So this study of translation surfaces, which I'll describe to you, I wanna sort of motivate a little bit by talking about billiard dynamics, okay? So you can get some sort of very beautiful mathematical objects right away by thinking about translation surfaces. This one is known as the necklace translation surface, which arrives from the the Pentagon. Alright? So when you play billiards, if you have played billiards or if you have studied say, basic physics and, and thought about movement of light against a mirror, um, you know that when you shoot a ball against an edge of a billiard table, the angle at which it comes back out is a reflection of the angle at which it came in. Okay? And so here I I truly stole this from something called like, like billiards, monthly <laugh> or something like this. Here is an example shot on a billiards table where we have the white ball, it hits an edge, edge and then pocket, okay? And these angles we can describe perfectly, um, using this reflection principle. Alright? So an easier way. So describing this path that the billiard takes is a little bit complicated as is I need to understand these angles. An easier way would be at least if you're one of us, to just reflect the billiards table instead of reflecting the path <laugh>, okay? So if I reflect the billiards table instead of the path, well I get a slightly more complicated billiards table, that's for sure. However, I get a straight line instead of an angle made. And so here I've done it with the original shot, okay? And why not just keep going, right?<laugh>. So I hit the top of the reflected table up here, let's reflect again and we now get a fully straight line path to our pocket, okay? We could reflect again if we wanted, but that would be a little silly. That's just another copy of this surface. Downstairs. More intelligent would be to identify this top edge with this bottom edge just like we did with the square when we were thinking about Tori. Okay? Notice by the way, that clearly whoever drew this illustration was not familiar <laugh> with reflecting things. We do not get a straight line <laugh>. Alright? Okay. So this process is known as unfolding. Okay? So we unfolded the billiards table to get a new shape, um, and identifying parallel edges here we obtain a surface. Okay? Alright, so translation surfaces generalize this idea very broadly. So the amazing thing is that we get objects, we recognize <laugh> when we do this right? So here for example, I have an octagon and a little cartoon of how it glues up if I identify parallel edges, okay? So if I identify top and bottom left and right, I've already done that with a square, right? I know that I get a Taurus, but of course I'm sort of missing the corners of my Taurus. And so there's a missing diamond here. Now those sides are supposed to be glued according to being parallel as well. So if I glue one pair of sides, I get this sort of Taurus with a, with an extra bit and if I glue the final pair, then I end up with an extra handle on my Taurus, which is a genus two surface. Okay? So this probably makes your brain hurt if you have not seen it before <laugh>. However, this cartoon does a reasonably good job of explaining the, the sort of folding and gluing, okay? So we have seen these objects before, however, <laugh> only in shape because translation surfaces are these surfaces, but not equipped with geometries in the sense that I mentioned before. Rather what we're going to do is to define a sort of pseudo geometry on the surface, which everywhere except for the corner points looks just like this flat geometry that came from the polygon, okay? At the corner points we have sort of bad behavior, not in the case of the Taurus, cuz the angles all add up to 360. But in the case of the octagon, these corner points all glue up to a single point and that has angle, if you sort of run around, it has angle, which is eight copies of 135 <laugh>, which I don't do math very well in my head <laugh>, but that's definitely more than 360 <laugh>. Alright? So these translation surfaces are surfaces which have some finite number of bad points and flat geometry everywhere else, okay? And please believe me when I say that they arise in other parts of mathematics than billiard dynamics <laugh>. Alright? So what we're going to think about is just like in the remont surfaces setting, we're gonna think about the space of all possible translation surfaces, okay? And the nice thing is that there's a very simple way to take a translation surface and get a new one. Anything that preserves lines and preserves the property of being parallel <laugh>, we'll take one translation surface and give you a new one. Okay? So any linear transformation as we call them applied to a polygon, will give a new translation surface and that new translation surface will share a lot of properties with the old one. For example, if we glue this thing up, yellow sides together, pink sides together, we still get something that looks like a Taurus, right? So the shape of the surface is the same, but this sort of almost geometry <laugh> might be very different. So for example, in this square, the length of the longest diagonal, indeed the only diagonal length is say if this is a unit square, route two, right? But if I apply a transformation to it, that length can get longer in one direction and shorter, sorry, longer in both directions since I'm thinking about diagonals. Okay? But say vertical line for example gets shorter, okay? So not all properties are preserved by these kinds of changes, but some of them are And so again, what we get in the space of all possible translation surfaces is a way to group them, okay? Where we say, okay, two of these surfaces are in the same group. They get the same color if they're related by one of these transformations. This kind of collection of all things that I've colored green, all things related to a given surface is what's called an orbit of translation surfaces, okay? And the study of this kind of object and dynamics is really ubiquitous. Trying to understand orbits, right? Things that can be related to another by a simple motion, um, is, is the found sort of foundation of a lot of dynamical systems study. Okay? And in general, describing orbits, or more precisely if you're familiar with the mathematics orbit closures, uh, is a very hard problem. Okay? So you might have heard of something called a, a strange attractor, okay? If you've ever encountered this kind of picture before, um, often people sort of just generally speak of this in qualitative terms about, you know, chaos and small changes leading to big changes and that kind of thing. A strange attractor in mathematics is a very precise object, <laugh>, it's an orbit of a certain dynamical system which has the property that if you sort of cut a cross section, even though this is all a nice smooth, you know, curve looks totally straightforward, if you cut a cross section, you actually get a very complicated fractal type set known as a cantor set. Okay? So these orbits can be very complicated and describing them is, is a very difficult pursuit. Just to give you another analogy, which helps describe the complexity of this question, I don't know if this will appeal to anyone, but I found it sort of pleasing. Um, it might seem hard at first instance to describe if, if you play chess, you know how a night moves, so I've got it up in the corner here. If you don't play chess, um, it can move sort of two up and one over or any rotation of that or reflection. And if you wanna try and understand the spaces a night can reach moving that way. It's not that hard to show your, or convince yourself that a night can reach any space. Eventually figuring out how many moves it takes is maybe complicated. That's what this diagram is indicating. Fine, that's not so bad. However, <laugh>, what if I told you that I'm not just thinking about this say infinite chess board, which has squares everywhere, but I'm taking any shape of any chess board that I could possibly wanna write down. And I want you to tell me exactly what all possible configurations of spaces that the night could hit look like. How would you even attack such a problem?<laugh>, right? It's an immensely complicated problem. Okay? Alright. So nonetheless, in a, in a monumental series of work, uh, comprising hundreds of pages, um, eskin and, and mishk, and together with their collaborator, Amir Mohamed, uh, were able to answer this question for this space of translation surfaces. Okay? So they were able to describe what do these orbit closures actually look like? Alright? All right. So here is a fun application of this work. Um, it's known as the illumination problem. Okay? Um, so you might have actually seen if you live in this lovely city, uh, a beautiful installation that was at the tape modern a couple of years ago. Um, so this is, uh, there were two I think, installations of Kosama, which was, uh, this sort of infinite mirrored room filled with lights <laugh>. So this is a good description of what the illumination problem is. So the illumination problem asks, if I have a room that is covered in mirrors on every wall and I have a light source, is it possible that there is anywhere that stays dark? Okay? Can we describe the set of points in the room which are lit because of course the light will reflect off the mirrors, right? And hit a lot of different points in the room. So in the billiard setting, you can think of this as taking a billiard ball and asking, is there anywhere on the billiard table that I could put a pocket that was never going to be able to be hit? Okay? Now here I've got a rectangular table <laugh>. The answer is definitely no, there is no such point on this rectangular table because rectangles are convex <laugh>, which is exactly the, the property I would need, which is that any two points in a rectangle can be a connected by a line that is still in the rectangle. Okay? So this straight line shot can reach any point on this rectangle. I don't even need bounces off the off the sides, okay? But for more general polygons, it turns out that this is not always the case. It's possible to configure or to construct a polygon with points that are not mutually mable, which is to say if I have a light source, say at point A, or I'm hit a B, hit this billiard ball at point A, it can never no matter which direction I point it hit this point B. Okay? So there are rooms or polygons which with points that cannot be connected by straight line pass. And in fact, now that we've learned what unfolding is, this example is not so mysterious, okay? It is complicated <laugh>, but not so mysterious. This example is an unfolding of a, of a triangle of this ISOs triangle. So it's not too hard to show, it takes somewhere, but it's not too hard to show that if you hit a billiard ball from this corner point A, it can never return to that corner point A, okay? No matter which direction you hit it. If you keep that in mind and you hit a billiard ball, if you take any unfolding of the isosceles triangle with the feature that no interior vertex is anything except a, which is to say that any B and C vertex is always on the outside of this polygon, then in fact you will get something that can never go from any a point to any other a point. Okay? And as you can hopefully see in this picture, that's precisely how these two points were constructed, okay? Cuz if we could get a straight line from here to here, then it would descend to a path from a to itself when we folded back up. Okay? So as a consequence of the work of Ashkin and Mu and Mohamed, a ball on a rational polygonal billiard table like this actually has only finitely many inaccessible points. Okay? I can actually write down the list of points on this polygon that a cannot reach, okay? Now, when I showed you the unfolding construction, that seems somehow eman believable, right? Like I did some finite number of unfoldings, there's only finitely many a points. But please be advised there are many, many non unfolded. You can get translation surfaces not from unfolding polygons. There are more complicated translation surfaces, okay? And the key feature is that this illumination property is actually preserved by that notion of linear translation transformation that I discussed, right? Describing a line here and then transforming this thing by a linear transformation. I still get a line in the new object. And so this is really a statement about an infinite collection of billiard tables, right? And you can rephrase it in terms of the modularized space, and that's where Eskin Muhammad's work comes in, alright? It's worth noting that this is really false, by the way, if you allow general shapes, okay? So if you don't require your shape to be a polygon with rational angles, then they're definitely possible. Not just infinitely many points, but entire sections of the room, which can fail to be illuminated. So these examples give, so this is, uh, penrose's room. So these examples give, um, a red point, which is a light source, and the unilluminated is in gray. So here, almost all the room is unilluminated, in fact, even though the walls are fully mirrored. Okay? Alright, so a second application of their work, which is somehow easy to phrase, uh, is known as the blocking problem. And I'd actually like to, to play a quick video for you of a talk that Maryam gave in Princeton, uh, describing <laugh> this blocking problem as an application of her work. Uh, this was described at a Princeton seminar in 2012. Two put two points. So the first point is maybe your child trying to escape, and this is your iPhone <laugh> she's trying to reach. And so light starting from this point A and then going point B. And you wanna block point A from reaching B by finite many points <laugh>. So assuming that your child going goes only straight and bounces off the edges, <laugh>, <laugh>. But in any case, it's, so, it's like starting from A, can I block B by putting finally many points, which means any path that intervene many, many, possibly infinitely, many paths from A to B, imagine that your billiard can be illuminated. So can I put finely many points here so that any path between A and B, you put the dad here and the grandma here. Maybe it was like the other people <laugh> anyway going this way, um, from A to B, this is not such a good picture. We'll go through one of these uh, points. So if you didn't hear the beginning of that, by the way, she, rather than taking a billiards description or a light source description was describing, if your child sees your iPhone <laugh> in the room and only takes straight line pass <laugh>, is it possible to block them from getting the iPhone by placing a finite number of family members in the way?<laugh>? Alright, um, Mary and her husband Jan, welcome their daughter Anta. Uh, about a year before that video was taken and a year after that seminar, she was diagnosed with breast cancer. Uh, at the time she successfully treated it into remission. And, uh, a year after that, <laugh>, she was awarded the fields medal, the highest honor in mathematics, um, and the prestigious clay award for her research. Um, but the cancer recurred relatively quickly. And Maryam died on the 14th of July, uh, 2017 at just 40 years old. Miriam's loss was deepest, of course, for her family and friends, but it was felt across the international scientific community, uh, not just for her stunning career, cut short, but for the loss of a woman whose life and work inspired us, um, with her ability to turn ambition into collegiality, <laugh> and, and collaboration instead of ego. Um, she was willing to share her passion for mathematics with students and with colleagues alike. And she was brilliant and ambitious, but she was also kind and humble, and she was gone too soon. There are a number of wonderful tributes to Miriam's life. Uh, the international women in Mass day was the 12th of May Miriam's birthday, a newly founded annual celebration, and there are many scholarships and prizes and schools in her name promoting the work of young and future mathematicians, particularly young women in mathematics. So I was lucky enough today to share some of Miriam's mathematics with you. Um, this talk actually shares a part of a title with a wonderful documentary that, that focuses also on her life from the perspective of her friends and family, her, her previous teachers. And, um, and so I wanted to share with you the access to that document, uh, that documentary if you haven't seen it before. With that, I think I will thank you all for coming and, uh, and maybe open up for questions.<laugh>. Hello. Thank you very much for your talk. I appreciate how difficult it is to explain what must be an incredibly complicated mathematics to mere mortals <laugh>. Um, could you summarize, maybe this is an impossible question. Roughly how your work made. Does it relate all to Miriam's work or could You tell us a little bit about your Work? Yeah, I'm very happy to, this is not a person that I know.<laugh> <laugh>. Um, yeah, so my own work is in complex dynamics primarily. And so, uh, I have a lot of common features with Miriam's work in the sense that I study iteration of systems. So this sort of, what I call these orbits, that Maryam studies, the, the objects that can be reached by a simple rule. Um, I study this similar kind of thing, but the systems that I study are just defined by functions. And so you can apply a function is just a rule that takes you from one number to another number <laugh>. And so you can apply a function is just a rule that takes you from one number to another number <laugh>. And you can apply it once or you can apply it twice or you can apply it three times. And you can apply it once or you can apply it twice or you can apply it three times. And what I work on considers what happens when you do that somehow, infinitely often. And what I work on considers what happens when you do that somehow, infinitely often. And, um, the type of work that I do, the, the common thread a little bit with what Maryam did and, and what I did, is that both of these fields have a slightly unexpected connection to number theory, um, that there's, because the simplicity of the system <laugh> means that you really only sort of need integers to describe what you're doing <laugh>, if that's, if that's a sensible thing to say, and that means that you can make sense of it. Whether the numbers you put in are complex numbers or real numbers or whole numbers or more exotic things like periodic numbers or numbers over some more complicated mathematical object, like a finite field or, or even even more complicated than that. And so, um, those common features mean that there are some arithmatic tools and arithmatic consequences of, of the type of dynamics that I do. And that's, and that's what I particularly focus on. And while that's not really what, what myam focused on in her own work, um, it is, it is, it is a consequence of her work that, that it is possible to do such things. Thank you for a question. Um, I'm going to pass over now to Dr. Kevin Houston from the London Mathematical Society, Right? Hello? Uh, hopefully this is coming across, is it? Yes. Yeah, that's good. Right. Okay. Um, I'm Kevin Huston. I'm the Education Secretary of the London Mathematical Society. And if you don't know, uh, what the math the London Math Society is, uh, despite its name, it is a national organization. It's involved in dispensing grants and helping people with their research funding, that sort of thing. We also have public lectures. Um, if you want more information, then please go online, lms doac.uk and you'll find loads of stuff there and maybe you'll even want to join. Um, but anyway, I'd like to finish tonight by thanking you all for coming, coming out tonight and, uh, enjoying such a, a great talk. Uh, I'd like to thank the people at Gresham for the wonderful organization and, uh, I'd like you to join me in thanking our speaker, Ali Krieger, for such a fantastic talk.