foreign begin I do want to 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 Rami and talk lubikash and Laura DeMarco who are both good friends of mariams and who have um communicated to me over the past couple of decades the experience of working with her and knowing her the others are through their writings Anton zorich and Kurt McMullen and Alex Wright among many others have written very beautiful pieces on Miriam's work that perhaps are not as accessible as I hope that this lecture is but inspired a lot of the directions that I decided to take when I was deciding what to tell you today okay all right so oh and of course thank you to Gresham college for hosting me and the London mathematical society as well all right so we are here today to discuss the work of one of mathematics brightest modern starskani Mariam was an Iranian mathematician who's deep and creative mathematics receives the highest accolades in our field and a warm and enthusiastic woman whose life and work were cut short at just the age of 40 when she's succumbed to breast cancer what I would like to do today is to communicate to you some of maryam's work not just the mathematical content though that is the main goal but also a sense of why other mathematicians find her work so impressive and so compelling um really what I'd like to share with you all is the appreciation that I feel when I read mariam'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 you a little bit of her mathematical Legacy as best I can um so maryam's work considered the connection between two subfields of mathematics the first is geometry which is familiar to us basically from Nursery right so geometry is is the study of of not just shapes which we in mathematics more often call topology but also Notions of say size or distance or curvature that we can talk about on shapes so even in Nursery we learn things like Circle or triangle these geometric objects but we also learn some some more some deeper Notions in Geometry like bigger and smaller or sort of we don't learn concave in convex and Nursery of course but we we landed some things are round and some things are flat right 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 but it also connects to many other subfields inside of mathematics like number Theory the study of the whole numbers and prime numbers or algebra so the second subfield that involved that Miriam's work involved is the field of Dynamics which is roughly speaking it's the study of 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 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 well there will be some very simple ones in fact we'll talk quite a bit about the dynamical system which is determined by walking in a straight line 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 mariam's work focused on a family of mathematical objects known as Riemann 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 going to take a tour of two of her most important contributions in that direction but I actually want to start with a quick warm-up problem that she herself considered in her teenage years 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 want to say I want to say the warm-up problem will help you sort of understand how mathematics was treated in her work as a an experimental and and visual science which is perhaps not how we think of mathematics okay all right so just a little bit of background since I'll sort of discuss this problem that she that she thought about when she was 16. um Ryan was born in Tehran in 1977 which was just before the cultural revolution uh the beginning of the cultural revolution which islamicized the Iranian Academia 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 she attended a very high quality and prestigious School for Girls in Tehran and by by the seventh grade it was clear that she she had really a standout talent in mathematics with the support of her school principal she and her friend Roya bahashti um were the first women permitted to participate in the the Preparatory National mathematical competition and and joined the national team for What's called the international math Olympiad okay so the IMO the international mathematical olympiadis is a competition held every year that is a competition in problem solving for pre-university students and it really draws immense mathematical Talent from around the world countries around the world and it is somehow famously difficult despite drawing together some of the best trained and most talented mathematicians of their age well you can see I have the points distribution from zero to perfect score here is not looking so good at the high end 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 or something which is very rare so uh Mariam attended the 1994 and the 1995 IMO as part of the Iranian national team here's a picture of the 95 team you can tell which one she is um and scored a rather astonishing two gold medals one point off of a perfect score in her first performance and a perfect score in her second I mentioned this not because it's somehow the bar of mathematical caliber but it gives you an indication of her level of Engagement already at that age and I think it's also worth noting that being a woman accomplished in mathematics and science in Iran at the time itself is not so unusual while gender segregation was absolutely a feature of Education there there were quite a few opportunities still available to women working in science at the time and as Maryam herself mentioned in an interview in 2008 she had to explain while she was living in the U.S quite a few times that she was in fact allowed to attend well she rather humbly says a university in fact it was the best university in Iran all right so I'd like to start with this this warm-up example comes from a summer course that Mario attended as a teenager it was held at the University for gifted math students and I will read this you don't have to somehow squint if you're in the back row here so this is 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 maryam's life and work and what he mentions is that in this summer course one of the topics that the professor had talked about was about decomposing graphs I will mention what all these words mean decomposing graphs into disjoint unions of Cycles including this rather curious examples of tripartite graphs into unions of five Cycles 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 Miriam 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 decomposability 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 Miriam was obviously smart but not smart enough to only Dole out one example a day foreign so I'd like to discuss this problem a little bit because it gives us a wonderful glimpse into the sense of exploration and persistence also that Maryam 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 mathematical mathematicals mathematicians use to model relationships okay so graphs are models for a huge array of relations and processes so say network analysis and neural networks are certainly popular applications these days there are applications in biology I have here a network of protein protein interactions you can see in the middle as well as say computational Neuroscience or social science and so on but all of these fields are sort of using graphs to to Encompass a simple idea 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 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 want to escalate any attention so I chose groupie 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 in this stage the 32 teams are split into their eight groups four teams in each group and inside any group so here I've got Group B with Spain Costa Rica Germany and Japan we can represent the idea of playing a match by connecting the two teams by a line right 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 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 all right so this property so what is a round robin a round robin is 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 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 just 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 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 want to go ahead and schedule the matches right and we need them subject to the to the Restriction that no team plays a two matches on the same day right so what I'm going to do in order to do that is I'm going to 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 and this decomposition although it's sort of simple especially since I started with a graphic that looks very simple 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 is part of the decomposition for okay all right so that's just a 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 Mariam 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 so it 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 first of all I've told you what complete means well complete tripartite is slightly different so this will take a little bit of explanation a tripartite graph is one whose vertices whose points can be grouped into three groups okay I have indicated that here by color which I'm hoping these colors are sort of different enough from each angle that 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 I'll start with the simplest one blue green and red 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 coloring correctly you'll see that that's actually exactly what I've got this is a table here of a lot of complete tripartite graphs with small numbers of vertices okay so that's what it means for a 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 these tripartite graphs can come in handy all right so what does it mean to be a five cycle then well it's exactly somehow what it sounds like if you started 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 of tripart 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 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 all right 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 can which if I count them there's a total which is divisible by five okay this graph has five edges this graph has 15 edges okay you can either count by hand or you can cook up a nice formula for the total number of edges based on how many vertices have each color okay all right so the question is here's at least two candidates for grabs 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 and at the back where we started if we start say the green vertex and we travel around it doesn't take long to see that we're going to run into trouble all 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 and similarly if we started this red we're going to have exactly the same problem you can try starting at a blue but actually the red and green vertices are still going to 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 graph 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 because 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 okay all right 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 to construct these infinitely many family or studies 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 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 it a hundred different times right this is very little memorization there's no technique you already know how to solve the problem there's persistence to draw 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 but there's also a sense of play and visualization and experimentation and as a professional mathematician had 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 an amazing sort of uh it's amazing that we can already see that in her work as a teenager all right are you all warmed up because it's kind of all right so maryam's work after University was focused on riemann's surfaces so Riemann surfaces are geometric objects and what defines a riemann's surface is that if you look closely if you put a Riemann 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 Riemann surface so keep in mind I'm taking like just the surface of the sort of Hollow Earth is a Riemann surface if I zoom in if I take a map which really zooms probably I want to do even better than this zoom in on London or zoom in on this block then it's something which really just looks like sort of the flat normal euclidean plan that we're used to working with but Riemann surfaces have an additional feature which is that 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 iconic things are really large and the streets don't match up to reality anything like that 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 Riemann surface okay a Riemann 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 all right so Riemann 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 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 there's no boundary to to really talk about 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 so here's one with no holes here's a donut or Taurus with one hole here's two holes and so on okay this number number of holes is known as the genus of the surface holes or handles however you prefer so if you have 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 I mean I guess yeah this is what they mean that's what both of those things are right if you're a topologist if all you care about is shape then all that matters is the number number of handles that every month surface has okay however we don't just care about shape we care about geometry and Riemann 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 angles or curvature all of this kind of mathematical description in a way that is intrinsic to the surface okay so 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 the 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 want to measure the length from one point on the sphere to another one all I need is a good bit of string right if I want to measure the angles between them then I just need two strings right the curvature of the sphere sort of Curves outwards that's real 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 want to focus on so the sphere has positive curvature it 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-euclidean 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 some places it has a negative curvature may be flat somewhere in between that is not true 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 torus 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 3 4 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 are all negatively curved so people often describe this as sort of 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 and 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 these pictures do an okay job of some things um so for example foreign these pictures don't do a terrible job of representing distance okay they do a very bad job of representing curvature they don't do a terrible job of representing distance and if all this makes your head hurt I mean sort of rest assured that that's also true for professional mathematicians in fact there's basically an entire subfield of geometry is about good models for these kinds of Riemann surfaces however it is in fact a theorem 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 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 Riemann surfaces admit geometries these Notions of distance or length but Riemann 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 2 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 want to 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 whole numbers and the same as it was for understanding graphs which is the idea of decomposition right and if I told you I'd like you to take this genus2 surface and give me some easier to understand surfaces what you might do is grab the nearest pair of scissors 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 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 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 Riemann surface and so if we're going to use this technique to understand Riemann surfaces really we need to understand Riemann surfaces where we allow these kinds of boundaries okay and so this is a lot of what maryam's work focused on in the case of these hyperbolic geometries 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 but by the way I asked I often ask my students about like americanisms and if they will embarrass me in front of crowds here I ask my students about this and because of course I know the the difference in American English and British English for pants and they told me first of all that it would be fine but second of all that 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 but of a Cambridge Professor who is so unwilling to say pair of pants while he was teaching the geometry class that he called them t-shapes I don't know if that's like T for trousers or t because 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 so there you go um all right so these surfaces are built out of pairs of pants and um there's right so right I did want to make the the point that if you are 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 Riemann 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 all right so what is a geodesic in particular closed geodesics are what we're going to be talking about mostly in this talk so at 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 Earth 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 higher 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 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 here 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 ladybird following a geodesic on a tour 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 ladybird 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 euclidean space you might expect her to just sort of cut off the top of 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 okay all right so understanding a surface is really somehow tightly bound because it's 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 close geodesics um so Mario's PhD work focused on precisely describing these things essentially from various sort of angles um so she completed her PhD at Harvard by the way in 2004 under the supervision of Kurt McMullen and I sort of already highlighted this accomplishment and as a teenager she accomplished something which any professional mathematician would tell you is much even harder than two gold medals at the IMO 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 very impressive indeed um but so what did Maryam discover about these geodesics so I'm going to focus on as you might see from sort of the words involved in these titles I'm going to focus on one of these three topics because it's somehow the easiest to get a feel for the the friendliest to describe so let me first start by analogy I want to 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 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 100 digits or less or fewer at most 100 digits then I can actually write down and count how many there are right so here I've hopefully written down correct counts for prime the how many primes there are of one two three 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 is 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 bound 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 close 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 gross matches 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 Miriam considered was a much more difficult it's very easy to say out loud 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 genus2 surface here is a non-simple geodesic on a genus2 surface okay and in a sense of course these are even more these simple geodesics are even more basic building blocks 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 going to think about the Taurus because it's a good example for explaining this so the Taurus I mentioned before has a flat geometry and here's that 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 I meant to have a piece of paper out to do this 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 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 all right 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 if I translate it over to this Square all I'm doing is going in a straight line keeping in mind that I have to sort of when I hit the top emerge from the bottom and so on until I hit another Red Dot okay 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 want to think about sort of like this Union of segments then you can think about taking lots of copies of the square just repeated 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 want to do this to think about all the possible geodesics on the Taurus then you need to take infinitely many copies of the square all right 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 they live inside some disk and so it's the same question as counting integer points that is to say coordinates with integer X and integer y um inside of the inside of the disk it's counting simple closed geodesics 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 disk in this sense well sorry it's actually the same primitive but I'm going to 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 because if you were to look in a straight line there's no other integer points blocking your way okay all right so it turns out the second thing is a much harder problem so counting integer points in a disk is basically the same as Computing the area of the disk right you just associate say I have a point here just associate it with this unit square that it's the right upper right corner of okay that gives a pretty good approximation of counting integer points inside the disk counting primitive integer points inside the disk this visibility thing this primitivity that I mentioned is really the same statement that the two integers defining the point have no common factors okay they're co-prime 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 so that's to give you an idea of course we're going to work in a hyperbolic world but the Taurus really captures a lot of the difficulty that in fact new mathematics has to come in in order to translate to the simple closure it does a counting problem okay all right so now we're in a position to understand one of the significant results of mariam's thesis work so she provided the first precise counts of the number of simple closed geodesics on a hyperbolic surface okay 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 this function to grow okay all right 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 is somehow decreasing as the length of the ghs that gets large okay so more striking than even the results 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 you 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 going to just use a 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 parity 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 talk 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 all right so this is a sort of toy model although very actually pretty reasonably descriptive of the technique that Mariam 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 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 all right so here is a fun application so let's think about a Genus 2 surface I mean sorry when I say my 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 are 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 mariam'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 and it turns out that the probability so it's an asymptotic result so you want to 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 all right which is not something people thought you should be able to get your hands up just to put it in subcontext all right so the technical heart of my item's early work this PhD work was understanding this space of all possible geometries that you can put on a particular surface so mathematicians call this a moduli space so it's an abstract idea but here's a very concrete example a modulus basic is basically a map okay each point in the space corresponds to a mathematical object instead of a place so for example the moduli 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 and up here we have a Taurus with a skinnier handle okay each point in this region corresponds to a different geometry more or less that we can put on the Taurus okay we study these spaces because they help us understand the objects they contain they're often interesting themselves but also they help us understand what does it mean for for tutorial 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 mariam's work that I wanted to understand or to communicate to you so after finishing her PhD at Harvard Marion went to Princeton under a clay fellowship and then accepted a professorship at Stanford University and in that time she began a sort of long-standing and very celebrated collaboration with Alex Eskin and what they were interested in studying was a different type of moduli space which keeps track of objects known as translation surfaces so these objects in mathematics arise one of the ways they arise is about thinking about billiard Dynamics and so I want to describe a little bit of that to you now 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 moment okay so this study of translation surfaces which I'll describe to you I want to 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 Pentagon all right 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 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 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 using this reflection principle all right 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 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 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 with reflecting things we do not get a straight line all right okay so this process is known as unfolding okay so we unfolded the billiards table to get a new shape and identifying parallel edges here we obtain a surface okay all right so translation surfaces generalize this idea very broadly so the amazing thing is that we get objects we recognize 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 a 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 2 surface okay so this probably makes your brain hurt if you have not seen it before however this cartoon does a reasonably good job of explaining the sort of folding and gluing okay so we have seen these objects before however 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 because 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 which I don't do math very well in my head but that's definitely more than 360. all right 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 all right so what we're going to think about is just like in the Riemann surfaces setting we're going to 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 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 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 root 2 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 it's a 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 in Dynamics is really ubiquitous trying to understand orbits right things that can be related to another by a simple motion um is is the fan 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 is a very hard problem okay so you might have heard of something called a strange attractor okay if you've ever encountered this kind of picture before 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 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 knight moves so I've got it up in the corner here if you don't play chess it can move sort of two up and one over or any rotation of that or reflection and if you want to 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 knight 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 what if I told you that I'm not just thinking about this say infinite chessboard which has squares everywhere but I'm taking any shape of any chessboard that I could possibly want to write down and I want you to tell me exactly what all possible configurations of spaces that the Knight could hit look like how would you even attack such a problem right it's an immensely complicated problem okay all right so nonetheless in a Monumental series of work comprising hundreds of pages escanon and mirishkani and together with their collaborator Amir mohammadi we're 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 and in fact what they found is that they are not complicated and fractal they're actually described by simple polynomial equations okay and this is a really ambitious piece of work that sort of draws on on almost every field of mathematics that exists or at least pure mathematics and really a beautiful piece of math all right so here is a fun application of this work it's known as the illumination problem okay so you might have actually seen if you live in this lovely city a beautiful installation that was at the Tate modern a couple of years ago um so this is uh there were two I think installations of kusama which was uh this sort of infinite mirrored room filled with lights 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 the answer is definitely no there is no such point on this rectangular table because rectangles are convex 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 illuminable which is to say if I have a light source say at Point a where I'm hit up 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 but not so mysterious this example is an unfolding of a of a triangle of this isosceles triangle so it's not too hard to show it takes some work 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 this 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 because if we could get a straight line from here to here then it would descend to a path from a to itself when we fold it back up okay so as a consequence of the work of ashken and mirzkani and muhammadi 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 imminently 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 they're more complicated translation surfaces okay and the key feature is that this illuminability 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 moduli space and that's where eschgani muhammadi's work comes in all right 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 penrose's room so these examples give 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 all right so a second application of their work which is somehow easy to phrase 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 describing this blocking problem as an application of her work this was described at a Princeton seminar in 2012. two you put two points so the first point is maybe your child trying to escape and this is your iPhone she's trying to reach and so or lights starting from this point a and then going point B and you want to block point a from reaching B by finitely making points so assuming that your child goes only straight it bounces off and then just but in any case it's so it's like starting from a can I block b by putting finally points which means any path the interim many possibly infinitely many parts from A to B imagine that your billiard can be illuminated so can I put finally many points here so that any path between A and B you put the dad here and the grandma here videos like the other people anyway going this way um from A to B this is not such a good picture we'll go through one of these uh 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 in the room and only take straight line pass is it possible to block them from getting the iPhone by placing a finite number of family members in the way foreign and her husband John welcomed their daughter anahita about a year before that video was taken and a year after that seminar she was diagnosed with breast cancer at the time she successfully treated it into remission and a year after that she was awarded the fields medal the highest honor in mathematics and the prestigious clay award for her research but the cancer recurred relatively quickly and Maryam died on the 14th of July 2017 at just 40 years old Miami's loss was deepest of course for her family and friends but it was felt across the International scientific Community not just for her stunning career cut short but for the loss of a woman whose life and work inspired us with her ability to turn ambition into collegiality and and collaboration instead of ego 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 maryam's life the international women in Mass day was the 12th of May Mario's Birthday 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 my mathematics with you 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 and so I wanted to share with you the access to that document that documentary if you haven't seen it before hmm with that I think I will thank you all for coming and uh and maybe open up for questions [Applause] 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 um could you summarize maybe this is an impossible question roughly how your work made does it relate at all to mariam's work or could you tell us a little bit about your work yeah I'm very happy too this is not a person that I know 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 there's sort of what I call these orbits that Maryam studies the objects that can be reached by a simple rule 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 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 the type of work that I do the the common thread a little bit with what Mariam did and what I did is that both of these fields have a slightly unexpected connection to number theory that there's because the Simplicity of the system means that you really only sort of need integers to describe what you're doing 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 arithmetic tools and arithmetic 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 Marion focused on in her own work it is it is it is a consequence of her work that that it is possible to do such things thank you for your question I'm going to pass over now to Dr Kevin Houston from the London mathematical Society right hello uh it's hopefully this is coming across is it yes yeah it's good right okay um I'm Kevin Houston I'm the education secretary of the London mathematical society and if you don't know uh what the London math Society is 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 if you want more information then please go online lms.ac.uk and you'll find loads of stuff there and maybe you'll even want to join um but anyway like to finish tonight by thanking you all for coming coming out tonight and enjoying such a a great talk I'd like to thank the people that's Gresham for the wonderful organization and I'd like you to join me in thanking our speaker Ollie Krieger for such a fantastic talk thank you