Enseignement 2023-2024 : Spectres en géométrie hyperbolique
Séminaire du 24 novembre 2023 : Curves, Surfaces and Intersection
Intervenant : Hugo Parlier, Université du Luxembourg
Understanding curves on surfaces has become a primary tool for understanding their hyperbolic structures and associated moduli spaces. This talk will be on understanding curves through their intersection with other curves and themselves.
For instance, through classical work of Dehn, simple closed curves can be described using intersection numbers with other simple curves. An underlying theme will be to figure out to what extent you can describe all curves in a similar fashion. More generally, curves are fabulous objects to experience the interplay between the topology and geometry of hyperbolic surfaces.
Part of the talk will be based on joint work with Binbin Xu.
Chaire Géométrie spectrale
Professeure : Nalini Anantharaman
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Okay welcome back everybody uh for the seminar it’s a pleasure to welcome Ugo from University of Luxembourg and he will give a talk in hyperbolic geometry and the title is curves surfaces and intersection uh if you wish to ask questions during the seminar it’s better to ask them in the microphone but it’s
Not always easy to circulate it but still let’s let’s try to to to ask the questions in the microphone and the floor is yours thank you very much so thank you you very much to everybody for being here thank you very much to Nini and the czra for the
Invitation it’s um it’s very nice to come from so far so when n says that I’m coming from Luxembourg she’s not referring to the park but to to the country um and what I want to talk to you about today is curves surfaces and [Applause] intersection [Applause] and despite the the the logistical
Difficulties in handing the microphone around please uh don’t hesitate and interrupt me if you have questions a lot of what I’m going to be talking about today is joint [Applause] with binu and some of it is with Peter Boozer as well okay but really my goal or at least
One of my goals for the talk today is to try to explain this picture to you which you see on the left and is to try to understand in some sense the space of all closed curves on a surface and what I mean by that I will explain but um
It’s about one of many different ways you can start to expl FL the Dynamics of closed curves on a surface and their relationship to hyperbolic geometry and lots of other things but all right so keep in mind that at the end if you don’t understand that
Picture then I haven’t done my job okay so what’s my basic setup it’s going to be similar to what we’ve just seen so throughout I’m going to look at a topological [Applause] surface [Applause] of genus G greater or equal to 2 okay so in my picture here we should be thinking of something
Like this so the genus is the number of connected uh compon it’s the number of Tor which I’m taking the connected sum so in this case here my genus G is equal to 3 my surfaces will always be closed and orientable um unless I say otherwise and
What I’m interested in is curves on surfaces so what is a curve as far as I’m concerned a curve is simply a map okay the image of a continuous map [Applause] um from the circle to my Surface okay so I take a a circle and I map it continuously to my
Surface and when I say a curve the way I’m going to be thinking of a curve I’m going to be thinking of Curves as being what am I going to be think so I want them to be non-trivial right by which I mean that they’re not homotopic to a
Point okay I want them to be [Applause] primitive which means so Nini mentioned this during her her course this means that once I have a the image of a curve okay I can take that curve and I can do it several times right so I’m not allowing that I’m only looking at curves
With the image one single image right the Primitive version and I’m thinking of them as unoriented objects okay unless I specifically say otherwise and the most important bit is I’m thinking of Curves up to homotopy so in this case it’s up to free homotopy okay so from the topological
Point of view if I take two curves for instance this very beautiful curve okay that’s a very nice curve or this one from the point of view of free homotopy classes those two curves are the same okay and I’m going to say something very controversial right now is that a multicurve [Applause]
Is a collection of [Applause] [Applause] curves so those of you who are not used to this business won’t understand how controversial that sentence is but I’ve noticed that there are three things one should never talk about if you don’t want to argue with somebody it’s religion politics and the definition of what a
Multicurve is okay so for me a multicurve is just a collection of Curves and I can see frowns in the audience already so I will not look at the audience for this all right so what are my guiding questions throughout the talk my guiding questions are going to
Be how do I find a parameter set so questions I’m interested [Applause] in so is there a good par parameter set for the space of all [Applause] [Applause] curves okay and there’s many ways to answer this question um but the way that I’m going to treat it or look at it is through
Intersection okay and what I mean by intersection is intersection with other curves okay so so let me so let me get back to that in just a second but the other thing I want to say and this is also important is that to the space of all curves there’s a parallel Theory
And that’s the study of closed geodesics on hyperbolic [Applause] surfaces so let me just say one word about this so when you look at a curve or a closed curve on a Surface this is a topological object none of this has anything to do with geometry but so here’s my Surface Sigma
Here’s my topological surface Sigma and I’m looking at curves up to free homotopy on this object but Associated to this because I’ve carefully chosen my Surface to have negative Oiler characteristic Associated to this I could choose a hyperbolic metric and put it on this I could get a hyperbolic [Applause]
Surface which I’ll call X right hyperbolic surface that topologically looks the same but this time it has a metric so this time I take for instance in the hyperbolic plane I take some sort of polygon which I I won’t continue to draw and in the hyperbolic plane I can
Take a polygon and then start pacing it together exactly like I would if I was doing a topological construction but this time I do it in a geometric way just that this side here is the same length as this side here such as for instance this side here is the same
Length as this side here and in a way such that when I paast it up I get a smooth hyperbolic surface so this means the surface whose metric locally looks exactly like a piece of the hyperbolic plane and what is the parallel Theory here the parallel theory is that if I
Have a closed curve here it corresponds exactly to a unique closed [Applause] geodesic but this time the closed geodesic is on the hyperbolic surface right so I have a closed curve on the topological surface this corresponds to the closed geodesic on the hyperbolic surface and the cartoon image of this which has been
Pointed out to me many times is not a proof but is at least an image of what’s going on is that a hyperbolic surface so it’s a surface that’s negatively curved what does that mean negatively curved means that if you think of this in terms of Romanian geometry you have a
Principal curvature that’s going in One Direction and you have the other principal curvature that’s going in the other direction so you have something that’s going this way and something that’s going that way okay so locally your surface looks like some sort of uh horse saddle right and so you have this
Negative curve phenomenon and here’s your homotopy class of closed curve that looks something like this and you’re in this negative curvature thing and in this homotopy class is a unique geodesic maybe the better image for this is really in the hyperbolic plane where this closed curve is going
To be something that looks something like this and then in the hyperbolic plane you’re going to have a unique geodesic that looks something like that right and so that’s the that’s how these two theories have something to do one with another but as much as I I’m going to really try in
Some sense to avoid the story of hyperbolic surfaces for for lots of what I say okay if I don’t need them I won’t use them but you’ll see I’ll keep jumping back to them because I like them so anyways all right and fact I think the real goal for my
Talk is to avoid erasing this drawing I think that’s going to be the hardest part okay all right so what I was talking about was intersection [Applause] okay so let me Define what I mean by intersection so we have Alpha and beta that are closed [Applause] curves so let me Define the intersection
Between Alpha and beta all right so this is the intersection number between the two curves and because I’m thinking of them as homotopy classes the way I want to Define them is some sort of minimal quantity like this so what I do is I look at the all
Possible ways that I can represent Alpha and beta and I try to minimize the number of points in which they intersect so this is the minimal cardinality of the intersection between all po possible representations of Alpha and beta so Alpha Prime will be a curve that’s freely homotopic to this Alpha and beta
Prime will be a curve that’s freely homotopic to this beta right this is the way I’m defining intersection but there’s a caveat here the way I’m doing this is that I’m supposing that there are no triple points okay what this means is I want all my intersection points if this is
Alpha and this is beta this is their representative I want them all to look exactly like this so this is allowed but what is not allowed is something like this where three three uh I get three local arcs that intersect through the same point so this is absolutely not
Allowed okay so there’s no triple points here all right so with this definition I have a question for you the audience so here’s a surface here’s a curve which I will call Alpha and my question for you is what is the intersection between Alpha and Alpha One One any other guesses
Two very good any other guesses so the answer is two actually all right so so why so so why is this so this this you want it to be one you want the answer to be one right but the way that I’ve defined it the answer is really two so why is this
Because you try to minimize I’m trying to minimize the intersection between one of Alpha’s representatives and one of Alpha’s representatives and in fact the way to minimize this is to do something like this so Alpha is a curve that looks like that I’m going to try to draw
Another curve that’s homotopic to it and the way I can minimize it is to do something like this and if I look carefully at this drawing in fact there are two intersection points here and here all right so it’s it’s a bit of a convention so in fact the answer from the audience
Is absolutely right if I had said the number what is the number of of minimal self intersections that would have been correct the answer would have been one but in fact the the way I’ve defined it when I take the intersection between a curve Alpha and itself this will always
Be an even number the way I’ve defined it and all right so so that’s that’s just a small confusing caveat to confuse you because it’s important to confuse people as much as possible when you try to tell them something okay so so the set of curves that self intersect 2K times are going
To play a particular part in what I’m going to say and we’re going to call them K curves so let me write that down so I don’t forget what the definition is so Alpha is a k curve if the intersection between Alpha and Alpha is equal to 2K okay so so this is
This is a definition and the set of K curves I will denote by CK this is the set of K [Applause] curves and in particular c0 these are called simple closed [Applause] curves and and simple curves um simple curves are very nice because the reason they’re called simple closed curves is
Because they’re very simple they they leave simple existences and they’re very nice to look at so simple curves are simple and you can describe them and this is what I’m going to tell you in a very nice way using intersection so these are called Ben thiren
Coordinates and let me tell you how they work so you take a surface let’s take a surface here of genus 3 like this and you start by taking this surface and breaking it down into so-called pairs of pants by considering collections of simple curves that break this surface into pairs of
Pants so I just keep taking curves until I can’t take any more curves that are disjoint from the ones I had and this is called this is a pants decomposition I’ll call these curves alpha 1 and if you count them correctly just using the oiler characteristic you can
Find that they’re exactly 3G minus 3 of them and you take these curves and now you look at a simple curve that’s not in this set and you look at what it looks like on the surface and you can start looking at its intersections with exactly the Curves in your pants
Decomposition right so in my example here I took that red curve I look at the intersection with this curve this gives me one intersection point this curve I get two intersection points this curve I get one intersection point this curve I get one intersection point and this
Curve here I get one intersection point and these intersection points intersections with these alpha eyes these determine a curve [Applause] almost and I’ll tell you what they determined it up to in just a second but let me just tell you what they determine so when you have a pair of
Pants like this so now I look at one of my pairs of pants and well I have the intersection numbers that the curve has to form with these different elements okay so maybe here I have four three and here I have to be careful three something like this and now if I
Have these intersection numbers there’s only one way I can complete this picture right so so the the fact of the matter is that of course I’m going to say this and then not manage to do it but well in this case I can right so the picture necessarily looks like the
Picture I’m drawing here there’s no other way to complete this picture so I know that whenever I have these intersection numbers this is what my curve has to look like on the pair of pants so from a topological point of view there’s no other way to complete the
Picture okay of course this doesn’t tell me what’s happening exactly around these curves so these are called Dent twists or twist along these curves so what this picture does not tell me is what is this curve doing for instance it does not it’s not able to tell me it tells me
That it looks sort of like this and it looks sort of like this but it doesn’t tell me that this thing isn’t wrapping around here multiple times okay and and this is what so these intersection numbers give me a pretty good idea of what’s going on
But they don’t tell me how the curves wrap around these Curves in the pants decomposition and so for that I need other curves but I can still do it and so what I can do is I can look at this pants decomposition so this is one of my
Curves Alpha I and I add in two other curves which I’ll [Applause] call beta [Applause] I and Gamma I and beta I and Gamma I tell me the [Applause] twists all right it tells me the behavior of the curves as it wraps around the as it wraps around the pants
Curves and all in all if I put this all together I have alpha 1 to Alpha 3G minus 3 I have beta 1 to Beta 3G minus 3 I have Gamma 1 to gamma 3G – 3 and the intersections with this set of 9 Gus n curves tells me
Exactly what the simple curve is right so so the map if you give me a curve what I’m saying in other terms if you give me a curve and I look at an intersection with these 9 G minus 9 curves then that map is injective right so no two curves can have different
Intersection numbers and this gives me a nice parameter set for the set of Curves in some sense mhm okay all right as as I pointed out um I’m going to be jumping to the hyperbolic case for just a second so for the study of hyperbolic surfaces simple curves and their geodesic cousins
Have played an important part so in the hyperbolic case Okay simple curves simple Clos geodesics are interesting but rare okay so so so so let me give you a few uh a few results which which illustrate this point so so here’s a first result it’s a theorem due to
Huber which is the following thing so what you can do is you can you can count the number of closed geodesics s such as their length is bounded by some constant l so these are closed the number of closed geodesics of length bounded by L and what Hooper showed is
That I’m going to be using this sign for lots of things in this case it’s the just the ASM totic the number of these curves is ASM totic to exponential L / L and in contrast you could ask the same thing for simple Curves right so um this time we ask simple close geodesics and we look at exactly the same function okay and so this case we have exponential growth I mean something very particular and pretty that doesn’t depend on the geometry of the surface and in this case you get polinomial
Growth so this time get a polinomial it’s degree is 6G minus 6 and there’s a constant in front that depends on the geometry of the surface so this is this is a result from I don’t know I guess published in 2007 or or or roughly this is you know I
Don’t know this is 1513 I have no idea this all right so it’s it was before I was born everything was before I was born will be 15 13 right so um I mean I I know that’s not true but um I forgot that this talk is recorded okay I do
This all the time anyways so that’s that’s one way in which you you sort of see that that simple curves are very rare in a in a geometric sense among the set of all curves another sense in which they’re very rare is that closed geodesics on X are
Dense okay if you’re on a hyperbolic surface and you do exactly the the the the exponential flow that we were talking about for graphs before but this time on a hyperbolic surface then you’re always arbitrarily close to a closed geodesic okay you’re arbitrarily closed to coming back to where you were right
And this is in strong contrast to what happens for simple curves this is a beautiful theorem of Burman in series which in some sense was at the origin of lots of these results here is that simple geodesics are nowhere dense simple geodesics are nowhere Den okay so so let me take this
Opportunity to use the excuse of talking about simple geodesics to show you a few pictures there we go looks like I’m going to show you my vacation pictures but it’s not true okay so what you see here on the first slide this is um this is the best
Picture this is the most accurate one this is the set of the closure of all closed geodesics on a hyperbolic surface okay so this is this is a very this is a very important slide okay you’re going to see in contrast to the simple ones all right so this is exactly the part
That’s joint with Peter Boozer so let me tell you a little bit about what we’re seeing here so what you’re seeing here is a fundamental domain for hyperbolic surface so this is a polygon in the hyperbolic plane and what you see is a single closed geodesic right so so because the sides
Are associated when you reach one side for instance if you are going along here and you go up here then that side there is associated to this bottom side here then you will continue to go along here for instance okay and so so what you see
Here is is a single closed geodic and I’m just this is sort of a confusing thing but what you see now is a closed geodesic along which we’ve performed a dent twist so you see the effect of a dent twist along a single closed geodesic and the dent twist that we’ve
Performed is exactly along something that’s here so you sort of you see the effect of a dent twist along a single closure jesic okay so this is just to give you an idea of what one curve looks like and now we perform another dent twist and you see that there’s this sort of
Limiting Behavior all right I I really do feel like I’m showing my vacation pictures okay this is this here is an example this these are also these are more multiple simple closed geodesics but this time we’re on a octagon which is regular and this is the most symmetric genus two surface
That you can imagine and what you see here are all the shortest closed geodesics on that surface so one of these it goes from here to here is the shortest closed AIC on the surface you have one that goes from here to here you have one
That goes from here to here and then over here and from here to here and they’re all the same length in this particular case and now I’m going to start showing you the next ones and the next ones and then okay lots and lots more
And you see that they sort of have this beautiful structure this is particular to the case of this surface because the surface that I’ve chosen to to illustrate this with is a perfectly symmetric surface the most symmetric surface you can imagine in Genus 2 so you see a lot of the symmetry
So you first showed the shortest one then the second shortest one then okay yeah I mean more or less I I I I’m cheating a little bit so I left some of them out that were less pretty but you know yeah but more or less
Yeah but remind I mean keep in mind the theorem of of Burman and series which says that if you put all of them together then you still get to know where DSE set I mean you can sort of see that this these surfaces here are becoming they look like they’re becoming
More and more dense but in fact so let’s let’s look at this they’re they’re becoming more and more dense but in fact the Burman series theorem tells you that they’re they’re they’re never completely dense even though they they look like they sort of have they have this this density that’s sort of
Appearing so now we take this surface here this symmetric and we deform it so we deform it we look at a hyperbolic surface that’s homeomorphic to it but not isometric we’ve deformed the surface somehow and you see how the Burman series said or in this case this set of
Geodesics deforms as you deform the surface okay and it becomes this is one should take these pictures with a grain of salt in the sense that some of these phenomenon are really due to Genus 2 so you have this concentration in these points and these points and these points and these are
Due to the fact that these are particular points in Genus 2 which are called vir stress points in which infinitely many closed geodesics intersect so this phenomenon you wouldn’t necessarily see in higher genus but nonetheless because I get to show whatever I want this is what I’m showing
You and here you can see as you go as you go move up you can really see that concentration these points I’m telling you about these concentration points these virra points that are here and here you can really sort of see them as you look into more and more
Geodesics and again you can deform it and you can sort of see what’s going on okay so I me this is just as as as a as a remark this is something we proved with Peter Boozer a few years ago is that in fact on a surface of genus G
You can always find a dis that has a certain size that’s disjointed from all simple closed geodesics okay and all right so I mean some would argue that this is barely a positive constant even for genus equals 2 but nonetheless it is a positive constant I mean not that
Anybody in this universe would be able to see it but anyways but as you can see sort of from the picture at least and as you we’re going to zoom in in that picture you can see that I mean there’s a reason that this constant isn’t huge
Is that in fact for a set that’s nowhere dense I find them pretty dense right as far as nowhere dense sets go they’re they’re they’re they’re they’re quite dense um I also want to point out that this is an more important part is first we made the pictures and then we proved
That the pictures actually are correct so the reason is I mean I’m not going to get into this because I have other things to tell you but the reason is the way we make the pictures is that we go out we take very very long strands and
We make sure that there’s no self there’s no intersection points we haven’t created any intersection points and then after a while our computers give up right and they say okay enough right and then we show what we’ve produced but in fact this this theorem that’s up here that we are able to prove
Tells you that that’s fine in fact if you’re able to do that if you’re able to shoot out for long enough and you haven’t created any intersection points then in fact you’re very close to being what you’ve really described at a local level is really very close to
Being on an actual simple closed jesic and so so that’s that’s what this theorem says okay all right let me go back to the again compare this with with this beautiful picture of all closed geodesics okay okay so there’s lots of things that we could talk about with the the theory of
Simple closed geodesics one of the things that happens with simple closed geodesics and that distinguishes it from closed geodesics is the following phenomenon that does not happen for simple closed geodesics so let me tell you what this phenomenon is which is sort of one of the origins of
Why we were interested in this in the first place so these are called length equivalent curves and it’s the following phenomena okay so what is this so we have a hyperbolic metric so On Any X hyperbolic there exist Alpha I’m saying this the wrong way okay okay
Um okay so I got the order all wrong but I will fix the order in just a second all right so the phenomenon is the following for any Alpha and beta so let me erase that and put that over here and now any hyperbolic metric there exists so there exist
Curves closed curves such that their geodesic Representatives always have the same length right so so instead of telling you what that means let me show you a picture so here’s a pair of pants that I can find on any hyperbolic surface and my claim is that the following two
Curves always have the same length uhoh hold on I’m going to do something different I’m going to draw the pair of pants like this otherwise it’ll feel like I’m drawing it upside down I sorry all right so here’s a curve I leave from here I wrap around the boundary and then I come
Back in the back and then here I wrap around here I wrap around here I come back around the back and then here I come up here in front right do people see sort of the curve that I drew and here I’m going to draw another
Curve it’s the same pair of pants I’m just drawing another picture and this time whenever I was in front I’m going to go in back and every time I was in back I will go in front right so this time I’m going to wrap around like this wrap around like
This here in the back I go in the back and here I go in the front I go in the back sorry and I wrap around here and I wrap around here I look like that right all right so these are clearly two different homat classes of Curves well
Clearly let’s suppose that we all agreed that these are two different homotopy classes of Curves okay and they always have the same length no matter what hyperbolic metric I put on the pair of pants so why is that that’s because a hyperbolic pair of pants if I look at it like
This I can do the following trick I can look at an orthog geodesic so the shortest geodesic that goes between one boundary curve and and another boundary curve and here another Ortho geodesic and another Ortho jesic and when I cut along these this gives me two hexagons one in the
Front and one in the back oops okay I’ve cut along these two things if I if I took my pair of pants and I cut along exactly that seam there and the two seams there I would show you two hexagons and if my pants were hyperbolic which they are not my two
Hexagons would be isometric so why is that it’s because in the hyperbolic plane if you know the length of this this these sides a b and c then in fact you know exactly what the hyperbolic hexagon looks like it’s a small computation in hyperbolic trigonometry
And so that means that the front of my pair of pants looks like the back of my pair of pants so there’s a front back symmetry that brings the front of the pair of pants to the back of the pair of pants and what does it due to these homotopy classes
Everything that was in front goes in back and everything that was in back goes in front so in fact it takes these two homotopic classes and it reverses them and so that’s more or less modulo details a full proof that you can find curves that have the same length for any
Hyperbolic metric and so in a sophisticated version this is a result that’s due to horvitz and put in geometric terms by Randall and um and you can repeat this operation and you can do all sorts of things with this there’s lots of interesting questions about these curves
That are still not known but one of the questions that was raised at the time was that there’s a funny fact which is that um length equivalent curves so these are curves that have the same length no matter what hyperbolic metric you have intersect simple [Applause] curves so all simple [Applause]
Curves the same number of times [Applause] and this is a troubling phenomenon because it tells you in particular that knowing the intersection with all simple curves no matter how dense they look on the pictures that I’m showing you in fact will not help you distinguish between two length
Equivalent curves if your goal in life is to distinguish between the red curve and the yellow curve knowing the inter section with all simple curves will not help you okay and why is this this is not a difficult phenomenon to see this is simply because if I look at a
Hyperbolic metric and here’s a simple curve here’s a sort of a local picture of a hyperbolic metric something I can do to the hyperbolic metric one of the deformations I can do is I can take it I can squeeze it right unlike my own pair of pants I can
Squeeze this curve to be very very very thin and when I squeeze this curve the area has to stay constant so what happens is that this becomes very very long as I squeeze the curve but what happens to a curve that intersects it so if my curve intersected if the yellow
Curve intersected it once and the red curve intersected it twice this intersection will persist the yellow Curve will become longer but the red Curve will become twice as long roughly right you can’t see it because they’re they’re both they’re they’re they’re parallel traveling at one point okay and so this this phenomenon here
Here tells us exactly this fact right and so so that’s that’s an interesting phenomenon so you can have curves that intersect all the simple curves the same number of times and so immediately what do you guess I what did people guess at the time when people figured this out they
Immediately guessed Hallelujah that’s exactly what length equivalent curves are those are exactly curves that intersect all simple curves the same number of times and people believe this for some time this was sort of you know at the end of the Middle Ages and and so people would believe this and went
Around saying this repeatedly that this is exactly what length equivalent curves were until Chris leninger came around and said no that’s not true okay so a result of Chris linger it’s 2003 I think or something like like that it says the converse is [Applause] false and I won’t get into Lininger’s
Proof of this but I will show you a picture which is this picture uhuh so what is this picture now I’ve gotten to the picture which is good news for me and maybe bad news for those of you who can’t see red chock but this picture shows two
Curves and my claim is that these curves intersect all simple curves the same number of times right so that’s a very difficult claim to prove because I’d have to show the intersection with all simple closed curves so you have to take my word for you have to take my word for
It for that but I will show you that they’re not length equivalent right so why are they not length equivalent these two homotopy classes look very similar in fact the way I drew them was exactly the same until I got to here right if I went around here and I
Look at the homotopy class here here what I did was I crossed over once and here I didn’t cross over so in particular if I want to get from this homotopy class to this homotopy class if I look at this picture here and I do a
Zoom right what I see here is two red strands that are like this this is the zoomed picture what I can do is I can take those two red strands and I can resolve the the intersection by instead removing that red strand and going here and then going
Here and so in particular what that means is I can find a representative and if see if if I curve the if I curve the the the the curve a little bit then this curve is going to become shorter so I can find a shorter representative for
The yellow one than I can for the red one and that means that the hyperbolic Representatives necessarily have an order and length that’s strictly that’s a strict inequality so the length of Alpha and beta the length of alpha for any hyperbolic metric will be strictly less than the length of beta in
Particular they’re not length equivalent they always have this length relationship so this is a funny phenomenon so what do most people do when they see this type of situation well they give up right they give up okay that that wasn’t the right answer fine let’s go home do something else you know
Watch a nice movie or whatever but what do what did bin binu and I do we didn’t give up we we we on the contrary we thought this was this was the beginning of something else like oh so that means in some sense that there’s a there’s
Clearly a zoo of curves that intersect simple curves the same number of times but that aren’t length equivalent right and so so this prompted us to look at the following [Applause] definition okay so we say that two curves Alpha and beta are K equivalent if what happens so what’s the
So zero equivalent would be you intersect the simple curves the same number of times K equivalent is now you intersect the K curves the same number of times so K equivalent if the intersection between Alpha and Gamma is equal to the intersection between beta and gamma for all K
Curves okay so not the intersection with simple curves but the intersection with all curves that self intersect K times or such that its intersection with itself is equal to 2K as we defined before M and we’ll just write this we’ll write that in this case Alpha is K equivalent to Beta I’ll use
This notation here okay all right so in particular length equivalent curves are zero equivalent right so that’s an example of zero equivalent but there are examples of zero equivalent curves that are not length equivalent that’s the examples I showed you before all right so what did we show so our first result is
That in fact if you are K equivalent then you are also zero equivalent meaning that if you intersect all K curves the same number of times and in fact you intersect all simple curves the same number of times so K equivalent curves are zero equivalent right so so how would you
Want to prove something like that just from um you know if if you’re I don’t know a first year analysis student or whatever you you you you want to prove something like this by induction you say okay so zero curves that makes sense right so what you really want to
Prove is that if you’re K equivalent then you’re also K minus one equivalent and if you’re K minus one equivalent then you’re kus2 equivalent and so on and so forth right the problem is that that’s completely wrong because all other implications are wrong so all other implications [Applause] fail [Applause]
Okay meaning that if I take K and K Prime that belong to the nonzero natural numbers and K is different from K Prime then there exists Alpha and beta that are K equivalent [Applause] but [Applause] not tape Prime [Applause] equivalent okay all right does this uh does the statement at least make sense
So so as you can see this this is um this is a completely topological phenomenon it’s it has nothing to do with hyperbolic geometry even though it’s inspired in some sense by hyperbolic geometry it’s really a topological phenomenon and I’m just going to tell you one ingredient of how do how it
Works which in some sense was the the most surprising bit for me and is something that we called the pants Lemma and it works as follows so I’m going to write it in a certain way that’s going to look wrong and then people will complain and then I will
Explain what I mean and I’ll say that it was all in the subtext okay so if I have a closed curve then there exists a pany [Applause] composition P such [Applause] that in some sense let’s see so gamma on the surface on which I’ve cut along the pantsy composition is a collection of
Disjoint simple [Applause] arcs okay so so so so what do I mean what I mean is this I can choose a pants decomposition so of course my my curve this only makes sense if the Curve gamma is not simple if the curve gamma is simple then there’s nothing to do but if
My curve gamma is not simple then it means it has self intersection points that are not resolvable so it means that when I cut along this panty composition I have to still see these intersection points right so I’m not claiming that they’ve disappeared but what I mean is that they
Look like something like this this is what a pair of pants will look like once I’ve cut along them so this is I had one single curve and maybe this is what I see right so you say well that doesn’t look like a collection of simple
Disjoint arcs and I say well it does to a topologist because if I can change this picture here I can take the free homotopy class of arcs where I’m allowing myself to Glide the end points along the boundary and I can turn this into something that looks like exactly this [Applause]
So I’ve made them all disjoint by taking the free homotopic class of arcs so what you call what what you call Sigma minus p is p is the collection of Curves yeah p is the collection of Curves and sigma minus p is the collection of pair of
Pants okay so what so what I’m saying is that on each pair of pants I’m going to see a picture like this which means that I can sort of resolve these intersections into something like this this is so you might it might look like every everything no matter what
Pants stti composition I take I can do this but that’s not the case so let me give you a non-example if this is the picture I see on my pants DEC [Applause] composition oops right this is the picture I see on my pants de composition there’s nothing
I can do to resolve these intersection points right I can’t Slide the arcs in a homotopy class and get rid of those intersection points whereas here I can which in some sense tells me that if the curve looks always like this the intersections in some sense we’re pushing towards the neighborhood
Of the curves so what we really see is something like this this is one of my Curves in the pants DEC composition and my curve looks simple up until here and then it looks simple after here and all of the intersection is coming from some form of permutation of
These strands so maybe this one up here and comes here maybe this one goes across this one goes here and this one goes here okay so intersection comes from permutation in the [Applause] strands okay let let me let me just somebody was supposed to complain no yeah is is all right yes
Okay so would you like to complain so maybe something is missing in your statement or like it’s not it’s it’s just that when I say is a collection of disjoint simple arcs I should put up to homotopy where you were allowing the end points of the arcs to
Glide yeah so that’s that’s what’s missing in the statement okay um let me just end with some consequences of this and the techniques that we do okay so I’m going to say a couple of things all right I I’ll leave that because I don’t want to all right
So okay so the first consequence of this in some sense is that closed curves are really in some sense simple curves or multi- Curves this is why I’m getting myself into trouble with my own words but with [Applause] added crosses okay and what I mean here is that you’re you’re getting all the
Intersection from the permutations in the cylinder and this is this is related to a whole slightly different but related point of view by or lansen and suo when they explored generalizations of muraki’s curve counting functions in other contexts and this is their theory of what are called ralas for those who
Have actually looked at their their work um that’s one thing so it’s it’s one way to sort of model Close Curves the other thing is that you need lots of Curves to parameterize all [Applause] curves okay so there’s a theorem of OT from the 1980s which says the following thing that
Intersection with all curves determines a [Applause] curve so in the case of simple curves we saw that a finite number of Curves suffices to determine all simple curves the intersection with the finite number shows that you have to look at all I mean doesn’t show that you have to look
At he shows that you can look at the intersection with all curves this is related in his his point of view his theorem on the uh bonah Han’s geodesic currence this it’s more or less the same more or less the same statement and the last thing I want to point out is that
Sort of from our work what we get is that um you need infinitely many curves to determine K curves for K is greater than zero so more precisely if you look at the map if mu1 to Mu n are a collection of Curves then the map that goes from the collection of K
Curves to the intersection with exactly these curves for any finite set of Curves okay no matter what your size of your finite set is is always non- injective meaning that you have two curves that are not the same but that have the same intersection with all these curvs that you’ve chosen okay and
With that I thank you very much [Applause]