nicolascurien: The left and the right wings.
Courant Events Right: asked during my talk. But it's a lively talk, Okay, so.
nicolascurien: So, France said that, when a constant.
Courant Events Right: curvature, and when the curvature.
nicolascurien: during posting boundary, this is…
Courant Events Right: Old stuff, finite-dimensional, everything.
nicolascurien: My goal is to try to convince.
Courant Events Right: It's true that.
nicolascurien: This is not.
Courant Events Right: Old stuff. Almost nothing is known.
nicolascurien: Also, the geometric point of view.
Courant Events Right: Alright, so there are three,
nicolascurien: Words in the title.
Courant Events Right: random hyperbolic surfaces, let me try to introduce first hyperbolic surface.
Courant Events Right: So, what do I mean by surface? I always mean, 2D lightfall, which is orientable, so,
nicolascurien: No.
Courant Events Right: Mobius trip in this talk. And, have a.
nicolascurien: surface, they have many, many definitions, that's why they're.
Courant Events Right: to many areas of mathematics. So here are a couple of definitions. I won't go through all of those definitions, even if you don't know what a high public surface is.
nicolascurien: It will give you.
Courant Events Right: you were… depends on.
nicolascurien: definition in West Forest, but.
Courant Events Right: Yeah, that's.
nicolascurien: the metric definition as constant curvature Riemannian manifold, or model dependence, meaning that you have a manifold and you have that class of.
Courant Events Right: shorts.
nicolascurien: Where, coordinate changes are isometry of,
Courant Events Right: Also, hyperbolic plane.
nicolascurien: repeat them if they were nearby H2.
Courant Events Right: There is a purely a generic.
nicolascurien: definition of,
Courant Events Right: hyperbolic surface as quotient of PSL2 by some nice groups. There is also a uniformization point of view, because hyperbolic surface, in each conformer class, you can find one.
nicolascurien: One, focus on the future, is.
Courant Events Right: Only one opens.
nicolascurien: quantity, which is a resurface.
Courant Events Right: so on, but I will find.
nicolascurien: Focus on the hand.
Courant Events Right: definition where a public surface will be obtained by grooming elementary.
nicolascurien: and pieces.
Courant Events Right: Either pair of pencils.
nicolascurien: form, ideal handguns.
nicolascurien: Okay, so let's do that.
Courant Events Right: We speak up, well.
nicolascurien: Force 101 on.
Courant Events Right: I covered.
nicolascurien: spring?
Courant Events Right: So, here at Amazon, I thought we're playing.
nicolascurien: with the disk.
Courant Events Right: dimension.
nicolascurien: Trick is displayed, and.
Courant Events Right: I remind you that geodesics are arcs in the.
nicolascurien: for them.
Courant Events Right: Which are.
nicolascurien: authored.
Courant Events Right: equivalent to the binary. The binary.
nicolascurien: integer of the circle. And, so that's the play.
Courant Events Right: of the plane, and if you are looking at the topology of.
nicolascurien: disk…
Courant Events Right: with 3 moles, it's called a pair of pens.
nicolascurien: And, to step in the right one.
nicolascurien: Sorry for that. We go to the left.
Courant Events Right: More than to the right. So, a pair of pants is.
nicolascurien: Is this…
Courant Events Right: Disc with 3 holes, And then.
nicolascurien: It's a unique metric.
Courant Events Right: That you can put on this pair of pants if you specify the boundary length.
nicolascurien: So if the boundaries.
Courant Events Right: are forced to be geodesic boundaries of.
nicolascurien: then it's A, B, and C, then there is a.
Courant Events Right: unit, temperature.
nicolascurien: Everybody's my turn.
Courant Events Right: in there, then.
nicolascurien: gives you this length AP, P, and C for the length of the bets, if you wish.
Courant Events Right: Notice that one of these, or even the three numbers.
nicolascurien: can be zero, and when it's zero, it's called a puncture, and it.
Courant Events Right: could be a… If the.
nicolascurien: in December.
Courant Events Right: So this guy is.
nicolascurien: non-compact is if one of these three numbers is equal to zero, it's a counter.
Courant Events Right: Okay?
nicolascurien: And so, once I told you that those pieces exist and are unique for each triplet of a number, then you can try to pass them together to build a hyperbolic surface.
nicolascurien: How do you do that? Well, it's easy… oops.
Courant Events Right: You can imagine that.
nicolascurien: as you…
Courant Events Right: take those pair of pads and improve them
Courant Events Right: along their boundaries. Of course, boundaries' lengths have to match if you want.
nicolascurien: to do something.
Courant Events Right: This has to be… sorry, let's me… let's do that here. So this has to be.
nicolascurien: You can do this if you want to glue them together.
Courant Events Right: And you could then…
nicolascurien: Along the combinatorics of a graph, and the combinatorics of this graph.
Courant Events Right: We'll also fix for you the topology. Here, the topology.
nicolascurien: is you've got Genius 3, because you've got 3 holes.
Courant Events Right: And there are two boundaries, one here and one there.
nicolascurien: Okay? So, it's purely.
Courant Events Right: combinatorial.
Courant Events Right: I fixed for you just the boundary length, and there is also an additional parameter, which is the amount of twist
Courant Events Right: You perform before gluing the surface, the pair of pants together.
Courant Events Right: So, the nice thing is that if you fix such a combinatorial decomposition, so-called a pair of tense decomposition, then by.
nicolascurien: varying.
Courant Events Right: All those numbers, they're.
nicolascurien: Thanks and the twist.
Courant Events Right: You can get all hyperbolic serves.
nicolascurien: Profess with this topological type.
nicolascurien: Is it true?
Courant Events Right: So a constant curvature minus 1, yes.
Courant Events Right: So, as Frank said.
nicolascurien: It's…
Courant Events Right: It's really a finite dimensional stuff.
nicolascurien: So…
Courant Events Right: End of the story. You can get all hyperlix surface by gluing
Courant Events Right: Those pair of pants together, with the prescribed length and the twist.
nicolascurien: But it's the end, the end, but also the beginning, because.
Courant Events Right: I told you, you can, but there is actually many ways you can obtain the same hyperlink surface by gluing pair.
nicolascurien: There are many,
Courant Events Right: that gives the same hyperbolic surface. So it's an overparametrization.
Courant Events Right: That's the big problem, because if you want to understand the geometry of those surface, you want to somehow to get.
nicolascurien: better catalog.
Courant Events Right: way of,
Courant Events Right: is hiding your surface, and this identification is, is, honest.
Courant Events Right: Okay? But still, we can denote by…
Courant Events Right: This is better MGL1 up to LN, that's the moduli space, it's the space of all genius G, so G, the genius.
Courant Events Right: hyperbolic surface with geodesic boundaries, though, so the boundaries are the thing you see at the exterior, of lengths L1 to LN, geodesic boundaries. And again, when one of these Li is equal to zero, it's… It's just a puncture.
Courant Events Right: Okay, so that's this guy.
Courant Events Right: Is a finite dimensional space, if you wish.
Courant Events Right: And we want to, understand it.
Courant Events Right: So, it'll count.
Courant Events Right: It tells you that if you fix the genius, and if you fix the number of boundaries, the number of parameters you need to specify the space
Courant Events Right: 6G minus 6 plus 2N, and you need 2G plus n minus 2 pair of n, and we will focus more specifically on 2,
Courant Events Right: modular space, so the one that I denied by MG, which is the modular space of JUSG, no boundaries, but the JUS will be larger than two high public surface, and M0M, so now the JUS is 0, it's a planar surface, but I have N punctures, so…
Courant Events Right: 000 n times, and n must be bigger than 3. Okay, so I focus on those two extremes, high genius, MG.
nicolascurien: will be large.
Courant Events Right: N0N, 0 is fixed, the genius is 0, but the number of functions goes to infinity.
Courant Events Right: Question so far?
Courant Events Right: 0, 0.
Courant Events Right: If you want to go back to your nap, here is the take-home message. The take-home message, roughly speaking, is that this space…
nicolascurien: Presidents.
Courant Events Right: morally speaking, a compact manifold of finite dimension, and it's morally the same as the space of prevalent map with 2G minus 2 low. So, prevalent map is just, you start with.
nicolascurien: That's two nodes of potential.
Courant Events Right: Relax, and you pair them uniformly at random.
Courant Events Right: uniformly. There is no one on the net gate. That's a big space. You can enumerate how many maps there are there.
Courant Events Right: And there's a lot, lot of beautiful theorems in the combinatorics and polymeristic literature on those guys, Olabash, Erdogan, and so on, and so on and so on. And what we're trying to do now is to take all those beautiful theorems and to import them in the… in the hyperpublic setup.
Courant Events Right: So, same spirit for the low genius manipuncture. This is still also.
nicolascurien: Compact Medford of Finance.
Courant Events Right: dimension, which… should be thought of as the space of triggerment planar maps, now planar, with,
Courant Events Right: And those profits.
Courant Events Right: And then also, lots of beautiful results that we want to import to the hyperbolic third.
nicolascurien: Okay?
Courant Events Right: Everybody's happy?
nicolascurien: Yes!
Courant Events Right: You're… Specify yourself left or right.
nicolascurien: I mean, all…
Courant Events Right: No, it's this line?
nicolascurien: And what's… right. Okay, good friend.
nicolascurien: I thought the M0M would be by gluing the desert on the tree. Yes. It's truly, truly.
Courant Events Right: a picture of that, it's growing along a tree. So the tree-valent map behind.
nicolascurien: It's a tree.
Courant Events Right: You… competition aside.
nicolascurien: positive, because, so…
Courant Events Right: Did you hear the question?
Courant Events Right: Basically, for this guy.
nicolascurien: And it drove blue.
Courant Events Right: pair of pens.
nicolascurien: the police country.
nicolascurien: So it's not the plan of trying to… If you do…
Courant Events Right: Well, that's.
nicolascurien: to teach you.
Courant Events Right: blue. Ideally.
nicolascurien: equivalent.
Courant Events Right: ideal triangle, not equilateral, then it would be mass status.
nicolascurien: behind. Yes?
nicolascurien: You're gonna tell us about the interview databases?
Courant Events Right: But that we can't do.
nicolascurien: Just true. Let's make the competition more interesting.
Courant Events Right: If you have a good question.
nicolascurien: Yay!
nicolascurien: But that one for me once, huh?
Courant Events Right: Okay, so first thing…
nicolascurien: That you won'.
Courant Events Right: to understand is.
nicolascurien: No vulnerability just purely determinates depression, you fix, and observe that one.
Courant Events Right: A generic quantity code is phi, and you want to.
nicolascurien: what are the values that Phi can take.
Courant Events Right: Over the modularized space.
nicolascurien: Okay, so please…
Courant Events Right: What?
nicolascurien: And try to understand, for example, extreme.
Courant Events Right: property, what are the lowest.
nicolascurien: principal value.
Courant Events Right: for a genius key high-level self-este.
nicolascurien: Although not just in the service.
Courant Events Right: purely deterministic.
Courant Events Right: Then there are round-up questions. Now, you won't be.
nicolascurien: also the behavior of this observable for roundups for this SG. If SG is picked.
Courant Events Right: in your space, but for this, you need.
nicolascurien: We need to specify the.
Courant Events Right: measure.
nicolascurien: the form of information that you will use. So there are many choices at Rio.
Courant Events Right: Okay, so let's start first with this, deterministic extremal property.
Courant Events Right: So the…
nicolascurien: So, I give you a list of some options.
Courant Events Right: Sables that you can.
nicolascurien: do we have?
Courant Events Right: The first one that comes to mind is the diameter.
nicolascurien: So if your surface is compact, the diameter is the maximal distance between.
Courant Events Right: the two points, of course.
nicolascurien: Of course, if you have a cusp.
Courant Events Right: Then it's infinite.
Courant Events Right: One which is really important for hyperbolic.
nicolascurien: speak German.
Courant Events Right: is the system. It's the length of the smallest.
nicolascurien: Non-trivial loop.
Courant Events Right: Here.
Courant Events Right: You also have an injectivity radius, so if you have a point on your surface, it's the largest point that you can put around this point.
nicolascurien: That is, that is, exactly the same as in a hyperbolic plane.
Courant Events Right: Building upon the sister you.
nicolascurien: I also want to ask about.
Courant Events Right: All the possible lengths.
nicolascurien: Of all loops, not only the smallest one, so it's.
Courant Events Right: all the length spectrum, I will come back to that.
Courant Events Right: The Jigger constant, also, which measures.
nicolascurien: The expansion property of your surface is, of.
Courant Events Right: for high interest.
Courant Events Right: The total pants length, so this is the… smallest… Length of curve you cut.
Courant Events Right: You'll suffer.
nicolascurien: best for me.
Courant Events Right: On your pair of pants, so it's…
nicolascurien: The minimum of all pan decomposition of.
Courant Events Right: The length, some of the.
nicolascurien: all of the curves.
Courant Events Right: And there is the spectral gap, so the spectral gap is.
nicolascurien: Good.
Courant Events Right: There's no trivial eigenvalue for the…
nicolascurien: aggressions that come on the surface. How to display that? So, this is my risk.
Courant Events Right: Representation of the spectra gap.
Courant Events Right: Alright, so there are many geometric and spectral observable that you can look at.
Courant Events Right: And just to tell you this is still active research, let's look at what we can say.
nicolascurien: but those…
Courant Events Right: Quantities, again, this is purely deterministic.
Courant Events Right: So what about the diameter? The diameter, you've got trivial bounds,
Courant Events Right: The diameter must be at least Logi, for example, But, but…
nicolascurien: Must be finite.
Courant Events Right: The system, it's easy to construct examples.
nicolascurien: Where the system is advertised for when the changes.
Courant Events Right: not…
nicolascurien: But the system is less than 2HG, it's a bit the same argument as this one.
Courant Events Right: The Jigo constant, also here you've got trillion bounds, you can construct surface with small Jigo constant, but the fact that there are a surface with a Jiga constant bonded away from.
nicolascurien: ruined.
Courant Events Right: G goes to infinity. This is a highly non-trillion. It's the construction of expanded.
nicolascurien: surfaces will go back.
Courant Events Right: to Margulis, Margulis, and so on. So this, this…
Courant Events Right: The fact that this guy is strictly positive, it's already non-trigger.
nicolascurien: The same false big tide gap.
Courant Events Right: The Jigger constant, by the way, is always less than 1 by comparing to the hyperbolic planes.
nicolascurien: Because there's one question.
Courant Events Right: by comparing to the universal cover as well, and so on and so forth. Yes!
Courant Events Right: Is that the same one quarter, that the…
Courant Events Right: No, no, so…
Courant Events Right: So go over here.
Courant Events Right: But still, it's a question.
Courant Events Right: You're lagging, guys.
Courant Events Right: Okay, so this was morally the state-of-the-art in the.
nicolascurien: 80s?
Courant Events Right: And now, just for the systol, the baseball up to now is still due to Fuso and Sarnac94. It's larger than 3 quarters, and it's not known whether this is the good,
Courant Events Right: bound, or the two locations, between 3 quarters
Courant Events Right: 4 thirds only. 4 thirds and 2 lucky.
Courant Events Right: What the ?
Courant Events Right: has been improved a little bit from the trigger bounds by Good Marier Young to, G to the 7 of a 6.
Courant Events Right: We recently proved with Domavilarsky and Brian Petri that actually the diameter, the lower bound, is the good answer, asymptically, log G. And it's quite recent, 2019.
Courant Events Right: So, here…
Courant Events Right: This was a breakthrough by Hyde and Meigi a couple of years ago that constructs hyperbolic surface whose spectral gap matches the asymptotically the upper bound, one quarter, and I'll come back to that in a minute.
Courant Events Right: And we, proved two years ago, or 1 year ago with this, my, like I said, friends, Thomas and Ram, that the Jigger constant, actually, is less than 2 over pi, 2 over pi is…
Courant Events Right: It's not the correct answer, I mean, prove the upper bound, but what's nice is it's not the upper bound, so it's strictly less than what it is in the hyperbolic plane.
Courant Events Right: So you see, this is still active, and we don't have yet the full answer for many canonical observables, geometric observables, for hyperbolic surfaces.
Courant Events Right: Questions?
Courant Events Right: So, although that seems very deterministic statement, that they are, deterministic theorem, in all of these groups.
Courant Events Right: Maybe not this one.
Courant Events Right: But all the rest.
Courant Events Right: Hondomeness is involved by… randomness is used to construct examples here of minimizing diameter of, here also for the Jigo constant.
Courant Events Right: expanders also, these examples are produced by taking random hyperbolic surfaces and so on. So, even though these are deterministic results, the core of the proof uses randomness.
Courant Events Right: So now… On the next, if you want to sample a hyperbread surface at random, I actually.
nicolascurien: Can you…
Courant Events Right: which measure do I use?
Courant Events Right: So they'll.
nicolascurien: How many, many.
Courant Events Right: Many, many, combinatorial, combinatorial,
Courant Events Right: definition of construction of random surf… of hydraulic surfaces. Hence, there are a lot of discrete measures that you can build using those decompositions. For example, I told you about the path decomposition. You can just imagine that you put the… your length at random, or the trivalent graph
Courant Events Right: beneath the consumption at random, and then this will produce for you, random.
nicolascurien: boom.
Courant Events Right: hyperlink surface. Yes, you really want to win the challenge.
Courant Events Right: So, in the case.
nicolascurien: So these graphs instead of surfaces, like, so there was the standard, there was a very difficult question about what's heat and what kind of… Yes, but…
Courant Events Right: Before that, actually, the first construction of expanders goes back to Kolmogorov, and it was random.
nicolascurien: So in the surface is…
Courant Events Right: There's a random.
nicolascurien: construction, but is there also a more additional study for realistic one?
Courant Events Right: So the… those Lubotsky, Philips Sarnac constructions, they are… they are, they are deterministic, but they are more complicated to explain than the random construction.
Courant Events Right: At least 20.
nicolascurien: Yes, sir?
Courant Events Right: You're taking the lead. Okay, so these are a lot of discrete measures, I would say, which are used.
Courant Events Right: scheduled use theorem, I told you, to prove the dynamic result of the extremal properties. But the measure I will focus on now is.
nicolascurien: measure, that's… Most people over here know that we're Indians.
Courant Events Right: as a measure on the molecular space, it could be thought of as the Levesque measure on this space.
Courant Events Right: Why is it so? So there are many ways to say it's.
nicolascurien: measure, comment.
Courant Events Right: the safest point.
nicolascurien: it's…
Courant Events Right: using pendant.
nicolascurien: composition?
Courant Events Right: So I told you that if you fix this, this combinator.
nicolascurien: pair of pads.
Courant Events Right: If I vary the length here.
nicolascurien: And the twist, and put them as real parameters.
Courant Events Right: Then…
nicolascurien: This…
Courant Events Right: Our role is subjective, by…
nicolascurien: angles.
Courant Events Right: 6 zigma.
nicolascurien: those six parameters, I can get any hyperbolic surface.
Courant Events Right: So what I can do is I can take a little big measure here, and take the push forward. Of course, I mod out by identifying two guys, if they are the same hyperbolic surface.
nicolascurien: And… non-trivial.
Courant Events Right: In fact, that after.
nicolascurien: those identity.
Courant Events Right: if I modify.
nicolascurien: by,
Courant Events Right: working class group, then I get a finite measure done down.
nicolascurien: on human.
Courant Events Right: the modularized space, which is the perpetestant measure, and so.
nicolascurien: So, it is.
Courant Events Right: forward of the.
nicolascurien: Liebeck measure.
Courant Events Right: modding out by this… this.
nicolascurien: section.
nicolascurien: for any And the competition.
nicolascurien: Good.
nicolascurien: So, how about.
Courant Events Right: probably, you all heard about, the breakthrough of Marianne Merzakani, who was probably the first to, to study those, those, surface from a probabilistic point of view, and the key ingredient for the…
Courant Events Right: random statements that will come are estimates on the volumes. So what do I mean by volumes? I mean, you just… DB is a measure, right? And so you just want to understand the volume, the measure of this space, the genius T with end punctures.
Courant Events Right: content, village N, and it turns out that this…
Courant Events Right: So, here we are produce new regulation for this variant, who enables her and the graph to, for example, make those type of assemblies. Those ascentities are for N0 fixed, and the genius goes to infinity. And these are the fundamental breaks, or the fundamental.
nicolascurien: improvement in the coming Sera.
Courant Events Right: From those new recursions, she was able to get a new proof of Beaten's conjecture, but more important for us here, she gave the first indication that this random vector acid surface, so what do I mean by random vector cell? I sampled.
nicolascurien: What do you mean?
Courant Events Right: energy according to the well-Meterson measurement.
Courant Events Right: So, typically, a guy like that.
Courant Events Right: We'll have a Jiga constant, which is strictly positive, with high probability, when G goes to infinity. We'll have a spectral gap, so there will be expanders, but the diameter will be less than 40 degrees.
Courant Events Right: It's just a bomb, we don't know whether it's 2 of G, one of G, or even if there is a low attached number, we don't know. And so, between the ideas, it's fine.
Courant Events Right: Okay, so this was breakthrough from 2007 to, 2007.
nicolascurien: 13.
Courant Events Right: But…
Courant Events Right: As a take-home message, let me give you a glimpse of what we are doing with Timothy Bird, Don Martinsowski, and Brad Petrie. It's somehow reinterpreting those
Courant Events Right: famous recursion of Mirzakani into the realm of this appealing exploration of random surface.
Courant Events Right: Don't worry!
Courant Events Right: what I'm going to say here is purely deterministic.
Courant Events Right: my reinterpretation will just be the fact that… so this T is nothing but R to the 6G minus 6 diabetic space.
Courant Events Right: This is just a pair of pads, and I told you there are many, many ways of gluing pair of pads to get the same erotic surface, and so what I will produce for
Courant Events Right: building on the academic side is to.
nicolascurien: intersection.
Courant Events Right: Can't go over the.
nicolascurien: That's it, but…
Courant Events Right: give you a hyperbolic surface, I will give for you one, only one, per decomposition of that surface. Canonical one.
Courant Events Right: So this canonical fan decomposition can be obtained as follows.
Courant Events Right: Here is your surface that you don't know, you want to encode it, put it to your, to your best friend.
Courant Events Right: So, for that, you will have… if you can get an algorithm, an algorithm is just here a function, and this function will tell you, given this function array, I want to start… is it an ant?
Courant Events Right: A little ant. I want to start a little ant at that point here. And this little ant.
nicolascurien: to ENTER my surface.
Courant Events Right: orthogonally to this boundary at that point, okay?
Courant Events Right: And this ant will,
nicolascurien: I can.
Courant Events Right: produce a little bit of a pheromone, you know? And when she crosses her past. Path.
Courant Events Right: or above.
nicolascurien: She will notice it.
nicolascurien: Okay, so she…
Courant Events Right: gets into the surface until it crosses its past.
nicolascurien: Ugh.
Courant Events Right: At this point. Then I claim that at this point, if the ant has noticed something strange.
Courant Events Right: its path or the boundary, then it must have discovered some new terminology. What do I mean by new topology? I mean that what you see here, the union of the blue curve
Courant Events Right: End of the white curve.
nicolascurien: It's topologically a pair of pants.
Courant Events Right: Indeed, imagine that you put Sprint!
nicolascurien: This is a little.
Courant Events Right: spring around here, and another spring around there. Those springs, they are not geodesic, but when you wiggle them a little bit, they will contract.
nicolascurien: Until they find.
Courant Events Right: their geodesic components.
nicolascurien: Do you?
nicolascurien: babies, Dad!
Courant Events Right: When they do so, the door spring will define a pair of pants.
Courant Events Right: In which the end was in. The end has not exceeded this boundary.
Courant Events Right: Okay? So that's a purely local way of exploring myself.
nicolascurien: Reference by an end?
Courant Events Right: and discovering one pair of pads. Then you cut at those points, at those.
nicolascurien: Those beds here, and you'.
Courant Events Right: You got a new surface. Yes!
nicolascurien: You can get lost in, but with very, very.
Courant Events Right: Probability 0, it's generic.
Courant Events Right: what I'm saying is, is.
nicolascurien: to say.
Courant Events Right: is the.
nicolascurien: in the extension. We'll probably.
Courant Events Right: You may… you should say, oh.
nicolascurien: It's a very.
Courant Events Right: when you need double penetration measure. No, no, no, no, it's generic for anything that will be…
Courant Events Right: would have, it's generic.
Courant Events Right: Yes.
nicolascurien: You take the shortest root curve?
Courant Events Right: Terrible.
Courant Events Right: No, no, the end is just a geodesic that has been shut orthogonally to the binary.
Courant Events Right: Okay.
Courant Events Right: Sorry? Sure.
Courant Events Right: That's my algorithm that tells me that. My algorithm is a function, so actually, when you start.
nicolascurien: There is a mark quote.
Courant Events Right: And you are presented. If you see a mark point, then you start 2 meters on the right of this mark point.
Courant Events Right: That's your algorithm, it's a function that you have your work for.
Courant Events Right: pure instrument.
Courant Events Right: Then, you go along with what's remaining. So.
nicolascurien: Okay?
Courant Events Right: You look in your pocket, in your pocket, you see, oh, if my surface is made of two holes of that length, blah blah blah blah blah blah, then I want to start from this point.
Courant Events Right: It's in your function. Then you shoot your new end at that point.
Courant Events Right: Do the same thing, she will grow.
nicolascurien: After opening, until it is.
Courant Events Right: Either the battery or its, path.
Courant Events Right: It defines a pair of pants in which the ant was in, you cut, and you go on. Your integration tells you, now I want to shoot my hand from this point up, new pair of pants.
Courant Events Right: And this drawing took me 1 hour. This one.
Courant Events Right: This is a pair of pants.
Courant Events Right: You cut?
Courant Events Right: So I'll let you 20 seconds to see what is going on.
Courant Events Right: So you imagine that you put a little spring here.
nicolascurien: here?
Courant Events Right: And a spring around there, here, Shhh.
Courant Events Right: They will find their… Gender Z representative, and you cut.
Courant Events Right: Okay, everybody got it?
Courant Events Right: Do we agree?
nicolascurien: And so on and so forth.
nicolascurien: So…
Courant Events Right: In this way, I have decomposed my surface, In two weeks.
Courant Events Right: A pair of panic decomposition, only using my algorithm, and this algorithm was purely.
nicolascurien: Local, and… Based only on the legitimity of the.
Courant Events Right: surface, not its stash number.
nicolascurien: I wasn't that easy.
Courant Events Right: It is something that is defined on modularized space. It's.
nicolascurien: was… I was just using the isomer.
Courant Events Right: Because… of,
nicolascurien: the surface.
nicolascurien: Okay?
Courant Events Right: But what is important in this
Courant Events Right: special decomposition, is that those things, they are special.
Courant Events Right: It's not any pent. It's a pent in which, if I start.
nicolascurien: this ant, and then I know that the ant will exit.
Courant Events Right: Horses pass or exit the boundary inside this path. That's a very special property.
Courant Events Right: For example, here, if I start the end here, then she cannot do that.
Courant Events Right: Because by definition, the PEMP was defined, by the first crossing of the end of its path, or the Pandarios.
Courant Events Right: Okay? That's a very special property. So that's the determination.
nicolascurien: Do you think…
Courant Events Right: block, it mirrorsaken execution, and from that, now if you put product
Courant Events Right: measure. Then, the fact that those plants are special is the so-called machine.
nicolascurien: identity.
Courant Events Right: And then this decomposition will immediately give you the…
nicolascurien: But… You can take this as a deterministic.
Courant Events Right: Way of encoding your suffix.
Courant Events Right: Yes!
Courant Events Right: Don't pray anymore. Stop. Yes?
Courant Events Right: So I see the length of the cast, but where are the twists? Yeah, the twist…
Courant Events Right: It's, it's here, I, you know, you have to remember this information.
nicolascurien: But this…
Courant Events Right: input is glued to this one. So the twist here is value met with 0 and 2 pi, let's say, because I don't need more than 2 pi.
Courant Events Right: Goodbye.
Courant Events Right: Oops.
nicolascurien: The twist here is… okay, let's go back here, maybe.
Courant Events Right: Those two points, I tell you I need to glue this one to this one.
Courant Events Right: And when I discover this, I have this point, and I tell you, that this point is the same as this one. And this will encode the… this will encode the twist. How do you… Yeah, that's the initial three-point.
nicolascurien: But then, when…
Courant Events Right: When you… once you discover that from… okay, cut out your survey, this is not relevant, but…
Courant Events Right: the ant, or the family of ants here, she knows… they know his point, and they know this point, and the fact that you get here, you have an additional.
nicolascurien: Should this work be the same as this one?
Courant Events Right: And this will encode.
nicolascurien: Oh, your, your proof.
Courant Events Right: Okay, yes?
nicolascurien: Are you asserting that with this decomposition, for each band, there is no compatible
nicolascurien: Yeah, so this was the question that has been mentioned already here today in Spain.
nicolascurien: Very tiny probability, then, that your ant is doing.
Courant Events Right: something.
nicolascurien: Like this, but that's a zero probability event, actually.
Courant Events Right: So if you're.
nicolascurien: Jake!
nicolascurien: If you have fixed your algorithm in your pocket.
Courant Events Right: If you'll.
nicolascurien: surface is generic, you will never encounter this.
nicolascurien: Your end will always, with probability 1.
Courant Events Right: Always cross this path.
nicolascurien: Or, come back to a new vulnerability.
Courant Events Right: If… There's something I missed, which is, how do you keep your…
Courant Events Right: What's your algorithm telling you about Run 30.
Courant Events Right: It takes you, if you have 3 more, I've got 1, 2, 3 components, but they put their points on them.
Courant Events Right: Yeah. Let's say you have one monitor here, also, you have one point at the right.
nicolascurien: you know.
Courant Events Right: Is that Mark? Yes, it's not covered.
Courant Events Right: Okay?
Courant Events Right: Any other questions from that side?
Courant Events Right: Good!
Courant Events Right: So that's the deterministic… algorithm.
nicolascurien: Underlying…
Courant Events Right: Once you have these numbers, those very Peterson volumes, then you can do a lot of probability. Very recently, so I summarized this in one slide, but there has been a lot of work on the spectral gap.
Courant Events Right: And also those, those community teams, those, those other teams.
Courant Events Right: And manage now to show this nearly amineering phenomenon that a typical Ben-Peterson surface, SG, will have.
nicolascurien: spectral gap, which is almost maximum.
Courant Events Right: Ultimate, etc.
nicolascurien: With very high probability.
Courant Events Right: All of those works.
nicolascurien: are based on.
Courant Events Right: of course, on the Selbert Press formula, and you need to put in this formula, so if you don't know, it relates,
Courant Events Right: spectral information to length of curves. So, to put… to have some control on the length of closed geodesics, you need to understand those numbers, volume of modularized space with
Courant Events Right: Genius G and say boundary of lengths N1 and.
nicolascurien: So it's… it's an ingredient in their proof.
Courant Events Right: Okay?
Courant Events Right: So by now.
nicolascurien: How do we know that?
nicolascurien: Sun surface is nearly a million.
Courant Events Right: Clearly, the optimal spectral,
nicolascurien: Yep.
Courant Events Right: So that was for the specs.
nicolascurien: Listen Trump.
Courant Events Right: Now, the length central, So what is the length spectrum? If you have a surface.
Courant Events Right: Hi, Beverly self is here.
nicolascurien: I can…
Courant Events Right: Draw all possible close curves on this one, and take their geodesic representative.
Courant Events Right: So, you have a curve like that, you have one like that, and so on.
Courant Events Right: There is only a discrete set of those guys, and you can measure that.
nicolascurien: your legs.
Courant Events Right: And actually, their legs will never… the red one here is probably the shortest.
nicolascurien: The orange is the length of this one, which you see.
Courant Events Right: go around, below, and come back. You also, you may.
nicolascurien: may do several turns, also, that's a green one, and so on. So, imagine all possible curves.
Courant Events Right: topological type of curves on those surfaces. Take their jersey.
nicolascurien: quantitative?
Courant Events Right: their length, this will give you a.
nicolascurien: prediction of numbers.
Courant Events Right: Positive numbers.
Courant Events Right: In fact, they will not accumulate near zero.
Courant Events Right: They will…
nicolascurien: only accumulate towards infinity, and so that's a random.
Courant Events Right: Collection of number.
nicolascurien: Random, because the surface itself is random.
nicolascurien: Then, again.
Courant Events Right: By putting in the machine those, estimates.
nicolascurien: volumes, mes incognitrie, where.
Courant Events Right: forgive him?
nicolascurien: that this…
Courant Events Right: collection of numbers, when the genius goes.
nicolascurien: for infinity.
Courant Events Right: Convergence is.
nicolascurien: distribution?
Courant Events Right: In other words, the Poisson point process with an explicit intensity, this cosh-1.
nicolascurien: But…
Courant Events Right: In particular, they get,
nicolascurien: I don't.
Courant Events Right: Distribution of the system.
nicolascurien: decreases.
Courant Events Right: The sun is just the first of this point.
Courant Events Right: And what's… Still kind of unknown, or not understood.
nicolascurien: It's the universality of this Poisson-Poly process.
Courant Events Right: Because it also pops up as, the…
nicolascurien: aspect.
Courant Events Right: of other quantum surfacing hygienus.
nicolascurien: like, Honda.
Courant Events Right: maps, or tight.
nicolascurien: Thank you.
Courant Events Right: spectrum of,
nicolascurien: Elbita sun surface.
Courant Events Right: This!
Courant Events Right: No, no, no, no, no, because there is only one place where the system is small. The system actually converges towards a random number, but which is simply positive.
Courant Events Right: So you want… For fixed genius, for fixed genus, but… Yeah, yes, that's… no, no, even if the genus still infinity. That's… that's, so for fifth, G does not calculate, but their theorem is for G equals to infinity.
Courant Events Right: So you have to have a place where the system goes to experience.
Courant Events Right: It is, it is. But this comes from, asymptotic estimates on theson volume with large G and a small, small bounding.
Courant Events Right: That is surprising, but… You can do a computation for random triggers as well.
Courant Events Right: So, for random three-round graph, you know that if you take a 3-gram graph with many LEDs, you pair them at random.
Courant Events Right: Then… It's possible that you won't see smoke cycles.
Courant Events Right: It's possible.
Courant Events Right: Of the smallest cycle converges the distribution.
Courant Events Right: Symphony.
Courant Events Right: If I forget to count the points, please tell me.
Courant Events Right: It's Rick against… Writing.
Courant Events Right: Okay.
Courant Events Right: So that was for high genius. Now let's… Let's move on to the…
Courant Events Right: Genius Zero planar case with punctures. So it's a hyperbolic surface.
Courant Events Right: Sample according to Will Peterson, genius zero, and punctures. So, as you said, it looks like a tree, so it depends decomposition.
nicolascurien: If you were to decompose.
Courant Events Right: Put this in a pair of pants, you will get a tree.
Courant Events Right: But there are other ways to decompose.
nicolascurien: Who's…
Courant Events Right: IDF Tigers, for example.
nicolascurien: And you would get more, like, trivalent planar map heading there.
Courant Events Right: So let's denote…
nicolascurien: SN with this dot.
nicolascurien: A valPatterson surface with JS0N punctures, sampled uniformly.
nicolascurien: And this point, 0 is just a reference point picture.
Courant Events Right: Uniformly in the surface, according to.
nicolascurien: pizza.
Courant Events Right: hyper.
nicolascurien: ignition.
Courant Events Right: Then we proved that last year with Timothy that
Courant Events Right: We have two types of convergence of those surfaces. You can first sit on this reference point and let n go to infinity, and look at what you see around you. This is a local convergence with retronormalization, and this would converge towards an infinite high public surface.
Courant Events Right: Which is homomorphic to the blind minus discrete punctures.
Courant Events Right: But it's random, something random in the limit, with a simulation.
Courant Events Right: But you can also see this SN0 as somehow.
nicolascurien: Compact, excuse me, it's not compact because of the punctures, but you cut the punctures.
Courant Events Right: Then you would get.
nicolascurien: compact metric space.
Courant Events Right: And if you renormalize all distances by N to the 1 quarter, then you get a convergence towards universal variance here.
Courant Events Right: Which is the random fractal.
nicolascurien: space that appears.
Courant Events Right: also as a scale.
nicolascurien: a bit of… Totality, and so on.
nicolascurien: So, let me give you…
Courant Events Right: aesthetic flavor of that. How much time do I have?
Courant Events Right: Kendrick?
nicolascurien: Including questions.
nicolascurien: Okay.
nicolascurien: Digital type.
Courant Events Right: Okay, so we have this…
nicolascurien: It's illuminum.
Courant Events Right: And this…
nicolascurien: Mr.
Courant Events Right: Totally.
nicolascurien: Maybe in the.
Courant Events Right: representation of SN. So, what is this? So SN is hyperbolic.
Courant Events Right: surface, KSD, or hand punctures.
Courant Events Right: So, it can be seen as a holy crap in the high public street space where the vultures are at the boundary.
Courant Events Right: That's a theorem of really, and there is a huge such representation, so that what you see here in 3… in the 3D hyperbolic space, is exactly the hyperbolic surface as
Courant Events Right: Okay, so this guy, if you sit at the random point here.
Courant Events Right: and then n to infinity, then you will converge as the number of punctures goes to infinity towards a random landscape of punctures. You're sitting inside, and you see something around. Yes?
nicolascurien: Whose cells are the, like, the journalistic sphere?
Courant Events Right: Yes, yes.
Courant Events Right: These, these are, half spheres, so they are just… And…
Courant Events Right: Although this picture is misleading, because this is not what we prove. We… we cannot, would be very happy to understand what is… what is doing with this simulation, with the repetition of points on the binary. What we prove is the metric aspect of this.
Courant Events Right: We have metric space here, we sit at a point, and we converge towards the random metric space that we see here.
Courant Events Right: It's only metric, nothing…
Courant Events Right: But this convergence tells actually a lot, it tells a lot about local statistics. For example, if you sit at a random point.
Courant Events Right: and look at the ball of radius 15 around you, how many punctures can you capture within this ball of radius 15? That's a random number, and as n goes through 15, converges over the same random number defined here for this guy.
Courant Events Right: Yes? Quick question, I missed something on the previous slide. So, you said that in the local topology, this is displayed by this countable set of points. Is it true that there is nothing specific about Z2?
Courant Events Right: Well, I said public. Okay, excuse me. It is a high public service with some government functions, which is on the market to, to… Yeah, yeah, yeah, you could take whatever you wish. Thank you.
Courant Events Right: It's time to wake up!
Courant Events Right: Yep.
Courant Events Right: Any other questions from that side?
Courant Events Right: Nope.
Courant Events Right: All right, and on the other side, so that was the benchmark local convergence point of view, on the other side, it's this scaling point of view, where now we see that as a metric space, and you rescale the metric by the end of the one quarter, and you converge towards this buoyancy
Courant Events Right: So the one you see here, I took a very beautiful simulation. If you don't know what to do during my talk, just Google Schauffeur and look at his amazing simulations. They are turning.
nicolascurien: Beautiful.
Courant Events Right: And this season.
Courant Events Right: Also, again, nothing… our conversions has nothing to do with the embedding that you see here, it's just purely metric, and this… this metric space here is a compact, random compact metric space.
Courant Events Right: So what… what does this conversion tell you about this? It tells you about, for example, distance between k points. If you sample k points uniformly in there.
Courant Events Right: in there, K points. Then, the metric.
nicolascurien: So, for the.
Courant Events Right: distances is a random matrix. If you rescale by n to the 1 quarter, it will convert towards the same random matrix here that you would have between any k points.
nicolascurien: convoluted for me.
Courant Events Right: For those specialists in the room, whoa, whoa, whoa, whoa, whoa, how do you get to grow.
nicolascurien: of Hosdorf convergency, because this object is non-compact.
Courant Events Right: True. Luke?
nicolascurien: written grammar.
Courant Events Right: So there are two ways to cope with this problem. Either you just cut, if you have a problem, you cut the problem. Either you cut the.
nicolascurien: It's possible to do this in a canonical.
Courant Events Right: And if you cut the punctures, then you are back to.
nicolascurien: We'll compact thing, and then we have got a.
Courant Events Right: of house, convergence. But if.
nicolascurien: You don't want to.
Courant Events Right: Look at the punctures, this gum of poker off captures
Courant Events Right: But the measure is telling you what's the measure.
nicolascurien: It's a high probability.
Courant Events Right: measure, and then.
nicolascurien: There's very little mass around the cusp, around the punctures.
Courant Events Right: So, actually, they kind of disappear in the limit.
nicolascurien: I don't see that here.
Courant Events Right: object is non-compact in the limit. This is a compact object, or if you wish, the measure here will accumulate on a compact
Courant Events Right: of metric space.
Courant Events Right: So, what type of observable also you can,
Courant Events Right: you can, put in this Gromof, Hausdorf or Gromof-proof convergences, length of separating, curves. You won't say a curve.
Courant Events Right: In your metric space.
nicolascurien: that would…
Courant Events Right: That would separate you?
nicolascurien: put my face.
Courant Events Right: into two pieces containing at least
Courant Events Right: N of the 4 punctures on each side.
Courant Events Right: Then you want to minimize over the length of all those curves. This is an observation that you.
nicolascurien: You can put this.
Courant Events Right: convergence.
Courant Events Right: Okay, so what are the methods we're using?
nicolascurien: used to prove this theorem.
Courant Events Right: In principle, this is actually.
nicolascurien: accessible.
Courant Events Right: If the peeling,
Courant Events Right: method I described to you with this end process and so on, which is easier in this.
nicolascurien: the…
Courant Events Right: Genius Zero…
nicolascurien: Business, because they are…
Courant Events Right: It's no genius.
nicolascurien: It's just a tree.
Courant Events Right: So, in principle.
nicolascurien: Where is it?
Courant Events Right: The bridge initial, conversions would follow from the peeling.
Courant Events Right: The grommos.
nicolascurien: golf.
Courant Events Right: Here, with opening, it's more complicated, and I.
nicolascurien: We don't.
Courant Events Right: do a…
nicolascurien: We, rely on the…
Courant Events Right: Organization?
nicolascurien: of those objects?
Courant Events Right: with label trees, it's something classical in, random planar map theory.
nicolascurien: That you encode for you.
Courant Events Right: planar graph or map by simple objects, which are label trees. It's this famous, Schaeffer or Plutiere de Francesco bijection.
Courant Events Right: And actually, what's funny is that those bijections.
nicolascurien: was including reef.
Courant Events Right: label trees.
nicolascurien: were discovered.
Courant Events Right: in this hyperbolic context, before the discrete analog for planar maps. Nobody noticed, except Timothy, maybe, but this was done before.
nicolascurien: by Penaire and British.
Courant Events Right: So the… actually, I did something more general, and the specification of this construction would give.
Courant Events Right: the… the encoding.
nicolascurien: use.
Courant Events Right: So the idea, rough idea is… the following. You have your.
nicolascurien: surface here, down here.
Courant Events Right: And you have a specific puncture.
Courant Events Right: In orange here. Now, you will consider the cutlocus. So the cut locus is something geometric.
nicolascurien: It's the set of all points on your surface from which you can.
Courant Events Right: Stop.
nicolascurien: Two different geodesics.
Courant Events Right: towards this puncture. Of course, this puncture is at infinity, but you can make sense of adjusting towards this puncture.
Courant Events Right: And, two different.
nicolascurien: conjun deser.
Courant Events Right: Having the same length.
Courant Events Right: Again.
nicolascurien: The length is infinite, but you can make sense of that. So…
Courant Events Right: Makes sense, right? If you will start from here.
Courant Events Right: If you want to go to this puncture, let's say there is a.
nicolascurien: only one way.
Courant Events Right: go down. If you're on the.
nicolascurien: The other side of your surface, you go down.
Courant Events Right: the other side, but there is… like a path?
Courant Events Right: where you could, you know, go other… two different ways to this function. And there are those special points.
Courant Events Right: Where it's, punching of other three, where you've got three different possible ways to go to this punch.
Courant Events Right: So, generically, the structure you see emerging here in red is an embedded
Courant Events Right: a regular tree, plain tree, and if you decorate this tree with angles or lengths of geodesics, then this completely encodes your surface. If I give you this guy.
nicolascurien: And…
Courant Events Right: You have your surface.
Courant Events Right: And now what you do is you take the limit measure
Courant Events Right: It's very Peterson measure on surface. You push it forward.
Courant Events Right: On this tree, and it turns out that what you see here is.
nicolascurien: reasonably well-behaved.
Courant Events Right: It's not far from being a uniform tree plus some labels, which evolve a bit like a random work along the path. And then.
Courant Events Right: From this, you're back to…
nicolascurien: safe business.
Courant Events Right: where Gregory and Jean-Francois prove the convergence to the Boolean sphere, and you try to adapt the.
nicolascurien: the proofs.
Courant Events Right: Adapting doesn't mean,
nicolascurien: Copy and paste.
Courant Events Right: Of course.
Courant Events Right: Let me mention that those digestions have been extended to a much more general setting by Bern and Zollenveld very recently, and I really recommend those.
Courant Events Right: So this…
nicolascurien: paper.
Courant Events Right: Okay, so that's… that's all for today on the G is equal to 0, so the planar case.
Courant Events Right: But after all, there are many perspectives, so,
Courant Events Right: We understood a piece of what is going on for the large genius.
Courant Events Right: The log genius now, we kind of understand this, Boylean sphere limit, but the universality phenomenon is not yet understood, because we have for this model was trick a little bit the model. For example, you have geodesic boundaries, and so on, so there should be,
Courant Events Right: two universities, those, or the bonus here.
nicolascurien: and many classes of the stable maps.
Courant Events Right: use.
Courant Events Right: Genius 1 already is not understood from this person.
nicolascurien: 15.
Courant Events Right: We have no.
nicolascurien: understood.
Courant Events Right: why the length spectrum.
nicolascurien: is universal, and the Laplaceanche spectrum, I told you.
Courant Events Right: about this one quarter, this was a big achievement of many people.
Courant Events Right: But it's only for the Val Peterson.
nicolascurien: Bill Peterson.
Courant Events Right: Guy with, large teeth, no punctures, and,
nicolascurien: engine.
Courant Events Right: hygienist.
Courant Events Right: We don't know what's the diameter of the sun in large denius.
Courant Events Right: We don't know what happens if we add punctures, and say, even if we add one puncture, do we have to redo the proof, or is it somehow,
Courant Events Right: resistant to adding one puncture. But what if the genius is proportional to the number of punctures?
nicolascurien: something.
Courant Events Right: What about other measures that are.
nicolascurien: other…
Courant Events Right: Natural measures, like, measures, which is not at all the same as the Velvetesan measure, so this is
Courant Events Right: Olop.
Courant Events Right: Thanks a lot!
Courant Events Right: It's still time for the left part.
nicolascurien: to make it up.
Courant Events Right: this!
Courant Events Right: Well, maybe you said something that I missed, in this, convergence to the Brownian sphere.
Courant Events Right: I have an impression that you should also have a certain point process, experimented sphere, which is the image of the cusps.
Courant Events Right: They ask about what's happening to these cusp on the buoyant sphere. So, mathematically, those cusps on the buoyant sphere, first you don't see them, so they are.
Courant Events Right: Then, this flowing process for the Boylean sphere, just gives the mass measure on the Boylean sphere.
Courant Events Right: So it's correct to say that I see the Brownian sphere, and on top of that, the possible process… It's not the possible process. You were considering a planulation, and you asked what the vertices in the planar angulation. Yeah, so you can consider that as a measure. It will give after normalization the last measure at the point.
Courant Events Right: Yes.
Courant Events Right: Yelinda.
Courant Events Right: And I think also, Gerald on us.
Courant Events Right: G is the right font, but I think they…
Courant Events Right: You know much better than me. So, Yelim said that, for the spectrum gap.
Courant Events Right: So not the tri-hydon, but the spinal gap, this one partial phenomenon should hold also when you add little o, let's say, or square root G, punctures.
Courant Events Right: But, what if… what if you had long boundaries?
Courant Events Right: Did I count your… so, the left is ahead, You miss some fragments.
Courant Events Right: I missed?
Courant Events Right: How many in?
Courant Events Right: I wasn't counting, but… Did I?
Courant Events Right: Nope.
Courant Events Right: The left and the right.
Courant Events Right: Okay, so this is why?
Courant Events Right: Any other questions?
Courant Events Right: Okay, so I don't think we can find anything else.
Courant Events Right: Is that your throwing, the very aggressive, Connie?
nicolascurien: Good deal.
Courant Events Right: But, on top of, A picture.
nicolascurien: It's easy.
Courant Events Right: Good luck.
nicolascurien: Thanks.