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nicolascurien: The left and the right wings.

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Courant Events Right: asked during my talk. But it's a lively talk, Okay, so.

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nicolascurien: So, France said that, when a constant.

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Courant Events Right: curvature, and when the curvature.

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nicolascurien: during posting boundary, this is…

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Courant Events Right: Old stuff, finite-dimensional, everything.

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nicolascurien: My goal is to try to convince.

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Courant Events Right: It's true that.

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nicolascurien: This is not.

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Courant Events Right: Old stuff. Almost nothing is known.

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nicolascurien: Also, the geometric point of view.

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Courant Events Right: Alright, so there are three,

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nicolascurien: Words in the title.

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Courant Events Right: random hyperbolic surfaces, let me try to introduce first hyperbolic surface.

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Courant Events Right: So, what do I mean by surface? I always mean, 2D lightfall, which is orientable, so,

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nicolascurien: No.

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Courant Events Right: Mobius trip in this talk. And, have a.

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nicolascurien: surface, they have many, many definitions, that's why they're.

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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.

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nicolascurien: It will give you.

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Courant Events Right: you were… depends on.

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nicolascurien: definition in West Forest, but.

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Courant Events Right: Yeah, that's.

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nicolascurien: the metric definition as constant curvature Riemannian manifold, or model dependence, meaning that you have a manifold and you have that class of.

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Courant Events Right: shorts.

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nicolascurien: Where, coordinate changes are isometry of,

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Courant Events Right: Also, hyperbolic plane.

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nicolascurien: repeat them if they were nearby H2.

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Courant Events Right: There is a purely a generic.

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nicolascurien: definition of,

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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.

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nicolascurien: One, focus on the future, is.

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Courant Events Right: Only one opens.

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nicolascurien: quantity, which is a resurface.

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Courant Events Right: so on, but I will find.

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nicolascurien: Focus on the hand.

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Courant Events Right: definition where a public surface will be obtained by grooming elementary.

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nicolascurien: and pieces.

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Courant Events Right: Either pair of pencils.

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nicolascurien: form, ideal handguns.

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nicolascurien: Okay, so let's do that.

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Courant Events Right: We speak up, well.

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nicolascurien: Force 101 on.

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Courant Events Right: I covered.

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nicolascurien: spring?

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Courant Events Right: So, here at Amazon, I thought we're playing.

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nicolascurien: with the disk.

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Courant Events Right: dimension.

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nicolascurien: Trick is displayed, and.

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Courant Events Right: I remind you that geodesics are arcs in the.

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nicolascurien: for them.

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Courant Events Right: Which are.

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nicolascurien: authored.

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Courant Events Right: equivalent to the binary. The binary.

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nicolascurien: integer of the circle. And, so that's the play.

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Courant Events Right: of the plane, and if you are looking at the topology of.

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nicolascurien: disk…

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Courant Events Right: with 3 moles, it's called a pair of pens.

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nicolascurien: And, to step in the right one.

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nicolascurien: Sorry for that. We go to the left.

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Courant Events Right: More than to the right. So, a pair of pants is.

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nicolascurien: Is this…

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Courant Events Right: Disc with 3 holes, And then.

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nicolascurien: It's a unique metric.

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Courant Events Right: That you can put on this pair of pants if you specify the boundary length.

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nicolascurien: So if the boundaries.

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Courant Events Right: are forced to be geodesic boundaries of.

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nicolascurien: then it's A, B, and C, then there is a.

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Courant Events Right: unit, temperature.

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nicolascurien: Everybody's my turn.

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Courant Events Right: in there, then.

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nicolascurien: gives you this length AP, P, and C for the length of the bets, if you wish.

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Courant Events Right: Notice that one of these, or even the three numbers.

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nicolascurien: can be zero, and when it's zero, it's called a puncture, and it.

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Courant Events Right: could be a… If the.

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nicolascurien: in December.

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Courant Events Right: So this guy is.

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nicolascurien: non-compact is if one of these three numbers is equal to zero, it's a counter.

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Courant Events Right: Okay?

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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.

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nicolascurien: How do you do that? Well, it's easy… oops.

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Courant Events Right: You can imagine that.

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nicolascurien: as you…

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Courant Events Right: take those pair of pads and improve them

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Courant Events Right: along their boundaries. Of course, boundaries' lengths have to match if you want.

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nicolascurien: to do something.

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Courant Events Right: This has to be… sorry, let's me… let's do that here. So this has to be.

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nicolascurien: You can do this if you want to glue them together.

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Courant Events Right: And you could then…

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nicolascurien: Along the combinatorics of a graph, and the combinatorics of this graph.

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Courant Events Right: We'll also fix for you the topology. Here, the topology.

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nicolascurien: is you've got Genius 3, because you've got 3 holes.

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Courant Events Right: And there are two boundaries, one here and one there.

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nicolascurien: Okay? So, it's purely.

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Courant Events Right: combinatorial.

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Courant Events Right: I fixed for you just the boundary length, and there is also an additional parameter, which is the amount of twist

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Courant Events Right: You perform before gluing the surface, the pair of pants together.

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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.

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nicolascurien: varying.

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Courant Events Right: All those numbers, they're.

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nicolascurien: Thanks and the twist.

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Courant Events Right: You can get all hyperbolic serves.

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nicolascurien: Profess with this topological type.

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nicolascurien: Is it true?

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Courant Events Right: So a constant curvature minus 1, yes.

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Courant Events Right: So, as Frank said.

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nicolascurien: It's…

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Courant Events Right: It's really a finite dimensional stuff.

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nicolascurien: So…

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Courant Events Right: End of the story. You can get all hyperlix surface by gluing

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Courant Events Right: Those pair of pants together, with the prescribed length and the twist.

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nicolascurien: But it's the end, the end, but also the beginning, because.

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Courant Events Right: I told you, you can, but there is actually many ways you can obtain the same hyperlink surface by gluing pair.

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nicolascurien: There are many,

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Courant Events Right: that gives the same hyperbolic surface. So it's an overparametrization.

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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.

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nicolascurien: better catalog.

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Courant Events Right: way of,

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Courant Events Right: is hiding your surface, and this identification is, is, honest.

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Courant Events Right: Okay? But still, we can denote by…

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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.

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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.

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Courant Events Right: Okay, so that's this guy.

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Courant Events Right: Is a finite dimensional space, if you wish.

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Courant Events Right: And we want to, understand it.

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Courant Events Right: So, it'll count.

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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

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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,

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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…

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Courant Events Right: 000 n times, and n must be bigger than 3. Okay, so I focus on those two extremes, high genius, MG.

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nicolascurien: will be large.

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Courant Events Right: N0N, 0 is fixed, the genius is 0, but the number of functions goes to infinity.

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Courant Events Right: Question so far?

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Courant Events Right: 0, 0.

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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…

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nicolascurien: Presidents.

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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.

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nicolascurien: That's two nodes of potential.

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Courant Events Right: Relax, and you pair them uniformly at random.

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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.

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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.

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Courant Events Right: So, same spirit for the low genius manipuncture. This is still also.

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nicolascurien: Compact Medford of Finance.

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Courant Events Right: dimension, which… should be thought of as the space of triggerment planar maps, now planar, with,

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Courant Events Right: And those profits.

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Courant Events Right: And then also, lots of beautiful results that we want to import to the hyperbolic third.

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nicolascurien: Okay?

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Courant Events Right: Everybody's happy?

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nicolascurien: Yes!

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Courant Events Right: You're… Specify yourself left or right.

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nicolascurien: I mean, all…

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Courant Events Right: No, it's this line?

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nicolascurien: And what's… right. Okay, good friend.

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nicolascurien: I thought the M0M would be by gluing the desert on the tree. Yes. It's truly, truly.

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Courant Events Right: a picture of that, it's growing along a tree. So the tree-valent map behind.

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nicolascurien: It's a tree.

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Courant Events Right: You… competition aside.

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nicolascurien: positive, because, so…

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Courant Events Right: Did you hear the question?

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Courant Events Right: Basically, for this guy.

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nicolascurien: And it drove blue.

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Courant Events Right: pair of pens.

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nicolascurien: the police country.

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nicolascurien: So it's not the plan of trying to… If you do…

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Courant Events Right: Well, that's.

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nicolascurien: to teach you.

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Courant Events Right: blue. Ideally.

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nicolascurien: equivalent.

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Courant Events Right: ideal triangle, not equilateral, then it would be mass status.

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nicolascurien: behind. Yes?

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nicolascurien: You're gonna tell us about the interview databases?

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Courant Events Right: But that we can't do.

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nicolascurien: Just true. Let's make the competition more interesting.

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Courant Events Right: If you have a good question.

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nicolascurien: Yay!

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nicolascurien: But that one for me once, huh?

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Courant Events Right: Okay, so first thing…

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nicolascurien: That you won'.

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Courant Events Right: to understand is.

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nicolascurien: No vulnerability just purely determinates depression, you fix, and observe that one.

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Courant Events Right: A generic quantity code is phi, and you want to.

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nicolascurien: what are the values that Phi can take.

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Courant Events Right: Over the modularized space.

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nicolascurien: Okay, so please…

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Courant Events Right: What?

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nicolascurien: And try to understand, for example, extreme.

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Courant Events Right: property, what are the lowest.

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nicolascurien: principal value.

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Courant Events Right: for a genius key high-level self-este.

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nicolascurien: Although not just in the service.

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Courant Events Right: purely deterministic.

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Courant Events Right: Then there are round-up questions. Now, you won't be.

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nicolascurien: also the behavior of this observable for roundups for this SG. If SG is picked.

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Courant Events Right: in your space, but for this, you need.

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nicolascurien: We need to specify the.

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Courant Events Right: measure.

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nicolascurien: the form of information that you will use. So there are many choices at Rio.

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Courant Events Right: Okay, so let's start first with this, deterministic extremal property.

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Courant Events Right: So the…

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nicolascurien: So, I give you a list of some options.

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Courant Events Right: Sables that you can.

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nicolascurien: do we have?

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Courant Events Right: The first one that comes to mind is the diameter.

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nicolascurien: So if your surface is compact, the diameter is the maximal distance between.

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Courant Events Right: the two points, of course.

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nicolascurien: Of course, if you have a cusp.

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Courant Events Right: Then it's infinite.

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Courant Events Right: One which is really important for hyperbolic.

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nicolascurien: speak German.

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Courant Events Right: is the system. It's the length of the smallest.

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nicolascurien: Non-trivial loop.

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Courant Events Right: Here.

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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.

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nicolascurien: That is, that is, exactly the same as in a hyperbolic plane.

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Courant Events Right: Building upon the sister you.

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nicolascurien: I also want to ask about.

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Courant Events Right: All the possible lengths.

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nicolascurien: Of all loops, not only the smallest one, so it's.

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Courant Events Right: all the length spectrum, I will come back to that.

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Courant Events Right: The Jigger constant, also, which measures.

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nicolascurien: The expansion property of your surface is, of.

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Courant Events Right: for high interest.

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Courant Events Right: The total pants length, so this is the… smallest… Length of curve you cut.

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Courant Events Right: You'll suffer.

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nicolascurien: best for me.

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Courant Events Right: On your pair of pants, so it's…

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nicolascurien: The minimum of all pan decomposition of.

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Courant Events Right: The length, some of the.

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nicolascurien: all of the curves.

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Courant Events Right: And there is the spectral gap, so the spectral gap is.

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nicolascurien: Good.

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Courant Events Right: There's no trivial eigenvalue for the…

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nicolascurien: aggressions that come on the surface. How to display that? So, this is my risk.

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Courant Events Right: Representation of the spectra gap.

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Courant Events Right: Alright, so there are many geometric and spectral observable that you can look at.

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Courant Events Right: And just to tell you this is still active research, let's look at what we can say.

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nicolascurien: but those…

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Courant Events Right: Quantities, again, this is purely deterministic.

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Courant Events Right: So what about the diameter? The diameter, you've got trivial bounds,

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Courant Events Right: The diameter must be at least Logi, for example, But, but…

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nicolascurien: Must be finite.

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Courant Events Right: The system, it's easy to construct examples.

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nicolascurien: Where the system is advertised for when the changes.

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Courant Events Right: not…

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nicolascurien: But the system is less than 2HG, it's a bit the same argument as this one.

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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.

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nicolascurien: ruined.

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Courant Events Right: G goes to infinity. This is a highly non-trillion. It's the construction of expanded.

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nicolascurien: surfaces will go back.

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Courant Events Right: to Margulis, Margulis, and so on. So this, this…

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Courant Events Right: The fact that this guy is strictly positive, it's already non-trigger.

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nicolascurien: The same false big tide gap.

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Courant Events Right: The Jigger constant, by the way, is always less than 1 by comparing to the hyperbolic planes.

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nicolascurien: Because there's one question.

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Courant Events Right: by comparing to the universal cover as well, and so on and so forth. Yes!

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Courant Events Right: Is that the same one quarter, that the…

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Courant Events Right: No, no, so…

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Courant Events Right: So go over here.

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Courant Events Right: But still, it's a question.

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Courant Events Right: You're lagging, guys.

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Courant Events Right: Okay, so this was morally the state-of-the-art in the.

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nicolascurien: 80s?

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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,

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Courant Events Right: bound, or the two locations, between 3 quarters

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Courant Events Right: 4 thirds only. 4 thirds and 2 lucky.

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Courant Events Right: What the ?

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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.

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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.

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Courant Events Right: So, here…

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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.

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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…

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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.

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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.

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Courant Events Right: Questions?

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Courant Events Right: So, although that seems very deterministic statement, that they are, deterministic theorem, in all of these groups.

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Courant Events Right: Maybe not this one.

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Courant Events Right: But all the rest.

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Courant Events Right: Hondomeness is involved by… randomness is used to construct examples here of minimizing diameter of, here also for the Jigo constant.

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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.

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Courant Events Right: So now… On the next, if you want to sample a hyperbread surface at random, I actually.

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nicolascurien: Can you…

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Courant Events Right: which measure do I use?

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Courant Events Right: So they'll.

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nicolascurien: How many, many.

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Courant Events Right: Many, many, combinatorial, combinatorial,

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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

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Courant Events Right: beneath the consumption at random, and then this will produce for you, random.

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nicolascurien: boom.

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Courant Events Right: hyperlink surface. Yes, you really want to win the challenge.

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Courant Events Right: So, in the case.

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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…

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Courant Events Right: Before that, actually, the first construction of expanders goes back to Kolmogorov, and it was random.

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nicolascurien: So in the surface is…

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Courant Events Right: There's a random.

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nicolascurien: construction, but is there also a more additional study for realistic one?

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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.

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Courant Events Right: At least 20.

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nicolascurien: Yes, sir?

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Courant Events Right: You're taking the lead. Okay, so these are a lot of discrete measures, I would say, which are used.

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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.

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nicolascurien: measure, that's… Most people over here know that we're Indians.

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Courant Events Right: as a measure on the molecular space, it could be thought of as the Levesque measure on this space.

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Courant Events Right: Why is it so? So there are many ways to say it's.

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nicolascurien: measure, comment.

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Courant Events Right: the safest point.

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nicolascurien: it's…

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Courant Events Right: using pendant.

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nicolascurien: composition?

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Courant Events Right: So I told you that if you fix this, this combinator.

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nicolascurien: pair of pads.

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Courant Events Right: If I vary the length here.

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nicolascurien: And the twist, and put them as real parameters.

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Courant Events Right: Then…

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nicolascurien: This…

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Courant Events Right: Our role is subjective, by…

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nicolascurien: angles.

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Courant Events Right: 6 zigma.

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nicolascurien: those six parameters, I can get any hyperbolic surface.

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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.

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nicolascurien: And… non-trivial.

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Courant Events Right: In fact, that after.

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nicolascurien: those identity.

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Courant Events Right: if I modify.

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nicolascurien: by,

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Courant Events Right: working class group, then I get a finite measure done down.

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nicolascurien: on human.

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Courant Events Right: the modularized space, which is the perpetestant measure, and so.

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nicolascurien: So, it is.

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Courant Events Right: forward of the.

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nicolascurien: Liebeck measure.

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Courant Events Right: modding out by this… this.

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nicolascurien: section.

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nicolascurien: for any And the competition.

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nicolascurien: Good.

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nicolascurien: So, how about.

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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…

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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.

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Courant Events Right: content, village N, and it turns out that this…

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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.

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nicolascurien: improvement in the coming Sera.

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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.

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nicolascurien: What do you mean?

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Courant Events Right: energy according to the well-Meterson measurement.

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Courant Events Right: So, typically, a guy like that.

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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.

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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.

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Courant Events Right: Okay, so this was breakthrough from 2007 to, 2007.

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nicolascurien: 13.

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Courant Events Right: But…

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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

351
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Courant Events Right: famous recursion of Mirzakani into the realm of this appealing exploration of random surface.

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Courant Events Right: Don't worry!

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Courant Events Right: what I'm going to say here is purely deterministic.

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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.

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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

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Courant Events Right: building on the academic side is to.

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nicolascurien: intersection.

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Courant Events Right: Can't go over the.

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nicolascurien: That's it, but…

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Courant Events Right: give you a hyperbolic surface, I will give for you one, only one, per decomposition of that surface. Canonical one.

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Courant Events Right: So this canonical fan decomposition can be obtained as follows.

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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.

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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?

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Courant Events Right: A little ant. I want to start a little ant at that point here. And this little ant.

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nicolascurien: to ENTER my surface.

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Courant Events Right: orthogonally to this boundary at that point, okay?

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Courant Events Right: And this ant will,

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nicolascurien: I can.

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Courant Events Right: produce a little bit of a pheromone, you know? And when she crosses her past. Path.

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Courant Events Right: or above.

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nicolascurien: She will notice it.

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nicolascurien: Okay, so she…

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Courant Events Right: gets into the surface until it crosses its past.

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nicolascurien: Ugh.

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Courant Events Right: At this point. Then I claim that at this point, if the ant has noticed something strange.

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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

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Courant Events Right: End of the white curve.

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nicolascurien: It's topologically a pair of pants.

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Courant Events Right: Indeed, imagine that you put Sprint!

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nicolascurien: This is a little.

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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.

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nicolascurien: Until they find.

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Courant Events Right: their geodesic components.

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nicolascurien: Do you?

385
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nicolascurien: babies, Dad!

386
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Courant Events Right: When they do so, the door spring will define a pair of pants.

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Courant Events Right: In which the end was in. The end has not exceeded this boundary.

388
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Courant Events Right: Okay? So that's a purely local way of exploring myself.

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nicolascurien: Reference by an end?

390
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Courant Events Right: and discovering one pair of pads. Then you cut at those points, at those.

391
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nicolascurien: Those beds here, and you'.

392
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Courant Events Right: You got a new surface. Yes!

393
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nicolascurien: You can get lost in, but with very, very.

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Courant Events Right: Probability 0, it's generic.

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Courant Events Right: what I'm saying is, is.

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nicolascurien: to say.

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Courant Events Right: is the.

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nicolascurien: in the extension. We'll probably.

399
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Courant Events Right: You may… you should say, oh.

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nicolascurien: It's a very.

401
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Courant Events Right: when you need double penetration measure. No, no, no, no, it's generic for anything that will be…

402
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Courant Events Right: would have, it's generic.

403
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Courant Events Right: Yes.

404
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nicolascurien: You take the shortest root curve?

405
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Courant Events Right: Terrible.

406
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Courant Events Right: No, no, the end is just a geodesic that has been shut orthogonally to the binary.

407
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Courant Events Right: Okay.

408
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Courant Events Right: Sorry? Sure.

409
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Courant Events Right: That's my algorithm that tells me that. My algorithm is a function, so actually, when you start.

410
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nicolascurien: There is a mark quote.

411
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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.

412
00:25:14.230 --> 00:25:16.809
Courant Events Right: That's your algorithm, it's a function that you have your work for.

413
00:25:17.780 --> 00:25:18.970
Courant Events Right: pure instrument.

414
00:25:19.660 --> 00:25:24.740
Courant Events Right: Then, you go along with what's remaining. So.

415
00:25:24.740 --> 00:25:25.550
nicolascurien: Okay?

416
00:25:25.550 --> 00:25:34.550
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.

417
00:25:35.030 --> 00:25:39.189
Courant Events Right: It's in your function. Then you shoot your new end at that point.

418
00:25:40.000 --> 00:25:42.169
Courant Events Right: Do the same thing, she will grow.

419
00:25:42.170 --> 00:25:44.720
nicolascurien: After opening, until it is.

420
00:25:44.720 --> 00:25:47.460
Courant Events Right: Either the battery or its, path.

421
00:25:49.450 --> 00:25:59.430
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.

422
00:26:00.480 --> 00:26:04.160
Courant Events Right: And this drawing took me 1 hour. This one.

423
00:26:04.820 --> 00:26:06.699
Courant Events Right: This is a pair of pants.

424
00:26:07.170 --> 00:26:08.140
Courant Events Right: You cut?

425
00:26:08.970 --> 00:26:13.209
Courant Events Right: So I'll let you 20 seconds to see what is going on.

426
00:26:14.330 --> 00:26:17.460
Courant Events Right: So you imagine that you put a little spring here.

427
00:26:17.460 --> 00:26:18.110
nicolascurien: here?

428
00:26:18.330 --> 00:26:22.480
Courant Events Right: And a spring around there, here, Shhh.

429
00:26:22.740 --> 00:26:26.349
Courant Events Right: They will find their… Gender Z representative, and you cut.

430
00:26:26.930 --> 00:26:28.959
Courant Events Right: Okay, everybody got it?

431
00:26:34.360 --> 00:26:35.070
Courant Events Right: Do we agree?

432
00:26:39.310 --> 00:26:40.840
nicolascurien: And so on and so forth.

433
00:26:41.130 --> 00:26:42.130
nicolascurien: So…

434
00:26:42.130 --> 00:26:47.720
Courant Events Right: In this way, I have decomposed my surface, In two weeks.

435
00:26:47.890 --> 00:26:53.749
Courant Events Right: A pair of panic decomposition, only using my algorithm, and this algorithm was purely.

436
00:26:53.830 --> 00:26:58.290
nicolascurien: Local, and… Based only on the legitimity of the.

437
00:26:58.290 --> 00:26:59.939
Courant Events Right: surface, not its stash number.

438
00:26:59.940 --> 00:27:01.280
nicolascurien: I wasn't that easy.

439
00:27:01.280 --> 00:27:04.859
Courant Events Right: It is something that is defined on modularized space. It's.

440
00:27:05.470 --> 00:27:08.189
nicolascurien: was… I was just using the isomer.

441
00:27:08.190 --> 00:27:09.730
Courant Events Right: Because… of,

442
00:27:09.730 --> 00:27:11.130
nicolascurien: the surface.

443
00:27:12.870 --> 00:27:13.700
nicolascurien: Okay?

444
00:27:13.720 --> 00:27:17.319
Courant Events Right: But what is important in this

445
00:27:17.480 --> 00:27:22.410
Courant Events Right: special decomposition, is that those things, they are special.

446
00:27:22.610 --> 00:27:26.820
Courant Events Right: It's not any pent. It's a pent in which, if I start.

447
00:27:26.910 --> 00:27:31.440
nicolascurien: this ant, and then I know that the ant will exit.

448
00:27:32.140 --> 00:27:37.170
Courant Events Right: Horses pass or exit the boundary inside this path. That's a very special property.

449
00:27:37.550 --> 00:27:41.990
Courant Events Right: For example, here, if I start the end here, then she cannot do that.

450
00:27:45.130 --> 00:27:56.720
Courant Events Right: Because by definition, the PEMP was defined, by the first crossing of the end of its path, or the Pandarios.

451
00:27:57.050 --> 00:28:00.570
Courant Events Right: Okay? That's a very special property. So that's the determination.

452
00:28:00.570 --> 00:28:01.260
nicolascurien: Do you think…

453
00:28:01.520 --> 00:28:05.119
Courant Events Right: block, it mirrorsaken execution, and from that, now if you put product

454
00:28:06.480 --> 00:28:10.800
Courant Events Right: measure. Then, the fact that those plants are special is the so-called machine.

455
00:28:10.800 --> 00:28:11.700
nicolascurien: identity.

456
00:28:11.700 --> 00:28:14.879
Courant Events Right: And then this decomposition will immediately give you the…

457
00:28:14.880 --> 00:28:20.219
nicolascurien: But… You can take this as a deterministic.

458
00:28:20.390 --> 00:28:22.120
Courant Events Right: Way of encoding your suffix.

459
00:28:22.720 --> 00:28:23.730
Courant Events Right: Yes!

460
00:28:24.400 --> 00:28:27.349
Courant Events Right: Don't pray anymore. Stop. Yes?

461
00:28:27.470 --> 00:28:33.699
Courant Events Right: So I see the length of the cast, but where are the twists? Yeah, the twist…

462
00:28:34.390 --> 00:28:38.190
Courant Events Right: It's, it's here, I, you know, you have to remember this information.

463
00:28:38.190 --> 00:28:38.820
nicolascurien: But this…

464
00:28:38.820 --> 00:28:46.750
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.

465
00:28:48.230 --> 00:28:49.030
Courant Events Right: Goodbye.

466
00:28:50.700 --> 00:28:51.420
Courant Events Right: Oops.

467
00:28:51.420 --> 00:28:55.309
nicolascurien: The twist here is… okay, let's go back here, maybe.

468
00:28:55.670 --> 00:28:59.620
Courant Events Right: Those two points, I tell you I need to glue this one to this one.

469
00:29:00.830 --> 00:29:18.880
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.

470
00:29:18.880 --> 00:29:20.069
nicolascurien: But then, when…

471
00:29:20.070 --> 00:29:25.579
Courant Events Right: When you… once you discover that from… okay, cut out your survey, this is not relevant, but…

472
00:29:25.690 --> 00:29:35.349
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.

473
00:29:35.350 --> 00:29:38.060
nicolascurien: Should this work be the same as this one?

474
00:29:39.970 --> 00:29:41.510
Courant Events Right: And this will encode.

475
00:29:41.510 --> 00:29:42.579
nicolascurien: Oh, your, your proof.

476
00:29:44.080 --> 00:29:45.060
Courant Events Right: Okay, yes?

477
00:29:45.060 --> 00:29:51.709
nicolascurien: Are you asserting that with this decomposition, for each band, there is no compatible

478
00:29:52.090 --> 00:29:59.530
nicolascurien: Yeah, so this was the question that has been mentioned already here today in Spain.

479
00:30:00.350 --> 00:30:03.310
nicolascurien: Very tiny probability, then, that your ant is doing.

480
00:30:03.310 --> 00:30:03.710
Courant Events Right: something.

481
00:30:03.710 --> 00:30:06.799
nicolascurien: Like this, but that's a zero probability event, actually.

482
00:30:06.800 --> 00:30:07.300
Courant Events Right: So if you're.

483
00:30:07.300 --> 00:30:09.070
nicolascurien: Jake!

484
00:30:09.200 --> 00:30:11.679
nicolascurien: If you have fixed your algorithm in your pocket.

485
00:30:11.680 --> 00:30:12.090
Courant Events Right: If you'll.

486
00:30:12.090 --> 00:30:14.829
nicolascurien: surface is generic, you will never encounter this.

487
00:30:15.090 --> 00:30:18.370
nicolascurien: Your end will always, with probability 1.

488
00:30:18.370 --> 00:30:19.960
Courant Events Right: Always cross this path.

489
00:30:19.960 --> 00:30:22.299
nicolascurien: Or, come back to a new vulnerability.

490
00:30:25.220 --> 00:30:30.069
Courant Events Right: If… There's something I missed, which is, how do you keep your…

491
00:30:30.310 --> 00:30:35.400
Courant Events Right: What's your algorithm telling you about Run 30.

492
00:30:37.470 --> 00:30:42.349
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.

493
00:30:45.090 --> 00:30:54.599
Courant Events Right: Yeah. Let's say you have one monitor here, also, you have one point at the right.

494
00:30:54.600 --> 00:30:55.179
nicolascurien: you know.

495
00:30:55.560 --> 00:30:58.440
Courant Events Right: Is that Mark? Yes, it's not covered.

496
00:31:04.480 --> 00:31:05.500
Courant Events Right: Okay?

497
00:31:09.570 --> 00:31:11.389
Courant Events Right: Any other questions from that side?

498
00:31:13.030 --> 00:31:13.980
Courant Events Right: Good!

499
00:31:16.140 --> 00:31:20.370
Courant Events Right: So that's the deterministic… algorithm.

500
00:31:20.370 --> 00:31:23.220
nicolascurien: Underlying…

501
00:31:23.680 --> 00:31:35.719
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.

502
00:31:40.450 --> 00:31:45.430
Courant Events Right: And also those, those community teams, those, those other teams.

503
00:31:45.700 --> 00:31:55.509
Courant Events Right: And manage now to show this nearly amineering phenomenon that a typical Ben-Peterson surface, SG, will have.

504
00:31:55.510 --> 00:31:58.540
nicolascurien: spectral gap, which is almost maximum.

505
00:31:58.540 --> 00:31:59.319
Courant Events Right: Ultimate, etc.

506
00:31:59.320 --> 00:32:01.380
nicolascurien: With very high probability.

507
00:32:01.750 --> 00:32:02.899
Courant Events Right: All of those works.

508
00:32:02.900 --> 00:32:03.520
nicolascurien: are based on.

509
00:32:03.520 --> 00:32:11.289
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,

510
00:32:11.770 --> 00:32:25.920
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

511
00:32:26.120 --> 00:32:28.960
Courant Events Right: Genius G and say boundary of lengths N1 and.

512
00:32:28.960 --> 00:32:31.640
nicolascurien: So it's… it's an ingredient in their proof.

513
00:32:32.770 --> 00:32:33.700
Courant Events Right: Okay?

514
00:32:34.040 --> 00:32:34.440
Courant Events Right: So by now.

515
00:32:34.440 --> 00:32:35.540
nicolascurien: How do we know that?

516
00:32:36.560 --> 00:32:38.790
nicolascurien: Sun surface is nearly a million.

517
00:32:38.790 --> 00:32:40.760
Courant Events Right: Clearly, the optimal spectral,

518
00:32:40.760 --> 00:32:41.460
nicolascurien: Yep.

519
00:32:42.900 --> 00:32:44.300
Courant Events Right: So that was for the specs.

520
00:32:44.300 --> 00:32:45.320
nicolascurien: Listen Trump.

521
00:32:45.600 --> 00:32:50.959
Courant Events Right: Now, the length central, So what is the length spectrum? If you have a surface.

522
00:32:51.760 --> 00:32:53.549
Courant Events Right: Hi, Beverly self is here.

523
00:32:53.550 --> 00:32:54.890
nicolascurien: I can…

524
00:32:55.650 --> 00:33:01.259
Courant Events Right: Draw all possible close curves on this one, and take their geodesic representative.

525
00:33:01.670 --> 00:33:05.460
Courant Events Right: So, you have a curve like that, you have one like that, and so on.

526
00:33:05.850 --> 00:33:09.789
Courant Events Right: There is only a discrete set of those guys, and you can measure that.

527
00:33:09.790 --> 00:33:10.640
nicolascurien: your legs.

528
00:33:10.840 --> 00:33:17.110
Courant Events Right: And actually, their legs will never… the red one here is probably the shortest.

529
00:33:17.490 --> 00:33:21.229
nicolascurien: The orange is the length of this one, which you see.

530
00:33:21.230 --> 00:33:24.819
Courant Events Right: go around, below, and come back. You also, you may.

531
00:33:24.820 --> 00:33:30.309
nicolascurien: may do several turns, also, that's a green one, and so on. So, imagine all possible curves.

532
00:33:30.640 --> 00:33:35.130
Courant Events Right: topological type of curves on those surfaces. Take their jersey.

533
00:33:35.130 --> 00:33:36.040
nicolascurien: quantitative?

534
00:33:36.290 --> 00:33:38.410
Courant Events Right: their length, this will give you a.

535
00:33:38.410 --> 00:33:39.810
nicolascurien: prediction of numbers.

536
00:33:39.870 --> 00:33:41.279
Courant Events Right: Positive numbers.

537
00:33:42.090 --> 00:33:44.559
Courant Events Right: In fact, they will not accumulate near zero.

538
00:33:45.240 --> 00:33:46.000
Courant Events Right: They will…

539
00:33:46.000 --> 00:33:49.489
nicolascurien: only accumulate towards infinity, and so that's a random.

540
00:33:49.490 --> 00:33:50.770
Courant Events Right: Collection of number.

541
00:33:50.770 --> 00:33:53.330
nicolascurien: Random, because the surface itself is random.

542
00:33:54.370 --> 00:33:56.140
nicolascurien: Then, again.

543
00:33:56.510 --> 00:33:59.649
Courant Events Right: By putting in the machine those, estimates.

544
00:33:59.650 --> 00:34:02.889
nicolascurien: volumes, mes incognitrie, where.

545
00:34:02.890 --> 00:34:03.470
Courant Events Right: forgive him?

546
00:34:03.470 --> 00:34:04.920
nicolascurien: that this…

547
00:34:04.920 --> 00:34:07.220
Courant Events Right: collection of numbers, when the genius goes.

548
00:34:07.220 --> 00:34:08.270
nicolascurien: for infinity.

549
00:34:08.270 --> 00:34:08.940
Courant Events Right: Convergence is.

550
00:34:08.940 --> 00:34:09.420
nicolascurien: distribution?

551
00:34:09.420 --> 00:34:14.310
Courant Events Right: In other words, the Poisson point process with an explicit intensity, this cosh-1.

552
00:34:14.310 --> 00:34:14.969
nicolascurien: But…

553
00:34:16.590 --> 00:34:18.629
Courant Events Right: In particular, they get,

554
00:34:18.630 --> 00:34:18.980
nicolascurien: I don't.

555
00:34:18.989 --> 00:34:19.939
Courant Events Right: Distribution of the system.

556
00:34:19.940 --> 00:34:20.440
nicolascurien: decreases.

557
00:34:20.440 --> 00:34:22.489
Courant Events Right: The sun is just the first of this point.

558
00:34:23.050 --> 00:34:29.129
Courant Events Right: And what's… Still kind of unknown, or not understood.

559
00:34:29.510 --> 00:34:32.500
nicolascurien: It's the universality of this Poisson-Poly process.

560
00:34:32.500 --> 00:34:35.830
Courant Events Right: Because it also pops up as, the…

561
00:34:35.830 --> 00:34:36.290
nicolascurien: aspect.

562
00:34:36.290 --> 00:34:39.589
Courant Events Right: of other quantum surfacing hygienus.

563
00:34:39.590 --> 00:34:40.069
nicolascurien: like, Honda.

564
00:34:40.070 --> 00:34:43.050
Courant Events Right: maps, or tight.

565
00:34:43.219 --> 00:34:43.799
nicolascurien: Thank you.

566
00:34:43.800 --> 00:34:45.249
Courant Events Right: spectrum of,

567
00:34:45.250 --> 00:34:46.499
nicolascurien: Elbita sun surface.

568
00:34:49.230 --> 00:34:49.940
Courant Events Right: This!

569
00:34:51.610 --> 00:35:09.909
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.

570
00:35:10.450 --> 00:35:22.900
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.

571
00:35:23.480 --> 00:35:28.029
Courant Events Right: So you have to have a place where the system goes to experience.

572
00:35:30.170 --> 00:35:41.669
Courant Events Right: It is, it is. But this comes from, asymptotic estimates on theson volume with large G and a small, small bounding.

573
00:35:43.780 --> 00:35:49.630
Courant Events Right: That is surprising, but… You can do a computation for random triggers as well.

574
00:35:50.170 --> 00:35:56.450
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.

575
00:35:56.550 --> 00:36:00.729
Courant Events Right: Then… It's possible that you won't see smoke cycles.

576
00:36:00.860 --> 00:36:01.850
Courant Events Right: It's possible.

577
00:36:02.470 --> 00:36:05.110
Courant Events Right: Of the smallest cycle converges the distribution.

578
00:36:05.690 --> 00:36:06.790
Courant Events Right: Symphony.

579
00:36:11.040 --> 00:36:14.640
Courant Events Right: If I forget to count the points, please tell me.

580
00:36:19.670 --> 00:36:23.810
Courant Events Right: It's Rick against… Writing.

581
00:36:25.270 --> 00:36:26.510
Courant Events Right: Okay.

582
00:36:27.640 --> 00:36:33.589
Courant Events Right: So that was for high genius. Now let's… Let's move on to the…

583
00:36:33.820 --> 00:36:39.140
Courant Events Right: Genius Zero planar case with punctures. So it's a hyperbolic surface.

584
00:36:39.580 --> 00:36:47.279
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.

585
00:36:47.280 --> 00:36:48.369
nicolascurien: If you were to decompose.

586
00:36:48.370 --> 00:36:50.580
Courant Events Right: Put this in a pair of pants, you will get a tree.

587
00:36:50.950 --> 00:36:52.480
Courant Events Right: But there are other ways to decompose.

588
00:36:52.480 --> 00:36:53.320
nicolascurien: Who's…

589
00:36:53.480 --> 00:36:55.370
Courant Events Right: IDF Tigers, for example.

590
00:36:55.370 --> 00:37:00.339
nicolascurien: And you would get more, like, trivalent planar map heading there.

591
00:37:00.480 --> 00:37:01.960
Courant Events Right: So let's denote…

592
00:37:01.960 --> 00:37:04.050
nicolascurien: SN with this dot.

593
00:37:04.730 --> 00:37:09.640
nicolascurien: A valPatterson surface with JS0N punctures, sampled uniformly.

594
00:37:09.900 --> 00:37:13.109
nicolascurien: And this point, 0 is just a reference point picture.

595
00:37:13.110 --> 00:37:15.720
Courant Events Right: Uniformly in the surface, according to.

596
00:37:15.720 --> 00:37:16.650
nicolascurien: pizza.

597
00:37:16.860 --> 00:37:17.620
Courant Events Right: hyper.

598
00:37:17.620 --> 00:37:18.270
nicolascurien: ignition.

599
00:37:19.180 --> 00:37:22.139
Courant Events Right: Then we proved that last year with Timothy that

600
00:37:23.650 --> 00:37:39.179
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.

601
00:37:39.190 --> 00:37:43.940
Courant Events Right: Which is homomorphic to the blind minus discrete punctures.

602
00:37:44.860 --> 00:37:49.379
Courant Events Right: But it's random, something random in the limit, with a simulation.

603
00:37:49.540 --> 00:37:54.269
Courant Events Right: But you can also see this SN0 as somehow.

604
00:37:54.270 --> 00:37:59.930
nicolascurien: Compact, excuse me, it's not compact because of the punctures, but you cut the punctures.

605
00:37:59.930 --> 00:38:00.659
Courant Events Right: Then you would get.

606
00:38:00.660 --> 00:38:02.070
nicolascurien: compact metric space.

607
00:38:02.460 --> 00:38:08.829
Courant Events Right: And if you renormalize all distances by N to the 1 quarter, then you get a convergence towards universal variance here.

608
00:38:09.130 --> 00:38:11.340
Courant Events Right: Which is the random fractal.

609
00:38:11.340 --> 00:38:12.239
nicolascurien: space that appears.

610
00:38:12.240 --> 00:38:12.879
Courant Events Right: also as a scale.

611
00:38:12.880 --> 00:38:17.240
nicolascurien: a bit of… Totality, and so on.

612
00:38:18.440 --> 00:38:20.379
nicolascurien: So, let me give you…

613
00:38:21.120 --> 00:38:23.949
Courant Events Right: aesthetic flavor of that. How much time do I have?

614
00:38:24.660 --> 00:38:25.550
Courant Events Right: Kendrick?

615
00:38:26.380 --> 00:38:29.040
nicolascurien: Including questions.

616
00:38:29.730 --> 00:38:30.680
nicolascurien: Okay.

617
00:38:31.070 --> 00:38:31.970
nicolascurien: Digital type.

618
00:38:32.190 --> 00:38:33.549
Courant Events Right: Okay, so we have this…

619
00:38:33.550 --> 00:38:34.450
nicolascurien: It's illuminum.

620
00:38:36.290 --> 00:38:37.930
Courant Events Right: And this…

621
00:38:37.930 --> 00:38:38.350
nicolascurien: Mr.

622
00:38:38.350 --> 00:38:38.800
Courant Events Right: Totally.

623
00:38:38.800 --> 00:38:39.330
nicolascurien: Maybe in the.

624
00:38:39.330 --> 00:38:43.990
Courant Events Right: representation of SN. So, what is this? So SN is hyperbolic.

625
00:38:44.840 --> 00:38:47.690
Courant Events Right: surface, KSD, or hand punctures.

626
00:38:48.150 --> 00:38:56.889
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.

627
00:38:57.160 --> 00:39:11.469
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

628
00:39:12.730 --> 00:39:16.949
Courant Events Right: Okay, so this guy, if you sit at the random point here.

629
00:39:17.140 --> 00:39:31.979
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?

630
00:39:34.130 --> 00:39:36.900
nicolascurien: Whose cells are the, like, the journalistic sphere?

631
00:39:37.180 --> 00:39:38.550
Courant Events Right: Yes, yes.

632
00:39:38.690 --> 00:39:46.780
Courant Events Right: These, these are, half spheres, so they are just… And…

633
00:39:47.520 --> 00:40:04.090
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.

634
00:40:04.250 --> 00:40:10.909
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.

635
00:40:11.060 --> 00:40:13.679
Courant Events Right: It's only metric, nothing…

636
00:40:13.960 --> 00:40:22.550
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.

637
00:40:22.820 --> 00:40:38.390
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.

638
00:40:38.680 --> 00:40:50.509
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?

639
00:40:50.590 --> 00:41:07.670
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.

640
00:41:13.960 --> 00:41:15.260
Courant Events Right: It's time to wake up!

641
00:41:16.120 --> 00:41:16.860
Courant Events Right: Yep.

642
00:41:17.040 --> 00:41:18.939
Courant Events Right: Any other questions from that side?

643
00:41:22.540 --> 00:41:23.390
Courant Events Right: Nope.

644
00:41:24.410 --> 00:41:40.850
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

645
00:41:40.880 --> 00:41:52.590
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.

646
00:41:52.610 --> 00:41:53.670
nicolascurien: Beautiful.

647
00:41:54.150 --> 00:41:55.790
Courant Events Right: And this season.

648
00:41:55.970 --> 00:42:07.139
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.

649
00:42:07.300 --> 00:42:22.320
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.

650
00:42:22.490 --> 00:42:25.949
Courant Events Right: in there, K points. Then, the metric.

651
00:42:25.950 --> 00:42:26.590
nicolascurien: So, for the.

652
00:42:26.590 --> 00:42:35.110
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.

653
00:42:35.110 --> 00:42:36.210
nicolascurien: convoluted for me.

654
00:42:36.550 --> 00:42:41.739
Courant Events Right: For those specialists in the room, whoa, whoa, whoa, whoa, whoa, how do you get to grow.

655
00:42:41.740 --> 00:42:45.109
nicolascurien: of Hosdorf convergency, because this object is non-compact.

656
00:42:46.640 --> 00:42:48.610
Courant Events Right: True. Luke?

657
00:42:48.610 --> 00:42:49.860
nicolascurien: written grammar.

658
00:42:49.860 --> 00:43:00.320
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.

659
00:43:00.320 --> 00:43:03.980
nicolascurien: It's possible to do this in a canonical.

660
00:43:03.980 --> 00:43:07.380
Courant Events Right: And if you cut the punctures, then you are back to.

661
00:43:07.380 --> 00:43:10.109
nicolascurien: We'll compact thing, and then we have got a.

662
00:43:10.110 --> 00:43:13.070
Courant Events Right: of house, convergence. But if.

663
00:43:13.070 --> 00:43:13.820
nicolascurien: You don't want to.

664
00:43:13.820 --> 00:43:17.249
Courant Events Right: Look at the punctures, this gum of poker off captures

665
00:43:17.610 --> 00:43:20.110
Courant Events Right: But the measure is telling you what's the measure.

666
00:43:20.110 --> 00:43:21.110
nicolascurien: It's a high probability.

667
00:43:21.110 --> 00:43:21.740
Courant Events Right: measure, and then.

668
00:43:21.740 --> 00:43:24.850
nicolascurien: There's very little mass around the cusp, around the punctures.

669
00:43:24.850 --> 00:43:27.360
Courant Events Right: So, actually, they kind of disappear in the limit.

670
00:43:27.360 --> 00:43:28.079
nicolascurien: I don't see that here.

671
00:43:28.080 --> 00:43:35.220
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

672
00:43:36.010 --> 00:43:37.110
Courant Events Right: of metric space.

673
00:43:37.940 --> 00:43:41.480
Courant Events Right: So, what type of observable also you can,

674
00:43:41.740 --> 00:43:51.639
Courant Events Right: you can, put in this Gromof, Hausdorf or Gromof-proof convergences, length of separating, curves. You won't say a curve.

675
00:43:52.410 --> 00:43:53.430
Courant Events Right: In your metric space.

676
00:43:53.430 --> 00:43:54.520
nicolascurien: that would…

677
00:43:54.520 --> 00:43:55.430
Courant Events Right: That would separate you?

678
00:43:55.430 --> 00:43:56.000
nicolascurien: put my face.

679
00:43:56.000 --> 00:43:58.929
Courant Events Right: into two pieces containing at least

680
00:43:59.080 --> 00:44:01.460
Courant Events Right: N of the 4 punctures on each side.

681
00:44:01.970 --> 00:44:06.669
Courant Events Right: Then you want to minimize over the length of all those curves. This is an observation that you.

682
00:44:06.670 --> 00:44:08.370
nicolascurien: You can put this.

683
00:44:08.370 --> 00:44:09.110
Courant Events Right: convergence.

684
00:44:11.580 --> 00:44:14.149
Courant Events Right: Okay, so what are the methods we're using?

685
00:44:14.150 --> 00:44:16.170
nicolascurien: used to prove this theorem.

686
00:44:16.170 --> 00:44:17.580
Courant Events Right: In principle, this is actually.

687
00:44:17.580 --> 00:44:18.180
nicolascurien: accessible.

688
00:44:18.180 --> 00:44:20.239
Courant Events Right: If the peeling,

689
00:44:20.440 --> 00:44:26.349
Courant Events Right: method I described to you with this end process and so on, which is easier in this.

690
00:44:26.350 --> 00:44:26.680
nicolascurien: the…

691
00:44:26.680 --> 00:44:28.440
Courant Events Right: Genius Zero…

692
00:44:28.600 --> 00:44:30.249
nicolascurien: Business, because they are…

693
00:44:30.250 --> 00:44:32.389
Courant Events Right: It's no genius.

694
00:44:32.410 --> 00:44:33.700
nicolascurien: It's just a tree.

695
00:44:34.050 --> 00:44:34.849
Courant Events Right: So, in principle.

696
00:44:34.850 --> 00:44:35.520
nicolascurien: Where is it?

697
00:44:36.340 --> 00:44:40.470
Courant Events Right: The bridge initial, conversions would follow from the peeling.

698
00:44:41.120 --> 00:44:42.309
Courant Events Right: The grommos.

699
00:44:42.310 --> 00:44:42.700
nicolascurien: golf.

700
00:44:42.700 --> 00:44:45.560
Courant Events Right: Here, with opening, it's more complicated, and I.

701
00:44:45.560 --> 00:44:46.060
nicolascurien: We don't.

702
00:44:46.060 --> 00:44:46.740
Courant Events Right: do a…

703
00:44:46.740 --> 00:44:48.850
nicolascurien: We, rely on the…

704
00:44:48.850 --> 00:44:49.330
Courant Events Right: Organization?

705
00:44:49.330 --> 00:44:50.020
nicolascurien: of those objects?

706
00:44:50.020 --> 00:44:55.510
Courant Events Right: with label trees, it's something classical in, random planar map theory.

707
00:44:55.510 --> 00:44:57.140
nicolascurien: That you encode for you.

708
00:44:57.580 --> 00:45:06.960
Courant Events Right: planar graph or map by simple objects, which are label trees. It's this famous, Schaeffer or Plutiere de Francesco bijection.

709
00:45:07.200 --> 00:45:10.450
Courant Events Right: And actually, what's funny is that those bijections.

710
00:45:10.450 --> 00:45:11.599
nicolascurien: was including reef.

711
00:45:11.600 --> 00:45:12.690
Courant Events Right: label trees.

712
00:45:12.950 --> 00:45:14.320
nicolascurien: were discovered.

713
00:45:14.320 --> 00:45:24.290
Courant Events Right: in this hyperbolic context, before the discrete analog for planar maps. Nobody noticed, except Timothy, maybe, but this was done before.

714
00:45:24.290 --> 00:45:25.830
nicolascurien: by Penaire and British.

715
00:45:25.830 --> 00:45:32.179
Courant Events Right: So the… actually, I did something more general, and the specification of this construction would give.

716
00:45:32.890 --> 00:45:34.609
Courant Events Right: the… the encoding.

717
00:45:34.610 --> 00:45:35.690
nicolascurien: use.

718
00:45:35.690 --> 00:45:40.300
Courant Events Right: So the idea, rough idea is… the following. You have your.

719
00:45:40.300 --> 00:45:42.000
nicolascurien: surface here, down here.

720
00:45:42.000 --> 00:45:43.779
Courant Events Right: And you have a specific puncture.

721
00:45:44.190 --> 00:45:49.759
Courant Events Right: In orange here. Now, you will consider the cutlocus. So the cut locus is something geometric.

722
00:45:49.760 --> 00:45:53.789
nicolascurien: It's the set of all points on your surface from which you can.

723
00:45:53.790 --> 00:45:54.390
Courant Events Right: Stop.

724
00:45:54.390 --> 00:45:55.960
nicolascurien: Two different geodesics.

725
00:45:56.250 --> 00:46:03.999
Courant Events Right: towards this puncture. Of course, this puncture is at infinity, but you can make sense of adjusting towards this puncture.

726
00:46:04.680 --> 00:46:05.919
Courant Events Right: And, two different.

727
00:46:05.920 --> 00:46:06.399
nicolascurien: conjun deser.

728
00:46:06.400 --> 00:46:07.850
Courant Events Right: Having the same length.

729
00:46:07.970 --> 00:46:08.740
Courant Events Right: Again.

730
00:46:08.740 --> 00:46:12.589
nicolascurien: The length is infinite, but you can make sense of that. So…

731
00:46:12.590 --> 00:46:14.889
Courant Events Right: Makes sense, right? If you will start from here.

732
00:46:15.200 --> 00:46:17.399
Courant Events Right: If you want to go to this puncture, let's say there is a.

733
00:46:17.400 --> 00:46:18.310
nicolascurien: only one way.

734
00:46:18.310 --> 00:46:19.499
Courant Events Right: go down. If you're on the.

735
00:46:19.500 --> 00:46:21.820
nicolascurien: The other side of your surface, you go down.

736
00:46:21.820 --> 00:46:24.980
Courant Events Right: the other side, but there is… like a path?

737
00:46:25.430 --> 00:46:33.979
Courant Events Right: where you could, you know, go other… two different ways to this function. And there are those special points.

738
00:46:34.700 --> 00:46:40.820
Courant Events Right: Where it's, punching of other three, where you've got three different possible ways to go to this punch.

739
00:46:41.510 --> 00:46:47.930
Courant Events Right: So, generically, the structure you see emerging here in red is an embedded

740
00:46:48.150 --> 00:46:59.090
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.

741
00:46:59.090 --> 00:46:59.610
nicolascurien: And…

742
00:46:59.610 --> 00:47:00.810
Courant Events Right: You have your surface.

743
00:47:02.310 --> 00:47:05.240
Courant Events Right: And now what you do is you take the limit measure

744
00:47:05.700 --> 00:47:08.979
Courant Events Right: It's very Peterson measure on surface. You push it forward.

745
00:47:09.310 --> 00:47:12.869
Courant Events Right: On this tree, and it turns out that what you see here is.

746
00:47:12.870 --> 00:47:14.950
nicolascurien: reasonably well-behaved.

747
00:47:14.950 --> 00:47:25.379
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.

748
00:47:25.560 --> 00:47:27.709
Courant Events Right: From this, you're back to…

749
00:47:27.710 --> 00:47:28.769
nicolascurien: safe business.

750
00:47:28.770 --> 00:47:34.310
Courant Events Right: where Gregory and Jean-Francois prove the convergence to the Boolean sphere, and you try to adapt the.

751
00:47:34.310 --> 00:47:35.320
nicolascurien: the proofs.

752
00:47:35.520 --> 00:47:37.450
Courant Events Right: Adapting doesn't mean,

753
00:47:37.450 --> 00:47:38.080
nicolascurien: Copy and paste.

754
00:47:38.080 --> 00:47:38.860
Courant Events Right: Of course.

755
00:47:39.950 --> 00:47:48.869
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.

756
00:47:49.680 --> 00:47:50.240
Courant Events Right: So this…

757
00:47:50.240 --> 00:47:50.840
nicolascurien: paper.

758
00:47:52.270 --> 00:47:57.410
Courant Events Right: Okay, so that's… that's all for today on the G is equal to 0, so the planar case.

759
00:47:58.950 --> 00:48:02.149
Courant Events Right: But after all, there are many perspectives, so,

760
00:48:02.360 --> 00:48:07.669
Courant Events Right: We understood a piece of what is going on for the large genius.

761
00:48:08.320 --> 00:48:26.079
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,

762
00:48:27.440 --> 00:48:31.000
Courant Events Right: two universities, those, or the bonus here.

763
00:48:31.000 --> 00:48:34.119
nicolascurien: and many classes of the stable maps.

764
00:48:34.120 --> 00:48:35.080
Courant Events Right: use.

765
00:48:35.220 --> 00:48:38.739
Courant Events Right: Genius 1 already is not understood from this person.

766
00:48:38.740 --> 00:48:39.490
nicolascurien: 15.

767
00:48:40.070 --> 00:48:40.800
Courant Events Right: We have no.

768
00:48:40.800 --> 00:48:41.200
nicolascurien: understood.

769
00:48:41.200 --> 00:48:42.720
Courant Events Right: why the length spectrum.

770
00:48:42.720 --> 00:48:45.879
nicolascurien: is universal, and the Laplaceanche spectrum, I told you.

771
00:48:45.880 --> 00:48:50.010
Courant Events Right: about this one quarter, this was a big achievement of many people.

772
00:48:50.470 --> 00:48:53.269
Courant Events Right: But it's only for the Val Peterson.

773
00:48:53.270 --> 00:48:55.389
nicolascurien: Bill Peterson.

774
00:48:57.130 --> 00:49:01.510
Courant Events Right: Guy with, large teeth, no punctures, and,

775
00:49:01.510 --> 00:49:02.110
nicolascurien: engine.

776
00:49:02.110 --> 00:49:03.190
Courant Events Right: hygienist.

777
00:49:03.550 --> 00:49:07.189
Courant Events Right: We don't know what's the diameter of the sun in large denius.

778
00:49:07.460 --> 00:49:17.259
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,

779
00:49:17.960 --> 00:49:24.320
Courant Events Right: resistant to adding one puncture. But what if the genius is proportional to the number of punctures?

780
00:49:24.320 --> 00:49:25.390
nicolascurien: something.

781
00:49:25.520 --> 00:49:27.629
Courant Events Right: What about other measures that are.

782
00:49:27.630 --> 00:49:27.950
nicolascurien: other…

783
00:49:27.950 --> 00:49:35.600
Courant Events Right: Natural measures, like, measures, which is not at all the same as the Velvetesan measure, so this is

784
00:49:35.750 --> 00:49:36.680
Courant Events Right: Olop.

785
00:49:37.410 --> 00:49:38.470
Courant Events Right: Thanks a lot!

786
00:49:45.460 --> 00:49:47.830
Courant Events Right: It's still time for the left part.

787
00:49:47.830 --> 00:49:49.210
nicolascurien: to make it up.

788
00:49:49.450 --> 00:49:50.050
Courant Events Right: this!

789
00:49:51.040 --> 00:49:58.189
Courant Events Right: Well, maybe you said something that I missed, in this, convergence to the Brownian sphere.

790
00:49:58.490 --> 00:50:04.700
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.

791
00:50:04.800 --> 00:50:20.899
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.

792
00:50:21.230 --> 00:50:27.690
Courant Events Right: Then, this flowing process for the Boylean sphere, just gives the mass measure on the Boylean sphere.

793
00:50:28.600 --> 00:50:50.799
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.

794
00:50:53.000 --> 00:50:54.010
Courant Events Right: Yes.

795
00:50:54.430 --> 00:50:55.450
Courant Events Right: Yelinda.

796
00:50:56.510 --> 00:51:12.280
Courant Events Right: And I think also, Gerald on us.

797
00:51:12.490 --> 00:51:26.320
Courant Events Right: G is the right font, but I think they…

798
00:51:26.440 --> 00:51:36.120
Courant Events Right: You know much better than me. So, Yelim said that, for the spectrum gap.

799
00:51:36.420 --> 00:51:47.350
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.

800
00:51:48.080 --> 00:51:51.869
Courant Events Right: But, what if… what if you had long boundaries?

801
00:51:54.630 --> 00:52:01.760
Courant Events Right: Did I count your… so, the left is ahead, You miss some fragments.

802
00:52:01.890 --> 00:52:02.840
Courant Events Right: I missed?

803
00:52:02.960 --> 00:52:03.960
Courant Events Right: How many in?

804
00:52:04.140 --> 00:52:08.259
Courant Events Right: I wasn't counting, but… Did I?

805
00:52:12.150 --> 00:52:12.970
Courant Events Right: Nope.

806
00:52:13.620 --> 00:52:15.069
Courant Events Right: The left and the right.

807
00:52:15.540 --> 00:52:19.979
Courant Events Right: Okay, so this is why?

808
00:52:22.940 --> 00:52:24.400
Courant Events Right: Any other questions?

809
00:52:25.590 --> 00:52:28.190
Courant Events Right: Okay, so I don't think we can find anything else.

810
00:52:33.520 --> 00:52:36.089
Courant Events Right: Is that your throwing, the very aggressive, Connie?

811
00:52:36.100 --> 00:52:36.950
nicolascurien: Good deal.

812
00:52:37.420 --> 00:52:41.670
Courant Events Right: But, on top of, A picture.

813
00:52:41.670 --> 00:52:44.590
nicolascurien: It's easy.

814
00:52:45.100 --> 00:52:45.930
Courant Events Right: Good luck.

815
00:52:46.110 --> 00:52:46.910
nicolascurien: Thanks.