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# Moduli spaces of surfaces: Curves, paths, and counting

Counting meanders and square-tilings on surfaces- Eduard Duryev (Institut de mathématiques de Jussieu in Paris)

Abstract: Meander is a homotopy class of a pair of transversal simple closed curves on a sphere. They appear, in particular, as natural enumeration of polymer foldings. Statistics of meanders as the number of intersection grows is one question about meanders people have been interested in. We will show how meanders are related to a particular class of square tilings on surfaces and explain an approach to this more general counting problem that uses ribbon graphs and intersection theory on moduli space.

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Extremal length systole of the Bolza surface- Didac Martinez-Granado (Indiana University Bloomington)

Abstract: In Bers-Teichmueller theory there is a rich interaction between the hyperbolic geometry and the conformal geometry, as a result of the uniformization theorem. If the length notion in the hyperbolic side of this dichotomy is the hyperbolic length, on the conformal side the counterpart is extremal length, a conformal invariant notion of homotopy classes of curves. For a choice of hyperbolic structure, there is an essential curve of minimal extremal length and one of minimal hyperbolic length. We call both the curve and its length, respectively, the extremal length and hyperbolic length systoles. These are functions on moduli space. Bounds for maximal values of the hyperbolic length systole for some geni are known, and for genus 2, its absolute maximum is the Bolza surface, a triangle surface. In this talk we show that the Bolza surface realizes a local maximum of the extremal length systole, and compute its value: square root of 2. This is work in progress with Maxime Fortier Bourque and Franco Vargas Pallete.

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Gaps of saddle connection directions for some branched covers of tori- Anthony Sanchez (University of Washington Seattle)

Abstract: Translation surfaces given by gluing two identical tori along a slit have genus two and two cone-type singularities of angle $4\pi$. There is a distinguished set of trajectories called saddle connections that are the straight lines trajectories between cone points. We can associate a holonomy vector in the plane to each saddle connection whose components are the horizontal and vertical displacement of the saddle connection. How random is the planar set of holonomy of saddle connections? We study this question by computing the gap distribution for slopes of saddle connections for these and other related classes of translation surfaces.

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The topology and geometry of random square-tiled surfaces- Sunrose Thapa Shrestha (Tufts University)

Abstract: A square-tiled surface (STS) is a branched cover of the standard square torus with branching over exactly one point. They are concrete examples of translation surfaces which are an important class of singular flat metrics on 2-manifolds with applications in Teichmüller theory and polygonal billiards. In this talk, we will consider a randomizing model for STSs based on permutation pairs and see how to use it to compute the genus distribution. We will also look at holonomy vectors (Euclidean displacement vectors between cone points) on a random STS. Holonomy vectors of translation surfaces provide coordinates on the space of translation surfaces and their enumeration up to a fixed length has been studied by various authors such as Eskin and Masur. We will obtain finer information about the set of holonomy vectors, Hol(S), of a random STS. In particular, we will see how often Hol(S) contains the set of primitive integer vectors and find how often these sets are exactly equal.

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## 12 replies on “Moduli spaces of surfaces: Curves, paths, and counting”

[…] Curves, paths, and counting: […]

DidacMGsays:

Hi, my name is Didac Martinez-Granado.
I just finished my Ph.D. at Indiana University, under guidance of Dylan Thurston, and I will start a postdoc position in the Fall at University of California, Davis, mentored by Misha Kapovich.
My talk is joint work in progress with Maxime Fortier Bourque and Franco Vargas Pallete.
We show that the extremal length systole of Riemann surfaces of genus two attains a local maximum at the Bolza surface, where it takes the value square root of 2.
The extremal length of a curve on a surface is a conformal invariant notion of length. Given a curve length on a surface, a systole is a non-trivial curve of minimal length.
The extremal length systole is a function on moduli space that achieves a global maximum. The Bolza surface is a hyperbolic surface in the moduli space of surfaces of genus 2, which happens to be an extremal point on the genus 2 moduli space for other natural notions of length.
I will be happy to answer questions via the comment section here. If you prefer to ask them “in person”, my Zoom office hours will be 2-3pm on Thursday, June 11, joint with Chandrika Sadanand.
In the Zoom office hour, priority will be given to questions from graduate students.

Hi Didac,

That was a very clear talk, thank you!
Can you give a quick hint how the elliptic integral identities come into play here?

DidacMGsays:

Hi Anja,

Thanks for watching, and for your question! I’m glad you enjoyed the talk.

The elliptic integral identities appear when computing the extremal length of $latex \alpha$, the candidate extremal length systole curve depicted in the Step 2 of the proof in my talk.
The extremal length should be equal to the length of the curve in the flat metric divided by the height of the cylinder.
The length of the curve is equal to twice the flat length of the edge connecting 0 and 1. In the planar octahedron model given by P, the length of this segment is given by the integral $latex \int_0^1 \sqrt{|q|}.$
Also from the model P, it follows that the height of the cylinder is given by $latex \int_1^{\infty} \sqrt{|q|}$, where $latex q$ is the quadratic differential realizing the extremal metric of $latex \alpha$.
Therefore, showing that $latex EL(\alpha; P)$ is equal to $latex 2\sqrt{2}$ amounts to show that

$latex \int_0^1 \sqrt{|q|} =\sqrt{2} \int_1^{\infty} \sqrt{|q|}.$

This is an identity between elliptic integrals which can be proven geometrically using the symmetries of $latex P$ (I can give more details, if you want), but also follows from identities for elliptic integrals of the first kind, https://mathworld.wolfram.com/EllipticIntegralSingularValue.html, which apply in our case when the aspect ratio above is square root of an integer. So one by-product of our proof is a geometric interpretation of some of these elliptic integral identities as geometric identities on the pillowcase. Similar geometric interpretations of other elliptic integral identities might also be possible, perhaps considering the extremal length of other curves on the octahedron, or other low complexity surfaces, and we would like to explore that avenue in the future.
I am happy to expand on any of this here, or give more details in the Zoom discussion.

Thanks for elaborating. This is a nice byproduct and I hope it will work out for other surfaces!
I plan to stop by at your office hour to ask some more things about the explicit quadratic differential that you have here.

sunrose_shresthasays:

Hi Everyone,

My name is Sunrose Shrestha. I just finished my PhD at Tufts University under Moon Duchin. If you have any questions regarding my talk, please absolutely feel free to stop by my virtual office hours on Wednesday June 10 at 10am.

Hi Sunrose,

This bridge between square-tiled surfaces and group theory is really beautiful (and so nicely put in your glossary).
I’m wondering how well your model is compatible with the Masur-Veech volume. For example, your holonomy theorem means that a fixed proportion of the surfaces have no removable singularities, right? I wouldn’t expect to say something like that (whatever “like that” is for a measure zero set) in Masur-Veech volume. Or should I?

sunrose_shresthasays:

Hi Anja,

Thanks for the chat and the comment here! Sorry I am just getting to it right now after having spoken with you.

Yes, I wouldn’t expect the model to be showing what one would see in the Masur Veech measure, unfortunately, since we are not fixing the stratum. It would be nice to have an easy model for a fixed stratum itself, but I don’t have one yet.

eduardduryevsays:

Hi, I am Eduard Duryev. I am a postdoc in Paris 7 Diderot, working under the supervision of Anton Zorich. My thesis was about SL(2,Z)-orbits of square-tiled surfaces. In general, I am interested in geometric structures on surfaces, combinatorics and enumeration problems. I like flat surfaces in general.

My talk is mostly an elementary introduction to square-tiled surfaces and related questions, it should be accessible to mathematicians of all levels. Please, leave comments, ask questions, I will be happy to answer and discuss.

Hi Eduard,

Thank you for explaining this paper to the world and making me understand some ingredients more in there!
Could you summarize again what’s the relation between ribbon graphs and square-tiled surfaces? And where are the stable graphs in that part of the picture?

eduardduryevsays:

Hi Anja, thanks for watching and thanks for this question. I indeed had a bit of a hasty ending, so let me elaborate on that.

Here is how to obtain a stable graph from a square-tiled surface. For every maximal horizontal cylinder (horizontal strip) of a square-tiled surface take a core curve. The union of such curves is a multicurve. As discussed in the second video, by contracting a multicurve and passing to the dual graph one obtains a stable graph. Recall that in that construction the multicurve turned into the nodes of a singular surfaces, which in turn turned into the edges of the stable graph. Therefore an edge e_C of the resulting stable graph corresponds to some horizontal cylinder C of the square-tiled surface, while a vertex corresponds to the union of certain boundaries of cylinders, a.k.a. singular leaves of horizontal foliation. Precisely, each cylinder C has two circular boundaries corresponding to the two end points of the edge e_C, the union of the boundaries corresponding to a vertex v over all edges adjacent to v is in fact a ribbon graph. The vertices of this ribbon graph are the conical points of the square-tiled surface. The edges are the saddle connections, and combined together they form faces of the ribbon graph and the boundaries of the cylinders. And this is how metric ribbon graphs appear on the picture.

You can think of a stable graph associated to a square-tiled surface as of its ‘combinatorial type’. Fix a combinatorial type and think of all possible square-tiled surfaces that have it. What are the parameters that enumerate those? First, we can choose the widths and the heights of the cylinders. Second, we can choose the lengths of the saddle connections, i.e. of the edges of ribbon graphs, with one restriction that the lengths of faces agree with the widths of the cylinders. This sheds some light on how we are going to count square-tiled surfaces of area bounded by N. We will take the:
1) sum over all stable graphs, i.e. combinatorial types, of the
2) sum over all heights and widths, so that the total area doesn’t exceed N, of the
3) product of the numbers of all stable graphs of a given genus and lengths of faces, over all vertices of the stable graph, times
4) the square-tiled surfaces with all the above parameters fixed.
It turns out that the latter is simply the product of widths, as the only degrees of freedom left are the twists.

And from (3) it becomes clear why do we care about Kontsevich polynomials. Recall that each vertex of a stable graph was decorated by a Kontsevich polynomial N_{g,n}(w_1,…,w_n), where g is the genus at the vertex and n is its valency. Such polynomial counts the asymptotics (as w_i -> infinity) of the number of ribbon graphs of genus g and with n faces of lengths w_1, …, w_n. That is exactly what we are looking for in (3).

That is what the end of my talk should have been, so thanks for the opportunity to clear things up. Did it help?

Yes, that helps enormously! Thank you for writing it all up.