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Our main point of focus is the set of closed geodesics on hyperbolic surfaces. For any fixed integer k, we are interested in the set of all closed geodesics with at least k (but possibly more) self-intersections. Among these, we consider those of minimal length and investigate their self-intersection numbers. We prove that their intersection numbers are upper bounded by a universal linear function in k (which holds for any hyperbolic surface). Moreover, in the presence of cusps, we get bounds which imply that the self-intersection numbers behave asymptotically like k for growing k.
A meander is a topological configuration of a line and a simple closed curve in the plane (or a pair of simple closed curves on the 2-sphere) intersecting transversally. Meanders can be traced back to H. Poincaré and naturally appear in various areas of mathematics, theoretical physics and computational biology (in particular, they provide a model of polymer folding). Enumeration of meanders is an important open problem. The number of meanders with
crossings grows exponentially when
grows, but the long-standing problem on the precise asymptotics is still out of reach.
We show that the situation becomes more tractable if one additionally fixes the topological type (or the total number of minimal arcs) of a meander. Then we are able to derive simple asymptotic formulas for the numbers of meanders as
tends to infinity. We also compute the asymptotic probability of getting a simple closed curve on a sphere by identifying the endpoints of two arc systems (one on each of the two hemispheres) along the common equator.
The new tools we bring to bear are based on interpretation of meanders as square-tiled surfaces with one horizontal and one vertical cylinder. The proofs combine recent results on Masur–Veech volumes of moduli spaces of meromorphic quadratic differentials in genus zero with our new observation that horizontal and vertical separatrix diagrams of integer quadratic differentials are asymptotically uncorrelated. The additional combinatorial constraints we impose in this article yield explicit polynomial asymptotics.
be the moduli space of connected, oriented, complete, finite area hyperbolic surfaces of genus
cusps. We study the global behavior of the Mirzakhani function
which assigns to
the Thurston measure of the set of measured geodesic laminations on
of hyperbolic length
. We improve bounds of Mirzakhani describing the behavior of this function near the cusp of
and deduce that
is square-integrable with respect to the Weil–Petersson volume form. We relate this knowledge of
to statistics of counting problems for simple closed hyperbolic geodesics.
J.-C. Yoccoz proposed a natural extension of Selberg’s eigenvalue conjecture to moduli spaces of abelian differentials. We prove an approximation to this conjecture. This gives a qualitative generalization of Selberg’s
theorem to moduli spaces of abelian differentials on surfaces of genus
Using Roelcke’s formula for the Green function, we explicitly construct a basis in the kernel of the adjoint Laplacian on a compact polyhedral surface
and compute the
at the zero value of the spectral parameter. We apply these results to study various self-adjoint extensions of a symmetric Laplacian on a compact polyhedral surface of genus two with a single conical point. It turns out that the behaviour of the
-matrix at the zero value of the spectral parameter is sensitive to the geometry of the polyhedron.
In this we exploit the arithmeticity criterion of Oh and Benoist–Miquel to exhibit an origami in the principal stratum of the moduli space of translation surfaces of genus three whose Kontsevich–Zorich monodromy is not thin in the sense of Sarnak.
By use of a natural extension map and a power series method, we obtain a local stability theorem for
-Kähler structures with the
-lemma under small differentiable deformations.
We describe in this article the dynamics of a one-parameter family of affine interval exchange transformations. This amounts to studying the directional foliations of a particular dilatation surface introduced in Duryev et al [Affine surfaces and their Veech groups. Preprint, 2016, arXiv:1609.02130], the Disco surface. We show that this family displays various dynamical behaviours: it is generically dynamically trivial but for a Cantor set of parameters the leaves of the foliations accumulate to a (transversely) Cantor set. This study is achieved through analysis of the dynamics of the Veech group of this surface combined with a modified version of Rauzy induction in the context of affine interval exchange transformations.
be the irreducible Hermitian symmetric domain of type
. There exists a canonical Hermitian variation of real Hodge structure
of Calabi–Yau type over
. This short note concerns the problem of giving motivic realizations for
. Namely, we specify a descent of
and ask whether the
can be realized as sub-variation of rational Hodge structure of those coming from families of algebraic varieties. When
, we give a motivic realization for
, we show that the unique irreducible factor of Calabi–Yau type in
can be realized motivically.
This paper contains two results on Hodge loci in
. The first concerns fibrations over curves with a non-trivial flat part in the Fujita decomposition. If local Torelli theorem holds for the fibers and the fibration is non-trivial, an appropriate exterior power of the cohomology of the fiber admits a Hodge substructure. In the case of curves it follows that the moduli image of the fiber is contained in a proper Hodge locus. The second result deals with divisors in
. It is proved that the image under the period map of a divisor in
is not contained in a proper totally geodesic subvariety of
. It follows that a Hodge locus in
has codimension at least 2.
We consider compact Kählerian manifolds
of even dimension 4 or more, endowed with a log-symplectic holomorphic Poisson structure
which is sufficiently general, in a precise linear sense, with respect to its (normal-crossing) degeneracy divisor
. We prove that
has unobstructed deformations, that the tangent space to its deformation space can be identified in terms of the mixed Hodge structure on
of the open symplectic manifold
, and in fact coincides with this
provided the Hodge number
, and finally that the degeneracy locus
deforms locally trivially under deformations of
We will develop a theory of multi-pointed non-commutative deformations of a simple collection in an abelian category, and construct relative exceptional objects and relative spherical objects in some cases. This is inspired by a work by Donovan and Wemyss.
be an irreducible holomorphic symplectic (hyperkähler) manifold. If
, we construct a deformation
which admits a symplectic automorphism of infinite order. This automorphism is hyperbolic, that is, its action on the space of real
-classes is hyperbolic. If
, similarly, we construct a deformation which admits a parabolic automorphism (and many other automorphisms as well).
We apply the spectral curve topological recursion to Dubrovin’s universal Landau–Ginzburg superpotential associated to a semi-simple point of any conformal Frobenius manifold. We show that under some conditions the expansion of the correlation differentials reproduces the cohomological field theory associated with the same point of the initial Frobenius manifold.
Zhou et al. [‘On weakly non-decreasable quasiconformal mappings’, J. Math. Anal. Appl.386 (2012), 842–847] proved that, in a Teichmüller equivalence class, there exists an extremal quasiconformal mapping with a weakly nondecreasable dilatation. They asked whether a weakly nondecreasable dilatation is a nondecreasable dilatation. The aim of this paper is to give a negative answer to their problem. We also construct a Teichmüller class such that it contains an infinite number of weakly nondecreasable extremal representatives, only one of which is nondecreasable.
We outline an algorithm to compute
in genus two in quasi-linear time, borrowing ideas from the algorithm for theta constants and the one for
in genus one. Our implementation shows a large speed-up for precisions as low as a few thousand decimal digits. We also lay out a strategy to generalize this algorithm to genus
We compute equations for real multiplication on the divisor classes of genus-2 curves via algebraic correspondences. We do so by implementing van Wamelen’s method for computing equations for endomorphisms of Jacobians on examples drawn from the algebraic models for Hilbert modular surfaces computed by Elkies and Kumar. We also compute a correspondence over the universal family for the Hilbert modular surface of discriminant
and use our equations to prove a conjecture of A. Wright on dynamics over the moduli space of Riemann surfaces.