Permalinkhttps://doi.org/10.5802/ahl.82

Keywordsmapping class group, character variety, measured lamination

Abstract

We show that the algebraic automorphism group of the ${\mathrm{SL}}_2\left(ℂ\right)$ character variety of a closed orientable surface with negative Euler characteristic is a finite extension of its mapping class group. Along the way, we provide a simple characterization of the valuations on the character algebra coming from measured laminations.

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Permalinkhttps://doi.org/10.5802/ahl.81

KeywordsWeierstrass points, Riemann surfaces, stable curves, Weierstrass gap sequences

Abstract

Cet article donne la description des nœuds des courbes stables qui sont limites de points de Weierstraß. Cela résout le problème posé par Esteves de caractériser les courbes stables dont aucun nœud n’est limite de points de Weierstraß. De plus, nous étendons ce résultat au cas des points de $k$-Weierstraß pour tout $k\ge 1$. Enfin, nous donnons la description des lacunes de points de $k$-Weierstraß réalisées sur des surfaces de Riemann de genre $2$. Les preuves reposent sur la compactification des strates de différentielles introduites par Bainbridge–Chen–Gendron–Grushevsky–Möller.

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Permalinkhttps://doi.org/10.5802/ahl.80

KeywordsLagrangian torus, SYZ fibration, dual complex, negative vertex

Abstract

We give a construction of Lagrangian torus fibrations with controlled discriminant locus on certain affine varieties. In particular, we apply our construction in the following ways:

• We find a Lagrangian torus fibration on the 3-fold negative vertex whose discriminant locus has codimension 2; this provides a local model for finding torus fibrations on compact Calabi–Yau 3-folds with codimension 2 discriminant locus.
• We find a Lagrangian torus fibration on a neighbourhood of the one-dimensional stratum of a simple normal crossing divisor (satisfying certain conditions) such that the base of the fibration is an open subset of the cone over the dual complex of the divisor. This can be used to construct an analogue of the non-archimedean SYZ fibration constructed by Nicaise, Xu and Yu.

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Permalinkhttps://doi.org/10.5802/ahl.79

Keywordsreductive algebraic groups, characters, $p$-canonical basis

Abstract

In this paper we prove a character formula expressing the classes of simple representations in the principal block of a simply-connected semisimple algebraic group $G$ in terms of baby Verma modules, under the assumption that the characteristic of the base field is bigger than $2h-1$, where $h$ is the Coxeter number of $G$. This provides a replacement for Lusztig’s conjecture, valid under a reasonable assumption on the characteristic.

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Permalinkhttps://doi.org/10.5802/ahl.78

Keywordsfoliation, locally product

Abstract

In the main result of this paper we prove that a codimension one foliation of ${\mathbb{P}}^n$, which is locally a product near every point of some codimension two component of the singular set, has a Kupka component. In particular, we obtain a generalization of a known result of Calvo Andrade and Brunella about foliations with a Kupka component.

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Permalinkhttps://doi.org/10.5802/ahl.77

Keywordsspin glass, statistical inference, Hamilton–Jacobi equation

Abstract

We argue that Hamilton–Jacobi equations provide a convenient and intuitive approach for studying the large-scale behavior of mean-field disordered systems. This point of view is illustrated on the problem of inference of a rank-one matrix. We compute the large-scale limit of the free energy by showing that it satisfies an approximate Hamilton–Jacobi equation with asymptotically vanishing viscosity parameter and error term.

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Permalinkhttps://doi.org/10.5802/ahl.76

KeywordsSharp mixing rates, nonuniform hyperbolicity, billiards, multidimensional intermittent maps, operator renewal theory

Abstract

Gouëzel and Sarig introduced operator renewal theory as a method to prove sharp results on polynomial decay of correlations for certain classes of nonuniformly expanding maps. In this paper, we apply the method to planar dispersing billiards and multidimensional nonMarkovian intermittent maps.

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Permalinkhttps://doi.org/10.5802/ahl.75

KeywordsFokker–Planck equation, hypoelliptic, Schauder estimate, nonlinear kinetic equation

Abstract

This article is concerned with the Schauder estimate for linear kinetic Fokker–Planck equations with Höder continuous coefficients. This equation has an hypoelliptic structure. As an application of this Schauder estimate, we prove the global well-posedness of a toy nonlinear model in kinetic theory. This nonlinear model consists in a non-linear kinetic Fokker–Planck equation whose steady states are Maxwellian and whose diffusion in the velocity variable is proportional to the mass of the solution.

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Permalinkhttps://doi.org/10.5802/ahl.74

KeywordsBayesian nonparametrics, exchangeability, hierarchical exchangeability, Aldous–Hoover representation, de Finetti representation

Abstract

Motivated by the problem of designing inference-friendly Bayesian nonparametric models in probabilistic programming languages, we introduce a general class of partially exchangeable random arrays which generalizes the notion of hierarchical exchangeability introduced in Austin and Panchenko (2014). We say that our partially exchangeable arrays are DAG-exchangeable since their partially exchangeable structure is governed by a collection of Directed Acyclic Graphs. More specifically, such a random array is indexed by ${\mathbb{N}}^{\left|V\right|}$ for some DAG $G=(V,E)$, and its exchangeability structure is governed by the edge set $E$. We prove a representation theorem for such arrays which generalizes the Aldous-Hoover and Austin–Panchenko representation theorems.

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Permalinkhttps://doi.org/10.5802/ahl.73

KeywordsWang tilings, aperiodic, rotation, Markov partition, cut and project

Abstract

We define a partition ${𝒫}_0$ and a ${\mathbb{Z}}^2$-rotation (${\mathbb{Z}}^2$-action defined by rotations) on a 2-dimensional torus whose associated symbolic dynamical system is a minimal proper subshift of the Jeandel–Rao aperiodic Wang shift defined by 11 Wang tiles. We define another partition ${𝒫}_{𝒰}$ and a ${\mathbb{Z}}^2$-rotation on ${𝕋}^2$ whose associated symbolic dynamical system is equal to a minimal and aperiodic Wang shift defined by 19 Wang tiles. This proves that ${𝒫}_{𝒰}$ is a Markov partition for the ${\mathbb{Z}}^2$-rotation on ${𝕋}^2$. We prove in both cases that the toral ${\mathbb{Z}}^2$-rotation is the maximal equicontinuous factor of the minimal subshifts and that the set of fiber cardinalities of the factor map is $\{1,2,8\}$. The two minimal subshifts are uniquely ergodic and are isomorphic as measure-preserving dynamical systems to the toral ${\mathbb{Z}}^2$-rotations. It provides a construction of these Wang shifts as model sets of 4-to-2 cut and project schemes. A do-it-yourself puzzle is available in the appendix to illustrate the results.

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Permalinkhttps://doi.org/10.5802/ahl.72

KeywordsLieb–Thirring inequalities, uncertainty principle, exclusion principle, Poincaré inequality

Abstract

We propose a general strategy to derive Lieb–Thirring inequalities for scale-covariant quantum many-body systems. As an application, we obtain a generalization of the Lieb–Thirring inequality to wave functions vanishing on the diagonal set of the configuration space, without any statistical assumption on the particles.

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Permalinkhttps://doi.org/10.5802/ahl.71

KeywordsMarkoff surface, Kesten–McKay law, cubic surfaces, graphs and groups

Abstract

For each prime $p$, we study the eigenvalues of a 3-regular graph on roughly $p^2$ vertices constructed from the Markoff surface. We show they asymptotically follow the Kesten–McKay law, which also describes the eigenvalues of a random regular graph. The proof is based on the method of moments and takes advantage of a natural group action on the Markoff surface.

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Permalinkhttps://doi.org/10.5802/ahl.70

Keywordsself-intersection, random tessellation, geodesic, hyperbolic surface, Poisson line process

Abstract

We show that the tessellation of a compact, negatively curved surface induced by a long random geodesic segment, when properly scaled, looks locally like a Poisson line process. This implies that the global statistics of the tessellation – for instance, the fraction of triangles – approach those of the limiting Poisson line process.

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Permalinkhttps://doi.org/10.5802/ahl.69

KeywordsCanonical relations, symplectic reduction, Poisson sigma model, symplectic groupoids

Abstract

A Lagrangian subspace $L$ of a weak symplectic vector space is called split Lagrangian if it has an isotropic (hence Lagrangian) complement. When the symplectic structure is strong, it is sufficient for $L$ to have a closed complement, which can then be moved to become isotropic. The purpose of this note is to develop the theory of compositions and reductions of split canonical relations for symplectic vector spaces. We give conditions on a coisotropic subspace $C$ of a weak symplectic space $V$ which imply that the induced canonical relation $L_C$ from $V$ to $C/C^{\omega }$ is split, and, from these, we find sufficient conditions for split canonical relations to compose well. We prove that the canonical relations arising in the Poisson sigma model from the Lagrangian field theoretical approach are split, giving a description of symplectic groupoids integrating Poisson manifolds in terms of split canonical relations.

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Permalinkhttps://doi.org/10.5802/ahl.68

KeywordsBerry–Esseen bound, central limit theorem, cumulants, high dimensions, mod-$\varphi$ convergence, moderate deviations, large deviations, random simplex, Poisson–Delaunay tessellation, stochastic geometry

Abstract

Typical weighted random simplices $Z_{\mu }$, $\mu \in (-2,\infty )$, in a Poisson–Delaunay tessellation in ${ℝ}^n$ are considered, where the weight is given by the $(\mu +1)$st power of the volume. As special cases this includes the typical ($\mu =-1$) and the usual volume-weighted ($\mu =0$) Poisson–Delaunay simplex. By proving sharp bounds on cumulants it is shown that the logarithmic volume of $Z_{\mu }$ satisfies a central limit theorem in high dimensions, that is, as $n\to \infty$. In addition, rates of convergence are provided. In parallel, concentration inequalities as well as moderate deviations are studied. The set-up allows the weight $\mu =\mu \left(n\right)$ to depend on the dimension $n$ as well. A number of special cases are discussed separately. For fixed $\mu$ also mod-$\varphi$ convergence and the large deviations behaviour of the logarithmic volume of $Z_{\mu }$ are investigated.

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