We study the Cauchy problem for the nonlinear wave equations (NLW) with random data and/or stochastic forcing on a two-dimensional compact Riemannian manifold without boundary. (i) We first study the defocusing stochastic damped NLW driven by additive space-time white noise, and with initial data distributed according to the Gibbs measure. By introducing a suitable space-dependent renormalization, we prove local well-posedness of the renormalized equation. Bourgain’s invariant measure argument then allows us to establish almost sure global well-posedness and invariance of the Gibbs measure for the renormalized stochastic damped NLW. (ii) Similarly, we study the random data defocusing NLW (without stochastic forcing or damping), and establish the same results as in the previous setting. (iii) Lastly, we study the stochastic NLW without damping. By introducing a space-time dependent renormalization, we prove its local well-posedness with deterministic initial data in all subcritical spaces.

These results extend the corresponding recent results on the two-dimensional torus obtained by (i) Gubinelli–Koch–Oh–Tolomeo (2021), (ii) Oh–Thomann (2020), and (iii) Gubinelli–Koch–Oh (2018), to a general class of compact manifolds. The main ingredient is the Green’s function estimate for the Laplace–Beltrami operator in this setting to study regularity properties of stochastic terms appearing in each of the problems.

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We give a new proof of the finiteness of maximal arithmetic reflection groups. Our proof is novel in that it makes no use of trace formulas or other tools from the theory of automorphic forms and instead relies on the arithmetic Margulis lemma of Fraczyk, Hurtado and Raimbault.

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The second author has shown that existence of extremal Kähler metrics on semisimple principal toric fibrations is equivalent to a notion of weighted uniform K-stability, read off from the moment polytope. The purpose of this article is to prove various sufficient conditions of weighted uniform K-stability which can be checked effectively and explore the low dimensional new examples of extremal Kähler metrics it provides.

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We give two explicit sets of generators of the group of invertible regular functions over $\mathbf{Q}$ on the modular curve $Y^1\left(N\right)$.

The first set of generators is very surprising. It is essentially the set of defining equations of $Y^1\left(k\right)$ for $k\le N/2$ when all these modular curves are simultaneously embedded into the affine plane, and this proves a conjecture of Derickx and Van Hoeij [DvH14]. This set of generators is an elliptic divisibility sequence in the sense that it satisfies the same recurrence relation as the elliptic division polynomials.

The second set of generators is explicit in terms of classical analytic functions known as Siegel functions. This is both a generalization and a converse of a result of Yang [Yan04, Yan09].

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S. Gersten announced an algorithm that takes as input two finite sequences $\overrightarrow{K}=(K_1,\cdots ,K_N)$ and $\overrightarrow{K^{\prime }}=(K_1^{\prime },\cdots ,K_N^{\prime })$ of conjugacy classes of finitely generated subgroups of $F_n$ and outputs:

• (1) YES or NO depending on whether or not there is an element $\theta \in \mathsf{Out}\left(F_n\right)$ such that $\theta \left(\overrightarrow{K}\right)={\overrightarrow{K}}^{\prime }$ together with one such $\theta$ if it exists and
• (2) a finite presentation for the subgroup of $\mathsf{Out}\left(F_n\right)$ fixing $\overrightarrow{K}$.

S. Kalajdžievski published a verification of this algorithm. We present a different algorithm from the point of view of Culler–Vogtmann’s Outer space. New results include that the subgroup of $\mathsf{Out}\left(F_n\right)$ fixing $\overrightarrow{K}$ is of type $\mathsf{VF}$, an equivariant version of these results, an application, and a unified approach to such questions.

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Following the 1984 seminal work of Belavin, Polyakov and Zamolodchikov on two-dimensional conformal field theories, Toda conformal field theories were introduced in the physics literature as a family of two-dimensional conformal field theories that enjoy, in addition to conformal symmetry, an extended level of symmetry usually referred to as W-symmetry or higher-spin symmetry. More precisely Toda conformal field theories provide a natural way to associate to a finite-dimensional simple and complex Lie algebra a conformal field theory for which the algebra of symmetry contains the Virasoro algebra. In this document we use the path integral formulation of these models to provide a rigorous mathematical construction of Toda conformal field theories based on probability theory. By doing so we recover expected properties of the theory such as the Weyl anomaly formula with respect to the change of background metric by a conformal factor and the existence of Seiberg bounds for the correlation functions.

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We establish the existence of solutions to path-dependent rough differential equations with non-anticipative coefficients. Regularity assumptions on the coefficients are formulated in terms of horizontal and vertical Dupire derivatives.

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We present new Poisson process approximation results for stabilizing functionals of Poisson and binomial point processes. These functionals are allowed to have an unbounded range of interaction and encompass many examples in stochastic geometry. Our bounds are derived for the Kantorovich–Rubinstein distance using the generator approach to Stein’s method. We give different types of bounds for different point processes. While some of our bounds are given in terms of coupling of the point process with its Palm version, the others are in terms of the local dependence structure formalized via the notion of stabilization. We provide two supporting examples for our new framework – one is for Morse critical points of the distance function, and the other is for large $k$-nearest neighbor balls. Our bounds considerably extend the results in Barbour and Brown (1992), Decreusefond, Schulte and Thäle (2016) and Otto (2020).

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We initiate a quantitative study of measure equivalence (and orbit equivalence) between finitely generated groups, which extends the classical setting of ${\mathrm{L}}^p$ measure equivalence. In this paper, our main focus will be on amenable groups, for which we prove both rigidity and flexibility results.

On the rigidity side, we prove a general monotonicity property satisfied by the isoperimetric profile, which implies in particular its invariance under ${\mathrm{L}}^1$ measure equivalence. This yields explicit “lower bounds” on how integrable a measure coupling between two amenable groups can be. This result also has an unexpected application to geometric group theory: the isoperimetric profile turns out to be monotonous under coarse embedding between amenable groups. This has various applications, among which the existence of an uncountable family of $3$-solvable groups which pairwise do not coarsely embed into one another.

On the flexibility side, we construct explicit orbit equivalences between amenable groups with prescribed integrability conditions. Our main tool is a new notion of Følner tiling sequences. We show in a number of instances that the bounds derived from the isoperimetric profile are sharp up to a logarithmic factor. We also deduce from this study that two important quasi-isometry invariants are not preserved under ${\mathrm{L}}^1$ orbit equivalence: the asymptotic dimension and finite presentability.

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We establish abstract limit theorems which provide sufficient conditions for a sequence $\left(A_l\right)$ of rare events in an ergodic probability preserving dynamical system to exhibit Poisson asymptotics, and for the consecutive positions inside the $A_l$ to be asymptotically iid (spatiotemporal Poisson limits). The limit theorems only use information on what happens to $A_l$ before some time ${\tau }_l$ which is of order $o(1/\mu \left(A_l\right))$. In particular, no assumptions on the asymptotic behavior of the system akin to classical mixing conditions are used. We also discuss some general questions about the asymptotic behaviour of spatial and spatiotemporal processes, and illustrate our results in a setup of simple prototypical systems.

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We compute the expectation of the number of linear spaces on a random complete intersection in $p$-adic projective space. Here “random” means that the coefficients of the polynomials defining the complete intersections are sampled uniformly from the $p$-adic integers. We show that as the prime $p$ tends to infinity the expected number of linear spaces on a random complete intersection tends to $1$. In the case of the number of lines on a random cubic in three-space and on the intersection of two random quadrics in four-space, we give an explicit formula for this expectation.

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In this article we define and study a stochastic process on Galoisian covers of compact manifolds. The successive positions of the process are defined recursively by picking a point uniformly in the Dirichlet domain of the previous one. We prove a theorem à la Kesten for such a process: the escape rate of the random walk is positive if and only if the cover is non amenable. We also investigate more in details the case where the deck group is Gromov hyperbolic, showing the almost sure convergence to the boundary of the trajectory as well as a central limit theorem for the escape rate.

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We investigate the effects of noise reinforcement on a Bessel process of dimension $d\in (0,2)$, and more specifically on the asymptotic behavior of its additive functionals. This leads us to introduce a local time process and its inverse. We identify the latter as an increasing self-similar (time-homogeneous) Markov process, and from this, several explicit results can be deduced.

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Let $E$ be a field of characteristic $p$. The group ${\mathbf{Z}}_p^{\times }$ acts on $E\left(\right(X\left)\right)$ by $a·f\left(X\right)=f({(1+X)}^a-1)$. This action extends to the $X$-adic completion $\tilde{\mathbf{E}}$ of ${\cup }_{n\ge 0}E\left(\left(X^{1/p^n}\right)\right)$. We show how to recover $E\left(\right(X\left)\right)$ from the valued $E$-vector space $\tilde{\mathbf{E}}$ endowed with its action of ${\mathbf{Z}}_p^{\times }$. To do this, we introduce the notion of super-Hölder vector in certain $E$-linear representations of ${\mathbf{Z}}_p$. This is a characteristic $p$ analogue of the notion of locally analytic vector in $p$-adic Banach representations of $p$-adic Lie groups.

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We classify completely the infinite, planar triangulations satisfying a weak spatial Markov property, without assuming one-endedness nor finiteness of vertex degrees. In particular, the Uniform Infinite Planar Triangulation (UIPT) is the only such triangulation with average degree $6$. As a consequence, we prove that the convergence of uniform triangulations of the sphere to the UIPT is robust, in the sense that it is preserved under various perturbations of the uniform measure. As another application, we obtain large deviation estimates for the number of occurencies of a pattern in uniform triangulations.

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