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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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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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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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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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 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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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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