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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It was shown by Gersten that a central extension of a finitely generated group is quasi-isometrically trivial provided that its Euler class is bounded. We say that a finitely generated group $G$ satisfies Property QITB (quasi-isometrically trivial implies bounded) if the Euler class of any quasi-isometrically trivial central extension of $G$ is bounded. We exhibit a finitely generated group $G$ which does not satisfy Property QITB. This answers a question by Neumann and Reeves, and provides partial answers to related questions by Wienhard and Blank. We also prove that Property QITB holds for a large class of groups, including amenable groups, right-angled Artin groups, relatively hyperbolic groups with amenable peripheral subgroups, and 3-manifold groups.

Finally, we show that Property QITB holds for every finitely presented group if a conjecture by Gromov on bounded primitives of differential forms holds as well.

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We give a new proof of a theorem of Giordano, Putnam and Skau characterizing orbit equivalence of minimal homeomorphisms of the Cantor space in terms of their sets of invariant Borel probability measures. The proof is based on a strengthening of a theorem of Krieger concerning minimal actions of certain locally finite groups of homeomorphisms, and we also give a new proof of the Giordano–Putnam–Skau characterization of orbit equivalence for these actions.

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We complete the study of characters on higher rank semisimple lattices initiated in [BH21, BBHP22], the missing case being the case of lattices in higher rank simple algebraic groups in arbitrary characteristics. More precisely, we investigate dynamical properties of the conjugation action of such lattices on their space of positive definite functions. Our main results deal with the existence and the classification of characters from which we derive applications to topological dynamics, ergodic theory, unitary representations and operator algebras. Our key theorem is an extension of the noncommutative Nevo–Zimmer structure theorem obtained in [BH21] to the case of simple algebraic groups defined over arbitrary local fields. We also deduce a noncommutative analogue of Margulis’ factor theorem for von Neumann subalgebras of the noncommutative Poisson boundary of higher rank arithmetic groups.

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On a closed manifold $M$, we consider a smooth vector field $X$ that generates an Anosov flow. Let $V\in C^{\infty }(M;ℝ)$ be a smooth function called potential. It is known that for any $C>0$, there exists some anisotropic Sobolev space ${\mathscr{H}}_C$ such that the operator $A=-X+V$ has intrinsic discrete spectrum on $\mathrm{Re}\left(z\right)>-C$ called Ruelle resonances. In this paper, we show a “Fractal Weyl law”: the density of resonances is bounded by $O\left({\langle \omega \rangle }^{\frac{n}{1+{\beta }_0}}\right)$ where $\omega =\mathrm{Im}\left(z\right)$, $n=dimM-1$ and $0<{\beta }_0\le 1$ is the Hölder exponent of the distribution $E_u\oplus E_s$ (strong stable and unstable). We also obtain some more precise results concerning the wave front set of the resonances and the invertibility of the transfer operator. Since the dynamical distributions $E_u,E_s$ are non smooth, we use some semi-classical analysis based on wave packet transform associated to an adapted metric $g$ on $T^{*}M$ and construct some specific anisotropic Sobolev spaces.

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Let $|·|$ be the standard Euclidean norm on ${ℝ}^n$ and let $X=({ℝ}^n,\parallel ·\parallel )$ be a normed space. A subspace $Y\subset X$ is strongly $\alpha$-Euclidean if there is a constant $t$ such that $t\left|y\right|\le \parallel y\parallel \le \alpha t\left|y\right|$ for every $y\in Y$, and say that it is strongly $\alpha$-complemented if $\parallel P_Y\parallel \le \alpha$, where $P_Y$ is the orthogonal projection from $X$ to $Y$ and $\parallel P_Y\parallel$ denotes the operator norm of $P_Y$ with respect to the norm on $X$. We give an example of a normed space $X$ of arbitrarily high dimension that is strongly 2-Euclidean but contains no 2-dimensional subspace that is both strongly $(1+ϵ)$-Euclidean and strongly $(1+ϵ)$-complemented, where $ϵ>0$ is an absolute constant. This property is closely related to an old question of Vitali Milman. The example is probabilistic in nature.

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Using semi-classical analysis in ${ℝ}^n$ we present a quite general model for which the topological index formula of Atiyah–Singer predicts a spectral flow with the transition of a finite number of eigenvalues between clusters (energy bands). This model corresponds to physical phenomena that are well observed for quantum energy levels of small molecules [FZ00, FZ01], also in geophysics for the oceanic or atmospheric equatorial waves [DMV17, Mat66] and expected to be observed in plasma physics [QF22].

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We construct extensions of the pure-jump $\Lambda$-Wright–Fisher processes with frequency-dependent selection ($\Lambda$-WF with selection) with different behaviors at their boundary $1$. Those processes satisfy some duality relationships with the block counting process of simple exchangeable fragmentation-coagulation processes (EFC processes). One-to-one correspondences are established between the nature of the boundaries $1$ and $\infty$ of the processes involved. They provide new information on these two classes of processes. Sufficient conditions are provided for boundary $1$ to be an exit boundary or an entrance boundary. When the coalescence measure $\Lambda$ and the selection mechanism verify some regular variation properties, conditions are found in order that the extended $\Lambda$-WF process with selection makes excursions out from the boundary $1$ before getting absorbed at $0$. In this case, $1$ is a transient regular reflecting boundary. This corresponds to a new phenomenon for the deleterious allele, which can be carried by the whole population for a set of times of zero Lebesgue measure, before vanishing in finite time almost surely.

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We study the Dirac equation coupled to scalar and vector Klein–Gordon fields in the limit of strong coupling and large masses of the fields. We prove convergence of the solutions to those of a cubic non-linear Dirac equation, given that the initial spinors coincide. This shows that in this parameter regime, which is relevant to the relativistic mean-field theory of nuclei, the retarded interaction is well approximated by an instantaneous, local self-interaction. We generalize this result to a many-body Dirac–Fock equation on the space of Hilbert–Schmidt operators.

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In this paper, we establish an exponential inequality for random fields, which is applied in the context of convergence rates in the law of large numbers and weak invariance principle in some Hölder spaces.

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This paper, which is the last of a series of three papers, studies dynamical properties of elements of $\mathrm{Out}\left(F_N\right)$, the outer automorphism group of a nonabelian free group $F_N$. We prove that, for every subgroup $H$ of $\mathrm{Out}\left(F_N\right)$, there exists an element $\varphi \in H$ such that, for every element $g$ of $F_N$, the conjugacy class $\left[g\right]$ has polynomial growth under iteration of $\varphi$ if and only if $\left[g\right]$ has polynomial growth under iteration of every element of $H$.

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We prove distality of quantifier-free relations on valued fields with finite residue field. By a result of Chernikov–Galvin–Starchenko, this yields Szemerédi–Trotter-like incidence bounds for function fields over finite fields. We deduce a version of the Elekes–Szabó theorem for such fields.

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We prove a radial source estimate in Hölder–Zygmund spaces for uniformly hyperbolic dynamics (also known as Anosov flows), in the spirit of Dyatlov–Zworski [DZ16]. The main consequence is a new linear stability estimate for the marked length spectrum rigidity conjecture, also known as the Burns–Katok [BK85] conjecture. We show in particular that in any dimension $\ge 2$, in the space of negatively-curved metrics, $C^{3+\epsilon }$-close metrics with same marked length spectrum are isometric. This improves recent works of Guillarmou–Knieper and the second author [GKL22, GL19]. As a byproduct, this approach also allows to retrieve various regularity statements known in hyperbolic dynamics and usually based on Journé’s lemma: the smooth Livšic Theorem of de La Llave–Marco–Moriyón [LMM86], the smooth Livšic cocycle theorem of Niticā–Török [NT98] for general (finite-dimensional) Lie groups, the rigidity of the regularity of the foliation obtained by Hasselblatt [Has92] and others.

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In this paper, we give estimates of the quadratic transportation cost in the conditional central limit theorem for a large class of dependent sequences. Applications to irreducible Markov chains, dynamical systems generated by intermittent maps and $\tau$-mixing sequences are given.

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We compute symplectic cohomology for Milnor fibres of certain compound Du Val singularities that admit small resolution by using homological mirror symmetry. Our computations suggest a new conjecture that the existence of a small resolution has strong implications for the symplectic cohomology and conversely. We also use our computations to give a contact invariant of the link of the singularities and thereby distinguish many contact structures on connected sums of $S^2\times S^3$.

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We are interested in the construction of a smooth branch of travelling waves to the Nonlinear Schrödinger Equation and the Euler–Korteweg system for capillary fluids with nonzero condition at infinity. This branch is defined for speeds close to the speed of sound and looks qualitatively, after rescaling, as a rarefaction pulse described by the Kadomtsev–Petviashvili equation. The proof relies on a fixed point theorem based on the nondegeneracy of the lump solitary wave of the Kadomtsev–Petviashvili equation.

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Bounded weak solutions are constructed for a degenerate parabolic system with a full diffusion matrix, which is a generalized version of the thin film Muskat system. Boundedness is achieved with the help of a sequence ${\left({ℰ}_n\right)}_{n\ge 2}$ of Liapunov functionals such that ${ℰ}_n$ is equivalent to the $L_n$-norm for each $n\ge 2$ and ${ℰ}_n^{1/n}$ controls the $L_{\infty }$-norm in the limit $n\to \infty$. Weak solutions are built by a compactness approach, special care being needed in the construction of the approximation in order to preserve the availability of the above-mentioned Liapunov functionals.

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Fix a container polygon $P$ in the plane and consider the convex hull $P_n$ of $n\ge 3$ independent and uniformly distributed in $P$ random points. In the focus of this paper is the vertex number of the random polygon $P_n$. The precise variance expansion for the vertex number is determined up to the constant-order term, a result which can be considered as a second-order analogue of the classical expansion for the expectation of Rényi and Sulanke (1963). Moreover, a sharp Berry–Esseen bound is derived for the vertex number of the random polygon $P_n$, which is of the same order as one over the square-root of the variance. The latter is optimal and improves the earlier result of Bárány and Reitzner (2006) by removing the factor ${(loglogn)}^{60}$ in the planar case. The main idea behind the proof of both results is a decomposition of the boundary of the random polygon $P_n$ into random convex chains and a careful merging of the variance expansions and Berry–Esseen bounds for the vertex numbers of the individual chains. In the course of the proof, we derive similar results for the Poissonized model.

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The broad motivation of this work is a rigorous understanding of reversible, local Markov dynamics of interfaces, and in particular their speed of convergence to equilibrium, measured via the mixing time $T_{mix}$. In the $(d+1)$-dimensional setting, $d\ge 2$, this is to a large extent mathematically unexplored territory, especially for discrete interfaces. On the other hand, on the basis of a mean-curvature motion heuristics [Hen97, Spo93] and simulations (see [Des02] and the references in [Hen97, Wil04]), one expects convergence to equilibrium to occur on time-scales of order $\approx {\delta }^{-2}$ in any dimension, with $\delta \to 0$ the lattice mesh.

We study the single-flip Glauber dynamics for lozenge tilings of a finite domain of the plane, viewed as $(2+1)$-dimensional surfaces. The stationary measure is the uniform measure on admissible tilings. At equilibrium, by the limit shape theorem [CKP01], the height function concentrates as $\delta \to 0$ around a deterministic profile $\varphi$, the unique minimizer of a surface tension functional. Despite some partial mathematical results [LT15a, LT15b, Wil04], the conjecture $T_{mix}={\delta }^{-2+o\left(1\right)}$ had been proven, so far, only in the situation where $\varphi$ is an affine function [CMT12]. In this work, we prove the conjecture under the sole assumption that the limit shape $\varphi$ contains no frozen regions (facets).

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