We introduce the concept of directed orbifold, namely triples $(X,V,D)$ formed by a directed algebraic or analytic variety $(X,V)$, and a ramification divisor $D$, where $V$ is a coherent subsheaf of the tangent bundle $T_X$. In this context, we introduce an algebra of orbifold jet differentials and their sections. These jet sections can be seen as algebraic differential operators acting on germs of curves, with meromorphic coefficients, whose poles are supported by $D$ and multiplicities are bounded by the ramification indices of the components of $D$. We estimate precisely the curvature tensor of the corresponding directed structure $V\langle D\rangle$ in the general orbifold case – with a special attention to the compact case $D=0$ and to the logarithmic situation where the ramification indices are infinite. Using holomorphic Morse inequalities on the tautological line bundle of the projectivized orbifold Green–Griffiths bundle, we finally obtain effective sufficient conditions for the existence of global orbifold jet differentials.

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We consider a magnetic Laplacian on a compact manifold, with a constant non-degenerate magnetic field. In the large field limit, it is known that the eigenvalues are grouped in clusters, the corresponding sums of eigenspaces being called the Landau levels. The first level has been studied in-depth as a natural generalization of the Kähler quantization. The current paper is devoted to the higher levels: we compute their dimensions as Riemann–Roch numbers, study the associated Toeplitz algebras and prove that each level is isomorphic with a quantization twisted by a convenient auxiliary bundle.

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We study the sizes of the Voronoi cells of $k$ uniformly chosen vertices in a random split tree of size $n$. We prove that, for $n$ large, the largest of these $k$ Voronoi cells contains most of the vertices, while the sizes of the remaining ones are essentially all of order $nexp(-\mathrm{const}\sqrt{logn})$. This “winner-takes-all” phenomenon persists if we modify the definition of the Voronoi cells by (a) introducing random edge lengths (with suitable moment assumptions), and (b) assigning different “influence” parameters (called “speeds” in the paper) to each of the $k$ vertices. Our findings are in contrast to corresponding results on random uniform trees and on the continuum random tree, where it is known that the vector of the relative sizes of the $k$ Voronoi cells is asymptotically uniformly distributed on the $(k-1)$-dimensional simplex.

Two intermediary steps in the proof of our main result may be of independent interest because of the information they give on the typical shape of large random split trees: we prove convergence in probability of their “profile”, and we prove asymptotic results for the size of fringe trees (trees rooted at an ancestor of a uniform random node).

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We show the existence of a trace process at infinity for random walks on hyperbolic groups of conformal dimension $<2$ and relate it to the existence of a reflected random walk. To do so, we employ the theory of Dirichlet forms which connects the theory of symmetric Markov processes to functional analytic perspectives. We introduce a family of Besov spaces associated to random walks and prove that they are isomorphic to some of the Besov spaces constructed from the co-homology of the group studied in Bourdon–Pajot (2003). We also study the regularity of harmonic measures of random walks on hyperbolic groups using the potential theory associated to Dirichlet forms.

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We prove that the closed orbit of the Eierlegende Wollmilchsau is the only ${\text{SL}}_2\left(ℝ\right)$-orbit closure in genus three with a zero Lyapunov exponent in its Kontsevich–Zorich spectrum. The result recovers previous partial results in this direction by Bainbridge–Habegger–Möller and the first named author. The main new contribution is the identification of the differentials in the Hodge bundle corresponding to the Forni subspace in terms of the degenerations of the surface. We use this description of the differentials in the Forni subspace to evaluate them on absolute homology curves and apply the jump problem from the work of Hu and the third named author to the differentials near the boundary of the orbit closure. This results in a simple geometric criterion that excludes the existence of a Forni subspace.

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The cutoff phenomenon is an abrupt transition from out of equilibrium to equilibrium undergone by certain Markov processes in the limit where the size of the state space tends to infinity: instead of decaying gradually over time, their distance to equilibrium remains close to the maximal value for a while and suddenly drops to zero as the time parameter reaches a critical threshold. Despite the accumulation of many examples, this phenomenon is still far from being understood, and identifying the general conditions that trigger it has become one of the biggest challenges in the quantitative analysis of finite Markov chains. Very recently, the author proposed a general sufficient condition for the occurrence of a cutoff, based on a certain information-theoretical statistics called varentropy. In the present paper, we demonstrate the sharpness of this approach by showing that the cutoff phenomenon is actually equivalent to the varentropy criterion for all sparse, fast-mixing chains. Reversibility is not required.

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Dolgopyat and Sarig showed that for any piecewise smooth function $f:𝕋\to ℝ$ and almost every pair $(\alpha ,x_0)\in 𝕋\times 𝕋$, $S_N(f,\alpha ,x_0):={\sum }_{n=1}^{N}f(n\alpha +x_0)$ fails to fulfill a temporal distributional limit theorem. In this article, we show that the doubly metric statement can be sharpened to a single metric one: For almost every $\alpha \in 𝕋$ and all $x_0\in 𝕋$, $S_N(f,\alpha ,x_0)$ does not satisfy a temporal distributional limit theorem, regardless of centering and scaling. The obtained results additionally lead to progress in a question posed by Dolgopyat and Sarig.

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We consider the inverse scattering problems for two types of Schrödinger operators on locally perturbed periodic lattices. For the discrete Hamiltonian, the knowledge of the S-matrix for all energies determines the graph structure and the coefficients of the Hamiltonian. For locally perturbed equilateral metric graphs, the knowledge of the S-matrix for all energies determines the graph structure.

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Motivated by Mather theory of minimizing measures for symplectic twist dynamics, we study conformally symplectic flows on a cotangent bundle. These dynamics are the most general dynamics for which it makes sense to look at (asymptotic) dynamical Maslov index. Our main result is the existence of invariant measures with vanishing index without any convexity hypothesis, in the general framework of conformally symplectic flows. A degenerate twist-condition hypothesis implies the existence of ergodic invariant measures with zero dynamical Maslov index and thus the existence of points with zero dynamical Maslov index.

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We study the $k$-forms of almost homogeneous varieties over perfect base fields $k$. First, we discuss criteria for the existence of $k$-forms in the homogeneous case. Then, we extend the Luna-Vust theory from algebraically closed fields to perfect fields to determine when a given $k$-form of the open orbit of an almost homogeneous variety extends to a $k$-form of the entire variety. Finally, in the last section, we apply these results to determine the real forms of complex almost homogeneous ${SL}_2$-threefolds.

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For groups of a topological origin, such as braid groups and mapping class groups, an important source of interesting and highly non-trivial representations is given by their actions on the twisted homology of associated spaces; these are known as homological representations. Representations of this kind have proved themselves especially important for the question of linearity, a key example being the family of topologically-defined representations introduced by Lawrence and Bigelow, and used by Bigelow and Krammer to prove that braid groups are linear. In this paper, we give a unified foundation for the construction of homological representations using a functorial approach. Namely, we introduce homological representation functors encoding a large class of homological representations, defined on categories containing all mapping class groups and motion groups in a fixed dimension. These source categories are defined using a topological enrichment of the Quillen bracket construction applied to categories of decorated manifolds. This approach unifies many previously-known constructions, including those of Lawrence–Bigelow, and yields many new representations.

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In this article we show that the braid type of a set of $1$-periodic orbits of a non-degenerate Hamiltonian diffeomorphism on a surface is stable under perturbations which are sufficiently small with respect to the Hofer metric $d_{\mathrm{Hofer}}$. We call this new phenomenon braid stability for the Hofer metric.

We apply braid stability to study the stability of the topological entropy $h_{\mathrm{top}}$ of Hamiltonian diffeomorphisms on surfaces under small perturbations with respect to $d_{\mathrm{Hofer}}$. We show that $h_{\mathrm{top}}$ is lower semicontinuous on the space of Hamiltonian diffeomorphisms of a closed surface endowed with the Hofer metric, and on the space of compactly supported diffeomorphisms of the two-dimensional disk $𝔻$ endowed with the Hofer metric. This answers the two-dimensional case of a question of Polterovich.

En route to proving the lower semicontinuity of $h_{\mathrm{top}}$ with respect to $d_{\mathrm{Hofer}}$, we prove that the topological entropy of a diffeomorphism $\phi$ on a compact surface can be recovered from the braid types realized by the periodic orbits of $\phi$.

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The investigation of the behaviour for geometric functionals of random fields on manifolds has drawn recently considerable attention. In this paper, we extend this framework by considering fluctuations over time for the level curves of general isotropic Gaussian spherical random fields. We focus on both long and short memory assumptions; in the former case, we show that the fluctuations of $u$-level curves are dominated by a single component, corresponding to a second-order chaos evaluated on a subset of the multipole components for the random field. We prove the existence of cancellation points where the variance is asymptotically of smaller order; these points do not include the nodal case $u=0$, in marked contrast with recent results on the high-frequency behaviour of nodal lines for random eigenfunctions with no temporal dependence. In the short memory case, we show that all chaoses contribute in the limit, no cancellation occurs and a Central Limit Theorem can be established by Fourth-Moment Theorems and a Breuer–Major argument.

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Given a phenomenon described by a self-similar fragmentation equation, how to infer the fragmentation kernel from experimental measurements of the solution? To answer this question at the basis of our work, a formal asymptotic expansion suggested us that using short-time observations and initial data close to a Dirac measure should be a well-adapted strategy. As a necessary preliminary step, we study the direct problem, i.e. we prove existence, uniqueness and stability with respect to the initial data of non negative measure-valued solutions when the initial data is a compactly supported, bounded, non negative measure. A representation of the solution as a power series in the space of Radon measures is also shown. This representation is used to propose a reconstruction formula for the fragmentation kernel, using short-time experimental measurements when the initial data is close to a Dirac measure. We prove error estimates in Total Variation and Bounded Lipshitz norms; this gives a quantitative meaning to what a “short” time observation is. For general initial data in the space of compactly supported measures, we provide estimates on how the short-time measurements approximate the convolution of the fragmentation kernel with a suitably-scaled version of the initial data. The series representation also yields a reconstruction formula for the Mellin transform of the fragmentation kernel $\kappa$ and an error estimate for such an approximation. Our analysis is complemented by a numerical investigation.

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We prove an upper bound for the number of Ruelle resonances for Koopman operators associated to real-analytic Anosov diffeomorphisms: in dimension $d$, the number of resonances larger than $r$ is a $𝒪\left(\right|logr{|}^d)$ when $r$ goes to $0$. For each connected component of the space of real-analytic Anosov diffeomorphisms on a real-analytic manifold, we prove a dichotomy: either the exponent $d$ in our bound is never optimal, or it is optimal on a dense subset. Using examples constructed by Bandtlow, Just and Slipantschuk, we see that we are always in the latter situation for connected components of the space of real-analytic Anosov diffeomorphisms on the $2$-dimensional torus.

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Using recent work of Carrand on equilibrium states for the billiard map, and adapting techniques from Baladi and Demers, we construct the unique measure of maximal entropy (MME) for two-dimensional finite horizon Sinai (dispersive) billiard flows ${\Phi }^1$ (and show it is Bernoulli), assuming the bound $h_{\text{top}}\left({\Phi }^1\right){\tau }_{min}>s_{0}log2$, where $s_0\in (0,1)$ quantifies the recurrence to singularities. This bound holds in many examples (it is expected to hold generically).

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Let $G$ be a reductive group over a number field or $p$-adic field, and let $V$ be a faithful representation of $G$. A lattice $\Lambda$ in $V$ induces an integral model ${\mathtt{mdl}}_G\left(\Lambda \right)$ of $G$. The first main result of this paper states that up to the action of the normalizer of $G$, there are only finitely many $\Lambda$ yielding the same ${\mathtt{mdl}}_G\left(\Lambda \right)$. We first prove this for split $G$ via the theory of Lie algebra representations, then for nonsplit $G$ via Bruhat–Tits theory. The second main result shows that in a moduli space of principally polarized abelian varieties, a special subvariety is determined, up to finite ambiguity, by its integral Mumford–Tate group. We obtain this result by applying the first main result to the symplectic representations underlying special subvarieties.

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We study some properties of the function $\mathrm{KVol}$ defined by

on the moduli space of translation surfaces. For the Teichmüller discs ${𝒯}_n$ of the original Veech surfaces arising from the right-angled triangles $(\pi /2,\pi /n,(n-2)\pi /2n)$ for odd $n\ge 5$, we establish the first known explicit formula for $\mathrm{KVol}$ (beyond the case of the moduli space of flat tori).

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We show that the signature of a positive braid link is bounded from below by one-quarter of its first Betti number. This equates to one-half of the optimal bound conjectured by Feller, who previously provided a bound of one-eighth.

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We study the compactly supported rational cohomology of configuration spaces of points on wedges of spheres, equipped with natural actions of the symmetric group and the group $Out\left(F_g\right)$ of outer automorphisms of the free group. These representations show up in seemingly unrelated parts of mathematics, from cohomology of moduli spaces of curves to polynomial functors on free groups and Hochschild–Pirashvili cohomology.

We show that these cohomology representations form a polynomial functor, and use various geometric models to compute many of its composition factors. We further compute the composition factors completely for all configurations of $n\le 10$ points. An application of this analysis is a new super-exponential lower bound on the symmetric group action on the weight $0$ component of $H_c^{*}\left({ℳ}_{2,n}\right)$.

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We develop a calculus based on zonoids – a special class of convex bodies – for the expectation of functionals related to a random submanifold $Z$ defined as the zero set of a smooth vector valued random field on a Riemannian manifold. We identify a convenient set of hypotheses on the random field under which we define its zonoid section, an assignment of a zonoid $\zeta \left(p\right)$ in the exterior algebra of the cotangent space at each point $p$ of the manifold. We prove that the first intrinsic volume of $\zeta \left(p\right)$ is the Kac–Rice density of the expected volume of $Z$, while its center computes the expected current of integration over $Z$. We show that the intersection of random submanifolds corresponds to the wedge product of the zonoid sections and that the preimage corresponds to the pull-back.

Combining this with the recently developed zonoid algebra, it allows to give a multiplication structure to the Kac–Rice formulas, resembling that of the cohomology ring of a manifold. Moreover, it establishes a connection with the theory of convex bodies and valuations, which includes deep results such as the Alexandrov–Fenchel inequality and the Brunn–Minkowski inequality. We export them to this context to prove two analogous new inequalities for random submanifolds. Applying our results in the context of Finsler geometry, we prove some new Crofton formulas for the length of curves and the Holmes–Thompson volumes of submanifolds in a Finsler manifold.

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Triggered by the fact that, in the hydrodynamic limit, several different kinetic equations of physical interest all lead to the same Navier–Stokes–Fourier system, we develop in the paper an abstract framework which allows to explain this phenomenon. The method we develop can be seen as a significant improvement of known approaches for which we fully exploit some structural assumptions on the linear and nonlinear collision operators as well as a good knowledge of the Cauchy theory for the limiting equation. In particular, we fully exploit the fact that the collision operator is preserving both momentum and kinetic energy. We adopt a perturbative framework in a Hilbert space setting and first develop a general and fine spectral analysis of the linearized operator and its associated semigroup. Then, we introduce a splitting adapted to the various regimes (kinetic, acoustic, hydrodynamic) present in the kinetic equation which allows, by a fixed point argument, to construct a solution to the kinetic equation and prove the convergence towards suitable solutions to the Navier–Stokes–Fourier system. Our approach is robust enough to treat, in the same formalism, the case of the Boltzmann equation with hard and moderately soft potentials, with and without cut-off assumptions, as well as the Landau equation for hard and moderately soft potentials in presence of a spectral gap. New well-posedness and strong convergence results are obtained within this framework. In particular, for initial data with algebraic decay with respect to the velocity variable, our approach provides the first result concerning the strong Navier–Stokes limit from Boltzmann equation without Grad cut-off assumption or Landau equation. The method developed in the paper is also robust enough to apply, at least at the linear level, to quantum kinetic equations for Fermi–Dirac or Bose–Einstein particles.

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Utilizing KAM theory, we show that there are certain levels in relative $\mathrm{SU}\left(2\right)$ and $\mathrm{SU}\left(3\right)$ character varieties of the once-punctured torus where the action of a single hyperbolic element is not ergodic.

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We show that the space of (reversible) Finsler metrics on the two-torus ${𝕋}^2$ whose geodesic flow is conjugate to the geodesic flow of a flat Finsler metric strongly deformation retracts to the space of flat Finsler metrics with respect to the uniform convergence topology. Along the proof, we also show that two Finsler metrics on ${𝕋}^2$ without conjugate points, whose Heber foliations are smooth and with the same marked length spectrum, have conjugate geodesic flows.

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This paper is dedicated to the study of a one-dimensional congestion model, consisting of two different phases. In the congested phase, the pressure is free and the dynamics is incompressible, whereas in the non-congested phase, the fluid obeys a pressureless compressible dynamics.

We investigate the Cauchy problem for initial data which are small perturbations in the non-congested zone of traveling wave profiles. We prove two different results. First, we show that for arbitrarily large perturbations, the Cauchy problem is locally well-posed in weighted Sobolev spaces. The solution we obtain takes the form $(v_s,u_s)(t,x-\tilde{x}\left(t\right))$, where $x<\tilde{x}\left(t\right)$ is the congested zone and $x>\tilde{x}\left(t\right)$ is the non-congested zone. The set $\{x=\tilde{x}\left(t\right)\}$ is a free boundary, whose evolution is coupled with the one of the solution. Second, we prove that if the initial perturbation is sufficiently small, then the solution is global. This stability result relies on coercivity properties of the linearized operator around a traveling wave, and on the introduction of a new unknown which satisfies better estimates than the original one. In this case, we also prove that traveling waves are asymptotically stable.

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We study a nonlinear recombination model from population genetics as a combinatorial version of the Kac–Boltzmann equation from kinetic theory. Following Kac’s approach, the nonlinear model is approximated by a mean field linear evolution with a large number of particles. In our setting, the latter takes the form of a generalized random transposition dynamics. Our main results establish a uniform in time propagation of chaos with quantitative bounds, and a tight entropy production estimate for the generalized random transpositions, which holds uniformly in the number of particles. As a byproduct of our analysis we obtain sharp estimates on the speed of convergence to stationarity for the nonlinear equation, both in terms of relative entropy and total variation norm.

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We study infinite analogues of expander graphs, namely graphs whose subgraphs weighted by heat kernels form an expander family. Our main result is that there does not exist any infinite expander in this sense. This proves the analogue for random walks of Benjamini’s conjecture that there is no infinite graph whose metric balls are uniformly expanders. The proof relies on a study of stationary random graphs, in particular proving non-expansion of heat kernels in that setting. A key result is that any stationary random graph is stationary hyperfinite, which is a new property of independent interest.

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We obtain the existence, uniqueness and regularity results for solutions to kinetic Fokker–Planck equations with bounded measurable coefficients in the presence of boundary conditions, including the inflow, diffuse reflection and specular reflection cases.

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We consider the so-called transport-Stokes system which describes sedimentation of inertialess suspensions in a viscous flow and couples a transport equation and the steady Stokes equations in the full three-dimensional space. First we present a global existence and uniqueness result for $L^1\cap L^p$ initial densities where $p\ge 3$. Secondly, we prove that, in the case where $p>3$, the flow map which describes the trajectories of these solutions is analytic with respect to time. Finally we establish the small-time global exact controllability of the transport-Stokes system. These results extend to the transport-Stokes system some results obtained for the incompressible Euler system respectively by Yudovich in [Yud63], by Chemin in [Che92, Che95] and by Coron, and Glass, in [Cor96, Gla00].

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We prove that the extremal length systole of genus two surfaces attains a strict local maximum at the Bolza surface, where it takes the value $\sqrt{2}$.

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In this article, the Hodge decomposition for any degree of differential forms is investigated on the whole space ${ℝ}^n$ and the half-space ${ℝ}_{+}^n$ on different scales of function spaces namely the homogeneous and inhomogeneous Besov and Sobolev spaces, ${\dot{\mathrm{H}}}^{s,p}$, ${\dot{\mathrm{B}}}_{p,q}^s$, ${\mathrm{H}}^{s,p}$ and ${\mathrm{B}}_{p,q}^s$, for $p\in (1,+\infty )$, $s\in (-1+\frac{1}{p},\frac{1}{p})$. The bounded holomorphic functional calculus, and other functional analytic properties, of Hodge Laplacians is also investigated in the half-space, and yields similar results for Hodge–Stokes and other related operators via the proven Hodge decomposition. As consequences, the homogeneous operator and interpolation theory revisited by Danchin, Hieber, Mucha and Tolksdorf is applied to homogeneous function spaces subject to boundary conditions and leads to various maximal regularity results with global-in-time estimates that could be of use in fluid dynamics. Moreover, the bond between the Hodge Laplacian and the Hodge decomposition will even enable us to state the Hodge decomposition for higher order Sobolev and Besov spaces with additional compatibility conditions, for regularity index $s\in (-1+\frac{1}{p},2+\frac{1}{p})$. In order to make sense of all those properties in desired function spaces, we also give appropriate meaning of partial traces on the boundary in the appendix.

“La raison d’être” of this paper lies in the fact that the chosen realization of homogeneous function spaces is suitable for non-linear and boundary value problems, but requires a careful approach to reprove results that are already morally known.

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We introduce the shuffle of deformed permutahedra (a.k.a. generalized permutahedra), a simple associative operation obtained as the Cartesian product followed by the Minkowski sum with the graphical zonotope of a complete bipartite graph. Besides preserving the class of graphical zonotopes (the shuffle of two graphical zonotopes is the graphical zonotope of the join of the graphs), this operation is particularly relevant when applied to the classical permutahedra and associahedra. First, the shuffle of an $m$-permutahedron with an $n$-associahedron gives the $(m,n)$-multiplihedron, whose face structure is encoded by $m$-painted $n$-trees, generalizing the classical multiplihedron. We show in particular that the graph of the $(m,n)$-multiplihedron is the Hasse diagram of a lattice generalizing the weak order on permutations and the Tamari lattice on binary trees. Second, the shuffle of an $m$-associahedron with an $n$-associahedron gives the $(m,n)$-constrainahedron, whose face structure is encoded by $(m,n)$-cotrees, and reflects collisions of particles constrained on a grid. Third, the shuffle of an $m$-anti-associahedron with an $n$-associahedron gives the $(m,n)$-biassociahedron, whose face structure is encoded by $(m,n)$-bitrees, with relevant connections to bialgebras up to homotopy. We provide explicit vertex, facet, and Minkowski sum descriptions of these polytopes, as well as summation formulas for their $f$-polynomials based on generating functionology of decorated trees.

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The aim of this work is to prove global $L^p$ Carleman estimates for the Laplace operator in dimension $d\ge 3$. Our strategy relies on precise Carleman estimates in strips and a suitable gluing of local and boundary estimates obtained through a change of variables. The delicate point and most of the work thus consists in proving Carleman estimates in the strip with a linear weight function for a second-order operator with coefficients depending linearly on the normal variable. This is done by constructing an explicit parametrix for the conjugated operator, which is estimated through the use of Stein–Tomas restriction theorems. As an application, we deduce quantified versions of the unique continuation property for solutions of $\Delta u=Vu+W_1·\nabla u+÷\left(W_{2}u\right)$ in terms of the norms of $V$ in $L^{q_0}\left(\Omega \right)$, of $W_1$ in $L^{q_1}\left(\Omega \right)$ and of $W_2$ in $L^{q_2}\left(\Omega \right)$ for $q_0\in (\frac{d}{2},\infty ]$ and $q_1$ and $q_2$ satisfying either $q_1,q_2>\frac{3d-2}{2}$ and $\frac{1}{q_1}+\frac{1}{q_2}<4(1-\frac{1}{d})/(3d-2)$, or $q_1,q_2>\frac{3d}{2}$.

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