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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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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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 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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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 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 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 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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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 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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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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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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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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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 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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