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We consider a pseudo-Riemannian metric that changes signature along a smooth curve on a surface, called the discriminant curve. The discriminant curve separates the surface locally into a Riemannian and a Lorentzian domain. We study the…
This paper provides rates of convergence for empirical (generalised) barycenters on compact geodesic metric spaces under general conditions using empirical processes techniques. Our main assumption is termed a variance inequality and…
In the present work we obtain rigidity results analysing the set of regular points, in the sense of Oseledec's Theorem. It is presented a study on the possibility of an Anosov diffeomorphisms having all Lyapunov exponents defined…
In this paper, we study the ergodicity of a one-parameter diagonalizable subgroup of a connected semisimple real algebraic group $G$ acting on a homogeneous space or, more generally, a homogeneous-like space, equipped with a…
In this paper, we establish new geometric rigidity results through the study of Lyapunov exponent level sets via invariant measures. First, we prove that for a manifold $M$ without focal points, if the zero Lyapunov exponent level set has…
Tight geodesics were introduced by Masur-Minsky in [17]. They and their hierarchies have been a powerful tool in the study of the curve complex, mapping class groups, Teichm\"uller spaces, and hyperbolic 3-manifolds. In the same paper, they…
Scaling laws and intermittency in the wall region of a turbulent flow are addressed by analyzing moderate Reynolds number data obtained by single component hot wire anemometry in the boundary layer of a flat plate. The paper aims in…
We survey the recent advance in the rigorous qualitative theory of the 2d stochastic Navier-Stokes system that are relevant to the description of turbulence in two-dimensional fluids. After discussing briefly the initial-boundary value…
We classify quasiconformal Anosov flows whose strong stable and unstable distributions are at least two dimensional and the sum of these two distributions is smooth. We deduce from this classification result the complete classification of…
Consider the geodesic flow on a real-analytic closed hypersurface $M$ of $\mathbb{R}^n$, equipped with the standard Euclidean metric. The flow is entirely determined by the manifold and the Riemannian metric. Typically, geodesic flows are…
We present numerical studies of stretching of Hookean dumbbells in a turbulent Navier-Stokes flow with a linear mean profile, <u_x>=Sy. In addition to the turbulence features beyond the viscous Kolmogorov scale \eta, the dynamics at the…
We establish a rigorous asymptotic theory for the joint estimation of roughness and scale parameters in two-dimensional Gaussian random fields with power-law generalized covariances \cite{Matheron1973, Stein1999, Yaglom1987}. Our main…
For a finitely generated group $G$, we introduce an asymmetric pseudometric on projectivized deformation spaces of $G$-trees, using stretching factors of $G$-equivariant Lipschitz maps, that generalizes the Lipschitz metric on Outer space…
We study the geometry of geodesics on $\mathsf{SL}(n)$, equipped with the Hilbert-Schmidt metric which makes it a Riemannian manifold. These geodesics are known to be related to affine motions of incompressible ideal fluids. The $n = 2$…
We prove that if a geodesic flow on a closed orientable $C^\infty$ surface is transitive and has positive topological entropy, then it has a unique measure of maximal entropy. This covers all previous results of the literature on the…
We apply topological methods and a Lusternik-Schnirelmann-type approach to prove existence results for closed geodesics of Finsler metrics on spheres and projective spaces. The main tool in the proofs are spherical complexities, which have…
We consider $L^2$ minimizing geodesics along the group of volume preserving maps $SDiff(D)$ of a given 3-dimensional domain $D$. The corresponding curves describe the motion of an ideal incompressible fluid inside $D$ and are (formally)…
This paper presents a mathematical analysis of a one-dimensional model of turbulence based on a stochastic generalized Constantin-Lax-Majda-DeGregorio (gCLMG) equation. We focus on the specific case where the nonlinearity in the equation…
In a recent paper [15], Giardin{\`a}, Giberti, Hofstad, Prioriello have proved a law of large number and a central limit theorem with respect to the annealed measure for the magnetization of the Ising model on some random graphs including…
The goal of this article is to present -- in a cohesive, and somewhat self-contained fashion -- several recent results revealing an experimentally, numerically, and mathematical analysis-supported \emph{geometric scenario} manifesting…