arXiv research

Combines geometric hydrodynamics with magnetic systems to derive new equations and prove well-posedness.

Geometric approach solves Euler equations with random forces.

The paper studies geodesic completeness for Lie groups and their metrics.

Study shows $H^{3/2}$ metric on circle diffeomorphisms has long-time solutions.

The paper extends Hamiltonian Monte Carlo to Lie groups and constrained mechanics.

Study extremals on Lie groups with asymmetric polyhedral Finsler structures using Pontryagin's Maximal Principle.

Fractional Sobolev metrics on immersions are well-posed.

Study on well-posedness of EPDiff equations with pseudo-differential inertia.

We consider the group of sense-preserving diffeomorphisms $\Diff S^1$ of the unit circle and its central extension, the Virasoro-Bott group, with their respective horizontal distributions chosen to be Ehresmann connections with respect to a projection to the smooth universal Teichmüller space and the universal Teichmül…

Develops methods for constructing exact, non-stationary solutions to Euler equations.

We derive an analytic formula for the hydrodynamic Green function and the Robin function on every orientable surface admitting a hydrodynamic Killing vector field. Closed-form expressions are provided for all fourteen canonical Riemann surfaces, covering both compact and non-compact cases; the formulae satisfy the slip…

We study the geometry of the space of densities $\VolM$, which is the quotient space $\Diff(M)/\Diff_μ(M)$ of the diffeomorphism group of a compact manifold $M$ by the subgroup of volume-preserving diffemorphisms, endowed with a right-invariant homogeneous Sobolev $\dot{H}^1$-metric. We construct an explicit isometry f…

In this article we write the equations of barotropic compressible fluid mechanics as a geodesic equation on an infinite-dimensional manifold. The equations are given by \begin{align} u_t + \nabla_uu = -\frac{1}ρ \grad p \\ ρ_t + \diver{(ρu)} = 0, \end{align} where the fluid fills up a compact manifold $M$, $u$ is a tim…

The geometric approach to optimal transport and information theory has triggered the interpretation of probability densities as an infinite-dimensional Riemannian manifold. The most studied Riemannian structures are Otto's metric, yielding the $L^2$-Wasserstein distance of optimal mass transport, and the Fisher--Rao me…

We propose multidimensional versions of the Painlevé VI equation and its degenerations. These field theories are related to the isomonodromy problems of flat holomorphic infinite rank bundles over elliptic curves and take the form of non-autonomous Hamiltonian equations. The modular parameter of curves plays the role o…

We are interested in the geometry of the group $\mathcal{D}_q(M)$ of diffeomorphisms preserving a contact form $θ$ on a manifold $M$. We define a Riemannian metric on $\mathcal{D}_q(M)$, compute the corresponding geodesic equation, and show that solutions exist for all time and depend smoothly on initial conditions. In…

The group of diffeomorphisms of a compact manifold endowed with the L^2 metric acting on the space of probability densities gives a unifying framework for the incompressible Euler equation and the theory of optimal mass transport. Recently, several authors have extended optimal transport to the space of positive Radon …

New geometric interpretation of Amari-Cencov α-connections on probability densities.