Related papers: Geometry of diffeomorphism groups, complete integr…

The paper considers the Euler system of PDE on a smooth compact Riemannian manifold of positive curvature without boundary, and the sphere ${\mathbb{S}}^2$ in particular. The paper interprets the Euler equations as a transport problem for…

We study a possibly integrable model of abelian gauge fields on a two-dimensional surface M, with volume form mu. It has the same phase space as ideal hydrodynamics, a coadjoint orbit of the volume-preserving diffeomorphism group of M,…

Given a closed surface endowed with a volume form, we equip the space of compatible Riemannian structures with the structure of an infinite-dimensional symplectic manifold. We show that the natural action of the group of volume-preserving…

We start by constructing a Hilbert manifold T of orientation preserving diffeomorphisms of the circle (modulo the group of bi-holomorphic self-mappings of the disc). This space, which could be thought of as a completion of the universal…

We study the class of norms on the space of smooth functions on a closed symplectic manifold, which are invariant under the action of the group of Hamiltonian diffeomorphisms. Our main result shows that any such norm that is continuous with…

We describe first integrals of geostrophic equations, which are similar to the enstrophy invariants of the Euler equation for an ideal incompressible fluid. We explain the geometry behind this similarity, give several equivalent definitions…

Our monograph presents the foundations of the theory of groups and semigroups acting isometrically on Gromov hyperbolic metric spaces. Our work unifies and extends a long list of results by many authors. We make it a point to avoid any…

In this paper, I shall demonstrate that sufficiently high-dimensional closed positively-curved Riemannian manifolds are either diffeomorphic to a spherical space form, or isometric to a locally compact rank one symmetric space. This…

We establish a simple relation between curvatures of the group of volume-preserving diffeomorphisms and the lifespan of potential solutions to the inviscid Burgers equation before the appearance of shocks. We show that shock formation…

This paper establishes robust obstructions to representing Hamiltonian diffeomorphisms as $k$-th powers ($k \geq 2$) or embedding them in flows for certain higher-dimensional symplectic manifolds $(M,\omega)$, including surface bundles. We…

Diffeological and differential spaces are generalisations of smooth structures on manifolds. We show that the "intersection" of these two categories is isomorphic to Fr\"olicher spaces, another generalisation of smooth structures. We then…

The main result of this note is that every closed Hamiltonian S^1 manifold is uniruled, i.e. it has a nonzero Gromov--Witten invariant one of whose constraints is a point. The proof uses the Seidel representation of \pi_1 of the Hamiltonian…

We study the exponential map of group of volume-preserving diffeomorphisms on closed orientable surfaces via the vorticity formulation of the incompressible Euler equation. We present an alternative, fluid dynamical proof of the theorem of…

Given a finite dimensional manifold $N$, the group $\operatorname{Diff}_{\mathcal S}(N)$ of diffeomorphism of $N$ which fall suitably rapidly to the identity, acts on the manifold $B(M,N)$ of submanifolds on $N$ of diffeomorphism type $M$…

We develop sheaf-theoretic methods to deal with non-smooth objects in symplectic geometry. We show the completeness of a derived category of sheaves with respect to the interleaving distance and construct a sheaf quantization of a…

In this book chapter we study the Riemannian Geometry of the density registration problem: Given two densities (not necessarily probability densities) defined on a smooth finite dimensional manifold find a diffeomorphism which transforms…

We study the optimal transport problem in sub-Riemannian manifolds where the cost function is given by the square of the sub-Riemannian distance. Under appropriate assumptions, we generalize Brenier-McCann's Theorem proving existence and…

We develop a family of infinite-dimensional Banach manifolds of measures on an abstract measurable space, employing charts that are "balanced" between the density and log-density functions. The manifolds, $(\tilde{M}_{\lambda},\lambda\in…

In this work, we establish the existence of solutions to stochastic differential equations on the Wasserstein space over a closed Riemannian manifold, under suitable regularity assumptions on the driving vector fields. Interpreting the…

In this paper we define and study a notion of discrete homology theory for metric spaces. Instead of working with simplicial homology, our chain complexes are given by Lipschitz maps from an $n$-dimensional cube to a fixed metric space. We…